Every single thing he said it is perfectly understandable, but it requires time. As he said in the video, all mathematics is based on a small set of axioms we recognize as true (A=B , B=C, it means A=C), you do not require proof, but from it you build an entire universe piece by piece. Most people would be confused, and it is not their fault, because he is pointing really far. He knows the entire path. But at the same time, because of what he said; most people would be able to make the same path (made of small perfectly understandable steps). The hard truth, sadly, it is that most will not put the time and effort it requires, and they will choose to remain confused, as if it was insurmountable, when it isn't.
If I may to simplify, he's staying that mathematics is invented, and that the axioms or assumptions we use to develop mathematics are not necessarily absolutes.
A superb discussion on this topic. One of the best I've heard, and for me it contained some ideas I never considered before -- like, for example, that one of the reasons we see mathematics as soon good at describing the natural world, is because we tend to use it on just those kinds of questions it is ideally suited to answering. Very thought provoking.
Completely agree with you and him. Nevertheless, I would have liked it if the following question was posed to him: How we would be able to reconognize as (or call) a form of "mathematics" some theory or system of tought completely different from what we call "mathematics"? He seems to have a definition of what "mathematics" should be even if it is different from what we call "mathematics". I hope I made myself clear, since I am a mathematician, but English is not my first language....
SW touches on this question several times in his references to "mathematicses" and of course to Gödel incompleteness. On one hand we can consider the universe of all possible axiomatic systems and ask if any are fundamentally "better" than others. (We could, for example, downrate systems which give rise to internal contradiction.) But apart from certain pathological exclusions, a metamathematical rating system seems intrinsically hard to prove complete, in purely abstract terms. So to think about mathematics as something which independently "exists" out there in its own right, waiting to be discovered, is an interesting conjecture, but not a very promising one. On the other hand, in practical terms, and with reference to the properties of the one physical universe that we know about, it may be possible to devise a more constraining test which only one (or a well-defined subset) of all possible axiomatic systems passes. And to this, on balance, SW gives a nod of approval. This postulated mathematics is an artifact, selected and crafted for its (circumstantial) fit to the particular physics in which we find ourselves (and which, by the way, we may never completely understand.) In other words, while we may in principle be withdrawing only one particular subset of books out of a vast abstract library, our relationship to our universe - a mere artifact of circumstance - constrains us to evaluate only that particular subset.
I in no means can give you an answer but maybe something close to the question your asking maybe it will make some sense if you give it some thought. I will use music as an example. In music or rather improvisation we have several sounds for instance two guitars winch are trying to reach a harmony or the tone that the first guitar produces answering the tone of the second forming a harmony. We try to reach harmony using mostly the feal of the tones and a set of rules for which tone to play next so it still sounds good simply sed. What I suspect is that logic is not that much involved in this process because I don't fell it to be a step by step one. What I am trying to say is that the way we recognise and follow harmony is strange. It's not mathematics but it's somehow close. This is more as an example to your question. I hope you find it interesting to think about.
Von Neumans article, "Danger Signals" is a nice explanation of how abstraction, that useful in math, is rooted in empirical thinking. Abstract ideas, are based on abstract models, which in turn come from the empirical organization of information. That is to say, those who believe in circles, owe it to the moon.
Von Neuman can be discredited here if mankind can develop AI which can, of its own accord, develop mathematics. No body = no empirical organizational structure since artificial intelligence is fundamentally based on quantum computing which takes empirically-based Boolean logic completely out of the equation, so-to-speak (or at least that's as far as I can understand it as a layman).
See, I think there's some equivocation going on here. Kuhn asks Wolfram whether the other systems of mathematics are self-contradictory and therefore unused and Wolfram answers no, they are self consistent and valid and their disuse is a just a consequence of the contingency of history. Kuhn should have asked, "but what is self-consistency, or validity without reference to our supposedly contingent mathematics?" That question would bring Wolfram back to Kuhn's original. The set of possible mathematics cannot in itself be fundamental if we can say things of them like they are possible, they are a set, they are valid and etc. Our math is the historically contingent instantiation of one self-consistent system out of the set of all possible self consistent systems, but self-consistency, validity, all those attributes which the set of all possible mathematical systems share and by which they count as such systems must be more fundamental and it's these fundamentals that may be necessary and eternal.
Very good point. How do you prove weather "other mathematics" are right? Probably some rules and axioms have to be respected universally to state such a thing. I mean the guy even mentioned that there could be 50.000 other mathematics. According to what? Our mathematics? :) And also: if those mathematics are based on some fundamental axioms, than why cant they be part or branch of our present mathematics?
@@kashmirha I've recently thought about the nature of logical implication, which is, in my opinion, a central object in mathematical logic. According to what i've briefly read mathematical logic as we know should be the product of studies about the human thinking which have roots in Aristotele, which then have been developed by mathematicians and philosophers. Logical implications are, in my opinion, inherently adopted by humans. From personal experience i can say for example that i didn't need to study propositional logic for understanding "if then statements", when i was younger. I claim that our way of determining logical consequences was the product of an evolution process of our species. Related events that repeatedly happened in the yearly stage of evolution (if i touch the fire then i will feel pain, etc..) shaped our mind and built our intuition of logic which then have been studied and formalized. The notion of consistency comes from this particular mental structure that we learned. My argument is that it seems too hard to state what even is "right" or "wrong" in other mathematics, in my point of view, we are too linked to the history of our evolution.
Yeah, at a certain point it’s like we’re asking “is validity valid” or “is reality real,” and it feels absurd to answer no to those questions. If our math is just “one possible math,” then the new exciting science of these possible mathematices will just become our new math. Not the axioms themselves, but the process of creating and validating those axioms. And asking “is THAT real or just a construct?” is like asking “is anything real?l
@@michaelchikos4551 I had not thought about it that way; I was in the same line as Greg Bechtel and I thought about the nature of implication when I studied Aristotle in secondary school; about what you are saying, I also ask myself if our brain is capable of a level of generality sufficient to grasp the "real" type of abstractions we need, or if those abstractions are beyond of concepts like "validity" or "reality" and we would need to answer "no" to those questions;
Sorry for the pedentatic point, but I think this series began in 2000, so this isn't before Wiles' proof. He said "mathematics has had all these unsolved problems, things like Fermat's Last Theorem" - the key word being "had".
It's interesting how he uses the idea of "the space of all mathematics's", in so doing using the notion of an abstract space, which comes from human mathematics. Presumably there are completely equivalent ways to describe this idea in all the other possible mathematics's. Does this not hint at the idea that there is some kind of universality across all the mathematics's? For example, what reason would there be for an alien race, with a totally different mathematics to our own, to not be able to consider the idea of other mathematics's (or the "space of all mathematics's"), in whatever way that their version of mathematics dictates? If this universailty is true, then perhaps human mathematics isn't entirely an artifact. Either that or there exists something deeper that we don't even consider to be mathematics anymore.
why do you think this "common denominator" is indeed relating all the elements? this could be just a semantic issue and be just a common container of unrelated things used for a similar purpose
Physics is mathematical not because we know so much about the physical world, but because we know so little. It is only its mathematical aspects that we can discover. - Bertrand Russell
what could be the non-mathematical aspects? Isn't maths the all supreme which makes other sciences useful and translates it into something we comprehend?
When discriminating between an invention and a discovery a useful rule of thumb is that an invention can be changed, a discovery cannot. We cannot change Pi, we can change Choleski decomposition. An alien civilisation somewhere will know about Pi, prime numbers etc.
Here's what I think Mr. Wolfram is trying to say: imagine throwing a piece of fabric up in the air, a limp and sheer fabric, more like silk than canvas. It falls to the ground an a heap with folds and manifolds, all twisted and turned but still one piece of unbroken fabric. Let that fabric be the most common, age old, tried and true axiom system we know of. That piece of fabric is in time and space. As we mathmatically explore where it goes, folds, twists and turns we think of it more as a complete and unbroken piece of fabric than something very irregular. Still, as this fabric is in time and space and intersects it many times it is useful for getting around and plotting out that which it intersects. But it's particular twists and turns might miss something in that physical world it occupies. Another piece of fabric which falls to the floor with different twists and turns will be another set of axioms, which can also plot the same space, but may allow other physical realities or structures to be "seen" or plotted, predicted, etc. If we allow that there is one physical reality, but it's more than our favorite axiomatic systems can work out, then the axiomatic system of each different piece of cloth and how is shaped after falling into place is a description of how it is shaped, described against the background of whatever actually is reality in it's totality. Explaining this well enough for anyone to take notice is a task beyond my pay scale. But to begin to think about it we might think of our straight number line as actually curved and warped. We might look at oddities, like transcendental numbers, as clues to where bends or warpages are extreme. Non-Euclidean geometry does something like I am suggesting. What if logics, in which something is or is not, represent places where fabrics intersect. "is or is not" hardly ever, perhaps never happens in the world we know, in the fabric we know, except when we identify an attribute that we know exists in one case but not the other, like on top/underneath, alive or dead, etc. We know these attributes have "is/is not" property because we define them that way. Our fabric is linear and everything changes gradually. But on the quantum scale it is said there are discreet states with nothing inbetween. It could be that everything is linear, but we mistake an intersection of axiomatic systems as jumps in the one fabric we know. Well I'm just speculating. The most impossible part of the idea of ever really grasping and using alternate axiomatic system is that it may not be physically possible for our brains/minds to ever work out the terrain of any intersecting or overlaid fabrics. In this pessimism I am taking Immanuel Kant's suspicions to the extreme in saying that we can only comprehend or paint a world picture in one way according to rules that are part of the construction of our brain/mind.
+rh001YT I just thought something similar. As all our perceptions are given by our senses, we can impossibly know what a "thing in itself" is, we can only paint a map, and the map is not the landscape - that's basically what Kant says I think. Now I met people who said: "Maybe an apple is something totally different than what we think it is", and than I answer: "No, an apple is exactly what we think it is - it corresponds to our definition of an apple, that's the way we percieve an apple." Our senses is all we have to explore and map the world, so if according to our maps an apple is an apple, than it's meaningless to say that it could be something totally different - we only have maps and conventions just to make an apple be an apple : ) So I also think that: "The most impossible part of the idea of ever really grasping and using alternate axiomatic system is that it may not be physically possible for our brains/minds to ever work out the terrain of any intersecting or overlaid fabrics." Of course theories and conventions (maps) can change, if they remain consistent - but I have a suspect that the mathematical axioms are something more basical than a theory? in the sense that it might be impossible for our brain/mind to ever work out different ones?
Gwunderi25 Yes, I think we agree. You wrote: " if they remain consistent - but I have a suspect that the mathematical axioms are something more basical than a theory? in the sense that it might be impossible for our brain/mind to ever work out different ones?" It could be that maths are very fundamental, which is why Kant pointed to Euclidean geometry as a possible fundamental for building up reasoning that results in truth. Non-Euclidean geometry is often cited as a refutation of all of Kant, but even he doubted the parallel postulate. Anyway, non-Euclidean geometries can still be expressed with Euclidean geometry but for many purposes it is easier to use the non-Euclidean maths to get the desired shape. Still any non-Euclidean shape is an xyz map, so not really a refutation of Kant. You may have noticed as have I that one part of what Kant claimed is given undue currency,namely, that our mind paints the world we see. This has led to rampant speculation that the real world might be different and that there may be some solutions there. That then becomes the justification for all sorts of nonsense. Kant's scary point, as you seem to have understood, is that we have no choice except to accept our mind's representation of the world. It is even due to the way our mind works that we demand when we can have it proof for claims. Here I am not talking about eyewitness accounts of who dun it, but we demand that the laws of physics all work together so that we can have proofs. This can be as simple as government certification of scales to expecting the cell phone to work as advertised when signal is available. And if it is not available, like when backpacking in the mountains, we want to be certain why the cell phone does not work...no towers anywhere...else we want a refund. So Kant's Critiques were not suggestions that other ways of seeing the world are possible, but that there is in fact only one way of seeing the world, at least in a way that representations all connect to create a somewhat intelligible world where stuff makes sense.
rh001YT Yes, because reality is that which governs. If we perceive incorrectly and stumble upon reality, we suffer the consequences. When our actions meet with success, we have matched reality.
It was a great pleasure to use that incredible set of tools developed by Wolfram Research ("Mathematica") in my undergraduate years as a student of Comp. Science, there in the nineties. I was writing some pieces of code in several programming styles for diverse areas, as Bayesian Calculus, Neural Networks, and things related. Even using those early CLI-based versions which I started with, previous to the first GUI ones, it was a very good intellectual and learning experience. Thanks to Mr. Stephen Wolfram for doing it possible (and obviously, my teachers! :) God bless him.
Every system of thought requires an associative reference frame. Change the basis, you change the rules of relationships. Take categorization - how objects are distinguished. Similarity is fundamental but ultimately limited and arbitrary.
I think Stephen refers to the specific mathematical system that we are using rather than the general definition of "Mathematics". The 'artefact' is the products of mathematics such as Geometry, Calculus or Topology, but not the methodology of Mathematics. I think there is only one 'Mathematics' though. Say if there are other possible 'Mathematics' which are different from our Mathematics, they are in the set of all possible mathematics. This set must be defined and depends on something else. Likely, that 'something' is also mathematics if anything that is depended by mathematics is also mathematics. So there is actually one mathematics in general.
Maybe our brains are evolved to reason in the context of their surrounding physical laws, so the axioms may be limited but at least they aren't arbitrary.
i dont think he implies it was arbitrary at all. in fact i understood the expression "historical accidents" to mean based on circumstance. but it's only because we just happened to have uses for specific tools that we developed them. we would have developed other sorts of tools if we had different needs or surroundings.
The idea of computational irreducibility is an interesting one.. sort of like conservation of energy . Getting to equivalent statements from different axioms is equally difficult.
Another comment I felt needed repeating: "See, I think there's some equivocation going on here. Kuhn asks Wolfram whether the other systems of mathematics are self-contradictory and therefore unused and Wolfram answers no, they are self consistent and valid and their disuse is a just a consequence of the contingency of history. Kuhn should have asked, "but what is self-consistency, or validity without reference to our supposedly contingent mathematics?" That question would bring Wolfram back to Kuhn's original. The set of possible mathematics cannot in itself be fundamental if we can say things of them like they are possible, they are a set, they are valid and etc. Our math is the historically contingent instantiation of one self-consistent system out of the set of all possible self consistent systems, but self-consistency, validity, all those attributes which the set of all possible mathematical systems share and by which they count as such systems must be more fundamental and it's these fundamentals that may be necessary and eternal."
I agree this comment, which you say you are merely repeating, is mostly right and very important. But if you are repeating this comment, where is the original? Is it your comment originally also?
@@allisterblue5523 Sorry, I was referring to the comment you were replying to, not your comment, and that comment announces that it is repeating some earlier comment. But regarding you comment, I agree also, that there is in effect room for pluralism in regular mathematics. However, I think at the ultimate level there is something in the nature of rational thought about the purely abstract that is not pluralistic, and you could call that logic perhaps. It is not easy to get at what this is, but I think we should be trying. Of course there are "alternative" and even "deviant" logics so the task is not easy. But I think the concepts involved are the sort that the original poster was referring to.
@@allisterblue5523 I don't think it would be easy to go into this at length in youtube comments unless you really want to. Anyway I took the phrase "deviant logic", with some humour intended, from the title of Susan Haack's book, but her subsequent and classic book "Philosophy of Logics" [sic] might be a good reference if you want to see how a so-called "deviant" logic (deviant from the standard classical first order logic) might lead to something "correct".
@@Benson_Bear I somewhat agree with Alex here. Though a deviant form of logic might be possible, I guess I will just have to read that book you mentioned to find out!
What I took from this video was that there is a difference between the concept of math and the math that we use as a society. Yes math (when you refer to what we have as a society) is indeed an artifact, HOWEVER Math itself is independent of our perception and existence... therefore can only be discovered and not invented... only the NOTATION and logic behind which is used to describe what little of it we can use and prove through application. Although there are arguments for things beyond this as well I believe that is another topic and it may transcend all possible human comprehension i.e. infinity, 1=2 etc. This is a move in that direction and I am trying diligently to pursue more information on this subject and it is very difficult. Would appreciate anyone who could shed more light on this topic.
Does Wolfram imply that someone could have invented a different Pythagoras' theorem for a right angled triangle in a plane? It would appear that is just another post-modern attempt at deconstruction that is not very constructive in itself.
Where might one find something like the two-page list of the axioms of our mathematics which Dr. Wolfram alludes to? Such a summary would be very interesting.
This argument seems to just push the question back a step. If we find our axioms are a special case in a much larger "universe of possible mathematicses" [sic], we start studying the more general system. As Luis Dias quipped below, "would this space be an artifact or existent?" Wolfram's argument doesn't seem to show that we aren't discovering something. It may be that the something is just bigger than we thought.
Raymond Brinzer _"If we find our axioms are a special case in a much larger "universe of possible mathematicses" [sic], we start studying the more general system."_ Well, yes and no. The "yes" part of the answer would be that we already do - there are tons of different axiomatic systems in mathematics - each having its own isolated properties and purposes and they are not something you would want to unify as a larger system even if you could, but in many cases they also contradict each other. The "no" part of the answer is that the vast majority of possible axiomatic systems are completely useless for any purpose and we have no desire to study them. An important part of mathematics is to create languages that are small and extremely useful for specific purposes. Not to create one gigantic, but practically useless, system that can deal with anything.
Holy crap! that was that a clear and concise description of the "landscape"/"space" of this discussion and a putting-into-context of many of the key arguments that have been brought up in it, like Hilbert's "unreasonable effectiveness of mathematics". I definitely think it's worthwhile to have a formalised meta-discussion of mathematics (in case the usefulness of a "philosophy of mathematics" was ever doubted), to sort out trivial or naive responses to the question from more sophisticated ones. Way to go Prof. Dr. Wolfram! Exciting to think of future research into "possible other maths" and how they might impact society and as a consequence human consciousness.
A wonderful talk, brings the two "opposing" views under a common umbrella, which is innovative in itself. The term mathematics is used to describe two different notions: the total of mathematical theories, a single mathematical theory. The question still remains.
It will be interesting to someday see what kind of mathematics an alien civilization has. If they have "invented" the same systems we have "invented" perhaps we may conclude that there is a commonality of mathematics that is universal, and hence discovered.
The value of this question is on the question itself rather than the partial answer that was given. What is it that we are doing when we question the nature and origin of mathematics? What are we looking for? What implications any answer are we expecting it to have? What are relevant the facts around this question? What is the truth that such facts bear out? What do we really mean when we try to answer the question about the nature of mathematics? Among such facts which ones are relevant and which ones are not? These are very important questions but I am tempted to believe that any useful answer to such questions will be the result of the coordinated effort over a long period of time of a large community of people.
I think the problem is in the question itself, not how other different people answer it, because whether you think it's invented or discovered, you have a good point. The issue is really just in defining what is meant by discovery or invention? If invention is what is meant by creating formal abstract systems that describe natural phenomena then yes maths and all of science for that matter is invented and the opposite if otherwise, to put it simply.
Formal systems like math do not describe natural phenomena any more than the rules of chess do. Rather, math is a formal extension of the *use* of conceptual frameworks (which are established in human language through practices, i.e. participation in the world) which fix what *counts* as a phenomenon of a certain type. Before we can have any formal mathematical systems, we must learn to use "ordinary language arithmetic" (OLA), that is, the way children learn to count, add, etc. before they encounter any formalisms. The following applies to OLA and formal math alike, but the emphasis is on the fact that it *already* applies to OLA: The case that 1+1 does NOT equal 2 is not even conceivable. How would a world look like in which 1+1=3 was true instead? We cannot say because the question makes no sense. It makes no sense because we do not learn what "1", "2", "+", and "=" mean independently of each other. Rather, we learn "1+1=2" as a rule of language, whose application *shows* (but does not *say* and hence not describe) how we perform certain transformations. (Whenever I may say "1+1", I may say "2" instead and vice versa.) "1+1=2" is therefore not a statement of an empirical, contingent fact. It cannot be verified by experiment. It is a conceptual fact, a linguistic norm which we use to classify and categorize the world around us. We can see in at least two ways that the function of arithmetic is to improve our abilities of categorization: First, by realizing that creating enriched extensions of concepts like "amount" and "order" enables us to go from "much" to "how much" and from "best" to "4th best". Secondly, by noticing that we apply "1+1=2" selectively. If two clouds merge into one, we do not take that as a cue to question the validity of "1+1=2" and replace it by "1+1=1". (Although we could use such a cloud-arithmetic. It would just not have the same domain of application as ordinary arithmetic.) The incredible success of *formal* arithmetic in science (as opposed, for example, to the uselessness of the rules of chess, which on a formal level are just like math) is only partly owed to abstraction. We don't see 2 cars, 2 houses and 2 cats and *infer* or suddenly understand that what they all have in common is "2". Rather, we learn the meaning of "2" as we learn the *practices* of counting, ordering, and adding, which depend on our natural ability to learn to make distinctions (visual, aural, tactile, etc.). For that, we need experiences in and with the world around us. Once we have learned to employ arithmetic and other mathematical concepts formally, we can use abstraction to talk about certain features of any phenomenon susceptible to numerical treatment. But it is not math itself that is derived by abstraction. Rather, math is a formal language created on top of ordinary language, which in turn is partially constituted by certain concept-establishing human practices, such as pointing at one thing after another while saying "1", "2", "3", etc. Thus there is no such thing as discovering mathematics (i.e. its conceptual framework). It's invented (and perhaps more importantly, *passed on*) all the way through. Once a formal system has been established, we may then discover (= find, notice) features of it we hadn't been previously aware of. (Theorems, proofs, etc.) But these are not discoveries about what the world is like, but about what our way of viewing the world is like. (Because they are properties of the conceptual framework which we use to *talk* about the world, in the particular ways that the framework's concepts afford us.)
Overly semantical explanation. All you're saying is that, "We use language and numbers to represent abstract ideas and values.". Well, no shit. Your assertion that these "formal systems" are not statements of empirical contingent fact, and cannot be verified by experiment is simply wrong. Obviously an experiment that measures anything with a value has to have some form of language to describe it and, if the language is strictly defined it can, in fact, accurately describe what we need it to.
You haven't understood a single word I wrote. 'Overly semantical explanation.' There is no such thing. Semantics is the study of meaning, so you're saying that my explanation has too much meaning, which is obviously an absurd claim. 'All you're saying is that, "We use language and numbers to represent abstract ideas and values.".' That's pretty much the opposite of what I'm saying. Linguistic/mathematical abstractions may (partially) CONSIST of numbers, but they are not represented by them. Representation is a relation between certain phenomena in the world and formal constructs in language (mathematical or not) which we can USE for certain purposes. For example, a map of Europe is a representation of Europe if and only if, and this is important, I know the method of projection employed in the map's creation, so that I am actually able to USE the map for the purposes it was created for. Thus, representation = the map + my ability to use it AS a map. The map by itself represents Europe no more than the surface of a lake represents, by way of reflection, the clouds in the sky above it. 'Obviously an experiment that measures anything with a value has to have some form of language to describe it and, if the language is strictly defined it can, in fact, accurately describe what we need it to.' A measured value does in no way *describe* what was measured, it IS what was measured. One can then go on to USE that measurement/value to do things one could not have done without it, but it is not the measurement/value alone that affords us whatever new possibilities arise from it. For example, one could measure how many horsepowers a motor has by empirically determining the maximum force it can exert on a fixed mass. Let's say it is found that the motor has 100 hps, which means we can now calculate what we can expect to happen (in terms of acceleration) if we let the motor exert its pull on any given mass m. Neither the 100 hps nor the expected accelerations *describe* anything. Rather, they *inform* us and help us to *figure out* what we may want or need to do. But there is nothing in the motor which *corresponds* to the value of 100 hps, just like there is nothing in my legs that describes or corresponds to my ability to walk. Thus "100 hps" does not represent or describe the motor. Rather, it informs us what the motor is capable of. It is a (dimensional) quantity, a useful abstraction, ASCRIBED to the motor. But ascription is not at all the same as description. Mathematical formalisms thus do not represent *anything*, just like the fact that the king in chess moves exactly one square at a time does not represent a chess rule. Rather, is IS a rule of chess. Rules in chess, just like rules in math, when taken by themselves, do not have any meaning. Mathematical rules only acquire meaning in an applied practical (participatory) context, i.e. within *practices* inspired by encountering certain phenomena on the one hand and humans being capable of abstraction on the other. But there is no descriptive correspondence between numbers and anything "out there", just like there is no descriptive correspondence between a hammer and a nail being driven into a wall with it. Mathematical formalisms are *instruments*, not representations. One can of course USE formalisms to create representations, for example, one could create a mathematical vector model of a building. The model can serve as a representation of the real thing. But, again, the model in and of itself does not represent or describe anything. Rather, the representation is immanent in our USE of the model when we employ it to construct or navigate the building.
"Your assertion that these "formal systems" are not statements of empirical contingent fact, and cannot be verified by experiment is simply wrong." There is no experiment which can verify that 1+1 is indeed the same as 2, because we do not learn what "2" means independently of what "1" means. Knowing how to use "2" includes knowing that it is the same as "1+1". Just ask yourself what it would look like if 1+1 was 3 instead of 2. You cannot even picture that without changing the meaning of "1", "2", "+", or "3". (As opposed to being able to picture "pigs can fly", which is false but at least imaginable.) All logical/conceptual facts are experimentally unverifiable by their very nature. There is no need to verify them, just like there is no need (and no way) to verify whether the king in chess REALLY moves only one square at a time. (Because that is simply how chess is played. It makes no sense to ask for a corroboration that the king rule is "correct". Someone who isn't sure whether he has learned the rule correctly may confirm or disconfirm that his *knowledge* of chess is in fact correct, but the rule itself can neither be correct nor incorrect. The category of "correctness" does not apply to definitions.)
Using your criteria the only thing that can represent something is the thing itself, but of course, that doesn't give us any meaningful information. That's like saying, "Only Europe is Europe.". It's true, but that doesn't really MEAN anything. You say a map of Europe is useless unless I know how to use it, since it is not, in fact, Europe. Yes obviously, but again, the fact that it is not Europe doesn't really mean anything. Obviously we have to use formal systems otherwise communication between us would be impossible and all we could do is just observe. You say that a mathematical vector of a building does not represent anything, but it represents the building within that system. System and terms can be interchanged, but the purpose is the same, communication. Systems exist and they don't do so to BE or REPLACE something, they are simply representations of what we think we are observing. You keep bolding words like DESCRIBE and MEANING, but you don't define them as it pertains to this subject. What are you arguing? That systems don't exist? That a numerical value has no meaning? Outside of a system it doesn't, but within a system it does. That the numerical value is not the thing itself? Obviously, but who said it was?
If mathematics are historical artifacts created by us, then how come unrelated theories end up been interderivable when they speak about completely unrelated subjects? E.g Lowenheim-Skolem-Tarski Theorem, Tychonoff's Theorem and Zorn's Lemma are all interderivable. If all mathematical entities are historical artifacts, how do mathematicians with absolutely no relation with each other and describing completely different and unrelated things end up with statements that can be derived from one another? This points to a deeper structure of mathematics that can not be explained by the "creation of mathematicians". It is more akin to a discovery of a deeper structure of the world than to a creation in the mind of mathematicians.
But one cannot discover anything, without having some interpretation. And it's precisely our human interpretations that provide math with much of it's content. This doesn't mean that their is nothing real about math (there certainty is), but the human apparatus does not direct access to what is real, and can only construct paradigms. So it is more so, that math largely reflects an approximation of reality.
Actually, it does not. You are talking about mathematical theories that try to explain reality. But that is physics, not mathematics. Example: Riemann's geometry is partially useful to physics in that it can explain most features of the General Theory of Relativity, but the question of "does physical space have a Riemannian geometry" is a question of physics and not a mathematical question. Riemannian geometry as well as Euclidean Geometry are both consistent and true (although possessing different axioms) and they are so irrespective of the question about physical space been actually more akin to one or the other, or even been akin to none of them.
Geometry deals with space. The parallel postulate , is in fact, a mathematical production, which originally involved the broad generalization, regarding the nature of parallel lines. However, this generalization was shown to be incorrect, upon the construction of non-Euclidean geometry. Now a-priorists attempt to defend the validity by distinguishing between 'pure' and 'applied' math, and sometimes euclidean vs non-euclidean geometry, but as Hillary Putnam says , that is simply ad-hoc. The original postulate had no specification that the postulate only holds for 'euclidean geometry' - it was in fact, an over-generalization , a result of our perceptual limitations, which was later shown ( empirically) to be in imprecise.
fergoesdayton That could not be further from the truth. Hillary Putnam got that wrong along with almost everything else he has defended even before Models and Reality. The parallel postulate was precisely one of the most problematic postulates in the history of Geometry. Ever since Euclid included it in the Elements it was clear that it wasn`t self-evident. Even Euclid had probably realized early on that he could not prove it or proceed without it and for the next 2,000 years many tried to provide a proof of it without any success. The problem was the constant belief that space had to be Euclidean (a question which already was beyond mathematics), but there was no proof of inconsistency if it was rejected. Finally, there was some light at the end of the tunnel. Poincaré (one of the fathers of non euclidean geometries) writes in "L'expériment et la gèometrie" that "no possible physical experiment can make the physicist abandon Euclidean geometry as the geometry of our physical space...(because)... it is not on the basis of experience that Euclidean geometry or Non Euclidean geometries can be refuted... Experiments can only teach us how the different objects relate in space, but not how they relate to space, nor how the different parts of space relate to each other nor, thus what the nature of physical space is". In fact, there are many properties usually attributed to Euclidean Geometry that do not correspond to our perceptual Space, namely: continuity, infinity, isotropy... etc... He concludes: "physical geometry is not an empirical science, since in such a case it would only be approximate and provisional." (1905) Even before Poincaré many other mathematicians had begun to doubt the truth about the parallel postulate and some of them, like Husserl even wrote early on (1892-1901) that the traditional conception of Euclidean geometry as been the same as physical space had to be revised since "on cannot proove that with respect to any straight line one can trace through each point only one straight line not intersecting it; or that parallel segments between parallels are equal;... thus, briefly, the parallel axiom is not true." In fact, in 1897 in a letter to Natorp, after arguing on behalf of the existence of Euclidean manifolds of more than three dimensions (in which space and time are subsumed under more general concepts) Husserl writes states that the thesis about the Euclidean structure of physical space is an unfounded hypothesis made by NATURAL SCIENTISTS (caps are mine), which can only be founded empirically. So it is perfectly clear that even for XIX century mathematicians and philosophers the three dimensionality or n-dimensionality of space, as well as the Euclidicity or non-Euclidicity of it, were EMPIRICAL ISSUES, not MATHEMATICAL ISSUES. And that has nothing to do with "experiments" showing that the parallel postulate is false. Because that has simply never happened. PS Another consideration apart from this one is that, contrary to what Putnam believes, there has never been an instance of a refutation of a purely mathematical or logical theory by experimental means. But that´s another, more broader topic, which I do not wish to entertain right now.
biokant Poincare's use of 'empirical' is different than Putnam's. In Putnam's quasi-empirical view (which generally rejects the a priori - though supports the 'contextual a priori'), mathematics is the result of 'empirical research'. And it is this research which ultimately lead to the rejection of the parallel postulate. But categorizing the issue of the Geometry/parallel postulate, as an 'empirical issue' is just a way to throw dirt on empiricism. The parallel postulate is unambiguously born from a priori (rationalist) intuition - and not simple observation. Had the parallel postulate been defined with exclusive respect to 'euclidean space' - then Husserl would have had a point. Instead, the case is that the parallel postulate, assuming Euclidean space, over-generalized the nature of parallel lines.
Wolfram's response is self-contradictory. He says our math is an artifact ...but then explains how our math is just one amongst a huge set of maths that exist in a concept space of various diverse maths. One way to resolve this contradiction is via an ontology where invention is *the same as* discovery. That is, the mind that discovers is actually inventing that discovery. It is only believed to be discovery because the self's awareness of its role of creation has been suppressed.
You've misunderstood his words. He CLEARLY states that math systems are essentially CREATIONS of axiomatic systems that are simply TOOLS to allow us to get something done. When we humans one day CREATE a better tool, we will leave the old one behind. Ex. Is a HAMMER a fundamental reality in our Universe? Were we created simply to bring about the HAMMER? Isn't computation essentially a hammer system in mathematics? Mr. W also CLEARLY states that there MIGHT BE a system of axioms we might create that IMMEDIATELY reveals the outcomes of that system. Ergo, no need for the hammer anymore. It was NEVER needed, except as a stepping stone in our brain development. Don't be a prisoner of your brain's inability to think outside of its artificial realities created to satisfy your desires, expectations, etc. You are ruled by biases, NOT fundamental realities of the real world. You CAN NOT know such things. The brain is a tool, NOTHING MORE. Think in terms of this saying: If your only tool is a hammer, everything looks like nails. Let go of the hammer/math.
We can only understand a type of mathematics based on the axioms that we can perceive as humans. Wolfram is suggesting there are other axiomatic systems which may not be apparent to us. Thus, our Math is not so much fundamental to reality, though indeed, in some respect, and to some extent, it may represent real things. Furthermore, empirical artifacts of math have already shown that some of our most intuitive beliefs about the world, have been defeated (eucleadean geometry, boolean logic, constancy of time). We don't have direct assess to truths.
Is it not utterly surprising that the rules of complex analysis,like conjugation, Harmitian operators etc., describes exactly, quantum states,and conforms to physical reality that existed outside the mind. How true "invention is the same as discovery". Socrates was hitting a prime cord when he claimed that Meno's slave new higher mathematics. Mind of Meno's slave,is programmed with god's knowledge-!!!
That's not a contradiction at all. From an ontological perspective he's not saying that these other possible math systems exist, but merely that they CAN or COULD exist if we had chosen differently. Our mathematics does not exist in a platonic ideal sense to be discovered. It developed over time, evolved if you will, was invented from our simple intuitive understandings of arithmetic and geometry through the methodology of theorem postulation and proving. In point of fact, mathematics is not a single monolithic entity, but rather a set of closely aligned sets of axioms and provable theorems and annoying paradoxes. Each theoretical mathematician tries to come up with a more fundamental set of concepts and axioms or tries to avoid paradoxes or places restrictions to make math more computable. Not all such innovations become generally accepted, but if the innovation is found to be useful in some field it will spread.
Damian Torn You are right, the brain is a tool. It acts as a "central processing unit" that handles a number of "inputs", producing a number of "outputs". Is mathematics a tool only? Consider the conclusion that "rate of change at every point of an exponential function have the same value", or "numbers are operators", these are truths that would have remained unknown if maths was all about "tool", is it not true?
The word mathematics comes from the Greek μάθημα (máthēma), which, in the ancient Greek language, means "that which is learnt", "what one gets to know," hence also "study" and "science", and in modern Greek just "lesson." The word máthēma is derived from μανθάνω (manthano), while the modern Greek equivalent is μαθαίνω (mathaino), both of which mean "to learn." In Greece, the word for "mathematics" came to have the narrower and more technical meaning "mathematical study" even in Classical times. Its adjective is μαθηματικός (mathēmatikós), meaning "related to learning" or "studious", which likewise further came to mean "mathematical". In particular, μαθηματικὴ τέχνη (mathēmatikḗ tékhnē), Latin: ars mathematica, meant "the mathematical art".
In Latin, and in English until around 1700, the term mathematics more commonly meant "astrology" (or sometimes "astronomy") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This has resulted in several mistranslations: a particularly notorious one is Saint Augustine's warning that Christians should beware of mathematici meaning astrologers, which is sometimes mistranslated as a condemnation of mathematicians.
The apparent plural form in English, like the French plural form les mathématiques (and the less commonly used singular derivative la mathématique), goes back to the Latin neuter plural mathematica (Cicero), based on the Greek plural τα μαθηματικά (ta mathēmatiká), used by Aristotle (384-322 BC), and meaning roughly "all things mathematical"; although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after the pattern of physics and metaphysics, which were inherited from the Greek. In English, the noun mathematics takes singular verb forms. It is often shortened to maths or, in English-speaking North America, math.
Good talk, the accumulation of history in practical usage has led to the next big realization that an expectation of "speaking in math" will eventually solve all speculation, but then came QM with unpredictable properties because not everything is measurable. The cosmological equivalent of a measuring device has to be tunable to the subject of study, and so we believe in only a few percent of what can be inferred to exist.
Does Plato's view of mathematics a mere illusion or a hard fact? I don't think this talk really touches much on this question. It has implications that mathematics is invented but it would be perfectly valid to say that there is an ideal mathematics and we only know the shadow of it cause we are blinded by so many constraints while having the exactly the same view with Stephen Wolfram. But I would have to add that I do not fully understand how he came up with all the possible mathematical systems so it is hard to say anything with definite certainty.
Led to wonder if the thing that's special about our mathematics, in contrast to the universe of all other possible axiomatic systems, is that the axioms of our mathematics are largely based on human experience of the meso-scale natural world. Taking a quick look at the Zermelo-Fraenkel axioms (thank you Wikipedia) this seems to be true; if you jumbled those axioms at random, you'd end up with a bunch of rules that have no connection to the physical world that we experience.
I wish he'd given an example of a couple different axioms from which a different math could be built. His explanation was 100% abstract, so I never really knew if I was actually understanding what he was trying to explain.
@@Wonderboywonderings I'd say the most famous example of a different set of axioms deriving different mathematics is non-Euclidean geometries. In his book of Elements, Euclid enumerates his set of axioms and then compilates a bunch of theorems with their proofs. Centuries later mathematicians started to question if Euclid's fifth postulate was really an axiom or if it should be proved and after trying and failing to prove it some mad lads denied it and started building theorems without it and such an effort was valid and ended up in what is now called hyperbolic and elliptic geometry. In elliptic geometry for example, triangles' internal angles always add up to a number greater than 180° instead of exactly 180° as it happens in Euclidean geometry. Guess what happens with triangles in hyperbolic geometry.
mathematics is not fundamental to reality. it describes reality. descriptions of reality are not fundamental to its existence. the universe does not have to carry the one in order that x happen. KEvron
What Wolfram was stating is that there are alternative systems of axioms which are distinct (ie. you can prove different theorems) from the standard ZFC system that most of our current mathematics is founded upon. The existence of these alternate systems makes ZFC a historic artifact. A possible consequence of their existence is that they may be able to model real-world phenomena that we currently can't, such as consciousness. And to directly answer your question. If we include mathematical proofs as part of our reality then mathematics can never model a ZFC proof of the continuum hypothesis.
it might have been in reference to fractal geometry which is easily found in nature, but is hard to define using euclidean geometry. usually, we need to move away from pure maths and rely on some sort of recursive computer algorithm for calculations or graphic renditions. in our "current maths", we have developed relevant descriptors like the fractal dimension, but this is just an index to quantify how unlike our usual shapes a geometry might be. if the building blocks of our maths were different, fractal objects as found in nature might be no more complex than our elementary shapes such as triangles or circles.
I should add that idealization, dreaming, and wishful thinking all have practical considerations going for it. It raises achievements in many disciplines, paradoxically, by setting unachievable goals. And sometimes, it's what gets us up in the morning. (Especially coupled with zone-out substances such as a hot cup of java ;))
As with most of these discussions, it seems to focus on the linguistic/symbolic description of mathematics rather than the fundamental underlying question of whether mathematics exists independently of human cognitive awareness. Of the many "mathematicses" possible, would any of them change the value of Pi or would they just use different words/symbols to identify it?
Is Wolfram proposing that there is a "landscape" of formal systems of math and logic each with a different set of axioms? If so, then within that landscape we don't know where our particular formal systems are situated, nor can we know how many diferent axiomatic formal systems are possible?
I know it's a broad generalization, but I find most mathematicians say math is discovered, and most scientists and engineers say math is invented. Wolfram started as a physicist.
the nature of our universe is, essentially, creative. like a blank page, many things can be drawn upon it. imagine being able to draw on an infinite page, with infinite imagination... what wouldn't arise?
Oh, what I say is of no importance, it was just an opinion. Who knows. I didn't mention randomness... But there may be something worth looking at in that word. Seems like the randomness is used in several ways which don't mean the same thing.
A very different view point that mathematics is an artifact. I remember textbooks on mathematics only showing the polished and beautiful results or theorems of mathematics and hiding the complex and rather hard and not so beautiful scaffolding that is the cause of the result. So maybe as he says text book mathematical results are just artifacts. Further, I like his point that there are Infinitely many mathematical systems with infinitely many unprovable results. Also, I like his point on reinforcement loops between maths and physics that can explain "incredible effectiveness of mathematics". A good and demystifying answer which in an unlikely manner has reinvigorated my mathematical pursuits.
Godel's incompleteness proof...even when math is awesome, it is more beautiful, since we dont know yet everything about it, and probably we will never have.
he still failed to convince me that the 'artifact', the mathematics human 'invented' or in his view one possible mathematics isn't just truly a discovery of part of the entirety of mathematics. in a word philosophically(or logically, which as part of maths may be hardly adequate for this topic any more), there's no way to affirm those single pieces of mathematics don't form a unified integral mathematics. and he was also far from convincing me that mathematics is not objective and has its objective existence. I agree that between mathematics, our physical world and our mind, there's something very deep about it. it may be that in the end of the day, when this ultimate problem is solved (if by any chance it is possible), maths turns out to be something non-objective and be immersed with the ultimate real 'physical law' and become one, but i guess this is way beyond what he said here and so far we have discovered.
So, it seems even if you say we're inventing math because we just conjured up some axioms there are meta-statements which are discovered and true of all formal systems.
Many claim that our mathematics is a historical artifact, a social construction, etc., but I’m yet to see a functioning “constructed” mathematics, that doesn’t borrow anything from our own (you could say, is “orthogonal” to our own) and has a similar explanatory power.
Mathematics was discovered. Pick any example you like and regardless of who discovered it the principle would be exactly the same. The angles in a triangle. The times tables, etc. Discoveries don't change, they are set in stone. Inventions, on the other hand, vary from one another.
Did the Wright Brother's discover flight? Just as they "discovered" flight someone "discovered" mathematics. They could've flown in many ways. But they used nature as guide. Just like the first people who invented/discovered maths did to invent whatever system or problem maths could solve for them. That's what Stephens getting at. We invented ways of using math to problem solve the issues around us. Had the problems been different... Harder, easier, more involved in one subject etc, would mathematics be fundamentally different as well?
Bertrand Russell said "even in the remotest depths of stellar space there are still three feat to the yard." I believe that mathematics and logic are absolute.
clovis Frank I wouldn’t consider the conversion from feet to yards as not mathematics since that’s a human definition. As an example of what he’s talking about: That two parallel lines will never intersect is only possible if space isn’t curved
Wolfram seems to miss the fact that some mathematical systems, of which logic is an example, are used in everyday life because they play a valuable role. Arithmetic is the same, but number theory isn't (though it does play a role in technical applications such as encryption, but it does not enter ordinary language the way that arithmetic does). So it's a bit misleading suggesting it is 50000th on the list of possibilities. However, if we regard logic as a problem-solving device, then different kind of problem might involve different logics, but I think this would be a different kind of beast to propositional logic, predicate logic etc.
In OTHER words, his elaborate answer can be reduced as mathematics being invented, developed from historical axioms of our civilization. This is perfectly reasonable, and I don't see a need for an argument. I liked this.
I feel like they ask this questions to people who are too abstract to see what’s right in front of their faces. They are focused too much on the symbology to simply recognize the concepts often are and have been discovered in parallel by multiple different cultures on our planet without an exchange of information. The babylonians, Chinese and Pythagorus all discovered the principal we call the “Pythagorean Theorem.” Yes, you could come up with a different way of notating the Hypotenuse^2 = Perpendicular^2 + Base^2, but the principal works the same in binary, hexadecimal or by throwing beads in dish. The principal is therefore fundamental but the notion is arbitrary.
Well, that wasn't the answer I expected. Very thought provoking. It's been a while since I took symbolic logic, but this makes me want to take it again. Obviously I'm not in my right mind.
Ahh... so... can God make a rock , but in order to make it he has to make half of it first and there are an infinite number of half steps, but if he goes back into the past and kills his grandfather before his father was born... Many of these just seem retoric that ends up to be a bit pointless.
There are things which we can neither prove nor disprove, and those things are the subject of philosophical debate. Many of them will forever remain mysterious. For example many questions which start with "Why?" Eg. Why does mass have the property of Gravity? Science can describe how gravity works but doesn't tell us why it is so. Incidentally that Zeno's paradox has got a solution since 1/2 + 1/4 + 1/8 + ... to infinity actually has a finite sum it equals 1
He is saying that within an axiomatic system there are things that are unprovable by that system therefore we need to explore other systems to explain those things which our current system is incapable of proving.
Excellent & fun video! Thanks! It really makes me think...If math is invented and can't explain everything in the universe does it mean we have to replace math with some kind of "computer language of the universe"? I mean, computer games have a physics engine but also have a lot of "IF-THAN" code (or bugs) that can break the rules.
It is more than an artefact, the entire modern society is based on mathematics, from simple vehicles to powerful computers, economics to mathematical models of many subjects (music etc.).
If there's anything to conclude from this spiel it certainly isn't going to be obvious without some guesses. I'd suggest shooting at the dark, as Wolfram implies, and trying to come up with new questions and new ways to find the answers, and see if any are totally incomprehensible under out set of mathematical axioms.
I'm afraid Platonic Idealism died with Aristotle. Extrapolating to perfect forms (simplifying by dismissing details;seeing something's ideal form) is something that comes naturally to us and that we do all the time. I think it's a part of our make-up that we need in order to function at all. It's the sort of imagining that we apply to what-if scenarios, sometimes going against what we actually know will happen. I suspect we need this kind of dreaming or wishful thinking to plan ahead at all.
I could listen to him talk about this for two hours... still I am misunderstanding because most random symbol sets quickly fall into either lacking breadth or depth because of self contradictions (I guess one could create another symbol set like a bridge and just add a dimension... still we would judge this symbol set with our perception of complexity, and conciseness etc... like programming languages made from random symbol sets that are actually useful.... seems like millions would just hit the recycle bin really quickly. What am I missing?
This will almost certainly get a tremendous amount of criticism (much of which will be motivated from an almost teenage-like angst which I find curious). It seems to me that mathematics represents the nature beyond nature. Not so much the physical universe itself as the underlying principle. Math is very much the scaffold upon which the physical universe, it would appear, seems to be built. I would say that from a Judeo-Christian world view, one would benefit by considering mathematics to be a manifestation of what is referred to in the Bible as ‘wisdom’. In a Greek or Roman influenced culture, one might call this ‘reason’. In Jewish and Christian tradition, wisdom is much more than ‘making good choices’ wisdom is a characteristic of God, part of his nature that is ‘Hevel’, or enigmatic. Something eternal and something we don’t fully experience or understand. Wisdom is a trait of God that we see as a shadow, or the way a two dimensional being would see a three dimensional object as it passes through a two dimensional plane. According to various psalms, proverbs, and other Jewish writings, God established creation (or at least the physical part of it that we interact with) upon a foundation of wisdom. Wisdom is often portrayed as a person, an entity, who was with God and through whom all things were made. Christians would say that ‘the word was god and the word was with god and through the word all things were made.’ Christianity goes further as it is revealed that ‘the word became flesh and dwelt among us’. Basically, as near as I can understand, God, who is wisdom, entered into his creation as a man (Christ). This is all detailed in the 66 books of the Bible, but I digress. What I posit is that Math/wisdom/reason are the same ‘thing’, or at least manifestations from a singular origin that is outside of the physical universe that is itself the source, or origin, or scaffolding of our universe. I wonder if mathematics/reason/wisdom has ‘leaked in’ to our universe in much the same way it has been theorized that gravity has ‘leaked’ in. It is foundational and seems somehow separate at the same time. Anyway, just some random ideas here. Regardless, it seems fallacious to me to assume that the physical universe is ‘it’, that there is nothing else. Not that one should assume otherwise of course, however, we must conclude that the pattern of reasonable thought and understanding to which we have become accustomed are infallible. I think it reasonable to consider that science as a framework for ‘good and productive’ thinking best practices must have its limits. The scientific method itself relies on the preservation of information and cause/effect relationships which begin to break down when we consider the singularity and origins of the universe. If the Big Bang is a door, and cause and effect relationships exist as they do on this side of the door, then if there is something beyond that door whose to say those relationships exist similarly on the other side? Language itself cannot communicate ideas effectively and we begin to realize that there is an end to our understanding. As such, unless the universe itself is infinite, then there must also be an end to scientific inquiry. But math, that may actually extend beyond the door. Or... maybe not.
Mathematics is discovered. Even given that the mathematics most explored by mathematicians today is based upon one of many possible sets of axioms which themselves result in many possible mathematical "universes," every such mathematical universe is nevertheless implied by its respective axioms. Mathematicians explore the implications of the fundamental axioms of a mathematical universe or even the nature of all possible mathematical universes themselves. In either case, mathematicians are discovering the direct implications of the axioms of a particular mathematical universe, or discovering the possible variations of mathematical universes based on different axiomatic structures. Mathematics is not invented, but discovered.
So are you saying that different sets of axioms are fundamental aspects of our and any other universe and that our stumbling upon a specific one is therefore a discovery? I tend to view it such as we as a species are able to identify and define various axiom systems which themselves are not fundamental parts of our universe, and in that way mathematics is an invention and not a discovery. Edit: if somehow mathematics was a fundamental truth to our (and any other) universe then it would be a discovery, however if it just a methodology or system that we have defined that would make it an invention.
There was a schism in mathematics in 1930. People who follow the schism believe that mathematics is made up. Their logical system has fewer contradictions than platonic mathematics, so there's a good chance they are closer to the truth than the mainstream mathematicians. Some well-known proof systems came from that community. Their teaching of mathematics makes more sense.
Are you suggested that if an alien species came to Earth, they may have never seen Math. They would look at all the startling coincides and predictions Math is able to predict about the universe and would say to us, translated, "Neat! Who would have though there was order to the universe!"
Once we humans redesign the brain for Homo sapiens 2, perhaps novel mathematics will be developed. Certainly after strong AI is enabled and loosed upon its runaway course, new mathematics will be developed, one after the other at shorter and shorter intervals, though they will become less and less understandable to mere humans.
i don't buy this idea that strong ai will be *so* transformative; humans do things beyond individuals all the time through division of labour.. shared knowledge.. books, and now the internet. (i think we have much further to go with that.. "the network is the computer", including human brains)
someones been reading his ludwig wittgenstein. IMO the existence of a space of possible mathematics has no bearing on physical reality even though they would be self consistent. the way objects come into existence & interact in the physical world is the basis for their quantitative treatment and abstraction thereof but the question of the place of mathematics in a possible space of axioms is different from its universality
I think all these mathematics systems have to collapse to a 'kernal' that all self consistent mathematics have. This kernal is discovered not invented. And it is embedded in physical reality. It doesn't matter if we use a given "construction", we are still doing discovery. To make an analogy, there are lots of ways to draw a map but how you draw the map doesn't change the structure of what you are mapping.
I think he's just suggesting that a different evolution of mathematics, say not based on ancestral knowledge like farming, would produce different results in terms of it's effects on furthering more mathematical discovery. If we had or hadn't focused on a particular area of maths or used it for tasks abstract to us now would it be fundamentally different? Would we have discovered more or less? Would it be more efficient? In what way? How? Why? Did it breed better strategies to answer our hardest questions?
Very interesting, up to this point in my life, I always thought there must be something "inherently" true about Mathematics; or at least mathematics can indeed lead us if not to a higher truth, at least to a higher understanding of of most phenomena without which it wouldn't be possible. I can't accept what Stephen Wolfram says at face value, however, it's something to think about .... a man-made artifact stemmed from axioms partly found by accident. =================================== A relevant question might be: "Would other mathematical systems have survived and evolved as the one human has created so far?" ... "Has the mathematics we know come out of a Darwinian competition with flying colors?" .... interesting indeed.
A. M. Goudarzi _"Would other mathematical systems have survived and evolved as the one human has created so far?"_ Well, humans have created many mathematical systems, so it's hard to understand why you represent it as a hypothetical. In fact, many alternative forms of algebra are being used in specialized areas. Modulo arithmetic is one backbone of cryptography for example.
Gnomefro Thanks for your comment. Modular arithmetic is indeed one strong tool and widely utilized in cryptography. However, modular arithmetic is not perceived nor it claims to be a substitute for the whole "mathematical system" Wolfram is referring to.
If maths is invented and not discovered, then it *seems* like all mathematical things can be described. But some mathematical things cannot be described, since the set of all numbers is uncountable but the set of all describable numbers is countable (since the set of all English sentences is countable). Contradiction?
He seems to suggest that the initial axioms of math were chosen at random and other axioms would have done just as well. Surely the initial axioms of math get altered if the system fails to be predictive of natural phenomenon. What we have is an artefact but an evolutionary model that has been constructed over time. In the same way we can think of all manner of alien life forms that bear no resemblance to life on Earth. This doesn't mean that they would all be just as likely to exist. I can imagine a creature that is nothing but eye balls, doesn't mean that is just as likely to evolve as a sparrow.
Definitely very interesting. I'm just waiting for the day when that which is mathematically irreducible becomes reducible. I am pretty sure that, given the right formal system of logic, all life processes can be modeled mathematically. This would allow us to predict all things with relative degrees of certainty--namely human behavior, animal behavior, social patterning, etc.
I’m still confused, but at a much higher level.
Every single thing he said it is perfectly understandable, but it requires time. As he said in the video, all mathematics is based on a small set of axioms we recognize as true (A=B , B=C, it means A=C), you do not require proof, but from it you build an entire universe piece by piece.
Most people would be confused, and it is not their fault, because he is pointing really far. He knows the entire path.
But at the same time, because of what he said; most people would be able to make the same path (made of small perfectly understandable steps).
The hard truth, sadly, it is that most will not put the time and effort it requires, and they will choose to remain confused, as if it was insurmountable, when it isn't.
Me either
That’s life, I guess
If I may to simplify, he's staying that mathematics is invented, and that the axioms or assumptions we use to develop mathematics are not necessarily absolutes.
So true....
A superb discussion on this topic. One of the best I've heard, and for me it contained some ideas I never considered before -- like, for example, that one of the reasons we see mathematics as soon good at describing the natural world, is because we tend to use it on just those kinds of questions it is ideally suited to answering. Very thought provoking.
Completely agree with you and him. Nevertheless, I would have liked it if the following question was posed to him: How we would be able to reconognize as (or call) a form of "mathematics" some theory or system of tought completely different from what we call "mathematics"? He seems to have a definition of what "mathematics" should be even if it is different from what we call "mathematics". I hope I made myself clear, since I am a mathematician, but English is not my first language....
SW touches on this question several times in his references to "mathematicses" and of course to Gödel incompleteness.
On one hand we can consider the universe of all possible axiomatic systems and ask if any are fundamentally "better" than others. (We could, for example, downrate systems which give rise to internal contradiction.) But apart from certain pathological exclusions, a metamathematical rating system seems intrinsically hard to prove complete, in purely abstract terms. So to think about mathematics as something which independently "exists" out there in its own right, waiting to be discovered, is an interesting conjecture, but not a very promising one.
On the other hand, in practical terms, and with reference to the properties of the one physical universe that we know about, it may be possible to devise a more constraining test which only one (or a well-defined subset) of all possible axiomatic systems passes. And to this, on balance, SW gives a nod of approval. This postulated mathematics is an artifact, selected and crafted for its (circumstantial) fit to the particular physics in which we find ourselves (and which, by the way, we may never completely understand.) In other words, while we may in principle be withdrawing only one particular subset of books out of a vast abstract library, our relationship to our universe - a mere artifact of circumstance - constrains us to evaluate only that particular subset.
”Humans see what they want or need to see.”
I in no means can give you an answer but maybe something close to the question your asking maybe it will make some sense if you give it some thought. I will use music as an example. In music or rather improvisation we have several sounds for instance two guitars winch are trying to reach a harmony or the tone that the first guitar produces answering the tone of the second forming a harmony. We try to reach harmony using mostly the feal of the tones and a set of rules for which tone to play next so it still sounds good simply sed. What I suspect is that logic is not that much involved in this process because I don't fell it to be a step by step one. What I am trying to say is that the way we recognise and follow harmony is strange. It's not mathematics but it's somehow close. This is more as an example to your question. I hope you find it interesting to think about.
Yes. That’s a point the philosopher Wittgenstein constantly made is his mathematical remarks.
Von Neumans article, "Danger Signals" is a nice explanation of how abstraction, that useful in math, is rooted in empirical thinking. Abstract ideas, are based on abstract models, which in turn come from the empirical organization of information. That is to say, those who believe in circles, owe it to the moon.
Von Neuman can be discredited here if mankind can develop AI which can, of its own accord, develop mathematics. No body = no empirical organizational structure since artificial intelligence is fundamentally based on quantum computing which takes empirically-based Boolean logic completely out of the equation, so-to-speak (or at least that's as far as I can understand it as a layman).
@@ChristAliveForevermore idealism vs materialist monism.
Observer is diffrent in Quantum state and simulated state.
See, I think there's some equivocation going on here. Kuhn asks Wolfram whether the other systems of mathematics are self-contradictory and therefore unused and Wolfram answers no, they are self consistent and valid and their disuse is a just a consequence of the contingency of history. Kuhn should have asked, "but what is self-consistency, or validity without reference to our supposedly contingent mathematics?" That question would bring Wolfram back to Kuhn's original. The set of possible mathematics cannot in itself be fundamental if we can say things of them like they are possible, they are a set, they are valid and etc. Our math is the historically contingent instantiation of one self-consistent system out of the set of all possible self consistent systems, but self-consistency, validity, all those attributes which the set of all possible mathematical systems share and by which they count as such systems must be more fundamental and it's these fundamentals that may be necessary and eternal.
Very good point. How do you prove weather "other mathematics" are right? Probably some rules and axioms have to be respected universally to state such a thing. I mean the guy even mentioned that there could be 50.000 other mathematics. According to what? Our mathematics? :) And also: if those mathematics are based on some fundamental axioms, than why cant they be part or branch of our present mathematics?
@@kashmirha I've recently thought about the nature of logical implication, which is, in my opinion, a central object in mathematical logic. According to what i've briefly read mathematical logic as we know should be the product of studies about the human thinking which have roots in Aristotele, which then have been developed by mathematicians and philosophers. Logical implications are, in my opinion, inherently adopted by humans. From personal experience i can say for example that i didn't need to study propositional logic for understanding "if then statements", when i was younger. I claim that our way of determining logical consequences was the product of an evolution process of our species. Related events that repeatedly happened in the yearly stage of evolution (if i touch the fire then i will feel pain, etc..) shaped our mind and built our intuition of logic which then have been studied and formalized. The notion of consistency comes from this particular mental structure that we learned. My argument is that it seems too hard to state what even is "right" or "wrong" in other mathematics, in my point of view, we are too linked to the history of our evolution.
Yeah, at a certain point it’s like we’re asking “is validity valid” or “is reality real,” and it feels absurd to answer no to those questions. If our math is just “one possible math,” then the new exciting science of these possible mathematices will just become our new math. Not the axioms themselves, but the process of creating and validating those axioms. And asking “is THAT real or just a construct?” is like asking “is anything real?l
@@michaelchikos4551 I had not thought about it that way; I was in the same line as Greg Bechtel and I thought about the nature of implication when I studied Aristotle in secondary school; about what you are saying, I also ask myself if our brain is capable of a level of generality sufficient to grasp the "real" type of abstractions we need, or if those abstractions are beyond of concepts like "validity" or "reality" and we would need to answer "no" to those questions;
you can tell a video is old when you see fermat still remained unsolved then
Sorry for the pedentatic point, but I think this series began in 2000, so this isn't before Wiles' proof. He said "mathematics has had all these unsolved problems, things like Fermat's Last Theorem" - the key word being "had".
@@jacderida What's "pedentatic" if not pedantic?
@@GeoCoppens Haha, fair enough! I didn't even notice that at the time!
I find unsolvibility in most mathematics
+Patrick Hickman It's everywhere in calculus, and it uses Riemann sums and derivatives to 'get around' the problem.
lol
LOL
lmao
huh
My new favorite phrase: "possible mathematicses"
It's interesting how he uses the idea of "the space of all mathematics's", in so doing using the notion of an abstract space, which comes from human mathematics. Presumably there are completely equivalent ways to describe this idea in all the other possible mathematics's. Does this not hint at the idea that there is some kind of universality across all the mathematics's? For example, what reason would there be for an alien race, with a totally different mathematics to our own, to not be able to consider the idea of other mathematics's (or the "space of all mathematics's"), in whatever way that their version of mathematics dictates?
If this universailty is true, then perhaps human mathematics isn't entirely an artifact. Either that or there exists something deeper that we don't even consider to be mathematics anymore.
why do you think this "common denominator" is indeed relating all the elements? this could be just a semantic issue and be just a common container of unrelated things used for a similar purpose
A dubious sounding channel that actually has amazing content.
Physics is mathematical not because we know so much about the physical world, but because we know so little. It is only its mathematical aspects that we can discover.
- Bertrand Russell
I suspect tis' the case
what could be the non-mathematical aspects? Isn't maths the all supreme which makes other sciences useful and translates it into something we comprehend?
When discriminating between an invention and a discovery a useful rule of thumb is that an invention can be changed, a discovery cannot. We cannot change Pi, we can change Choleski decomposition. An alien civilisation somewhere will know about Pi, prime numbers etc.
@Noah dean seriously? you correct people's punctuation ..on the internet!
Tf, Noah ? What are you doing ? Punctuation ? Really ?? 🤦🏽♂️
Here's what I think Mr. Wolfram is trying to say: imagine throwing a piece of fabric up in the air, a limp and sheer fabric, more like silk than canvas. It falls to the ground an a heap with folds and manifolds, all twisted and turned but still one piece of unbroken fabric. Let that fabric be the most common, age old, tried and true axiom system we know of. That piece of fabric is in time and space. As we mathmatically explore where it goes, folds, twists and turns we think of it more as a complete and unbroken piece of fabric than something very irregular. Still, as this fabric is in time and space and intersects it many times it is useful for getting around and plotting out that which it intersects. But it's particular twists and turns might miss something in that physical world it occupies. Another piece of fabric which falls to the floor with different twists and turns will be another set of axioms, which can also plot the same space, but may allow other physical realities or structures to be "seen" or plotted, predicted, etc. If we allow that there is one physical reality, but it's more than our favorite axiomatic systems can work out, then the axiomatic system of each different piece of cloth and how is shaped after falling into place is a description of how it is shaped, described against the background of whatever actually is reality in it's totality.
Explaining this well enough for anyone to take notice is a task beyond my pay scale. But to begin to think about it we might think of our straight number line as actually curved and warped. We might look at oddities, like transcendental numbers, as clues to where bends or warpages are extreme. Non-Euclidean geometry does something like I am suggesting.
What if logics, in which something is or is not, represent places where fabrics intersect. "is or is not" hardly ever, perhaps never happens in the world we know, in the fabric we know, except when we identify an attribute that we know exists in one case but not the other, like on top/underneath, alive or dead, etc. We know these attributes have "is/is not" property because we define them that way. Our fabric is linear and everything changes gradually. But on the quantum scale it is said there are discreet states with nothing inbetween. It could be that everything is linear, but we mistake an intersection of axiomatic systems as jumps in the one fabric we know.
Well I'm just speculating. The most impossible part of the idea of ever really grasping and using alternate axiomatic system is that it may not be physically possible for our brains/minds to ever work out the terrain of any intersecting or overlaid fabrics. In this pessimism I am taking Immanuel Kant's suspicions to the extreme in saying that we can only comprehend or paint a world picture in one way according to rules that are part of the construction of our brain/mind.
+rh001YT
I just thought something similar. As all our perceptions are given by our senses, we can impossibly know what a "thing in itself" is, we can only paint a map, and the map is not the landscape - that's basically what Kant says I think. Now I met people who said: "Maybe an apple is something totally different than what we think it is", and than I answer: "No, an apple is exactly what we think it is - it corresponds to our definition of an apple, that's the way we percieve an apple."
Our senses is all we have to explore and map the world, so if according to our maps an apple is an apple, than it's meaningless to say that it could be something totally different - we only have maps and conventions just to make an apple be an apple : )
So I also think that: "The most impossible part of the idea of ever really grasping and using alternate axiomatic system is that it may not be physically possible for our brains/minds to ever work out the terrain of any intersecting or overlaid fabrics."
Of course theories and conventions (maps) can change, if they remain consistent - but I have a suspect that the mathematical axioms are something more basical than a theory? in the sense that it might be impossible for our brain/mind to ever work out different ones?
Gwunderi25 Yes, I think we agree. You wrote: " if they remain consistent - but I have a suspect that the mathematical axioms are something more basical than a theory? in the sense that it might be impossible for our brain/mind to ever work out different ones?"
It could be that maths are very fundamental, which is why Kant pointed to Euclidean geometry as a possible fundamental for building up reasoning that results in truth. Non-Euclidean geometry is often cited as a refutation of all of Kant, but even he doubted the parallel postulate. Anyway, non-Euclidean geometries can still be expressed with Euclidean geometry but for many purposes it is easier to use the non-Euclidean maths to get the desired shape. Still any non-Euclidean shape is an xyz map, so not really a refutation of Kant.
You may have noticed as have I that one part of what Kant claimed is given undue currency,namely, that our mind paints the world we see. This has led to rampant speculation that the real world might be different and that there may be some solutions there. That then becomes the justification for all sorts of nonsense.
Kant's scary point, as you seem to have understood, is that we have no choice except to accept our mind's representation of the world. It is even due to the way our mind works that we demand when we can have it proof for claims. Here I am not talking about eyewitness accounts of who dun it, but we demand that the laws of physics all work together so that we can have proofs. This can be as simple as government certification of scales to expecting the cell phone to work as advertised when signal is available. And if it is not available, like when backpacking in the mountains, we want to be certain why the cell phone does not work...no towers anywhere...else we want a refund.
So Kant's Critiques were not suggestions that other ways of seeing the world are possible, but that there is in fact only one way of seeing the world, at least in a way that representations all connect to create a somewhat intelligible world where stuff makes sense.
rh001YT Yes, because reality is that which governs. If we perceive incorrectly and stumble upon reality, we suffer the consequences. When our actions meet with success, we have matched reality.
It was a great pleasure to use that incredible set of tools developed by Wolfram Research ("Mathematica") in my undergraduate years as a student of Comp. Science, there in the nineties. I was writing some pieces of code in several programming styles for diverse areas, as Bayesian Calculus, Neural Networks, and things related. Even using those early CLI-based versions which I started with, previous to the first GUI ones, it was a very good intellectual and learning experience. Thanks to Mr. Stephen Wolfram for doing it possible (and obviously, my teachers! :) God bless him.
Every system of thought requires an associative reference frame.
Change the basis, you change the rules of relationships. Take categorization - how objects are distinguished. Similarity is fundamental but ultimately limited and arbitrary.
This was totally clear to me. Very insightful, yet simple.
I think Stephen refers to the specific mathematical system that we are using rather than the general definition of "Mathematics". The 'artefact' is the products of mathematics such as Geometry, Calculus or Topology, but not the methodology of Mathematics.
I think there is only one 'Mathematics' though.
Say if there are other possible 'Mathematics' which are different from our Mathematics, they are in the set of all possible mathematics. This set must be defined and depends on something else. Likely, that 'something' is also mathematics if anything that is depended by mathematics is also mathematics. So there is actually one mathematics in general.
The set of all possible sets! Thanks Cantor your garden is live and kicking!
Excellent. One of my favorite interviews. Sold thoughts from Stephen Wolfram
Fantastic claims but did not hear evidence or sufficient examples to support those claims.
Maybe our brains are evolved to reason in the context of their surrounding physical laws, so the axioms may be limited but at least they aren't arbitrary.
i dont think he implies it was arbitrary at all. in fact i understood the expression "historical accidents" to mean based on circumstance. but it's only because we just happened to have uses for specific tools that we developed them. we would have developed other sorts of tools if we had different needs or surroundings.
Thanks for the note. We lightened the video in RUclips for now, and will look into reposting a better version soon.
The idea of computational irreducibility is an interesting one.. sort of like conservation of energy . Getting to equivalent statements from different axioms is equally difficult.
That was mindblowingly amazing, thank u for explaining it so well!!!!
i don't think that anything he said is well explained
@@cq33xx58 Good for you! It means your brain is working fine. :-)))
Another comment I felt needed repeating:
"See, I think there's some equivocation going on here. Kuhn asks Wolfram whether the other systems of mathematics are self-contradictory and therefore unused and Wolfram answers no, they are self consistent and valid and their disuse is a just a consequence of the contingency of history. Kuhn should have asked, "but what is self-consistency, or validity without reference to our supposedly contingent mathematics?" That question would bring Wolfram back to Kuhn's original. The set of possible mathematics cannot in itself be fundamental if we can say things of them like they are possible, they are a set, they are valid and etc. Our math is the historically contingent instantiation of one self-consistent system out of the set of all possible self consistent systems, but self-consistency, validity, all those attributes which the set of all possible mathematical systems share and by which they count as such systems must be more fundamental and it's these fundamentals that may be necessary and eternal."
I agree this comment, which you say you are merely repeating, is mostly right and very important. But if you are repeating this comment, where is the original? Is it your comment originally also?
@@allisterblue5523 Sorry, I was referring to the comment you were replying to, not your comment, and that comment announces that it is repeating some earlier comment. But regarding you comment, I agree also, that there is in effect room for pluralism in regular mathematics. However, I think at the ultimate level there is something in the nature of rational thought about the purely abstract that is not pluralistic, and you could call that logic perhaps. It is not easy to get at what this is, but I think we should be trying. Of course there are "alternative" and even "deviant" logics so the task is not easy. But I think the concepts involved are the sort that the original poster was referring to.
@@allisterblue5523 I don't think it would be easy to go into this at length in youtube comments unless you really want to. Anyway I took the phrase "deviant logic", with some humour intended, from the title of Susan Haack's book, but her subsequent and classic book "Philosophy of Logics" [sic] might be a good reference if you want to see how a so-called "deviant" logic (deviant from the standard classical first order logic) might lead to something "correct".
@@Benson_Bear I should have given a proper citation. Its from 6 years ago, by Greg Bechtel. You can find it in the comments here yes.
@@Benson_Bear I somewhat agree with Alex here. Though a deviant form of logic might be possible, I guess I will just have to read that book you mentioned to find out!
What I took from this video was that there is a difference between the concept of math and the math that we use as a society. Yes math (when you refer to what we have as a society) is indeed an artifact, HOWEVER Math itself is independent of our perception and existence... therefore can only be discovered and not invented... only the NOTATION and logic behind which is used to describe what little of it we can use and prove through application. Although there are arguments for things beyond this as well I believe that is another topic and it may transcend all possible human comprehension i.e. infinity, 1=2 etc.
This is a move in that direction and I am trying diligently to pursue more information on this subject and it is very difficult. Would appreciate anyone who could shed more light on this topic.
Brilliant thinker. Wish I had such a mind as my teacher when I was a young student of mathematics.
Does Wolfram imply that someone could have invented a different Pythagoras' theorem for a right angled triangle in a plane? It would appear that is just another post-modern attempt at deconstruction that is not very constructive in itself.
Where might one find something like the two-page list of the axioms of our mathematics which Dr. Wolfram alludes to? Such a summary would be very interesting.
This argument seems to just push the question back a step. If we find our axioms are a special case in a much larger "universe of possible mathematicses" [sic], we start studying the more general system. As Luis Dias quipped below, "would this space be an artifact or existent?" Wolfram's argument doesn't seem to show that we aren't discovering something. It may be that the something is just bigger than we thought.
Raymond Brinzer _"If we find our axioms are a special case in a much larger "universe of possible mathematicses" [sic], we start studying the more general system."_
Well, yes and no. The "yes" part of the answer would be that we already do - there are tons of different axiomatic systems in mathematics - each having its own isolated properties and purposes and they are not something you would want to unify as a larger system even if you could, but in many cases they also contradict each other.
The "no" part of the answer is that the vast majority of possible axiomatic systems are completely useless for any purpose and we have no desire to study them.
An important part of mathematics is to create languages that are small and extremely useful for specific purposes. Not to create one gigantic, but practically useless, system that can deal with anything.
Holy crap! that was that a clear and concise description of the "landscape"/"space" of this discussion and a putting-into-context of many of the key arguments that have been brought up in it, like Hilbert's "unreasonable effectiveness of mathematics". I definitely think it's worthwhile to have a formalised meta-discussion of mathematics (in case the usefulness of a "philosophy of mathematics" was ever doubted), to sort out trivial or naive responses to the question from more sophisticated ones. Way to go Prof. Dr. Wolfram! Exciting to think of future research into "possible other maths" and how they might impact society and as a consequence human consciousness.
Excellent conversation!
A wonderful talk, brings the two "opposing" views under a common umbrella, which is innovative in itself. The term mathematics is used to describe two different notions: the total of mathematical theories, a single mathematical theory. The question still remains.
as a language it is ivented but as principles it is discovered.
It will be interesting to someday see what kind of mathematics an alien civilization has. If they have "invented" the same systems we have "invented" perhaps we may conclude that there is a commonality of mathematics that is universal, and hence discovered.
The value of this question is on the question itself rather than the partial answer that was given. What is it that we are doing when we question the nature and origin of mathematics? What are we looking for? What implications any answer are we expecting it to have? What are relevant the facts around this question? What is the truth that such facts bear out? What do we really mean when we try to answer the question about the nature of mathematics? Among such facts which ones are relevant and which ones are not? These are very important questions but I am tempted to believe that any useful answer to such questions will be the result of the coordinated effort over a long period of time of a large community of people.
Out of all the interviews I think Stephen wolfram is the deepest thinker in fact he might have discovered the theory of everything
I think the problem is in the question itself, not how other different people answer it, because whether you think it's invented or discovered, you have a good point. The issue is really just in defining what is meant by discovery or invention? If invention is what is meant by creating formal abstract systems that describe natural phenomena then yes maths and all of science for that matter is invented and the opposite if otherwise, to put it simply.
Formal systems like math do not describe natural phenomena any more than the rules of chess do. Rather, math is a formal extension of the *use* of conceptual frameworks (which are established in human language through practices, i.e. participation in the world) which fix what *counts* as a phenomenon of a certain type.
Before we can have any formal mathematical systems, we must learn to use "ordinary language arithmetic" (OLA), that is, the way children learn to count, add, etc. before they encounter any formalisms. The following applies to OLA and formal math alike, but the emphasis is on the fact that it *already* applies to OLA:
The case that 1+1 does NOT equal 2 is not even conceivable. How would a world look like in which 1+1=3 was true instead? We cannot say because the question makes no sense. It makes no sense because we do not learn what "1", "2", "+", and "=" mean independently of each other. Rather, we learn "1+1=2" as a rule of language, whose application *shows* (but does not *say* and hence not describe) how we perform certain transformations. (Whenever I may say "1+1", I may say "2" instead and vice versa.) "1+1=2" is therefore not a statement of an empirical, contingent fact. It cannot be verified by experiment. It is a conceptual fact, a linguistic norm which we use to classify and categorize the world around us.
We can see in at least two ways that the function of arithmetic is to improve our abilities of categorization:
First, by realizing that creating enriched extensions of concepts like "amount" and "order" enables us to go from "much" to "how much" and from "best" to "4th best".
Secondly, by noticing that we apply "1+1=2" selectively. If two clouds merge into one, we do not take that as a cue to question the validity of "1+1=2" and replace it by "1+1=1". (Although we could use such a cloud-arithmetic. It would just not have the same domain of application as ordinary arithmetic.)
The incredible success of *formal* arithmetic in science (as opposed, for example, to the uselessness of the rules of chess, which on a formal level are just like math) is only partly owed to abstraction. We don't see 2 cars, 2 houses and 2 cats and *infer* or suddenly understand that what they all have in common is "2". Rather, we learn the meaning of "2" as we learn the *practices* of counting, ordering, and adding, which depend on our natural ability to learn to make distinctions (visual, aural, tactile, etc.). For that, we need experiences in and with the world around us.
Once we have learned to employ arithmetic and other mathematical concepts formally, we can use abstraction to talk about certain features of any phenomenon susceptible to numerical treatment. But it is not math itself that is derived by abstraction. Rather, math is a formal language created on top of ordinary language, which in turn is partially constituted by certain concept-establishing human practices, such as pointing at one thing after another while saying "1", "2", "3", etc.
Thus there is no such thing as discovering mathematics (i.e. its conceptual framework). It's invented (and perhaps more importantly, *passed on*) all the way through. Once a formal system has been established, we may then discover (= find, notice) features of it we hadn't been previously aware of. (Theorems, proofs, etc.) But these are not discoveries about what the world is like, but about what our way of viewing the world is like. (Because they are properties of the conceptual framework which we use to *talk* about the world, in the particular ways that the framework's concepts afford us.)
Overly semantical explanation.
All you're saying is that, "We use language and numbers to represent abstract ideas and values.".
Well, no shit. Your assertion that these "formal systems" are not statements of empirical contingent fact, and cannot be verified by experiment is simply wrong. Obviously an experiment that measures anything with a value has to have some form of language to describe it and, if the language is strictly defined it can, in fact, accurately describe what we need it to.
You haven't understood a single word I wrote.
'Overly semantical explanation.'
There is no such thing. Semantics is the study of meaning, so you're saying that my explanation has too much meaning, which is obviously an absurd claim.
'All you're saying is that, "We use language and numbers to represent abstract ideas and values.".'
That's pretty much the opposite of what I'm saying. Linguistic/mathematical abstractions may (partially) CONSIST of numbers, but they are not represented by them. Representation is a relation between certain phenomena in the world and formal constructs in language (mathematical or not) which we can USE for certain purposes.
For example, a map of Europe is a representation of Europe if and only if, and this is important, I know the method of projection employed in the map's creation, so that I am actually able to USE the map for the purposes it was created for. Thus, representation = the map + my ability to use it AS a map. The map by itself represents Europe no more than the surface of a lake represents, by way of reflection, the clouds in the sky above it.
'Obviously an experiment that measures anything with a value has to have some form of language to describe it and, if the language is strictly defined it can, in fact, accurately describe what we need it to.'
A measured value does in no way *describe* what was measured, it IS what was measured. One can then go on to USE that measurement/value to do things one could not have done without it, but it is not the measurement/value alone that affords us whatever new possibilities arise from it.
For example, one could measure how many horsepowers a motor has by empirically determining the maximum force it can exert on a fixed mass. Let's say it is found that the motor has 100 hps, which means we can now calculate what we can expect to happen (in terms of acceleration) if we let the motor exert its pull on any given mass m. Neither the 100 hps nor the expected accelerations *describe* anything. Rather, they *inform* us and help us to *figure out* what we may want or need to do. But there is nothing in the motor which *corresponds* to the value of 100 hps, just like there is nothing in my legs that describes or corresponds to my ability to walk. Thus "100 hps" does not represent or describe the motor. Rather, it informs us what the motor is capable of. It is a (dimensional) quantity, a useful abstraction, ASCRIBED to the motor. But ascription is not at all the same as description.
Mathematical formalisms thus do not represent *anything*, just like the fact that the king in chess moves exactly one square at a time does not represent a chess rule. Rather, is IS a rule of chess. Rules in chess, just like rules in math, when taken by themselves, do not have any meaning. Mathematical rules only acquire meaning in an applied practical (participatory) context, i.e. within *practices* inspired by encountering certain phenomena on the one hand and humans being capable of abstraction on the other. But there is no descriptive correspondence between numbers and anything "out there", just like there is no descriptive correspondence between a hammer and a nail being driven into a wall with it. Mathematical formalisms are *instruments*, not representations.
One can of course USE formalisms to create representations, for example, one could create a mathematical vector model of a building. The model can serve as a representation of the real thing. But, again, the model in and of itself does not represent or describe anything. Rather, the representation is immanent in our USE of the model when we employ it to construct or navigate the building.
"Your assertion that these "formal systems" are not statements of empirical contingent fact, and cannot be verified by experiment is simply wrong."
There is no experiment which can verify that 1+1 is indeed the same as 2, because we do not learn what "2" means independently of what "1" means. Knowing how to use "2" includes knowing that it is the same as "1+1". Just ask yourself what it would look like if 1+1 was 3 instead of 2. You cannot even picture that without changing the meaning of "1", "2", "+", or "3". (As opposed to being able to picture "pigs can fly", which is false but at least imaginable.) All logical/conceptual facts are experimentally unverifiable by their very nature. There is no need to verify them, just like there is no need (and no way) to verify whether the king in chess REALLY moves only one square at a time. (Because that is simply how chess is played. It makes no sense to ask for a corroboration that the king rule is "correct". Someone who isn't sure whether he has learned the rule correctly may confirm or disconfirm that his *knowledge* of chess is in fact correct, but the rule itself can neither be correct nor incorrect. The category of "correctness" does not apply to definitions.)
Using your criteria the only thing that can represent something is the thing itself, but of course, that doesn't give us any meaningful information. That's like saying, "Only Europe is Europe.". It's true, but that doesn't really MEAN anything. You say a map of Europe is useless unless I know how to use it, since it is not, in fact, Europe. Yes obviously, but again, the fact that it is not Europe doesn't really mean anything.
Obviously we have to use formal systems otherwise communication between us would be impossible and all we could do is just observe. You say that a mathematical vector of a building does not represent anything, but it represents the building within that system.
System and terms can be interchanged, but the purpose is the same, communication. Systems exist and they don't do so to BE or REPLACE something, they are simply representations of what we think we are observing.
You keep bolding words like DESCRIBE and MEANING, but you don't define them as it pertains to this subject. What are you arguing? That systems don't exist? That a numerical value has no meaning? Outside of a system it doesn't, but within a system it does. That the numerical value is not the thing itself? Obviously, but who said it was?
If mathematics are historical artifacts created by us, then how come unrelated theories end up been interderivable when they speak about completely unrelated subjects? E.g Lowenheim-Skolem-Tarski Theorem, Tychonoff's Theorem and Zorn's Lemma are all interderivable. If all mathematical entities are historical artifacts, how do mathematicians with absolutely no relation with each other and describing completely different and unrelated things end up with statements that can be derived from one another? This points to a deeper structure of mathematics that can not be explained by the "creation of mathematicians". It is more akin to a discovery of a deeper structure of the world than to a creation in the mind of mathematicians.
But one cannot discover anything, without having some interpretation. And it's precisely our human interpretations that provide math with much of it's content. This doesn't mean that their is nothing real about math (there certainty is), but the human apparatus does not direct access to what is real, and can only construct paradigms. So it is more so, that math largely reflects an approximation of reality.
Actually, it does not. You are talking about mathematical theories that try to explain reality. But that is physics, not mathematics. Example: Riemann's geometry is partially useful to physics in that it can explain most features of the General Theory of Relativity, but the question of "does physical space have a Riemannian geometry" is a question of physics and not a mathematical question. Riemannian geometry as well as Euclidean Geometry are both consistent and true (although possessing different axioms) and they are so irrespective of the question about physical space been actually more akin to one or the other, or even been akin to none of them.
Geometry deals with space. The parallel postulate , is in fact, a mathematical production, which originally involved the broad generalization, regarding the nature of parallel lines. However, this generalization was shown to be incorrect, upon the construction of non-Euclidean geometry. Now a-priorists attempt to defend the validity by distinguishing between 'pure' and 'applied' math, and sometimes euclidean vs non-euclidean geometry, but as Hillary Putnam says , that is simply ad-hoc. The original postulate had no specification that the postulate only holds for 'euclidean geometry' - it was in fact, an over-generalization , a result of our perceptual limitations, which was later shown ( empirically) to be in imprecise.
fergoesdayton
That could not be further from the truth. Hillary Putnam got that wrong along with almost everything else he has defended even before Models and Reality. The parallel postulate was precisely one of the most problematic postulates in the history of Geometry. Ever since Euclid included it in the Elements it was clear that it wasn`t self-evident. Even Euclid had probably realized early on that he could not prove it or proceed without it and for the next 2,000 years many tried to provide a proof of it without any success. The problem was the constant belief that space had to be Euclidean (a question which already was beyond mathematics), but there was no proof of inconsistency if it was rejected. Finally, there was some light at the end of the tunnel.
Poincaré (one of the fathers of non euclidean geometries) writes in "L'expériment et la gèometrie" that "no possible physical experiment can make the physicist abandon Euclidean geometry as the geometry of our physical space...(because)... it is not on the basis of experience that Euclidean geometry or Non Euclidean geometries can be refuted... Experiments can only teach us how the different objects relate in space, but not how they relate to space, nor how the different parts of space relate to each other nor, thus what the nature of physical space is".
In fact, there are many properties usually attributed to Euclidean Geometry that do not correspond to our perceptual Space, namely: continuity, infinity, isotropy... etc... He concludes: "physical geometry is not an empirical science, since in such a case it would only be approximate and provisional." (1905)
Even before Poincaré many other mathematicians had begun to doubt the truth about the parallel postulate and some of them, like Husserl even wrote early on (1892-1901) that the traditional conception of Euclidean geometry as been the same as physical space had to be revised since "on cannot proove that with respect to any straight line one can trace through each point only one straight line not intersecting it; or that parallel segments between parallels are equal;... thus, briefly, the parallel axiom is not true." In fact, in 1897 in a letter to Natorp, after arguing on behalf of the existence of Euclidean manifolds of more than three dimensions (in which space and time are subsumed under more general concepts) Husserl writes states that the thesis about the Euclidean structure of physical space is an unfounded hypothesis made by NATURAL SCIENTISTS (caps are mine), which can only be founded empirically. So it is perfectly clear that even for XIX century mathematicians and philosophers the three dimensionality or n-dimensionality of space, as well as the Euclidicity or non-Euclidicity of it, were EMPIRICAL ISSUES, not MATHEMATICAL ISSUES. And that has nothing to do with "experiments" showing that the parallel postulate is false. Because that has simply never happened.
PS
Another consideration apart from this one is that, contrary to what Putnam believes, there has never been an instance of a refutation of a purely mathematical or logical theory by experimental means. But that´s another, more broader topic, which I do not wish to entertain right now.
biokant Poincare's use of 'empirical' is different than Putnam's. In Putnam's quasi-empirical view (which generally rejects the a priori - though supports the 'contextual a priori'), mathematics is the result of 'empirical research'. And it is this research which ultimately lead to the rejection of the parallel postulate.
But categorizing the issue of the Geometry/parallel postulate, as an 'empirical issue' is just a way to throw dirt on empiricism. The parallel postulate is unambiguously born from a priori (rationalist) intuition - and not simple observation. Had the parallel postulate been defined with exclusive respect to 'euclidean space' - then Husserl would have had a point. Instead, the case is that the parallel postulate, assuming Euclidean space, over-generalized the nature of parallel lines.
Wolfram is brilliant!
Wolfram's response is self-contradictory. He says our math is an artifact ...but then explains how our math is just one amongst a huge set of maths that exist in a concept space of various diverse maths.
One way to resolve this contradiction is via an ontology where invention is *the same as* discovery. That is, the mind that discovers is actually inventing that discovery. It is only believed to be discovery because the self's awareness of its role of creation has been suppressed.
You've misunderstood his words. He CLEARLY states that math systems are essentially CREATIONS of axiomatic systems that are simply TOOLS to allow us to get something done. When we humans one day CREATE a better tool, we will leave the old one behind.
Ex. Is a HAMMER a fundamental reality in our Universe? Were we created simply to bring about the HAMMER? Isn't computation essentially a hammer system in mathematics? Mr. W also CLEARLY states that there MIGHT BE a system of axioms we might create that IMMEDIATELY reveals the outcomes of that system. Ergo, no need for the hammer anymore. It was NEVER needed, except as a stepping stone in our brain development.
Don't be a prisoner of your brain's inability to think outside of its artificial realities created to satisfy your desires, expectations, etc. You are ruled by biases, NOT fundamental realities of the real world. You CAN NOT know such things. The brain is a tool, NOTHING MORE. Think in terms of this saying: If your only tool is a hammer, everything looks like nails. Let go of the hammer/math.
We can only understand a type of mathematics based on the axioms that we can perceive as humans. Wolfram is suggesting there are other axiomatic systems which may not be apparent to us. Thus, our Math is not so much fundamental to reality, though indeed, in some respect, and to some extent, it may represent real things. Furthermore, empirical artifacts of math have already shown that some of our most intuitive beliefs about the world, have been defeated (eucleadean geometry, boolean logic, constancy of time). We don't have direct assess to truths.
Is it not utterly surprising that the rules of complex analysis,like conjugation, Harmitian operators etc., describes exactly, quantum states,and conforms to physical reality that existed outside the mind. How true "invention is the same as discovery". Socrates was hitting a prime cord when he claimed that Meno's slave new higher mathematics. Mind of Meno's slave,is programmed with god's knowledge-!!!
That's not a contradiction at all. From an ontological perspective he's not saying that these other possible math systems exist, but merely that they CAN or COULD exist if we had chosen differently.
Our mathematics does not exist in a platonic ideal sense to be discovered. It developed over time, evolved if you will, was invented from our simple intuitive understandings of arithmetic and geometry through the methodology of theorem postulation and proving.
In point of fact, mathematics is not a single monolithic entity, but rather a set of closely aligned sets of axioms and provable theorems and annoying paradoxes. Each theoretical mathematician tries to come up with a more fundamental set of concepts and axioms or tries to avoid paradoxes or places restrictions to make math more computable. Not all such innovations become generally accepted, but if the innovation is found to be useful in some field it will spread.
Damian Torn
You are right, the brain is a tool. It acts as a "central processing unit" that handles a number of "inputs", producing a number of "outputs".
Is mathematics a tool only?
Consider the conclusion that "rate of change at every point of an exponential function have the same value", or "numbers are operators", these are truths that would have remained unknown if maths was all about "tool", is it not true?
The word mathematics comes from the Greek μάθημα (máthēma), which, in the ancient Greek language, means "that which is learnt", "what one gets to know," hence also "study" and "science", and in modern Greek just "lesson." The word máthēma is derived from μανθάνω (manthano), while the modern Greek equivalent is μαθαίνω (mathaino), both of which mean "to learn." In Greece, the word for "mathematics" came to have the narrower and more technical meaning "mathematical study" even in Classical times. Its adjective is μαθηματικός (mathēmatikós), meaning "related to learning" or "studious", which likewise further came to mean "mathematical". In particular, μαθηματικὴ τέχνη (mathēmatikḗ tékhnē), Latin: ars mathematica, meant "the mathematical art".
In Latin, and in English until around 1700, the term mathematics more commonly meant "astrology" (or sometimes "astronomy") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This has resulted in several mistranslations: a particularly notorious one is Saint Augustine's warning that Christians should beware of mathematici meaning astrologers, which is sometimes mistranslated as a condemnation of mathematicians.
The apparent plural form in English, like the French plural form les mathématiques (and the less commonly used singular derivative la mathématique), goes back to the Latin neuter plural mathematica (Cicero), based on the Greek plural τα μαθηματικά (ta mathēmatiká), used by Aristotle (384-322 BC), and meaning roughly "all things mathematical"; although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after the pattern of physics and metaphysics, which were inherited from the Greek. In English, the noun mathematics takes singular verb forms. It is often shortened to maths or, in English-speaking North America, math.
Exceptionally well explained
Mr. Stephen Wolfram is definitely a genius!
*genius.
dlwatib
I have a shoe in my mouth as I write this reply. Thank you for correcting.
Good talk, the accumulation of history in practical usage has led to the next big realization that an expectation of "speaking in math" will eventually solve all speculation, but then came QM with unpredictable properties because not everything is measurable. The cosmological equivalent of a measuring device has to be tunable to the subject of study, and so we believe in only a few percent of what can be inferred to exist.
Does Plato's view of mathematics a mere illusion or a hard fact? I don't think this talk really touches much on this question. It has implications that mathematics is invented but it would be perfectly valid to say that there is an ideal mathematics and we only know the shadow of it cause we are blinded by so many constraints while having the exactly the same view with Stephen Wolfram. But I would have to add that I do not fully understand how he came up with all the possible mathematical systems so it is hard to say anything with definite certainty.
Led to wonder if the thing that's special about our mathematics, in contrast to the universe of all other possible axiomatic systems, is that the axioms of our mathematics are largely based on human experience of the meso-scale natural world. Taking a quick look at the Zermelo-Fraenkel axioms (thank you Wikipedia) this seems to be true; if you jumbled those axioms at random, you'd end up with a bunch of rules that have no connection to the physical world that we experience.
It’s only a language. As is all forms of communication that is used to describe anything.
This was a dream shattering and a mind opening, at the same time!
Wow, never thought that different variations of axioms can derive different mathematics..
I wish he'd given an example of a couple different axioms from which a different math could be built. His explanation was 100% abstract, so I never really knew if I was actually understanding what he was trying to explain.
@@Wonderboywonderings I'd say the most famous example of a different set of axioms deriving different mathematics is non-Euclidean geometries. In his book of Elements, Euclid enumerates his set of axioms and then compilates a bunch of theorems with their proofs. Centuries later mathematicians started to question if Euclid's fifth postulate was really an axiom or if it should be proved and after trying and failing to prove it some mad lads denied it and started building theorems without it and such an effort was valid and ended up in what is now called hyperbolic and elliptic geometry. In elliptic geometry for example, triangles' internal angles always add up to a number greater than 180° instead of exactly 180° as it happens in Euclidean geometry. Guess what happens with triangles in hyperbolic geometry.
mathematics is not fundamental to reality. it describes reality. descriptions of reality are not fundamental to its existence. the universe does not have to carry the one in order that x happen.
KEvron
What "things" in our reality can mathematics not model?
What Wolfram was stating is that there are alternative systems of axioms which are distinct (ie. you can prove different theorems) from the standard ZFC system that most of our current mathematics is founded upon. The existence of these alternate systems makes ZFC a historic artifact. A possible consequence of their existence is that they may be able to model real-world phenomena that we currently can't, such as consciousness. And to directly answer your question. If we include mathematical proofs as part of our reality then mathematics can never model a ZFC proof of the continuum hypothesis.
it might have been in reference to fractal geometry which is easily found in nature, but is hard to define using euclidean geometry. usually, we need to move away from pure maths and rely on some sort of recursive computer algorithm for calculations or graphic renditions. in our "current maths", we have developed relevant descriptors like the fractal dimension, but this is just an index to quantify how unlike our usual shapes a geometry might be. if the building blocks of our maths were different, fractal objects as found in nature might be no more complex than our elementary shapes such as triangles or circles.
I should add that idealization, dreaming, and wishful thinking all have practical considerations going for it. It raises achievements in many disciplines, paradoxically, by setting unachievable goals. And sometimes, it's what gets us up in the morning. (Especially coupled with zone-out substances such as a hot cup of java ;))
As with most of these discussions, it seems to focus on the linguistic/symbolic description of mathematics rather than the fundamental underlying question of whether mathematics exists independently of human cognitive awareness. Of the many "mathematicses" possible, would any of them change the value of Pi or would they just use different words/symbols to identify it?
Is Wolfram proposing that there is a "landscape" of formal systems of math and logic each with a different set of axioms? If so, then within that landscape we don't know where our particular formal systems are situated, nor can we know how many diferent axiomatic formal systems are possible?
I know it's a broad generalization, but I find most mathematicians say math is discovered, and most scientists and engineers say math is invented. Wolfram started as a physicist.
That's because mathematicians are the only people who truly understand math.
Best explanation about Math Invented Vs Discovered
the nature of our universe is, essentially, creative. like a blank page, many things can be drawn upon it. imagine being able to draw on an infinite page, with infinite imagination... what wouldn't arise?
Oh, what I say is of no importance, it was just an opinion. Who knows. I didn't mention randomness... But there may be something worth looking at in that word. Seems like the randomness is used in several ways which don't mean the same thing.
I meant the word randomness causes confusion due to several incompatible definitions.
mrtn zrzr everything would arise
@@OfficialShadowKing there is no such thing as randomness, only cause and effect
A very different view point that mathematics is an artifact. I remember textbooks on mathematics only showing the polished and beautiful results or theorems of mathematics and hiding the complex and rather hard and not so beautiful scaffolding that is the cause of the result. So maybe as he says text book mathematical results are just artifacts. Further, I like his point that there are Infinitely many mathematical systems with infinitely many unprovable results. Also, I like his point on reinforcement loops between maths and physics that can explain "incredible effectiveness of mathematics". A good and demystifying answer which in an unlikely manner has reinvigorated my mathematical pursuits.
Godel's incompleteness proof...even when math is awesome, it is more beautiful, since we dont know yet everything about it, and probably we will never have.
On this question, I prefer this guy's answer to this question to the one given by Sir Roger Penrose.
he still failed to convince me that the 'artifact', the mathematics human 'invented' or in his view one possible mathematics isn't just truly a discovery of part of the entirety of mathematics. in a word philosophically(or logically, which as part of maths may be hardly adequate for this topic any more), there's no way to affirm those single pieces of mathematics don't form a unified integral mathematics. and he was also far from convincing me that mathematics is not objective and has its objective existence.
I agree that between mathematics, our physical world and our mind, there's something very deep about it. it may be that in the end of the day, when this ultimate problem is solved (if by any chance it is possible), maths turns out to be something non-objective and be immersed with the ultimate real 'physical law' and become one, but i guess this is way beyond what he said here and so far we have discovered.
Mathematics is so much about making aesthetic decisions. That aesthetic represents something real, objective, and divine.
But what about paraconsistent logic? It seems that logic still matters.
So, it seems even if you say we're inventing math because we just conjured up some axioms there are meta-statements which are discovered and true of all formal systems.
Many claim that our mathematics is a historical artifact, a social construction, etc., but I’m yet to see a functioning “constructed” mathematics, that doesn’t borrow anything from our own (you could say, is “orthogonal” to our own) and has a similar explanatory power.
Mathematics was discovered. Pick any example you like and regardless of who discovered it the principle would be exactly the same. The angles in a triangle. The times tables, etc.
Discoveries don't change, they are set in stone.
Inventions, on the other hand, vary from one another.
Did the Wright Brother's discover flight? Just as they "discovered" flight someone "discovered" mathematics. They could've flown in many ways. But they used nature as guide. Just like the first people who invented/discovered maths did to invent whatever system or problem maths could solve for them. That's what Stephens getting at. We invented ways of using math to problem solve the issues around us. Had the problems been different... Harder, easier, more involved in one subject etc, would mathematics be fundamentally different as well?
Finally, one scientist who know what math and science is.
Bertrand Russell said "even in the remotest depths of stellar space there are still three feat to the yard." I believe that mathematics and logic are absolute.
clovis Frank I wouldn’t consider the conversion from feet to yards as not mathematics since that’s a human definition.
As an example of what he’s talking about: That two parallel lines will never intersect is only possible if space isn’t curved
Wolfram seems to miss the fact that some mathematical systems, of which logic is an example, are used in everyday life because they play a valuable role. Arithmetic is the same, but number theory isn't (though it does play a role in technical applications such as encryption, but it does not enter ordinary language the way that arithmetic does). So it's a bit misleading suggesting it is 50000th on the list of possibilities.
However, if we regard logic as a problem-solving device, then different kind of problem might involve different logics, but I think this would be a different kind of beast to propositional logic, predicate logic etc.
Examples of axioms would be helpful to illuminate this discussion.
In OTHER words, his elaborate answer can be reduced as mathematics being invented, developed from historical axioms of our civilization. This is perfectly reasonable, and I don't see a need for an argument. I liked this.
I feel like they ask this questions to people who are too abstract to see what’s right in front of their faces.
They are focused too much on the symbology to simply recognize the concepts often are and have been discovered in parallel by multiple different cultures on our planet without an exchange of information.
The babylonians, Chinese and Pythagorus all discovered the principal we call the “Pythagorean Theorem.” Yes, you could come up with a different way of notating the Hypotenuse^2 = Perpendicular^2 + Base^2, but the principal works the same in binary, hexadecimal or by throwing beads in dish.
The principal is therefore fundamental but the notion is arbitrary.
Well, that wasn't the answer I expected. Very thought provoking. It's been a while since I took symbolic logic, but this makes me want to take it again. Obviously I'm not in my right mind.
these are amazing videos, the audio is horribly low am afraid
Ok.... so... what the hell is the answer?? Anybody get that?
There is no definite answer, it's a topic of Philosophical debate.
Ahh... so... can God make a rock , but in order to make it he has to make half of it first and there are an infinite number of half steps, but if he goes back into the past and kills his grandfather before his father was born...
Many of these just seem retoric that ends up to be a bit pointless.
There are things which we can neither prove nor disprove, and those things are the subject of philosophical debate. Many of them will forever remain mysterious. For example many questions which start with "Why?"
Eg. Why does mass have the property of Gravity? Science can describe how gravity works but doesn't tell us why it is so.
Incidentally that Zeno's paradox has got a solution since 1/2 + 1/4 + 1/8 + ... to infinity actually has a finite sum it equals 1
He is saying that within an axiomatic system there are things that are unprovable by that system therefore we need to explore other systems to explain those things which our current system is incapable of proving.
Well, in that case, he could simply have started with "I don't have the answer, furthermore, bla bla bla..." :)
So, basically, the answer to whether mathematics is invented or discovered is, "Yes."
Godels theorem is most fascinating! Wolfram is great at articulating this wonderful topic!
Excellent & fun video! Thanks! It really makes me think...If math is invented and can't explain everything in the universe does it mean we have to replace math with some kind of "computer language of the universe"? I mean, computer games have a physics engine but also have a lot of "IF-THAN" code (or bugs) that can break the rules.
It is more than an artefact, the entire modern society is based on mathematics, from simple vehicles to powerful computers, economics to mathematical models of many subjects (music etc.).
If there's anything to conclude from this spiel it certainly isn't going to be obvious without some guesses. I'd suggest shooting at the dark, as Wolfram implies, and trying to come up with new questions and new ways to find the answers, and see if any are totally incomprehensible under out set of mathematical axioms.
The lighting is very low in this video. Maybe it can be edited and re-posted.
I'm afraid Platonic Idealism died with Aristotle. Extrapolating to perfect forms (simplifying by dismissing details;seeing something's ideal form) is something that comes naturally to us and that we do all the time. I think it's a part of our make-up that we need in order to function at all.
It's the sort of imagining that we apply to what-if scenarios, sometimes going against what we actually know will happen. I suspect we need this kind of dreaming or wishful thinking to plan ahead at all.
I could listen to him talk about this for two hours... still I am misunderstanding because most random symbol sets quickly fall into either lacking breadth or depth because of self contradictions (I guess one could create another symbol set like a bridge and just add a dimension... still we would judge this symbol set with our perception of complexity, and conciseness etc... like programming languages made from random symbol sets that are actually useful.... seems like millions would just hit the recycle bin really quickly. What am I missing?
Who is the interviewer?
Please can this be included in the description.
Where can I get this with english subtitles?
+vincentmack37 Click CC on the video
This will almost certainly get a tremendous amount of criticism (much of which will be motivated from an almost teenage-like angst which I find curious). It seems to me that mathematics represents the nature beyond nature. Not so much the physical universe itself as the underlying principle. Math is very much the scaffold upon which the physical universe, it would appear, seems to be built. I would say that from a Judeo-Christian world view, one would benefit by considering mathematics to be a manifestation of what is referred to in the Bible as ‘wisdom’. In a Greek or Roman influenced culture, one might call this ‘reason’. In Jewish and Christian tradition, wisdom is much more than ‘making good choices’ wisdom is a characteristic of God, part of his nature that is ‘Hevel’, or enigmatic. Something eternal and something we don’t fully experience or understand. Wisdom is a trait of God that we see as a shadow, or the way a two dimensional being would see a three dimensional object as it passes through a two dimensional plane. According to various psalms, proverbs, and other Jewish writings, God established creation (or at least the physical part of it that we interact with) upon a foundation of wisdom. Wisdom is often portrayed as a person, an entity, who was with God and through whom all things were made. Christians would say that ‘the word was god and the word was with god and through the word all things were made.’ Christianity goes further as it is revealed that ‘the word became flesh and dwelt among us’. Basically, as near as I can understand, God, who is wisdom, entered into his creation as a man (Christ). This is all detailed in the 66 books of the Bible, but I digress. What I posit is that Math/wisdom/reason are the same ‘thing’, or at least manifestations from a singular origin that is outside of the physical universe that is itself the source, or origin, or scaffolding of our universe. I wonder if mathematics/reason/wisdom has ‘leaked in’ to our universe in much the same way it has been theorized that gravity has ‘leaked’ in. It is foundational and seems somehow separate at the same time. Anyway, just some random ideas here. Regardless, it seems fallacious to me to assume that the physical universe is ‘it’, that there is nothing else. Not that one should assume otherwise of course, however, we must conclude that the pattern of reasonable thought and understanding to which we have become accustomed are infallible. I think it reasonable to consider that science as a framework for ‘good and productive’ thinking best practices must have its limits. The scientific method itself relies on the preservation of information and cause/effect relationships which begin to break down when we consider the singularity and origins of the universe. If the Big Bang is a door, and cause and effect relationships exist as they do on this side of the door, then if there is something beyond that door whose to say those relationships exist similarly on the other side? Language itself cannot communicate ideas effectively and we begin to realize that there is an end to our understanding. As such, unless the universe itself is infinite, then there must also be an end to scientific inquiry. But math, that may actually extend beyond the door. Or... maybe not.
Mathematics is discovered. Even given that the mathematics most explored by mathematicians today is based upon one of many possible sets of axioms which themselves result in many possible mathematical "universes," every such mathematical universe is nevertheless implied by its respective axioms. Mathematicians explore the implications of the fundamental axioms of a mathematical universe or even the nature of all possible mathematical universes themselves. In either case, mathematicians are discovering the direct implications of the axioms of a particular mathematical universe, or discovering the possible variations of mathematical universes based on different axiomatic structures. Mathematics is not invented, but discovered.
So are you saying that different sets of axioms are fundamental aspects of our and any other universe and that our stumbling upon a specific one is therefore a discovery? I tend to view it such as we as a species are able to identify and define various axiom systems which themselves are not fundamental parts of our universe, and in that way mathematics is an invention and not a discovery.
Edit: if somehow mathematics was a fundamental truth to our (and any other) universe then it would be a discovery, however if it just a methodology or system that we have defined that would make it an invention.
I agree. Any being that conceives three lines form a triangle in plane will deduce all the same facts. This may have already happened.
+Max Schwenke
cope
There was a schism in mathematics in 1930. People who follow the schism believe that mathematics is made up. Their logical system has fewer contradictions than platonic mathematics, so there's a good chance they are closer to the truth than the mainstream mathematicians. Some well-known proof systems came from that community. Their teaching of mathematics makes more sense.
Are you suggested that if an alien species came to Earth, they may have never seen Math. They would look at all the startling coincides and predictions Math is able to predict about the universe and would say to us, translated, "Neat! Who would have though there was order to the universe!"
Mathematics is the ultimate refutation of the materialist creed.
Once we humans redesign the brain for Homo sapiens 2, perhaps novel mathematics will be developed.
Certainly after strong AI is enabled and loosed upon its runaway course, new mathematics will be developed, one after the other at shorter and shorter intervals, though they will become less and less understandable to mere humans.
i don't buy this idea that strong ai will be *so* transformative; humans do things beyond individuals all the time through division of labour.. shared knowledge.. books, and now the internet. (i think we have much further to go with that.. "the network is the computer", including human brains)
He probably says quaternions. If you're interested in generalized number systems, you can also look up clifford algebra :)
The circularity conjecture is correct.
someones been reading his ludwig wittgenstein. IMO the existence of a space of possible mathematics has no bearing on physical reality even though they would be self consistent. the way objects come into existence & interact in the physical world is the basis for their quantitative treatment and abstraction thereof but the question of the place of mathematics in a possible space of axioms is different from its universality
I don't follow you, Wolfram seems to be expressing a Platonic description of Logic, while Wittgenstein was quite hostile to Platonic ideals.
what are the things hes talking about that were unable to model ?
I think all these mathematics systems have to collapse to a 'kernal' that all self consistent mathematics have.
This kernal is discovered not invented. And it is embedded in physical reality.
It doesn't matter if we use a given "construction", we are still doing discovery.
To make an analogy, there are lots of ways to draw a map but how you draw the map doesn't change the structure of what you are mapping.
I think he's just suggesting that a different evolution of mathematics, say not based on ancestral knowledge like farming, would produce different results in terms of it's effects on furthering more mathematical discovery. If we had or hadn't focused on a particular area of maths or used it for tasks abstract to us now would it be fundamentally different? Would we have discovered more or less? Would it be more efficient? In what way? How? Why? Did it breed better strategies to answer our hardest questions?
It wouldnt be the same maths we're use to but it wouldn't be unintelligible. As it too is based on the same nature.
"Is there something special about "our" mathematics? I don't think so." Well Stephen, the actual greatest mathematicians do think so.
I wonder if Wolfram's statements would hold if we change Mathematics for axiomatic systems.
BEST MOST PROFOUND ANSWERING TO THIS QUESTION YET!!!! "HISTORICAL ARTIFACT"
Very interesting, up to this point in my life, I always thought there must be something "inherently" true about Mathematics; or at least mathematics can indeed lead us if not to a higher truth, at least to a higher understanding of of most phenomena without which it wouldn't be possible.
I can't accept what Stephen Wolfram says at face value, however, it's something to think about .... a man-made artifact stemmed from axioms partly found by accident.
===================================
A relevant question might be: "Would other mathematical systems have survived and evolved as the one human has created so far?" ... "Has the mathematics we know come out of a Darwinian competition with flying colors?" ....
interesting indeed.
A. M. Goudarzi _"Would other mathematical systems have survived and evolved as the one human has created so far?"_
Well, humans have created many mathematical systems, so it's hard to understand why you represent it as a hypothetical. In fact, many alternative forms of algebra are being used in specialized areas. Modulo arithmetic is one backbone of cryptography for example.
Gnomefro Thanks for your comment. Modular arithmetic is indeed one strong tool and widely utilized in cryptography. However, modular arithmetic is not perceived nor it claims to be a substitute for the whole "mathematical system" Wolfram is referring to.
If maths is invented and not discovered, then it *seems* like all mathematical things can be described. But some mathematical things cannot be described, since the set of all numbers is uncountable but the set of all describable numbers is countable (since the set of all English sentences is countable). Contradiction?
He seems to suggest that the initial axioms of math were chosen at random and other axioms would have done just as well. Surely the initial axioms of math get altered if the system fails to be predictive of natural phenomenon. What we have is an artefact but an evolutionary model that has been constructed over time. In the same way we can think of all manner of alien life forms that bear no resemblance to life on Earth. This doesn't mean that they would all be just as likely to exist. I can imagine a creature that is nothing but eye balls, doesn't mean that is just as likely to evolve as a sparrow.
How can we find new mathematics if we teach students fully the mathematics we know?
You have misunderstand what he says. Studying propositional calculus helps to understand what he mean.
Definitely very interesting. I'm just waiting for the day when that which is mathematically irreducible becomes reducible. I am pretty sure that, given the right formal system of logic, all life processes can be modeled mathematically. This would allow us to predict all things with relative degrees of certainty--namely human behavior, animal behavior, social patterning, etc.