It's true, math is a social activity, but I bet a lot of it exists without a preponderance of mathematicians even being aware of it, much less agreeing it exists. — jgill

In your professional research, did you have the feeling that you were investigating aspects of truths outside yourself that you were trying to find out about? Or that you were merely pushing around symbols in a formal game? I'd wager the former but I'd be interested to know.

In your professional research, did you have the feeling that you were investigating aspects of truths outside yourself that you were trying to find out about? — fishfry

I'm more or less continuously involved in light, unimportant research these days. As an example, I've defined a variation of linear fractional transformations in which the normally constant fixed points are themselves functions of the underlying variable. Then I search for criteria that produce convergence upon iteration. I definitely look for "truths" outside myself, and it's this exploration that's fascinating. Knowing that such truths depend ultimately on axiomatic structures has no bearing on my minor investigations, and the very thought of moving around symbols in a formal game is anathema.

I was once a rock climber and it was the delight of exploration that was intensely compelling. :cool:

Have you ever seen the cauchy sequence of a non-computable real number? If I claim that that Cauchy sequence is for the number 42, how could you challenge that claim? — Ryan O'Connor

We can embed the rational numbers in the real numbers this way. Let f(n) = 42 for all n in N (a constant sequence). Then the real number 42 is the equivalence class that contains f and all g such that |f(n) - g(n)| --> 0 as n --> inf. Another representative of 42 in R would be f(n) = 42 - 2/n. Clearly there are infinitely many representatives for each real number, and there are proofs that show that it doesn't matter which representative is used in computing sums and products, etc.

It's a pretty good system. The interesting stuff is (as you hint) the gulf between computable and noncomputable numbers

You may be right, but I'm of the view that we don't know exactly what we're talking about because there's more work to be done. — Ryan O'Connor

The point I was getting at in this context is something like: we often think we are talking about numbers when we are really talking about talk about numbers.

I agree that there's more work to be done, but there's already a mountain of stuff out there. After years of formal study (proof writing), I still would argue that intuition is primary and that math is a language.

Wildberger is a nut, his math doctorate notwithstanding. He does have some very nice historical videos and some interesting ideas. But his views on the real numbers are pure crankery. You should not use him in support of your ideas, since that can only weaken your argument. — fishfry

I don't know much about Wildberger (I remember him being passionate and unorthodox), but you touch on 'the' issue here, social reality. Any philosophical conversation about numbers is perhaps bound to get back around to authority and who has it. For instance: obviously you have every right to call Wildberger a crank and indicate that you find him anti-persuasive in this context. But we are in the strange situation of talking anonymously. If Ryan can't trust a PhD who uses his own name and face, why should he trust you or me? There's just no substitute for looking around the space and seeing what those in power are actually saying. Being in power doesn't make anyone right, but typically what people actually want is recognition.

After years of formal study (proof writing), I still would argue that intuition is primary and that math is a language — norm

What's your background, Norm? Another math proof on the forum? If so, welcome. :smile:

Hi ! I'll private-message you about that.

Hi ! I'll private-message you about that. — norm

Oh boy they're gonna gossip about the rest of us!

obviously you have every right to call Wildberger a crank — norm

I'm not the only one, Google around. And FWIW, I'm a crankologist. I enjoy reading math cranks and am familiar with the work of most of the prominent ones.

Oh boy they're gonna gossip about the rest of us! — fishfry

Well, I hesitated when it came to posting that, but I thought it would be rude to not reply publicly to such a friendly hello. Seems silly now. At the same time, I really like anonymity. (Since I want to freedom to react quickly and maybe say something silly, that anonymity comes in handy.)

I'm not the only one, Google around. And FWIW, I'm a crankologist. I enjoy reading math cranks and am familiar with the work of most of the prominent ones. — fishfry

I believe you, and personally I think you know your math very well, significantly more than me, clearly. But still, it matters that all of this is anonymous. To profess really is to profess, to take the risk and burden of professing, subject to accusations of being a crank, for instance. That's his proper name, probably the one he was born with, and he's publicly called a crank. A little part of me cheers for the underdog, though I wouldn't want to be a crank myself.

Anyway, if information hygiene is really the issue, I don't see how we can unironically play doctor in our masks. I'm not at all suggesting that anyone unmask...quite the opposite. Academia already exists, so what's needed is a place where people can play with ideas, take some risks.

That's his proper name, probably the one he was born with, and he's publicly called a crank. A little part of me cheers for the underdog — norm

You're right, he's a professor of math and he puts his ideas out there under his own name, and the likes of me throws rocks from behind my anonymous handle. Can't deny it.

You're right, he's a professor of math and he puts his ideas out there under his own name, and the likes of me throws rocks from behind my anonymous handle. Can't deny it. — fishfry

I decided to look him up.

The pure mathematical community depends on these and other fancies to support a range of “theories” that appear pleasant but are not actually corresponding to reality, and “theorems” which are not logically correct. Measure theory is a good example –this is a subject in which the majority of “results” are without computational substantiation. And the Fundamental theorem of Algebra is a good example of a result which is in direct contradiction to direct experience: how do you factor x^7+x-2 into linear and quadratic factors? Answer: you can’t do this exactly — only approximately.

By removing ourselves from the seductive but false dreamings of modern pure mathematics, we open our eyes to a more computational, logical and attractive mathematics –where everything is above board, where computations actually finish in finite time, where examples can be laid out completely, and where we acknowledge the proper distinction between the exact and the only approximate. This is a pure mathematics which is closer to applied mathematics, and more likely to be able to support it. It also gives us many new insights, more precise definitions, and theorems which are actually …correct. — Wildberger

https://njwildberger.com/

My first reaction is that this guy is another Cantor crank! But if he taught at Stanford at one point and is about to retire at some other school, presumably as a full professor, then he must 'know better. ' He must know how crankish he sounds and how bold he is being to abandon traditional foundations in some kind of informal constructivism.

He must know how crankish he sounds — norm

This isn't the time or place to discuss Wildberger's crankitude and I'll leave you to your research. FWIW he's one of two PhD-level math cranks I know, the other being Edgar Escultura. As I mentioned, Wildberger has some very nice historical expositions on Youtube and is a perfectly sane and smart guy, just cranky about the real numbers. Then of course there was the late Alexander Abian, a perfectly respectable mathematician who advocated blowing up the moon.

So in the end you agree with the notion that existence is contingent on opinion, and you simply differ on which opinions count. You just lost the argument methinks.

And what if I find a metaphysician who, based on two years of dialog with me, clearly hasn't bothered to learn the most elementary facts of mathematics? Why should I trust that individual's judgment about anything? — fishfry

What argument have I lost? "Existence" is a word which is being used here as a predicate. So we need criteria to decide which referents have existence in order justify any proposed predication. Naturally we ought to turn to the field of study which considers the nature of existence, to derive this criteria, and this is metaphysics. Mathematics does not study the nature of existence, so mathematicians have no authority in this decision as to whether something exists or not, regardless of whether it is a common opinion in the society of mathematicians.

If you are arguing otherwise, then show me where mathematics provides criteria for "existence" rather than starting with an axiom which stipulates existence.

Where do they live? And what else lives there? The baby Jesus? The Flying Spaghetti Monster? Pegasus the flying horse? Platonism is untenable. There is no magical nonphysical realm of stuff. And if there is, I'd like to see someone make a coherent case for such a thing. — fishfry

The eternal truths that I am referring to are different from your eternal truths because mine are finite. I don't need to assert the existence of an actually infinite entity beyond our comprehension. Nevertheless I believe that all truths are contingent on a 'computer', it's just that there exists a 'computer' that lives outside of time. For example, I believe that the laws of nature exist outside of time and that these 'eternal' laws reflect the 'personality' of the grand computer. If on the other hand the laws of nature somehow did exist within time, what laws allowed them to pop into existence? If such deeper laws exist, then it is those laws which I'm referring to as external truths. I believe that the only object which can live outside of time is the unmeasured wave function of the universe...or more generally, a continuum filled with infinite potential.

Well I don't understand. Contingent on what? If there is a Platonic realm after all, surely mathematical truths live there if nothing else. — fishfry

I believe that everything in actual existence is finite, even 'the grand computer' in which eternal truths live. And so 'the grand computer' can not actualize the set of all natural numbers any better than us. The actual existence of 5 is contingent on a computer 'thinking' about it. When no computer is thinking about it, it does not actually exist and it is meaningless to say that it has definite properties (akin to Schrödinger's cat). With that being said, I am comfortable saying that when a computer is 'thinking' about 5 that it will certainly be prime.

Wildberger is a nut, his math doctorate notwithstanding. He does have some very nice historical videos and some interesting ideas. But his views on the real numbers are pure crankery. You should not use him in support of your ideas, since that can only weaken your argument. — fishfry

I believe that, like Zeno, Wildberger is able to take our 'whole-from-parts' view to its limit to suggest that there are fundamental problems with it. We should value people who identify paradoxes as paradox leads to progress. However, I don't agree with Wildberger's resolutions. They do not offer the richness that infinite math does. If there indeed are paradoxes, the resolution should not be to weaken mathematics.

The axiom of infinity lets us take all of the numbers given by the Peano axioms and put them in a set. That's the essential content of the axiom.

The Peano axioms gives us 0, 1, 2, 3, ...

The axiom of infinity gives us {0, 1, 2, 3, ...}

The former will not suffice as a substitute for the latter. For example we can form the powerset of {0, 1, 2, 3, ...} to get the theory of the real numbers off the ground. But we can't form the powerset of 0, 1, 2, 3, ... because there's no set. — fishfry

If there is no way to reinterpret the Axiom of Infinity to apply it to potential infinity, then I'm inclined to reject it on some level. However, that doesn't necessarily mean that I entirely reject ZF. When working with ZF, we are always dealing with finite statements. Is it possible that these statements are the mathematical objects, not the sets which they are talking about? By 'actually exist', I'm trying to identify the objects that computers are actually working with. A finite computer can never work with an actually infinite set, it can only work with finite objects and potentially infinite algorithms.

You're talking about how you'd like mathematics to be, but you do it entirely in a castles-in-the-air manner without regard to even a minimal understanding of the mathematical context. Philosophizing about mathematics is fine. But when the philosophizing concerns actual mathematical concepts, then, unless there is an understanding of the actual mathematics and the demands of deductive mathematics, that philosophizing is bound to end as heap of half-baked gibberish. — GrandMinnow

I agree that I can only half-bake a mathematical idea. But even when mathematicians 'bake' an original idea at some point it's half-baked. And why must I bake it all by myself? I understand that you may not want to invest your time in evaluating half-baked ideas, but isn't a forum like this a good place to discuss them?

Ordinary calculus does use infinite sets. — GrandMinnow

Do limits require the existence of infinite sets?

The set theoretic axiomatization of mathematics is very straightforward, easy to understand, and eventually yields precise formulations for the notions of the mathematics of the sciences. — GrandMinnow

Do you believe that there are any paradoxes related to the set theoretic axiomatization of mathematics, and if so, is it fair to conclude that it's straightforward? ZF Axioms are rarely if ever mentioned in applied math (science, engineering, etc.).

If you are sincerely interested in the subject, even from a philosophical point of view, you should learn the set theoretic foundations and then also you could learn about alternative foundations that bloom in the mathematical landscape. — GrandMinnow

I have learned a lot of math independently, but I certainly have much to learn and realistically not enough time to learn what is needed to do everything by myself. May I ask, how much education should a person have before initiating a discussion on a philosophy forum like this?

And neither is there a Zeno's paradox with set theoretic infinity. — GrandMinnow

What resolution of Zeno's paradox are you satisfied with? Limits can be used to describe a process of approaching a destination but they cannot describe arriving there. So how does one arrive at some new destination?

I wouldn't begrudge philosophical objections to the notion of infinity. My point though is that one does not have to be platonist to work with theorems that are "read off" in natural language as "there exists an infinite set". The axiom that is (nick)named 'the axiom of infinity' does not mention 'infinity' and, for formal purposes, use of the adjective 'is infinite' can just as well be dispensed in favor of a purely formally defined 1-place predicate symbol. — GrandMinnow

Are your and fishfry's posts in agreement? As I said to fishfry earlier in this post, I believe that 'there exists an infinite set' could be a valid mathematical object. But I think one does have to be a platonist if they think that such an infinite set exists.

as you mention what you consider to be flaws in classical mathematics, as I said before, you have not offered a specific alternative that we could examine for its own flaws. — GrandMinnow

I totally understand where you're coming from. I'm sure you've dealt with many infinity-cranks in the past and this probably feels like deja-vu. I get that. But I don't like that you are labelling my view incoherent. That is entirely my fault since I haven't communicated it well enough in my post. I've actually produced a collection of videos roughly explaining my views, would you consider reviewing a couple of them:

Derivative Paradox: https://youtu.be/PSzqoQ9J5yg
Dartboard Paradox: https://youtu.be/LQmwZZNUMNA
Zeno's Paradox: https://youtu.be/_96tczP_eaY

Please note that I'm not forwarding you these links in an attempt to generate views. I gave up on my channel last year and have no plans to do anything with it. I'm only linking you to it since it may better communicate my view. If you don't want to watch the videos I could try explaining my views further on this chat...

Also, even though you're providing much resistance, you (and others like fishfry) nevertheless have generously given me some of your time by reviewing my posts and writing responses so please note that I'm very appreciative of that.

Then the real number 42 is the equivalence class that contains f and all g such that |f(n) - g(n)| --> 0 as n --> inf. — norm

I understand your limit-based 'algorithm' but would there ever be an instant in time when you would be sure that it's not 42?

we often think we are talking about numbers when we are really talking about talk about numbers. — norm

Yes, and I think we do the same about actual infinity. We don't conceive of actual infinity, we conceive of conceiving of actual infinity (using potentially infinite algorithms).

I understand your limit-based 'algorithm' but would there ever be an instant in time when you would be sure that it's not 42? — Ryan O'Connor

I'd say you'd want to look into the details, but a couple points:

In the mainstream version, incomputable Cauchy-sequences of rationals are allowed.
There is a strict definition for < and > that includes something like that instant in time where 'not =' is established.

In (quite different ) computable analysis (which you'd probably like if you don't already), equality is not a computable function. It takes arbitrarily long to see whether two numbers are different. Just think of decimal expansions. I can't tell whether 0.999999...[?] is different than 1 until I finally find a non-9 in the expansion somewhere, so there's no bound on the check for equality. I'm far from an expert on computable analysis. It's just something I looked into and that's a piece I vaguely remember.
Also, have you looked into Zeilberger? He's a maverick too, a bit of a finitist.

Yes, and I think we do the same about actual infinity. We don't conceive of actual infinity, we conceive of conceiving of actual infinity (using potentially infinite algorithms). — Ryan O'Connor

Right! We are in some sense actually talking about talk. Wittgenstein's beetle in the box aphorism applies here, and it's significant that he spent so much time talking about math.

Hi ! I'll private-message you about that. — norm

Oh boy they're gonna gossip about the rest of us! — fishfry

Not so, my friend. Norm is mathematically authentic, as are you and fdrake, and I will probably learn something from his posts, as I have from the two of you.

I'm of two minds about revealing anything about the expertise of math people on this forum. I realize the knowledge may intimidate some others and dissuade them from contributing their ideas. Or it might have the opposite effect of encouraging attacks on academia. Oh well, not a big deal.

I can't tell whether 0.999999...[?] is different than 1 until I finally find a non-9 in the expansion somewhere, so there's no bound on the check for equality. — norm

You can't tell by inspecting the digits, but at least 0.999... is computable so you can make some assessments by comparing the algorithms used to generate 0.999... and 1. The same cannot be said about non-computable numbers, which is what I was getting at.

have you looked into Zeilberger? He's a maverick too, a bit of a finitist. — norm

I've read about him and listened to him being interviewed on a podcast but I have only very briefly skimmed his website. I can't recall why but I left the podcast not being interested in pursuing his ideas further.

If there is no way to reinterpret the Axiom of Infinity — Ryan O'Connor

As I alluded previously, your "reinterpret an axiom" has no apparent meaning (surely not rigorous) other than as a vague personal notion. Axioms are formal syntactic objects; you have not provided any meaningful sense of what in general a "reinterpretation" of an axiom is.

Moreover, do you even know what the axiom of infinity is? Do you even know the basics of the first order language of set theory in which the axiom of infinity is formulated?

When working with ZF, we are always dealing with finite statements. — Ryan O'Connor

Formulas of ZF are finite sequences of symbols. But ZF itself is a certain infinite set of formulas. And an axiomatization of ZF is a certain infinite set of formulas. Yes, any given formula of ZF is finite, and any given proof in ZF is finite (indeed, not just finite, but algorithmically checkable). But the study of ZF goes on to considerations of infinite sets of formulas (and while the set of axioms itself is infinite, it is algorithmically checkable whether any given formula is or is not an axiom). .

isn't a forum like this a good place to discuss [half-baked ideas]? — Ryan O'Connor

The personal context you present is based in gross ignorance of the subject. I don't opine as to all discussions, but this discussion engenders an unfortunate trail of waste product - gross misinformation, misunderstanding, and confusion. Sometimes people who have an appreciation of the subject wish not to see such otherwise beautiful scenic trails left without being de-littered.

If you were sincere about this subject, rather than blindly swinging about flimsy claptrap, you would familiarize yourself with the basics of the subject. I know nothing about marine molecular biology, so I don't go into a thread saying "the notion of cell structures in mollusks needs to be reinterpreted." I don't go into a thread about neuroepistemology and say "the very notion of a neural network is not acceptable to me; I propose instead an inside-out interpretation instead of the classical outside-in framework."

Do limits require the existence of infinite sets? — Ryan O'Connor

The study of limits in ordinary calculus involves, among other things, functions on real intervals, which are infinite. Infinite sets, intervals, domains, ranges, functions, et. al are all over calculus.

Do you believe that there are any paradoxes related to the set theoretic axiomatization of mathematics — Ryan O'Connor

Paradoxes are related to set theory and foundations, of course. But formal set theory does not have paradoxes; it has or does not have actual formal contradictions.

how much education should a person have before initiating a discussion on a philosophy forum like this? — Ryan O'Connor

I don't have a general answer to such a question. But I would suggest for you this undergraduate sequence:

Symbolic Logic (suggest: 'Logic: Techniques Of Formal Reasoning - Kalish, Montague & Mar; supplemented by chapter 8 ('Theory Of Definition') in 'Introduction To Logic' - Suppes)

Set Theory (suggest: 'Elements Of Set Theory' - Enderton, electively supplemented by 'Axiomatic Set Theory' - Suppes)

Mathematical Logic (suggest: 'A Mathematical Introduction To Logic - Enderton)

What resolution of Zeno's paradox are you satisfied with? Limits can be used to describe a process of approaching a destination but they cannot describe arriving there. So how does one arrive at some new destination? — Ryan O'Connor

Zeno's paradox (at least as it is usually presented) is not a formal mathematical problem, but instead is a challenge to certain intuitive explanations of certain observable phenomena. 'arriving' and 'destination' are not, in this context, mathematical terms. Set theory is not responsible for disentangling every everyday common notion. Meanwhile, in mathematics we speak of the limit of a function at a point. It is quite clear and, as far as I know, it works for solving certain scientific problems.

Are your and fishfry's posts in agreement? — Ryan O'Connor

You may reasonably ascribe to me only what I post myself. I imagine that holds for any poster.

videos — Ryan O'Connor

I watched the one about Zeno's paradox up to the point you said, "mathematicians begin with an assumption [that] [an] infinite process can be completed". What exact particular quote by a mathematician are you referring to? Please cite a quote and its context so that one may evaluate your representation of it in context, let alone your generalization about what "mathematicians [in general] begin with as an assumption",

Set theory itself (at least at the level of this discussion), as formal mathematics, does not say "an infinite process can be completed". Set theory doesn't even have vocabulary that mentions "completion of infinite processes". And the assumptions of set theory are the axioms. There is no axiom of set theory "an infinite process can be completed".

You can't tell by inspecting the digits, but at least 0.999... is computable so you can make some assessments by comparing the algorithms used to generate 0.999... and 1. The same cannot be said about non-computable numbers, which is what I was getting at. — Ryan O'Connor

FWIW, I agree with Chaitin that noncomputable numbers are suspicious. I can't even show you one. I can only talk about them indirectly. But if one does reject non-computable numbers, then R has measure 0, which completely breaks modern analysis.

For context, my overall view is that many alternatives are interesting (you might also like smooth infinitesimal analysis, which has a nice intersection with dual numbers for autodiff), but no foundation has ever seemed 'just right ' to me. There's always some ugly weeds. In the end I'm a relatively carefree antifoundationalist who enjoys math as an excellent if imperfect language. IMO, there's a 'know how' at the bottom of things that perhaps can never be formalized or made explicit. In some ways the quest for perfect mathematical foundations is a miniature version of the metaphysical quest. The impossible mission is to automate critical thinking, to capture that know-how in rules as clean as those for chess. Some of the philosophers I like have made strong cases against the possibility of this automation. They can't provide a decisive proof precisely because language is a soft machine.

A possibility occurs to me: When people who don't study the actual mathematics of set theory hear about such things as the axiom of infinity or encounter the notion of an infinite set, they don't grasp it except on their own terms. They can only grasp it by imposing their own explanation of what they think it means, and with that imposed explanation they declare that the notion, such as that of an infinite set, is wrong. Then they never move past that stubborn misunderstanding, no matter how many times one suggests starting out by actually reading the basics of the subject.

In particular, the notion of a 'process' is imposed on axioms that don't mention 'process' at all. Granted, some mathematicians do mention stages of set building and levels attained (or, for example, see Potter's book in which he actually makes an equivalent set theory axiomatization as a theory of levels), for the purpose of providing an intuitive or philosophical framework work for thinking about the mathematics, while the actual mathematics is not itself liable for whatever difficulties may be found in such intuitive or philosophical frameworks.

Again, set theory rises to the challenge of providing a formal system by which there is an algorithm for determining whether a sequence of formulas is indeed a proof in the system. So whatever you think its flaws are, that would have to be in context of comparison with the flaws of another system that itself rises to that challenge. — GrandMinnow

That is a central point that I have made twice now. You have not responded.

Not so, my friend. Norm is mathematically authentic, as are you and fdrake, and I will probably learn something from his posts, as I have from the two of you. — jgill

Thanks! You'll probably learn more from the others, since I'm a philosopher/comedian at heart.

I'm of two minds about revealing anything about the expertise of math people on this forum. I realize the knowledge may intimidate some others and dissuade them from contributing their ideas. Or it might have the opposite effect of encouraging attacks on academia. Oh well, not a big deal. — jgill

To explain my personal attachment to privacy: we live in polarized times and I'm still in the job market. Corporations and academia look very sensitive to me when it comes to exciting opinions. In other threads I talk about charged subjects like suicide, pessimism, war, etc. Even though I am a 'liberal,' I'm the Bill Maher type of liberal. He can get away with it, because he's a comedian. I don't want to limit my options. It's not only prudence though. There are other reasons for other threads that I value anonymity, which may be slipping away from us in general (and is only imperfect now, anyway.)

The personal context you present is based in gross ignorance of the subject. I don't opine as to all discussions, but this discussion engenders an unfortunate trail of waste product - gross misinformation, misunderstanding, and confusion. Sometimes people who have an appreciation of the subject wish not to see such otherwise beautiful scenic trails left without being de-littered. — GrandMinnow

Perhaps we are both making this 'scenic trail' unpleasant for the other in different ways. Do you want to live in a country where the 'scenic trails' are exclusive to the 'privileged rich'? You are trying to find a way to reject my ideas without understanding them. Don't waste your time and simply disregard my posts. 

But if one does reject non-computable numbers, then R has measure 0, which completely breaks modern analysis. — norm

Can you give me an example of what would break down without non-computable numbers?

but no foundation has ever seemed 'just right ' to me. There's always some ugly weeds. In the end I'm a relatively carefree antifoundationalist who enjoys math as an excellent if imperfect language. — norm

I like that you admit that there are ugly weeds. So you're just satisfied ignoring the weeds? But you must enjoy the philosophy to some extent, you're here after all? Actually, I'd love to hear what you think these weeds are...

IMO, there's a 'know how' at the bottom of things that perhaps can never be formalized or made explicit. — norm

Can you explain what lies at the bottom that you don't think can be explained?

Do you want to live in a country where the 'scenic trails' are exclusive to the 'privileged rich'? — Ryan O'Connor

I appreciate that threads are open to posting by both well informed and less informed posters. That doesn't entail that misinformation, misconception, and confusion should not be called out for what it is.

You are trying to find a way to reject my ideas without understanding them — Ryan O'Connor

I have explained exactly how certain of your ideas are ill-conceived and how you disservice the subject on which on which you opine while ignorant of its basics.

You asked me to look at your videos. Upon looking at one, I found that near the very start, you made a claim that "mathematicians begin with an assumption that an infinite process can be completed." I asked you to please say what specific statement by a mathematician you have in mind so that we can understand its context and to see how it fits your claim as to what mathematicians assume. Your critique of classical mathematics itself makes assumptions about classical mathematics - most of them quite ill-founded. You are the one who is critiquing ideas about mathematics without understanding them.

1. Is flush with critiques of a subject while he is unwilling to inform himself of the basics of that subject by even reading an introductory textbook on it. No matter how many times it is pointed out that he is terribly confused on basic points, he will never just pick up a book on the subject.

2. Keeps confusing technical points. But he keeps eliding the corrections presented by resorting to the cop-out "I'm only talking about it philosophically", even though the philosophizing is a critique of a technical subject. Or he just skips over the decisive corrections, as instead he replies by adding even more diverting tangents.

3. Projects onto others that they are not giving him a fair chance, that they don't try to understand him. Yet he keeps skipping over the actual explanations from others as to where he is confused, incorrect, and ill-informed.

4, Thinks he is presenting an innovative alternative in the subject. Yet he ignores the work already done in the subject over the recorded history of man - work by people who have dedicated truly incredible intellectual curiousness, creativity, rigor, and industriousness, while responsibly submitting their work to the most exacting standards of the peer-review method. This includes even ignoring serious work in alternatives to classical foundations - work that may be aligned with what he himself is ineptly stumbling to convey. The literature blooms with finitist, computationalist, constructivist and myriad other alternatives. Yet he won't inform himself about them.

5. Finally resorts to umbrage and the sophomoric instruction to ignore his posts, presumably then not to comment on them. Even though it was just explained that at least one motive in commenting on his posts is to not leave his falsehoods, misconceptions, and confusions uncorrected. Also, anyone should understand that it is the prerogative of posters arbitrarily to read and comment on whatever they want and that saying "then don't read my posts" is likely a doomed instruction anyway.

6. His misconceptions center on the usual crank bugbear: infinity.

So 1 through 6. But do we dare say 'crank'?

It's worth noting that the challenges in the first post of this thread have been met. But hell if I know whether the poster understands that by now.

Can you give me an example of what would break down without non-computable numbers? — Ryan O'Connor

You may already know these things, but just in case:

The set of computable real numbers is countably infinite.

Countably infinite subsets of R have measure 0.

If we take out the noncomputable numbers in R, we are left with m(R') = 0.

This means that all subsets of R' would have measure 0, so that measure theory on the line would be dead.

The Lebesgue integral depends on measure theory.

It's 'the' mainstream integral (not the Riemann, whatever its old-fashioned charms.)

The mainstream real line is a vast darkness speckled by bright computable numbers, numbers we can actually talk about, numbers with names, while most of them are lost in the darkness and inferred to exist only indirectly.

For instance, if R has positive measure, then most real numbers are uncomputable, because the computable numbers have the cardinality of N (because we can enumerate Turing machines.)

Let me emphasize again that I don't specialize in foundations. Like every math student, I learned measure theory and the Lebesgue integral, so I can speak to the mainstream. It's plain to me that some of the other folks on this thread know much more about the nitty-gritty of logic and foundations.

I like that you admit that there are ugly weeds. So you're just satisfied ignoring the weeds? But you must enjoy the philosophy to some extent, you're here after all? Actually, I'd love to hear what you think these weeds are... — Ryan O'Connor

Here's one example. If you follow the construction of the real numbers in set theory all the way from the construction of the natural numbers, you witness complexity stacked on complexity. You end up with something like equivalence classes of equivalence classes of equivalences classes. The process is spectacular really. I felt proud of myself when I could follow it all of the way. I even worked on a few of my own constructions of R starting from Q (nothing remarkable, but I was engrossed as if I were sculpting.)

But when I do math, I don't think of R in terms of that glorious set-theory mess at all (though I do think in terms of naive set theory and subsets of R), and of course these constructions of the real numbers came after many spectacular applications of the calculus. One of my favorite math books is Analysis By Its History. It's full of quotes from mathematicians on foundation issues in historical context along with early results. Altogether it's a living, breathing culture. Now we have more knowledge but at the cost of hyperspecialization.

In my POV, foundations is its own fascinating kind of math. It doesn't really hold up the edifice of applied calculus, IMO. It's a decorative foundation. Humans trust tools that work most of the time. Full stop. We could have taken a more empirical attitude toward math from the beginning. I'm not saying that we should have. It makes perfect sense that mathematicians want theorems and that results become more and more complex and presuppose more. It's a maddening mountain of knowledge, and it takes years of work to master a tiny piece of it, and only a few people understand what you are talking about (pretty lonely and dreary unless you fucking love the math,)

Can you explain what lies at the bottom that you don't think can be explained? — Ryan O'Connor

I don't want to derail the thread, but I'm talking about ideas in Wittgenstein, Heidegger,...others. Groundless Grounds is an excellent single book on the topic.