Logical difficulties in modern mathematics (2012)

https://njwildberger.com/2012/10/13/the-problem-of-rigour-in-modern-mathematics/

by lainon • 7 years ago

98 90 7 years ago

effie• 7 years ago

If real numbers get this guy's goat, I think he will go ballistic when he hears about bra-ket formalism that quantum theory instructors foist upon students of physics.

agumonkey• 7 years ago

#sedenions

btilly9• 7 years ago

According to classical mathematics, only a countable number of finite definitions of numbers exist. And an uncountable number of real numbers exist. Therefore almost all real numbers that exist do not correspond to any possible finite definition of a number.

Tell me. In what sense does an abstract concept exist that has no possible definition or unique description?

panic• 7 years ago

The amount of paper in the universe is finite, not countably infinite, which means almost all natural numbers can't be written down either. But despite its philosophical dubiousness, the set of natural numbers is still useful. If we can prove things about all natural numbers, it doesn't matter how much paper we have; the things we prove will still be true for any individual natural number we can write.

It's the same idea for the reals; we don't actually care about the vast majority of the real numbers, but since it's hard to know ahead of time which ones we will care about, we might as well prove things about all of them!

ginnungagap2• 7 years ago

So you're arguing that a set can exist even if its elements don't? Because we can surely give a unique description of the set of real numbers (the unique complete ordered field up to iso) and this description forces it to be uncountable

aeneasmackenzie• 7 years ago

He hates Dedekind cuts, not real numbers. Real numbers exist all right, here are some: https://arxiv.org/abs/0805.2438

btilly9• 7 years ago

There is a distinction between analytically correct expositions, and ones which build on naive intuition to teach students. It is inappropriate to expect beginning students to follow a logically rigorous exposition.

That said, there are philosophical questions about truth, infinity, unknowable statements and the like. Mathematicians have by and large settled on a set of answers to these. Every statement is true or false, regardless of whether we know the answer, or even whether we can know the answer. Infinite sets exist, and are described by a known set of axioms called ZFC. Almost all real numbers that exist can never, even in principle, be written down or described in any meaningful way. (In what sense do they exist again?)

All of these statements are part of classical mathematics. Almost every elementary exposition will implicitly assume that they are try. Yet they can all be questioned, and their truth can never be settled in any absolute sense. However woe betide the student who dares question these in a math class.

AlexCoventry2• 7 years ago

Almost all real numbers that exist can never, even in principle, be written down or described in any meaningful way.

I don't think many people see that as a matter of philosophy.

btilly9• 7 years ago

Most people are uninformed.

This question strikes at the heart of the debate between Constructivism and Formalism. A debate about what it means for things to exist, statements to be true, and so on. This is very much a matter of philosophy.

To a Constructivist, most of classical mathematics is nonsense. And Constructivism is at least as logically consistent as classical mathematics.

More precisely any contradiction found in Constructivism necessarily will lead to a contradiction in classical mathematics. The converse is only partially true. Gödel did prove that a logical contradiction in the classical handling of infinity will lead to a contradiction in Constructivism. But a flaw in a specific set of classical axioms, such as ZFC, need not lead to a flaw in usual Constructivism.

hnuser355• 7 years ago

Well I don’t mean to be rude but many logicians and set theorists I know will be sad to find themselves out of a job if what you said about these questions being settled is true

btilly9• 7 years ago

I disbelieve your claim.

In particular I maintain that any competent logician or set theorist should be able to explain to you the sense in which the statements that I made cannot logically be settled, and also explain to you the extent to which the rest of mainstream mathematics accepts these statements as true.

jacobolus3• 7 years ago

Infinite sets exist, and are described by a known set of axioms called ZFC

This is the assertion of many mathematicians, but the justification for it is “this is convenient” and/or “we take this as an article of faith” (or often “I never really thought about it, but it doesn’t much affect my work day to day one way or the other”).

There is no way to prove that infinite sets “exist” by reasonable definitions of “exist”. Indeed, by a conventional definition of “exist” infinite sets pretty clearly don’t qualify.

Instead, mathematicians have redefined the words “exist” and “true”. In mathematics it now means something like “if we accept a particular set of non-obvious and rather handwavey premises, we will also accept any conclusions that result from symbolic manipulations thereof following our established formal rules.” [This is not a full or precise definition of mathematical existence; folks interested can do a search for those keywords and find piles of material.]

* * *

Personally I am happy to accept ZFC or the like because it is convenient and I can’t be bothered to work up an alternative system from scratch and carefully examine all of the conclusions that might follow from that, and whether ZFC is “true” or not doesn’t really affect me. It seems intuitively wrong to me, but I remain agnostic.

harf• 7 years ago

(i.e. one of the standard set theory axioms amounts to "an infinite set exists")

fmap4• 7 years ago

“I never really thought about it, but it doesn’t much affect my work day to day one way or the other”

In my experience, people won't come out and say it, but this seems to be what everyone is thinking. :)

The problem with this is that it is wrong.

Classical ZFC in particular is a very strong and specific set of assumptions* with a very tenuous link to any practical application. If you actually want to develop a useful bit of mathematics it makes sense to consider the "foundations" as a moving piece. It's a part of the design space for modeling your problem domain, not some god-given notion of truth.

You can translate between different logical theories by building a model of one in another, so it's not like you loose anything. But it's cooky to insist that we should start with ZFC of all things.

---

*) I mean that second-order ZFC has basically no non-trivial models, so there is no real way of extending ZFC to talk about domain specific aspects of your problems.

roenxi3• 7 years ago

Almost all real numbers that exist can never, even in principle, be written down or described in any meaningful way. (In what sense do they exist again?)

A profound question to reflect on; and interesting to contrast with modern science as a philosophical foundation that there might not be any continuous objects present in reality. Might be discrete all the way down.

In a sense, it seems like the uncomputable reals are an artifact of assuming continuity, ie, between any two numbers located on a line there must be more numbers. Part of the reason it is so unintuitive is we don't have any real lines to play with at the physical human scale, they fall apart at the atomic level and turn out to be non-continuous approximations.

umanwizard4• 7 years ago

I don't think "between any two numbers located on a line there must be more numbers" is the property you're looking for, since it holds for the rationals (all of which are, of course, computable).

Maybe "connectedness" is the notion you're trying to get at -- the real numbers are topologically connected, but the rationals aren't. If "A" is the set of rational numbers x with x^2 < 2, and B is the set with x^2 > 2, then the rationals are the union of A and B, and there is a "hole where sqrt(2) should be", so the rationals are disconnected. It's possible to define the word "connected" in a way that makes this notion precise.

A related notion is what's called "(sequential) completeness". The infinite sequence whose terms are (2, 2 + 1/2, 2 + 1/2 + 1/6, 2 + 1/2 + 1/6 + 1/24, ...), where the nth term is obtained by adding 1/(n!) to the previous term, intuitively "should" converge, since its elements get arbitrarily close together as n gets arbitrarily large. Any such sequence converges to a real value (this one converges to the exponential constant "e"). But if our number system is only countably infinite, there must be some sequences that get arbitrarily close together but don't converge. For example, if we restrict ourselves to rational numbers, this is a valid infinite sequence (every element is rational), and its terms get arbitrarily close together as "n" is large, but it doesn't converge to anything.

roenxi3• 7 years ago

I know Norman; not personally.

His real complaint as far as I've ever been able to determine is that he is a highly symbolic thinker; and because of that won't accept certain assumptions that everyone else takes as a given - usually what the Reals are. I'm very happy to accept than any length in geometry is a number by definition (hence sqrt(2), constructed by a 1-1-sqrt(2) triangle, is clearly a number corresponding to that length). He won't accept sqrt(2) as a number because it can't be represented in Hindu-Arabic notation. This isn't really a logical issue, he just won't use everyone else's definitions.

He's worth listening too because he is good at maths despite that handicap and his perspective is interesting to provoke a bit of reflection on what your assumptions are and what does infinity really mean anyway. His complaints are otherwise unlikely to catch on.

nokcha• 7 years ago

He won't accept sqrt(2) as a number because it can't be represented in Hindu-Arabic notation.

You sure of that? Based on another of his other blog posts [1], his objection seems to be about uncomputable real numbers. Very roughly, a real number R is computable iff there exists a Turing machine that, given a natural number n on its initial tape, terminates with the nth digit of R. See [2] for a formal definition. Sqrt(2) and all familiar real numbers are computable. Of course, since there is only a countable infinity of Turing machines, but an uncountable infinity of reals, some reals must be uncomputable. Some versions of constructivist mathematics do differ from standard mathematics by rejecting the uncomputable reals and instead defining "real numbers" in such a way that they are essentially the computable reals.

[1] https://njwildberger.com/2016/01/01/uncomputable-decimals-an...

[2] https://en.wikipedia.org/wiki/Computable_number#Formal_defin...

roenxi3• 7 years ago

Yeah, pretty confident. He doesn't like the computable or uncomputable reals; although I suspect he would like the uncomputables less.

Eg, "These phoney real numbers that most of my colleagues pretend to deal with on a daily basis ... such as sqrt(2), and pi, and Euler’s number e." [0]

Even in the article you cite, the irony of a pure mathematician of all people complaining that a concept has no tangible link to reality is a bit of a give away that he is speaking from the heart rather than the head. That isn't a valid complaint about pure mathematics; the point is patterns for patterns sake. So what if there are no known examples of your pattern? Study it anyway!

Great case of the flaw maketh the masterpiece; apart from that one little quirk with infinite things he is a lovely character and a force to be reckoned with. And I expect his personality motivates a lot of interesting research from him regardless.

[0] https://njwildberger.com/2014/10/06/the-infinitely-real-delu...

zzless2• 7 years ago

I am curious as to what his view would be on, say, the positive solution to the 3n+1 problem (i.e. every 3n+1/2 sequence eventually becomes a cycle). Some of these sequences may be profoundly long, so long that in his view, they may not exist. Or what if someone proves that 3 appears only finitely many times in the decimal expansion of sqrt{2} (God forbid!). How is the proof of something simple, say the solution of the Koenigsberg bridge problem by Euler any different: after all he did not enumerate all the possible paths! Even if he could, it is easy to find a manageably sized graph with too many possible paths.

umanwizard4• 7 years ago

I'm very happy to accept than any length in geometry is a number by definition

Interestingly, that set of numbers is still very incomplete relative to what we expect to be able to talk about in modern math. It doesn't even include the roots of all polynomials (for example, the unique positive solution to x^3 - 2 = 0 isn't the length of any constructible segment in classical geometry).

nimonian• 7 years ago

What does 'in geometry' mean in this context? There are numbers which are not constructible with compass and straight-edge constructible but which are constructible by other means (e.g. 2^(1/3) is origami constructible).

RGamma• 7 years ago

I watched a couple of videos of his "Foundations of Maths A" series and wasn't very impressed (even with my limited undergraduate knowledge). For instance his arguments in his videos about set theoretic constructions aren't very rigorous or convincing. It's like he missed all the developments in category theory, type theory and logic w.r.t. those topics.

Then I watched a couple of his more advanced videos and (from my limited watching) saw that he seems not so crazy after all. It's just that he seems to like natural numbers and finite constructions a lot, although I didn't really fact check that much. Infinite and more abstract structures _abound_ (it's in their nature :P) in mathematics obviously and can be encoded symbolically just fine.

Seems fine by me, finite structures are very important as well and you can make reasoning about them very rigorous. It's just, maybe he shouldn't be teaching about all those other kinds of topics...

gizmo6862• 7 years ago

Sqrt(2) is algebraic, and can therefore be constructed fairly explocitly. Starting with integers, you can construct the rationals as an equivelence class of ordered pairs of integers with a particular definition of addition and multiplication. From their you can define polynomials with rational coeficients, and from there you can define quotient fields, Q[x]/<x^2-2> which contains two elements whose square is 2, and is isomorphic to a subset of R (eg, behaves as you would expect numbers to). The most problamatic step of the above is defining polynomials.

umanwizard4• 7 years ago

Can't you just define polynomials as finite sequences of rationals with nonzero last term? E.g. encode x^2 + 2x -1/2 as (-1/2, 2, 1), and define evaluation at a given point in the obvious way.

fanzhang2• 7 years ago

Did the guy ever deliver on his promise to create a better system? I thought the problems he was pointing out was motivation for a better system (that united the naive understanding and rigor better?), but the further into the article I got, the more it just seemed like a list of complaints.

Having 1) studied "naive statistics" when I was young and 2) learning the full-on measure-theory version a la Bourbaki, and now 3) using applied statistics every day, I can say the naive version is the most useful by far. Yes, there is hand waving, but the hand waving works, and you really don't get into that much trouble unless you try to be pathological.

I admit, it would be great to have a version where the naive intuition (many of which are motivated by empirics, physics, and real world situations) and the theoretical definition matched better. Even still, I don't think the formal treatment is appropriate as an introduction -- it's an unnecessary hazing on learning minds when they can get most of the value with a bit of hand waving.

wjn0• 7 years ago

Interesting that you chose statistics as your example. I've always considered statistics to be the one branch of math I've studied where intuition keeps pace with depth and rigour. The caveat being that I am by definition an amateur when it comes to any other branch of math.

psykotic4• 7 years ago

"Measure theory a la Bourbaki" is probably not what you meant if you did probability theory and statistics. Bourbaki famously sidestepped classical measure theory with sigma algebras by constructing Radon measures as functionals via functional analysis. This is sufficient for many purposes but not for probability theory. First and foremost, sigma algebras in probability theory are not just a technical device to avoid paradoxes with unmeasurable sets. Sigma algebras play a starring role in conditional expectation, stochastic processes and martingale theory.

The best way to appreciate the information-theoretic role of sigma algebras is to look at them in the simplest case, where you have a discrete-time, finite-valued process. Then a sigma algebra is equivalent to a partition of the state space and it represents the information that can be gained from an observation; it's like a random variable without specific values, just the discriminating information from different outcomes. To say that a random variable is measurable with respect to the sigma algebra is to say that its value may only depend on information that can be gained from an observation. A filtration of sigma algebras corresponds to a causal series of observations where the observer learns more information over time.

The conditional expectation of a random variable with respect to a sigma algebra (or partition or other random variable) is another random variable that tells you the expectation over the states consistent with a given observation; this new random variable is measurable with respect to the sigma algebra you conditioned on, which as mentioned earlier means it only depends on the information gained from an observation. The conditional expectation is the best least-squares estimator given the information from an observation in the same way that the usual expectation is the best least-squares estimator given no information.

heyitsguay4• 7 years ago

As someone who moved from pure mathematics -> applied mathematics -> machine learning, perhaps I can offer a similar perspective to GP that explains why the measure theory stuff might not seem so useful to some people. Basically, in application you never need to worry about anything but your simplest case - that of a discrete-time, finite-valued process. All of the subtleties of measure theory, which occupy most of what mathematical measure theorists/analysts work on, are introduced when you assume you've got an infinity or a continuum somewhere in a model, and that assumption is never borne out in practice. Unless you really want to do theoretical math, you're better off learning only the aspects that relate to finite functions, i.e. almost none of what a real mathematician would consider to be "math". And that's not just out of laziness or disinterest - once you've restricted yourself to finite functions for empirical applications, you encounter a whole new realm of engineering difficulties to deal with, that theoretical mathematics doesn't touch. You've got to learn how to solve a different class of problems that aren't as purely logical, but are every bit as challenging.

To put it another way - think about all the pathologies you may encounter working with even a "nice" function space like L^2(R). You'll never deal with those pathologies in reality, because every empirical function is much better behaved - finite domain, finite range, and even if you assume a continuous domain, you can choose a model that only has a finite number of discontinuities, is Lipschitz continuous in between them, has finite total variation, etc. And that's why the hand-wavey, "intuitive" approach works so well if you're not a theorist, at least in my opinion.

tnecniv2• 7 years ago

I've only studied measure theory in a probability context. Can you talk a little bit about why the Radon measure isn't sufficient there, and what it is useful for outside of probability?

alimw• 7 years ago

It's as if I was trying to get from Camden to Islington and was given directions via Edinburgh. Suffice it to say, if I had a vague idea in the first place it's gone now.

fanzhang2• 7 years ago

Correct on your first point. I do not mean I learned the specific version published by Bourbaki, but rather in the modern mathematical style (hence the "a la", as opposed to 19th century hand waving style that nearly all undergraduate first-courses do).

And by the modern style, I meant starting out with analysis, defining the axioms of what measures are, demonstrating the existence of nonmeasurable sets with the axiom of choice, etc. IIRC, the course did rely on Borel algebras for its buildup, but did not openly buildup from sigma algebra machinery.

hyperpallium• 7 years ago

He did at least do the MathFoundations youtube videos he mentioned (pity he didn't edit a link in) https://youtube.com/playlist?list=PL5A714C94D40392AB

spindle• 7 years ago

There are MANY foundations for set theory, some of them very different from each other. It's completely pointless to treat the subject as briefly as this post does.

zzless2• 7 years ago

He is obviously a well trained mathematician so reading his pieces (of which there are many) is akin to watching a tsunami come to shore---you cannot look away yet you know that this is a tragedy unfolding.

This said, his complaints about set theory would have been (probably) more convincing if he stated the axioms correctly. The Infinite Set axiom does not simply say that some nebulous infinite set exists it states the existence of a set with some very specific properties. His complaints about the undefined nature of a 'property' are also not above criticism: GB theory (equivalent to ZF) eliminates it completely.

gchpaco• 7 years ago

In general his complaints regarding logic are incoherent. A function is not a messy weird object in mathematics; it's a possibly infinite set of ordered pairs where there are no duplicates of the first element of the ordered pair. That's it all it is, formally. Or "class" is a concept that has highly specific meaning in NBG theory—in ZFC it's not a formal distinction and so there's a transformation that must be obeyed where "x \in Class" means "{ x | x satisfies class definition}"; which due to Foundation must be the subset of some other, preexisting, class (and thus is Russell's paradox defeated). Almost all of these things exist and are present and are extremely well defined in the theory, he just doesn't like them for some reason.

unimployed• 7 years ago

I'm curious to hear what this guy thinks of the esoteric math in higher level economics... how base assumptions there are built and extrapolated on to ultimately guide economic policy. He would have a field day with the assumptions used in DSGE (which there are empirics to contradict those assumptions, and economists keep using those assumptions anyway).

steve76• 7 years ago

I would start with identity, that's all everything boils down to, and that math is as much a language, a shorthand to express ideas and communicate across the world like some state department, as it is computational.

Then I would explain that imaginary numbers mean harmonics. exponential functions give you a derivative, one function can serve as input to another, PID control, ... ... and that's about it.

GolDDranks• 7 years ago

After listening to Wildberger's rants – some of which are very educational and some of which are just rants, I keep thinking that his actual problem is that he doesn't seem to believe in the implications of the axiom of the excluded middle. He goes on about infinities and such, but the deeper issue seems to be that he thinks in a constructive, intuitionistic way whereas the majority of mathematics uses classical logic and even accept the axiom of choice.

I wonder why he never talks about this – surely as a mathematician he should be aware of intuitionistic logic? (Or is it that as a computer science and linguistics wannabe, I am aware of it but many mathematicians aren't bothered to take a look?)

If you stop thinking of axioms as value judgements and instead as definitions of formal systems (or adopt a more general system that encompasses others, such as Gentzen style sequent calculus with the only axiom being modus ponens), you achieve a piece of mind. But obviously his beef is not only that; he would like other people to admit the value of the thinking he prefers. And I agree with that.

theaeolist• 7 years ago

While reading his rant I kept thinking it's a pitch for HOTT or Univalent Math. Many of the motivations are similar.

posterboy• 7 years ago

In what way can kinds of constructive logic play a role in linguistics, in your opinion?

ZoltanAK• 7 years ago

I keep thinking that his actual problem is that he doesn't seem to believe in the implications of the axiom of the excluded middle. [...] I wonder why he never talks about this – surely as a mathematician he should be aware of intuitionistic logic? (Or is it that as a computer science and linguistics wannabe, I am aware of it but many mathematicians aren't bothered to take a look?)

Heck, I know professional logicians (mostly model theorists) who have never seen intuitionistic logic, in any context, ever. That said, Norman does know a fair bit about intuitionistic logic and constructive mathematics, and even some type theory. He does not like the underlying philosophy any more than he likes classical mathematics.

Gentzen style sequent calculus with the only axiom being modus ponens

Sidenote: a calculus where modus ponens is an axiom is definitely not a Gentzen-style sequent calculus.

aragnoil• 7 years ago

I'm not sure if he's a finitist or ultrafinitist [1] but the uncountable reals are clearly a problem for him. I'd like to argue for their usage from a "soft" point of view as opposed to just giving axioms.

Mathematics and physics both began with the advent of astronomy: Babylonians and others were curious to trace star patterns and from there both physics and math developed in tandem and influenced each other greatly. Really, neither would have developed without the other.

Calculus was invented for calculations in physics. This gave rise to differential equations which we use to model so many nontrivial things. The differential eqns describe flow and continuity and arguably reality. Differentiation and smoothness can't be defined over finite sets in the same way. My philosophical counter-argument to finitists is that clearly we're on the right path to understanding the universe and nature when using the reals. It seems foolish to shy from this because computers have trouble computing some functions. Statements like "there are only finitely many atoms in the universe" don't improve our understanding of much but PDEs explain.

[1] https://en.wikipedia.org/wiki/Ultrafinitism

jacobolus3• 7 years ago

Mathematics and physics both began with the advent of astronomy

Arguably mathematics began with accounting and surveying and trade, and more generally with counting and measurement.

pfortuny• 7 years ago

Never underestimate the power of taxes. On land, especially, for Geometry.

kccqzy• 7 years ago

A large part of his article seems concerned with the deficiencies and the lack of rigor in first year calculus textbooks. But I think, pedagogically, isn't it appropriate to introduce a subject intuitively and hand-wavingly, before proceeding to a more rigorous treatment? There's a reason students learn calculus before analysis.

pron10• 7 years ago

When discussing a certain discipline, be it mathematics or any other, the issue of rigor and foundation are completely separate. Rigor is established if you are careful to always follow the axioms determined by the foundation. But the foundation, i.e. the choice of the axioms can only be established in another, lower level, discipline, which, in the case of mathematics is philosophy.

In the early decades of the 20th century there were furious philosophical debates about the philosophy of the foundation of mathematics (between Brouwer's Intuitionism and Russell's Logicism, and also Hilbert's compromise of Formalism), but they all stemmed from an underlying assumption that mathematics gains its validity from some notion of philosophical truth. One could argue to no end on how the truth of mathematics is established, but a different perspective later emerged that avoids this debate altogether: that mathematics takes its validity not from truth but from utility (which is always relative to a specific task). We cannot say that one of those views is unacceptable, and if your position is that mathematical validity stems from utility, we cannot tell you that your foundation is shaky if its utility is established.

MaxBarraclough8• 7 years ago

But axioms aren't assertions of truth. They're not declarations from on high about universal truths, they're how mathematicians ensure they're talking about the same things.

Mathematics is about exploring the consequences of your axioms. Axioms are chosen, not dictated by the universe. If your axioms turn out to be inconsistent, well, congratulations on successfully showing that consequence of those axioms.

Case in point: the axiom of choice. There is no objective truth value to it. The physical world gives us no 'answer'. Mathematicians can choose to adopt it, or not, and then explore the consequences. ( https://en.wikipedia.org/wiki/Axiom_of_choice#Statements_con... )

pron10• 7 years ago

Well, that certainly was not the common perception in the philosophy of mathematics in the first half of the 20th century. Sure, they believed mathematics is an exploration of the axioms, but also that the axioms must be true in some deep philosophical sense. Brouwer wrote: [A]n incorrect theory, even if it cannot be inhibited by any contradiction that would refute it, is none the less incorrect,[1] and Russell believed that all of mathematics could be deduced from laws of logic that are true in the most absolute sense.

It is also doubtful that this is the perception today. If we want to use mathematics to derive any result about the physical world -- and we most certainly do -- our axioms must be consistent with it. How do you ensure that that is the case?

But such questions are not part of mathematics itself, just as the scientific method is not part of physics, but rather belong in a more fundamental discipline, called foundations of mathematics, or the philosophy of mathematics, which is usually studied by philosophers/logicians.[2]

[1]: On the Significance of the Principle of Excluded Middle in Mathematics, 1923

[2] https://plato.stanford.edu/entries/philosophy-mathematics/

tome• 7 years ago

Did anyone understand this bit?

The approaches using equivalence classes of Cauchy sequences ... suffer from an inability to identify when two “real numbers” are the same

Perhaps there's no computable general method, but it seems like it highlights an even bigger problem which remained unspoken. There is an inability to tell when a sequence of numbers approaches zero!

zozbot123• 7 years ago

Yup, you can define distinctness of constructive real numbers as two algorithms returning rationals that will differ by more than the approximation error ε they were given as input, for some arbitrarily small ε - but equality involves proving that two algorithms will never be apart in this way - and this cannot be done in general. As you say, even comparison with zero is problematic. So instead of equivalence relations, you need to define the apartness relation - the notion that a supposed equivalence between two such numerical algorithms can be constructively refuted. This is nothing new - it's a well-known part of constructive math, and is also the sensible way to formalize numerical analysis.

quantum_state• 7 years ago

Tried watching his video on limit ... but what was said prevented me from wasting my time further. Contrary to the claim of math foundation, it is apparent that the guy does not understand math despite being a stats professor. It is not even wrong.

Logical difficulties in modern mathematics (2012)

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