The constructive reals aren't complete because there are too few of them, only countably many — fishfry

Too few...or too many? The subset of computable total functions that correspond to the provably convergent Cauchy sequences form a countable and complete ordered field, that is a proper subset of the provably total functions.

Here's my definition of infinity, and for simplicity I'm only referring to positive infinity: infinity is a number, but it has a characteristic that all real numbers do not possess. — Michael Lee

Good luck reading. MO as you know is a site for professional mathematicians so the best one can hope for is to understand a few of the words on the page. — fishfry

The standard reals are the Goldilocks model of the reals. Not too small and not too big to be Cauchy-complete. They're just right. And are therefore to be taken as the morally correct model of the reals. — fishfry

My current understanding is that there exists indeed a detailed description of the infinite model(s) for real numbers but at this point I am unable to pierce through the dense vocabulary and concepts in order to develop a correct mental picture on the matter.

Concerning the phrase "infinity is a number, but it has a characteristic that all real numbers do not possess", in my impression, it does not adequately reflect the breath and the depth of existing knowledge on real-number model(s). I personally feel that this summary is overly simplistic.

The constructive reals aren't complete because there are too few of them, only countably many
— fishfry

Too few...or too many? The subset of computable total functions that correspond to the provably convergent Cauchy sequences form a countable and complete ordered field, that is a proper subset of the provably total functions. — sime

Too few, clearly. There are only countably many of them.

I do apprehend the point that the computable reals are computably uncountable, since there is no computable bijection between the computable reals and the natural numbers.

So what? The moment after Turing defined what it means to be computable, he showed that there are naturally-stated problems that are not computable. Point being that even computer scientists recognize the existence of noncomputable phenomena. See Chaitin's Omega, for example.

So yeah, there's no computable bijection. But there is a bijection, just as sure as there are only countably many Turing machines.

And no countable ordered field can be complete. It's a theorem.

My current understanding is that there exists indeed a detailed description of the infinite model(s) for real numbers but at this point I am unable to pierce through the dense vocabulary and concepts in order to develop a correct mental picture on the matter. — alcontali

I'm not sure exactly what you're looking for. To my knowledge, and I'm no specialist in these matters, the second-order theory of the real numbers is categorical, which means there is only one unique model up to isomorphism.

On the other hand set theorists do study alternate models of the reals that arise if you change the axioms of set theory. For example there's a famous example of Solovay in which, in the absence of the axiom of choice and the presence of an inaccessible cardinal, all sets of reals are Lebesgue measurable. This kind of thing may be of interest to you if you're curious about alternative models of the reals.

https://en.wikipedia.org/wiki/Solovay_model

I shudder to think of what would happen here if the posters on this and other threads with minimal mathematical knowledge apart from set theory and logic were to launch investigations into subjects like functional integration or even metric spaces or advanced calculus. But maybe there is a hidden reservoir of mathematical understanding just waiting for opportunities for expression. I may try starting a thread and see what happens. I know several of you have significant mathematical depth. But others? Not so sure. :smile:

I shudder to think of what would happen here if the posters on this and other threads with minimal mathematical knowledge apart from set theory and logic were to launch investigations into subjects like functional integration or even metric spaces or advanced calculus. — jgill

Stephen Wolfram wrote something very relevant in that regard:

Curating the math corpus. So how big is the historical corpus of mathematics? There’ve probably been about 3 million mathematical papers published altogether—or about 100 million pages, growing at a rate of about 2 million pages per year. And in all of these papers, perhaps 5 million distinct theorems have been formally stated. — Stephen Wolfram on 'Curating the math corpus'

Being knowledgeable of say 1% of these 5 million theorems, i.e. of 50,000 theorems, is probably already overly ambitious.

Hence, that cannot possibly be what it is about.

Furthermore, it does not make sense to memorize these theorems along with their proofs, because it would turn such person into an incomplete and rather useless sub-database machine of the (curated) math corpus. Either you use the machine, or else you build the machine, because in all other cases you are just a slow, failed, useless, and sorry excuse for a machine.

People investigate what they are interested in.

There is no reason why that would necessarily include functional integration, metric spaces, or advanced calculus, none of which would give that person any understanding in other math sub-disciplines such as for example elliptic-curve cryptography or in zero-knowledge succinct arguments of knowledge.

Vitalik Buterin is a good example of what I mean:

He attended the University of Waterloo but dropped out in 2014, when he received the Thiel Fellowship in the amount of $100,000,[10] and went to work on Ethereum full-time.[10] — Wikipedia on Vitalik Buterin

Vitalik does not just write about top-level mathematics, as in his medium article series:

This is the third part of a series of articles explaining how the technology behind zk-SNARKs works; the previous articles on quadratic arithmetic programs and elliptic curve pairings are required reading, and this article will assume knowledge of both concepts. — Vitalik Buterin's article intro

Vitalik successfully implemented this mind-blowing math in the ethereum source code. As Linus Torvalds famously said:

Talk is cheap. Show me the code. — Linus Torvalds on 'cheap talk'

Why would someone like Vitalik Buterin even be interested in functional integration, metric spaces, or advanced calculus? At the age of 20 he had already become a multimillionaire from his deep understanding of the math subjects that truly mattered to him, while many PhD graduates in math are associate-lecturers living off food stamps:



These people are not just on food stamps, many of them are also widely despised for their runaway arrogance. They should take an example to Vitalik, in order to improve their lives, known to suck. Vitalik, on the other hand, is a very humble person. He is friendly and pleasant. He does not try to "prove" that other people "know nothing". He is just a good human being, albeit stinking rich too. ;-)

I shudder to think of what would happen here if the posters on this and other threads with minimal mathematical knowledge apart from set theory and logic were to launch investigations into subjects like functional integration or even metric spaces or advanced calculus. — jgill

Happy to put this into perspective.

First, this is a philosophical message board and not a mathematical or a general purpose on. It's natural that when math comes up, it's in the context of logic, mathematical logic, set theory, category theory, alternative foundations, constructivism, etc. Those are the parts of math that touch on philosophy.

Second, on a site like this each person brings their own knowledge and experience to the table. Many people these days come to mathematical topics through computer science or related disciplines. So they may know quite a bit about constructive math or category theory as applied to logic, or things like Boolean algebras and so forth, without necessarily having a traditional math major background in analysis, abstract algebra, and so forth. There's no reason to "shudder" at the fact that you know things others don't. Perhaps others know things you don't. I daresay your own mathematical orientation includes advanced knowledge of some things, and maybe not so much on others. You don't want us to shudder at what you know and what you don't, right? People who live in glass houses should live and let live, I say. Don't you agree?

But maybe there is a hidden reservoir of mathematical understanding just waiting for opportunities for expression. I may try starting a thread and see what happens. I know several of you have significant mathematical depth. But others? Not so sure. :smile: — jgill

You're free to start any kind of thread you like; and if the moderators, of which I'm not one, see fit to let it stand, then it was good. Else not.

But as much fun as it would be, this isn't really the place to talk math. There are some math-oriented sites, one of my favorites is https://mymathforum.com/ . If you go over there are start talking about philosophy you'll be off-topic; but your advanced math comments and questions will be welcome. Likewise there is the famous https://www.physicsforums.com/, which has some pretty decent mathematicians. They're strictly anti-philosophy. They delete anything even remotely philosophical. And being a physics forum, they're much stronger on differential geometry than abstract algebra. It all depends on the orientation of the board.

To use an analogy, say we're talking about life. On a philosophy forum we might ask, what is the meaning of life. On a biology forum we'd ask about the role of osmosis in the Krebs cycle. Likewise when we talk about the brain here we talk about the mind and consciousness; and not so much about the electrochemical mechanisms underlying neurotransmitter reuptake.

Make sense?

But still. In an online forum the only true rule is: Post whatever you want and let the mods take it down.

Too few, clearly. There are only countably many of them.
...
And no countable ordered field can be complete. It's a theorem. — fishfry

The computable total functions are sub-countable. An enumeration of all and only the constructively convergent cauchy sequences isn't possible as this is equivalent to deciding every mathematics proposition. Nevertheless we can construct a countable enumeration of a proper subset of the computable total functions, namely the provably convergent cauchy sequences with locateable limits, which collectively constitute a complete and ordered field, where by "complete" we mean with respect to a constructive least upper-bound principle.

The computable total functions are sub-countable. An enumeration of all and only the constructively convergent cauchy sequences isn't possible as this is equivalent to deciding every mathematics proposition. Nevertheless we can construct a countable enumeration of a proper subset of the computable total functions, namely the provably convergent cauchy sequences with locateable limits, which collectively constitute a complete and ordered field, where by "complete" we mean with respect to a constructive least upper-bound principle. — sime

The computable numbers are countable. That's because the set of Turing machines is countable. Over a countable alphabet there are countably many TMs of length 1, countably many of length 2, etc.; and the union of countable sets is countable. QE Freaking D.

"where by "complete" we mean with respect to a constructive least upper-bound principle."

Well sure, if you supply your own definition of complete then you can make anything you like conform to your made up definition.

Turing recognized the importance of non-computability. Too many Wiki pages, not enough math, that's my diagnosis of your posts.

The sequence of n-th truncations of the binary expansion of Chaitin's number is a Cauchy sequence that does not converge to a computable real. End of story. Then you say, "Oh but that sequence isn't computable," and I say, "So freaking what?" and this goes on till I get tired of talking to yet another disingenuous faux-constructivist.

Too many Wiki pages, not enough math, that's my diagnosis of your posts. — fishfry

Thanks for your post regarding mine, fishfry. Your quote above to sime is germane.

I’ve ruffled some feathers with my post, for which I apologize. I got a bit irritated last night and didn’t express my thoughts well.

First, I’m not coming from a feeling of superiority regarding math. As a retired prof my interests are in a sliver so small it’s barely visible, one low-interest page among 40,000 on Wikipedia. There are sophisticated discussions on this forum about math, computer science, and logic that I can only stand aside and watch. And most conversations about foundations are beyond me.

But sometimes posters will make statements about mathematics in general that are erroneous, but said with conviction. Such as claiming that math proofs are computer programs, or that there are no more geometrical proofs. Or saying that fiddling with axioms makes the entire body of mathematics flawed, when, in fact, most mathematicians wouldn’t even notice. Claiming that irrational numbers are a mistake and that this undercuts the entire structure of mathematics. Stating that calculus is largely manipulating symbols and that formal education is detrimental. That adding a symbol, a “number”, for infinity will undermine current mathematics. For misusing the expression “chaos theory” when discussing randomness. For claiming that much of what we know of math now was derived or discovered two thousand years ago. On and on. I've probably misinterpreted some of this. If so, apologies.

It’s this moving away from what one knows to speculative territory, but being convinced one is correct – that’s a little annoying to me. But this is a philosophy forum, so no harm done.

As for physics, well all is not well in that discipline. For example, there is an argument about the aether that seemingly goes as follows: The premise is that every wave must travel through a physical substance, and that the aether exists. Electromagnetic pulses are waves, therefore must be propagated through the aether. Hence, electromagnetic waves travel through a physical substance. Makes sense if the premise is true. It's conjecture stated as fact.

I took a year of physics in college, and as a math prof used some physics in my classes. But I would feel incompetent to engage in a discussion about anything beyond the simplest ideas. But here we have string theory, differentiable manifolds, general relativity, entanglement, Bell’s theorem, and on and on – all as if the poster is sure of what he is talking about and not merely parroting Wikipedia. Maybe it’s no more than a lack of modesty. If I have offended anyone, sorry.

The computable numbers are countable. That's because the set of Turing machines is countable. Over a countable alphabet there are countably many TMs of length 1, countably many of length 2, etc.; and the union of countable sets is countable. QE Freaking D. — fishfry

?? Perhaps I should have been clearer from the beginning, but i took everyone's understanding for granted that a computable number refers (in some way) to a computable total function. Apologies if that is the case. For surely you appreciate that the computable total functions aren't countable?

The computable total functions are a proper subset of the computable functions that also contain partial functions. i.e. that do not halt on a given input.

It is true to say that the whole set of computable functions is countable, for reasons you'e sketched. It is not true to say that the set of computable total functions are countable, for we cannot solve the halting problem. Hence the reason why we say the computable numbers are sub-countable: the only way we could 'effectively' enumerate the computable numbers is to simulate every Turing machine and wait forever, meaning that any 'candidate enumeration' we construct of our computable numbers after waiting a finite time is also going to contain computable functions that aren't total and hence are not numbers.

For the constructivist, this "subcountability" is all 'that 'uncountability' means. It is simply means that we can never construct a total surjective function from the natural numbers onto the computable numbers. It doesn't mean in any literal sense that we have more computable real numbers than natural numbers.

The sequence of n-th truncations of the binary expansion of Chaitin's number is a Cauchy sequence that does not converge to a computable real. End of story. Then you say, "Oh but that sequence isn't computable," and I say, "So freaking what?" and this goes on till I get tired of talking to yet another disingenuous faux-constructivist. — fishfry

We have to be careful there. We can run every Turing Machine and at any given time create a bar-chart of the ones which have halted, and this histogram comprises a sequence of computable functions whose limit isn't a computable function. To my understanding this sequence of functions isn't cauchy convergent, for we cannot construct a bound on the distance between successive histograms. Let's not forget that there are an infinite number of computer programs of every size.

Compare this situation to a computable total function f(n) representing the "values" of the Goldbach's Conjecture; Let's say that f(n) = 0 if every even number less than n is the sum of two primes, otherwise f(n)=1. Here we can also compute the individual digits in finite time. If GC is decidable, i.e. GC OR ~GC, then f(n) is Cauchy convergent to either 0 or 1. But if GC isn't decidable, then as with Chaitin's constant f(n) doesn't have a cauchy convergent limit, even though f(n) is a computable total function.

Therefore, in order to know that one has constructed a complete and ordered field of computable numbers, one must only use a set of provably Cauchy-convergent computable total functions, for which every cauchy-convergent sequence of these functions is also provably cauchy-convergent.

?? Perhaps I should have been clearer from the beginning, — sime

You were already clear. I reviewed the wiki article on subcountability and nothing you said caused me to change anything I wrote.

https://en.wikipedia.org/wiki/Subcountability

Now obviously, any countable list of Provably Cauchy-Convergent Total Functions is unfinished, in the sense that a further PCCTF can be built that is is not already in the list via a diagonal argument. No problem, we just shuffle along the existing enumeration to add the new function into the existing list. But then doesn't this contradict the notion that our previous list was complete?

There seems to be an ambiguity between two definitions of completeness. If Dedekind completeness is understood to be an axiom of construction then it is trivially satisfiable in the sense that the axiom itself can be used to assist in the generation of a real from an existing list of real numbers. After all, if there wasn't a countable model of the Axioms of the reals, then they would be inconsistent, since Second-order quantification can always be interpreted as referring only to the sets constructively definable in first-order logic.

On the other hand, if completeness is understood to refer to a finished list of PCCTFs, our list is not complete in that sense.

So it seems to me that countable model of reals, both first and second order, are especially useful ( not to mention the only models we use in practice),for clarifying the relationship between Dedekind completion, Cantor's theorem and ordered fields.

If one abandon's the second-order completeness axiom, and possibly cauchy convergence, then there are less constraints in the construction process, allowing one to define a potentially larger field of computable numbers that includes infinitesimals as is done with the (constructive) Hyperreals, and one can even include computable 'numbers' that are aren't provably total. In which case ones countable list is now finished, but now there are no more numbers to be added, because now the diagonal argument cannot be used to construct a new numbers in virtue of one's list including non-numbers that aren't guaranteed to halt on their inputs.

So i hope this had lead to a satisfactory conclusion.

the (constructive) Hyperreals, — sime

Now that's something I've never run across. Both too big and too small at the same time. But it takes a weak form of the axiom of choice to have a nonprincipal ultrafilter, which is needed to construct the hyperreals. Do constructivists allow that?

I cannot think of anything in mathematics or logic that is not a concept. — Michael Lee

First, your own logic results in the fair conclusion that infinity is a concept. If infinity is a number and all things mathematical are concepts, then infinity is a concept.

Second, even if all things within the realms of mathematics or logic are a concept, that does not mean that all concepts are within the realm of mathematics or logic.

Consequently, it you are correct and infinity is a number, then it is okay to call infinity a concept. On the other hand, if you are incorrect and infinity is not a number, then it is still okay to call infinity a concept.

Perhaps you should have settled for infinity as a mathematical concept and argued for a mathematical representation of the that mathematical concept.

Just saying.

Now that's something I've never run across. Both too big and too small at the same time. But it takes a weak form of the axiom of choice to have a nonprincipal ultrafilter, which is needed to construct the hyperreals. Do constructivists allow that? — fishfry

emmm......... Nope :) for the reason you've just mentioned. For where is the algorithm of construction? Of course , the trivial principle ultrafilter is permitted, which then produces a countable model..

By "constructive hyperreal" i was merely colloquially referring to using functions such as f(n)=1/n as numbers according to some constructive term-oriented method that didn't involve assuming or using cauchy limits.

emmm......... Nope :) for the reason you've just mentioned. For where is the algorithm of construction? Of course , the trivial principle ultrafilter is permitted, which then produces a countable model.. — sime

From what I've seen, constructivists typically allow weak forms of choice, for the reason that otherwise you can't get satisfactory math. So it wouldn't surprise me if some constructivists allow nonprinciple ultrafilters. The trivial ultrafilter of course doesn't give you the hyperreals but I can see how it might produce something that might be called constructive hyperreals.

By "constructive hyperreal" i was merely colloquially referring to using functions such as f(n)=1/n as numbers according to some constructive term-oriented method that didn't involve assuming or using cauchy limits. — sime

Wait so you just made that up? It's not a real thing? You had me convinced. Why not mod out the reals by the trivial ultrafilter and see what you get? What do you get?

Why are there so many die-hard constructivists on this forum? If you go to any serious math forum, the subject never comes up, unless one is specifically discussing constructive math. You never see constructivists claiming that their alternative definitions are right and standard math is wrong. Only here. It's a puzzler.

Math can deal with infinities. To say that math has it wrong is silly. You need the concept of infinities to do things like derivatives and integrals, and you need those in turn to do physics, which is empirically supported. The physics works and the physics is based on a mathematics of infinite, so we can't act as if math is wrong about infinite. It's not.

Why are there so many die-hard constructivists on this forum? If you go to any serious math forum, the subject never comes up, unless one is specifically discussing constructive math. You never see constructivists claiming that their alternative definitions are right and standard math is wrong. Only here. It's a puzzler. — fishfry

Just a guess, but I would imagine that one typically becomes a constructivist in the first place for primarily philosophical reasons--e.g., dissatisfaction with the philosophical basis of standard math, hence the desire for and advocacy of alternative definitions. Since this is a philosophical forum, rather than a mathematical forum, it is a natural place for committed constructivists to make their case.

Wait so you just made that up? It's not a real thing? You had me convinced. Why not mod out the reals by the trivial ultrafilter and see what you get? What do you get?

Why are there so many die-hard constructivists on this forum? If you go to any serious math forum, the subject never comes up, unless one is specifically discussing constructive math. You never see constructivists claiming that their alternative definitions are right and standard math is wrong. Only here. It's a puzzler. — fishfry

Well obviously from a pure mathematics perspective, every proof in ZFC is considered construction, in contrast to Computer Science that has traditionally had more natural affinity with ZF for obvious reasons, and there is a long historical precedent for using classical logic and mathematics. As a language, there is nothing of course that classical logic cannot express in virtue of being a "superset" of intuitionistic logic, but classical mathematics founded upon classical set theory IS a problem, because it is less useful, is intuitively confusing, false or contradictory, lacks clarity and encourages software bugs.

In my opinion, Constructive mathematics founded upon intuitionistic logic is going to become mainstream, thanks to it's relatively recent exposition by Errett Bishop and the Russian school of recursive mathematics. Constructive mathematics is practically more useful and less confusing for students in the long term. Consider the fact that the standard 'fiction' of classical real analysis doesn't prepare an engineering student for working in industry where he must work with numerical computing and deal with numerical underflow.

The original programme of Intuitionism on the other hand (which considers choice-sequences created by the free-willed subject to be the foundation of logic, rather than vice versa) doesn't seem to have developed at the same rate as the constructive programme it inspired. However, it's philosophically interesting imo, and might eventually find an applied niche somewhere, perhaps in communication theory or game theory.

BTW, i'm not actually a constructivist in the philosophical sense, since the constructive notion of a logical quantifier is too restrictive. In a real computer program, the witness to a logical quantifier isn't always an internally constructed object, but an external event the program receives on a port that it is listening. What's really needed is a logic with game semantics. Linear logic, which subsumes intuitionistic and classical logic is the clearest system i know of for expressing their distinction and their relation to games.

As for a trivial ultrafilter, its an interesting question. Perhaps a natural equivalence class of Turing Machine 'numbers' is in terms of their relative halting times. Although we already know that whatever reals we construct, they will be countable from "outside" the model, and will appear uncountable from "inside" the model.

Thanks for your post regarding mine, fishfry. Your quote above to sime is germane.

I’ve ruffled some feathers with my post, for which I apologize. I got a bit irritated last night and didn’t express my thoughts well. — jgill

Don't worry, nobody noticed or cares. You only ruffled a feather or two of mine, and I'm easily ruffled.

First, I’m not coming from a feeling of superiority regarding math. As a retired prof my interests are in a sliver so small it’s barely visible, one low-interest page among 40,000 on Wikipedia. There are sophisticated discussions on this forum about math, computer science, and logic that I can only stand aside and watch. And most conversations about foundations are beyond me. — jgill

Very few working mathematicians care about foundations. A lot of philosophers and pseudo-philosophers imagine that all mathematicians sit around writing proofs directly from the axioms of ZFC. As has been often noted, the average mathematicians couldn't write down the axioms of ZFC if challenged.

But sometimes posters will make statements about mathematics in general that are erroneous, but said with conviction. — jgill

Yes. A lot of that around in the public discourse as well, wouldn't you agree?

FWIW there are the strictly moderated forums that are no fun, and the more loosely moderated forums that allow a bit of give and take, but are thereby welcoming to people with varying degrees of knowledge and sanity. Among all the loosely moderated philosophy forums, this place is by far the best. It gets a lot worse on some other similar forums. It's just part of the fun of being online. This is not the proceedings of the Royal Society.

Such as claiming that math proofs are computer programs, — jgill

This is in fact true. It's the famous Curry-Howard correspondence.

If you think about it, it's quite sensible. Say we prove that some wildly non-constructive object has mathematical existence. Vitali's nonmeasurable set for example.

Nevertheless the proof of existence is a constructive object. It's a sequence of syntactic moves starting from a set of axioms, which are well-formed formulas of some formal language; and a set of inference rules. Given the axioms and the inference rules, a computer could calculate whether a given derivation is legal.

In effect we're ALL constructivists. We construct proofs, even if those proofs claim the existence of nonconstructive objects. Our proofs literally are translatable to computer programs in an abstract sense.

One of our resident constructivists got me to understand this a while back. I got some insight into constructivism from that.

or that there are no more geometrical proofs. — jgill

Haven't seen that one. But this is not a technical forum. It's a freewheeling discussion forum that is nevertheless far more intelligent in general than most other online discussion forums out there. And of course Wikipedia is partially to blame. A lot of people think they know things these days that they really don't know.

Or saying that fiddling with axioms makes the entire body of mathematics flawed, when, in fact, most mathematicians wouldn’t even notice. — jgill

Hard to explain to philosophers how little working mathematicians care about foundations. If ZFC were discovered inconsistent tomorrow morning, hardly anyone would care besides the specialists. Nobody ever heard of set theory before Cantor but a lot of great math was done. Attitudes towards foundations come and go as a matter of historical contingency.

Claiming that irrational numbers are a mistake and that this undercuts the entire structure of mathematics. — jgill

Our friend @Metaphysician Undercover is a special case. He is so sure of himself and he writes well; so the challenge on a forum like this is to try to engage him rationally and see how well one understands and can advance their own point of view. I've always found that when I'm debating someone online who has an unorthodox/alternative/cranky/crazy opinion, the real challenge is to see if I can be transcendentally clear and persuasive myself. Either that or just ignore what you don't like. That's what free speech is about IMO, and discussion forums are about community-moderated free speech.

Stating that calculus is largely manipulating symbols — jgill

To be fair, that's exactly how we teach it. "Bring down the exponent and subtract one." Calculus is a service course for the benefit of the engineering, physics, economics, pre-med, and other departments. It's got very little to do with math. You can't blame the kids for being confused. As my grad advisor put it to me once, when I was about to embark on being a calculus TA: "Freshman calculus is a futile exercise in mind fucking." Truer words were never spoken.

and that formal education is detrimental. — jgill

Also to be fair, many of the high and mighty in the land say the same. Isn't Elon Musk one of those masters of the universe telling kids to drop out of school and just get to work doing what they care about?

And from what I've heard about higher education these days, education's not what it used to be. I'm not sure any of us are in a good position to defend what passes for formal education these days.

That adding a symbol, a “number”, for infinity will undermine current mathematics. — jgill

Lot of confusion about the extended reals and their relation (which is none whatsoever) to the transfinite numbers of set theory.

For misusing the expression “chaos theory” when discussing randomness. — jgill

Now this is a malady common to science journalism in general. A lot of your concerns are better addressed to the miserable state of science journalism in general. AI hype, quantum computing hype, hype in general.

For claiming that much of what we know of math now was derived or discovered two thousand years ago. — jgill

Most people don't know much about math, even educated people. Might as well yell at the tides. The challenge of a venue like this is to state your case as clearly as you can and see if anyone's convinced. If all you want is technical questions and authoritative answers, that's what Stackexchange is for.

On and on. I've probably misinterpreted some of this. If so, apologies. — jgill

None needed, really. You do give the impression of not having been on the Internet much. The world has had this problem since Gutenberg. Once you give the public a voice, no telling what they'll say. Reminds me of something Churchill said. "The best argument against democracy is a five minute conversation with the average voter."

It’s this moving away from what one knows to speculative territory, but being convinced one is correct – that’s a little annoying to me. But this is a philosophy forum, so no harm done. — jgill

It's the nature of online discourse. And public discourse too. If you have no idea what you're talking about, say it real loud and with a sense of self-righteousness. Again, there's always Stackexchange. This place ain't that. And it's a good thing in general. Think of it more like the corner bar. Takes all kinds.

As for physics, well all is not well in that discipline. For example, there is an argument about the aether that seemingly goes as follows: The premise is that every wave must travel through a physical substance, and that the aether exists. Electromagnetic pulses are waves, therefore must be propagated through the aether. Hence, electromagnetic waves travel through a physical substance. Makes sense if the premise is true. It's conjecture stated as fact. — jgill

This was resolved in the 1900s. There is no luminiferous aether. I did not know this is still an issue. I don't think it is.

I took a year of physics in college, and as a math prof used some physics in my classes. But I would feel incompetent to engage in a discussion about anything beyond the simplest ideas. But here we have string theory, differentiable manifolds, general relativity, entanglement, Bell’s theorem, and on and on – all as if the poster is sure of what he is talking about and not merely parroting Wikipedia. Maybe it’s no more than a lack of modesty. If I have offended anyone, sorry. — jgill

Well of course it's the Wikipedia factor. Someone reads a Wiki page and they feel emboldened to vociferously promote their own mistaken understandings; even if they are talking to someone who actually knows what they're talking about. Nature of modern society.

Our friend Metaphysician Undercover is a special case. — fishfry

Thanks for the compliments. The biggest stumbling block between us is your concept of "mathematical existence". The proof that something has mathematical existence is really meaningless unless we have a rigorous definition, or convention, concerning what "mathematical existence" means. If it simply means to be consistent with some set of axioms, and we have no standard as to how an axiom might be justified, then all sorts of fictions may be proven to have mathematical existence. And, if mathematical existence is not consistent with "existence" in the more general, philosophical sense, then it's not even a type of existence at all, and use of the term "existence" is misleading. Then we'd be better to replace "mathematical existence" with "pseudo-existence", or "crackpot existence", so as to be less misleading with our terms.

Thanks for the compliments. The biggest stumbling block between us is your concept of "mathematical existence". The proof that something has mathematical existence is really meaningless unless we have a rigorous definition, or convention, concerning what "mathematical existence" means. — Metaphysician Undercover

Does the knight's move have chess existence? The other day you said you reject chess because it doesn't refer to anything in the real world. That's extreme nihilism. You can't get out of bed in the morning with a philosophy like that. How do you know it's your own bed? Property's an abstraction.

Such as claiming that math proofs are computer programs, — jgill

This is in fact true. It's the famous Curry-Howard correspondence — fishfry

This is certainly valid regarding the structure of a mathematical argument. But by itself it leaves the impression that mathematics is merely symbol manipulation and not what it really is: exercises in imagination and creativity. On the other hand, it may be that sometime in the future AI will explore and develop mathematics so convoluted and complicated that the results will be on the edge of human understanding, or beyond. Perhaps issues like the nature of time will be resolved, but humans will not be able to comprehend the results. Who knows? :chin:

Stating that calculus is largely manipulating symbols — jgill

To be fair, that's exactly how we teach it. — fishfry

Oh, I am well aware of that! :nerd: I taught calculus at all levels for 29 years. But my introduction to the subject was unusual: after taking an excellent course in analytic geometry as a freshman at Georgia Tech in 1955, I was recruited, along with about fourteen other students, into an experimental first quarter calculus course taught by two professors. Epsilons and deltas on the first day and no text book. Mostly we were bewildered at first, the exception being two brainiacs who caught on instantly. There was no attempt to continue the experiment into the second quarter, so we all migrated back to the standard curriculum for engineers, physicists, etc. What a huge difference!

At the University of Chicago in the fall of 1958, I was surprised to learn that the physics department was no longer allowing its students to enroll in courses from the math department and was teaching its own mathematics.

Long ago, what attracted me to mathematics was the same thing that attracted me to my avocation as a rock climber: exploration, discovery and creativity. The course work in grad school was someth9ing I needed to plow through, and doing assigned problems that had been solved by generations of students was a chore. Like repeating well established climbs. The original research at the end was a delight, however.

and that formal education is detrimental. — jgill

Also to be fair, many of the high and mighty in the land say the same. — fishfry

Yes, a few spectacular success do just that. I disagree. I've watched students of mine graduate and move on into successful careers.

You do give the impression of not having been on the Internet much — fishfry

:smile: Wrong impression, Dude! I was on an outstanding climbers' forum for years until it folded last May. Along with a great deal of climbing discussion, there were threads about other subjects. The one I particularly enjoyed delved into the nature of mind and consciousness.

By the time the forum ended, this thread had well over 20,000 posts. Among those participating were several mathematicians, a well-known physicist, a neuroscientist, an academic anthropologist, a retired management prof, several academic philosophers, a well-known author who has practiced Zen for decades, and many others at all educational levels who chimed in from time to time. In particular, the debates between the physicist and the Zen person were stimulating. Also, even though avatars were used we all knew the identities of the primary contributors. So you see where I'm coming from. :cool:

Thanks for your comments!

Does the knight's move have chess existence? The other day you said you reject chess because it doesn't refer to anything in the real world. That's extreme nihilism. You can't get out of bed in the morning with a philosophy like that. How do you know it's your own bed? Property's an abstraction. — fishfry

What I said, or at least meant, is that I refuse to play chess because I find it irrelevant to my endeavours, so it's a waste of time. What could you possibly mean by "chess existence"? Let's say that the game consists of some physical pieces, and some stated "rules". What you have referenced is "a rule". How do you think that a rule exists? Does it exist as the symbols on the paper as the stated "rules of the game"? If so then these rules require being read, and interpreted, understood, in order for someone to actually play the game. Then the person's play is dependent on the person's interpretation. If the rules exist as the interpretation, within someone's head, then who's interpretation is correctly called "the rule". Consider President Trump's, impeachment trial. Who's interpretation of what is required for impeachment constitute the actual existing "rules"? And if it's what's written on paper, that is the existing "rule", how does it have any meaning as a "rule" without being read and interpreted?

Because I have not seen any resolution to these questions, I would not say that a "rule" has any existence at all. I think it is a simplification of something we do not understand. There is a subject of human behaviour, habituation, etc., which is not well understood, and some people like to represent it as understood, so they say there are "rules" which human beings are following. The use of "rules" creates the illusion that human behaviour is understood. The human being follows rules, just like matter follows rules of physics. Use of the term "rule" is just a convenient fiction, used to hide the fact that this subject is not well understood. It's a fiction because it doesn't represent any real, existent thing, it just creates an illusion. So when I get out of bed in the morning (hopefully it's the morning), I am not following rules, I am acting on my own terms. That's what it means to be a free willing human being.

The one I particularly enjoyed delved into the nature of mind and consciousness. — jgill

What do you think "rule" signifies?

The proof that something has mathematical existence is really meaningless unless we have a rigorous definition, or convention, concerning what "mathematical existence" means. — Metaphysician Undercover

I take your point to heart.

The actual meaning of mathematical existence is that it's whatever working professional mathematicians say it is. You don't accept that, but that is how it works.

I do take the point that this is not sufficient for you; and that if ALL I mean by mathematical existence is something I can prove from arbitrary axioms, that's not much of a criterion for existence. I could posit the existence of purple flying elephants but that wouldn't mean I've proven their existence.

I would be willing to stipulate that although the criterion I gave: that mathematical existence is whatever professional mathematicians say it is; I do owe you a better explanation. I haven't got one at the moment that would be satisfactory to you. But I do want to say that I take your point and I'm mulling the question over in my mind.

What could you possibly mean by "chess existence"? — Metaphysician Undercover

You have the same objection to football, baseball, Chinese checkers, and whist? You reject playing poker because the only Queen you know is Elizabeth? Nihilism. Childish rejection of the very concept of abstraction.

This is certainly valid regarding the structure of a mathematical argument. But by itself it leaves the impression that mathematics is merely symbol manipulation and not what it really is: exercises in imagination and creativity. — jgill

I see this a lot among those who have seen a little category theory in the context of computer science, and think they understand the deeper meaning of math. I also see this among novices who find out for the first time that math is based on axioms. They immediately leap to the conclusion that math is about writing down the consequences of the axioms. On the contrary, the math itself precedes the axioms. We know what's true and then we try to formalize it. The formalization is distinctly secondary to the math.

My sense is that professional philosophers of math (Maddy et. al.) perfectly well understand this point. it's the amateurs on the online forums who don't.

At the University of Chicago in the fall of 1958, I was surprised to learn that the physics department was no longer allowing its students to enroll in courses from the math department and was teaching its own mathematics. — jgill

When I studied math at UC Berkeley I called up the Physics dept one day and asked them if they had a fast track intro to physics for math majors. The person I spoke to said no and was attitudinal about it. The only way to learn physics is from the official physics courses! At that time physicists had a genuine dislike of the math curriculum. I believe things aren't quite as bad as that today.

Also, even though avatars were used we all knew the identities of the primary contributors. — jgill

Bad sample space. Anonymous forums are entirely different.

On the other hand I entirely agree with you about certain aspects of this forum. I think I've just finally gotten used to it. And like I say, this forum is the best of the philosophy forums out there.

Well obviously from a pure mathematics perspective, every proof in ZFC is considered construction, — sime

Yes I take that point. But note that it's a theoretical result about abstract, idealized proofs. In actual every day professional mathematics, proofs are not only not programs -- they're not even proofs as a logician would recognize them. They're mostly informal arguments, as much prose text as symbology.

In other words if a proof is a sequence of statements, each one following from an axiom or a result of previous statements, then no working mathematician has ever seen a proof.

I wonder if that's part of the disconnect between philosophers and mathematicians. Working mathematicians don't write proofs the way philosophers and (some) computer scientists conceive proofs. @jgill has made this point.

in contrast to Computer Science that has traditionally had more natural affinity with ZF for obvious reasons, and there is a long historical precedent for using classical logic and mathematics. — sime

Less than ZF in fact. A Turing machine is an unbounded tape and not an infinite tape. The tape is as long as it needs to be but at any step is always finite. It's a potential infinity and not a completed one if you like. Computer science does not require the axiom of infinity. The Peano axioms will do. Except for those parts of CS that do require infinite sets.

As a language, there is nothing of course that classical logic cannot express in virtue of being a "superset" of intuitionistic logic, but classical mathematics founded upon classical set theory IS a problem, because it is less useful, is intuitively confusing, false or contradictory, lacks clarity and encourages software bugs. — sime

You and @Mephist are in agreement but again, the question isn't that one framework's better than another. They're all tools in the service of discovering higher truth. The mathematics that's being talked about is the same mathematics whether you represent it in type theory or category theory or intuitionist logic or classical logic. They're all tools to be used as appropriate. It is not a cage match to the death as some seem to believe.

As far as one approach or another being better for programming, there's a long history of one false panacea after another. "Common business-oriented language," or COBOL, was going to make it possible for business analysts to write code. Didn't work. Procedural programming would make software more reliable. Didn't work. Structured programming was the answer. Didn't work. Object-oriented programming, everyone is dumping on it these days. Inheritance is a lie, nobody ever ended up building useful industry-wide libraries of base classes. Now functional programming's the thing. It will solve all our problems.

I've seen a lot of this history first-hand and I'm not likely to be impressed by the latest proposed solution to the eternal software crisis. That to me seems like a very different discussion than the role of intuitionism in math.

In my opinion, Constructive mathematics founded upon intuitionistic logic is going to become mainstream, thanks to it's relatively recent exposition by Errett Bishop and the Russian school of recursive mathematics. Constructive mathematics is practically more useful and less confusing for students in the long term. Consider the fact that the standard 'fiction' of classical real analysis doesn't prepare an engineering student for working in industry where he must work with numerical computing and deal with numerical underflow. — sime

Well sure, it's all a matter of historical contingency and intellectual fashion. I've argued that point myself. I may have my own doubts about constructivism, but I don't deny that it's inevitably gaining mindshare in our age of computation.

The original programme of Intuitionism on the other hand (which considers choice-sequences created by the free-willed subject to be the foundation of logic, rather than vice versa) doesn't seem to have developed at the same rate as the constructive programme it inspired. However, it's philosophically interesting imo, and might eventually find an applied niche somewhere, perhaps in communication theory or game theory. — sime

I made an honest, good-faith attempt to understand free choice sequences once. I simply could not get past the idea of a "subject" that makes choices. Too woo-woo for me. And you're right, modern intuitionism became important when the computer became important in the world. Brouwer's revenge.

BTW, i'm not actually a constructivist in the philosophical sense, since the constructive notion of a logical quantifier is too restrictive. In a real computer program, the witness to a logical quantifier isn't always an internally constructed object, but an external event the program receives on a port that it is listening. — sime

I spent much of my professional life working with networked applications, but never thought much about the abstract semantics. What does it mean when Turing machines get external input. I gather it can't make too much of a fundamental theoretical difference otherwise I'd have heard about it, but I could be wrong.

But it sounds like your approaching programs from a proof-of-correctness point of view rather than a day-to-day software engineering perspective. Am I getting that right?

What's really needed is a logic with game semantics. Linear logic, which subsumes intuitionistic and classical logic is the clearest system i know of for expressing their distinction and their relation to games. — sime

Needed for what, exactly? You seem to be relating the math to programming theory. Surely little or none of this relates to the building of actual computer systems except at a theoretical level.

As for a trivial ultrafilter, its an interesting question. Perhaps a natural equivalence class of Turing Machine 'numbers' is in terms of their relative halting times. Although we already know that whatever reals we construct, they will be countable from "outside" the model, and will appear uncountable from "inside" the model. — sime

I have no idea. I remember about three years ago I spent some time coming up to speed on the technical aspects of the hypperreals and ultrafilters but I've forgotten most of it.