Math Dispute Breakdown: In Defense of Analysis

[@Rodhern]

Very well explained Daniel. It is interesting to get to know your thoughts on the topic. I am particularly happy that you are willing to give concrete examples of 'properties'/'principles' that may not be important to you and others that you deem valuable.

[@toddtrimble2555]

Well, it's great that you took up the cudgels with Wildberger, and your report on those discussions here is couched in super-polite terms. Your counterpoints seem very sensible and pragmatic. I've listened to a number of his videos, where he talks about these "crises". Sometimes it takes the form that there are these big gaping holes in (say) calculus textbooks which, after going down the line and consulting more advanced texts, he says are never patched up. But that's just not true (I'd use stronger words, but this is a family channel), as one can see in the huge and ever-growing compilations of theorems that have been formally verified (whether in Lean, or Mizar, or Coq, or whatever), going right down into the bones of axiomatic frameworks. (Mizar for example uses ZFC, maybe with Tarski-Grothendieck universes). A spectacular example is Tom Hales's computerized verification of his proof of Kepler's conjecture. It sure seems that his objections are philosophical, not mathematical, in which case: shrug. There is no crisis he can point to in the sense of a contradiction emerging from some standard axiomatic framework such as ZFC. These frameworks are robust and appeal powerfully and coherently to people's realist conceptions of mathematics. Or, along more formalist lines, one could shift attention from worrying whether the axioms are "true" or whether infinite sets "really exist", which might be a fruitless argument, to whether we indeed achieve certainty that our chains of inference are correct, whether mathematicians are playing by the rules of the game of mathematical reasoning. To repeat, here I think we have growing and powerful evidence from the world of formally verified mathematics that there is no crisis whatever. I've not seen him respond to this point.

[@mgostIH]

What do you think of a middle ground where infinite sets are allowed, but constructions can only happen iff computable? For example the natural numbers would form a well defined set because they are defined inductively, while real numbers would work a bit differently, I could define that x^3 - 15 has a real root because there exists an algorithm working in rational numbers that can approximate the root as much as we like, while rejecting the notion of uncomputable numbers, the axiom of choice and some even go as far as rejecting the law of excluded middle. I realize there's quite some issues in adopting theories so strict, computable numbers would be a countable set so we'd have to redefine measure theory and a lot of functional analysis with it, but I feel like there's a big distinction in the philosophy of computable vs uncomputable worth investigating further. This said, I enjoy your exploration of niche topics and will look at the debate I missed!

[@mgostIH]

@@markv785 I'm not not asking you to 👀

[@derendohoda3891]

Though the computable numbers are a field, they don't have most of the properties needed in analysis that the real numbers do.

[@roderictaylor]

" "What do you think of a middle ground where infinite sets are allowed, but constructions can only happen iff computable?" This sounds like it might be constructivism, where for example a real number is defined to be an algorithm producing a sequence of rational numbers, together with a proof that the sequence converges in some sense. In constructivism, there is no distinction made between a proposition being true, and a proposition being demonstrably true. I prefer classical mathematics, but constructivism is interesting and mathematicians have done serious work on it. I enjoyed reading Errett Bishop's Foundations of Constructive Analysis.

[@elcapitan6126]

call it personal taste, but i like the algebraic / combinatorial approach to building mathematics. it's constructive, concrete, and doesnt require a range of adhoc axioms to formalize. it's really easy to think about too!

[@nabla_mat]

I saw your conversation -- which I liked a lot -- and regarding the issue of "e + π + √2" I think you responded very well when you pointed out that you can readily calculate two rational numbers that limit that real number. I really think that is the Archimedean spirit behind Cauchy's & Co. work, and that is a better solution to the problem than just stating that those numbers don't exist.

[@SavaConrad]

But that’s not the actual real number

[@SavaConrad]

That’s the point. You can’t give me the actual square root of 2

[@Dominic_Muller]

​@@SavaConrad be careful not to confuse the representation of something with the thing itself. Both "the square root of two" and represent the exact same thing; the same concept. The square root of two is a real, actionable thing, and generating its decimal *representation* is no requirement for existence or sensibility. Math is not merely a manipulation of formal symbols on paper, it is reasoning about the concepts to which those symbols refer. Any line of thinking which confuses the two is not mathematics, and is deeply flawed in thinking and reasoning.

[@tomctutor]

@@SavaConrad I can, its a real solution to x^2-2=0, the curve crosses the real axis at that value, so that value exists at that point!

[@theflaggeddragon9472]

Some really interesting developments in the direction of "constructivist" or "finiteist" math are a recent push by Kevin Buzzard (and many others) to formalize mathematics, i.e. to put mathematical definitions, theorems, and proofs into computer checkable form. This has the formalizers encountering foundational problems constantly. In particular, one find that much of the "formal" math that is done in university is really highly informal and sketchy (thousands of isomorphisms, identifications, compatibilities are completely ignored as working that out by hand would be tedious and uninteresting). As such, sweeping a myriad of details under the rug isn't a serious problem for math research (mathematicians are experts at separating the "real content" of a proof from the "routine junk" and ignore the latter), but there could be many advantages to having formalized a large swath of modern mathematics (see Kevin Buzzard's recent article for some motivations). I think this formalization of mathematics can neatly reconcile the finitist view with infinite/nonconstructive/noncomputable things. A computer can check all of the logic of real and complex analysis and find that the theorems are correct, and note that anything proved constructively can be applied to specific, computable examples, and that those examples will simply time out when the computer runs out of memory. What's proven constructively can be worked with explicitly, and what's not proven constructively is still accepted logically by the proof checker.

[@jessewolf6806]

Sketchy?

[@KarmaPeny]

I think that Norman was talking about a sequence of better and better approximations to a real-world value as opposed to a 'real number'. Norman doesn't accept that real numbers, such as the cubed root of 15, actually exist. And if they don't exist then we can't have better and better approximations of them. For example, if we plot a square shape on a computer screen then we can't plot a 'perfect' diagonal of that square because the screen consists of a finite number of smallest parts that we call pixels. In the real world we might have a unit cube (of 1 inch per edge) but can the diagonal on any of its faces measure the 'perfect' length of the square root of 2 of an inch, or does real (& imaginary) space itself have to be granular, just like the finite amount of pixels on the computer screen? If you want a major crisis then look no further than the concept of an infinitely small point on an infinitely thin number line. If the square root of 2 can exist as such a point then so can all the other points in the sequence 1, 1.4, 1.41, 1.414, 1.4142, ... And if all these points exist as static points on the number line, then how can there not be a last one of these points before the √2 point? This problem of a last point is, for me (& many like me), an obvious contradiction to the concept of 'infinitely many' which requires there to be no last point. But you and your fellow mainstream followers simply refuse to accept this as a contradiction. You can say things like "there is no last point" and through your carefully worded system of (limit) definitions you avoid talking about this problem. But some of us don't accept that side-stepping the issue through word games is an acceptable solution. Since we are talking about static points that supposedly already exist on the line, not a process of choosing a point, then it doesn't make sense that there is not a last point. Some of us consider this to be a major crisis even if you don't. This brings us to another key point that you mentioned, which is how can we decide which opinion is correct when discussing foundational issues. If I claim something forms a contradiction and you say it does not then who decides? If maths is a science then we should examine empirical real-world things and use the approach that best fits the evidence. But if it is a belief system, like a religion, then we should all fall in line behind the consensus of opinion of the religious leaders. Sadly maths matches the second of these approaches. You appear to be choosing to dismiss these foundational issues on the basis that maths is 'successful' and it works. There are many cases in our history where fundamental theories were thought to be correct because they appeared to work such as a flat earth, Aristotle's theory of free-fall, and the geocentric model of the universe. The difference between these and mathematics is that these other things relate to the real-world rather than things in our imagination. So we can modify our theories as more evidence becomes available. The current mainstream approach to mathematics is akin to a religious believe because it is not evidence-based. It relies on a consensus of opinion, reinforced by strong indoctrination through mathematics courses in academia. You appear to be concerned that any overhaul of maths foundations would mean that all the delightfully complicated things in mathematics like infinite Fourier series would be thrown out with the baby's bathwater. But perhaps they wouldn't; perhaps they would simply be described in a way that could relate to physical reality rather than containing references to arguably unimaginable concepts such as actual infinities. When it comes down to it, all of maths is just a finite amount of symbols being used by a finite amount of brain cells, and so ultimately everything about it is finite. We just like to pretend we can conceive of the infinite, when the truth is we are not really doing that at all; we are deluding ourselves. It would be nice to remove this delusion from the language of mathematics.

[@craig4320]

I hope someone is working on the overhaul of maths foundations that you mentioned.

[@KarmaPeny]

​@@craig4320 I don't think anyone is working on the overhaul of maths foundations. I do my best to encourage an uprising via the videos on my channel. I've just released a new one about 0.999... but, like most of my efforts, it has not attracted much interest. People have been complaining about these issues from the time of the Ancient Greek's (notably Democritus) up to the present day. But the people in positions of authority in the field will have gone through the academic route. And so it is a self-perpetuating situation that the vast majority of people in positions of authority will believe that the existing approach to mathematics is the correct one. I find it very strange that almost everyone 1) accepts the concept of infinity as a valid concept and 2) believes they can conceive of things with no counterpart in physical reality like infinite sets. While the vast majority of people believe in these things, and while they are happy with maths just the way it is because it appears to be successful, then there will not be an overhaul of maths foundations.

[@craig4320]

@@KarmaPeny Is this overhaul truly a huge project? Does it really involve getting lots of people to change their minds about the reality of infinity? Do we have to convince the engineers and physicists that their calculations don't really work? Maybe it is possible to rebuild the foundations without toppling and ruining all that is built on them. Maybe infinities can be shown to be a calculation shortcut that works in certain situations.

[@turdferguson3400]

@@KarmaPeny most of the things you mentioned are nonsense objections. Like, obviously for every number in a sequence that converges to root 2 has another even better approximation later. So this objection that there "has to be a last one" is nonsense.

[@azimuth4850]

Why does there need to be a "last point"? If there's infinitely many, doesn't that imply there won't be a "last point" before ANY number?

[@flatisland]

I guess the most difficult thing to wrap your mind around is about whether something can be infinite or not. On the one hand you can't imagine that you can only go up to a certain point and then it stops. On the other hand you also can't imagine that you can go on for forever (either on a smaller and smaller scale or a bigger and bigger scale). It's a real dilemma. My personal opinion is that we always have to deal with finite obects. For example planets, stars, galaxies only contain a finite number of atoms. Then there are finite / discrete energy levels of electrons in atoms, discrete charge etc. Everywhere we look the universe seems to offer only finite / discrete stuff. Does that give us a hint that in the end there is no continuity in general? What bothers me though is the question of time. It's hard to believe that time has a beginning or an end. Because you can always ask: what was before that? what will come after? ... Unless time is just a figment of our thinking and doesn't really exist - or rather a sort of 4th spatial dimension / space-time.

[@maynardtrendle820]

As a layman, I love Mr. Wildberger's lectures, but I am left wondering how his arguments for constant concrete computability (so to speak- alliteratively☺️) are viewed by his peers. Where would I find good examples of others (both contemporarily and historically) making arguments similar to his, or is he relatively alone in his objections?

[@jonathancohen2351]

There has been some study in this area en.wikipedia.org/wiki/Intuitionism but it is not a thriving field. It became more interesting IMO when it was found that intuitionistic logic formed a very nice correspondence with computer science. (en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence) I think that what Prof Wildberger seeks to do is very interesting in that regard, even if he is not able to provide a more compelling argument than philosophy.

[@derendohoda3891]

you should start with the external links on wikipedia's entry for "ultrafinitism"

[@toddtrimble2555]

@@jonathancohen2351 On the contrary, intuitionistic mathematics is a vibrant and fascinating field; its restriction on certain rules of inference means, on the positive side, that it enjoys a much broader semantics than ordinary classical mathematics. For example, interpretations in toposes, categories of set-like objects whose internal logic is typically intuitionistic. Not out of some Brouwerian philosophical whim, but because that's simply how it is in such universes. However, intuitionistic mathematics certainly admits infinite objects. I doubt Wildberger has any engagement with intuitionistic mathematics as practiced by professionals today.

[@logos3522]

George berkeleys “Analyst: A discourse to an infidel mathematician” Thomas Hobbes “six lessons to the professors of geometry” Rene Guenon “metaphysical principles of the infinitesimal calculus”

[@emiliomatos6071]

Wildberger's willingness to accept that the graphs of y=15 and y=x^3 don't intersect is to me much more problematic than Infinity. It seems obvious to me that these curves must intersect and that mathematics must model this fact. It doesn't seem mathematically fruitful to simply dismiss analytic geometry altogether as a good representation of curves. Any theory with such positions seem to be in serious foundational trouble.

[@alijvn3783]

I think if you don't believe in existence of continues line of real numbers, then it is inevitable that y=15 and y=x^3 can have no intersection. I am not sure if we have anything in real world that is as continuous as real line of number. Actually everything seems to be discrete and finite. So he is not crazy

[@writerightmathnation9481]

The universe of hereditarily finite sets provides a way to discuss existence of arbitrarily large finite numbers without accepting the axiom of infinity, so I think that’s the clarification you seek at 28:02.

[@gurmeet0108]

20:27, that is in fact almost never the case. Even the case that you mentioned the series solution methods already existed before Arzelà-Ascoli theorem, that theorem finally gave rigor general foundation for such solutions. Calculus in general and Fourier series etc, that you mentioned in the video are another example. Most of the time, first the theory is developed, later people realize the foundations are shaky or not general enough then pure maths tries to fill that gap.

[@gurmeet0108]

And also, 27:39, I'm interested is what do you think why in so many years of computers, now we have ways to handle numbers, groups, algebra everything in languages but not proper real numbers? The set theory based theorem provers don't progress much and even in them real numbers are ridiculously hard to handle. Why do you think that is the case? We already have many equivalent definitions of reals based on the ideas of infinity and infinite sets like cauchy sequence and dedekind cuts, but these don't give you an easy way to programs them in languages and theorem provers. Is there another equivalent way to define reals which is maybe more type theoretic, so it's easy to deal with in functional languages?

[@gurmeet0108]

@Gennady Arshad Notowidigdo 1) Stop being so unnecessarily defensive. 2) I never claimed I am representing Wildberger's position, I was just commenting about what interests me regarding these issues (I specifically wrote "...I'm interested..."). 3) I don't care how you want to decide the line between computer science and mathematics as well. Anyways, type theory is as much part of mathematics as a part of computer science, I guess your PhD wasn't able you teach you anything about foundations of mathematics (modern approaches, not the ancient set theory based one). 4) AGAIN, I never said I am talking about a problem of mathematics, even if this is not a problem of mathematics, WHY THE HECK SHOULD I CARE? I am interested in this problem, whether it's mathematics or not (in your definition) doesn't matter a dime. I guess your PhD made you too narrowly focused. 5) You are the one who is making claims about Wildberger's position, in fact, from whatever I understand about Wildberger's position, you are VERY HEAVILY misrepresenting him, but I don't want to debate about this, I am not Wildberger. 6) I'm doing PhD too, but I wish I don't become like you.

[@gurmeet0108]

@Gennady Arshad Notowidigdo I'm sorry for the personal attacks. I'm not sure why I did that, I humbly apologize for that. Now to continue the discussion - 1) I am actually focusing on the practical only. In recent times, theorem provers are getting more and more popular, and they should as they ensure the correctness better than humans, it's easy to build search tools over them, this kind of digitization will help in getting better and better automated theorem provers and many other advantages. 2) I'm not *just "deep-dive" into the foundations*, but somebody have to get their hands dirty to reach a foundation which is better in terms of programming, theorem proving etc. And type theory has performed significantly better it that regard. 3) In fact, from the current research a very deep connection is developing between category theory (topos theory) and type theory which is resulting in better formulation (again in terms of automation/programming/theorem proving etc) of a lot of modern maths. 4) As far as analysis is concerned, type theories handle them just fine. As evidence, lean theorem prover handles reals very well. 5) You earlier wrote "His position is to assert that computer science and physics problems should be the ONLY valid mathematics problems" then when I challenged you, you changed the position to "only mathematics that can be written down and calculated exactly is considered valid mathematics". This is known as Motte-and-bailey fallacy. 6) Can you tell me any mathematical object which can't be written down? Literally to specify such an object you would actually be writing it down. Wildberger is for sure not being rigorous in expressing his thoughts but from whatever I can see, his views are not very different from the views which lead Brouwer and others to build intuitionistic and constructivistic schools. And the modern advent of topos theory is showing why and how these are connected to "normal" mathematics. And also these train of thoughts have been very advantageous in, again, theorem proving.

[@gurmeet0108]

@Gennady Arshad Notowidigdo In a mathematical and/or scientific argument this kind of language, respect, apology, sincerity these kind of things are just nice good-to-haves, these have no baring at all in the validity of the argument. So, it's alright that you don't accept my apology, the argument is independent of that. I was engaging in the argument in the hope of learning something new and if you choose to enlighten me by showing the flaw in my arguments, I'll be grateful as I should be, the best you can do for me is to show me I'm wrong but if you don't want to waste your time, that's alright too, thanks again though for your replies. In the end, by Aumann's theorem with scott's extension, two rational agents never have to agree to disagree and reach an agreement effectively, so as you have focused way more on sophistry instead of rational arguments, I'll urge you to reconsider your way of argument in general and focus on actual argument instead of sophistry. Thanks again. 😁

[@writerightmathnation9481]

It’s important to have a good foundation, so I disagree here with you (16:56). However, one must have some notion of what constitutes “good foundations”, and this is where I differ with the implicit notion of “good foundations” to which Norman ascribes. His view is subject to a kind of “cascade of collapse of the acceptability”, which I mentioned in a comment on your video of the debate itself. The criterion for “goodness” to which I ascribe includes a “sufficiency” portion and a “necessary” portion. I don’t require a metaphysical component but I consider that a sideline worthy of discussion in the philosophy of mathematics and the philosophy of science. The “cascade of collapses of acceptability” that I ascribe to Norman’s philosophy is that his objections led naturally, and I believe, inexorably, to a nihilistic philosophy of numbers, in which we may as well deny the existence of every natural number. This is a very strongly anti-platonic view of mathematics, and one who subscribes to such a view may as well also posit that nothing exists. I digress. Your view - that foundations are almost irrelevant - fails in the same way that the lack of secure foundations failed in the 1800s, in that it allows contradictions in our foundations, and hence leads to mathematics that is devoid of any certainty of validity at all. I think then that it’s clear that I’ve taken a middle road between these, in that we need foundations that at the very least are not provably inconsistent (the “necessary” condition for a philosophical stance in mathematics), but are, from a pragmatic standpoint, at least provide a way to develop the solutions of problems that may be formulated in our mathematical language. Inherently, this allows for at least the existence of arbitrarily large positive integers, if we are to consider the problems of analysis as relevant to either pure or applied mathematics. …

[@nychan2939]

To have a foundation sounds good, but it's not necessary sometimes. So don't waste too much time on the topic. It's ok as long as we can use it to prove something we want to. If someone has made up assumptions to prove 1+1=2 and turns out that 1+1 != 2, we don't have to worry because we know that 1+1=2 must be correct and there's just something wrong with the assumptions. By the way, distance and angle are simple, clear and useful concepts. They shouldn't be replaced by the much more difficult ones as suggested in Mr Wildberger's book.

[@darkwingduck42]

He seemed defensive from the start, with a hostile tone, often going on tangents instead of answering questions. You appeared uncomfortable and baffled. When you raised objections, he cut you off and changed the subject. The interview was difficult to listen to. He tries to present a new view of mathematics but is insulting, using ad hominem attacks and calling things religious. He comes off as hostile and emotionally driven, resembling a modern conspiracy theorist rather than a serious academic.

[@kyaume21]

But what if you run into a fundamental inconsistency, like Voevodsky mentioned in his Princeton lecture (available on KZbin). It made him go back to foundations, and query the whole structure of modern mathematics. Voevodsky told us how he could not carry on with 'math business as usual' and felt he had to recheck the foundations from the start.

[@santerisatama5409]

Yeah, I agree that FOL has been fundamentally inconsistent from the get go. Hilbert's program was nearly instantly falsified by Gödel, Church and Turing, and underlying reason is that the quantifier "forall" is applicable only to finite domains, as already Brouwer tried to make clear. Undecidability applies both to classical logic and intuitionist logic, but unlike classical logic which it destroys as incomplete and inconsistent, intuitionist logic feels more complete with it and cherishes it, as double negation has been defined as logically undefinable and more kin to the creative oppositions of syllogistic logic.

[@accountname1047]

In my view the way we build analysis up from the foundations is inherently approximate. "We can do some proposition P for any positive epsilon" says to me that you must first set your tolerance if you want to execute P in practice. That said, I can accept infinity and related concepts in the same way I accept formal power series and generating functions, as tools.

[@jessewolf6806]

Approximate?

[@RuthvenMurgatroyd]

I don't think that's approximate. In practice maybe but what's happening in mathematics land is that the proposition P is literally true for all epsilons that exist. In math land we don't have to look for smaller and smaller epsilons because we have them, all of them, and we this is true in principal. This is nothing like an approximation, it's just a fact one can use to approximate.

[@seanhunter111]

In an epsilon delta proof of a limit as x tends to a for example you show there exists a delta and as long as x is closer to a than delta then f(x) is closer to the limit than epsilon for any epsilon greater than zero. So what it's showing is that whatever tolerance on f(x) you choose, I can give you the tolerance on the input variable that will get you closer to the limit than that tolerance, and it doesn't matter how small your tolerance is. Executing the proposition P in practise isn't interesting generally - for example if I have the rational function 2(x-3)/(x-3)(x-4) the limit as that function tends to 3 is -3 even though the function is not defined at 3. That's not inexact and the proof of this limit literally shows that it's not inexact given it is arbitrarily more exact than any epsilon you can chose.

[@Thrashenizer]

That's not what Wildberger said Daniel! Please forgive the shallowness of my mind, but, what I understood from Norman ( _using the example of 15^(1/3)_ ) is as follows: *15^(1/3)* is a " *Symbolic Representation* " of an *_infinite Set of approximations_* . And an _infinite set_ *cannot* really be taken as the _definition_ of a _singular_ (Atomic) " *Number* " (Real or otherwise). The definition of a " *Number* " should be more fundamental than that! I believe that _Pi_ & *_e_* are more like _constant_ " *Functions* " whose arguments are not numbers but rather a _structure_ of (Geometric or Algebraic) _space_ wherein the function is evaluated.

[@Thrashenizer]

@Gennady Arshad Notowidigdo Thanks chief .. I really do appreciate your clarification. My question is : How could you be certain of the concreteness of your logical framework given that it's based on statements like " _The axiom of infinity, which allows us to talk about infinite sets, is an assumption_ " ? What is the definition of an " *Axiom* " in your Logico-Mathematical mental framework ? And when does an " *Assumption* " qualify to become an " *Axiom* " ? .... And unlike you, I care about your opinion :) p.s. The principle under treatise, here, is known as " *The Axiom of Choice* " (not the " The Axiom of Infinity _ " , a subtle difference there boss ... And Norman may be engaged in many activities, but "rambling" ain't one of them :)

[@derendohoda3891]

One way to look at NJW's position is not a denial of the activity of analysis as such but that the current method of definition and the purported objects of discourse are somehow together or individually improperly defined or interpreted (think back to Leibniz's infinitesimals versus epsilon-delta versus non-standard analysis). Simply, we can't do infinite work, so whatever we're doing, we can't be doing it with infinite objects or processes. You can see this from his critique of reals defined as cauchy sequences or dedekind cuts. (Famous Math Problems 19 a through d). These videos, I think the last two specifically, deal with your comments on requiring unambiguity. Another example would be his quasi-algebraic proof of a certain definite integral; you would define this definite integral, say, as a riemann sum and he partially shows why that isn't necessary (but, maybe not coincidentally, then leaves the entire definition of area undefined) (Famous Math Problems 10 a and b). I liked your discussion and this follow-up video. I've binged your channel hard tonight. Subscribed. Thanks.

[@DanielRubin1]

Yes, this is also how I see a strong interpretation of Norman's views. Working on a video that will prove the important points about the Cauchy sequence construction and remark on some foundational issues.

[@elcapitan6126]

the concept of infinity seems to only make sense when thought of as an unbounded process generating a sequence, meaning no finite bound is predetermined and its left to the "user" of the process to choose a suitable criteria leading to a finite approximate on a per use basis. that's more or less how limits are defined etc anyway with the pick-any-epsilon idea.

[@francescaerreia8859]

@@elcapitan6126 I would instead call those indefinite, not infinite.

[@elcapitan6126]

@@francescaerreia8859 can they not be shown to be isomorphic / algebraically equivalent ideas? what do we meaningfully gain to distinguish them?

[@francescaerreia8859]

@@elcapitan6126 idk enough about math foundations to say much honestly but philosophically they seem totally different. One difference in math I can think of is that an infinite limit seems logically totally different than an infinite set. An infinite set says it is or describes an actual infinity of objects whereas an infinite limit just says that however big you want the output (for any N), I can get you close enough to an input that maps to something that big or even bigger. Infinity never actually shows up anywhere as an assumption in the definition of infinite limits, we just happen to call them that, but the definition is really implicitly only offering an indefinite procedure, one which terminates at whatever approximation you desire.

[@DrScrutinator]

I disagree with your statement that the definition of Cauchy sequence doesn't depend on the axiom of choice; it does. And the axiom of choice is the source of all 'mathematical evil' so to say. If there is a contradiction in the foundations of mathematics, in the sense of Voevodsky (who suspected there was, and therefore he initiated his program of computer proofs), it must be in the AoC.

[@elcapitan6126]

might be in the axiom of infinity too. in terms of "infinite sets" as a complete object, but it's more subtle since one can talk computationally about a priori unbounded sequences without "going to infinity" or having an entire infinite sequence at hand to talk about

[@samb443]

no, he is actually right, the definition of cauchy sequence doesnt depend on choice, even if choice is evil. the axiom of infinity is fine, basically none of the axioms are evil. The issue is mathematicians ignoring the limitations of the axiom of specification. a subset must be specified by a formula. But mathematicians claim there are uncountably many subsets of N, despite there being only countably many formulae. Theyve forgotten that everything needs an explicit finite definition. This is the issue with cauchy sequences, they use "there exists", but non-constructively that doesnt give you a definition to work with, so its meaningless.

[@ryancreedon5100]

I'd say I agree on most of your points, except your claim that pure math research happens before applied math research (20:25). I'd say it's more nuanced than that. Look at calculus, for example: the subject was developed from a computational and almost experimental point of view to describe planetary motion, among other things. It was only 100-150 years later that those formal calculations could be more precisely pinned down, abstracted, and generalized into mathematical analysis. In contrast, if we look at number theory, its applications to error-correcting codes, cyber security, and cryptography came after the theory had been developed. Personally, I don't care for the distinction that's been drawn between pure and applied mathematics these days. I think they should be regarded as two halves of the same whole. Any grounded mathematician should have footing in both perspectives, just like any of the great mathematicians of the past.

[@roderictaylor]

If one rejects the idea of an actual infinite, one will object to the assumptions of modern mathematics. I don't share that objection, but that's a reasonable position. Constructivists and Intuitionists have proposed alternatives. I highly recommend Errett's Bishop's book "Foundations of Constructive Analysis," if you're interested in one possible way to approach the real numbers without appealing to actual infinities. Again, I'm not personally convinced of the constructivist or intuitionist positions, and I would not wish to have to do mathematics this way, but I think they are serious and philosophically interesting proposals. As for ideas of the infinite, we need to distinguish between the idea of a potential infinity and an actual infinity. The ancient greek mathematicians rejected actual infinities, and only used potential infinities. For example, a modern mathematician might think of a line as extending infinitely in both directions. To the ancient greeks, a line was a finite line segment that could be extended in either direction without limitation. There is no limit to how far it can be extended, but no matter how far we extend it, it's still finite. We're distinguishing between the ideas of something being unlimited versus actually infinite. Cantor introduced the idea of actual infinities in mathematics. We think of the set of natural numbers as an infinite set who's members all exist all at once. But if I think of the natural numbers as a potential infinity, then I can say 1 is a natural number, and I can add 1 to a natural number to get another natural number, and there is no limitation to how many times I can do this. But I don't think of the natural numbers as existing as a completed infinite whole. To a mathematician like Bishop, that would be philosophical nonsense. Consider the Goldbach conjecture, every positive even integer at least 4 can be written as the sum of two positive prime numbers. If I treat the natural numbers as an actual infinity, as an infinite set that exists in some Platonic reality, then I will say Goldbach's conjecture is either true or false (even though I don't know which it is). Either every even integer greater than 2 can be so written, or there exists one that can't. I imagine the natural numbers exist as a completed whole independently of my knowledge of them or reasoning about them, and so I may conclude either they all have certain properties or they don't. But if I'm a constructivist who rejects actual infinities, I will reject this. I will reject this idea of some infinite set of natural numbers existing out there somewhere in some dubious platonic reality. If I can find some even integer greater than 2 and confirm it cannot be written as the sum of two primes, then I may conclude Goldbach's conjecture is false. If I can prove somehow that every even integer greater than 2 can be written as the sum of two primes, then I may conclude it is true. But unlike the classical mathematician, I may not conclude the conjecture is either true or false without demonstrating it. I may not assume the law of the excluded middle. In his book, "Foundations of Constructive Analysis," Errett Biship attempts to show how we can do modern analysis in such a context. He rejects the law of trichotomy, we may not assume every real number is either positive, zero, or negative. We may not assume that any rational number is either rational or irrational. We may not just assume e+pi is either irrational or rational, unless we can prove which it is.

[@antpresentations]

I think the issue is the modern transition to a "consistent definition" from the traditional view of "actual representation of reality". The transcendental aspects of modern analysis sort of assume that an omniscient being can complete all the operations that we cannot do ourselves. People had issues with the calculus because of infinities and limits were viewed as unphysical so they were not used. Today we simply just take the infinity for granted and place ourselves on a pedestal, all proud of what "we" achieve, even though we are doing things that pre-19th century mathematicians would regard as fanciful. Furthermore, we do not prove the intermediate value theorem, by stating all the assumptions about rigour we need so it follows as a triviality. It's entirely circular, and a little disingenuous given that Cauchy's bisection algorithm was known for centuries.

[@billh17]

@Newton40 said "a little disingenuous given that Cauchy's bisection algorithm was known for centuries." But modern analysis did not consider that Cauchy's proof was rigorous enough (especially since real numbers were not defined rigorously enough). Modern analysis is essentially redoing Cauchy's bisection algorithm in a rigorous manner. It is not disingenuous since that was the explicit purpose of doing modern analysis: to put classical analysis of the 17th-19th centuries on a rigorous foundation.

[@antpresentations]

@@billh17 What do you mean by "rigorous definition" of a real number? The understanding since Stevin was that such a thing was just an infinite decimal expansion (ide). There are a couple of problems with arithmetic, but they can easily be glossed over in the "proceed to infinity" step (which we nevertheless must assume if ide's are to be taken as a workable object). What do we gain by using Cantor's "equivalence class of Cauchy sequences"? Are we not simply shifting the goalpost?

[@billh17]

@Newton40 asked "What do you mean by 'rigorous definition' of a real number?" Well, even though Wildberger disputes that ZFC is even valid, much less rigorous, the modern analysis uses ZFC as its foundation to provide a rigorous definition of real numbers. ZFC provides a rigorous definition of a real number (of course, only if you accept ZFC as a valid framework to work in). @Newton40 said "The understanding since Stevin was that such a thing was just an infinite decimal expansion (ide)." But there were objections to infinite decimal expansions (pretty much in the vein of what Wildberger says). There was also objections to the way infinitesimals were used in calculus and the way infinite series were handled. Cauchy provided a step in the right direction to handle these things rigorously, but there were still problems in doing it correctly (eg, how to define real numbers). @Newton40 said "There are a couple of problems with arithmetic, but they can easily be glossed over in the "proceed to infinity" step (which we nevertheless must assume if ide's are to be taken as a workable object)." But one doesn't just want to gloss over the problems. That is not rigorous (almost the opposite: it is just then hand waving). One wants to eliminate the "proceed to infinity" step and not assume that such a thing is possible (pretty much in the vein of what Wildberger says). @Newton40 asked "What do we gain by using Cantor's 'equivalence class of Cauchy sequences'? Are we not simply shifting the goalpost?" By using 'equivalence class of Cauchy sequences' (not sure if it is due to Cantor) in the framework of ZFC, one makes the definition and handling of real numbers rigorous. That is the point. We are not shifting the goalpost (one still wants to do calculus and real analysis), but rather one is shifting the starting point.

[@antpresentations]

@@billh17 By "gloss over" I meant only in the context of this conversation, not in the actual mathematics. The issue with infinite decimals is actually "doing" the arithmetic. You will not achieve the exact digits to the new number unless you proceed to the infinite limit. This is the same with Cauchy sequences, one does not obtain the whole sequence, only the specification of any term of the sequence we wish to evaluate. We do not actually do the arithmetic in any definition of real numbers, we just let the symbols do it for us and assume that this is enough. Wildberger's issue is that he believes infinity, as currently espoused, is an imprecise concept. From a formalist viewpoint, that is not true. But if we wish to attach real meaning to our words then he does have a point. Just so we are clear on each of our positions, I don't think ZFC is invalid as a theory (as pointed out by OP, we currently have no crisis), but I don't believe something that directly applies to the real world, i.e. analysis, should stand or fall depending on the validity of a highly transcendental theory of sets. Now, my point is that people in the past had issue with the notion of doing "infinite steps" such as in the case of evaluating x+y where x,y are infinite decimals (though the general opinion amongst most working mathematicians was if arbitrary precision is possible, this didn't matter). But placing everything on set theory does not solve it. We are just assuming we can, because if we couldn't things would get difficult. We then assume we can take power sets of such objects, or postulate the suprema of a set. Then after we put all these fantastic notions together, we get IVT as a formal corollary. That is what I mean by disingenuous, because we are assuming a lot more than just "infinite digits".

[@billh17]

@Newton40 said "but I don't believe something that directly applies to the real world, i.e. analysis, should stand or fall depending on the validity of a highly transcendental theory of sets." Neither the real world nor the theory of sets depend upon one another. This is not claimed. If the real world exhibited something we would want to understand mathematically but the theory of sets did not suffice, then some new mathematical framework would be developed. This actually happened with category theory. ZFC did not provide a framework that would show two things were "categorically" equivalent. Category theory was created to handle this. In quantum physics, the logic of particles seemed to obey different laws than the standard logical laws. No problem: just develop a new framework to capture what was happening at the quantum level. ZFC or set theory is not claimed to be the only framework to do mathematics in. @Newton40 said "We then assume we can take power sets of such objects, or postulate the suprema of a set. Then after we put all these fantastic notions together, we get IVT as a formal corollary. That is what I mean by disingenuous, because we are assuming a lot more than just 'infinite digits'." I think it would be hard to come up with a logical framework (using first order logic) to define real numbers and allow their manipulation without postulating a lot of fantastic notions and assumptions. Wildberger's approach is to forget about real numbers and just develop a framework where one can still get the standard results of classical mathematics (and maybe more). Also, the goal of ZFC is more than just to come up with a framework to handle "infinite digits". Dedekind was the chief promoter of the idea to use set theory to phrase the concepts used in mathematics (eg function). Dedekind promoted the idea that set theory helps the mathematician to avoid computations in proving theorems (computations were hard to do before there were computers in the 1970's). Using set theory (and ZFC in particular), it makes it easy to give formal definitions of certain concepts. Dedekind gave many examples of doing this (eg compare Dedekind definition of ideal numbers with Kummer's definition of ideal numbers). Also, in computer science, set theory is used to define many of their concepts. For example, a Turing machine is defined as a 7-tuple of certain sets (see Wikipedia).

[@writerightmathnation9481]

Here (@ 17:57), you’re disagreeing with Norman’s disavowal of the “added value” of post-19 th century analysis. I think that your argument is stated as overly subjective - opinion. It wiles benefit from specific examples, and I’m convinced that you can find many. For example, I’m convinced that there are useful algorithms in areas of mathematics to which Norman does not object which were developed specifically because analysts used mainstream mathematics to lead them to understand and create and implement such algorithms. Proving to Norman that an example you find has that property may be more difficult than you’d like, but providing more evidentiary support for your argument among your viewers, including the proverbial “choir” is probably even more important to you than getting the “blessings” of Norman or any of his followers.

[@Achrononmaster]

@2:50 yes. In science and mathematics having to ""deal with a problem" is a good thing, not a bad thing. At some points NJW get's close to admitting this, and seems to suggest that the issue is we need a firmer foundation for transfinite sets and defining ℝ in a nicer way, which is a nice open puzzle. He even suggests more use of "algebra" harking back to Gauss et al, but he seems to not recognize Category Theory is a firmer foundation he might seek. At other times he comes across as an ideologue, like Wolfram and others, which I find distasteful. Although _de gustibus non est disputandum_ I still find _such_ ideology to be a sign of knuckle-headedness

[@ashnur]

4:15 I agree it's a philosophical standpoint. "There is a real number", sounds like you say that the stuff we imagine weighs literally just as much, as just as much momentum, energy as physical material that we usually talk about as "existing". This to me is bad philosophy. The stuff we imagine is qualitatively different from the stuff that does the imagining. I can't wish into existence any amount of cakes, and no, ordering them online doesn't actually solve the problem, because that's still something physical, there are actual people actually baking that cake from actual stuff. Real numbers are not like that, actual people do actually imagine them, but when you write them down, they don't remain real numbers, they become symbols or writing or some other form of communication, or effect. So the good philosophy here I believe is to distinguish between the 3 realms, imagination/communication/World and admit that they have different limits.

[@InfiniteQuest86]

This is really interesting. I'm guessing he doesn't accept any proof using the axiom of choice. That would invalid a ton of mathematics. But a lot of it is very useful, even in real world applications. There might be some foundational issues, but as you point out if there are some explanatory and useful math, then we should be using it until someone comes along with something better.

[@cdrundles]

Having listened to some of his concerns, I believe that he has some valid points: the definition of real numbers is somewhat slippery: Dedekind cuts for some situations, then we switch to limits of sequences of rationals for others. Another concern he raises is one physicists would agree with: reality has no infinities (yet), so why should math? Daniel's point that no such requirement applies to math (which just provides useful models after all) is hard to foster given reactions to the Banach Paradox. I've heard math guys defenestrate the Axiom of Choice on uncountably infinite sets for just this reason. Others say that the paradox is obviated by removing the logical excluded middle: no proofs by contradiction! So, something is clearly assumed about math relating to the physical world! Another concern he raises has to do with finite computation and verification using algorithms that terminate. This objection conjures Xeno's ghost: how can we have endless digits and still have a quantity? As Daniel pointed out, this is a philosophical objection raise by finitism. Alternative approaches to analysis are already worked out with difference equations, etc. that one uses when computing with rational numbers. I think he calls this algebraic calculus or something. This is not new. Bottom line, I agree with the current trend of accepting infinite sets, despite the problems brought along (I suspect that axiom consistency is not possible with this). ( Banach's paradox more likely a slight-of-hand with measurability being preserved, and that the Axiom of Choice is just fine with uncountably infinite sets.) Besides, analysis is too much fun!

[@roderictaylor]

That we can construct the real numbers using various devices, Dedekind Cuts, sequences of rationals, even sequences of decimal digits, is no problem at all, and it is not at all slippery. When saying what the real numbers are, we don't give some construction; we give the axioms the real numbers satisfy, the axioms of a complete ordered field. Any system of entities that satisfy the axioms of a complete ordered field can be the real numbers. We prove that any two systems that satisfy the axioms of a complete ordered field are isomorphic; that they are functionally identical, so it doesn't matter which model we use. Of course we can't just offer an axiomitization, we have show that axiomitization is consistent. And we do that by giving a model that satisfies the axiomitization. For example, we give Dedekind Cuts, or we give sequences of rational numbers, or whatever (there are many possibilities). Once that is done, we throw away the model which has fulfilled its purpose, and use the axiomitization.

[@Achrononmaster]

@17:00 misguided young sprite? Foundations are important because we don't want houses built on quicksand. The issue is that most people trust classical mathematics, for good reason, no glaring inconsistencies. So "it's all right Jack" for applied mathematics. But the thing is, we other weirdos tend to look to the future of civilization when we ourselves will long since be dead, and yet still worry. We don't want a Trisolarian invasion to catch our descendants with their pants down when some strange esoteric piece of obscure mathematics turns out to be wrong because of an inconsistency in the foundations. I joke. But I am half serious. I noticed on your channel, too often for comfort, that some of your talk is individualistic. Things don't matter _to _*_you._* That's fine, but other people exist.

[@mikeolsze6776]

It is not necessarily a matter of analyzing infinities. It is more so actually a case of potentiating & encompassing rolling infinities, in our factitious systematizations. Its not about analyzing infinities. Its accepting that conducing them is in our best interests. As a true wholistic systematization innately manifests (particularized) infinities.

[@bendavis2234]

You should check out his videos on Foundational Mathematics where he spells out his views very clearly and unambiguously. Some of the questions that you addressed in this video are answered by him in his series of videos. Even if I don’t 100% agree with his view, I respect his transparency on the subject and thoroughness in covering the foundational views.

[@axe-vw6ek]

lol norman guy is a complete fraud. Anyone that has studied mathematics extensively knows he is lol

[@robfielding8566]

If nothing else, following Wildberger leads you to math that also works on finite fields; and works well on computers.

[@santerisatama5409]

Thanks for participating in very important and very nuanced discussion. I share many similar views with Norman, and also reject the notion of "completed infinite process" as inherentry contradictory. As we are discussing pure mathematics, arguments from utility don't apply, as they belong to the pragmatic truth theory of applied mathematics. I also reject the theological notion of "timeless being" of Platonia as an arbitrary and unnecessary axiom. So, I'm left with Intuitionism and coherence theory of truth, for which I argue as coherent ontology of mathematics and mathematical truth. The foundational crisis of mathematics has been going on unsolved since the Brouwer-Hilbert controvercy. The crisis was swiped under the rug by Acacemic sociology (Hilbert's cancel culture against Brouwer etc.), not solved by power of argumentation. Greek pure geometry made clear distinction between pure (no neusis) and applied (neusis allowed). We are basically continuing the sama debate of how neusis relates to foundations of mathematics, as was also Berkeleys criticism of infinitesimals in his "Against Analysis". Norman does at least in practice accept many kinds of infinities, including the kind that 1/0 refers to, but of course I can talk only for myself. I agree that the Stern-Brocot construct (SB) is very foundational, and IMHO the best candidate offer a coherent solution to our foundational problems. AFAIK standard analysis is generally committed to field arithmetics, and thus rejects 1/0 and consequently also SB as a form of analysis. A very crucial foundational problem is that we have no coherent theory nor philosophical consensus over what does "number" mean. I take conservative position is that regard, and define numbers as product of tally operations. Instead of numbers, I suggest that we call operators operators and algorithms algorithms. Much confusion can be clarified this way, and instead of "algebraic numbers" we can speak of algebraic algorithms. With the inclusion of origami in the tool pack of classical constructive geometry, the classical meaning of "transcendental number" has become outdated, and we need another term for algorithms which can't be solved by the method of straight edge, compass and origami. Naturally, this definition excludes "real numbers", especially "non-computable real numbers" which are not algebraic algorithms and don't have any closed form demonstration, which can function as in input for computation. What cannot be named, cannot have linguistic existence, and if mathematics is a "language game", as Formalists claim, then playing language games with what can't be named is inconsistent and dishonest wrong playing. If somebody claims that "real numbers form a field", then according to basic syllogism it is necessary to prove that claim by demonstrating that each and every real number can participate in arithmetic operations. If that requirement is not accepted, then we lose the ability to falsify a conjecture by a counterexample, and end up with the situation that mathematics as a whole is a logical Explosion where anything goes. I don't accept that as a foundationally coherent position, but of course heuristically we are free to assume what ever we like. Formalist method of arbitrary language games, assumining "finite infinities" and what ever can be heuristics, but not claim the status of mathematical truth and coherent foundation. *** Holding on the principle that mathematical coherence theory of truth includes also empirical truth conditions, the question arises, what is the temporal ontology of mathematics, if we reject Platonist ontology of timeless being? The bi-directional T-symmetry of quantum is already as mathematical as mathematics gets, so most certainly we are not limited only to classical consecutive unilinear time. The strong both mathematical and empirical proof of the tautochrone-brachistochroen property of the cycloid provides also a very fundamental notion of duration. We can't prove that a mathematical proof is eternal and immutable, but we can conclude that mathematical proofs and truths can have very large duration spreading both to past and future from a proof event, even if foundationally indefinite duration coherently with the Halting problem. Reversible computing in quantum duration might be able to also empirically support computational pure mathematics that purely classical ontology would forbid, and without assuming logical and empirical absurdities such as "completed infinity". Such view would mean that we and our pure mathematics are nested in and reflecting quantum cosmology, and new challenges would arise, such as mathematically more coherent formulation of holistic quantum mechanics. Formally we can present holistic quantum duration as the following nesting algorithm, in which the fundamental operators < and > (cf. arrows of time, relational operators, Bra Ket notation and Dyck language) symbolize continuous directed movement: < > < > < > etc. When we define that < and > have the numerical value 1/0, we can define that 1/0+1/0=0/1. Thus for the second row < > we get the numerical interpretation 1/0 0/1 1/0 which functions as numerical generater for numerical two-sided SB-construct, and we get the same numerical result by defining as countable elements and tallying them in each generated mediant word. The blanks between the mediant words form a binary tree, in which continued fractions can be defined as L/R paths. This gives all that Cauchy sequences does, but in foundationally coherent and fully computable manner (see also Gosper arithmetic.). From this holistic perspective, (mereological) fractions come first, and integers and naturals are mereological decompositions of fractions.

[@joaquincapellancruz7402]

If someone believes that rational numbers exist, there's no contradiction in asserting the existence of a largest natural number. Both with approximation algorithms(successor function, power series)and with infinite information.

[@Achrononmaster]

@4:00 but it is _not_ just "philosophical". If NJW is serious in rejecting a much of mathematics it should only be because it is inconsistent. There is a proof his side is "proper" mathematics if indeed it is the only proper math, which is to show anything more is inconsistent. You don't just assume there is no proof of inconsistency of a system, say of ZFC. That's getting Gödel the wrong way around, Le Chien.

[@epennrogers]

Norman seems to be completely ignorant of the fact that we have an entire branch of analysis (numerical analysis) that is interested in computing the things that are defined using various notions of infinities in a finite number of steps.

[@marcushellstrom1157]

I as a layman like both your positions and that is a problem. I think it comes down to degrees of freedom. You recognice two different degrees of freedom. You and the classical view of analysis recognice an admittance for mathematical existance of that which is merely definable because proved boundaries for what it can be defined as. The alternate position has a degree of freedom in choice of adding structure or not defining such, thereby not admitting to a boundary of structure, but choosing not to do so.

[@TheGloryofMusic]

Wildberger is an Aristotelian, not a Platonist, so he rejects the axiom of infinity. The philosopher Fichte said that a man's philosophical opinions are in the end a matter of taste.

[@jinks908]

If we are to do away with making statements such as "there are an infinite number of primes," then by implication, we must reject the proof that accompanies this statement. This is a slippery slope because rejecting that proof (there are actually several, but I'll refer particularly to Euclid's as it is the simplest and most convincing, imo) means rejecting hundreds of others which rely on this proof. Not only that, but what is to be said then about the logical methods of such a proof? If we cannot accept this proof as unequivocally true, then that must mean our logical framework is flawed. We would then need to tear down essentially the entire structure upon which mathematics is built. So until this professor (and those who share his beliefs) can build a new one, I don't know how we could benefit at all from this. I mean perhaps this is, as Daniel points out, simply a matter of mathematical philosophy, in which case we can argue about these kinds of things ad infinitum (pun intended :). But by rejecting propositions such as the infinitude of the primes, you would also need to reject literally hundreds of theorems, corollaries, and mathematical results, not to mention actual TANGIBLE byproducts that depend on it (e.g., many results and developments in computer science and basically all of modern cryptography). It seems hardly practical or beneficial to profit from these developments while asserting that their entire foundation is false unless we can effectively and completely replace such a system of mathematical logic to account for them...

[@martinepstein9826]

Euclid proved that for any finite set of primes there is a prime not in the set. We can accept this result but still reject the statement "there are an infinite number of primes" by rejecting the notion of "infinite number". If we reject the axiom of infinity then there simply is no set of all primes, just like how in ordinary set theory there is e.g. no set of all ordinal numbers.

[@jinks908]

​@@martinepstein9826 But that proof is equivalent to stating "there is no largest prime," since for any finite set of primes we can always find a larger prime. So how is rejecting the statement "there is an infinite number of primes" different than saying "there is a largest prime?" This seems to be a matter of pointless semantics since if there is no largest prime, then the primes must continue without end. It seems to me that saying there is no set of all primes is only philosophically different than saying there is an infinite number of them, not mathematically different. We seem to be arguing over what it means to be "infinite". To say "there is no set of all primes" is to say that they are _not_ finite, for if they were finite, there would be a set containing all of them.

[@martinepstein9826]

@@jinks908 It sounds like you're using unrestricted comprehension, the same principle that leads to Russell's paradox. A number is either prime or it isn't. Hence we can form the set of all primes. Likewise, a set either contains itself or it doesn't. Hence we can form the set of all sets that don't contain themselves... Judging from this comment section fans of Norman Wildberger think the existence of a set of all primes leads to similar paradoxes. Fortunately the "paradox" always turns out to be that infinite sets don't conform to their intuition about finite sets 😛

[@borisdatzar]

I think he would say that the proof is wrong and has a flaw. The classical proof of infinitely many primes is assuming you have a finite number of primes, you form a new number by multiplying these primes and adding one. Norman would object to this claiming that unless you can write down this number, you don't in fact have a legitimate number. By the way he has a point: if you asked a high school student what is 3x5 + 1 you would expect the student to write down the answer 16, not so simply repeat that the answer is 3x5+1

[@SavaConrad]

Suppose I had a Cauchy sequence defined by Newtons method that you’re talking about. Now tack on a term to the beginning. And another one after the one you just tacked on. Say I let that sequence converge to a different real number, and at some number n the sequence suddenly begins at 2 and to converge on sqrt15. Now since we like doing infinities, take the limit. That bundle of Cauchy sequences which converge to sqrt15 in the limit converge to, say, pi. You might say: hey! The infinitieth one doesn’t exist, and so it doesn’t apply to the set of Cauchy sequences which converge to sqrt15. Exactly. But wait… a Cauchy sequence is an infinite set of a list of an infinite number of elements. An element of the set of Cauchy sequences which converge to sqrt15 also belongs to the set for pi. And e. And every real number. Oops. Contradiction in definition. You are thinking: no that doesn’t apply because etc. the fact is, you’ve accepted choice and infinity so as to say that these constructions are possible. there’s an infinitieth element which is the real number specified, because you consider an infinite number of elements. Oops. Self defeating worldview.

[@SavaConrad]

Solution? Don’t talk about real numbers. Say: I can get successive rational numbers which satisfy this condition better and better.

[@billh17]

@Sasha said " An element of the set of Cauchy sequences which converge to sqrt15 also belongs to the set for pi. And e." This is not true. Let { a_i } be a sequence of rational numbers which converge to sqrt 15. Take any well known sequence { b_i } of rational numbers that converges to pi. Then, { b_i } belongs to the set pi (defined as a set of Cauchy sequences of rational numbers such that etc). But it easy to show that { a_i } is not Cauchy equivalent to { b_i }. By definition, this means that { a_ i } does not belong to the set for pi. Any two Cauchy sequence of rational numbers in the set pi are by definition equivalent Cauchy sequences.

[@portreemathstutor]

Rewritten with greater clarity.

[@vincentdiamond1707]

The Ancient Greeks understood that it is not possible to derive the continuous from the discrete. That is, a line (a continuous object) cannot consist of an infinite number of points (a discrete object). Whilst a line has size (called its length), a point has no size. Which implies that any definition of "real number" that requires "the completeness axiom or property" will always be flawed. One must begin with the continuous (e.g. Euclid's Axioms) and not with the discrete (e.g. Peano's Axioms). Today, most modern mathematicians are oblivious to this fallacy.

[@martinepstein9826]

Can you demonstrate this fallacy in some other way than calling on the authority of the Ancient Greeks?

[@peamutbubber]

Almost every measure in the real world is irrational, just because its easy to work with what we can compute doesnt mean we can just deny the existence of irrationality. If you want your true height in centimetres it will be an irrational real number, not a rational number. Being able to work with irrational numbers and infinities is especially important because real life is not rational in the slightest.

[@joaquincapellancruz7402]

Hey, I hope you're doing well. The problem is in my view, that irrational numbers in order to exist, need an infinite amount of information. If you can't understand/experience infinite information, given that what we call existence is all experience, therefore irrational numbers cannot exist. Although my argument is questionable, there is no way to represent an irrational numbers besides of naming it.

[@writerightmathnation9481]

At 24:42 you use the word “silly”. I’m afraid that I must not concur, as I think that that’s as disrespectful to his philosophy as is his claim that acceptance it use of the axiom of infinity is a” religion “. I suggest avoiding that kind of characterization of his position.

[@emiliomatos6071]

What I don't understand about his position is his necessity of dismissing as wrong areas of mathematics based on axioms he doesn't accept. It is a very acceptable position to unserstand that mathematics can accomodate for all these diferent views, and to conclude that all he needs to do is to develop his theory without Infinity and Choice to the maximum extent possible, what could possibly lead to interesting discoveries about computability theory, perhaps P vs NP, who knows? There is no need to dismiss everything else to do this, it simply doesn't follow.

[@dunai2012]

Defending existing theorizing is always easy to find flaws and evolve the theory.

[@elizabethharper9081]

Having read comments of fanboys, i have one thing to say: get some logic or set theory background, and then start your math revolution. Math is about solving interesting problems, not metaphysics.

[@DanielRubin1]

I agree about the primacy of solving interesting problems.

[@elizabethharper9081]

Maybe first part of comment sounds quite arrogant, but this is an actual advice i got from a logic professor at the start of my path.

[@ashnur]

14:15 I can tell you about such a problem: identity, continuity. It's a problem for privacy, for security. But it's a problem in literally every area of academic studies, and it's especially a problem in math, where such arcane ideas are kept alive simply to make it harder for people to understand the subject.

[@shashwatavasthi1423]

I guess the crux of the issue is whether you agree with infinite processes having an "end result" and as such this is something that has been thought about a lot, it is ironic that so many people talk about the "number line" when one has not even defined what that line means after you put in the rational numbers, and in fact that is the fundamental reason to my mind that one needs to allow formal completions to work, you cannot talk about the links between geometry and algebra without allowing for those limits. Wildberger to my view does not really disagree with analysis on the specifics given that in his world, all theorems with real numbers would be replaced by a sequence of rational number that is Cauchy. His real issue with the above would be generatikn of the sequence , which I don't know, why would one care about how it is generated , given that it is a formal object and defined as such to give a broad enough canvas to work with. His questions on" existence " in reality are not a concern since math is not at least in principle obligated to have any touch with reality, that is the job of the physicist

[@shashwatavasthi1423]

@Gennady Arshad Notowidigdo pardon me if this was not clear, but "he", "his" and all other second person pronouns refer to Wildberger and not to Dr Rubin.

[@shashwatavasthi1423]

@Gennady Arshad Notowidigdo >This is just like replacing a working computer with the exact same working computer. Any point he tries to make around this is just moot. My point exactly I do not think the idea of the line makes sense for the rationals because the geometric intuition that one wishes to capture, is that of the continuum, and that is not the case until you do the completion. and obtain the reals. I have never implied an inconsistency in the real number construction. What I am pointing out is that Norman takes offence to the idea that we are permitting infinite processes in such constructions, in the limited sense of limits and what he calls 'choice" sequences. My only submission is that one can have consistent, albeit maybe uninteresting math that comes about with these choices. Norman has not pointed any actual inconsistency with the construction, he has a problem at the level of definitions and axioms which are analogous to the rules of the game, he is not saying some rules have been broken, his claim is that the rules are themselves broken.

[@jlmassir]

The problem is not accepting the axiom of infinity. Axioms don't need to be accepted. They just define a game you may decide to play. The problem is a professional mathematician not being able to state clearly that he is proposing another game, and trying to convince people that there is something wrong with the previous game while offering only matters of taste as a justigication.

[@charliemoll5435]

Ok hilbert

[@samb443]

Axioms *must* be accepted, you, and many other people like you mistakenly call "assumptions" axioms. In the process you've stymied this unrelated discussion with your misunderstanding of English. Axiom certainly means "assumption", but its use case and connotation is reserved for THE fundamental assumptions underlying all math, that we must all agree upon. They're called the ZFC axioms, not the ZFC assumptions. If you have some other theory youre studying, it isn't based off of axioms, its got its own *assumptions*. Working mathematicians don't have the time to spend on deciding which game theyre playing every time they open a new article.

[@jlmassir]

@@samb443 I'm not sure what you mean. I said in my original post that if a mathematician (Wildberger) is proposing new axioms (i.e. alternatives to ZFC), he should be clear about that and tell what is wrong with ZFC, not just kinda say "I don't like them". Because one needs to acknowledge all the results these axioms brought to pure and applied mathematics, and by extension to all technology we have. Therefore, it seems we are more or less on the same page here on paying respect to ZFC. That said, I thing you're being a bit too dogmatic when you draw a distinction between axiom and assumption. The ZFC axioms had to be invented one day and may be replaced by something better in the future as "the" standard axioms. For a mathematician working on logic and foundation of mathematics, it is commonplace to propose new axioms, and she will sure have to understand which game is being played every time she opens an article in that field. Besides, it would be an error to lightly call "assumptions" the axioms of a new mathematical theory. It seems an excuse in case his theory fails. If Dr. Wildberger has another theory for the foundation of mathematics, he may very well talk about the axioms of his theory. I just find too bad that he doesn't do that, but instead keeps criticising ZFC and calling it a mere phenomenon of the "sociology" of mathematics.

[@samb443]

​@@jlmassir i was referring to >Axioms don't need to be accepted. They just define a game you may decide to play. the issue i took is purely linguistic: "Axioms" need to be accepted. Assumptions define games you may decide to play. Ive had numerous people (model theorists) make example "theories" wherein they call obviously ridiculous things "axioms" which is why im so sensitive to misusing the word. I agree wildberger doesnt do a sufficient job to replace ZFC with anything, though that doesnt make his criticisms invalid.

[@ashnur]

17:20 "foundations are not important so let's not deal with them" "you can't know which foundations are good without trying a lot of different ones"

[@anomos1611]

Surprised Godel didn't come up.

[@ashnur]

The interesting phenomena is fine, but trying to force a single framework over all such things is as dogmatic as it gets.

[@azzteke]

Phenomenon.

[@aaronmarchand999]

Norman seems to perhaps be afflicted with some degree of narcissistic personality disorder. At least that's the impression I get. It seems he doesn't really have anything of substance to back up his arguments, and accuses everyone else of being too philosophical when it is he himself the one arguing about the philosophical foundations of math

[@ashnur]

20:40 The problem with this is that you get stuff like Hilbert curves and Banach-Tarski, and also export ideas into physics that are incompatible with the real world and then basically handicap scientific thinking often for centuries, then turn around and claim success...

[@pmcate2]

How did those two theorems impede science? To my knowledge they didn’t.

[@ashnur]

@@pmcate2 Exactly.

[@pmcate2]

@@ashnur So which theories handicapped science?

[@ashnur]

@@pmcate2 i said ideas. and specifically those that promote thinking where we assume that the world behaves in a way which is incompatible with how it actually functions. A very good example is: time-travel to the past. But also that a segment (or a volume) is just an infinite set of infinitely small points.

[@pmcate2]

@@ashnur Ok so your argument is: analysis exists, therefore people waste their time of time-travel theories? 'But also that a segment (or a volume) is just an infinite set of infinitely small points' actual this property of analysis has proven very fruitful. How else, for example am I to calculate the behavior of light passing through a material if not to assume that material is continuous? Yes, every serious mathematician and physicist knows that it may not represent reality, so what is your point?

[@RichardAlsenz]

There is rational counting that a human can perform. There is no human method of counting irrationally because a human has a lifetime that ends, and each human will die. IF YOU DISAGREE, SHARE IT HEAR:?(

[@TheGloryofMusic]

I've heard Wildberger say that irrational numbers don't exist, but then claim that they might exist in the mind of God or the hereafter. So his real claim seems to be that they cannot be constructed. My question for him would be, take the closed set from 0 to 1 and remove the right endpoint. This seems like a "constructible" thing to do. Does the resultant half-open set have a greatest member?

[@santerisatama5409]

1/1 0/1 1/1 is a constructible interval Stern-Brocot style, as is also 1/0 0/1 1/0 in which the above row exists as partial generation. When we write 1/0 as both < and > and do the same mediant construction with operator language < > < > <

[@marcoghiotti7153]

I am a theoretical physicist, therefore most mathematical disputes usually do not bother me unless they are directly relevant to my field. In this instance, however, I quite firmly side with Prof. Wildberger, for at least one very reason: to me mathematics is fundamentally the study of how to count objects. If this premise is accepted, and it might not be for many of us, then I expect, or rather demand mathematics to give me exact solutions. Not approximations. If I were interested in the latter, I would rely upon physics or engineering. It seems to me that most modern mathematics is more concerned with axiomatic thinking than actual hard proofs. When I was a young student, I was taught that mathematics is synonym of exactness, rigour and hard logic. Well, apparently not really. Cauchy sequences, Dedekind cuts, real numbers are artificial constructs to provide us with a framework that 1) we can assume as correct 2) we can forget and move on to other topics. This approach would not be accepted in Physics for instance, as it would dangerously sound more like a religious belief than Science. A divine knowledge bestowed upon us never to be unveiled or questioned. Either you can prove your theory with an experiment, scalable and repeatable, or your theory is a conjecture and so it will be treated. Galilei taught us this method, Feynman elevated it to the highest intellectual peak humanly possible at this given time in our evolution. I however do appreciate the effort made by this channel to bring forth this important discussion, openly and honestly, regardless of your own stand on the matter. Keep up the good work

[@eccotom1]

Re: "demand mathematics to give me exact solutions. Not approximations". As far as I can tell, real analysis offers both? To even talk about an approximation to a number or a function, we need a well-defined notion of the number or function in the first place. Re: "axiomatic thinking than actual hard proofs": what does this mean? Those two go firmly hand in hand.

[@ashnur]

11:50 It is very rare for me to see someone so clearly state that they prefer cognitive dissonance all the time instead of trying to achieve a coherent state of mind.

[@turdferguson3400]

Coherence at the cost of stagnation is not desirable.

[@ashnur]

@@turdferguson3400 great name for such an elevated opinion, "stagnation", eh? since post modernism and believing that any choice you make creates some thing new is the only way to avoid stagnation, right? but we are the unreasonable ones?

[@turdferguson3400]

@@ashnur thanks, I got it from Norm MacDonald.

[@ashnur]

@@turdferguson3400 too bad it diverts attentions from the subject at hand :)

[@xjuhox]

Let Norman live in the darkness of *Plato's Cave* while we enjoy the infinite colours and fruits of the *Cantor's paradise* 😇 😇 😇

[@nedmerrill5705]

I feel the argument is akin to the number of angels on the head of a pin; i.e. highly philosophical.

[@MrLove8736]

There are some major problems with the axiom of infinity (besides it just being accepted as an assumption in set theory without any demonstration of its validity)- If natural numbers are defined as being finite, a collection of natural numbers has to be finite by definition, with any highest number under consideration being an enumeration of the set up to that point. If the numbers are finite, if the highest considered number is finite, the enumeration must be equal to this, being finite. If the axiom dictates that there is an infinite set of natural (finite) numbers, this is already a contradiction. A second contradiction comes from claiming that the successor function is closed, such that each time you add 1, the number must remain in the set, but that you can always keep adding 1. If you can keep adding 1 indefinitely, the set cannot be closed, and the set cannot be defined definitely and consistently. If the set is closed, it must have some kind of limit, whether or not it is something a human or computer can calculate. Another problem is that the meaning of "infinite" is never clearly defined in set theory. It is just taken for granted that some sets are "finite" and some are "infinite" but it is never clearly stated exactly what "infinite" or "finite" means other than just saying that infinite means it is not finite.

[@pmcate2]

Yes infinite mean not finite. And what does finite mean? That a bijection exists between a bounded subset of the naturals and that set. Now negate that statement and you have what infinite means.

[@MrLove8736]

@@pmcate2 That is not a very clear definition, though, since "finite" and "natural" are taken for granted without any explanation of what the higher numbers in the set of natural numbers are like and how they are different from infinite numbers, and it gives no explanation of the transition between the finite and infinite. Suppose that we are dealing with some very large number, "q", and we know that it is larger than any currently known finite number, but we are not certain if it is a very large finite number or if it is an infinite number. In principle, would there be any way to answer whether q is infinite using set theory?

[@MrLove8736]

@Gennady Arshad Notowidigdo What matters here is if these concepts are self-consistent and whether or not it is reasonable to accept them as being true. If advocates of set theory cannot address these questions- what happens in the transition between finite and infinite, how can the total amount of finite numbers be an infinite amount, if you have a number which you are not sure if it is infinite or finite, how can you tell, what distinguishing feature do infinite numbers have that finite numbers do not, etc, etc.. -then the distinction between finite and infinite is not clear to you and you don't really know what you're talking about. Suggesting there is or is not a bijection between N for this only works if N is defined in a consistent way, and only if you understand in detail which extremely large numbers are included in N and which are not and how to tell them apart.

[@MrLove8736]

@Gennady Arshad Notowidigdo As expected, as a typical set theory adherent, you didn't answer any of the questions I asked. What happens in the transition between finite and infinite? How can the total amount of finite numbers be an infinite amount? If you have a number which you are not sure if it is infinite or finite, how can you tell? What distinguishing feature do infinite numbers have that finite numbers do not? The axiom of infinity is stated in such a way that it contradicts itself in several ways. Godel's incompleteness theorems rely on Cantor's diagonalization arguments which are completely invalid. Godel was wrong. A good alternative to set theory is, at the very least, any theory that can address the questions I asked above without contradiction and does not rely on taking an axiom of infinity on faith.

[@MrLove8736]

@Gennady Arshad Notowidigdo You have chosen not to answer any of those questions about infinity because you believe only in set theory, and Set Theory is a weak, contradictory theory which DOES NOT PROVIDE AN ANSWER TO THOSE QUESTIONS. You are not open to alternatives, so you don't know the answers. The logic doesn't pan out on Cantor's diagonal argument or the axiom of infinity itself- If you start with axioms which are impossible and filled with contradiction, the logical conclusions you deduce from those axioms are pretty useless.

[@ashnur]

3:10 Usefulness, especially in abstract studies such as mathematics, where you rely on larger institutions embedded in even larger institutions, is dependent on your ability to clearly communicate. Usefulness in general is dependent on your ability to survive in the wild and to evolve. Neither has anything to do with truth, which I think is somewhat related to mathematics. Let me repeat: just because something is extremely useful, doesn't mean it's already validated it's own existence. It might be that it's a local maximum. Weird to see someone claiming such a thing as a serious argument.

[@ashnur]

0:35 Please, the word you are looking for is heterodox, not "unorthodox", "unorthodox" is basically admitting, that the worst thing you can say about it is that it's unconventional.

[@ashnur]

It's interesting that you want to call them numbers, but you don't want me to want to add them together. Every actual number in mathematics can be added together. In fact, we start learning about numbers usually by counting, that is adding one, then another, then another. How is it that your universal numbers are so important for you, but cannot even support the most basic operations. Like forget addition, you can't even compare real numbers. Actually, forget comparing, you can't even talk about real numbers only indirectly. If the procedures that we are using to produce these numbers are useful, which we agree they are, why is it so important for you to invent next to the procedures these fantasy land creatures that are called just like actual numbers, but bare no resemblance to them? It seems completely unnecessary to have real numbers for analysis to be just as useful as it is today. The only problem is that democratizing analysis would take away the pride of the mathematicians in their imagined superiority. Imagine if everyone could at least integrate and do multivariate function analysis on some intuitional level... but no, because we want real numbers more than smart people. It's useful for people to not understand math...

[@DanielRubin1]

I disagree with several assertions here. If you are willing to accept that equivalence classes of Cauchy sequences of rational numbers are well-defined, then you can show that the set of such objects forms an ordered field. I also disagree with the idea that this definition of a real number or the modern notion of a limit acts as an unnecessary barrier to students who want to learn math, as I've also heard Norman Wildberger suggest. My view is actually that Cauchy sequences, besides being well-defined, actually provide an intuitive, satisfying, and practical way of dealing with limits when the target is unknown or not guaranteed to be part of the same space. Will do a video on the construction of the reals soon!

[@ashnur]

​@@DanielRubin1 Thanks for the considerate response, I was worried that after the spam I did, I wouldn't warrant one. Cauchy sequences by themselves are not well defined, since I can't even compare them already. When you talk about Cauchy sequences that can be used for anything, I am claiming that you are doing something much simpler than real numbers. Then you introduce the idea of "equivalence class" as if that could be well defined, but it has all kinds of logical and philosophical problems by itself. What is a class? It's not well defined afaik. Looking forward to the video

[@dmitryezhov8352]

_Imagine if everyone could at least integrate and do multivariate function analysis on some intuitional level_. Yes, this is an interesting question: "What should be the analysis to make it intuitive?" The modern definition of a derivative is a logical operator in which too much mutually contradictory information is packed. The worst thing about this weird logical container is linearization. How can this strange logical operator be intuitively understood by students when they studying analysis? I think, for students to have an intuitive vision, the definition of the derivative must be completely and strictly algebraic without any linearizations and the use of limits. The level of natural intuition should be such that math students (and even more so practicing engineers) can easily see why the coefficients in the Taylor expansion are functions, and the coefficients in the Maclaurin expansion are fixed values. Also need a good answer to the question - what comes first, analysis and, as a his consequence, Taylor expansion. Or vice versa - mathematical analysis exists and work precisely because any function a priori has the form of Taylor expansion. My subjective vision is that the Taylor expansion is primary, even without math analysis and Humanity in general.

[@billh17]

@@ashnur asked "What is a class?" In the phrase "equivalence class", "class" is just a variation from using the word "set". That is, in this context, "class" is just "set".

[@ashnur]

@@billh17 but it's not, since sets elements are unique,and elements of a class don't have to

[@rigansmontes]

If you are a mathematician, you should be able clearly to answer questions like "What is mathematics?," "What is mathematics about?," "What method(s) should be used in doing mathematics and why?" I wonder whether you have answers to these simple questions. They are fundamental in settling the controversies raised by the dispute.

[@DanielRubin1]

Those are some broad questions subject to many interpretations. It can be fun to posit answers or think of what might constitute an answer, but I'm not sure anyone has convincing, all-encompassing answers. Although some amount of philosophy may be unavoidable for mathematicians, who need to justify their work and form preferences, it certainly isn't necessary and probably isn't a good use of their time to try to answer this kind of philosophical question. Rather than attempt to answer such questions, for people who are interested in this sort of thing, I think it's helpful to proceed descriptively, and identify what goes on in specific contexts.

[@rigansmontes]

@@DanielRubin1 Physicists can answer analogous questions. So can zoologists, biologists, anthropologists, psychologists, astronomers, and so on. Why should mathematicians be any different?

[@shashwatavasthi1423]

@@rigansmontes mathematicians are different because mathematical truth depends on sound deduction from axioms and as such this deduction can happen about a whole host of fairly disparate objects because the underlying framework (something like set theory maybe) is very broad. Physicists, and zoologists are concerned with understanding specific questions and as such have a much more constrained study: the oracle of truth for them is experimental success. Mathematics is fundamentally different. So I don't know if I am wrong but from what I understand : mathematics is about objects that can exist consistently in the framework of set theory, category theory and other such things, the methods allowed are literally anything that are consistent with the axioms, and the truths you are interested in, literally depend on the questions you are interested in. Mathematicians might impose much more structure on how they proceed with their enquiry, but mathematics itself is not as restrictive. Part of the thing with the physical or natural sciences is that their methods atleast on the experimental side are constrained by technology, since they are interested in first knowing the answer and then creating models consistent with those answers , mathematics is not typically concerned with observation, because the ground truth in mathematics does not require nature(a lot of great mathematics though comes from nature and it's analysis)

[@rigansmontes]

@@shashwatavasthi1423 Ancient Greek mathematicians could answer all of the above questions. In fact, mathematicians of all ages could--except for the last 100 years or so. This is a clear indication that there has been a significant rupture. What I find disturbing is the lack of reflection on the part of mathematicians. As Lewis Carroll put it in the mouth of the Cat, you're sure get somewhere if you only walk long enough. The trouble is, you don't know where you are going.

[@shashwatavasthi1423]

@@rigansmontes I mean just to push back a little, what I have described is the state of mathematics as I see it, a number theorist for instance would have a concrete answer(the following is my assumption, I am by no stretch a number theorist): they want to understand that structure of natural numbers and their extensions. They use methods from complex analysis, algebra, algebraic geometry, Fourier analysis etc for this. However might I ask why this clear answer needs to exist for all of mathematics in some unified voice? And why the lack of such an answer is so unsettling ?

[@infty1369]

continuity is an assumption.

[@infty1369]

OK, yes, replacement is underway

[@infty1369]

stopped listening at "fundamentally a philosophical problem"

[@marioalvarez4316]

They say that "those who don't learn from History are condemned to repeat it", well in this case those who don't learn from Philosophy are doomed to do poor Philosophy. You two are slaves to a philosophical disagreement about the nature of Mathematics, i.e. a philosophy of Mathematics. Norman's position is a classic position taken over and over again in the history of western philosophy. His position is very much a conservatism in Mathematics, via finitism, via "common sensical" ideas, via computable, via ..... I have answers to both of you children

[@ashnur]

For every integer there isn't another one, eventually you run out of space... if you assume you never run out of space then you are begging the question, your argument is circular. But also, you assume that any space can be infinitely divided, which is again, not possible, not how anything works. Like, you know that there is no global present moment in the universe, right? Right?

[@IsomerSoma]

What do you even mean by space? It's clear you have never studied pure mathematics. Not even one semester. Your insistence on mathematics having to be constructed to conform to some physical reality of finiteness is entirely arbitrary. Mathematics is the science of abstract structures. They don't at all need to be actually realizable in the physical world.

[@ashnur]

​@@IsomerSoma cognitive dissonance much? When you are hungry, do you eat your breakfast abstract?

[@IsomerSoma]

@@ashnur Mathematics is the study of abstract, idealized structures. How is this supposed to be cognitive dissonance? Your arguements make no sense.

[@ashnur]

@@IsomerSoma Maybe because it was not an argument. I only argue with people who come to me with an open mind. You seem to have decided that I have nothing to offer you, why should I waste time trying to convince you of anything. It would be insulting to your intellect.

[@IsomerSoma]

@@ashnur I have read some of your comments and you are greatly overestimating your own knowledge in mathematics. You project your inability to grasp some concepts or lack of enjoyment in some concepts onto the illusion that everyone else must be stupid or delusional. You accuse me of not having an open mind? Look at yourself. The computational part of mathematics is an important part and if you enjoy this part good for you but why do you feel the need to call anyone stupid who doesn't sees the alleged nonsensicalness of more abstract mathematics? You make flat out wrong statements such as that you can't compare or add real numbers. You claim that real Analysis is unintuitive - that may be true for you, but not for everyone. Also intuition can be acquired and learned.

[@writerightmathnation9481]

Here (@ 17:57), you’re disagreeing with Norman’s disavowal of the “added value” of post-19 th century analysis. I think that your argument is stated as overly subjective - opinion. It wiles benefit from specific examples, and I’m convinced that you can find many. For example, I’m convinced that there are useful algorithms in areas of mathematics to which Norman does not object which were developed specifically because analysts used mainstream mathematics to lead them to understand and create and implement such algorithms. Proving to Norman that an example you find has that property may be more difficult than you’d like, but providing more evidentiary support for your argument among your viewers, including the proverbial “choir” is probably even more important to you than getting the “blessings” of Norman or any of his followers.