You know, I've always found interesting how our 'critics' call us finitists, when in reality it seems we are the ones who actually believe in infinity. We respect the concept by accepting the fact that we can't do an infinite amount of steps, that the sequence of natural numbers is vas and complex and could never hope to reach to the end of it. Meanwhile ZFC practicioners, think that by formalist trickery, they can somehow "tame" infinity, resulting in complete fiction like infinite cardinals and ridiculous results that I'm sure most of us are familiar, which in an ironic twist of fate collapses infinity and math into a mere symbolic manipulation of finite strings. Please never stop releasing videos Professor Wildberger, you are doing wonders in the mathematical community especially with the young generation. True revolutions are subtle and take their time to happen, and slowly but surely they do happen.
we analyse finite quantities between zero and infinity.There are some limits for the series as you said in for e.g 1+1÷2+1×4+…..There may are likewise limits for pi,e,and 1/3.Here we approximate.But how is it justifiable to apply limits to derivatives?.It is ,I am sure,misplaced application of limits.The limit of f'(x) of f(x) is not logical.Consider f'(x) of f(x)=x².It is x²+2xh+h²-x²/h=2x+h.How can it be possible to reject h,for h can take any value in reality?.The difference between x² and (x+1)² is always 2x+1.If you take larger values for h,then the difference is 2x+larger values,and if h is less than 1 (but not zero) we have 2x+h (h or < make no precise sense.
@@irappapatil8621 I agree there is an issue with how limits are used to find the slope of the tangent line . A leap of faith is made when the limit is claimed to correspond to the slope of the tangent line.
Thank you so much for making this excellent lecture - for two reasons: First, it so very clearly explains the fundamental difference in quality of the type 2 and type 3 series you show - this was exactly the clarification I was seeking as a result of some of the earlier lectures on this topic. Secondly, this lecture identifies the type 3 series, and the related Cauchy Convergence Criterion applications as leading to some highly problematic issues, including some very problematic non-true facts, and this is very interesting to me for the following reason: As an undergraduate student I studied mathematics and physics. In the first year I attended a mixture of pure maths, applied maths and physics courses. In one of the courses aimed at pure maths I encountered the Cauchy Criterion, epsilon delta arguments etc and I found it very difficult to get comfortable that i really understood it. I concluded that at least one of two things had occurred: That I had somehow missed out on some elementary parts of mathematics education at school, back where advanced topics were introduced but in terms that "the real truth will be covered at university", and as a result of missing the relevant topics I now was befuddled by this analysis topic having got to university, at which there seemed to be an air that certain of things were "obvious" from school maths, and second that I lacked the mental equipment to truly appreciate and excel at these pure maths topics. As a result I specialised into Applied Mathematics and Theoretical Physics, which I enjoyed very much, and then after university I moved into computing, which I also enjoy very much. And now, thanks to your extensive lecture series on here aimed at putting various areas of pure maths on a rational footing , my interest in pure maths has also been rekindled!
So, the properties of the rule that generates the series are mistakenly forced into the category of numbers or number-like mental objects. In fact, a certain algorithm-analysis (based on arithmetics) is being dressed up as "real" analysis. Thank you, professor Wildberger. I get your point, I am woken up. I hope that some of your colleagues will join you soon in reconfiguring mathematics. You have a lot of work to do.
Thank you for another superb clarification of issues too many other mathematicians like to keep obscure! As far as I can see, many people, including many mathematicians, build vague 'notions' on other vague 'notions', whereby the 'existence' of the 'irrational' 'limit' is a special case of the more general lack of understanding what 'existence' is supposed to mean. In the more general case, quite possibly there are some actual sculptors who really believe that a particular future sculpture already exists inside a big rock, thereby foolishly denigrating their own work of bringing it into existence. Pure mathematicians, to their perceived advantage and the disadvantage of those seeking thorough understanding, have found a way of avoiding the hard work the honest sculpor does. They claim and usually really think, that if they have the equivalent of the rock they already have the equivalent of the sculpture and thus don't need to take the trouble of creating it. And if the actual creation of the sculpture should turn out to be impossible, they brush that aside. 'Building' further on that, soon they no longer need the actual rock, either, at least when it becomes too troublesome that for one mathematician's actual rock there will be a rival mathematician's plan for a sculpture that is too big to 'exist' inside the rock. If I 'have' an 'infinitely' large rock, Hilbert's hotel already 'exists' inside it, and if it is unfurnished an 'infinity' of furniture will fix that problem, too.
As far as I remember Cauchy originally only stated that converging sequences are Cauchy sequences. At that time there were no formal definitions of real numbers, so it was neither possible formulate nor prove that Cauchy sequences converge to rational numbers.
On a serious note...The more I listen to you about real numbers, the more it reminds me of Wolfram's "Computational Irreducibility", with the added proviso of requiring infinite computing power. Computational Irreducibility can occur in finite steps. I don't know if that is pertinent or not. Are finite and non-finite computationally irreducible systems equivalent? This also reminds me of your analysis of whole numbers and their fractal nature. There is something going on here with numbers that has not been clearly specified yet; some big insight to take things to a whole new understanding. I would guess that computational irreducibility will be an essential ingredient in whatever this next notion of numbers turns out to be. Thanks for all you do!
Thanks, Prof. W., for this interesting video. I remember when I first saw the claim made in a class that real numbers could be defined as equivalence classes of Cauchy sequences, followed of course by a hand wave over the issue of proving that those things comprise a field. As I understand it, that briefly is your objection to the real numbers - even if you accepted that those equivalence classes exist, would they provably form a field? I wonder if it would work to allow that yes, a Cauchy sequence of rationals has a limit, but almost always that limit is just a geometric point but not a number. Then pi would be something, a geometric object, but for engineering purposes we have to use partial sums of a series whose nonnumeric “sum” is pi.
The way the concept of 'real numbers' was introduced to me is quite similar to what is going on in this video. The remark that not every (Cauchy) sequence of rationals converges to a rational number is a very important observation. This means that if our concept of 'number' should include things like the length of a diagonal in a square and the circumference of the circle, then we have to consider some other number system. There are multiple ways of doing this, but the one that naturally follows from what was said in the video is to consider sequences of rational numbers themselves. You then notice that for any given rational, there are many sequences with that rational as a limit, so we better consider all those as equivalent to each other. We now define two sequences (converging to rationals or not) as equivalent if their difference converges to 0. We then define a 'real number' as an equivalence class of Cauchy sequences. We then see that any rational number is a real number, namely the equivalence class of the constant sequence. Next we can define addition and multiplication componentwise (we have to prove that the the sum of two Cauchy is Cauchy, and if equivalent sequences are added/multiplied we obtain equivalent sequences). Then we have to prove that it satisfies the standard calculation rules (associativity commutativity, distributivity, etc). Once we did all this we are justified to say that the 'real numbers' are a number system. We can then even consider sequences of real numbers (so sequences of equivalence classes of Cauchy sequences) and in that Context it makes sense to say that for every Cauchy sequence there is a real number, which is its limit. When analysts talk about limits, this is what they mean, no? I have never met an analyst saying, "no, real numbers are just decimals that go on and on"
That’s all fine as a philosophical system. Or perhaps as a religious system. But what do you do when you actually want to make computations? For example, how does one compute pi +e+ sqrt(2)? If you have an arithmetical system in which you can’t actually perform the operations then you’re kidding yourself.
@@njwildberger You haven't rebutted what @MrLordCoder wrote. Also, in what sense do you mean by "can't actually perform the operations?" Although I could in principle, I couldn't actually calculate 12^100 without a computer. On the other hand, in practice we can compute millions or billions of digits of pi + e + sqrt(2) and somebody like Mr Lord Coder has a grasp of the underlying definitions of addition of real numbers.
@@rupertmillard In the strictest sense imaginable: The order and equality relation among computable real numbers is Turing undecidable. This has been proven long ago. The same goes for Dedekind cuts. You cannot maintain all the properties of the reals without being impredicative.
>the one that follows naturally from what was said in the video is to consider sequences of rational numbers themselves no, that is not the natural one, and you have no reason to think they are besides the fact that they're the only choice you ever consider "Definable" sequences of rationals are inherently more natural than the ones you wish to include furthermore, the only rational sequences that can have a limit taken on them, without introducing meta-theoretical issues, are the computable rational sequences
Real numbers are often defined as (equivalence classes of) Cauchy sequences. With that definitition it is very easy to prove at the sum of 1/(n!)^3 is a real number.
Unfortunately though, that definition of real numbers as equivalence classes of Cauchy sequences is nonsense. See my Math Foundations lectures starting around ruclips.net/video/kcirwIwRIUw/видео.html
@@njwildberger It may still be useful that we are able to define the notion of an Cauchy sequence and it may also be useful that we can state that two Cauchy sequences are equivalent. The notion of a Cauchy sequence captures a mathematical property that can be studied.
@@njwildberger Now I saw your old video again, and you point to two main problems. The first problem is what one means by an infinite sequence. For me it is fine if we restrict our attention to recursive sequences (or computable sequences as you talk about in the video). The second problem is about equivalent classes. The problem with the equivalence classes already appear in the standard definition of rational numbers where a rational is defined as equivalence classes of pairs of integers. The equivalence is defined by (m,n)~(x,y) when m*y=x*n . The point in this "construction" of the rational numbers seems to be that rational numbers can be identified with sets and is a result of the project of identifying any mathematical object with a set. The problem is obviously that someting simple like the rational number 1/2 has to be identified with an infinite set! For me a rational number is not a set but is a pair of numbers separated by a bar, i.e. a fraction. For me the fraction 9/6 is not the same fraction as 3/2. When we write 9/6=3/2 it does not mean that we have the same fraction on both sides. It means that 9/6 and 3/2 has the same value, and what does that mean? it means that 9*2=3*6. Again, 9*2=3*6 does not mean that we have the same on both sides, but it means that 9*2 and 3*6 has the same value, and to verify this, some kind of calculation is needed. We should distinguish between sense and denotation. In my oppinion the real problem is the Platonic view mathematics, where the focus is entirely on denotation and where sense is reduced to some sort of subjective illusion.
@@PeterHarremoes- With your strange view of rational numbers, can you perform computations with them? In your view, what is, e.g., the result of adding 1 and 1/2? Is it 3/2? Why not 9/6? If both fractions are different, as you say, are there any situations where one can use one of them, but not the other?
@@PeterHarremoes You raise important issues. The best we got so far for recursive sequences (in the sense of repeating structures) are the quadratic periods of continued fractions. Cauchy sequencies have intuitive connection with continued fractions, especially their form of path informations along the binary tree of blanks nested in Stern-Brocot type structure of totally ordered rationals. Gosper arithmetic of continued fractions / continued logarithms is at least better defined effective method than the basically undefined decimal arithmetic. Our professor is first to admit that also the coordinate system neusis dependent definition of "point" as a pair or more "numbers" is also very problematic. The issue of continuum and continuous directed movement is why Greeks rejected neusis from pure geometry into the lower caste of mere "applied math", in which the rigorous criteria of mathematical truth can be relaxed for some pragmatic teleological purposes. The notion of "equivalence classes" is on the other hand very counterintuitive, and on the other hand very rigid and cumbersome. It's been abstracted away from intuitive clarity into what we can fairly consider the wrong direction. In the heuristic aspect of math we need to study also the wrong directions deeply enough to realize how and why they are wrong, we are supposed to do also mistakes and learn from them. What is intuitively strong and enduring keeps on reappearing. Greeks intuitively understood equivalence as modal negation of relational operators: When A is neither more nor less than B, then A =B (see method of exhaustion and the intuition behind many of Euclid's proofs). IMHO the greatest insight of Don Knuth's inspired book "Surreal Numbers" was the consciouss derivation of equivalence relation from relational operators, which AFAIK Greeks never formally define but just implied intuitively. It's been considered so intuitively self-evident, that it need not be formally defined. Of course the derivation of equivalence relation from relational operators is contextual, and applies coherently only to coherently comparable properties. It's much weaker than "Law of Identity", and that's a good thing. There are very good reasons why Whitehead chose the point-free path after the failure of PM. I see no good reasons to abandon Euclid's definition of point as the halting of the top-down process of mereological decomposing (Point is that which has no part). Because the post-modern set theory is incoherent with mereology (as already Gödel observed), many academically trained contemporary mathematicians have lost basic mereological intuitions, including standard vocabulary. Wildbergers foundational inquiry is deep down very mereological, as we can see e.g. from the basic inclusion relation of the Box Arithmetic. It's a big task, but as we have made also genuine progress since Euclid, we are now ready to research mereological foundations with fresh tools. In this regard, I see no other coherent alternative than to accept that continuous directed movement is holistically irreducible empirical foundation of mathematics. From this acceptance it's not very complicated to construct and accept that fractions/rationals are not "points" of coordinate system neusis or "number line", but continuous intervals, mereological partitions of the ontologically irreducible continuum. As we keep on intuitively teaching the kids... only to later betray their trust with absurd point-reductionism.
I don't think i have completely understood your objections to the notion of real numbers or infinity but i would like to add my two cents in the converstation for what is worth. I don't see what exactly is the issue with having infinity in mathematics. The fact that computations with reals are never truly achievable via the use of computers, to me, is just a limitation of computers, not evidence that our foundations are flawed. Furthermore, as a mathematician, I find the axiom of infinity and the axiom of choice very much intuitive. They are just axioms, though, meaning that you can either accept them or not. I myself dont see any reason why i should deny them , but i would love to see an argument in favor of that position! Thank you, and i hope my comment wasn't too incomprehensible!
Thanks, very illuminating. A further question. In the third case we can we can represent the decimal string as a continued fraction reading of a list of rational convergents. Can we thus define distinction between 2nd and 3rd case as continued fraction with a single final convergent vs. open ended reading of many convergents? Are the other complexities involved that make this definition incomplete?
Ok, thanks for a good analysis on limits. I think lots of things could be added though. The reason for real numbers in the first place must be to be able to successfully deal with continuum problems. You simply add "all" irrational points on the line and call them real. Rational numbers have the advantage of being point specific, as well as whole numbers, but for completeness reasons, that's not true of the reals. There is also the issue of mixing irrational numbers (irrational reals) with rational numbers or rational real numbers. This is not trivial, since the results of calculations will differ depending on which model you pick. Take the rational number 1/3 for example; is it really exactly equal in every way to 0.333...? Certainly not if you define 0.333... to be a real instead of a rational number. A real number has a sort of "fuzziness" to it, where values (or more specifically, points on a geomertical line) that are infinitesimally close count as the same real number, something that doesn't hold for rationals, since real and rational number objects have very different definitions. So, who's to blame for the confusion about these concepts?
Thanks for the comment, but to someone who has not seen a proper definition of the term "real number" and "irrational number", I am afraid that I can make little sense of what you are talking about.
@@njwildberger Well, we could make the geometrical definition of irrational numbers as corresponding to those points on the line that are not rational. That is not enough though to make them usable in calculations; we need more rules, so that every point in the infinitesimal interval between the two groups of bigger and smaller rational number points, for example as examplified as a Dedekind Cut, are defined as the same real number object. As long as we don't use an infinite number of additions (or other similar finite operations), the math will essentialy be the same as using rational number additions on the number line. We have to add the rule that infinite decimals that look different represent the same real number, but this is clear if we can identify the interval from both sides, for example pi: (3, 4) , (3.1, 3.2), (3.14, 3.15), (3.141, 3.142) and so on, going on to an infinitesimal interval where we can stop, and 1/3: (0.3, 0.4), (0.33, 0.34), (0.333, 0.334), and so on, where it is clear that the interval length will be smaller than finite but bigger than zero. We can take the interval length to be zero when calculating, because it will not change to a finite size if we keep doing a finite number of operations with these real number objects. The use of having access to all points on the line is of course very practical, and when we really need to perform an infinity of operations on those irrational objects, we need extra rules to make the results well behaved, which has led to calculus, where having a zero not always means to having a rational zero, since when you go to infinity adding it to itself, it can give you any finite real number depending on how the infinity works in that calculus formula. There are a number of different definitions of real numbers, picking one will represent them in a way that is more convenient for certain calculations and uses. We could for example instead use binary infinite decimals and they would work equally good as representations of reals as the base ten infinite decimals, we could switch between any base we like (2, 3, 4, ...) and still retain the real value representation relative to the geometrical line points representation. This still works converting to and from infinite decimals with irrational bases. When using a number system that allows for infinitesimal representation the problem with having reals representing an infinitesimal interval becomes obvious and you get a one-to-many relation between the real number and the infintiesimally enriched number system. More rules are invented in the case of translation between reals and such systems, such as between reals and hyperreals, where many options are open going to hyperreals, since choices have to be made between one real value and infinitelly many possible hyperreal values. Going to reals from hyperreals is easier, since we can just set the infinitesimal part to zero, at least if the hyperreal infinite part is equal to zero.
Rather than the issue being the limit, i see the issue as arising from the existential quantifier in the cauchy condition instead of saying "for any M, there is some k" if you say "there a computable function (in other words, a function, non-computables dont make sense) that takes M to k" which satisfies the normal conditions Then you dont get the real numbers, you get the very well behaved computables. This eliminates many issues you have the process of taking the limit of reals, since using that function, you now know ahead of time exactly how far down the sequence you need to look, in order to see what it equals And i take issue with your criticism at the end, that we cant define numbers to be a "process", they must somehow be something more fundamental than that. This just isn't true, even with your beloved rationals, the process which makes them is obvious every time you write them down. A rational is the process of dividing one integer by another, you cant actually break one thing into 2 but we say that we can. Likewise the naturals are the process of taking the successor.
I've implemented code that uses exact rational arithmetic. The numerators and denominators quickly grow to enormous sizes, making the calculations impractical.
If the argument is that it is wrong to define the number as being a result of a process wouldn't same reasoning also apply to simple rational numbers like 1/3. 1/3 also has an infinite decimal represntation but since we can define it as a result of a division operation it is a rational and thus fine. But this is only because we allowed ourselves the operation of division. Why is defining a number through a more convoluted process using sums and factorials really that different to this? 1/3 is in the same way incomputable and only exists because we allow purselves the "cheat" of keeping track of both nominator of denominator. But you can never actually write it down without that cheat.
To common sense it appears that a quantity is less than,close to,bigger than is imprecise.It must be said 'how many times'.Limits cannot satisfy precision which is a must in logic.WHEN WE SAY DELTA DIFFERENCE GIVES EPSILON DIFFERENCE MAKES ONE TO PUZZLE AROUND WHAT ARE THE MAGNITUDES OF DELTA AND EPSILON .
I don't really think that not being able to fully describe a number is a problem. There are just more numbers than ways to describe or compute them, but we can still find properties of numbers, even ones we can't define. The problem doesn't get much better when you move onto things like functions.
it is odd that we cannot even systematically *name* such general real numbers since their mere representation is without limit. this is very unusual compared to say rationals which, while may have unbounded decimal representation, have a systematic and _finite_ way of notating them (as fractions). real numbers really are a different sort of beast of an idea, and one that I think warrants more skepticism than it attracts.
How about the problem of fully DEFINING a number. That surely is a problem. That's exactly the situation we are in: despite all the mumbo jumbo re equivalence classes of Cauchy sequences and Dedekind cuts and infinite decimals etc, there is no proper DEFINITION of what a real number is, and what the operations on real numbers are.
@njwildberger an analogous situation in computer science terms would be to define a type for which there is no general implementation, only for specific subsets of that type. where the type would be "real number" but all theoretically possible implementations (instances) would only cover an almost infinitely small number of "implementations" that the type purports to allow. I.e. we would view such a type as too unconstrained, somewhat absurdly so, despite covering the valid (implementable) instances.
practically what we would do is instead define types that admit implementations, such as an algebraic type for rationals (perhaps as fractions composed of two integers) or more fancy types for algebraic objects that fulfil the role of roots, etc. all the polynomial objects can be defined algebraically without the non-computable notion of real numbers.
I understand the argument here. But what if we just say that it converges to "some number" but we will never truly know what that number is? We can only specify it up to a certain precision. At least I'm comfortable with that level of doubt.
It comes down then to our definition of “number” in the first place. If that is a precise definition, then it should not be allowing of doubt and imprecision. This is mathematics after all.
So if I understand correctly, if a series "converges" to pi, it is a "type 3" series? since pi only exists as an iterative computational process... We can call our 2.1297 number "zi" and say our series converges to zi.
You've touched on something Ive been pondering about for a few years, ever since I finished my bachelors of philosophy. I now study at UNSW a double major of physics and math. Just from my very nature, even from a young age, Ive never felt comfortable with the idea of infinity. For years Ive pondered on what in essence a number is. If the sqrt(2) is a number, I eventually concluded that sqrt(2) was a process (because it could never reach a defined end). Hence, I concluded the essence of a number is not as an end or final outcome but a process. Hence, we now have to go back to all the natural numbers and think of them as processes rather than ends. A number is no longer a computed result but the computational process itself. I found this conclusion absurd. Hence, Ive always privately rejected the idea of infinite real number decimals as anything which can exist or be real. Moreover, a natural number is clearly not a process but a defined end. So then its not in the essence of number that it is a process, and therefore because the natural numbers are logically and conceptually fundamental to any other type of number, if a natural number is not a process, then no process can be a number. Therefore, the sqrt(2) cannot be a number. There's so much more I want to say. I hope one day I can have a nice chat with you and share ideas on these topics.
@Continential Yeah, I would say so. One may say it's because both the process and number are distinct, that we cannot have both at the same time. However, our use of the word process is ambiguous. In one sense, a process is an epistemological procedure, that is, the methodological means by which we go from ignorance of a a number thats a solution, to knowledge of the solution, namely, the number. Another sense of process is as an ontological property, the currently accepted idea that a number can itself be entirely and ONLY a process. Now, for sqrt(2), in this sense of process, you'll have to accept the idea that an incompletable process is an object which is the under the same category (category of objects we consider numbers) of objects which are not processes (but which can be the result of an epistemological process), which is say a natural number. We may say that the result of the process of adding 1 to 1 is 2, isnt saying that 2 is an ontological process, that is, the essential definition of 2 is the process of computing 1+1, why?, because 2 can be the result of different computations of different numbers. Rather, this process of computation is an epistemological process of identifying that the solution to 1+1 is the result 2, or of learning that 2 can be identified with the outcome of the process of computing 1+1. Epistemological processes don't define a number but identify a number with a process, in as much as the process has a defined outcome. The epistemological process can tell us about the necessary properties of a number, namely, that the property of 1 additional to another 1 is identical to 2. However, an ontological process as I have defined it earlier doesn't simply identify the number with a process but defines the number as the process. However, this view disregards the essential distinction between process and outcome, that an object can be exactly both at the same time: this results in the contradiction that an endless process can be the same as a process that comes to an end. However, one can respond by saying when we state the sqrt(2) is a number, we are saying that the endless process which is the sqrt(2) is just essentially the DESCRIPTION that the sqrt(2) is every part of a process which, although endless, exist simulateously. So every part of the infinite process which is the sqrt(2) exists simultaneously and therefore any reference to the endless process not having an outcome is simply epistemological: we cannot know all that is in the sqrt(2) (all the infinite decimals) simultaneously (as we cannot know an infinite amount of things at the same instant) despite all its parts existing simultaneously. But, who's to say that epistemological limits determine the limits of what a number can be? Moreover, all parts of an endless process existing simultaneously does have the property of a process that ends: neither remains incomplete. However, here lies the conundrum. In an epistemological process we come to know what something is. If we admit the existence of an ontological process for numbers as I have specifically defined it, and that all parts of this process exist simultaneously, an essential property of a number remains still that it is a complete end. However, by definition a number counts things. Since, no thing has an infinite amount of parts, the sqrt(2), having an infinite amount of numerical parts, can never count anything. Therefore, sqrt(2) cannot be considered a number. It cannot exist as a number. But if it's not a number, then what is it? What are we referring to by the term sqrt(2)? Well, it's an amalgamation of different concepts that's given a single name: sqrt(2). This amalgamation consists of, for instance, the concepts of number, process, infinite, parts, convergence, existence etc, arranged in a certain way as to convey a single idea we term sqrt(2). In the same way, for instance, golden Mountain that speaks English is just an amalgamation of different concepts: gold, mountain, English, speech etc. However, I think to even say something exists in some sense entails that all the parts of the object exist. However, something with an infinite amount of parts cannot exist since there will always be a part which isn't accounted for which must exist for the object to exist. Therefore, all parts of an object with an infinite amount of parts cannot exist simultaneously since for whatever portion of the object that exists there will always be a subsequent part of it which hasn't been accounted for as existing. Therefore all parts of an object with an infinite amount of parts cannot exist since not all parts of the object can be accounted for as existing: it can never completely exist. Hence, I leave you with this final thought. The concept of the sqrt(2) is an idealisation which has use in an epistemological process in mathematics. It ensures we account for essential properties of a solution, depending on the mathematical environment we are working with. For instance, if a solution to the root of a function is the sqrt(2), then it tells me in an applied setting that the finite solution to problem in the applied setting is found in the process of computing the sqrt(2), that is, that the exact finite solution will be a finite number which is derived from the process of computing the sqrt(2). However, in a pure math setting, to say the exact solution is the sqrt(2) is nonsensical since the sqrt(2) cannot exist because all parts of it cannot be accounted for as existing. For any part of it which exists, there will always be a part of it which hasn't been accounted for as existing, therefore it's existence can never be complete: all its part cannot be accounted for.
Of the scenarios you've provided, whereas the first two posites "Does the series converge, and if so, onto what real numbered value ?", I view the third example series as an approximation problem, up to an arbitrary precision, unrelated to the two. However the third example has monstrous ramifications in the area of artificial neural networks verification, which largely base their operational characteristics on mathematical models! The third example series you've provided is an excellent example of the inherent problem involved in verifying most artificial neural network models, and also why most of the verification methods of these systems embedded in our automotive, healthcare, food industry, and military systems, even up to a 99.99% reliability, is wildly insufficient.
The wonders of circular reasoning. Academic "philosophers" are paid to spot logical fallacies in the reasoning of other academics in a disparate array of academic disciplines, but they never do. I prefer to call irrational numbers "notional" numbers rather than give them the elevated status of being within a class of numbers described as "real". Complete misuse of language and generations of professional mathematicians have been fooled in to thinking that algebraic irrational "notional" numbers and transcendental "notional" numbers exist in the same sense that rational numbers exist. They exist in an analogous sense to the "largest natural number" that can be imagined. There is always a larger number than the largest you can ever imagine because you can always add 1 to it. There are always more digits of a notional number to be found, just determine one more.
I think "readings" is a good term for algorithmic "irrational numbers", and then we can reserve the word "number" for products and names of tally operations.
Sort of easier in binary. 1.1, 1.11, 1.111, 1.1111, 1.11111, 1.111111,... but like .999... is the same as the number 1, that's a problem with the notation. There are multiple ways to represent the same number. Isn't it sort of a good thing? In theory any three element set _is_ the number 3.
The set {elephant, cloud, computing} has three elements. It's very hard to avoid a deep feeling of absurdity if we claim that the set in question is the number 3.
Hi Norman, thanks for this video! I was wondering what downstream problems arise or are being ignored by overlooking this problem of defining irrational numbers like pi, e root 2 etc using a rational base 10 or 2 number system? I remember you mentioned the problem of 0.9999 repeater resulting in a 1 or not which can't be known until it's repeated to infinity. What other computer science problems do you see occurring when these irrational numbers are defined as programs and are combined by addition or multiplication or other operations? For example, i imagine that the halting problem is a concern? Thanks
The reality is that it is impossible to properly define “irrational numbers” at this stage of the development of mathematics. Any attempt to define them in terms of suitable algorithms, and to define the operations with them in terms of suitable operations on those algorithms is currently doomed due to intractable problems at the level of computer programs, and recognising when they are equivalent. Now it may be that in the future, some bright people will come up with a more restrictive of irrationalities, using a restricted kind of algorithm. But this is not the situation we are currently in.
For a so-called real number to be called a 'computable number', all that is required is that we can write an algorithm that would determine its n-th digit. My main reservation about terms like "computable number" and "computable function" is that they don't match the common usage of "computable", as such, they appear to have slippery definitions. The common understanding of "computable", as I see it, is "capable of being computed". This implies a computation process leading to a definite and precise answer. The mathematical definitions are crafted with the clear intention to categorise real numbers like √2 as being 'computable'. But to say that √2 is computable suggests that we can compute √2 to infinite precision, which is absurd. Based on the common understanding of "computable", it's evident that √2 isn't computable. Much of mathematics, particularly concerning real numbers, seems rife with misleading expressions. I believe that if we employ a term like "computable", it should not be defined in a manner that suggests the computation of infinite values is feasible. It is frustrating to witness mathematics perpetuate such practices, which only serve to reinforce the mistaken belief that infinite processes can be completed. There is a more in-depth discussion about these matters at mathforums dot com slash computer-science. Look for my comments in the thread called: About why I believe that the "What Computers Can't Do" argument (i.e. the halting problem proof as applied to real world computers) is not valid.
As always in mathematics, names can be chosen arbitrarily and are given their meanig by the corresponding definition only. You could call the computable numbers "green numbers" as well. They're not greener than they are computable in any sense of that word in which you use it in other contexts. Similarly, the name "natural numbers" is likely to give rise to expectations that can't be satisfied. They are not growing somewhere out there on a tree. (Which would be a reason to call them "green".) Both those terms don't seem too bad a choice for the property they are to describe, imo. As to the term of "real number", it seems to generate such large expectations on their "reality" that a different name might indeed have been better, like the "continuous" or the "linear" numbers. Now which further information would you like to have for a number that can be computed, apart from all its digits? You might say that it's not "all its digits" but "every single digit" at most, but where is the difference? A finitist's computer can only process a finite number of digits anyway. (This last sentence is NOT the definition of "a finitist".)
There is much to be said for the approach of using infinitesimals, but not as in non-standard analysis, which also relies heavily on assumptions about being able to do an infinite number of things. There are however quite elementary approaches to calculus using algebraically defined (nilpotent) infinitesimals. See for example by Famous Math Problems 22 lectures, starting with ruclips.net/video/D8_BBoolMm8/видео.html.
Well explained Prof. Wildberger. If you are upsetting some quarters of mathematics academia then you can be sure you are onto something. I understand what you are saying. It's not much of a stretch to understand it really.
but the process of "creating" also applies to the series 1/2^n or not? it's just that we can specify an exact number 2 as the limit which ist 1.99999 .... infinite 9s. In both cases that limit will never be reached but arbitrarily gotten close to.
@@njwildberger in the sense that you cannot specify an exact value via a formula that contains only rationals? as far as I understand it the Cauchy condition only makes a statement about whether a limit exists or not and does not specify its value. so you say if the limit cannot be specified as an exact / rational value then it does not exist? If so, does the series not converge but if it doesn't it neither diverges? how would you call then the "number" that describes what happens when n goes to infinity? btw, I'm not a mathematician (probably obvious from what I say :D), just reading / studying some maths books and coincidentally reading on-topic :-) I also have my "issues" with real numbers and infinities. Especially I "like" open intervals... 🤪
The difference is that with 1/2^n it's possible to know what those significant digits will be without actually having to compute them. We can store all the information needed to define the number with finite resources. The reciprocal of factorials cubed example isn't like that. We will never know what most of the significant digits are because the universe doesn't contain enough time or space to compute them.
@@andrewcole9824 ok, got it. But does that also mean that the limit does not exist? iow, does a number only exist if we can write it down? I mean, π and e also exist, or do they? If they don't then "ideal" circles and functions whose derivatives are themselves also don't exist?
@flatisland This all comes back to what we mean when we say "exists". How can you claim something exists when it's literally impossible to exist in the physical world, and can only exist in our imaginations? How are any of these questions different than asking if God can lift a rock to heavy too move? None of it means anything. What difference would it mean if the limit "exists" or not? Please tell me a single calculation that would change in either case?
"real" numbers are simply representative of unbounded computations. indeed they're quite lazily defined as they allow for not only non-computable "numbers", but un-nameable ones. gregory chaitin is a closet skeptic of the idea of real numbers too, despite having studied them from philosophy/foundations of mathematics. personally they seem like a lazy way to avoid having to better model more complex algebraic objects by just throwing every possible unbounded stream of digits under the same name
Prof. Wildberger, on your Twitter account and on this RUclips channel, I have given to you the links to my videos on new Trivial High-order arithmetical root-approximating methods that lead the way for the best definition of irrational numbers. Trivial high-order methods that inexplicably shamefully mathematicians have missed since antiquity and consequently do not appear in the mathematics literature. You continue by remaining silent before all that. That is just another example of the reasons one cannot trust in the current educational system.
I always had a problem with pretending zero is a counting number, I could/can only accept it is a placeholder in orders of magnitude. Zero means no number and I think a lot of problemc in arithmetics are solved by acknowledging zero is not a number. Division by zero means nothing 3/0 is similar to 3/happy, does not have any obvious meaning. 3*0 also is different, maybe it means you have 3 nothings? I understand it is convinient to use zero like a number but I believe it can fool us in the long run if we forget 0 is not a number like others. So we bend the rules to include zero as a counting number but why must the noncounting number 0 be a counting number like the counting ones? I think zero is its own class and maybe we could use it differently? Maybe math is the laws of physics and zero does not exist per definition? I would appreciate if someone could show me why zero is carelessly thrown around like it had a counting meaning. Zero and infinity are special cases that causes problems, impossible magnitudes.
Hi, maybe I could help. Counting numbers and whole numbers are separated in such a way, that you actually can represent the counting using any whole numbers in any order you like. "First", "Second", "Third" etc. can thus be represented with 1, 2, 3, ..., or 0, 1, 2, ..., or 1000, 1001, 1002, ..., or -200, -400, -600, ... and so on. As long as you have the relation clear, there will be no problems. Zero could thus, as merely a label, or symbol, represent anything you like, while it as a whole number represents the value of nothing. It comes natural to start counting with zero if what you count is continuous, like time, where you define a certain starting point and can relate to time point before and after that, and start with 1 when you deal with whole positive units, like people.
@@MisterrLi thank you. But my mind is stubborn and counting with zero to me is like counting with infinity. Zero simply means we have nothing to count, that is the opposite of something to count. Likewise infinitly many means you have stopoed counting. I am not turning math upside down just feel we should be more careful when operating with zero. Say we have equation 4*0=x. I dont think it is correct to say that is the same as x=0. I think you should stop at 4*0 and not simplify further to not mix counting and non counting numbers unless we are sure it is ok. But we should stop and think. 4*0 realky means we have four instanses of no counting. If it is my monthly salary that has not been paid four months that means I have not been paid 4 times, that is important. I have a hunch we loose clarity when mixing counting and non counting just by not paying attention. Keep them separated until problem is solved and see if it telks us something more about what is going on on a deeper level. Thats how I feel but I could be totally wrong. But I have never seen any mathematician pay attention to this fundamental issue and that in it self bothers me becuse I dont see this as a yrivial consern that is self evident. I have dabbled in number theory and arithmetics, comouter science but never resolved this zero problem. Fundamentally nothing is not something, they are different ideas
Well, there is no need for zero especially to be used in counting. Zero also have different meanings in different systems. Just like there are different kinds of infinite sets, there are different kinds of zero. For example, in calculus, you often encounter the 0/0 form, which often can be useful. Zero as a word is just a label, a symbol, that gets different meanings depending on the context and the system you use. I can't agree with the proposition that zero is not a number, rather it is a number that stands for nothing in value, in certain systems. In other systems it stands for other things, it can be a value infinitesimally close to absolute nothing for example (as in calculus). Division by zero also has to be related to in which context and systems you are using that operation, in some cases it is perfectly accepted, in other illegal. So, using symbols such as "zero" must be taken into a certain context to have any meaning at all. That context is up to you to set, since there are a plethora of meanings that symbol can have. A symbol is just a symbol if you define it, if not you can't possibly know what it means.
It dawned on me that you probably meant "calculating" rather than "counting" with regard to zero. Language translation error? In that case, the answer is similar. It depends on the context and number system in use. If you don't specify that, you don't know what zero you're dealing with. Zero came into math pretty late, for reasons that you needed to represent nothing in a lot of situations. It obviously made calculations much easier. But it means very different things in different contexts, so you need to be careful not to divide by zero in arithmetic for example, while in other mathematical branches (analysis...) it is standard to be able to express a value where 0/0 is the form (with more details to it of course, such that both zeros have a more complex defining algebraic expression).
Zero is nothing.Something multiplied by zero is nothing.If zero is a number infinitely close to zero then we have to agree with 2×0=4×0,where in 0 cancelled out from both sides leaving behind 2=4,which is absurd.So infinitesimals is ambiguous and vague which should not be entertained in science.It is not objective.And so is the concept of limits which necessarily built up on that concept.
Actually imaginary numbers are not very problematic: this is just an example of a quadratic extension of a field, which can be concretely represented by suitable matrices, and the arithmetic is completely cut and dried. In fact complex numbers over (some) finite fields similarly make sense. The critical issue is that we have mathematical objects which can be specified in their entirety, and that the operations of arithmetic are well-defined and computable (meaning that their evaluation requires finite run time and memory on a computer).
@@njwildberger thanks for the response. That's very well stated and Im glad I asked. Never thought about it like that. However, in the way you have framed it, it's a lot easier to grasp conceptually than how I was previously. Cheers!
If our understanding of real numbers is logically deficient then rather than dismiss real numbers as a myth perhaps we need a new understanding of real numbers.
No we do not need to prop up our myths even though we may be heartily attached to them. Let’s rather take a big breath, accept the reality for what it is, and move on.
don't know why some describe you as a finitist. you seem to be more of a "computablist", which is a very reasonable mathematical position to take these days, probably increasingly so as we continue to see developments in computation as foundation of mathematics.
Of course there are ways to define real numbers rigorously. The only way to make the claim that "real numbers do not exist" is to use a different notion of existence to the rest of the mathematical community. This may legitimate, but I'm sad that Wildberger is not upfront about this. It is not clear to me that Wildberger's notion of existence is philosophically consistent, but that may just be a lack of understanding on my part.
I am so glad to hear that of course there are "ways to define real numbers rigorously". Could you please share with us one of these illustrious paths forward, and demonstrate its use by computing "pi + e + sqrt(2)".
@@njwildberger Here are a couple of paths, that I am sure you are aware of: - Starting from the rationals, you can take the Dedekind-MacNeille completion (which consists of the downwards complete subsets) - Starting from the rationals you can consider equivalence classes of Cauchy sequences - You could let the reals be any complete ordered field (which is a perfectly fine definition, if not constructive) I will not go any further, because you are of course aware of these paths. With some work any any these can give a reasonable and rigorous definition in any formal system that supports classical logic with ZFC set theory, for example. The question of 'computing' (whatever that means for you, I guess a decimal expansion is not enough) is not particularly relevant. This is all well-developed, good mathematics (if very technical) and can definitely be made rigorous. It might not be to your philosophical tastes, however. This is where your definitions may differ from the rest of the mathematical world's. Some notes: - I am making no metaphysical claim about "physical existence" of numbers. Existence of a real number is, for me, a formal statement in a formal system, which is really not a very strong claim. What I must admit I do not fully understand about your position is how it is possible that natural numbers exist in any way that real numbers do not. For example 8^9^10^11^12 can no more be computed than pi + e +sqrt(2), at least to my mind. The difference is that one is too precise for our universe and one is too large to fit in our universe. In both cases there is not enough room, I would have thought, but I am interested in your thoughts. - One could wonder what the use is of such a purely formal system. I can think of at least a couple. On the one hand the use of reals can simplify proofs involving one finite discrete objects (the proof of Sperner's lemma using Brouwer's fixed point theorem comes to mind, but there are many, many examples). The Gödel speedup theorem gives some explanation to this phenomenon. Secondly real numbers have turned out to be very useful to model physics. I do not claim that this means that the reals exist in our universe, just that it is easy to build good models with them. This is, for me, the most compelling reason to study these objects.
none of those definitions are constructive. constructive means implementable / computable ultimately. rationals are algebraically constructible. the reals by their definition either a) require an infinite computation to _define_ them, or b) algebraically admit non constructible instances.
@@elcapitan6126 You must then be using a non-standard definition of "constructive". Typically this refers to several approaches to mathematics (e.g. following Brouwer, Bishop or Martin-Löf). Normally you can define reals using either some version of Cauchy sequences or some version of Dedekind cuts, but these definitions may not meaningfully coincide. So yes, it is reasonable to call the definitions constructive. What is not clear to me is whether your notion of "constructive" can be made rigorous. In what way would you construct 9^10^11^12^13? There are not enough particles in the universe. Is this not the same difficulty as constructing e+pi+sqrt(2)?
@@josephcunningham5882 Given there are no ethereal realms of numbers then the only way to make sense of numbers is through computational approaches. There is no number, as in a decimal expansion, until you write it down or make it explicitly clear. Some sequences are physically possible to write down while others aren't. Those that aren't don't express anything meaningful as they literally can't give a finished product. You can't assume they are meaningful as if there is some actually infinite decimal expansion of it out there floating in the aether and our mind can only grasp it. There are no numbers outside the Human mind and it's capabilities. To think otherwise is to reify abstractions and speak in vagueness rather than preciseness.
You know, I've always found interesting how our 'critics' call us finitists, when in reality it seems we are the ones who actually believe in infinity. We respect the concept by accepting the fact that we can't do an infinite amount of steps, that the sequence of natural numbers is vas and complex and could never hope to reach to the end of it. Meanwhile ZFC practicioners, think that by formalist trickery, they can somehow "tame" infinity, resulting in complete fiction like infinite cardinals and ridiculous results that I'm sure most of us are familiar, which in an ironic twist of fate collapses infinity and math into a mere symbolic manipulation of finite strings. Please never stop releasing videos Professor Wildberger, you are doing wonders in the mathematical community especially with the young generation. True revolutions are subtle and take their time to happen, and slowly but surely they do happen.
Thanks for the great comment. I believe you are making a valid and important point.
@@njwildberger it could be the theory limits needs to be rethought.
we analyse finite quantities between zero and infinity.There are some limits for the series as you said in for e.g 1+1÷2+1×4+…..There may are likewise limits for pi,e,and 1/3.Here we approximate.But how is it justifiable to apply limits to derivatives?.It is ,I am sure,misplaced application of limits.The limit of f'(x) of f(x) is not logical.Consider f'(x) of f(x)=x².It is x²+2xh+h²-x²/h=2x+h.How can it be possible to reject h,for h can take any value in reality?.The difference between x² and (x+1)² is always 2x+1.If you take larger values for h,then the difference is 2x+larger values,and if h is less than 1 (but not zero) we have 2x+h (h or < make no precise sense.
@@irappapatil8621 I agree there is an issue with how limits are used to find the slope of the tangent line . A leap of faith is made when the limit is claimed to correspond to the slope of the tangent line.
Thank you so much for making this excellent lecture - for two reasons: First, it so very clearly explains the fundamental difference in quality of the type 2 and type 3 series you show - this was exactly the clarification I was seeking as a result of some of the earlier lectures on this topic. Secondly, this lecture identifies the type 3 series, and the related Cauchy Convergence Criterion applications as leading to some highly problematic issues, including some very problematic non-true facts, and this is very interesting to me for the following reason: As an undergraduate student I studied mathematics and physics. In the first year I attended a mixture of pure maths, applied maths and physics courses. In one of the courses aimed at pure maths I encountered the Cauchy Criterion, epsilon delta arguments etc and I found it very difficult to get comfortable that i really understood it. I concluded that at least one of two things had occurred: That I had somehow missed out on some elementary parts of mathematics education at school, back where advanced topics were introduced but in terms that "the real truth will be
covered at university", and as a result of missing the relevant topics I now was befuddled by this analysis topic having got to
university, at which there seemed to be an air that certain of things were "obvious" from school maths, and second that I lacked the mental equipment to truly appreciate and excel at these pure maths topics. As a result I specialised into Applied Mathematics and Theoretical Physics, which I enjoyed very much, and then after university I moved into computing, which I also enjoy very much. And now, thanks to your extensive lecture series on here aimed at putting various areas of pure maths on a rational footing , my interest in pure maths has also been rekindled!
What’s wrong with the Cauchy criterion? I think it’s pretty intuitive.
So, the properties of the rule that generates the series are mistakenly forced into the category of numbers or number-like mental objects. In fact, a certain algorithm-analysis (based on arithmetics) is being dressed up as "real" analysis. Thank you, professor Wildberger. I get your point, I am woken up. I hope that some of your colleagues will join you soon in reconfiguring mathematics. You have a lot of work to do.
This is the most important understanding in pure and informational mathematics at least in the last 100 hundred, and maybe in the last 200 years.
Thank you for another superb clarification of issues too many other mathematicians like to keep obscure!
As far as I can see, many people, including many mathematicians, build vague 'notions' on other vague 'notions', whereby the 'existence' of the 'irrational' 'limit' is a special case of the more general lack of understanding what 'existence' is supposed to mean.
In the more general case, quite possibly there are some actual sculptors who really believe that a particular future sculpture already exists inside a big rock, thereby foolishly denigrating their own work of bringing it into existence.
Pure mathematicians, to their perceived advantage and the disadvantage of those seeking thorough understanding, have found a way of avoiding the hard work the honest sculpor does. They claim and usually really think, that if they have the equivalent of the rock they already have the equivalent of the sculpture and thus don't need to take the trouble of creating it. And if the actual creation of the sculpture should turn out to be impossible, they brush that aside.
'Building' further on that, soon they no longer need the actual rock, either, at least when it becomes too troublesome that for one mathematician's actual rock there will be a rival mathematician's plan for a sculpture that is too big to 'exist' inside the rock. If I 'have' an 'infinitely' large rock, Hilbert's hotel already 'exists' inside it, and if it is unfurnished an 'infinity' of furniture will fix that problem, too.
As far as I remember Cauchy originally only stated that converging sequences are Cauchy sequences. At that time there were no formal definitions of real numbers, so it was neither possible formulate nor prove that Cauchy sequences converge to rational numbers.
On a serious note...The more I listen to you about real numbers, the more it reminds me of Wolfram's "Computational Irreducibility", with the added proviso of requiring infinite computing power. Computational Irreducibility can occur in finite steps. I don't know if that is pertinent or not. Are finite and non-finite computationally irreducible systems equivalent? This also reminds me of your analysis of whole numbers and their fractal nature. There is something going on here with numbers that has not been clearly specified yet; some big insight to take things to a whole new understanding. I would guess that computational irreducibility will be an essential ingredient in whatever this next notion of numbers turns out to be. Thanks for all you do!
Thanks, Prof. W., for this interesting video. I remember when I first saw the claim made in a class that real numbers could be defined as equivalence classes of Cauchy sequences, followed of course by a hand wave over the issue of proving that those things comprise a field. As I understand it, that briefly is your objection to the real numbers - even if you accepted that those equivalence classes exist, would they provably form a field? I wonder if it would work to allow that yes, a Cauchy sequence of rationals has a limit, but almost always that limit is just a geometric point but not a number. Then pi would be something, a geometric object, but for engineering purposes we have to use partial sums of a series whose nonnumeric “sum” is pi.
The way the concept of 'real numbers' was introduced to me is quite similar to what is going on in this video. The remark that not every (Cauchy) sequence of rationals converges to a rational number is a very important observation. This means that if our concept of 'number' should include things like the length of a diagonal in a square and the circumference of the circle, then we have to consider some other number system. There are multiple ways of doing this, but the one that naturally follows from what was said in the video is to consider sequences of rational numbers themselves. You then notice that for any given rational, there are many sequences with that rational as a limit, so we better consider all those as equivalent to each other. We now define two sequences (converging to rationals or not) as equivalent if their difference converges to 0. We then define a 'real number' as an equivalence class of Cauchy sequences. We then see that any rational number is a real number, namely the equivalence class of the constant sequence. Next we can define addition and multiplication componentwise (we have to prove that the the sum of two Cauchy is Cauchy, and if equivalent sequences are added/multiplied we obtain equivalent sequences). Then we have to prove that it satisfies the standard calculation rules (associativity commutativity, distributivity, etc). Once we did all this we are justified to say that the 'real numbers' are a number system. We can then even consider sequences of real numbers (so sequences of equivalence classes of Cauchy sequences) and in that Context it makes sense to say that for every Cauchy sequence there is a real number, which is its limit.
When analysts talk about limits, this is what they mean, no? I have never met an analyst saying, "no, real numbers are just decimals that go on and on"
That’s all fine as a philosophical system. Or perhaps as a religious system. But what do you do when you actually want to make computations? For example, how does one compute pi +e+ sqrt(2)?
If you have an arithmetical system in which you can’t actually perform the operations then you’re kidding yourself.
@@njwildberger You haven't rebutted what @MrLordCoder wrote.
Also, in what sense do you mean by "can't actually perform the operations?" Although I could in principle, I couldn't actually calculate 12^100 without a computer. On the other hand, in practice we can compute millions or billions of digits of pi + e + sqrt(2) and somebody like Mr Lord Coder has a grasp of the underlying definitions of addition of real numbers.
@@rupertmillard you say he didn't rebutted him but don't even get what Norman says? figures.
@@rupertmillard In the strictest sense imaginable: The order and equality relation among computable real numbers is Turing undecidable. This has been proven long ago. The same goes for Dedekind cuts. You cannot maintain all the properties of the reals without being impredicative.
>the one that follows naturally from what was said in the video is to consider sequences of rational numbers themselves
no, that is not the natural one, and you have no reason to think they are besides the fact that they're the only choice you ever consider
"Definable" sequences of rationals are inherently more natural than the ones you wish to include
furthermore, the only rational sequences that can have a limit taken on them, without introducing meta-theoretical issues, are the computable rational sequences
Real numbers are often defined as (equivalence classes of) Cauchy sequences. With that definitition it is very easy to prove at the sum of 1/(n!)^3 is a real number.
Unfortunately though, that definition of real numbers as equivalence classes of Cauchy sequences is nonsense. See my Math Foundations lectures starting around ruclips.net/video/kcirwIwRIUw/видео.html
@@njwildberger It may still be useful that we are able to define the notion of an Cauchy sequence and it may also be useful that we can state that two Cauchy sequences are equivalent. The notion of a Cauchy sequence captures a mathematical property that can be studied.
@@njwildberger Now I saw your old video again, and you point to two main problems.
The first problem is what one means by an infinite sequence. For me it is fine if we restrict our attention to recursive sequences (or computable sequences as you talk about in the video).
The second problem is about equivalent classes. The problem with the equivalence classes already appear in the standard definition of rational numbers where a rational is defined as equivalence classes of pairs of integers. The equivalence is defined by (m,n)~(x,y) when m*y=x*n . The point in this "construction" of the rational numbers seems to be that rational numbers can be identified with sets and is a result of the project of identifying any mathematical object with a set. The problem is obviously that someting simple like the rational number 1/2 has to be identified with an infinite set!
For me a rational number is not a set but is a pair of numbers separated by a bar, i.e. a fraction. For me the fraction 9/6 is not the same fraction as 3/2. When we write 9/6=3/2 it does not mean that we have the same fraction on both sides. It means that 9/6 and 3/2 has the same value, and what does that mean? it means that 9*2=3*6. Again, 9*2=3*6 does not mean that we have the same on both sides, but it means that 9*2 and 3*6 has the same value, and to verify this, some kind of calculation is needed.
We should distinguish between sense and denotation. In my oppinion the real problem is the Platonic view mathematics, where the focus is entirely on denotation and where sense is reduced to some sort of subjective illusion.
@@PeterHarremoes- With your strange view of rational numbers, can you perform computations with them? In your view, what is, e.g., the result of adding 1 and 1/2? Is it 3/2? Why not 9/6? If both fractions are different, as you say, are there any situations where one can use one of them, but not the other?
@@PeterHarremoes You raise important issues. The best we got so far for recursive sequences (in the sense of repeating structures) are the quadratic periods of continued fractions. Cauchy sequencies have intuitive connection with continued fractions, especially their form of path informations along the binary tree of blanks nested in Stern-Brocot type structure of totally ordered rationals. Gosper arithmetic of continued fractions / continued logarithms is at least better defined effective method than the basically undefined decimal arithmetic.
Our professor is first to admit that also the coordinate system neusis dependent definition of "point" as a pair or more "numbers" is also very problematic. The issue of continuum and continuous directed movement is why Greeks rejected neusis from pure geometry into the lower caste of mere "applied math", in which the rigorous criteria of mathematical truth can be relaxed for some pragmatic teleological purposes.
The notion of "equivalence classes" is on the other hand very counterintuitive, and on the other hand very rigid and cumbersome. It's been abstracted away from intuitive clarity into what we can fairly consider the wrong direction. In the heuristic aspect of math we need to study also the wrong directions deeply enough to realize how and why they are wrong, we are supposed to do also mistakes and learn from them. What is intuitively strong and enduring keeps on reappearing. Greeks intuitively understood equivalence as modal negation of relational operators: When A is neither more nor less than B, then A =B (see method of exhaustion and the intuition behind many of Euclid's proofs). IMHO the greatest insight of Don Knuth's inspired book "Surreal Numbers" was the consciouss derivation of equivalence relation from relational operators, which AFAIK Greeks never formally define but just implied intuitively. It's been considered so intuitively self-evident, that it need not be formally defined. Of course the derivation of equivalence relation from relational operators is contextual, and applies coherently only to coherently comparable properties. It's much weaker than "Law of Identity", and that's a good thing.
There are very good reasons why Whitehead chose the point-free path after the failure of PM. I see no good reasons to abandon Euclid's definition of point as the halting of the top-down process of mereological decomposing (Point is that which has no part). Because the post-modern set theory is incoherent with mereology (as already Gödel observed), many academically trained contemporary mathematicians have lost basic mereological intuitions, including standard vocabulary. Wildbergers foundational inquiry is deep down very mereological, as we can see e.g. from the basic inclusion relation of the Box Arithmetic. It's a big task, but as we have made also genuine progress since Euclid, we are now ready to research mereological foundations with fresh tools.
In this regard, I see no other coherent alternative than to accept that continuous directed movement is holistically irreducible empirical foundation of mathematics. From this acceptance it's not very complicated to construct and accept that fractions/rationals are not "points" of coordinate system neusis or "number line", but continuous intervals, mereological partitions of the ontologically irreducible continuum. As we keep on intuitively teaching the kids... only to later betray their trust with absurd point-reductionism.
I don't think i have completely understood your objections to the notion of real numbers or infinity but i would like to add my two cents in the converstation for what is worth.
I don't see what exactly is the issue with having infinity in mathematics.
The fact that computations with reals are never truly achievable via the use of computers, to me, is just a limitation of computers, not evidence that our foundations are flawed.
Furthermore, as a mathematician, I find the axiom of infinity and the axiom of choice very much intuitive. They are just axioms, though, meaning that you can either accept them or not.
I myself dont see any reason why i should deny them , but i would love to see an argument in favor of that position!
Thank you, and i hope my comment wasn't too incomprehensible!
"...in to another universe where there's an infinite amount of room" LOL!
Thanks, very illuminating. A further question. In the third case we can we can represent the decimal string as a continued fraction reading of a list of rational convergents. Can we thus define distinction between 2nd and 3rd case as continued fraction with a single final convergent vs. open ended reading of many convergents? Are the other complexities involved that make this definition incomplete?
Ok, thanks for a good analysis on limits. I think lots of things could be added though. The reason for real numbers in the first place must be to be able to successfully deal with continuum problems. You simply add "all" irrational points on the line and call them real. Rational numbers have the advantage of being point specific, as well as whole numbers, but for completeness reasons, that's not true of the reals. There is also the issue of mixing irrational numbers (irrational reals) with rational numbers or rational real numbers. This is not trivial, since the results of calculations will differ depending on which model you pick. Take the rational number 1/3 for example; is it really exactly equal in every way to 0.333...? Certainly not if you define 0.333... to be a real instead of a rational number. A real number has a sort of "fuzziness" to it, where values (or more specifically, points on a geomertical line) that are infinitesimally close count as the same real number, something that doesn't hold for rationals, since real and rational number objects have very different definitions. So, who's to blame for the confusion about these concepts?
Thanks for the comment, but to someone who has not seen a proper definition of the term "real number" and "irrational number", I am afraid that I can make little sense of what you are talking about.
@@njwildberger Well, we could make the geometrical definition of irrational numbers as corresponding to those points on the line that are not rational. That is not enough though to make them usable in calculations; we need more rules, so that every point in the infinitesimal interval between the two groups of bigger and smaller rational number points, for example as examplified as a Dedekind Cut, are defined as the same real number object. As long as we don't use an infinite number of additions (or other similar finite operations), the math will essentialy be the same as using rational number additions on the number line. We have to add the rule that infinite decimals that look different represent the same real number, but this is clear if we can identify the interval from both sides, for example pi: (3, 4) , (3.1, 3.2), (3.14, 3.15), (3.141, 3.142) and so on, going on to an infinitesimal interval where we can stop, and 1/3: (0.3, 0.4), (0.33, 0.34), (0.333, 0.334), and so on, where it is clear that the interval length will be smaller than finite but bigger than zero. We can take the interval length to be zero when calculating, because it will not change to a finite size if we keep doing a finite number of operations with these real number objects. The use of having access to all points on the line is of course very practical, and when we really need to perform an infinity of operations on those irrational objects, we need extra rules to make the results well behaved, which has led to calculus, where having a zero not always means to having a rational zero, since when you go to infinity adding it to itself, it can give you any finite real number depending on how the infinity works in that calculus formula.
There are a number of different definitions of real numbers, picking one will represent them in a way that is more convenient for certain calculations and uses. We could for example instead use binary infinite decimals and they would work equally good as representations of reals as the base ten infinite decimals, we could switch between any base we like (2, 3, 4, ...) and still retain the real value representation relative to the geometrical line points representation. This still works converting to and from infinite decimals with irrational bases.
When using a number system that allows for infinitesimal representation the problem with having reals representing an infinitesimal interval becomes obvious and you get a one-to-many relation between the real number and the infintiesimally enriched number system. More rules are invented in the case of translation between reals and such systems, such as between reals and hyperreals, where many options are open going to hyperreals, since choices have to be made between one real value and infinitelly many possible hyperreal values. Going to reals from hyperreals is easier, since we can just set the infinitesimal part to zero, at least if the hyperreal infinite part is equal to zero.
Rather than the issue being the limit, i see the issue as arising from the existential quantifier in the cauchy condition
instead of saying "for any M, there is some k"
if you say "there a computable function (in other words, a function, non-computables dont make sense) that takes M to k" which satisfies the normal conditions
Then you dont get the real numbers, you get the very well behaved computables.
This eliminates many issues you have the process of taking the limit of reals, since using that function, you now know ahead of time exactly how far down the sequence you need to look, in order to see what it equals
And i take issue with your criticism at the end, that we cant define numbers to be a "process", they must somehow be something more fundamental than that.
This just isn't true, even with your beloved rationals, the process which makes them is obvious every time you write them down.
A rational is the process of dividing one integer by another, you cant actually break one thing into 2 but we say that we can.
Likewise the naturals are the process of taking the successor.
so true
I've implemented code that uses exact rational arithmetic. The numerators and denominators quickly grow to enormous sizes, making the calculations impractical.
Pure and applied mathematics are not the same.
If the argument is that it is wrong to define the number as being a result of a process wouldn't same reasoning also apply to simple rational numbers like 1/3.
1/3 also has an infinite decimal represntation but since we can define it as a result of a division operation it is a rational and thus fine. But this is only because we allowed ourselves the operation of division.
Why is defining a number through a more convoluted process using sums and factorials really that different to this?
1/3 is in the same way incomputable and only exists because we allow purselves the "cheat" of keeping track of both nominator of denominator. But you can never actually write it down without that cheat.
To common sense it appears that a quantity is less than,close to,bigger than is imprecise.It must be said 'how many times'.Limits cannot satisfy precision which is a must in logic.WHEN WE SAY DELTA DIFFERENCE GIVES EPSILON DIFFERENCE MAKES ONE TO PUZZLE AROUND WHAT ARE THE MAGNITUDES OF DELTA AND EPSILON .
I don't really think that not being able to fully describe a number is a problem. There are just more numbers than ways to describe or compute them, but we can still find properties of numbers, even ones we can't define. The problem doesn't get much better when you move onto things like functions.
it is odd that we cannot even systematically *name* such general real numbers since their mere representation is without limit. this is very unusual compared to say rationals which, while may have unbounded decimal representation, have a systematic and _finite_ way of notating them (as fractions). real numbers really are a different sort of beast of an idea, and one that I think warrants more skepticism than it attracts.
@@elcapitan6126Well there are a countable number of rationals and an uncountable number of reals, so it makes sense.
How about the problem of fully DEFINING a number. That surely is a problem. That's exactly the situation we are in: despite all the mumbo jumbo re equivalence classes of Cauchy sequences and Dedekind cuts and infinite decimals etc, there is no proper DEFINITION of what a real number is, and what the operations on real numbers are.
@njwildberger an analogous situation in computer science terms would be to define a type for which there is no general implementation, only for specific subsets of that type. where the type would be "real number" but all theoretically possible implementations (instances) would only cover an almost infinitely small number of "implementations" that the type purports to allow. I.e. we would view such a type as too unconstrained, somewhat absurdly so, despite covering the valid (implementable) instances.
practically what we would do is instead define types that admit implementations, such as an algebraic type for rationals (perhaps as fractions composed of two integers) or more fancy types for algebraic objects that fulfil the role of roots, etc. all the polynomial objects can be defined algebraically without the non-computable notion of real numbers.
I understand the argument here. But what if we just say that it converges to "some number" but we will never truly know what that number is? We can only specify it up to a certain precision. At least I'm comfortable with that level of doubt.
It comes down then to our definition of “number” in the first place. If that is a precise definition, then it should not be allowing of doubt and imprecision. This is mathematics after all.
So if I understand correctly, if a series "converges" to pi, it is a "type 3" series? since pi only exists as an iterative computational process... We can call our 2.1297 number "zi" and say our series converges to zi.
That's right
You've touched on something Ive been pondering about for a few years, ever since I finished my bachelors of philosophy. I now study at UNSW a double major of physics and math. Just from my very nature, even from a young age, Ive never felt comfortable with the idea of infinity. For years Ive pondered on what in essence a number is. If the sqrt(2) is a number, I eventually concluded that sqrt(2) was a process (because it could never reach a defined end). Hence, I concluded the essence of a number is not as an end or final outcome but a process. Hence, we now have to go back to all the natural numbers and think of them as processes rather than ends. A number is no longer a computed result but the computational process itself. I found this conclusion absurd. Hence, Ive always privately rejected the idea of infinite real number decimals as anything which can exist or be real.
Moreover, a natural number is clearly not a process but a defined end. So then its not in the essence of number that it is a process, and therefore because the natural numbers are logically and conceptually fundamental to any other type of number, if a natural number is not a process, then no process can be a number. Therefore, the sqrt(2) cannot be a number.
There's so much more I want to say. I hope one day I can have a nice chat with you and share ideas on these topics.
Don't you think there is a difference between the process we use to find the number out and that which we are finding out?
@Continential Yeah, I would say so. One may say it's because both the process and number are distinct, that we cannot have both at the same time. However, our use of the word process is ambiguous. In one sense, a process is an epistemological procedure, that is, the methodological means by which we go from ignorance of a a number thats a solution, to knowledge of the solution, namely, the number. Another sense of process is as an ontological property, the currently accepted idea that a number can itself be entirely and ONLY a process. Now, for sqrt(2), in this sense of process, you'll have to accept the idea that an incompletable process is an object which is the under the same category (category of objects we consider numbers) of objects which are not processes (but which can be the result of an epistemological process), which is say a natural number. We may say that the result of the process of adding 1 to 1 is 2, isnt saying that 2 is an ontological process, that is, the essential definition of 2 is the process of computing 1+1, why?, because 2 can be the result of different computations of different numbers. Rather, this process of computation is an epistemological process of identifying that the solution to 1+1 is the result 2, or of learning that 2 can be identified with the outcome of the process of computing 1+1. Epistemological processes don't define a number but identify a number with a process, in as much as the process has a defined outcome. The epistemological process can tell us about the necessary properties of a number, namely, that the property of 1 additional to another 1 is identical to 2.
However, an ontological process as I have defined it earlier doesn't simply identify the number with a process but defines the number as the process. However, this view disregards the essential distinction between process and outcome, that an object can be exactly both at the same time: this results in the contradiction that an endless process can be the same as a process that comes to an end.
However, one can respond by saying when we state the sqrt(2) is a number, we are saying that the endless process which is the sqrt(2) is just essentially the DESCRIPTION that the sqrt(2) is every part of a process which, although endless, exist simulateously. So every part of the infinite process which is the sqrt(2) exists simultaneously and therefore any reference to the endless process not having an outcome is simply epistemological: we cannot know all that is in the sqrt(2) (all the infinite decimals) simultaneously (as we cannot know an infinite amount of things at the same instant) despite all its parts existing simultaneously. But, who's to say that epistemological limits determine the limits of what a number can be? Moreover, all parts of an endless process existing simultaneously does have the property of a process that ends: neither remains incomplete.
However, here lies the conundrum. In an epistemological process we come to know what something is. If we admit the existence of an ontological process for numbers as I have specifically defined it, and that all parts of this process exist simultaneously, an essential property of a number remains still that it is a complete end. However, by definition a number counts things. Since, no thing has an infinite amount of parts, the sqrt(2), having an infinite amount of numerical parts, can never count anything. Therefore, sqrt(2) cannot be considered a number. It cannot exist as a number.
But if it's not a number, then what is it? What are we referring to by the term sqrt(2)? Well, it's an amalgamation of different concepts that's given a single name: sqrt(2). This amalgamation consists of, for instance, the concepts of number, process, infinite, parts, convergence, existence etc, arranged in a certain way as to convey a single idea we term sqrt(2). In the same way, for instance, golden Mountain that speaks English is just an amalgamation of different concepts: gold, mountain, English, speech etc.
However, I think to even say something exists in some sense entails that all the parts of the object exist. However, something with an infinite amount of parts cannot exist since there will always be a part which isn't accounted for which must exist for the object to exist. Therefore, all parts of an object with an infinite amount of parts cannot exist simultaneously since for whatever portion of the object that exists there will always be a subsequent part of it which hasn't been accounted for as existing. Therefore all parts of an object with an infinite amount of parts cannot exist since not all parts of the object can be accounted for as existing: it can never completely exist.
Hence, I leave you with this final thought. The concept of the sqrt(2) is an idealisation which has use in an epistemological process in mathematics. It ensures we account for essential properties of a solution, depending on the mathematical environment we are working with. For instance, if a solution to the root of a function is the sqrt(2), then it tells me in an applied setting that the finite solution to problem in the applied setting is found in the process of computing the sqrt(2), that is, that the exact finite solution will be a finite number which is derived from the process of computing the sqrt(2). However, in a pure math setting, to say the exact solution is the sqrt(2) is nonsensical since the sqrt(2) cannot exist because all parts of it cannot be accounted for as existing. For any part of it which exists, there will always be a part of it which hasn't been accounted for as existing, therefore it's existence can never be complete: all its part cannot be accounted for.
Of the scenarios you've provided, whereas the first two posites "Does the series converge, and if so, onto what real numbered value ?", I view the third example series as an approximation problem, up to an arbitrary precision, unrelated to the two.
However the third example has monstrous ramifications in the area of artificial neural networks verification, which largely base their operational characteristics on mathematical models!
The third example series you've provided is an excellent example of the inherent problem involved in verifying most artificial neural network models, and also why most of the verification methods of these systems embedded in our automotive, healthcare, food industry, and military systems, even up to a 99.99% reliability, is wildly insufficient.
Love your videos! It makes my day
The wonders of circular reasoning. Academic "philosophers" are paid to spot logical fallacies in the reasoning of other academics in a disparate array of academic disciplines, but they never do. I prefer to call irrational numbers "notional" numbers rather than give them the elevated status of being within a class of numbers described as "real". Complete misuse of language and generations of professional mathematicians have been fooled in to thinking that algebraic irrational "notional" numbers and transcendental "notional" numbers exist in the same sense that rational numbers exist. They exist in an analogous sense to the "largest natural number" that can be imagined. There is always a larger number than the largest you can ever imagine because you can always add 1 to it. There are always more digits of a notional number to be found, just determine one more.
I think "readings" is a good term for algorithmic "irrational numbers", and then we can reserve the word "number" for products and names of tally operations.
Sort of easier in binary. 1.1, 1.11, 1.111, 1.1111, 1.11111, 1.111111,... but like .999... is the same as the number 1, that's a problem with the notation. There are multiple ways to represent the same number. Isn't it sort of a good thing? In theory any three element set _is_ the number 3.
The set {elephant, cloud, computing} has three elements. It's very hard to avoid a deep feeling of absurdity if we claim that the set in question is the number 3.
Hi Norman, thanks for this video! I was wondering what downstream problems arise or are being ignored by overlooking this problem of defining irrational numbers like pi, e root 2 etc using a rational base 10 or 2 number system? I remember you mentioned the problem of 0.9999 repeater resulting in a 1 or not which can't be known until it's repeated to infinity. What other computer science problems do you see occurring when these irrational numbers are defined as programs and are combined by addition or multiplication or other operations? For example, i imagine that the halting problem is a concern? Thanks
The reality is that it is impossible to properly define “irrational numbers” at this stage of the development of mathematics. Any attempt to define them in terms of suitable algorithms, and to define the operations with them in terms of suitable operations on those algorithms is currently doomed due to intractable problems at the level of computer programs, and recognising when they are equivalent. Now it may be that in the future, some bright people will come up with a more restrictive of irrationalities, using a restricted kind of algorithm. But this is not the situation we are currently in.
For a so-called real number to be called a 'computable number', all that is required is that we can write an algorithm that would determine its n-th digit.
My main reservation about terms like "computable number" and "computable function" is that they don't match the common usage of "computable", as such, they appear to have slippery definitions. The common understanding of "computable", as I see it, is "capable of being computed". This implies a computation process leading to a definite and precise answer.
The mathematical definitions are crafted with the clear intention to categorise real numbers like √2 as being 'computable'. But to say that √2 is computable suggests that we can compute √2 to infinite precision, which is absurd. Based on the common understanding of "computable", it's evident that √2 isn't computable.
Much of mathematics, particularly concerning real numbers, seems rife with misleading expressions. I believe that if we employ a term like "computable", it should not be defined in a manner that suggests the computation of infinite values is feasible. It is frustrating to witness mathematics perpetuate such practices, which only serve to reinforce the mistaken belief that infinite processes can be completed.
There is a more in-depth discussion about these matters at mathforums dot com slash computer-science. Look for my comments in the thread called:
About why I believe that the "What Computers Can't Do" argument (i.e. the halting problem proof as applied to real world computers) is not valid.
As always in mathematics, names can be chosen arbitrarily and are given their meanig by the corresponding definition only. You could call the computable numbers "green numbers" as well. They're not greener than they are computable in any sense of that word in which you use it in other contexts. Similarly, the name "natural numbers" is likely to give rise to expectations that can't be satisfied. They are not growing somewhere out there on a tree. (Which would be a reason to call them "green".)
Both those terms don't seem too bad a choice for the property they are to describe, imo.
As to the term of "real number", it seems to generate such large expectations on their "reality" that a different name might indeed have been better, like the "continuous" or the "linear" numbers.
Now which further information would you like to have for a number that can be computed, apart from all its digits? You might say that it's not "all its digits" but "every single digit" at most, but where is the difference? A finitist's computer can only process a finite number of digits anyway. (This last sentence is NOT the definition of "a finitist".)
Your phrase is not a natural number. It’s just a phrase that sounds like it describes a natural number
What do you think of using infinitesimals as in nonstandard analysis to define calculus, instead of the epsilon-delta limit approach?
There is much to be said for the approach of using infinitesimals, but not as in non-standard analysis, which also relies heavily on assumptions about being able to do an infinite number of things. There are however quite elementary approaches to calculus using algebraically defined (nilpotent) infinitesimals. See for example by Famous Math Problems 22 lectures, starting with ruclips.net/video/D8_BBoolMm8/видео.html.
Well explained Prof. Wildberger. If you are upsetting some quarters of mathematics academia then you can be sure you are onto something. I understand what you are saying. It's not much of a stretch to understand it really.
Have you heard about Terrance Howard and do you agree with his geometry?
terrence howard thinks 1x1=2.
but the process of "creating" also applies to the series 1/2^n or not? it's just that we can specify an exact number 2 as the limit which ist 1.99999 .... infinite 9s. In both cases that limit will never be reached but arbitrarily gotten close to.
In the third case, “that limit” is a phrase with no meaning
@@njwildberger in the sense that you cannot specify an exact value via a formula that contains only rationals? as far as I understand it the Cauchy condition only makes a statement about whether a limit exists or not and does not specify its value. so you say if the limit cannot be specified as an exact / rational value then it does not exist? If so, does the series not converge but if it doesn't it neither diverges? how would you call then the "number" that describes what happens when n goes to infinity?
btw, I'm not a mathematician (probably obvious from what I say :D), just reading / studying some maths books and coincidentally reading on-topic :-) I also have my "issues" with real numbers and infinities. Especially I "like" open intervals...
🤪
The difference is that with 1/2^n it's possible to know what those significant digits will be without actually having to compute them. We can store all the information needed to define the number with finite resources. The reciprocal of factorials cubed example isn't like that. We will never know what most of the significant digits are because the universe doesn't contain enough time or space to compute them.
@@andrewcole9824 ok, got it. But does that also mean that the limit does not exist? iow, does a number only exist if we can write it down? I mean, π and e also exist, or do they? If they don't then "ideal" circles and functions whose derivatives are themselves also don't exist?
@flatisland This all comes back to what we mean when we say "exists". How can you claim something exists when it's literally impossible to exist in the physical world, and can only exist in our imaginations? How are any of these questions different than asking if God can lift a rock to heavy too move? None of it means anything. What difference would it mean if the limit "exists" or not? Please tell me a single calculation that would change in either case?
"real" numbers are simply representative of unbounded computations. indeed they're quite lazily defined as they allow for not only non-computable "numbers", but un-nameable ones. gregory chaitin is a closet skeptic of the idea of real numbers too, despite having studied them from philosophy/foundations of mathematics. personally they seem like a lazy way to avoid having to better model more complex algebraic objects by just throwing every possible unbounded stream of digits under the same name
Prof. Wildberger, on your Twitter account and on this RUclips channel, I have given to you the links to my videos on new Trivial High-order arithmetical root-approximating methods that lead the way for the best definition of irrational numbers. Trivial high-order methods that inexplicably shamefully mathematicians have missed since antiquity and consequently do not appear in the mathematics literature. You continue by remaining silent before all that. That is just another example of the reasons one cannot trust in the current educational system.
What is the site address?
I'm interested
I always had a problem with pretending zero is a counting number, I could/can only accept it is a placeholder in orders of magnitude. Zero means no number and I think a lot of problemc in arithmetics are solved by acknowledging zero is not a number. Division by zero means nothing 3/0 is similar to 3/happy, does not have any obvious meaning. 3*0 also is different, maybe it means you have 3 nothings?
I understand it is convinient to use zero like a number but I believe it can fool us in the long run if we forget 0 is not a number like others. So we bend the rules to include zero as a counting number but why must the noncounting number 0 be a counting number like the counting ones? I think zero is its own class and maybe we could use it differently? Maybe math is the laws of physics and zero does not exist per definition? I would appreciate if someone could show me why zero is carelessly thrown around like it had a counting meaning. Zero and infinity are special cases that causes problems, impossible magnitudes.
Hi, maybe I could help. Counting numbers and whole numbers are separated in such a way, that you actually can represent the counting using any whole numbers in any order you like. "First", "Second", "Third" etc. can thus be represented with 1, 2, 3, ..., or 0, 1, 2, ..., or 1000, 1001, 1002, ..., or -200, -400, -600, ... and so on. As long as you have the relation clear, there will be no problems. Zero could thus, as merely a label, or symbol, represent anything you like, while it as a whole number represents the value of nothing. It comes natural to start counting with zero if what you count is continuous, like time, where you define a certain starting point and can relate to time point before and after that, and start with 1 when you deal with whole positive units, like people.
@@MisterrLi thank you. But my mind is stubborn and counting with zero to me is like counting with infinity. Zero simply means we have nothing to count, that is the opposite of something to count. Likewise infinitly many means you have stopoed counting. I am not turning math upside down just feel we should be more careful when operating with zero. Say we have equation 4*0=x. I dont think it is correct to say that is the same as x=0. I think you should stop at 4*0 and not simplify further to not mix counting and non counting numbers unless we are sure it is ok. But we should stop and think. 4*0 realky means we have four instanses of no counting. If it is my monthly salary that has not been paid four months that means I have not been paid 4 times, that is important. I have a hunch we loose clarity when mixing counting and non counting just by not paying attention. Keep them separated until problem is solved and see if it telks us something more about what is going on on a deeper level. Thats how I feel but I could be totally wrong. But I have never seen any mathematician pay attention to this fundamental issue and that in it self bothers me becuse I dont see this as a yrivial consern that is self evident. I have dabbled in number theory and arithmetics, comouter science but never resolved this zero problem. Fundamentally nothing is not something, they are different ideas
Well, there is no need for zero especially to be used in counting. Zero also have different meanings in different systems. Just like there are different kinds of infinite sets, there are different kinds of zero. For example, in calculus, you often encounter the 0/0 form, which often can be useful. Zero as a word is just a label, a symbol, that gets different meanings depending on the context and the system you use. I can't agree with the proposition that zero is not a number, rather it is a number that stands for nothing in value, in certain systems. In other systems it stands for other things, it can be a value infinitesimally close to absolute nothing for example (as in calculus). Division by zero also has to be related to in which context and systems you are using that operation, in some cases it is perfectly accepted, in other illegal. So, using symbols such as "zero" must be taken into a certain context to have any meaning at all. That context is up to you to set, since there are a plethora of meanings that symbol can have. A symbol is just a symbol if you define it, if not you can't possibly know what it means.
It dawned on me that you probably meant "calculating" rather than "counting" with regard to zero. Language translation error? In that case, the answer is similar. It depends on the context and number system in use. If you don't specify that, you don't know what zero you're dealing with. Zero came into math pretty late, for reasons that you needed to represent nothing in a lot of situations. It obviously made calculations much easier. But it means very different things in different contexts, so you need to be careful not to divide by zero in arithmetic for example, while in other mathematical branches (analysis...) it is standard to be able to express a value where 0/0 is the form (with more details to it of course, such that both zeros have a more complex defining algebraic expression).
Zero is nothing.Something multiplied by zero is nothing.If zero is a number infinitely close to zero then we have to agree with 2×0=4×0,where in 0 cancelled out from both sides leaving behind 2=4,which is absurd.So infinitesimals is ambiguous and vague which should not be entertained in science.It is not objective.And so is the concept of limits which necessarily built up on that concept.
I would like to hear your take on imaginary numbers and hypercomplex numbers in general
Actually imaginary numbers are not very problematic: this is just an example of a quadratic extension of a field, which can be concretely represented by suitable matrices, and the arithmetic is completely cut and dried. In fact complex numbers over (some) finite fields similarly make sense. The critical issue is that we have mathematical objects which can be specified in their entirety, and that the operations of arithmetic are well-defined and computable (meaning that their evaluation requires finite run time and memory on a computer).
@@njwildberger thanks for the response. That's very well stated and Im glad I asked. Never thought about it like that. However, in the way you have framed it, it's a lot easier to grasp conceptually than how I was previously. Cheers!
1+1 cant equal 2 bc there are infinite numbers between 1 and 2
If our understanding of real numbers is logically deficient then rather than dismiss real numbers as a myth perhaps we need a new understanding of real numbers.
No we do not need to prop up our myths even though we may be heartily attached to them. Let’s rather take a big breath, accept the reality for what it is, and move on.
@@njwildberger it could be the theory limits needs to be rethought.
I am no sociologist but it looks like this is related to finite semiotics.
don't know why some describe you as a finitist. you seem to be more of a "computablist", which is a very reasonable mathematical position to take these days, probably increasingly so as we continue to see developments in computation as foundation of mathematics.
Of course there are ways to define real numbers rigorously. The only way to make the claim that "real numbers do not exist" is to use a different notion of existence to the rest of the mathematical community. This may legitimate, but I'm sad that Wildberger is not upfront about this.
It is not clear to me that Wildberger's notion of existence is philosophically consistent, but that may just be a lack of understanding on my part.
I am so glad to hear that of course there are "ways to define real numbers rigorously". Could you please share with us one of these illustrious paths forward, and demonstrate its use by computing "pi + e + sqrt(2)".
@@njwildberger Here are a couple of paths, that I am sure you are aware of:
- Starting from the rationals, you can take the Dedekind-MacNeille completion (which consists of the downwards complete subsets)
- Starting from the rationals you can consider equivalence classes of Cauchy sequences
- You could let the reals be any complete ordered field (which is a perfectly fine definition, if not constructive)
I will not go any further, because you are of course aware of these paths. With some work any any these can give a reasonable and rigorous definition in any formal system that supports classical logic with ZFC set theory, for example. The question of 'computing' (whatever that means for you, I guess a decimal expansion is not enough) is not particularly relevant. This is all well-developed, good mathematics (if very technical) and can definitely be made rigorous. It might not be to your philosophical tastes, however. This is where your definitions may differ from the rest of the mathematical world's.
Some notes:
- I am making no metaphysical claim about "physical existence" of numbers. Existence of a real number is, for me, a formal statement in a formal system, which is really not a very strong claim. What I must admit I do not fully understand about your position is how it is possible that natural numbers exist in any way that real numbers do not. For example 8^9^10^11^12 can no more be computed than pi + e +sqrt(2), at least to my mind. The difference is that one is too precise for our universe and one is too large to fit in our universe. In both cases there is not enough room, I would have thought, but I am interested in your thoughts.
- One could wonder what the use is of such a purely formal system. I can think of at least a couple. On the one hand the use of reals can simplify proofs involving one finite discrete objects (the proof of Sperner's lemma using Brouwer's fixed point theorem comes to mind, but there are many, many examples). The Gödel speedup theorem gives some explanation to this phenomenon.
Secondly real numbers have turned out to be very useful to model physics. I do not claim that this means that the reals exist in our universe, just that it is easy to build good models with them. This is, for me, the most compelling reason to study these objects.
none of those definitions are constructive. constructive means implementable / computable ultimately. rationals are algebraically constructible. the reals by their definition either a) require an infinite computation to _define_ them, or b) algebraically admit non constructible instances.
@@elcapitan6126 You must then be using a non-standard definition of "constructive". Typically this refers to several approaches to mathematics (e.g. following Brouwer, Bishop or Martin-Löf). Normally you can define reals using either some version of Cauchy sequences or some version of Dedekind cuts, but these definitions may not meaningfully coincide.
So yes, it is reasonable to call the definitions constructive. What is not clear to me is whether your notion of "constructive" can be made rigorous. In what way would you construct 9^10^11^12^13? There are not enough particles in the universe. Is this not the same difficulty as constructing e+pi+sqrt(2)?
@@josephcunningham5882 Given there are no ethereal realms of numbers then the only way to make sense of numbers is through computational approaches.
There is no number, as in a decimal expansion, until you write it down or make it explicitly clear. Some sequences are physically possible to write down while others aren't. Those that aren't don't express anything meaningful as they literally can't give a finished product.
You can't assume they are meaningful as if there is some actually infinite decimal expansion of it out there floating in the aether and our mind can only grasp it. There are no numbers outside the Human mind and it's capabilities. To think otherwise is to reify abstractions and speak in vagueness rather than preciseness.
Truth was replaced by convenience for the social structure of academia.
Isn’t all structure applicable to some problem. A lively lecture. Thankyou