Great job Mr. Wildberger! I'm a recreational maths guy, and I remember how surprised I was when I first encountered your views on these subjects. I didn't know it was okay to object to such things, but even as a layman, I've sometimes felt suspicious of some of the faith expressed in the exacting nature of many a 'simple' construction. Always s point where the def'n giver just 'trails off', or points at something else... More study required here🐢 Keep it up, good sir!
@@maynardtrendle820 I just hope you don't base your opinion on math on what Wildberger is saying. When you look at the argument from Wildberger about infinity in math, they seem to be from someone who doesn't completly understanding formal language and set theory.
@@DanielBWilliams I refuse to believe wildberger has ever taken a set theory course. He is so insistent that addition of real numbers has never been proven to work. He never defends this position and anyone whose taken set theory 101 can tell he's wrong.
@@almightysapling He has literally dozens of videos spending hours dissecting the various problems with all forms of 'real' numbers, from axiomatics to infinite decimals, to Dedekind cuts, to Cauchy sequences, and on and on. When you criticize someone for 'never defending' their point of view and being 'ignorant' -- while being ignorant of their exhaustive defense of their point of view -- you come across as embarrassingly hypocritical. This interview was only an hour long and it's next to impossible to go into depth on the topic in the time available. And I doubt anyone who has only taken 'set theory 101' could give a proper definition of the 'real' numbers. 🙄
This was really cool! Mr. Wildberger probably never gets to talk about this subject to anybody worth their salt. You can tell he was excited to present these ideas. Maybe do it again (soon) with some of the foreknowledge of the other's arguments. ☺️ Great job guys!
Thanks for hosting Prof. Wildberger in this discussion/debate, Daniel. I can imagine it must have been very interesting and/or discombobulating to face these kinds of ideas 'fresh', so to speak. It's a lot to take in at once! 😅 You were a very gracious host, and it would be interesting if you were to do a follow-up video with your further thoughts, or perhaps a second interview after digesting this one. Cheers!
@@DanielRubin1 if you wish to do so he is wrong about ∞ essentially noone uses ∞ but rather many finite concepts, except for set theory which is religious infinitism. (Finite concepts such as: arbitrarily large finite number or it's axiomatic idealisation, all finites, or infinite increase rather than infinity, using a noun when only verb would be correct, or infinite (such as √(πe2) which actually does exist per gamma function estimate formulas) from finite algorithm thus, finite algorithms.) He is angry not over ∞ but religious ∞ per set theory, but does not attack it's errors as I do. The places set theorists refer to nonfinite ∞ are not so. Thus they have nothing to learn ∞ that is infinite from. So for example the set of real numbers equals the set of integer numbers in size, etc etc etc. So very many errors in set theory infinitism.
Religion refers here to incoherent babbling held in esteem, not to a point of view. Infinite number (in the literal sense) is nonsense. Finite number is a pointless term in a technical context, as all numbers are limited, and so therefore is "finitism." There are no isms, just clearly communicated concepts and incoherent ones.
@@ThePallidor no, per ∞ I can prove the size of c is not strictly larger than N, the ∞te sets don't have sizes, the proofs actually prove all the ∞te sets considered are =, in size to all each other, and by many functions such as +2, ×2, square, exponential, etc Per finite he doesn't demonstrate any harm from ∞ as it actually is used, axiomatic idealisation of sufficiently large number such that larger doesn't help thus effectively finite number, or another all finites (such as proof of four color theorem for every finite, or no win to the Mathematician
Aren’t there rational numbers whose full decimal expansions can’t be calculated with currently available computing power? Or with any computing power that will ever fit inside the universe at any point in the future?
This was such an enjoyable way to spend the hour! I think we do ourselves a disservice by limiting theory to what is computable as computational reality is dynamic and continuously evolving - however the thought experiment surrounding how real the set of R is will have me thinking for some time. Wonderful content… subscribed and looking forward to diving deeper into your channel!
By Wildberger’s argument, if a criterion for numbers being real is being necessarily related to “write down-ability” then, since a “point” in geometry is a dimensionless object, it is therefore not “write down-able” and therefore not real.
Sort of I think. He does explain that you have to prove the properties you wish to have, not just assume them. It is not the dimensionless that causes the problem, but can you prove that you can add these 2 points and not end up with some Banach-tarski paradox. In my limited understanding that is what I think he tries to say.
@Gennady Arshad Notowidigdo ages? Like how many years is an "age" ago? And what if it is "solved" ages ago, that does not mean there is not a better way to teach it, or that there is not a better foundation from which to "build" upon.
@@tomasmatias4109 well if you manage to show ZFC is inconsistent, I don't doubt there is a Fields medal coming your way He doesn't do that, or even attempt to do that though...
There needs to be a distinction between the actual physical world and the ideal Platonic world In the Platonic world ideal (perfect ) circles can exist, in the actual world not so much In the Platonic world ideas involving infinity are accessible, in the actual world not so much
The debate was basically about (computable) definable objects versus ideal objects. These are generally equivalent, but they don't differ on anything fundamental. Opinion: Ideal objects are less hassle (and thus better), but just use whatever makes you more productive.
I really enjoyed that. I’m philosophy trained, and therefore basically on NW’s “side”, but… 1) Analysis is far more efficient than discrete mathematics. Example: if you wish to find the thousandth term of the Fibonacci sequence, step by step addition takes a long time. There is an analytic extension of the series - the Pi function - which will do the job much more efficiently. But it will also calculate results for values of “n”, or rather x, which are not whole numbers. There is no square-root-7th member of the series, for example. The Pi function does not generate the Fibonacci sequence. The sequence generates - via some mathematical hard work - the analytic extension. 2) You see it as obvious that if every natural number has a successor, then there is a logical final result, an actual infinity. But in the real world a rich man may always own one dollar more, and then another… he can never have infinite wealth. 3) Von Neumann: “Young man: in mathematics you don’t understand things - you just get used to them.”
These are interesting discussions, but ultimately they are philosophical more than mathematical (which is not a bad thing at all). As such, the aim is for maximizing internal consistency, while recognizing there may be many equally internal consistent views. And its never clear what unknown path might exist that exposes inconsistencies in any other path, and replaces them with a better framework. There are certainly consensus views in math, as in any discipline, that could later be challenged from an alternate viewpoint. It’s useful to consider historical naive certainties that later appear quaint, and then consider one’s own views as a future history that will be viewed in a similar light.
The discussion around 40:00 about how in combinatorics the power series functions a treated as unbounded generating sequences is quite a good contrast from the analytical perspective that these functions produce a final real-valued output (from an infinite sum). I don't think analysis isn't worthy of analysis (heh) but it shows how one can choose two interpretations of the same concept and come up with a constructive one and a non-constructive one. I wonder if this kind of dualism is universal...
I think there is a big issue for which this question can not be sorted out, eg that the question to "keep in pure mathematics only what's science and not philosophy" is a philosophical question!
This is such a great video. NJ Wilderberger is such an interesting person. It would be so easy to dismiss his opinion and write him off. But you actually take him seriously and really have to think about why you disagree. I disagree with him too but as someone who is far less knowledgeable in mathematics I have a much harder time figuring out what I disagree with and why, especially because I don't want to just dismiss someone because their views challenge the established I understanding of the subject. I agree this seems to be largely a philosophical disagreement, even if he himself doesn't see it that way. And honestly that's part of why I find it so interesting.
Honestly I found your channel searching about him just because I find his view of mathematics so interesting. It's so radically different from mine. I kind of wish more mathematicians would have done something like this. I have a great deal of respect for you for doing this. Definitely subscribing.
"I myself am not a set theorist or logician" can someone please get one on? Since that's who needs to be going head to head with this guy because that's his issue. Definitions and philosophy. Not analysis.
All fields have to assume some philosophy, including mathematics. Wildberger just has a different Philosophy. Trying to separate analysis from philosophy is pointless though.
@@presence0420 @Presence 0 Philosophy shouldn't mean a bunch of points of view; it should mean (utilitarianly: would be useful to have it mean) the foundations of every field. The philosophy of mathematics should thus refer not to various people's perspectives on how to do math, but to the basic underlying epistemological foundations of the subject, which need to be the most rigorous of all. It's merely that mainstream mathematicians have neglected these foundations and relegated them to the philosophy department, which is even more clueless about rigor than most mathematicians are.
There is one philosophy, not many. There is one correct way to apprehend the action of mind that underlies what we call "doing math" and many confused ways. It's not a matter of arguing various opinions; it's a matter of speaking clearly enough that everyone recognizes what exactly we are talking about.
"Write it down and then we talk about it" is my take away. Sounds like it is better if we construct the objects that are going to be used in any mathematical theory or construct,,,and it is just fair. If the object can not be constructed, then the process of constructing the object of the discussion is lacking from the discussion. Therefore, the process to construct the object needs to be developed, no matter how coarse (is that an algorithm?). It is the process of constructing an object that keeps the focus on the discovery and construction of such an object
At 58:31 Daniel mentions some concept of being able to compare it to the size of any other real number, and claims that you could. But I think, that, is incorrect. Real numbers are in general *not* comparable to other real numbers, in that there is no algorithm to determine that. It's like saying you can compare any computer program to see whether it will halt or not - but in fact we know this is impossible. It's the same for the real numbers. There is a subset of real numbers which are comparable (vaguely corresponding to "computable numbers") but in the general formulation it can't always be compared.
The requirements to meet definitions surrounding real numbers involve computation. So whenever you're in the situation where a thing can be approximated better and better, say by some proven formula, then you may as well designate a symbol and call it a number (or class, etc). For all there exist kinds of statements involve ideas of infinity. But you'll have that formula there which may be used on any number you like.
@@anthonyymm511 It is an axiom, but the presentation of it is disingenuous. You can prove the least upper bound axiom for decidable sets. In order to "prove" it in general, you must assume perfect knowledge of undecidable sets, basically assuming you can solve the halting problem, which is clearly an issue. That assumption is the hidden axiom.
"Pure mathematics is nowhere near as old as applied mathematics", maybe, but what Euclid and the Greeks were doing was pure mathematics. Look at the compass and ruler constructions; why did they care that you couldn't exactly trisect an angle or square the cube using a compass and ruler if all they cared about were approximate applications.
You miss the point hidden in the meanings of the word « approximate ». Greek geometry was not about numbers! But about GEOMETRIC OBJECTS AND MAGNITUDES. Euclide knew that Pythagoricians proved square root of two was not rational. But he didn’t say it was « irrational ». He stayed on the contrary in purpose on the other side of the Stix, looking at geometric objects as geometric objects, not confusing them with dubious « measures » of them, and creating a name, magnitude. And they didn’t look at the diagonal of the unit square as having square root of two as a lenght or measure, but as a magnitude. They thus restricted very carefully quotients and ratios to quantities that had a common measure, like 2 and 3, by taking the magnitude of the trisection of the arbitrary starting unity, to be the new unity of rationalisation. In this regard pi was never a number for them. Never. It was on the contrary à magnitude but that was not measuring the « commensurability » of the perimeter to the diameter of any random circle, but on the contrary naming as « pi » the NON COMMENSURABILTY. In other words Geometry was for them an exact science. While arithmetic was just an inferior approximate one, because it could not handle the measure of the diagonal of a unit square or the commensurability between perimeter and diameter of circles. Totally opposit point of view as the one which is taken nowadays
I have a hard time with Wildbergers flavour of finitism, since I prefer frameworks for which at least in principle a full first-order logic axiomatization or similar is on the table - so that everybody knows all the legal formal rules "of the game." That said, this discussion was more about championing and questioning the early 19th century analysis, anyway. Now on the other hand, Daniel, I think you trivialize the reals when you respond to Wildberger by proposing they are mostly a useful formalizing wrapper for in-principle approximatable objects - since this is not the case. If you fix an alphabet, formalize a Turing complete framework, enumerate all algorithms a_n that can be written down (a subset of all strings) and let S = {n > 0 | a_n() eventually halts} be the subset of natural number indices for which the n'th program halts when ran on empty input, then sum_{k in S} 2^-k defines real number in the interval (0, 1) which however can't be approximated arbitrarily well (since the Halting problem is undecidable, and thus eventually there's a a_N the non-halting of which you can't establish and that leads to a knowledge gap of size 2^-N in your computation). The so called Specker sequences show that the computable reals are not even closed w.r.t. limits (there's sequences of rationals who's limit point isn't computable like the rationals.) And next to the uncomputable definable reals, in the standard interpretation (with more than one infinite cardinal, which when adopting Excluded Middle they are ordered in a sequence of "sizes") there's of course all the undefinable reals (since what can be defined is a subset of all strings, which in turn have the smallest infinity size). Modern analysis does more than filling the gaps between elements in Q via Dedekind cut's that give object that can be arbitrarily approximated. That step also adds objects very different from sequences of digits (definable uncomputable reals or arguably worse).
See his video on axiomatics. Euclid gave us 5 requirements that were self-evident. Modern mathematicians instead use the notion of axioms to ordain that the ill-defined is well-defined.
@@ThePallidor I'm happy with Euclid's definition of point (def. 1 and 3, mereological "has no part" and discontinuity "end of line"). The second definition "Line has no width" seems problematic". If line has no width, can it have depth instead? In which case there's just a difference of perspective. Archimedes calculus can be interpreted as denying Euclid's 2nd definition.
Potential infinities (with/under Halting problem) does not equal strict finitism. It's rather the middle way between finitism and Cantor's paradise/joke. As for logicism, do we really need to keep on playing that game after Gödel and Turing? Coherence theory of truth (which guides intuitionist constructions) is not without its challenges and problems, but at least it's not outright debunked like the logicism foundation of formalism.
@@santerisatama5409 Euclid wasn't perfect either. A line is long, thin rectangle. Better to say it has negligible width. A point is a dot of negligible size.
@@ThePallidor I like mereological approach, so I have no problem with 1st definition, when taken together with 3rd. Affine parallelism can be intuitively derived from those. I have no good solution for line, some sort of infinitesimal approach, perhaps, but not easy.
@@chjxb oh yeah? What is its value? Huh??? Do you know how to calculate sqrt(2)+sqrt(3)??? This should be even easier than pi+e+sqrt(2)!!! Do it! You can't.
Nice discussion. Quite humorous at times. Celestial Sphere ... haha ... nice. At any rate, I'd be very interested to hear more about your objections to Category Theory. Could you make a video delineating your main qualms with the subject?
Whether we like the real numbers framework or not, I think it is good to look at and push forward other frameworks for Analysis. Especially if we need less axioms with them. Similarly, in Geometry it has been very fruitful to remove some axioms and try with less, say for example to go from euclidean geometry to affine and projective geometry. Finally if a new framework gives some interesting insights and results then it probably worth to investigate it further and to challenge the more established one. Thanks to both of you for this interesting discussion.
The idea of axioms is broken and generally now used to mislead. Euclid didn't use axioms. He simply stated in clear terms what were already self-evident observations.
@@DanielBWilliams An axiom is at best a premise by another name. At worst - and this is its usual use - an axiom is an attempt to smuggle undefineable nonsense into a position of unquestionability. Cf. "There exists an infinite set."
@@ThePallidor There is no axiom in ZFC that said "there exists an infinite set". But even if that was the case, "infinite set" is perfectly defined in ZFC so I don't understand why would you said that. Can you explain why did you said isn't well defined ?
I agree with NJ Wildberg that a distinction between computable mathematics and imaginary mathematics is a good idea. Having said that, I like the blackboard of the mind. The blackboard in your mind is edgeless, you don't have to imagine the edges at infinity, instead, you can not bother imagining the edges at all.
Having a computer program that approximates cube root of 15 with the property Tha the approximation gets better if the program is allowed to run for longer is basically equivalent to saying that there is a cauchy sequence of rational numbers for which the limit is cube root. That's why you can't make a program that approximates root of negative one with rational numbers. He's basically using the concept of a limit to assert the existence of such program.
there is no limit because the program never stops, and never actually reaches the cube root of 15. in practice, we simply stop the program at some point (in modern computers, this is usually the saturation of the 64 bit floating point data structure) and take the approximate result so far. wildberger's philosophy mirrors what our machines actually do, and he sets his definitions that way, rather than pondering the philosophy of what a certain algorithm could be at its fundamental level in some alternative universe that cannot be proven to exist.
Daniel have you looked into algebraic analysis? It as well as hyperfunction theory may be more palatable to you as an analyst than Grothendieck. The ideas turned out to be very close developed in the mid 1900's by Grothendieck and Mikio Sato independently. I also feel it is a way to settle Wildberger's concerns particularly from a functional analysis point of view thinking about the algebraic structure of topological vector spaces, as Wildberger accepted in this debate you can have such a thing as cube root of 2 in a system of matrices representing an extended rational numbers to include it. Similarly you can have a system which extends to all the irrationals, complex algebraically, but that's why we have the infinite decimal notation to try and simplify things. Also on a more philosophical note 1/3 + 1/4 + 1/5 = 47/60, is just a simplification we don't do anything here actually solving something or finding anything. So I do not see any problem with defining sqrt(2) + pi + e as itself, math is tautological to begin with. There is no foundation which gets around this.
On the subject of mathematics being compared to religion (and one could easily apply this reasoning to mainstream physics as well), I will leave you with a quote: "If a ‘religion’ is defined to be a system of ideas that contains unprovable statements, then Godel taught us that mathematics is not only a religion, it is the only religion that can prove itself to be one." - John Barrow
At 15:15 they are talking about adjoining sqrt(15) to rational numbers, and Wildberger says that we once we do that, we don't have an order between numbers. This is just false. Those numbers have a natural order that can easily be defined.
If we go with Dr Wildberger and look at Phythagoras's theorem a ^2 +b^2= c^2 is therefore only applicable is certain situations - eg 2 ^2 +2^2 = 4 +4 =8 therefore c=square root of 8 = a 'real number' 2.828427.....P's theorem doesn't work if we reject 'real numbers'..... Do we then 'adjust' the Pythagorean theorem- maybe instead of the 'equal' sign we use some symbol which means only conditional on the values of a b and c do not produce a 'real number' as the result the computation is true only when the result is not e 'real number'?
Just to answer what some people in the comments are confused about: 1/3 having the same issue as square root of 2 because 1/3 is an infinite sequence of 0.3333.. Unlike roots, rational numbers can be constructed using Euclid methods to bisect a line into any (integer) amount of segments
what? computational materialism? that's not a real word or concept. Pi can't ever be calculated. You approach it indefinitely. Wildberger's statement is not philosophic
dunno about that... I think pure maths might be a shortcut to studying mathematics that could (perhaps with more creativity) be found though applied / constructive foundations.
at 58:00 , why is 31/30 a final answer? It's something you can write down, and you can also write down 1/2+1/3+1/5, which is a valid answer, and 1 1/30 is also a finite sequence of strokes that you can write. So is pi+e+sqrt(2). it can be written down in finite time and space.
Technically we could name anything and write it down, using that name. but I think his point is more subtle: representing 1/3 as 1/3 rather than the infinite expansion 0.3333... is composed of integers and the division operator, all remaining within the rationals, using a finite set of symbols and rules, which he is taking the position are enough to model continuity (with some creativity in coming up with theorems that work around the rationals not being complete if square roots (etc) are allowed). Honestly I see this as a design choice rather than a philosophical position. That is, he's championing avoiding defining a different class of objects that, while algebraically have the properties we want, admit many instances that aren't just non-computable, but can't even be indirectly expressed in other forms (as I understand it, most real numbers that "exist" fall into this category). Instead he's getting creative in redefining some areas of maths that ostensibly (at least how they're commonly taught) depend on continuity based on completeness (or equivalent, such as IVT, etc). Personally I think it's a pretty cool project. But I come from a computational background where computability is highly desirable. It seems almost too easy to design algebraic objects that, while being simple to declare in terms of algebraic properties/laws, admit rather absurd instances that we can never encounter or even refer to.
this is perfectly true, but mathematicians claim there are real numbers which cannot be defined in finitely many symbols. Chaitins constant requires a solution to the halting problem, which is the same as having infinitely many symbols, one to determine every Turing machines end state. If there are "more real numbers than natural numbers" then there are undefinable real numbers, since there are only countably many definitions. Granted, wildberger takes issue before we've reached this point, but this is a clear issue with standard mathematical assumptions.
I think he would say that the hypotenuse has a physical length, but that length cannot be expressed as a rational multiple of the sides. So it is incommensurable to the sides, and can only be approximated when using them as your unit.
if i’m understanding wildberger, he’s saying that numbers like the cube root of 15, when computed to a form that most people use, contain an infinite amount of non-repeating decimals, which requires an infinite number of information to encode. it’s an interesting pov.
The cube root of 15 itself doesn’t require an infinite amount of information, only its expansion in base 10, 2, etc. But that expansion is not an intrinsic property of the cube root of 15, it’s a relational property of that number and other numbers (10, 2, etc.)
@@synaestheziac Does that mean anytime we "write down" a number, we have to choose a representation of it, and such representation may or may not come at the cost of precision? Furthermore, any number that we can "write down" cannot be the number itself, but merely a representation of it, since there are no direct ways of handling numbers, concrete or otherwise?
@@justanother240 as long as we realize that a "representation" of a number is really just a relation between that number and other mathematical objects (such as powers of 10 and their coefficients, or the cube root operation and the integer 15, etc.)
This is a great discussion! I think an important distinction should be made between something like say 1) calculating with essentially 100% absolute certainty to the precision of a nanometer how long the radius of a circle 1 million times the size of the observable Universe would be whether or not such a circle could ever conceivably be formed or viewed, versus something much more fundamental and foundational like 2) the idea that the process we think we can continue to use may not be correct, or, though we have gotten accurate results so far, it may have never really been working for the reason we thought it was in the first place, like the way Professor Wildberger pointed out that computers do not store a set of all natural numbers up to a certain point and access them to make a calculation.. We think of the set of natural numbers as sort of already existing in some way, and that we can access them, but the idea that maybe mathematics actually works nothing like that, just that the results match that way of thinking at a certain scale, that such a way of thinking only holds up, up to a certain point where eventually it crashes.
@about 19:30, Norman states that pure mathematics didn’t develop until the 19th century, but I disagree. He’s ignoring the fact that recreational mathematics is actually a part of pure mathematics, and it has been around almost as long as applied mathematics, notwithstanding the fact that many questions in recreational mathematics have a “flavor “ of being related to or derived from certain specific applications, but with some aspect of entertainment injected, often via a modification of hypotheses to something that is motivated by a philosophical conversation about what we can imagine, instead of what we actually see and touch and use.
@@santerisatama5409 What do you mean there is no any answer. If I give you the sequence (1;1/2;1/3;...;1/n;..) and the sequence (-1;-1/2;-1/3;...;-1/n;...) it's easy to see that they point to the same number 0. So as you can see, with enough information on both of you sequence, you can conclude.
What does it mean for such things as reals or rationals to exist or to be 'write-down-able'? NJW defers to 'computer programs'. Why doesn't writing down 3^.1 in the context of a computer algebra system like Mathematica fulfill this criterion? Such a system provides a fully realized idea of what the concrete computations that address 3^.1 look like. Also, the definition of a limit relies on providing an exact (algorithmic) means of producing a delta for every epsilon. The idea of 'at infinity' is perhaps a nonsensical phrase out of context, but it is defined as a technical term in a fully finite way via epsilon-delta.
@Gennady Arshad Notowidigdo It seems like you took my criticism of NJW's position to be a criticism of the conventional (Weirstrauss/Cauchy) system, and it seems like you agree with me. I take your point that maybe the NJW critique belongs more to the philosophy of math than math itself because it doesn't make any discrepant claims. I will say that I do think it substantially rephrases the old claims, much as the 19th century analysis reframed the previous generation in response to Berkeley's 'ghosts of departed quantities' critiques, and that lead to a lot of new prospects in mathematical thought. I think the big weakness of the NJW position is that it relies on an idea of (philosophical or ontological) existence that is never defined. I think, though, it would be great if students were inculcated with the doubt that the cube root of 2 is a real thing from an early age, if for no other reason than it would give them a great foundation for understanding what real numbers are, haha!
Also, the definition of a limit relies on providing an exact (algorithmic) means of producing a delta for every epsilon this is the only meaningful definition of limit, but its not what mathematicians mean. they believe there are "non-computable" limits, which have literally 0 utility compared to the algorithmic version. If thats your definition of the real numbers, then youre actually talking about the computable real numbers.
@about 9:03, Norman introduces “a slightly more difficult” problem, but it’s actually too difficult to illustrate his concerns. He goes from considering an example that’s a linear equation with integer coefficients, to a cubic equation, x^3=15. A simple example is the following: Let e denote the smallest positive rational number representable in my desktop computer using Mathematica today (2024-06-09), and consider the equation 2x=e. Then no solution to this equation exists in the set of numbers that I can compute today using Mathematica in my desktop computer. This even more directly represents Norman’s viewpoint as he presents it in his videos, because he points out that he actually believes there is a largest positive integer. Here’s an ultra-finitist. Daniel, I would suggest that you re-watch Norman’s videos very carefully if you wish to enjoin him in a new “battle” or debate. …
I've been following Wildberger's thought-provoking and entertaining channel since he started on YT. I've often wondered: just from a pedagogical point of view, how is he able to teach a lot of undergraduate mathematics? He might have good personal reasons for not believing in the real numbers, but surely undergraduate maths students have to be exposed to, and understand the standard constructions of the reals, Cauchy sequences, Dedekind cuts and so on. How does he deal with that if he's teaching introductory Real Analysis?
He is still trying to find anyone who can write a passable definition of what a real number is. He did a video of it once which was hilarious. Its a bit like Alice in Wonderland stumbling across the mad hatters tea party or the boy who said "the emperor is wearing no clothes!"
@@gidi5779 you need to look at Normans video where he goes on a quest to find anything that actually explains what a real number is. Personally I think the number of digits is pretty clear if you are going to use an approximation of a real number then it should be a significant part of your definition. If you are going to be rigorous. How do you do arithmetic with real numbers other than simple restate them?
Computer floating point adition and multiplication are not even associative. Are we going to formulate mathematics on that? Even if real numbers are "only" abstractions, they constitute a language with which we can prove theorems which are known to be approximately true in the realm of computational floating point calculations. If we substitute this by concrete computer operations, we wil end up with some form of mathematics whose rules are complicated and computer-architecture dependent. Current real number mathematics is already computational. It says that the theorems are approximately true in any computational architecture, and the goodness of this approximation can be made arbitrarily good if we change the architecture, say by increasing the number of bits for example.
The thing is, these sequences really ARE about our imagination, and I think mathematicians should own that. If you can imagine it, then it exists. As long as there is some pattern or structure to the sequence, then it can be assumed to go on forever. What's wrong with not being able to approximate pi exactly? That's what makes life interesting. Somehow it's a failure because we can only approximate it to 60 trillion digits?
The main beef Prof Wildberger seems to have is w a faulty definition of the continuum and his disbelief in the existence of real numbers from a computational standpoint. As an engineer and computer scientist (and 30 yr user of Mathematica), I reject the notion that real numbers need to be computed to exist. I am perfectly content to defining their existence by some induction or limit process, e.g., bounding them from above and below at arbitrary precision. The number itself is an ideal, its existence results from the defintion of the bounding process, not from its computation. For two centuries, no practical "crisis" has yet emerged from this type of definition. Consider the method of proof by induction. It suffices for one to show a base case and the inductive step are true. One need not proceed to infinity. The idealized truth about some object found in infinity is established anyway. So it should be w the real numbers and continuum. We should listen to Gödel. Mathematics is provably incomplete and broken. But it is hell of useful. Let's focus on what works. P.S. -- later edit: I love Prof Wildberger's series of lectures on hyperbolic geometry. There he makes wide use of intersections of lines with a circle (Apollonius' construction for the tangents, the polars, etc.). If he doesn't think the circumference of a disk exists (it is an idealization after all), how can he (or Apollonius) compute intersections of lines with it?
Exactly. And in fact we learn this in 1st year calculus. It is sufficient to show that if a function is bound by N and monotonically increasing then there is no reason I cant say that IF I continue the calculation I can get as close to N as required. In other words as x increases indefinitely, f(x) = N . I do not need to compute what f( very large number) is to show its getting closer and closer to N.
@Gennady Arshad Notowidigdo interesting how u superimposed the meaning of ideal from an aspirative term (as used above) to a description of perfection. Indeed, ideal can mean both. I view this more as a wordplay than a contribution.
@31:19, Again, for the nth time, where n is larger than 3, Norman has indicated that he’s limiting his philosophy of mathematics to computer science and algorithmic foundations, which didn’t really exist as a philosophical consideration; this is in spite of his frequent claim that he’s wanting to avoid a philosophical discussion. For example, he clearly wants to deny that what he’s calling “choice sequences” of natural numbers exist or that such a restriction is a philosophical issue. Daniel has trouble addressing this, and I think the reason is that he’s not addressed these kinds of questions in holds own mind previously. It helps me to realize that I’m a human being in a world that has many things in it that I’ve never observed. When I do observe one of those things, it’s extremely unlikely that I’ll be able to view it as having been “generated” using an “algorithm” that I know or will either know or understand very soon after observing it. I cannot deny the existence of such a thing at the moment that I first observe it; that may even be dangerous. To live in this actual universe, I am compelled to be able to imagine certain things as being possible by virtue of not having introduced into my system of thought an obvious contradiction. Mathematically, this leads me to accept as potentially existing, arbitrarily large positive integers, and sets of them, and
Norman doesn't understand continuum (real numbers). It's just continuous quantities. There is nothing inexact or approximate about any real number. Sqrt(2), for example, is just as exact as any other number on the number line, 2 for example.
This is not correct. Any "computable real number" is arbitrarily precise. But uncomputables, like Chaitins constant, cannot be determined to arbitrary precision.
I wish Wildberger would have given you more time to talk, especially earlier on in the video. Being "write-down-able" as Wildberger argues, keeps mathematics more honest. It is an absolute necessity if we want computers to check and discover theorems for us, which will become more and more important going forward. But we might recognize different levels of write down ability. Rational numbers have canonical forms. Many limits have been written down exactly using limit notation, although they may not have canonical form for them and no easy generalizable way to check whether two differently-written limits are actually equal. And the alleged "non-computable" numbers can't be written in any manner at all. Anyway, he might consider that write-down-ability is not a black and white criteria, but a matter of degree. I agree with Wildberger's suggestion that how computers do math should guide how humans think about math. But with that in mind, we should try to better understand how computers actually do math. Many computer languages, such as Haskell, support a feature called "lazy-evaluation" where at runtime the resulting code keeps intermediate results in a data structure that exactly represents the expression in symbolic form. Conversion to a canonical numeric form (typically as a binary or decimal approximation) is delayed until the result is actually needed. Some systems, like Mathematica, can even simplify expressions algebraically prior to approximating them numerically. These are arguably the same that things mathematicians are doing when they write expressions in limit notation, or in a form like "π+e+√2". But in the end, we should realize that symbols like π, e, √2 are just names for useful algorithms for generating successive approximations that come arbitrarily close to satisfying certain criteria. They do not represent actual numbers that can be stored in a finite length of bits, or have operations like addition or multiplication exactly performed on them in finite time; in other words, we can't use them as fundamental building blocks of computation in the way that we can use small integers and rational numbers. Wildberger should be more careful in how he talks so as to better distinguish the different ways in which a solution can "fail to exist". A solution failing to exist because two curves fail to intersect at all is a fundamentally different situation compared to a solution "failing to exist" because the intersection point simply isn't amenable to canonical representation and can only be written approximately. The concept of real numbers may provide a useful way of thinking about how values that can't exactly be written down can nevertheless result from well-defined mathematical relationships. Also, it was weird to hear him talk about approximations, while at the same time denying the very existence of an actual value being approximated.
I agree with a lot of what you're saying. That's a very good point it did not occur to me to mention at the time that even in NJ's framework you'd want to distinguish between when curves don't intersect as real curves and when they don't intersect because the intersection point would not be rational.
@Wayne --- In other videos I like to distinguish between actually three types of "solution possibilities": 1) There is a solution, as in x^2= 25 2) There is no solution, but there are approximate solutions, as in x^2=7 3) There is no solution, not even approximately, as in x^2=-13.
"Write down" should mean "in code". I think of Pi as not a number, but a bit of code that generates digits for as long as you need to pull them. ie: e + pi + sqrt[2], is a function that you can pull digits from. It does not terminate though. I think continued fractions can handle roots in a way that terminates. So, there are issues in how you choose the number system. That's the only way that you can use the equality symbol. Simple rational numbers are functions that terminate. One of the things that recommends this approach, is to reformulate things to work on finite fields. And in a computer that can only do 64-bit integers; that would be such a field.
The problem is that your algorithm will never be able to reach produce the exact number. That is one of the main criticisms of conventional math. Math started as a logic way of understanding the world and ended depending on concepts that can't exist in our universe. That generated all kinds fo paradoxes and the theory itself ends not being consistent in many places.
An *actual infinity* is an infinite entity as a given, actual, completed mathematical object. Actual infinity is to be contrasted with *potential infinity,* in which a non-terminating process (such as "add 1 to the previous number") produces a sequence with no last element. Aristotle made a distinction between the two types (though he did not believe in an actual infinity). It is with the foundations of Set Theory that actual infinities came to be largely accepted and used in mathematics. Cantor had another type of infinity he presented, a type of actual infinity called Absolute infinity. so we have 4 types of numbers in this regard- two types of *potential infinity:* *finite potential infinity*- an (apparently) non-terminating process which produces no last element, and always remain finite- an indefinite finite value. a finite potential infinity is not a definite finite amount. *infinite potential infinity*- a non-terminating process which produces no last element, but does not have to be finite- an indefinite infinite value- not an actual amount. *actual infinity*- a completed infinity with a definite value. two types: *non-absolute actual infinity*- a complete infinity with a definite value, but more can be added to it to get a new number which is a different actual infinity. (adding more is something that usually has to be done outside of a given process, for example: 1+/12+1/4+1/8.. until 2 is reached will result in an actual infinity if you treat the last term as a 1 and convert the rest to 1's, but, although the original process has to end when 2 is reached, this does not mean that, outside of this process, 1 cannot be added to the total we go when the smallest unit was converted to 1 and added up. But so there are times where a sequence or process must terminate at a certain value, but that the value terminated at can still be added to in an other separate process.) *Absolute infinity*- an actual infinity that cannot be added to. It has reached the ultimate limit. If there is such a thing as this, no number can be higher than Absolute infinity. [In my personal opinion, if there is an Absolute infinity, potential infinities, although not actual amounts, can be conceptually MUCH larger than Absolute infinity, besides the definition that nothing can be larger. But for example, if ω^ω was an absolute limit that cannot be surpassed, conceptually, indefinite potential infinities can be described in ways that go way beyond that. But, like all potential infinities, they would be invalid.] some examples: *actual infinity:* 1+1+1+1... an infinite amount of times, until the first infinite number, ω, is reached, but no further. (ω cannot be reached in a finite amount of steps) *finite potential infinity:* 1+1+1+1.. a very large yet indefinite finite "number" of times. (This is not an actual number, of course.) *infinite potential infinity:* 1+1+1+1... that never stops or 1+1+1+1.. to infinity and beyond without stopping at ω or ω+1+1+1.. that never stops *Examples of actual infinities:* ω, ω+1, 2ω, ω^2 The number of discrete units on the smallest actually infinite discrete straight line The number of discrete units making up two different discrete lines like the one mentioned above. The number of discrete units on ω discrete lines like the above. The number of sides of a regular apeirogon or infinity-sided polygon, which reaches a limit of 180 degrees. The number of regular apeirogons that can fit in a 2-dimensional plane. The area of a discrete infinite two dimensional space The volume of a discrete infinite three dimensional space. etc *Examples of potential infinities:* All Aleph numbers, all Beth numbers, the indefinite amount of rooms in Hilbert's Hotel, the indefinite amounts used in one-to-one correspondence arguments, and in the diagonal argument, etc., c, the continuum... The indefinite amount of points on a Euclidean line. The amount of points on a disk, cube, sphere, ball, etc in a continuous space The area of an infinite plane in a continuous space The indefinite amount of "real numbers" between 0 and 1 The indefinite amount of "numbers" on a "real number" line The indefinite volume of the balls in the Banach-Tarski paradox. @Russell Sharpe
@Russell Sharpe This is why an apeirogon is so important to understanding all of this. The sequence of regular polygons {3},{4},{5},{6},{7},{8}... etc, tends toward {∞}, an infinity sided polygon called an 'Apeirogon' made up of countably infinity sides, which has a 180 degree internal angle. At which point, there cannot be another side added, since there is no space beyond 180 degrees (= totally flat) which it can curve into. We can deduce from this that space cannot be continuous. For an octagon, starting at some edge in the middle (call it edge 1) lets move to the right, so edge 1, edge 2, edge 3, etc. We move from edge 1 to the right and eventually get to edge 8 and then if we move right some more, we get back to edge 1 again. Now consider the apeirogon- Start with some edge in the middle, called edge 1 and we move to the right to edge 2, edge 3, edge 4, etc. moving to the right an infinite amount of steps will eventually lead to THE OTHER SIDE of the apeirogon, where we eventually reach edge infinity and moving once more to the right we are back to edge 1. Now remember that this polytope is connected edge to edge at 180 degree angles. In other words, it is identical to a segmented infinite line. So take a very small line segment, call this edge 1, and add another to the right of it at 180 degrees, call this edge two. If you continue doing this, you will start to see the edges showing up on the other side. When it reaches edge infinity, edge infinity will be next to edge one and we can no longer add another edge without either overlapping or leaving this dimension. If we compare this sequence with an identical sequence with large edges, the sequence with the larger edges will get back to where they started before the other sequence, and the last edge will not be # infinity. To get to infinity and get back to where we started, then, the line segment has to be of some exact smallest size. No larger, no smaller. It has to be discrete. The notion of a continuous space leads to contradiction. This modular quality of getting back to where it started is very helpful in this type of discussion because it sets a strict limit on these sequences without overlapping themselves or jumping up to a new dimension. This is similar to the other sequence we discussed: Take one entire apeirogon to be "1". Take a second apeirgon, and count one half of the sides, then count one fourth of the rest of the remaining sides. Then one eighth, and so on. If we continue doing this, eventually there will be a smallest unit. Another way of thinking of this is infinity + infinity/2 + infinity/4 + infinity/8... + 8 +4+2+1+1. And the sequence has reached the end. (The result is 2 x infinity) You could imagine a scenario where you split that last 1 into 1/2, 1/4, 1/8 etc, but this would be getting into a whole new area, and in a discrete setting this would not be possible unless we set up a scenario with even bigger infinities to split. But I think if you were to say that you can get to 1 then split it into 1/2, and 1/4, this would be similar to saying that N and R are equal. There is a way to show from all of this that the first infinite number has an exact sequence of digits associated with it. This is too advanced for you to comprehend at this stage, though, if you still are not following about what an apeirogon is, the difference between actual and potential, about a last digit in the sequence, about omega not being arbitrary, about discrete geometry, etc.
Great conversation, and very though provoking. About the "e + π + √2 = ?" issue, I wonder if I ask mr. Wildberger "what is 1 divided by 3" his answer would be "1/3", because that answer would not comply with his standard of not giving the same question as an answer. So, is 1/3 = 0.333... (an infinite decimal)? I see limits not as an infinite process, but as a direct consequence of the Archimedean property of R, so I have no issue with them. Of course, not being a Mathematician, I have a more naive approach to that concept.
To me the distinction is whether the quantities are expressible in proportion to one-another using natural numbers; 1/3 : 1 :: 1 : 3. Naturals are the only true numbers. To take a quantity 'e' times does not really mean anything to me, unless I am thinking of it as 2.718, but then I am thinking in terms of the proportion 2.718 : 1 :: 2718 : 1000 which is in terms of natural numbers. Also remember that 0.333... is itself an infinite sum of fractions, 3/10 + 3/100 + 3/1000...
@@livef0rever_147 That's exactly the reason of my thoughts... 1/3 cannot be explained exactly as a decimal, it can only be approximated, or understood as an infinite series, and Prof. Wildberger is a finitist. In my concept, "1/3" is just a symbol, and it's as valid as "e", "π", "√2" are.
@@nabla_mat what’s so special about decimals? 1/4 and 0.25 are both fractions. The only difference is that in one the denominator is supplied by the imagination (25/100). Pretty much the only reason we use the system of decimal fractions is because we have ten fingers. What if we had arbitrarily decided to use, say, base 9. Then it would be 1/3=3/9=0.3 and 1/2 approximately 0.444… would you then say 1/2 can only be understood by an infinite series? Your argument is absurd.
Couldn't a real analysis apologist simply argue that real analysis is outside the scope of computation? Because that fact on its own doesn't "disprove" real analysis or its conclusions. The question would then be the usual Wittgenstein question of whether the terms being used - real number and computation - are useful as used. And that's really what I think he is trying to say - some aspects of real analysis are not useful in a way that computation is. And I don't think anyone would disagree with that. But then why not vice versa?
@17:00 the ℂ numbers are not "points in a plane". They are algebraic, as he alludes to a minute earlier. Moreover, the setting for ℂ numbers really is the Clifford algebra, the bivectors or even subalgebra. And the proper setting for _that_ in turn is a Category. Such mathematical objects are not "numbers", they might however be homomorphic to a number system. The whole idea of "number" NJW is trying to motivate here is ill-conceived, or impoverished. You can restrict your notion to "what is computational in finite memory" but that's not all there is to play with in _Cantor's paradise._ When you play, the demand is relative consistency. When you eventually find inconsistency it is perfectly fine, you know you were not doing mathematics, which is a good thing to know, and move on from (or do whatever else you want with, like art). Mathematics, the sociology thereof, has to be such art, for if everyone instantly took to perfection when beginning mathematics it'd be impossible to make mistakes and we'd be gods, that's NJW's "religion". You have to be able to be wrong to be right.
Prof. Wildberger has a realy realy bedagogical method of teaching math i just watch i can repeat what he said from memory i wished if he could teach some of the mainstream math .
Mathematics is a philosophical endeavour at its core. It has been this way since its inception and will most likely remain so. Just because it’s philosophical, doesn’t mean it’s wrong - the real numbers have strong arguments in its favour and more than one method of arriving at them.
Thank you for this stimulating discussion! I wanted to comment on the "cube root of 15" portion. When dealing with the Real numbers, I think of them as extension fields of the Rational numbers, just as was mentioned in the video. However, not just a single extension, but a countable infinity of extensions, one for each Real number that I can name (pi, e, sqrt(n) for some Integer n, etc). The problem is that each extension in a sense introduces a new dimension to the field. (Example: extending the Rationals with with sqrt(-1) is two-dimensional). So the we are left with a countably-infinite dimensional object that we are treating as a 1-dimensional line. The concern is that there may be all sorts of logical inconsistency lurking in that embedding. While I cannot give a direct example of such a demon lurking there, an analogy would be how you can "show" that the sum of all positive Integers = -1/12 which demonstrates the logical inconsistencies that may be encountered when dealing with "completed infinities".
Yes, introduce "actual infinity" and rigor is gone. Anything at all can then be proven and it becomes more of a cultural exercise than an intellectual one.
@@DanielBWilliams From the Wikipedia entry on ZFC, in the section 7: The Axiom of Infinity: Let {\displaystyle S(w)}S(w) abbreviate {\displaystyle w\cup \{w\},}{\displaystyle w\cup \{w\},} where {\displaystyle w}w is some set. (We can see that {\displaystyle \{w\}}\{w\} is a valid set by applying the Axiom of Pairing with {\displaystyle x=y=w}{\displaystyle x=y=w} so that the set z is {\displaystyle \{w\}}\{w\}). Then there exists a set X such that the empty set {\displaystyle \varnothing }\varnothing is a member of X and, whenever a set y is a member of X then {\displaystyle S(y)}S(y) is also a member of X. {\displaystyle \exists X\left[\varnothing \in X\land \forall y(y\in X\Rightarrow S(y)\in X) ight].}{\displaystyle \exists X\left[\varnothing \in X\land \forall y(y\in X\Rightarrow S(y)\in X) ight].} More colloquially, there exists a set X having infinitely many members. (It must be established, however, that these members are all different, because if two elements are the same, the sequence will loop around in a finite cycle of sets. The axiom of regularity prevents this from happening.) The minimal set X satisfying the axiom of infinity is the von Neumann ordinal ω which can also be thought of as the set of natural numbers {\displaystyle \mathbb {N} .}\mathbb {N} .
@@WildEggmathematicscourses After "more colloquially", it's just an interpretation of what the axiom is saying, but If you look at the axiom, never in the axiom in itself it is written that there is a set with an infinite amount of things in it. More generaly, sets in ZFC aren't some "box" or "bags" in which we can store things : there nowhere in the axioms that idea. It's just a way of visualing sets in order to work faster and better, and a lot of mathematicians believe in this interpretation because it can be very usefull for them, but that's all. That's the problem with math education when it is about sets : this interpretation is taught as if it was what's really going on, because it's more easy to explain, but so many people think ZFC it talking about bags or box in which you can store things.
This was really fun, but I did get the impression that you were a little caught off guard by Prof Wildberger’s arguments here. Is there going to be a round 2?
@Tomas Matias answered "I would for example say when Wildberger asked about which sequences he believes in." Daniel Rubin replied that he "believes" in sequences that can be justified by the axioms of ZFC. There is not a distinction between "computed" or "choice" sequences in ZFC. One could limit oneself to "computed" sequences, but then one would be outside of ZFC. There is nothing wrong with that, but that is not the framework that Rubin is working in or discussing. Rubin answered the question: sequences that can be justified by the axioms of ZFC. He didn't seem to me to be caught off guard by Wildberger's question. It seems to me that Wildberger seemed to be the one caught off guard by that answer. Wildberger then drops the question on the distinction and then proposes the question of whether such sequences can be written down in the universe (within a finite amount of time and within a finite amount of memory).
@@billh17 I don't know if you are knowingly misrepresenting set theory by claiming there is no distinction between computable sequences and choice sequences but you definitely are mistaken if you think computable sequences lie "outside" ZFC.
@Gennady Arshad Notowidigdo I gave you a like but I disagree about some points. "Nobody should believe in any idea in mathematics", ok. "respecting what has already been learnt ..." unnecessary. The problem with Wildberger is not that one. I don't care that much about he disrespecting some idea and I have to recognise he builds up from previous knowledge, he actually does that. The problem is that what he is saying about how math people understand real numbers is clearly wrong. I mean, it seems he doesn't understand that saying "sqrt(2) is equal to 1,414..., with infinite decimals" just means "the only way to have sqrt(2) is by approximation". That's it. He insists there is something wrong because of "infinite". This is silly. The infinite in this case just indicates the process to calculate sqrt(2) will never make us get an exact rational number x satisfying x²=2, but we can come close to 2 as far as we want". Because of that, the definition of the real number sqrt(2) is inevitable theoretically, impossible computationally. There is no one making an argument different from this but he invokes these ghosts.
@Gennady Arshad Notowidigdo That's strange, math isn't about believing or not believing in something. What does that mean that the real numbers exist or don't exist ? When I do math in ZFC, I don't believe in anything particularly. I just say to myself "if ZFC axioms are taken as true, what can I conclude", that's all. I'm curious about where do the mathematicians you are refering to use belief. That would be like to say "I don't believe in the existence of the points in Monopoly game". That's a not about believing they exist or not, but about following rules of the game. ZFC is a list of rules, as the rules of the Monopoly game.
@Gennady Arshad Notowidigdo Oh yes I understand what you mean ! Wildberger should discuss with somebody who doesn't claim that so it would be more interseting that with somebody claming that's the only set of rules.
@Gennady Arshad Notowidigdo *"the exact same tactics that I have pointed out"* Which tactics are you referring to ? *"that both sets of rules are equally valid"* Can you remind me what is the other set of rule (other than ZFC) you are talking about ? I agree that there are other set of rules (other than ZFC) thare are valid, but as I don't know what are the other set of rules you are refering to, I can't say if it is valid or not. *"someone who is clearly not being up front and honest about his support of ZFC"* I use ZFC everyday because I think it is beautiful and useful, so I love it. That doesn't mean I don't agree with the fact that there are other set of rules that are valid too. Why are you thinking otherwise ?
The discussion would have been a good idea if the level of discussion was much higher, deeper, precise. Such foundational questions should be addressed to real gods of mathematic, for instance ALAIN CONNES. Otherwise it stays a « youtube » bavardage
I dont understed the problem with limits, its a finite process that model an infinite process and we work in that context Real nums need axioms so they are meta objects Rational nums only need defenitions and logic so they are pure objects When its about math/philosophy i can continue writing all day so im gonna stop here
If I have understood him, Wildberger says that the cube root of 15 cannot be specified exactly, since the decimal expansion is infinite, and we only have an algorithm which generates the digits and never terminates. But in the sense in which this is true, exactly the same is true of one third. The only difference is that the algorithm is a bit simpler for the latter (do{write('3')}). If it be objected that we _do_ have a way of specifying one third in a way which does not require potentially infinite processes (just express it as a rational) then we can say the exact same thing about the cube root of 15: just express it as a root.
In fact, it is not only real numbers which Wildberger is sceptical about. He spends a lot of time in his videos constructing larger and larger natural numbers to make the point that there are scarcely any for which we even have - or could have - a feasible notation. This is obviously true: there are so to speak vast expanses of inaccessibility between the extremely sparse integers we can construct names for. With this in mind he (for example) denies that all integers have prime factorisations, for the primes required are in general not expressible in any notation we can invent and use, and he rejects any notion that something might exist but be beyond human capacity to (as he likes to say) "write down". What this means is that he doesn't even believe in large integers, let alone reals, but only in sufficiently small and/or tractable integers. Or maybe it is not that he believes in some but not others, in a binary way. Perhaps he believes that some integers are more real than others, their reality diminishing with the feasibility of their minimal-complexity notation. (Perhaps 843639613031849 and 10^(10^(10^10)) are roughly as real as one another, and significantly more real than the unimaginably many integers lying between them which have no representation less than 10^(10^(10^9) characters long) This seems to make the sequence of natural numbers inconceivably complex and difficult to handle, as well as rather vague and subjective. To insist that you should believe in only what you can actually observe (or 'write down') reminds me of Berkeley's idealist metaphysics: esse est percipi. But does the world blink out of existence when one closes one's eyes?
@@WildEggmathematicscourses Why should the only things that exist be things which we can write down? Cosmologists tell us that as the universe expands galaxy clusters are continually passing beyond the limits of the observable universe. We don't assume that as they become unobservable they actually pass out of existence altogether. That would be crazily solipsistic. I don't see why we should be mathematical solipsists anymore than cosmological ones.
@@russellsharpe288 There ought to be a distinction between those phenomenon which are unobservable because of limitations on our direct instruments and those things which are completely and even theoretically beyond our observation. In physics observable has a broad interpretation: a galaxy might be beyond our radio telescopes, but perhaps its gravitational effects on something else is still recognizable. Something like how exo planets were first identified by their wobble effects on their stars. But please explain how P = {primes larger than 10^10^10^10} manifests itself in any fashion in our universe.
@@WildEggmathematicscourses First you'd have to explain how any number or set of numbers at all "manifests itself in our universe". Do you mean eg "is the number of a certain kind of thing in our universe", like say the number of electrons, or the number of ways such particles can stand in certain relations with one another? (Moving from the number of objects to the number of ways they can be combined produces very large numbers indeed of course) Or do you mean "occurs in our best physical theory of the universe". It's a bit odd to demand such a thing, isn't it? Does eg 32856592026365145058662950572640671230337000377644034712965711054753 manifest itself in our universe? How, exactly? Whether it does or it doesn't, can this really have any bearing on whether we should regard it as real as, say, 137? As far as mathematics goes, much larger numbers than 10^(10^(10^10))) crop up all the time. I see the Goodstein sequence starting at 5 has length greater than 10^(10^(10^19728))), and the one starting at 12 has length greater than Graham's number. But of course all Goodstein sequences do terminate and so have finite length. Do we really want to say that despite this, their lengths do not exist just because we cannot write them down? What we can and can't write down is not a well-defined notion anyway: we are either faced with a sorites paradox or else have to admit different levels of reality to numbers according to their relative tractability; level of reality which themselves will not be well-defined. This is a can of worms, surely? Does it really seem like a good way to go?
@Gennady Arshad Notowidigdo Do you mean flawed or just unclear. If this was the first time I came across these arguments, I would have skipped this video for sure.
@Gennady Arshad Notowidigdo I agree and have no affinity to compare mathematics with computation let alone computers, but I also think that not wanting to discuss a problem because one might find that the solution is to much work is not a good thing. Look what is happening in quantum physics and the standard model. They are suffering and progress is slow or even un-existing.
@Gennady Arshad Notowidigdo It's enough that he points to the problem and what the solution would require. He can't reinvent all of mathematics himself, possibly not even any of it, but that doesn't mean it doesn't need to be done. A more rigorous mathematics should yield easier computation.
@@ga35am What base is your tea leaf and fruit cake sitting on or are you saying a tea leaf is a one and the fruit cake is a 0 - a sort of picture maths representation? I get that fruit cakes are circular and tea leaves grow on stems but how do you add them together? When I was young I was told you can't add oranges with lemons. But you can drink lemon tea. Lol
Writing down something like 1/3 is no better than √2 because you will never be able to write down 1/3 fully. It's going on and on 0.3333... The same holds for e + π + √2. And all these numbers do exist because π is the length of the circle with a radius of 0.5, and similar geometric interpretations can be made for √2 and e. What's more, these numbers exist no matter what our universe is or what the laws of physics are. It doesn't matter if we have 3 or 4 dimensions or whatever. Mathematically, there exist many dimensions in Euclidean space. It also doesn't matter if our space is not Euclidean. Mathematically Euclidean space does exist. Triangles of infinitely small thickness exist. Not in real life but in math or in logic. It's not a philosophical claim. It's a logical claim. It's just logic. That said, I express my respect to the both mathematicians in this debate. They are both right. A huge like for the video. One point though: the point of disagreement was not really fully developed. Probably there's no real disagreement. I'm sorry but the whole thing looked to me like this: f(x) = x^2/x is the line y = x with a hole at point 0. An infinitesimally small hole! The opponent: there's no infinitesimally small hole there or maybe there is but it's all nonsense as you can't write it down or feel the infinitesimally small things. Perhaps it's not even there as we ourselves prohibited division by zero but allowed multiplication by 0. So we simply don't know. I agree but I also agree with the opposite: there's a hole there; we defined it that way (no division by zero is allowed) and that's why there's infinitesimally small hole which we also define recursively (I'm fine with it). Yes, the theory of limits came long after calculus. It's an artificial logical construct. I'm fine with it though. But I understand and respect the opposite position too. What's more, I think excessive formalism and too much limit theory in calculus might be bad and confusing, especially for students in engineering. Yes, it's an abstract artificial construct but it's okay as far as I'm concerned. There's a duality in math and real life, i.e. √4 = 2 and √4 = -2 (in the field of complex numbers) but it doesn't mean we have to conclude 2 = -2. Or do we? We have opposite things as a result. We have duality. In real life this duality is well represented by elementary particles. They are particles and/or waves. Is it _and_ or _or?_ This can be debated till hell freezes over. Can they be particles and waves at the same time? Can √4 be 2 and -2 at the same time? How can a particle be a wave? How can √4 be -2? How can 1+2+3+... =-1/12? It approaches plus infinity. Hence, it's not -1/12? Well, how about Casimir effect? How about that crazy duality in real life? How about this in real life: 1+2+3+... =-1/12 on the one hand and on the other hand plus infinity. I know this series infuriates lay mathematicians the most. I don't have problems with it either just like with the opposite stances of the mathematicians in this video.
Yeah, and I assume Wildberger would say that they don’t exist either. But I don’t understand why he thinks that mathematical existence requires observability or computability.
You cant just claim that everything random thing backs up your position. You sound like a religious nut saying that trees or the beauty of a sunset proves god exists. Points lines and planes do not construct the real numbers, nor do they require them. The constructible numbers do not contain every real number, I cant even tell what point you were trying to make.
I’ve often noticed that, for a lot of people, when they first hear that “Wildberger rejects infinity and infinite sets!”, they develop an immediate idea of what it is he must mean. They imagine, as does the host here, that he would concur with statements like: “There are no algorithms which begin and then never end!”, or “Newton’s method to approximate pi is wrong, simply because you cannot approximate the ratio of radius and circumference at all, period!”, or “No one has ever stumbled on ANY useful scientific results using e or sine waves!” Yet none of these is what Wildberger is saying! A key point Norman makes here (though it seems the pure mathematics community of today has happily forgotten it & is quite content to imagine it was never so), is that the broad base of analysis got built long ago, without this modern foundation on top of infinite sets & limits, and indeed it WAS successfully used to solve a grand part of the same sorts of practical problems for which it’s still deployed in the sciences today- but based on wholly different foundations! A problem then and a problem now is not that the approximate results that are obtained by these methods are not useful, or are not more or less accurate, but rather that the alleged building blocks used to explain and build the theory of WHY they work, from the ground up, are misty and dubious, and so an alternative explanation of WHY analysis works remains as badly needed as ever. As to crises of the Kuhnian variety that the host mentioned, this inability to determine the sum of the three most common irrationals, as “real numbers”- that it isn’t considered a sufficient crisis to merit scrutiny of their foundational underpinnings- especially given the history of this- is actually pretty remarkable. What could make the real number concept, as it’s built up today, more useless than its own perpetual supplantation, for practical purposes, with rational approximations? Maybe the crisis will demonstrate itself more plainly once a generation of new scholars has utilized non-infinitist frameworks to make some big leaps past current understanding, with the help of computers, as Wildberger suggests, without resort to infinite sets, axioms of choice, etc. Another key distinction, which I’d hope that the host and other skeptics of his perspective would consider for a moment, is between the idea of, say, an integer as a specific “type” of mathematical object vis a vis the set theory thing, where all the integers have to be put into some virtual bag or basket... which requires we first invent these bottomless virtual bags or baskets. Norman has videos where he shows how to build these numbers from the ground up without resort to an infinite set. IMO the math comes out all the clearer for it. I’d strongly suggest people take a look at this alternative methodology with an open mind, because they might end up quite surprised.
Yes, an alternative explanation of why analysis works is much more important than knowing exactly about sum of the irrational and transcendental constants. sin(x), exp(x), sqrt(2), pi - just a short record (name) on paper of the algorithm for calculating them. For practical constructive computability - everything is an algorithm. The simplest algorithm in mathematics is a polynomial. The most important fundamental block of analysis is the analytical function. What is an analytical function in analysis - only and strictly only a polynomial. For analysis oriented towards practical computability - no exist non-polynomial functions. My answer why the analysis works - because these are the algebraic properties of polynomial functions, without the need to use any linearizations and limits. Just consider the full form of polynomial functions - coefficients in all terms must be variable (not fixed). It is both simple and difficult at the same time. At first, it is difficult to turn on such vision for functions, but when you turn it on, it will become very easy to look at the analysis.
More than 1 hour of discussion, just to hear one single argument over, and over, and over, and over again: Wildberger likes to suggest that every mathematical object must be computable and must have an analogue in the "real world" (whatever this means). How tiring.
Frankly I don’t understand Wildberger’s insistence on rationals. The number 1/3 is just as “infinitary” as π is. Have you ever tried to a cut a pie exactly into 3 equal slices?? It’s totally impossible. Should we reject the concept “1/3” then??
@am p My point is that "1/3" also requires an infinite process to think of conceptually. If you object to my example with a circle then think of how impossible it is to cut a rectangle perfectly into thirds. Sure you can think of fractions as these algebraic symbols satisfying certain rules, but that's not really the point in my opinion. From a conceptual point of view, 1/3 is just as complicated as pi.
@@anthonyymm511 Not really. Take a stick: ─ Then measure out a length of 1 stick: ─ ─ The ratio of these is 1:1, or 1/1. Then measure out a length of 3 sticks: ─ ─── The ratio of these is 1:3, or 1/3. They're commensurable, whereas the length of the hypotenuse of □ is not commensurable with the length of its sides; there's no way to measure out 1, 2, 3, 4, etc. side lengths to reach the hypotenuse length. Calling it an argument by notional convenience would make yours an argument by notational inconvenience. In a way, you may as well choose one very small unit length and build everything out of that. 1 will be 1000000000000000/1000000000000000, etc.
Then you can build pi, e, root2, etc. but of course it's approximate not exact. Still I think pure math is a phantom. Choose a minimal tolerance and work with that.
@@ThePallidorBut in your argument you have 3 sticks of equal length. This is already impossible, but even pretending it was possible you can simply take two of the sticks and form a right triangle. Does the hypotenuse not have a length??
What Wildberger doesn't (want to) get(s) is that his rational domain of mathematics also isn't anything else but a human construction - just like the reals. The none explicitly computability of irrational numbers isn't a significant distinction - especially not for pure mathematics. It's telling how he calls reals "religious fantasy", which shows that his actual problem is that he is believing in a "religious" signficance of the rational domain which he wants to protect from something he deems to be unpure as it isn't explicitly computable. He's projecting his own error - some platonistic view of mathematics. He should discuss this with an actual set theorist or mathematical logician (not analyst). He has little ground to stand on but on stubborness in some arbitrary insistence on explicit computability. All you need to know about his critique of reals is who he attacks while critising: introductory real analysis textbooks. Such text books arent ment to go in depth on reals. Reals arent studied in analysis but taken for granted. They are studied in more foundational mathematics. He knows that his arguments arent good enough to critique the fields actually dealing with this topic.
There are a number of videos that Norman has done in which he examines the topic in greater detail. Separating numbers into their different types is a good start cos then we can say (& understand) what we can do with them and what their limits are.
@Jim Yocom, Actually I would like to defend Daniel on this point -- he did indeed ask me about some references prior to the debate, and I gave him a few videos to watch and also my paper, and he summarized some of my points quite well. It is a big ask to get someone to watch the dozens of videos I have on the weaknesses of the foundations of "real number arithmetic" and "modern set theory" etc.
Is mr. Wildberger an intuitionist? A constructivist? A finitist? Does he accept the law of the excluded middle? He seems to reason "by computer algorithms". What is his framework when he talks "about computer algorithms"? The discussion would be easier if mr. Wildberger would not only say what foundations he opposes, but also whose -- I want to hear names -- foundations he supports. Is L.E.J. Brouwer too vague in his opinion?
Norman Wildberger gets a lot of stick but he has a point? Is π/e a rational number? Because if it were, we could use the fact to determine the n'th digit of one of them knowing the digits of the other and so on. Notice that e×(1/e), which I assume is transcendental × transcendental = rational, so my conjecture is a possibility. Saying that he is not happy at all with ∞ appearing in a limit, claiming in fact it is an absurdity, basically stating that ∞ ∉ ℝ. So lets define a new number set, call it W = {ℝ \∞ }, and see what rules of arithmetic can be rescued using it. Of course we would loose concept of limit in the general sense, like ℓim x-> ∞, that would not be acceptable in a _Wildberger_ field.
@55:35 or so, Norman objects to adding 3 irrational numbers, but when he points out that it’s analogous to asking a grade 6 student to add three fractions. This leads to the following, which is my “domino theory” approach to Norman’s philosophical objections to irrational numbers: I claim that this philosophical objection reduces to an objection to the validity of the equation 0+0=0, since his objection is metaphysical, and the formalist objection to the Platonic view of the metaphysics of mathematics already objects to “actual” existence of any mathematical objects, including the number zero. If you actually object to analysis as a form of religion, then you might as well object to all of mathematics as a form of religion. If you want to object to mathematical concepts using that metaphysical concern as your foundation of objecting to things, then why do you call yourself a mathematician? Is that not a form of hypocrisy?
...does the theory of real analysis anywhere state that it is possible to calculate everything exactly using a finite decimal representation? If not, then I don't understand the arguments by Wildberger *against* real analysis. He also barely lets Rubin respond.
...does the theory of real analysis anywhere state that it is possible to calculate everything exactly using a finite decimal representation? no, only computable numbers can be calculated exactly. Every number youve ever heard of is computable, so including non-computables is a mistake on the part of real analysis.
@@bjornsundin5820 then im not sure what point you were getting at even rationals dont have finite decimal representations, before even considering reals
@@samb443 I should probably have said "finite decimal representation or fraction" or "decimal representation with a repeating or terminating pattern" then. But his definition of what is allowed as "as long as it can be written down" is vague. 10/11 is a label for a number and pi is a label for a number. Both can be written on a piece of paper. Both can be rigorously defined. I respect the attempt to remove e.g. the axiom of choice and still end up with a useful mathematical theory. But I do not understand the criticism of real analysis based on the fact that real numbers cannot be written using a finite or repeating decimal representation, as if the theory requires this arbitrary constraint.
The problem with settling for saying whether "π+e+sqrt(2)" is greater or less than some rational number is that the "number" believed in here is not a definite value. You can do more and more steps of some iterative calculation to get a more and more precise value, but there is no definite value that is being approximated- every new term in the iteration is a new value, a slightly different number. To believe in 'real numbers' is to pretend that each new term is just giving you higher and higher resolution of some definite value, but by definition, irrational numbers cannot have definite values. There is no square root of 2, two does not have a square root.. there is a sequence of steps of numbers that are very close to what the value would be if it was a definite value. Here is an analogy- suppose you have 1 blue marble, then you are given an additional red marble and blue marble, and keep adding more pairs of red and blue marbles. There is always one more blue marble. The more pairs of marbles you add, you will be closer and closer to 50% red and 50% blue, but you will never have exactly 50/50 since there is always one more blue marble.
In ZFC, what you call "a value" is just a name among a lot of other names for the same set, it is called "decimal representation". But that is just a name, not the only way of defining numbers, so even if we don't know the decimal representation of π+e, that doesn't mean it isn't defined.
@@DanielBWilliams Hi Daniel- For the point I'm making you don't have to get all the way into π+e+sqrt(2), that is just the example that was brought up in the video. My point is that the "square root of 2" (for example) is not a definite value, (not that it is not defined)..that is, it is not an exact amount. It can be convincing that rational numbers can be given which are greater than or less than any nth iteration of some calculation, but that does not make it a definite value or an actual exact amount.
@@ArtCreatorsChannel In ZFC, decimal representation of every real number exists and is defined, it is just that for instance for √2 we don't know the list of the digits. What makes you think it's value isn't defined ? (maybe what you call its value isn't what I call decimal representation)
@@DanielBWilliams Right- what I am calling "value" or "amount" is not the same as decimal representation. An example of a valid decimal representation would be 0.0625, since 625 x 16 = 10000, and the relation between 625 and 10000 is the same as the relation between 1 and 16, in other words, "1/16". The same cannot be said for √2. There is an indefinite sequence 1, 1.4, 1.41, 1.414, 1.4142.. etc which gets more precise with each new term, each new term being a slightly different value, a different number, a different amount. The symbol for this sequence, or some imaginary completion of the sequence is "√2", but this is fictional. There is no actual completion of this sequence, and "√2" does not have a definite value- each term in the sequence is a slightly different amount, and there is no actual value to "√2" since 2 does not and can not have a square root. There is not a number, that, when it is squared, equals 2. We do a sequence which gives us numbers getting closer and closer to 2 when it is squared, but like my marble analogy how there will always be one more blue marble, no number will ever be the square root of two, it can't be, it is not possible. The imagined value for "√2" is not what we are taught to think it is. Even with infinity digits to it's indefinite decimal expansion, it will never be a single value, and any truncated approximation at any given term will never be a number that, when squared, equals 2.
@@ArtCreatorsChannel But in the set R, there is an element x such that x²=2 and x>0. It is unique, and so we call it √2. Even if we can't list all of the digits of its decimal representation.
His first argument was that modern math is rooted in philosophy. His argument is also rooted in philosophy. So if being rooted in philosophy is bad then so is his argument.
He shouldn't attack philosophy, just the philosophy departments of today. Philosophy is better characterized as the neglected foundations of every field. In fact the division of academia into fields introduces many errors itself.
Hi, I think if you look at what he is genuinely saying, you may find that his point is that modern mathematics is based on beliefs. What he is saying is that math should be based on what can be proven. What can be proven should be considered true. So I think his distinction is truth vs belief, not your philosophy bad mine good.
I really wish Daniel Rubin cared a little bit more about computing because it's pretty much impossible to be a hobbyist computer programmer and pluck your favorite real function off the textbook into a tiny little program to play around with. You should be asking what is needed to create an exactly computable real number or somethibng,
The simple fact that the addition of a few irrational numbers cannot *_in principle_* produce an exact answer *without infinite work* seems worthy of far more pause than was exhibited. Great discussion!
@@coffeyjjj As every mathematical objects, a number has a lot of ways to be written. For instance, the number 24 can be written as 2³×3. So 24 and 2³×3 are two differents ways of writing the same number : one is the decimal representation, the other is the prime decomposition. If I know one representation, and not the other, that doesn't mean I don't know exactly the number. For instance, 3+4 is exactly defined, and I don't need to give the decimal representation to do it. Of course the decimal representation is 7, but I didn't need it to say it is exact. Generally speaking, a mathematical object is (exactly) defined when you have given a property verified by only one object of that kind. For instance, we can show that in R there is a unique element x positive such that x²=2. So we can define √2 to be that number, and as : 1) in R there is at least one number like that 2) in R there is no other number like that The number is exactly defined. After that, we can of course search for its decimal representation, but that would just give us more information, the number wouldn't be more defined.
@@DanielBWilliams - Sophistry. My comment *explicitly* mentions "exact answers" not "definitions", and was obviously in reference to this specific Wildberger video containing his *explicit* objection clearly noting the fact that *infinite work is required* in order to generate the DECIMAL REPRESENTATION of most real numbers. Your claim that I'm misunderstanding something appears to rest on very, very shaky ground, friend. How may the reasoning of your "definition" be used to *explicitly determine* each of the N decimal digits 0-9 located at the 10⁸⁰⁺ᴺ place in the decimal representation of √2, where N is all integers from 1 to 63? Or how about each of the N digits located at the 10⁸⁰⁰⁺ᴺ place of the decimal representation of √2? The "exact answer" I originally mentioned would provide those digits. I don't think your "definition" can identify those N digits, not even in principle. Do you think it can?
@@coffeyjjj That's what I was saying. For you having an exact answer means to know the decimal representation, but that's just a representation. Your problem (and Wildberger problem) is to think not knowing the decimal representation is not knowing the number. That's not how maths works. Knowing a number is just knowing a property that it's the only mathematical object to verify, and as you said it is the same thing as defining it. Why do you care so much about decimal representation ? Yes it is usefull, but as I said, it is *just* a reprensation among a lot of other. Yes ZFC can't provide decimal representation to all of its numbers, so what ?
@Gennady Arshad Notowidigdo Norman would do well to take a look at basic epistemology, David Hume being the most revealing, because once one realizes that first-person epistemological phrasing is the most foundational phrasing, it quickly becomes clear that labeling a mathematical object that is not visualizable (or otherwise sensualizable) _even in principle_ is labeling nonsense.
He dismisses philosophy as an airy subject, which it is today in modern academia, but the proper role of philosophy is to serve as the foundations of every field and therefore it should be the most rigorous of all. There is a rigorous way to do philosophy, but it requires clear definitions used consistently, which academic careerists have a heck of a time doing.
When Norm invokes "what we can write down" or "what our computers are telling us," he is reaching due to this lack of clarity on basic epistemology. As he says, a proof must in principle be an argument all the way from the foundations to the final result, but the real foundations of mathematics are in epistemology. Steve Patterson's Square One is a good easy intro to this, though David Hume's _A Treatise on Human Understanding_ is more thorough. Otherwise he opens himself up to the objection that his computational view is just another view or approach.
@@ThePallidor Maybe there is a logical way to do philosophy, maybe not, but it's a big stretch to say that David Hume is a necessary prerequisite to mathematics. Personally I'm still convinced that real numbers are a fantasy and I don't see how anything you've said about epistemology proves otherwise
@5:55 nonono... real numbers, like π 𝑒, and all those you'll never know, are not _defined_ by decimal expansion. The representations of concepts are not the concepts themselves. The whole point NJW misses (it seems) is that mathematics is not physics and is not computational, if it were it'd be the lesser. It is _because_ we can entertain nonphysical concepts with mathematics that allows us to refine physics (and a lot more besides). Science becomes feeble if we have no ability to entertain what is nonphysical. That's because so much of science is _negativa_ --- it's often what is _not true_ that you know the best.
@15:00 he admits this when he pivots to "algebraic concept ok". But "point on the number line" or "point in the plane" is not a mathematical concept, it's physical, or invoking physical intuition. There is no one true notion of a "continuum".
@1:03:00 you can perfomr transfinite arithmetic in some software. That fact you cannot do more is because no one's yet written the general code, but they have not written general code even for a tiny subset of ℕ either, so this "write-downable" is ridiculous. Everytime what we can write down increases he is saying mathematics expands. The mathematical expanding universe. But I doubt FRWL would've approved. It's an impoverished view of the word "mathematics" if you you constrain it by "what we can write down". Moreover, if he is saying "what we *_could_* write down" then it is completely undefined, since he has no idea what in the future is possible.
40:12 This is simply the principle of mathematical induction which Prof. Wildberger is assuming, which is one of the core assumptions for constructing the natural numbers, which leads to the construction of real numbers and real analysis.
Thanks Daniel for posting this interesting discussion! It was a lot of fun and I look forward to another chat in the future.
Great job Mr. Wildberger! I'm a recreational maths guy, and I remember how surprised I was when I first encountered your views on these subjects. I didn't know it was okay to object to such things, but even as a layman, I've sometimes felt suspicious of some of the faith expressed in the exacting nature of many a 'simple' construction. Always s point where the def'n giver just 'trails off', or points at something else... More study required here🐢 Keep it up, good sir!
@@maynardtrendle820 I just hope you don't base your opinion on math on what Wildberger is saying. When you look at the argument from Wildberger about infinity in math, they seem to be from someone who doesn't completly understanding formal language and set theory.
@@DanielBWilliams I refuse to believe wildberger has ever taken a set theory course. He is so insistent that addition of real numbers has never been proven to work. He never defends this position and anyone whose taken set theory 101 can tell he's wrong.
@@almightysapling He has literally dozens of videos spending hours dissecting the various problems with all forms of 'real' numbers, from axiomatics to infinite decimals, to Dedekind cuts, to Cauchy sequences, and on and on.
When you criticize someone for 'never defending' their point of view and being 'ignorant' -- while being ignorant of their exhaustive defense of their point of view -- you come across as embarrassingly hypocritical.
This interview was only an hour long and it's next to impossible to go into depth on the topic in the time available.
And I doubt anyone who has only taken 'set theory 101' could give a proper definition of the 'real' numbers. 🙄
@@DanielBWilliams I confess to the charges: I "don't completely understand formal language and set theory".
This was really cool! Mr. Wildberger probably never gets to talk about this subject to anybody worth their salt. You can tell he was excited to present these ideas. Maybe do it again (soon) with some of the foreknowledge of the other's arguments. ☺️ Great job guys!
Thanks for hosting Prof. Wildberger in this discussion/debate, Daniel. I can imagine it must have been very interesting and/or discombobulating to face these kinds of ideas 'fresh', so to speak. It's a lot to take in at once! 😅 You were a very gracious host, and it would be interesting if you were to do a follow-up video with your further thoughts, or perhaps a second interview after digesting this one. Cheers!
I think I will post a follow-up, and I hope to have more discussions with Norman in the future.
@@DanielRubin1 if you wish to do so he is wrong about ∞ essentially noone uses ∞ but rather many finite concepts, except for set theory which is religious infinitism.
(Finite concepts such as:
arbitrarily large finite number
or it's axiomatic idealisation,
all finites,
or infinite increase rather than infinity, using a noun when only verb would be correct,
or infinite (such as √(πe2) which actually does exist per gamma function estimate formulas) from finite algorithm thus, finite algorithms.)
He is angry not over ∞ but religious ∞ per set theory, but does not attack it's errors as I do.
The places set theorists refer to nonfinite ∞ are not so. Thus they have nothing to learn ∞ that is infinite from.
So for example the set of real numbers equals the set of integer numbers in size, etc etc etc. So very many errors in set theory infinitism.
Religion refers here to incoherent babbling held in esteem, not to a point of view. Infinite number (in the literal sense) is nonsense. Finite number is a pointless term in a technical context, as all numbers are limited, and so therefore is "finitism." There are no isms, just clearly communicated concepts and incoherent ones.
@@ThePallidor no, per ∞ I can prove the size of c is not strictly larger than N, the ∞te sets don't have sizes, the proofs actually prove all the ∞te sets considered are =, in size to all each other, and by many functions such as +2, ×2, square, exponential, etc
Per finite he doesn't demonstrate any harm from ∞ as it actually is used, axiomatic idealisation of sufficiently large number such that larger doesn't help thus effectively finite number, or another all finites (such as proof of four color theorem for every finite, or no win to the Mathematician
Aren’t there rational numbers whose full decimal expansions can’t be calculated with currently available computing power? Or with any computing power that will ever fit inside the universe at any point in the future?
This was such an enjoyable way to spend the hour! I think we do ourselves a disservice by limiting theory to what is computable as computational reality is dynamic and continuously evolving - however the thought experiment surrounding how real the set of R is will have me thinking for some time. Wonderful content… subscribed and looking forward to diving deeper into your channel!
To me this wasn’t a debate. It was a one-sided exposition
By Wildberger’s argument, if a criterion for numbers being real is being necessarily related to “write down-ability” then, since a “point” in geometry is a dimensionless object, it is therefore not “write down-able” and therefore not real.
Sort of I think. He does explain that you have to prove the properties you wish to have, not just assume them.
It is not the dimensionless that causes the problem, but can you prove that you can add these 2 points and not end up with some Banach-tarski paradox.
In my limited understanding that is what I think he tries to say.
@Gennady Arshad Notowidigdo ages? Like how many years is an "age" ago? And what if it is "solved" ages ago, that does not mean there is not a better way to teach it, or that there is not a better foundation from which to "build" upon.
Yes or a circle, which is an infinite set of points equidistant from a given central point. We have no problem working with circles though.
True, an Euklidian point is not real.
@@tomasmatias4109 well if you manage to show ZFC is inconsistent, I don't doubt there is a Fields medal coming your way
He doesn't do that, or even attempt to do that though...
There needs to be a distinction between the actual physical world and the ideal Platonic world
In the Platonic world ideal (perfect ) circles can exist, in the actual world not so much
In the Platonic world ideas involving infinity are accessible, in the actual world not so much
philosophy of mathematics is more sophistecated than your crude, simplistic view proper of two thousand years ago... 😄
@@MrKidgavilan insightful
The debate was basically about (computable) definable objects versus ideal objects.
These are generally equivalent, but they don't differ on anything fundamental.
Opinion:
Ideal objects are less hassle (and thus better), but just use whatever makes you more productive.
But computable and definable objects differ: for example en.wikipedia.org/wiki/Chaitin%27s_constant
Amazing conversation
Mathematicians duelling at 40 paces and Wildberger was left standing. Love it!
I love this. I'm gong to have to check out more of these insights into mathematics.
I really enjoyed that. I’m philosophy trained, and therefore basically on NW’s “side”, but…
1) Analysis is far more efficient than discrete mathematics. Example: if you wish to find the thousandth term of the Fibonacci sequence, step by step addition takes a long time. There is an analytic extension of the series - the Pi function - which will do the job much more efficiently. But it will also calculate results for values of “n”, or rather x, which are not whole numbers. There is no square-root-7th member of the series, for example. The Pi function does not generate the Fibonacci sequence. The sequence generates - via some mathematical hard work - the analytic extension.
2) You see it as obvious that if every natural number has a successor, then there is a logical final result, an actual infinity. But in the real world a rich man may always own one dollar more, and then another… he can never have infinite wealth.
3) Von Neumann: “Young man: in mathematics you don’t understand things - you just get used to them.”
These are interesting discussions, but ultimately they are philosophical more than mathematical (which is not a bad thing at all). As such, the aim is for maximizing internal consistency, while recognizing there may be many equally internal consistent views. And its never clear what unknown path might exist that exposes inconsistencies in any other path, and replaces them with a better framework. There are certainly consensus views in math, as in any discipline, that could later be challenged from an alternate viewpoint. It’s useful to consider historical naive certainties that later appear quaint, and then consider one’s own views as a future history that will be viewed in a similar light.
@DanielRubin1, Thank you for hosting and being such a patient and graceful host. @wildberger has good points and is clearly excited to share.
The discussion around 40:00 about how in combinatorics the power series functions a treated as unbounded generating sequences is quite a good contrast from the analytical perspective that these functions produce a final real-valued output (from an infinite sum). I don't think analysis isn't worthy of analysis (heh) but it shows how one can choose two interpretations of the same concept and come up with a constructive one and a non-constructive one. I wonder if this kind of dualism is universal...
I think there is a big issue for which this question can not be sorted out, eg that the question to "keep in pure mathematics only what's science and not philosophy" is a philosophical question!
This is such a great video. NJ Wilderberger is such an interesting person. It would be so easy to dismiss his opinion and write him off. But you actually take him seriously and really have to think about why you disagree. I disagree with him too but as someone who is far less knowledgeable in mathematics I have a much harder time figuring out what I disagree with and why, especially because I don't want to just dismiss someone because their views challenge the established I understanding of the subject.
I agree this seems to be largely a philosophical disagreement, even if he himself doesn't see it that way. And honestly that's part of why I find it so interesting.
Honestly I found your channel searching about him just because I find his view of mathematics so interesting. It's so radically different from mine. I kind of wish more mathematicians would have done something like this. I have a great deal of respect for you for doing this. Definitely subscribing.
"I myself am not a set theorist or logician" can someone please get one on? Since that's who needs to be going head to head with this guy because that's his issue. Definitions and philosophy. Not analysis.
All fields have to assume some philosophy, including mathematics. Wildberger just has a different Philosophy. Trying to separate analysis from philosophy is pointless though.
@@presence0420 @Presence 0 Philosophy shouldn't mean a bunch of points of view; it should mean (utilitarianly: would be useful to have it mean) the foundations of every field. The philosophy of mathematics should thus refer not to various people's perspectives on how to do math, but to the basic underlying epistemological foundations of the subject, which need to be the most rigorous of all.
It's merely that mainstream mathematicians have neglected these foundations and relegated them to the philosophy department, which is even more clueless about rigor than most mathematicians are.
There is one philosophy, not many. There is one correct way to apprehend the action of mind that underlies what we call "doing math" and many confused ways. It's not a matter of arguing various opinions; it's a matter of speaking clearly enough that everyone recognizes what exactly we are talking about.
@@ThePallidor And that's why ZFC axioms are clear enough that when you have learnt it correctly, there is no ambiguity at all.
@@DanielBWilliams ZFC axioms are the poster child for disclarity in mathematics and a mockery of rigor.
"Write it down and then we talk about it" is my take away. Sounds like it is better if we construct the objects that are going to be used in any mathematical theory or construct,,,and it is just fair. If the object can not be constructed, then the process of constructing the object of the discussion is lacking from the discussion. Therefore, the process to construct the object needs to be developed, no matter how coarse (is that an algorithm?). It is the process of constructing an object that keeps the focus on the discovery and construction of such an object
Great talk. Loved the smiles at the end :)
At 58:31 Daniel mentions some concept of being able to compare it to the size of any other real number, and claims that you could. But I think, that, is incorrect. Real numbers are in general *not* comparable to other real numbers, in that there is no algorithm to determine that. It's like saying you can compare any computer program to see whether it will halt or not - but in fact we know this is impossible. It's the same for the real numbers. There is a subset of real numbers which are comparable (vaguely corresponding to "computable numbers") but in the general formulation it can't always be compared.
The requirements to meet definitions surrounding real numbers involve computation. So whenever you're in the situation where a thing can be approximated better and better, say by some proven formula, then you may as well designate a symbol and call it a number (or class, etc).
For all there exist kinds of statements involve ideas of infinity. But you'll have that formula there which may be used on any number you like.
where do you see a computation in the least upper bound axiom for the existence ofof reals???
That isn’t an axiom. That’s something all the standard texts prove as a consequence of the definition of the reals.
@@anthonyymm511 It is an axiom, but the presentation of it is disingenuous. You can prove the least upper bound axiom for decidable sets.
In order to "prove" it in general, you must assume perfect knowledge of undecidable sets, basically assuming you can solve the halting problem, which is clearly an issue.
That assumption is the hidden axiom.
"Pure mathematics is nowhere near as old as applied mathematics", maybe, but what Euclid and the Greeks were doing was pure mathematics. Look at the compass and ruler constructions; why did they care that you couldn't exactly trisect an angle or square the cube using a compass and ruler if all they cared about were approximate applications.
You miss the point hidden in the meanings of the word « approximate ». Greek geometry was not about numbers! But about GEOMETRIC OBJECTS AND MAGNITUDES. Euclide knew that Pythagoricians proved square root of two was not rational. But he didn’t say it was « irrational ». He stayed on the contrary in purpose on the other side of the Stix, looking at geometric objects as geometric objects, not confusing them with dubious « measures » of them, and creating a name, magnitude. And they didn’t look at the diagonal of the unit square as having square root of two as a lenght or measure, but as a magnitude. They thus restricted very carefully quotients and ratios to quantities that had a common measure, like 2 and 3, by taking the magnitude of the trisection of the arbitrary starting unity, to be the new unity of rationalisation. In this regard pi was never a number for them. Never. It was on the contrary à magnitude but that was not measuring the « commensurability » of the perimeter to the diameter of any random circle, but on the contrary naming as « pi » the NON COMMENSURABILTY. In other words Geometry was for them an exact science. While arithmetic was just an inferior approximate one, because it could not handle the measure of the diagonal of a unit square or the commensurability between perimeter and diameter of circles. Totally opposit point of view as the one which is taken nowadays
I have a hard time with Wildbergers flavour of finitism, since I prefer frameworks for which at least in principle a full first-order logic axiomatization or similar is on the table - so that everybody knows all the legal formal rules "of the game." That said, this discussion was more about championing and questioning the early 19th century analysis, anyway. Now on the other hand, Daniel, I think you trivialize the reals when you respond to Wildberger by proposing they are mostly a useful formalizing wrapper for in-principle approximatable objects - since this is not the case. If you fix an alphabet, formalize a Turing complete framework, enumerate all algorithms a_n that can be written down (a subset of all strings) and let S = {n > 0 | a_n() eventually halts} be the subset of natural number indices for which the n'th program halts when ran on empty input, then sum_{k in S} 2^-k defines real number in the interval (0, 1) which however can't be approximated arbitrarily well (since the Halting problem is undecidable, and thus eventually there's a a_N the non-halting of which you can't establish and that leads to a knowledge gap of size 2^-N in your computation). The so called Specker sequences show that the computable reals are not even closed w.r.t. limits (there's sequences of rationals who's limit point isn't computable like the rationals.) And next to the uncomputable definable reals, in the standard interpretation (with more than one infinite cardinal, which when adopting Excluded Middle they are ordered in a sequence of "sizes") there's of course all the undefinable reals (since what can be defined is a subset of all strings, which in turn have the smallest infinity size). Modern analysis does more than filling the gaps between elements in Q via Dedekind cut's that give object that can be arbitrarily approximated. That step also adds objects very different from sequences of digits (definable uncomputable reals or arguably worse).
See his video on axiomatics. Euclid gave us 5 requirements that were self-evident. Modern mathematicians instead use the notion of axioms to ordain that the ill-defined is well-defined.
@@ThePallidor I'm happy with Euclid's definition of point (def. 1 and 3, mereological "has no part" and discontinuity "end of line"). The second definition "Line has no width" seems problematic". If line has no width, can it have depth instead? In which case there's just a difference of perspective. Archimedes calculus can be interpreted as denying Euclid's 2nd definition.
Potential infinities (with/under Halting problem) does not equal strict finitism. It's rather the middle way between finitism and Cantor's paradise/joke. As for logicism, do we really need to keep on playing that game after Gödel and Turing? Coherence theory of truth (which guides intuitionist constructions) is not without its challenges and problems, but at least it's not outright debunked like the logicism foundation of formalism.
@@santerisatama5409 Euclid wasn't perfect either. A line is long, thin rectangle. Better to say it has negligible width. A point is a dot of negligible size.
@@ThePallidor I like mereological approach, so I have no problem with 1st definition, when taken together with 3rd. Affine parallelism can be intuitively derived from those.
I have no good solution for line, some sort of infinitesimal approach, perhaps, but not easy.
I suggest to distinguish between "computable" and "constructible".
It would make no difference for Wildberger. He doesn't believe square root of 2 exists.
@@ga35am according to norman root 2 can be constructed as extended field.
@@chjxb oh yeah? What is its value? Huh??? Do you know how to calculate sqrt(2)+sqrt(3)??? This should be even easier than pi+e+sqrt(2)!!! Do it! You can't.
Newtons method is an approximation, which supports his argument right???
Nice discussion. Quite humorous at times. Celestial Sphere ... haha ... nice. At any rate, I'd be very interested to hear more about your objections to Category Theory. Could you make a video delineating your main qualms with the subject?
Whether we like the real numbers framework or not, I think it is good to look at and push forward other frameworks for Analysis.
Especially if we need less axioms with them.
Similarly, in Geometry it has been very fruitful to remove some axioms and try with less, say for example to go from euclidean geometry to affine and projective geometry.
Finally if a new framework gives some interesting insights and results then it probably worth to investigate it further and to challenge the more established one. Thanks to both of you for this interesting discussion.
The idea of axioms is broken and generally now used to mislead. Euclid didn't use axioms. He simply stated in clear terms what were already self-evident observations.
@@ThePallidor How is the idea of axioms broken ?
@@DanielBWilliams An axiom is at best a premise by another name. At worst - and this is its usual use - an axiom is an attempt to smuggle undefineable nonsense into a position of unquestionability. Cf. "There exists an infinite set."
@@ThePallidor There is no axiom in ZFC that said "there exists an infinite set".
But even if that was the case, "infinite set" is perfectly defined in ZFC so I don't understand why would you said that. Can you explain why did you said isn't well defined ?
@@ThePallidor "An axiom is at best a premise by another name." Hahahahahaha.
I agree with NJ Wildberg that a distinction between computable mathematics and imaginary mathematics is a good idea. Having said that, I like the blackboard of the mind. The blackboard in your mind is edgeless, you don't have to imagine the edges at infinity, instead, you can not bother imagining the edges at all.
Having a computer program that approximates cube root of 15 with the property Tha the approximation gets better if the program is allowed to run for longer is basically equivalent to saying that there is a cauchy sequence of rational numbers for which the limit is cube root. That's why you can't make a program that approximates root of negative one with rational numbers. He's basically using the concept of a limit to assert the existence of such program.
there is no limit because the program never stops, and never actually reaches the cube root of 15. in practice, we simply stop the program at some point (in modern computers, this is usually the saturation of the 64 bit floating point data structure) and take the approximate result so far. wildberger's philosophy mirrors what our machines actually do, and he sets his definitions that way, rather than pondering the philosophy of what a certain algorithm could be at its fundamental level in some alternative universe that cannot be proven to exist.
Daniel have you looked into algebraic analysis? It as well as hyperfunction theory may be more palatable to you as an analyst than Grothendieck. The ideas turned out to be very close developed in the mid 1900's by Grothendieck and Mikio Sato independently. I also feel it is a way to settle Wildberger's concerns particularly from a functional analysis point of view thinking about the algebraic structure of topological vector spaces, as Wildberger accepted in this debate you can have such a thing as cube root of 2 in a system of matrices representing an extended rational numbers to include it. Similarly you can have a system which extends to all the irrationals, complex algebraically, but that's why we have the infinite decimal notation to try and simplify things. Also on a more philosophical note 1/3 + 1/4 + 1/5 = 47/60, is just a simplification we don't do anything here actually solving something or finding anything. So I do not see any problem with defining sqrt(2) + pi + e as itself, math is tautological to begin with. There is no foundation which gets around this.
On the subject of mathematics being compared to religion (and one could easily apply this reasoning to mainstream physics as well), I will leave you with a quote:
"If a ‘religion’ is defined to be a system of ideas that contains unprovable statements, then Godel taught us that mathematics is not only a religion, it is the only religion that can prove itself to be one." - John Barrow
At 15:15 they are talking about adjoining sqrt(15) to rational numbers, and Wildberger says that we once we do that, we don't have an order between numbers. This is just false. Those numbers have a natural order that can easily be defined.
If we go with Dr Wildberger and look at Phythagoras's theorem a ^2 +b^2= c^2 is therefore only applicable is certain situations - eg 2 ^2 +2^2 = 4 +4 =8 therefore c=square root of 8 = a 'real number' 2.828427.....P's theorem doesn't work if we reject 'real numbers'..... Do we then 'adjust' the Pythagorean theorem- maybe instead of the 'equal' sign we use some symbol which means only conditional on the values of a b and c do not produce a 'real number' as the result the computation is true only when the result is not e 'real number'?
Just to answer what some people in the comments are confused about: 1/3 having the same issue as square root of 2 because 1/3 is an infinite sequence of 0.3333..
Unlike roots, rational numbers can be constructed using Euclid methods to bisect a line into any (integer) amount of segments
dislikes philosophy then critiques real numbers with a philosophic attack of computational materialism.
what? computational materialism? that's not a real word or concept. Pi can't ever be calculated. You approach it indefinitely. Wildberger's statement is not philosophic
Using Applied Mathematics solely as the basis for further mathematecal research seems to me will stymie research into mathematics (pure or applied).
dunno about that... I think pure maths might be a shortcut to studying mathematics that could (perhaps with more creativity) be found though applied / constructive foundations.
Great conversation!
at 58:00 , why is 31/30 a final answer? It's something you can write down, and you can also write down 1/2+1/3+1/5, which is a valid answer, and 1 1/30 is also a finite sequence of strokes that you can write. So is pi+e+sqrt(2). it can be written down in finite time and space.
Technically we could name anything and write it down, using that name. but I think his point is more subtle: representing 1/3 as 1/3 rather than the infinite expansion 0.3333... is composed of integers and the division operator, all remaining within the rationals, using a finite set of symbols and rules, which he is taking the position are enough to model continuity (with some creativity in coming up with theorems that work around the rationals not being complete if square roots (etc) are allowed). Honestly I see this as a design choice rather than a philosophical position. That is, he's championing avoiding defining a different class of objects that, while algebraically have the properties we want, admit many instances that aren't just non-computable, but can't even be indirectly expressed in other forms (as I understand it, most real numbers that "exist" fall into this category). Instead he's getting creative in redefining some areas of maths that ostensibly (at least how they're commonly taught) depend on continuity based on completeness (or equivalent, such as IVT, etc). Personally I think it's a pretty cool project. But I come from a computational background where computability is highly desirable. It seems almost too easy to design algebraic objects that, while being simple to declare in terms of algebraic properties/laws, admit rather absurd instances that we can never encounter or even refer to.
this is perfectly true, but mathematicians claim there are real numbers which cannot be defined in finitely many symbols.
Chaitins constant requires a solution to the halting problem, which is the same as having infinitely many symbols, one to determine every Turing machines end state.
If there are "more real numbers than natural numbers" then there are undefinable real numbers, since there are only countably many definitions.
Granted, wildberger takes issue before we've reached this point, but this is a clear issue with standard mathematical assumptions.
So does he think a right triangle with sides of length 1 is somehow an absurdity, or not an object of interest in pure maths?
I think he would say that the hypotenuse has a physical length, but that length cannot be expressed as a rational multiple of the sides.
So it is incommensurable to the sides, and can only be approximated when using them as your unit.
if i’m understanding wildberger, he’s saying that numbers like the cube root of 15, when computed to a form that most people use, contain an infinite amount of non-repeating decimals, which requires an infinite number of information to encode. it’s an interesting pov.
The cube root of 15 itself doesn’t require an infinite amount of information, only its expansion in base 10, 2, etc. But that expansion is not an intrinsic property of the cube root of 15, it’s a relational property of that number and other numbers (10, 2, etc.)
@@synaestheziac Does that mean anytime we "write down" a number, we have to choose a representation of it, and such representation may or may not come at the cost of precision? Furthermore, any number that we can "write down" cannot be the number itself, but merely a representation of it, since there are no direct ways of handling numbers, concrete or otherwise?
@@justanother240 yeah I think you could put it that way
@@justanother240 as long as we realize that a "representation" of a number is really just a relation between that number and other mathematical objects (such as powers of 10 and their coefficients, or the cube root operation and the integer 15, etc.)
@@synaestheziac Can objects be defined without relations to other objects? And are relations between objects also objects?
Nice discussion @Daniel Rubin. Subscribed!!
but you can model x^^3 on an analogue computer and display the output on a CRO
This is a great discussion! I think an important distinction should be made between something like say 1) calculating with essentially 100% absolute certainty to the precision of a nanometer how long the radius of a circle 1 million times the size of the observable Universe would be whether or not such a circle could ever conceivably be formed or viewed, versus something much more fundamental and foundational like 2) the idea that the process we think we can continue to use may not be correct, or, though we have gotten accurate results so far, it may have never really been working for the reason we thought it was in the first place, like the way Professor Wildberger pointed out that computers do not store a set of all natural numbers up to a certain point and access them to make a calculation.. We think of the set of natural numbers as sort of already existing in some way, and that we can access them, but the idea that maybe mathematics actually works nothing like that, just that the results match that way of thinking at a certain scale, that such a way of thinking only holds up, up to a certain point where eventually it crashes.
@about 19:30,
Norman states that pure mathematics didn’t develop until the 19th century, but I disagree. He’s ignoring the fact that recreational mathematics is actually a part of pure mathematics, and it has been around almost as long as applied mathematics, notwithstanding the fact that many questions in recreational mathematics have a “flavor “ of being related to or derived from certain specific applications, but with some aspect of entertainment injected, often via a modification of hypotheses to something that is motivated by a philosophical conversation about what we can imagine, instead of what we actually see and touch and use.
at 23:57 he put his cards up in the table... a mathematics without any philosophical issue underneath???!!!! as if that would be possible 😄
How to compare two cauchy sequence to see if they are equal i.e. point to the same real number?
That depends on what information you were given at the first place, there is no general answer.
@@DanielBWilliams Given that the propability that a real number is non-algorithmic and non-computable is 1, seems there's no any answer.
@@santerisatama5409 What do you mean there is no any answer.
If I give you the sequence (1;1/2;1/3;...;1/n;..) and the sequence (-1;-1/2;-1/3;...;-1/n;...) it's easy to see that they point to the same number 0.
So as you can see, with enough information on both of you sequence, you can conclude.
@@DanielBWilliams Hmm... does your example satisfy the criteria of a real number as specified in the wiki article of cauchy sequence?
@@santerisatama5409 Yes I think : those sequence converge to 0, so they are convergentes sequences, so they are Cauchy sequence.
What does it mean for such things as reals or rationals to exist or to be 'write-down-able'? NJW defers to 'computer programs'. Why doesn't writing down 3^.1 in the context of a computer algebra system like Mathematica fulfill this criterion? Such a system provides a fully realized idea of what the concrete computations that address 3^.1 look like.
Also, the definition of a limit relies on providing an exact (algorithmic) means of producing a delta for every epsilon. The idea of 'at infinity' is perhaps a nonsensical phrase out of context, but it is defined as a technical term in a fully finite way via epsilon-delta.
@Gennady Arshad Notowidigdo It seems like you took my criticism of NJW's position to be a criticism of the conventional (Weirstrauss/Cauchy) system, and it seems like you agree with me. I take your point that maybe the NJW critique belongs more to the philosophy of math than math itself because it doesn't make any discrepant claims. I will say that I do think it substantially rephrases the old claims, much as the 19th century analysis reframed the previous generation in response to Berkeley's 'ghosts of departed quantities' critiques, and that lead to a lot of new prospects in mathematical thought. I think the big weakness of the NJW position is that it relies on an idea of (philosophical or ontological) existence that is never defined. I think, though, it would be great if students were inculcated with the doubt that the cube root of 2 is a real thing from an early age, if for no other reason than it would give them a great foundation for understanding what real numbers are, haha!
Also, the definition of a limit relies on providing an exact (algorithmic) means of producing a delta for every epsilon
this is the only meaningful definition of limit, but its not what mathematicians mean.
they believe there are "non-computable" limits, which have literally 0 utility compared to the algorithmic version.
If thats your definition of the real numbers, then youre actually talking about the computable real numbers.
10:12 “we would like to believe” that’s just the incompleteness theorem 😊😊😊
Fireworks!
Making math fun again
@about 9:03,
Norman introduces “a slightly more difficult” problem, but it’s actually too difficult to illustrate his concerns. He goes from considering an example that’s a linear equation with integer coefficients, to a cubic equation, x^3=15. A simple example is the following: Let e denote the smallest positive rational number representable in my desktop computer using Mathematica today (2024-06-09), and consider the equation 2x=e. Then no solution to this equation exists in the set of numbers that I can compute today using Mathematica in my desktop computer.
This even more directly represents Norman’s viewpoint as he presents it in his videos, because he points out that he actually believes there is a largest positive integer. Here’s an ultra-finitist.
Daniel, I would suggest that you re-watch Norman’s videos very carefully if you wish to enjoin him in a new “battle” or debate.
…
I think it is a good idea to have a fence like Wildberger proposes.
Your channel is truly a gem.
I've been following Wildberger's thought-provoking and entertaining channel since he started on YT. I've often wondered: just from a pedagogical point of view, how is he able to teach a lot of undergraduate mathematics? He might have good personal reasons for not believing in the real numbers, but surely undergraduate maths students have to be exposed to, and understand the standard constructions of the reals, Cauchy sequences, Dedekind cuts and so on. How does he deal with that if he's teaching introductory Real Analysis?
He is still trying to find anyone who can write a passable definition of what a real number is. He did a video of it once which was hilarious. Its a bit like Alice in Wonderland stumbling across the mad hatters tea party or the boy who said "the emperor is wearing no clothes!"
@@sharonjuniorchess What is the issue with " a real number is an equivalence class of cauchy sequences of rational numbers ..."?
@@gidi5779 That is nowhere near a proper definition. How many digits are there in a real number?
@@sharonjuniorchess How does the number of digits in a real number have anything to do with whether the definition above is valid or not?
@@gidi5779 you need to look at Normans video where he goes on a quest to find anything that actually explains what a real number is. Personally I think the number of digits is pretty clear if you are going to use an approximation of a real number then it should be a significant part of your definition. If you are going to be rigorous. How do you do arithmetic with real numbers other than simple restate them?
Computer floating point adition and multiplication are not even associative. Are we going to formulate mathematics on that? Even if real numbers are "only" abstractions, they constitute a language with which we can prove theorems which are known to be approximately true in the realm of computational floating point calculations. If we substitute this by concrete computer operations, we wil end up with some form of mathematics whose rules are complicated and computer-architecture dependent. Current real number mathematics is already computational. It says that the theorems are approximately true in any computational architecture, and the goodness of this approximation can be made arbitrarily good if we change the architecture, say by increasing the number of bits for example.
The thing is, these sequences really ARE about our imagination, and I think mathematicians should own that. If you can imagine it, then it exists. As long as there is some pattern or structure to the sequence, then it can be assumed to go on forever. What's wrong with not being able to approximate pi exactly? That's what makes life interesting. Somehow it's a failure because we can only approximate it to 60 trillion digits?
This is a really unmathematical way of looking at the controversy
The main beef Prof Wildberger seems to have is w a faulty definition of the continuum and his disbelief in the existence of real numbers from a computational standpoint.
As an engineer and computer scientist (and 30 yr user of Mathematica), I reject the notion that real numbers need to be computed to exist. I am perfectly content to defining their existence by some induction or limit process, e.g., bounding them from above and below at arbitrary precision. The number itself is an ideal, its existence results from the defintion of the bounding process, not from its computation. For two centuries, no practical "crisis" has yet emerged from this type of definition.
Consider the method of proof by induction. It suffices for one to show a base case and the inductive step are true. One need not proceed to infinity. The idealized truth about some object found in infinity is established anyway. So it should be w the real numbers and continuum.
We should listen to Gödel. Mathematics is provably incomplete and broken. But it is hell of useful. Let's focus on what works.
P.S. -- later edit: I love Prof Wildberger's series of lectures on hyperbolic geometry. There he makes wide use of intersections of lines with a circle (Apollonius' construction for the tangents, the polars, etc.). If he doesn't think the circumference of a disk exists (it is an idealization after all), how can he (or Apollonius) compute intersections of lines with it?
Exactly. And in fact we learn this in 1st year calculus. It is sufficient to show that if a function is bound by N and monotonically increasing then there is no reason I cant say that IF I continue the calculation I can get as close to N as required. In other words as x increases indefinitely, f(x) = N . I do not need to compute what f( very large number) is to show its getting closer and closer to N.
If you bound the computation, it is not infinite. You thus sidestep the nonsense and effectively agree with Wildberger.
Godel merely showed that axiomatics is broken. See Norm's videos on axiomatics.
I am not sure if you chose your words wisely. An ideal pretty much encompasses non-existence, else it would not be an ideal.
@Gennady Arshad Notowidigdo interesting how u superimposed the meaning of ideal from an aspirative term (as used above) to a description of perfection. Indeed, ideal can mean both. I view this more as a wordplay than a contribution.
@31:19,
Again, for the nth time, where n is larger than 3, Norman has indicated that he’s limiting his philosophy of mathematics to computer science and algorithmic foundations, which didn’t really exist as a philosophical consideration; this is in spite of his frequent claim that he’s wanting to avoid a philosophical discussion.
For example, he clearly wants to deny that what he’s calling “choice sequences” of natural numbers exist or that such a restriction is a philosophical issue. Daniel has trouble addressing this, and I think the reason is that he’s not addressed these kinds of questions in holds own mind previously. It helps me to realize that I’m a human being in a world that has many things in it that I’ve never observed. When I do observe one of those things, it’s extremely unlikely that I’ll be able to view it as having been “generated” using an “algorithm” that I know or will either know or understand very soon after observing it. I cannot deny the existence of such a thing at the moment that I first observe it; that may even be dangerous. To live in this actual universe, I am compelled to be able to imagine certain things as being possible by virtue of not having introduced into my system of thought an obvious contradiction.
Mathematically, this leads me to accept as potentially existing, arbitrarily large positive integers, and sets of them, and
Norman doesn't understand continuum (real numbers). It's just continuous quantities. There is nothing inexact or approximate about any real number. Sqrt(2), for example, is just as exact as any other number on the number line, 2 for example.
This is not correct. Any "computable real number" is arbitrarily precise.
But uncomputables, like Chaitins constant, cannot be determined to arbitrary precision.
I wish Wildberger would have given you more time to talk, especially earlier on in the video.
Being "write-down-able" as Wildberger argues, keeps mathematics more honest. It is an absolute necessity if we want computers to check and discover theorems for us, which will become more and more important going forward. But we might recognize different levels of write down ability. Rational numbers have canonical forms. Many limits have been written down exactly using limit notation, although they may not have canonical form for them and no easy generalizable way to check whether two differently-written limits are actually equal. And the alleged "non-computable" numbers can't be written in any manner at all. Anyway, he might consider that write-down-ability is not a black and white criteria, but a matter of degree.
I agree with Wildberger's suggestion that how computers do math should guide how humans think about math. But with that in mind, we should try to better understand how computers actually do math. Many computer languages, such as Haskell, support a feature called "lazy-evaluation" where at runtime the resulting code keeps intermediate results in a data structure that exactly represents the expression in symbolic form. Conversion to a canonical numeric form (typically as a binary or decimal approximation) is delayed until the result is actually needed. Some systems, like Mathematica, can even simplify expressions algebraically prior to approximating them numerically. These are arguably the same that things mathematicians are doing when they write expressions in limit notation, or in a form like "π+e+√2". But in the end, we should realize that symbols like π, e, √2 are just names for useful algorithms for generating successive approximations that come arbitrarily close to satisfying certain criteria. They do not represent actual numbers that can be stored in a finite length of bits, or have operations like addition or multiplication exactly performed on them in finite time; in other words, we can't use them as fundamental building blocks of computation in the way that we can use small integers and rational numbers.
Wildberger should be more careful in how he talks so as to better distinguish the different ways in which a solution can "fail to exist". A solution failing to exist because two curves fail to intersect at all is a fundamentally different situation compared to a solution "failing to exist" because the intersection point simply isn't amenable to canonical representation and can only be written approximately. The concept of real numbers may provide a useful way of thinking about how values that can't exactly be written down can nevertheless result from well-defined mathematical relationships. Also, it was weird to hear him talk about approximations, while at the same time denying the very existence of an actual value being approximated.
I agree with a lot of what you're saying. That's a very good point it did not occur to me to mention at the time that even in NJ's framework you'd want to distinguish between when curves don't intersect as real curves and when they don't intersect because the intersection point would not be rational.
@Wayne --- In other videos I like to distinguish between actually three types of "solution possibilities": 1) There is a solution, as in x^2= 25 2) There is no solution, but there are approximate solutions, as in x^2=7 3) There is no solution, not even approximately, as in x^2=-13.
"Write down" should mean "in code".
I think of Pi as not a number, but a bit of code that generates digits for as long as you need to pull them. ie: e + pi + sqrt[2], is a function that you can pull digits from. It does not terminate though. I think continued fractions can handle roots in a way that terminates. So, there are issues in how you choose the number system.
That's the only way that you can use the equality symbol. Simple rational numbers are functions that terminate. One of the things that recommends this approach, is to reformulate things to work on finite fields. And in a computer that can only do 64-bit integers; that would be such a field.
The problem is that your algorithm will never be able to reach produce the exact number. That is one of the main criticisms of conventional math. Math started as a logic way of understanding the world and ended depending on concepts that can't exist in our universe. That generated all kinds fo paradoxes and the theory itself ends not being consistent in many places.
@@conelord1984 algorithm is a number :v
I need to debate Norm, I’ve seen him pull this shtick on so many people, who aren’t equipped to respond adequately..
An *actual infinity* is an infinite entity as a given, actual, completed mathematical object. Actual infinity is to be contrasted with *potential infinity,* in which a non-terminating process (such as "add 1 to the previous number") produces a sequence with no last element.
Aristotle made a distinction between the two types (though he did not believe in an actual infinity). It is with the foundations of Set Theory that actual infinities came to be largely accepted and used in mathematics. Cantor had another type of infinity he presented, a type of actual infinity called Absolute infinity.
so we have 4 types of numbers in this regard-
two types of *potential infinity:*
*finite potential infinity*- an (apparently) non-terminating process which produces no last element, and always remain finite- an indefinite finite value. a finite potential infinity is not a definite finite amount.
*infinite potential infinity*- a non-terminating process which produces no last element, but does not have to be finite- an indefinite infinite value- not an actual amount.
*actual infinity*- a completed infinity with a definite value. two types:
*non-absolute actual infinity*- a complete infinity with a definite value, but more can be added to it to get a new number which is a different actual infinity. (adding more is something that usually has to be done outside of a given process, for example: 1+/12+1/4+1/8.. until 2 is reached will result in an actual infinity if you treat the last term as a 1 and convert the rest to 1's, but, although the original process has to end when 2 is reached, this does not mean that, outside of this process, 1 cannot be added to the total we go when the smallest unit was converted to 1 and added up. But so there are times where a sequence or process must terminate at a certain value, but that the value terminated at can still be added to in an other separate process.)
*Absolute infinity*- an actual infinity that cannot be added to. It has reached the ultimate limit. If there is such a thing as this, no number can be higher than Absolute infinity.
[In my personal opinion, if there is an Absolute infinity, potential infinities, although not actual amounts, can be conceptually MUCH larger than Absolute infinity, besides the definition that nothing can be larger. But for example, if ω^ω was an absolute limit that cannot be surpassed, conceptually, indefinite potential infinities can be described in ways that go way beyond that. But, like all potential infinities, they would be invalid.]
some examples:
*actual infinity:*
1+1+1+1... an infinite amount of times, until the first infinite number, ω, is reached, but no further. (ω cannot be reached in a finite amount of steps)
*finite potential infinity:*
1+1+1+1.. a very large yet indefinite finite "number" of times. (This is not an actual number, of course.)
*infinite potential infinity:*
1+1+1+1... that never stops
or
1+1+1+1.. to infinity and beyond without stopping at ω
or
ω+1+1+1.. that never stops
*Examples of actual infinities:*
ω, ω+1, 2ω, ω^2
The number of discrete units on the smallest actually infinite discrete straight line
The number of discrete units making up two different discrete lines like the one mentioned above.
The number of discrete units on ω discrete lines like the above.
The number of sides of a regular apeirogon or infinity-sided polygon, which reaches a limit of 180 degrees.
The number of regular apeirogons that can fit in a 2-dimensional plane.
The area of a discrete infinite two dimensional space
The volume of a discrete infinite three dimensional space. etc
*Examples of potential infinities:*
All Aleph numbers, all Beth numbers, the indefinite amount of rooms in Hilbert's Hotel, the indefinite amounts used in one-to-one correspondence arguments, and in the diagonal argument, etc., c, the continuum...
The indefinite amount of points on a Euclidean line.
The amount of points on a disk, cube, sphere, ball, etc in a continuous space
The area of an infinite plane in a continuous space
The indefinite amount of "real numbers" between 0 and 1
The indefinite amount of "numbers" on a "real number" line
The indefinite volume of the balls in the Banach-Tarski paradox.
@Russell Sharpe
@Russell Sharpe
This is why an apeirogon is so important to understanding all of this.
The sequence of regular polygons {3},{4},{5},{6},{7},{8}... etc, tends toward {∞}, an infinity sided polygon called an 'Apeirogon' made up of countably infinity sides, which has a 180 degree internal angle. At which point, there cannot be another side added, since there is no space beyond 180 degrees (= totally flat) which it can curve into.
We can deduce from this that space cannot be continuous.
For an octagon, starting at some edge in the middle (call it edge 1) lets move to the right, so edge 1, edge 2, edge 3, etc. We move from edge 1 to the right and eventually get to edge 8 and then if we move right some more, we get back to edge 1 again.
Now consider the apeirogon- Start with some edge in the middle, called edge 1 and we move to the right to edge 2, edge 3, edge 4, etc. moving to the right an infinite amount of steps will eventually lead to THE OTHER SIDE of the apeirogon, where we eventually reach edge infinity and moving once more to the right we are back to edge 1.
Now remember that this polytope is connected edge to edge at 180 degree angles. In other words, it is identical to a segmented infinite line.
So take a very small line segment, call this edge 1, and add another to the right of it at 180 degrees, call this edge two. If you continue doing this, you will start to see the edges showing up on the other side. When it reaches edge infinity, edge infinity will be next to edge one and we can no longer add another edge without either overlapping or leaving this dimension.
If we compare this sequence with an identical sequence with large edges, the sequence with the larger edges will get back to where they started before the other sequence, and the last edge will not be # infinity.
To get to infinity and get back to where we started, then, the line segment has to be of some exact smallest size. No larger, no smaller. It has to be discrete. The notion of a continuous space leads to contradiction.
This modular quality of getting back to where it started is very helpful in this type of discussion because it sets a strict limit on these sequences without overlapping themselves or jumping up to a new dimension.
This is similar to the other sequence we discussed:
Take one entire apeirogon to be "1".
Take a second apeirgon, and count one half of the sides, then count one fourth of the rest of the remaining sides. Then one eighth, and so on.
If we continue doing this, eventually there will be a smallest unit.
Another way of thinking of this is infinity + infinity/2 + infinity/4 + infinity/8... + 8 +4+2+1+1. And the sequence has reached the end. (The result is 2 x infinity) You could imagine a scenario where you split that last 1 into 1/2, 1/4, 1/8 etc, but this would be getting into a whole new area, and in a discrete setting this would not be possible unless we set up a scenario with even bigger infinities to split. But I think if you were to say that you can get to 1 then split it into 1/2, and 1/4, this would be similar to saying that N and R are equal.
There is a way to show from all of this that the first infinite number has an exact sequence of digits associated with it. This is too advanced for you to comprehend at this stage, though, if you still are not following about what an apeirogon is, the difference between actual and potential, about a last digit in the sequence, about omega not being arbitrary, about discrete geometry, etc.
Wildberger totally won on this one
Great conversation, and very though provoking.
About the "e + π + √2 = ?" issue, I wonder if I ask mr. Wildberger "what is 1 divided by 3" his answer would be "1/3", because that answer would not comply with his standard of not giving the same question as an answer. So, is 1/3 = 0.333... (an infinite decimal)?
I see limits not as an infinite process, but as a direct consequence of the Archimedean property of R, so I have no issue with them. Of course, not being a Mathematician, I have a more naive approach to that concept.
To me the distinction is whether the quantities are expressible in proportion to one-another using natural numbers; 1/3 : 1 :: 1 : 3. Naturals are the only true numbers. To take a quantity 'e' times does not really mean anything to me, unless I am thinking of it as 2.718, but then I am thinking in terms of the proportion 2.718 : 1 :: 2718 : 1000 which is in terms of natural numbers.
Also remember that 0.333... is itself an infinite sum of fractions, 3/10 + 3/100 + 3/1000...
@@livef0rever_147 That's exactly the reason of my thoughts... 1/3 cannot be explained exactly as a decimal, it can only be approximated, or understood as an infinite series, and Prof. Wildberger is a finitist.
In my concept, "1/3" is just a symbol, and it's as valid as "e", "π", "√2" are.
@@nabla_mat what’s so special about decimals? 1/4 and 0.25 are both fractions. The only difference is that in one the denominator is supplied by the imagination (25/100). Pretty much the only reason we use the system of decimal fractions is because we have ten fingers. What if we had arbitrarily decided to use, say, base 9. Then it would be 1/3=3/9=0.3 and 1/2 approximately 0.444… would you then say 1/2 can only be understood by an infinite series? Your argument is absurd.
Couldn't a real analysis apologist simply argue that real analysis is outside the scope of computation? Because that fact on its own doesn't "disprove" real analysis or its conclusions. The question would then be the usual Wittgenstein question of whether the terms being used - real number and computation - are useful as used. And that's really what I think he is trying to say - some aspects of real analysis are not useful in a way that computation is. And I don't think anyone would disagree with that. But then why not vice versa?
@17:00 the ℂ numbers are not "points in a plane". They are algebraic, as he alludes to a minute earlier. Moreover, the setting for ℂ numbers really is the Clifford algebra, the bivectors or even subalgebra. And the proper setting for _that_ in turn is a Category. Such mathematical objects are not "numbers", they might however be homomorphic to a number system. The whole idea of "number" NJW is trying to motivate here is ill-conceived, or impoverished. You can restrict your notion to "what is computational in finite memory" but that's not all there is to play with in _Cantor's paradise._ When you play, the demand is relative consistency. When you eventually find inconsistency it is perfectly fine, you know you were not doing mathematics, which is a good thing to know, and move on from (or do whatever else you want with, like art). Mathematics, the sociology thereof, has to be such art, for if everyone instantly took to perfection when beginning mathematics it'd be impossible to make mistakes and we'd be gods, that's NJW's "religion".
You have to be able to be wrong to be right.
Very interesting discussion!
48:10 There is a crisis in physics. Can Algebraic Calculus solve quantum gravity?
@Gennady Arshad Notowidigdo Current methods can't unify quantum mechanics and general relativity.
Prof. Wildberger has a realy realy bedagogical method of teaching math i just watch i can repeat what he said from memory i wished if he could teach some of the mainstream math .
Mathematics is a philosophical endeavour at its core. It has been this way since its inception and will most likely remain so.
Just because it’s philosophical, doesn’t mean it’s wrong - the real numbers have strong arguments in its favour and more than one method of arriving at them.
Thank you for this stimulating discussion!
I wanted to comment on the "cube root of 15" portion. When dealing with the Real numbers, I think of them as extension fields of the Rational numbers, just as was mentioned in the video. However, not just a single extension, but a countable infinity of extensions, one for each Real number that I can name (pi, e, sqrt(n) for some Integer n, etc). The problem is that each extension in a sense introduces a new dimension to the field. (Example: extending the Rationals with with sqrt(-1) is two-dimensional). So the we are left with a countably-infinite dimensional object that we are treating as a 1-dimensional line.
The concern is that there may be all sorts of logical inconsistency lurking in that embedding. While I cannot give a direct example of such a demon lurking there, an analogy would be how you can "show" that the sum of all positive Integers = -1/12 which demonstrates the logical inconsistencies that may be encountered when dealing with "completed infinities".
Yes, introduce "actual infinity" and rigor is gone. Anything at all can then be proven and it becomes more of a cultural exercise than an intellectual one.
@@ThePallidor If you look at ZFC axioms, never you will see an axiom claiming that "there exists an actual infinity".
@@DanielBWilliams From the Wikipedia entry on ZFC, in the section 7: The Axiom of Infinity:
Let {\displaystyle S(w)}S(w) abbreviate {\displaystyle w\cup \{w\},}{\displaystyle w\cup \{w\},} where {\displaystyle w}w is some set. (We can see that {\displaystyle \{w\}}\{w\} is a valid set by applying the Axiom of Pairing with {\displaystyle x=y=w}{\displaystyle x=y=w} so that the set z is {\displaystyle \{w\}}\{w\}). Then there exists a set X such that the empty set {\displaystyle \varnothing }\varnothing is a member of X and, whenever a set y is a member of X then {\displaystyle S(y)}S(y) is also a member of X.
{\displaystyle \exists X\left[\varnothing \in X\land \forall y(y\in X\Rightarrow S(y)\in X)
ight].}{\displaystyle \exists X\left[\varnothing \in X\land \forall y(y\in X\Rightarrow S(y)\in X)
ight].}
More colloquially, there exists a set X having infinitely many members. (It must be established, however, that these members are all different, because if two elements are the same, the sequence will loop around in a finite cycle of sets. The axiom of regularity prevents this from happening.) The minimal set X satisfying the axiom of infinity is the von Neumann ordinal ω which can also be thought of as the set of natural numbers {\displaystyle \mathbb {N} .}\mathbb {N} .
@@WildEggmathematicscourses After "more colloquially", it's just an interpretation of what the axiom is saying, but If you look at the axiom, never in the axiom in itself it is written that there is a set with an infinite amount of things in it.
More generaly, sets in ZFC aren't some "box" or "bags" in which we can store things : there nowhere in the axioms that idea. It's just a way of visualing sets in order to work faster and better, and a lot of mathematicians believe in this interpretation because it can be very usefull for them, but that's all.
That's the problem with math education when it is about sets : this interpretation is taught as if it was what's really going on, because it's more easy to explain, but so many people think ZFC it talking about bags or box in which you can store things.
This was really fun, but I did get the impression that you were a little caught off guard by Prof Wildberger’s arguments here. Is there going to be a round 2?
Which arguments?
@@ga35am I would for example say when wildberger asked about which sequences he believes in.
@Tomas Matias answered "I would for example say when Wildberger asked about which sequences he believes in."
Daniel Rubin replied that he "believes" in sequences that can be justified by the axioms of ZFC.
There is not a distinction between "computed" or "choice" sequences in ZFC. One could limit oneself
to "computed" sequences, but then one would be outside of ZFC. There is nothing wrong with that, but
that is not the framework that Rubin is working in or discussing. Rubin answered the question:
sequences that can be justified by the axioms of ZFC. He didn't seem to me to be caught off guard by
Wildberger's question.
It seems to me that Wildberger seemed to be the one caught off guard by that answer. Wildberger then
drops the question on the distinction and then proposes the question of whether such sequences can
be written down in the universe (within a finite amount of time and within a finite amount of
memory).
@@billh17 I don't know if you are knowingly misrepresenting set theory by claiming there is no distinction between computable sequences and choice sequences but you definitely are mistaken if you think computable sequences lie "outside" ZFC.
@Gennady Arshad Notowidigdo I gave you a like but I disagree about some points. "Nobody should believe in any idea in mathematics", ok. "respecting what has already been learnt ..." unnecessary. The problem with Wildberger is not that one. I don't care that much about he disrespecting some idea and I have to recognise he builds up from previous knowledge, he actually does that. The problem is that what he is saying about how math people understand real numbers is clearly wrong. I mean, it seems he doesn't understand that saying "sqrt(2) is equal to 1,414..., with infinite decimals" just means "the only way to have sqrt(2) is by approximation". That's it. He insists there is something wrong because of "infinite". This is silly. The infinite in this case just indicates the process to calculate sqrt(2) will never make us get an exact rational number x satisfying x²=2, but we can come close to 2 as far as we want". Because of that, the definition of the real number sqrt(2) is inevitable theoretically, impossible computationally. There is no one making an argument different from this but he invokes these ghosts.
I'm here from NJ Wildberger's channel. Thumbs up if you don't believe in actual infinity.
I don't even know what actual infinity is supposed to mean.
Since when math is about believing in something ?
@Gennady Arshad Notowidigdo That's strange, math isn't about believing or not believing in something. What does that mean that the real numbers exist or don't exist ?
When I do math in ZFC, I don't believe in anything particularly. I just say to myself "if ZFC axioms are taken as true, what can I conclude", that's all. I'm curious about where do the mathematicians you are refering to use belief.
That would be like to say "I don't believe in the existence of the points in Monopoly game". That's a not about believing they exist or not, but about following rules of the game. ZFC is a list of rules, as the rules of the Monopoly game.
@Gennady Arshad Notowidigdo Oh yes I understand what you mean !
Wildberger should discuss with somebody who doesn't claim that so it would be more interseting that with somebody claming that's the only set of rules.
@Gennady Arshad Notowidigdo
*"the exact same tactics that I have pointed out"*
Which tactics are you referring to ?
*"that both sets of rules are equally valid"*
Can you remind me what is the other set of rule (other than ZFC) you are talking about ? I agree that there are other set of rules (other than ZFC) thare are valid, but as I don't know what are the other set of rules you are refering to, I can't say if it is valid or not.
*"someone who is clearly not being up front and honest about his support of ZFC"*
I use ZFC everyday because I think it is beautiful and useful, so I love it. That doesn't mean I don't agree with the fact that there are other set of rules that are valid too. Why are you thinking otherwise ?
The discussion would have been a good idea if the level of discussion was much higher, deeper, precise. Such foundational questions should be addressed to real gods of mathematic, for instance ALAIN CONNES. Otherwise it stays a « youtube » bavardage
I dont understed the problem with limits, its a finite process that model an infinite process and we work in that context
Real nums need axioms so they are meta objects
Rational nums only need defenitions and logic so they are pure objects
When its about math/philosophy i can continue writing all day so im gonna stop here
I recommend Normans Math Foundations 106 video which lays out some of the problems he sees with the epsilon delta formulation of a limit.
@@nazlfrag i have seen norman content thank you
If I have understood him, Wildberger says that the cube root of 15 cannot be specified exactly, since the decimal expansion is infinite, and we only have an algorithm which generates the digits and never terminates. But in the sense in which this is true, exactly the same is true of one third. The only difference is that the algorithm is a bit simpler for the latter (do{write('3')}). If it be objected that we _do_ have a way of specifying one third in a way which does not require potentially infinite processes (just express it as a rational) then we can say the exact same thing about the cube root of 15: just express it as a root.
In fact, it is not only real numbers which Wildberger is sceptical about. He spends a lot of time in his videos constructing larger and larger natural numbers to make the point that there are scarcely any for which we even have - or could have - a feasible notation. This is obviously true: there are so to speak vast expanses of inaccessibility between the extremely sparse integers we can construct names for. With this in mind he (for example) denies that all integers have prime factorisations, for the primes required are in general not expressible in any notation we can invent and use, and he rejects any notion that something might exist but be beyond human capacity to (as he likes to say) "write down".
What this means is that he doesn't even believe in large integers, let alone reals, but only in sufficiently small and/or tractable integers. Or maybe it is not that he believes in some but not others, in a binary way. Perhaps he believes that some integers are more real than others, their reality diminishing with the feasibility of their minimal-complexity notation. (Perhaps 843639613031849 and 10^(10^(10^10)) are roughly as real as one another, and significantly more real than the unimaginably many integers lying between them which have no representation less than 10^(10^(10^9) characters long)
This seems to make the sequence of natural numbers inconceivably complex and difficult to handle, as well as rather vague and subjective. To insist that you should believe in only what you can actually observe (or 'write down') reminds me of Berkeley's idealist metaphysics: esse est percipi. But does the world blink out of existence when one closes one's eyes?
@@russellsharpe288 We could also forget about what you or I "believe", and just stick to what we can write down.
@@WildEggmathematicscourses Why should the only things that exist be things which we can write down? Cosmologists tell us that as the universe expands galaxy clusters are continually passing beyond the limits of the observable universe. We don't assume that as they become unobservable they actually pass out of existence altogether. That would be crazily solipsistic. I don't see why we should be mathematical solipsists anymore than cosmological ones.
@@russellsharpe288 There ought to be a distinction between those phenomenon which are unobservable because of limitations on our direct instruments and those things which are completely and even theoretically beyond our observation. In physics observable has a broad interpretation: a galaxy might be beyond our radio telescopes, but perhaps its gravitational effects on something else is still recognizable. Something like how exo planets were first identified by their wobble effects on their stars. But please explain how P = {primes larger than 10^10^10^10} manifests itself in any fashion in our universe.
@@WildEggmathematicscourses First you'd have to explain how any number or set of numbers at all "manifests itself in our universe". Do you mean eg "is the number of a certain kind of thing in our universe", like say the number of electrons, or the number of ways such particles can stand in certain relations with one another? (Moving from the number of objects to the number of ways they can be combined produces very large numbers indeed of course) Or do you mean "occurs in our best physical theory of the universe". It's a bit odd to demand such a thing, isn't it? Does eg 32856592026365145058662950572640671230337000377644034712965711054753 manifest itself in our universe? How, exactly? Whether it does or it doesn't, can this really have any bearing on whether we should regard it as real as, say, 137?
As far as mathematics goes, much larger numbers than 10^(10^(10^10))) crop up all the time. I see the Goodstein sequence starting at 5 has length greater than 10^(10^(10^19728))), and the one starting at 12 has length greater than Graham's number. But of course all Goodstein sequences do terminate and so have finite length. Do we really want to say that despite this, their lengths do not exist just because we cannot write them down? What we can and can't write down is not a well-defined notion anyway: we are either faced with a sorites paradox or else have to admit different levels of reality to numbers according to their relative tractability; level of reality which themselves will not be well-defined. This is a can of worms, surely? Does it really seem like a good way to go?
Very polite. “I don’t think everything you say is total nonsense”. I think you mean just approximate nonsense then :-)
@Gennady Arshad Notowidigdo Do you mean flawed or just unclear. If this was the first time I came across these arguments, I would have skipped this video for sure.
@Gennady Arshad Notowidigdo I agree and have no affinity to compare mathematics with computation let alone computers, but I also think that not wanting to discuss a problem because one might find that the solution is to much work is not a good thing. Look what is happening in quantum physics and the standard model. They are suffering and progress is slow or even un-existing.
@Gennady Arshad Notowidigdo It's enough that he points to the problem and what the solution would require. He can't reinvent all of mathematics himself, possibly not even any of it, but that doesn't mean it doesn't need to be done. A more rigorous mathematics should yield easier computation.
@Gennady Arshad Notowidigdo We are completely on the same page on this.
@Gennady Arshad Notowidigdo Haha. Sure, lack of depth, experienced mathematician. Hahahaha. Best Stand Up comedy ever.
The sum of a tea leaf & a fruit cake may be theoretically possible but I can't write it down as a number.
I can. It's a 10! Haha. Yummy.
@@ga35am What base is your tea leaf and fruit cake sitting on or are you saying a tea leaf is a one and the fruit cake is a 0 - a sort of picture maths representation? I get that fruit cakes are circular and tea leaves grow on stems but how do you add them together? When I was young I was told you can't add oranges with lemons. But you can drink lemon tea. Lol
Writing down something like 1/3 is no better than √2 because you will never be able to write down 1/3 fully. It's going on and on 0.3333... The same holds for e + π + √2. And all these numbers do exist because π is the length of the circle with a radius of 0.5, and similar geometric interpretations can be made for √2 and e. What's more, these numbers exist no matter what our universe is or what the laws of physics are. It doesn't matter if we have 3 or 4 dimensions or whatever. Mathematically, there exist many dimensions in Euclidean space. It also doesn't matter if our space is not Euclidean. Mathematically Euclidean space does exist. Triangles of infinitely small thickness exist. Not in real life but in math or in logic. It's not a philosophical claim. It's a logical claim. It's just logic.
That said, I express my respect to the both mathematicians in this debate. They are both right. A huge like for the video. One point though: the point of disagreement was not really fully developed. Probably there's no real disagreement. I'm sorry but the whole thing looked to me like this:
f(x) = x^2/x is the line y = x with a hole at point 0. An infinitesimally small hole! The opponent: there's no infinitesimally small hole there or maybe there is but it's all nonsense as you can't write it down or feel the infinitesimally small things. Perhaps it's not even there as we ourselves prohibited division by zero but allowed multiplication by 0. So we simply don't know. I agree but I also agree with the opposite: there's a hole there; we defined it that way (no division by zero is allowed) and that's why there's infinitesimally small hole which we also define recursively (I'm fine with it). Yes, the theory of limits came long after calculus. It's an artificial logical construct. I'm fine with it though. But I understand and respect the opposite position too. What's more, I think excessive formalism and too much limit theory in calculus might be bad and confusing, especially for students in engineering. Yes, it's an abstract artificial construct but it's okay as far as I'm concerned. There's a duality in math and real life, i.e. √4 = 2 and √4 = -2 (in the field of complex numbers) but it doesn't mean we have to conclude 2 = -2. Or do we? We have opposite things as a result. We have duality. In real life this duality is well represented by elementary particles. They are particles and/or waves. Is it _and_ or _or?_ This can be debated till hell freezes over. Can they be particles and waves at the same time? Can √4 be 2 and -2 at the same time? How can a particle be a wave? How can √4 be -2? How can 1+2+3+... =-1/12? It approaches plus infinity. Hence, it's not -1/12? Well, how about Casimir effect? How about that crazy duality in real life? How about this in real life: 1+2+3+... =-1/12 on the one hand and on the other hand plus infinity. I know this series infuriates lay mathematicians the most. I don't have problems with it either just like with the opposite stances of the mathematicians in this video.
Are not points, lines, planes, etc., all pure abstractions not realizable in the real -- sorry, _rational_ -- world?
Yeah, and I assume Wildberger would say that they don’t exist either. But I don’t understand why he thinks that mathematical existence requires observability or computability.
You cant just claim that everything random thing backs up your position.
You sound like a religious nut saying that trees or the beauty of a sunset proves god exists.
Points lines and planes do not construct the real numbers, nor do they require them.
The constructible numbers do not contain every real number, I cant even tell what point you were trying to make.
I’ve often noticed that, for a lot of people, when they first hear that “Wildberger rejects infinity and infinite sets!”, they develop an immediate idea of what it is he must mean. They imagine, as does the host here, that he would concur with statements like:
“There are no algorithms which begin and then never end!”, or
“Newton’s method to approximate pi is wrong, simply because you cannot approximate the ratio of radius and circumference at all, period!”, or
“No one has ever stumbled on ANY useful scientific results using e or sine waves!”
Yet none of these is what Wildberger is saying!
A key point Norman makes here (though it seems the pure mathematics community of today has happily forgotten it & is quite content to imagine it was never so), is that the broad base of analysis got built long ago, without this modern foundation on top of infinite sets & limits, and indeed it WAS successfully used to solve a grand part of the same sorts of practical problems for which it’s still deployed in the sciences today- but based on wholly different foundations!
A problem then and a problem now is not that the approximate results that are obtained by these methods are not useful, or are not more or less accurate, but rather that the alleged building blocks used to explain and build the theory of WHY they work, from the ground up, are misty and dubious, and so an alternative explanation of WHY analysis works remains as badly needed as ever.
As to crises of the Kuhnian variety that the host mentioned, this inability to determine the sum of the three most common irrationals, as “real numbers”- that it isn’t considered a sufficient crisis to merit scrutiny of their foundational underpinnings- especially given the history of this- is actually pretty remarkable. What could make the real number concept, as it’s built up today, more useless than its own perpetual supplantation, for practical purposes, with rational approximations?
Maybe the crisis will demonstrate itself more plainly once a generation of new scholars has utilized non-infinitist frameworks to make some big leaps past current understanding, with the help of computers, as Wildberger suggests, without resort to infinite sets, axioms of choice, etc.
Another key distinction, which I’d hope that the host and other skeptics of his perspective would consider for a moment, is between the idea of, say, an integer as a specific “type” of mathematical object vis a vis the set theory thing, where all the integers have to be put into some virtual bag or basket... which requires we first invent these bottomless virtual bags or baskets.
Norman has videos where he shows how to build these numbers from the ground up without resort to an infinite set. IMO the math comes out all
the clearer for it. I’d strongly suggest people take a look at this alternative methodology with an open mind, because they might end up quite surprised.
Yes, an alternative explanation of why analysis works is much more important than knowing exactly about sum of the irrational and transcendental constants. sin(x), exp(x), sqrt(2), pi - just a short record (name) on paper of the algorithm for calculating them. For practical constructive computability - everything is an algorithm. The simplest algorithm in mathematics is a polynomial. The most important fundamental block of analysis is the analytical function. What is an analytical function in analysis - only and strictly only a polynomial. For analysis oriented towards practical computability - no exist non-polynomial functions. My answer why the analysis works - because these are the algebraic properties of polynomial functions, without the need to use any linearizations and limits. Just consider the full form of polynomial functions - coefficients in all terms must be variable (not fixed). It is both simple and difficult at the same time. At first, it is difficult to turn on such vision for functions, but when you turn it on, it will become very easy to look at the analysis.
At 21:45 he seems to not understand that real numbers are usually defined from set theory, not just "assumed".
More than 1 hour of discussion, just to hear one single argument over, and over, and over, and over again: Wildberger likes to suggest that every mathematical object must be computable and must have an analogue in the "real world" (whatever this means). How tiring.
Frankly I don’t understand Wildberger’s insistence on rationals. The number 1/3 is just as “infinitary” as π is. Have you ever tried to a cut a pie exactly into 3 equal slices?? It’s totally impossible. Should we reject the concept “1/3” then??
@am p Argument by notational convenience.
@am p My point is that "1/3" also requires an infinite process to think of conceptually. If you object to my example with a circle then think of how impossible it is to cut a rectangle perfectly into thirds. Sure you can think of fractions as these algebraic symbols satisfying certain rules, but that's not really the point in my opinion. From a conceptual point of view, 1/3 is just as complicated as pi.
@@anthonyymm511 Not really. Take a stick:
─
Then measure out a length of 1 stick:
─
─
The ratio of these is 1:1, or 1/1.
Then measure out a length of 3 sticks:
─
───
The ratio of these is 1:3, or 1/3. They're commensurable, whereas the length of the hypotenuse of □ is not commensurable with the length of its sides; there's no way to measure out 1, 2, 3, 4, etc. side lengths to reach the hypotenuse length.
Calling it an argument by notional convenience would make yours an argument by notational inconvenience. In a way, you may as well choose one very small unit length and build everything out of that. 1 will be 1000000000000000/1000000000000000, etc.
Then you can build pi, e, root2, etc. but of course it's approximate not exact. Still I think pure math is a phantom. Choose a minimal tolerance and work with that.
@@ThePallidorBut in your argument you have 3 sticks of equal length. This is already impossible, but even pretending it was possible you can simply take two of the sticks and form a right triangle. Does the hypotenuse not have a length??
What Wildberger doesn't (want to) get(s) is that his rational domain of mathematics also isn't anything else but a human construction - just like the reals. The none explicitly computability of irrational numbers isn't a significant distinction - especially not for pure mathematics. It's telling how he calls reals "religious fantasy", which shows that his actual problem is that he is believing in a "religious" signficance of the rational domain which he wants to protect from something he deems to be unpure as it isn't explicitly computable. He's projecting his own error - some platonistic view of mathematics. He should discuss this with an actual set theorist or mathematical logician (not analyst). He has little ground to stand on but on stubborness in some arbitrary insistence on explicit computability. All you need to know about his critique of reals is who he attacks while critising: introductory real analysis textbooks. Such text books arent ment to go in depth on reals. Reals arent studied in analysis but taken for granted. They are studied in more foundational mathematics. He knows that his arguments arent good enough to critique the fields actually dealing with this topic.
The interviewer did not prepare for this interview.
Wildberger could have kept it simple and just focused on infinity. Without that notion there are no real numbers.
Agree with Jim and Gennady.
There are a number of videos that Norman has done in which he examines the topic in greater detail. Separating numbers into their different types is a good start cos then we can say (& understand) what we can do with them and what their limits are.
@Jim Yocom, Actually I would like to defend Daniel on this point -- he did indeed ask me about some references prior to the debate, and I gave him a few videos to watch and also my paper, and he summarized some of my points quite well. It is a big ask to get someone to watch the dozens of videos I have on the weaknesses of the foundations of "real number arithmetic" and "modern set theory" etc.
Is mr. Wildberger an intuitionist? A constructivist? A finitist? Does he accept the law of the excluded middle? He seems to reason "by computer algorithms". What is his framework when he talks "about computer algorithms"? The discussion would be easier if mr. Wildberger would not only say what foundations he opposes, but also whose -- I want to hear names -- foundations he supports. Is L.E.J. Brouwer too vague in his opinion?
I don't think you are ever going to get answers to these questions.
non of those, he is a simplistic conservative 😄
Norman Wildberger gets a lot of stick but he has a point?
Is π/e a rational number?
Because if it were, we could use the fact to determine the n'th digit of one of them knowing the digits of the other and so on.
Notice that e×(1/e), which I assume is transcendental × transcendental = rational, so my conjecture is a possibility.
Saying that he is not happy at all with ∞ appearing in a limit, claiming in fact it is an absurdity, basically stating that ∞ ∉ ℝ.
So lets define a new number set, call it W = {ℝ \∞ }, and see what rules of arithmetic can be rescued using it. Of course we would loose concept of limit in the general sense, like ℓim x-> ∞, that would not be acceptable in a _Wildberger_ field.
@55:35 or so,
Norman objects to adding 3 irrational numbers, but when he points out that it’s analogous to asking a grade 6 student to add three fractions. This leads to the following, which is my “domino theory” approach to Norman’s philosophical objections to irrational numbers:
I claim that this philosophical objection reduces to an objection to the validity of the equation 0+0=0, since his objection is metaphysical, and the formalist objection to the Platonic view of the metaphysics of mathematics already objects to “actual” existence of any mathematical objects, including the number zero.
If you actually object to analysis as a form of religion, then you might as well object to all of mathematics as a form of religion. If you want to object to mathematical concepts using that metaphysical concern as your foundation of objecting to things, then why do you call yourself a mathematician? Is that not a form of hypocrisy?
...does the theory of real analysis anywhere state that it is possible to calculate everything exactly using a finite decimal representation? If not, then I don't understand the arguments by Wildberger *against* real analysis. He also barely lets Rubin respond.
...does the theory of real analysis anywhere state that it is possible to calculate everything exactly using a finite decimal representation?
no, only computable numbers can be calculated exactly. Every number youve ever heard of is computable, so including non-computables is a mistake on the part of real analysis.
@@samb443 I have heard of pi
@@samb443I have heard of pi. The definition of "number" as something which can be written using a finite decimal representation is a personal one.
@@bjornsundin5820 then im not sure what point you were getting at
even rationals dont have finite decimal representations, before even considering reals
@@samb443 I should probably have said "finite decimal representation or fraction" or "decimal representation with a repeating or terminating pattern" then. But his definition of what is allowed as "as long as it can be written down" is vague. 10/11 is a label for a number and pi is a label for a number. Both can be written on a piece of paper. Both can be rigorously defined. I respect the attempt to remove e.g. the axiom of choice and still end up with a useful mathematical theory. But I do not understand the criticism of real analysis based on the fact that real numbers cannot be written using a finite or repeating decimal representation, as if the theory requires this arbitrary constraint.
The problem with settling for saying whether "π+e+sqrt(2)" is greater or less than some rational number is that the "number" believed in here is not a definite value. You can do more and more steps of some iterative calculation to get a more and more precise value, but there is no definite value that is being approximated- every new term in the iteration is a new value, a slightly different number. To believe in 'real numbers' is to pretend that each new term is just giving you higher and higher resolution of some definite value, but by definition, irrational numbers cannot have definite values. There is no square root of 2, two does not have a square root.. there is a sequence of steps of numbers that are very close to what the value would be if it was a definite value. Here is an analogy- suppose you have 1 blue marble, then you are given an additional red marble and blue marble, and keep adding more pairs of red and blue marbles. There is always one more blue marble. The more pairs of marbles you add, you will be closer and closer to 50% red and 50% blue, but you will never have exactly 50/50 since there is always one more blue marble.
In ZFC, what you call "a value" is just a name among a lot of other names for the same set, it is called "decimal representation". But that is just a name, not the only way of defining numbers, so even if we don't know the decimal representation of π+e, that doesn't mean it isn't defined.
@@DanielBWilliams Hi Daniel- For the point I'm making you don't have to get all the way into π+e+sqrt(2), that is just the example that was brought up in the video. My point is that the "square root of 2" (for example) is not a definite value, (not that it is not defined)..that is, it is not an exact amount. It can be convincing that rational numbers can be given which are greater than or less than any nth iteration of some calculation, but that does not make it a definite value or an actual exact amount.
@@ArtCreatorsChannel In ZFC, decimal representation of every real number exists and is defined, it is just that for instance for √2 we don't know the list of the digits.
What makes you think it's value isn't defined ? (maybe what you call its value isn't what I call decimal representation)
@@DanielBWilliams Right- what I am calling "value" or "amount" is not the same as decimal representation. An example of a valid decimal representation would be 0.0625, since 625 x 16 = 10000, and the relation between 625 and 10000 is the same as the relation between 1 and 16, in other words, "1/16". The same cannot be said for √2. There is an indefinite sequence 1, 1.4, 1.41, 1.414, 1.4142.. etc which gets more precise with each new term, each new term being a slightly different value, a different number, a different amount. The symbol for this sequence, or some imaginary completion of the sequence is "√2", but this is fictional. There is no actual completion of this sequence, and "√2" does not have a definite value- each term in the sequence is a slightly different amount, and there is no actual value to "√2" since 2 does not and can not have a square root. There is not a number, that, when it is squared, equals 2. We do a sequence which gives us numbers getting closer and closer to 2 when it is squared, but like my marble analogy how there will always be one more blue marble, no number will ever be the square root of two, it can't be, it is not possible. The imagined value for "√2" is not what we are taught to think it is. Even with infinity digits to it's indefinite decimal expansion, it will never be a single value, and any truncated approximation at any given term will never be a number that, when squared, equals 2.
@@ArtCreatorsChannel But in the set R, there is an element x such that x²=2 and x>0. It is unique, and so we call it √2. Even if we can't list all of the digits of its decimal representation.
His first argument was that modern math is rooted in philosophy. His argument is also rooted in philosophy. So if being rooted in philosophy is bad then so is his argument.
He shouldn't attack philosophy, just the philosophy departments of today. Philosophy is better characterized as the neglected foundations of every field. In fact the division of academia into fields introduces many errors itself.
Exactly
Hi, I think if you look at what he is genuinely saying, you may find that his point is that modern mathematics is based on beliefs. What he is saying is that math should be based on what can be proven. What can be proven should be considered true. So I think his distinction is truth vs belief, not your philosophy bad mine good.
@@robocop30301 how would he prove axioms?
26:09 “mathematica” why? why not limit our discussion to just our head? 😊😊😊😊
Hola, si hay alguien que esté estudiando, investigando estos temas, de habla hispana, y quiera charlar de estos temas, nos contactemos!
I really wish Daniel Rubin cared a little bit more about computing because it's pretty much impossible to be a hobbyist computer programmer and pluck your favorite real function off the textbook into a tiny little program to play around with. You should be asking what is needed to create an exactly computable real number or somethibng,
The simple fact that the addition of a few irrational numbers cannot *_in principle_* produce an exact answer *without infinite work* seems worthy of far more pause than was exhibited. Great discussion!
You are confusing "cannot produce an exact answer" with "cannot produce the entire decimal representation".
@@DanielBWilliams - then please explain the difference.
@@coffeyjjj As every mathematical objects, a number has a lot of ways to be written. For instance, the number 24 can be written as 2³×3. So 24 and 2³×3 are two differents ways of writing the same number : one is the decimal representation, the other is the prime decomposition.
If I know one representation, and not the other, that doesn't mean I don't know exactly the number.
For instance, 3+4 is exactly defined, and I don't need to give the decimal representation to do it. Of course the decimal representation is 7, but I didn't need it to say it is exact.
Generally speaking, a mathematical object is (exactly) defined when you have given a property verified by only one object of that kind.
For instance, we can show that in R there is a unique element x positive such that x²=2.
So we can define √2 to be that number, and as :
1) in R there is at least one number like that
2) in R there is no other number like that
The number is exactly defined.
After that, we can of course search for its decimal representation, but that would just give us more information, the number wouldn't be more defined.
@@DanielBWilliams - Sophistry. My comment *explicitly* mentions "exact answers" not "definitions", and was obviously in reference to this specific Wildberger video containing his *explicit* objection clearly noting the fact that *infinite work is required* in order to generate the DECIMAL REPRESENTATION of most real numbers. Your claim that I'm misunderstanding something appears to rest on very, very shaky ground, friend.
How may the reasoning of your "definition" be used to *explicitly determine* each of the N decimal digits 0-9 located at the 10⁸⁰⁺ᴺ place in the decimal representation of √2, where N is all integers from 1 to 63? Or how about each of the N digits located at the 10⁸⁰⁰⁺ᴺ place of the decimal representation of √2?
The "exact answer" I originally mentioned would provide those digits. I don't think your "definition" can identify those N digits, not even in principle. Do you think it can?
@@coffeyjjj That's what I was saying. For you having an exact answer means to know the decimal representation, but that's just a representation. Your problem (and Wildberger problem) is to think not knowing the decimal representation is not knowing the number. That's not how maths works. Knowing a number is just knowing a property that it's the only mathematical object to verify, and as you said it is the same thing as defining it. Why do you care so much about decimal representation ? Yes it is usefull, but as I said, it is *just* a reprensation among a lot of other.
Yes ZFC can't provide decimal representation to all of its numbers, so what ?
Props to Daniel for doing this debate, but it's a little embarrassing how unprepared he was for some of Norman's basic arguments
@Gennady Arshad Notowidigdo Norman would do well to take a look at basic epistemology, David Hume being the most revealing, because once one realizes that first-person epistemological phrasing is the most foundational phrasing, it quickly becomes clear that labeling a mathematical object that is not visualizable (or otherwise sensualizable) _even in principle_ is labeling nonsense.
He dismisses philosophy as an airy subject, which it is today in modern academia, but the proper role of philosophy is to serve as the foundations of every field and therefore it should be the most rigorous of all. There is a rigorous way to do philosophy, but it requires clear definitions used consistently, which academic careerists have a heck of a time doing.
When Norm invokes "what we can write down" or "what our computers are telling us," he is reaching due to this lack of clarity on basic epistemology.
As he says, a proof must in principle be an argument all the way from the foundations to the final result, but the real foundations of mathematics are in epistemology. Steve Patterson's Square One is a good easy intro to this, though David Hume's _A Treatise on Human Understanding_ is more thorough.
Otherwise he opens himself up to the objection that his computational view is just another view or approach.
@@ThePallidor Maybe there is a logical way to do philosophy, maybe not, but it's a big stretch to say that David Hume is a necessary prerequisite to mathematics. Personally I'm still convinced that real numbers are a fantasy and I don't see how anything you've said about epistemology proves otherwise
@Gennady Arshad Notowidigdo I don't know if that's a hole or just something that needs to be expounded upon
@5:55 nonono... real numbers, like π 𝑒, and all those you'll never know, are not _defined_ by decimal expansion. The representations of concepts are not the concepts themselves. The whole point NJW misses (it seems) is that mathematics is not physics and is not computational, if it were it'd be the lesser. It is _because_ we can entertain nonphysical concepts with mathematics that allows us to refine physics (and a lot more besides). Science becomes feeble if we have no ability to entertain what is nonphysical. That's because so much of science is _negativa_ --- it's often what is _not true_ that you know the best.
@15:00 he admits this when he pivots to "algebraic concept ok". But "point on the number line" or "point in the plane" is not a mathematical concept, it's physical, or invoking physical intuition. There is no one true notion of a "continuum".
@1:03:00 you can perfomr transfinite arithmetic in some software. That fact you cannot do more is because no one's yet written the general code, but they have not written general code even for a tiny subset of ℕ either, so this "write-downable" is ridiculous. Everytime what we can write down increases he is saying mathematics expands. The mathematical expanding universe. But I doubt FRWL would've approved. It's an impoverished view of the word "mathematics" if you you constrain it by "what we can write down". Moreover, if he is saying "what we *_could_* write down" then it is completely undefined, since he has no idea what in the future is possible.
40:12 This is simply the principle of mathematical induction which Prof. Wildberger is assuming, which is one of the core assumptions for constructing the natural numbers, which leads to the construction of real numbers and real analysis.