As applied to mathematics, yes. Charles Peirce adopted his father Benjamin's definition of it as the science that draws necessary conclusions about hypothetical states of things. — aletheist

Ok. Not too far from Bertie's quip that math is the subject where we never know what we're talking about or if what we say is true.

There are no restrictions on conceiving such hypothetical states, other than that they be suited to the purposes for which we wish to analyze them. Applied mathematics seeks to formulate hypothetical states that resemble reality in certain significant ways, but pure mathematics does not have that particular limitation. — aletheist

Ok I perfectly well agree with that. Completely. Have explained it to many people on many forums over the years. May be coming to understand it myself!

But now Maddy's pragmatism (in the everyday sense) goes further. According to Peirce, we can accept or reject AC at will, which of course we can. But that does not explain why mainstream math accepts it and never gives it a second thought. For that you need Maddy's Maximize principle, which says that assuming AC gives a better or richer theory. I'm paraphrasing my understanding of Maddy and didn't double-check her article so I might be mangling her ideas a bit but I think I'm in the ballpark.

Point being that Maddy says that given two equally logically consistent but mutually inconsistent axioms; we adopt the one that's more fun, more interesting, gives more good theorems. That's a pragmatic justification for choosing AC over not-AC.

Is that an example of Peircean pragmatism or is that an expression of something else?

Also ... given that I'm perfectly in agreement with Peirce's point of view here, and Maddy's as well; can you help me to understand @Metaphysician Undercover's point of view? He rejects the idea of taking math on its own terms; insists that it must refer to something outside of itself. Nobody believes that, not about pure math. Why does @Metaphysician Undercover believe that? What is the basis for his ideas? Do they have a name? Is there a reference? I asked him this once and did not get a satisfactory answer. I can't tell if he is asserting the ideas of a particular school of thought, or just venting over a bad experience with his screechy third-grade math teacher.

I hope you see the problem with this. You're saying, if we (mathematicians) agree that it exists then it exists, without any definition of what it means to exist. In any other field, no one would agree that such and such "exists", unless there was a definition of "exists" and some evidence to show that the thing actually exists. For example, would some biologist come in with a fictitious life form and ask the other biologists, can we agree that this life form exists, so that it can be a real existent life form? Or would a physicist propose the existence of a fictitious particle? — Metaphysician Undercover

But yes and yes.

Physicists thought one day there must be atoms. Then they discovered the atoms are made of protons and electrons and neutrons. Then they discovered the protons are made of quarks. Now they think the quarks are made of strings. Do any of these abstractions exist? Yes they do, in the sense that they are part of an abstract mathematical theory that explains the experiments we're capable of doing at any moment in history.

Physicist invent new existing things all the time. And de-exist things to. The luminiferous aether was once regarded as existing, till Michelson and Morley couldn't find it and Einstein did away with its necessity.

A scientific entity has existence when it's a necessary ingredient of a successful physical theory. Nobody can say whether a quark or an electron "really" exists; only that positing their existence gives a good theory. That is the definition of scientific existence. And mathematical existence too. I'll go with that, since I challenged myself to define mathematical existence for you.

Biology? Once, disease was caused by ill humours in the blood. Then they came up with the germ theory of disease. Germs are an abstract thing that gives a good theory of disease. Now we can study germs under a microscope, but really, what are they? Bundles of biological material. More abstractions. In the end, they're all quarks and the properties that emerge from various organizations of quarks. But now we treat infections with antibiotics and not leeches, so there is slow progress towards the good. Our abstractions become real because they work. In the future some of the things we think are real will turn out not to be (like the force of gravity) and other things we didn't think were real will turn out to be (electrons, quarks, strings, loops ...)

I gather you call "real" only what is "really out there." But if the 20th century taught us anything, it's that the existence of such a thing as "real things out there" is an assumption and not a fact. I believe if I'm not mistaken this is called scientific realism. It's only an idea. We could kick it around. But you have no logical basis for claiming it's true and everybody else is wrong. The days of Euclidean geometry and Newtonian physics are gone. Now we know the world consists of probability waves that are everywhere at once till we measure them. What can that mean? We don't know. But you claiming that you personally know what things are real, is a delusion on your part. Since you called me delusional the other day, which I can live without.

Math precedes foundations. Not the other way 'round.
— fishfry

How could you conceive of this? — Metaphysician Undercover

Knowledge of the history of math.

But it's the same in any discipline. There's science, and then there's the philosophy of science. One can and does do science without regard for its philosophy. That's true in every field. X precedes the philosophy of X.

I say "there's no empty set therefore math is flawed". — Metaphysician Undercover

Yes I understand this. I get that you reject math and, when pressed, physical science as well. I'm happy for you. I can't take such a point of view seriously. The task of the philosopher is to explain how it comes to be that math and science are abstract yet useful. You can't solve the problem by rejecting math and science; not unless you live in a cave. Without Internet access. Even then you'd draw a square in the sand and eventually discover the square root of two; and from that, abstract algebra and group theory and modern physics. You haven't got a serious philosophy, just sophistry.

I am just pulling these out to highlight them as excellent observations. After all, Peirce--the founder of pragmatism--was decidedly non-foundationalist in his own philosophical system. SDG/SIA captures certain aspects of true continuity that modern analysis using the standard real numbers does not. — aletheist

Tell me something of pragmatism. Let me say first where I'm coming from.

I've read Maddy's great papers Believing the Axioms part I and part II. These papers provide a historical overview of how and why the various axioms of set theory came to be adopted; along with a number of pragmatic (in the everyday sense of the word) criteria for deciding whether to adopt an axiom. For example one principle is Maximize, which says that we choose the axioms that give us the richest theory.

I'll give an example. The axiom of choice (AC) may be taken or denied with equal logical consistency. Both AC and its negation are consistent with ZF.

Why do mathematicians simply accept AC? I'm not talking about specialists who investigate the consequences of the negation of AC, but rather everyday working mathematicians who never give a thought to foundations. They're taught in grad school to freely apply Zorn's lemma, the Hahn-Banach theorem, and other applications of AC in their work, and they do so as a matter of course.

Philosophically, we accept AC because it gives us more and better theorems. That is the reason. This is a very pragmatic (in the everyday sense, not necessarily the technical sense) way to view the axioms. We adopt the axioms that give us a good theory.

Is that Peircean pragmatism? Or what exactly is it, given what I already know about the practical or pragmatic (everyday sense) reasons to adopt or reject axioms?

I came to this forum about 6 months ago because I wanted to present some ideas that in my opinion are original but are not completely formalized (until now, maybe they will be one day). I discovered that nobody understood what I meant. So probably that's the wrong approach. I am not interested in discussing ZFC. I don't know ZFC and I am not really interested in studying it.

I think I'll have to complete my work in a more formal way and present it to a math forum.

Sorry for bringing trouble. You can continue without me. — Mephist

Ah ... was it something I said? I have no sense of having suppressed any thoughts you may have. I had no idea you were presenting anything original. If I somehow crushed your creativity I apologize. Is that what I did?

Just for my understanding, can you point to a post where you presented your original ideas?

Surely you may have noticed that you can write the most simple, commonly understood things here and have many people not understand you. Happens a lot to me.

I think I'll have to complete my work in a more formal way and present it to a math forum. — Mephist

Just so I have some idea what this is about ... can you just summarize your idea? I can't for the life of me remember any interaction where you presented an original idea and I crushed your spirit. Truly sorry if anything came across that way.

ps -- I've reviewed your earliest posts from 9 months ago. I see no evidence that you ever put forth an original idea and got shot down or discouraged by anyone. What the heck is this about? I simply can find no evidence of your assertion.

Here's a formal proof in Coq that the Calculus of Constructions is sound: http://www.lix.polytechnique.fr/~barras/publi/coqincoq.pdf — Mephist

I don't care. If anything whatsoever can prove its own consistency, it's useless as a foundation for math. If it requires something external to itself to prove its consistency, it's no better off than ZFC, which can easily (not complex or convoluted, but naturally) do the same.

But I am curious. Which of these is the case? Is the Calculus of Constructions useless as a foundation for most of modern math? Or relies on something outside itself for a proof of consistency?

If it's not one of these, then I stand to learn something. So please explain how this thing, whatever it is, defies Gödelian incompleteness.

I'll not discuss about the empty set any more. Yes, you are right. The empty set exists. You win! — Mephist

I was just surprised the other day when you agreed with @Metaphysician Undercover that it doesn't. I'm glad I was able to nip that in the bud, if in fact you mean it and are not just placating me. I should also point out that the reason denying the empty set entails denying so much more of modern math, is because the formalization of pretty much all of modern math depends on the existence of the empty set! This is a fact. Not of math itself, which is agnostic as to foundations. But to our formalizations of math. You can't do without the empty set. It exists on pragmatic grounds.

But now @Metaphysician Undercover has a good question. What does it mean that the empty set exists? Is it ONLY that its existence formally follows from somewhat arbitrary axioms? If so, he has a point. Can we do better in terms of defining mathematical existence?

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.

I don't understand what I am wrong about. — Mephist

First let me put this in context. You said the empty set doesn't exist. I gave a short existence proof from the axiom schema of specification. That's a valid proof in ZF of the existence of the empty set. You then objected to my proof by saying ZFC can't prove itself consistent. Which would result in your rejecting the entirety of modern mathematics.

So I'll explain why you're wrong First, your response is a total deflection, changing the subject. Second, your response has the same slippery slope problem as @Metaphysician Undercover's response to the same question. Namely, that it's not only the empty set that's not deserving of being called existent. Rather it's the entire enterprise of modern mathematics. Surely you must realize that such reasoning is untenable because it's so broad. You both want to reject the empty set on narrow terms -- "it makes no sense to have a collection that doesn't collect anything," etc. -- but you each end up saying that math itself is flawed therefore there's no empty set. There must be a name for such an argument. You want to argue a very narrow technical point and your only argument is to blow up the entire enterprise.

Third, your overall understanding of what math is about is inverted, in exactly the sense @jgill notes. Many people who come to math through foundations believe math is about the foundations. It's the other way 'round. Mathematics comes first and foundations are just our halting and historically contingent attempts to formalize accepted mathematical practice. Archimedes, Newton, Euler, and Gauss never heard of set theory. Were they not doing math? You see the absurdity of trying to put foundations logically prior to mathematics.

First we discover the math; then we make up the axioms that let us formalize it.

That's how math works. My sense is that professional philosophers of math (Maddy et. al.) perfectly well understand this; and that it's only the amateur enthusiasts on the message boards who believe otherwise.

And fourth, you're wrong on the math and logic of the situation.

So let me lay out some talking points in support of my four reasons you are wrong.

I said there is no proof that ZFC is inconsistent (meaning: nobody has never derived a contradiction from ZFC's axioms), but there is even no proof that ZFC is consistent. — Mephist

The horrors. I suppose when Andrew Wiles solved Fermat's last theorem you said, "Harrumph, poppycock, we don't even know if ZFC is consistent." I hope you see the absurdity of your own position. For that matter it might interest you to know that Wiles's proof is done in the framework of Grothendieck's approach to modern algebraic geometry; which as I mentioned to you in another thread is done within a Grothendieck universe, a model of set theory that (a) assumes ZFC is consistent; and (2) posits the existence of an inaccessible cardinal, a transfinite cardinal whose existence is independent of ZFC. There's a lengthy and famous Mathoverflow thread about whether or not an inaccessible cardinal is necessary to Wiles's proof. Consensus is that it's not.

Likewise when Maryam Mirzakhani became the first woman and the first Iranian to win the Fields medal for "the dynamics and geometry of Riemann surfaces and their moduli spaces," you of course shouted, "Doesn't she know ZFC hasn't been proven consistent? She shouldn't have bothered."

If you are arguing anything different than this please let me know. Else retract your nonsensical point that since ZFC can't prove itself consistent, it must be fatally flawed. And that you can use this as a trump card to win any mathematical argument "The empty set exists." "No it doesn't, ZFC can't be proven consistent."

Man is this what you are arguing to me?

I want to add that when @Metaphysician Undercover makes the same argument, I have less of a problem with it; because he at least openly admits he does not engage with symbolic arguments. Please correct me if I have mischaracterized that in any way.

@Mephist, on the other hand, you seem perfectly willing to claim mathematical and symbolic knowledge. So your argument here is just awful. The empty set doesn't exist because ZFC can't prove itself consistent. Said by someone claiming math sophistication.

Am I missing your point here? Please tell me if I'm going off on the wrong thing. Because if that's your argument then you are a nihilistic as @Metaphysician Undercover, but with less of an excuse. You both want to throw out the entirety of modern mathematics just to defend your point that the empty set is not deserving of existence. You must not have much of an argument, either of you.

* Note that even if ZFC is inconsistent, then the empty set exists! The derivation from the axiom schema of specification is valid. So your own logic is screwed up. If ZFC is consistent the empty set exists, and if ZFC is inconsistent the empty set exists. Or rather in either case, the proof of its existence is valid. And what more do you ask for in terms of mathematical existence? You both want to reify the empty set. What nonsense. That's sophistry, to pretend to reject mathematical abstraction.

That's why I prefer type theory to ZFC. — Mephist

You thanked me for posting the Stackexchange thread the other day but I'm not sure you got its message. The example of synthetic differential geometry was given to show that the point of alternative foundations is to shed light on problems, not to brag about which foundation is more fundamental.

Likewise he gave the example of someone saying that set theory's more fundamental than topology so they don't need to study topology. That's silly, right?

So when you say, "I prefer type theory" because of a spurious understanding of ZFC's inability to prove its own consistency, you sound like you're clinging to what you know because you can't understand what you don't know. So far your logic is "the empty set doesn't exist because ZFC can't prove its own consistency and that's why type theory is better."

You're making a poor argument and only showing the limitations of your own understanding.

Type theory is weaker but is provably consistent. — Mephist

I responded to this in more detail in another post. This claim cannot possibly mean what you say it does. If type theory or any other theory can prove itself consistent, then à la Gödel it's useless for doing modern math.

On the other hand if you mean it can be proven consistent using means outside of itself, so can ZFC, as is commonly and standardly done in the modern categorical approach to algebraic geometry as pioneered by Mac Lane and perfected by Grothendieck.

Didn't they mention any of this in your category theory book? This is what I mean by your having a lack of overall understanding of math. It's part of the wrongness of your reply. Category theory and type theory don't invalidate 20th century math. They view it from another perspective. The math itself is the thing represented by the representations. You're trying to privilege one particular representation over another simply because you know one and not the other and don't get that the representation is not the thing itself. Not a good argument, not making points with me.

Again: Math precedes foundations. Not the other way 'round.

Can you show me what I said wrong? — Mephist

I've said my piece, and if it was too long, it's because "I didn't have the time to make it shorter," as some clever person said once.

I think the sets that are defined in ZFC are a hierarchical tree-like structure that can be used to model the relation "belongs to" at the same way as the leaves of a tree "belong to" it's root. — Mephist

Maybe, but not what I was looking for. A set is anything that obeys the axioms of set theory; in exactly the same way that point, line, and plane are undefined terms in Euclidean geometry. As Hilbert noted: "One must be able to say at all times--instead of points, straight lines, and planes--tables, chairs, and beer mugs." That is how we regard sets.

You want to somehow reify sets. You think a set should refer to the real world. I for one don't believe that. There's no set containing the empty set and the set containing the empty set in the real world. In set theory we call that set '2'.

It lacks symmetry and is too complex. — Mephist

I'd argue the contrary. Sets as an abstraction of collections are very natural. You can teach sets to school kids in terms of unions and intersections of small finite sets. Type theory and category theory are more sophisticated concepts that require some mathematical training to appreciate.

But so what? Are you honestly rejecting the entirety of contemporary math because you have some kind of personal issue with set theory? That makes no sense. Set theory, type theory, and category theory are various tools in the toolkit for exploring the world of mathematical entities.

Math precedes foundations. Not the other way 'round.

I think in the future it will be substituted by a more elegant and simpler definition. — Mephist

But of course. Foundations are always historically contingent. Set theory in its current form is less than a century old dating from Zermelo's 1922 axiomitization. By the way Cantor always gets the credit, but it's Zermelo who did the heavy lifting in the development of modern set theory. Before Cantor there was no set theory. A few decades from now category theory and type theory will be much better known and perhaps set theory will fade into history. It won't be wrong, just out of fashion. That's inevitable.

I think it does not correspond to anything in the physical world, so basically yes: it's just an imaginary gadget that obeys the rules of set theory, ad it could be substituted by other similar gadgets that logically equivalent to it. — Mephist

Well of course. Was someone thinking set theory refers to the physical world? It's a formal game. It's the chess analogy I constantly use (to no effect) with @Metaphysician Undercover. You are standing on a soapbox fervently preaching something so obvious it barely needs to be said. Set theory is an attempt at a formalization of math. What of it?

Can you show me a proof of consistency of ZFC set theory that doesn't make use of another even more complex and convoluted set theory? — Mephist

Not complex or convoluted? Sure. Grothendieck universes are very plausible and straightforward, and are the standard everyday mathematical framework in much of modern math. Wiles's proof of Fermat's last theorem is presented in the framework of universes, even though that's probably not strictly necessary. The proof of the consistency of ZFC via assuming an inaccessible cardinal is part and parcel of modern math. Of course we DO have to assume an axiom in addition to ZFC; but that axiom is by no means unintuitive or unbelievable. It's rather natural.

And what of it? You are making a TERRIBLE argument. That because you have some technical objection to the empty set (which you have not articulated) therefore the entirety of modern math is rejected because, "Nyah nyah type theory is better." And this to a simple technical question, does the empty set exist. And you go, "Well no, because the entirety of contemporary mathematics is bullshit."

That's your argument?

If I may make an analogy, it's like a beginning programmer arguing that their favorite language is better, just because it's the only one they know. And you, an experienced developer with a dozen languages under your professional belt, can only shake your head and remember when you were that young and dumb.

Oh and Columbo would say, One More Thing.

The empty set is the unique initial object in the category Set. You do believe in the category Set, don't you?

https://en.wikipedia.org/wiki/Initial_and_terminal_objects#Examples

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.

By the way, dependent type theory - at least a subset of the version used in coq - has been proved to be consistent (but of course it is not complete - no way to avoid Godel's incompleteness theorem). — Mephist

This point must be profoundly wrong or disingenuous on your part. If anything at all -- dependent type theory, Coq, a tuna sandwich on rye -- can prove its own consistency, then it must necessarily be entirely useless for representing modern mathematics.

If on the other hand you mean that it's been proven consistent using assumptions outside of itself, then the same is true of set theory. In 1936 Gerhard Gentzen proved the consistency of Peano arithmetic by assuming the consistency of transfinite induction up to the ordinal $\varepsilon_0$.

And in algebraic geometry, the branch of math that led to the original discovery of category theory by Mac Lane in the 1940's, the existence of an inaccessible cardinal is assumed. This in effect amounts to the assumption that ZFC is consistent, since an inaccessible cardinal is a model of set theory; that is, a set that satisfies the axioms of set theory.

In category theory and algebraic geometry the inaccessible cardinal shows up in the definition of a Grothendieck universe. The reason universes are the natural setting for categorical algebraic geometry is that they ensure that the categories in question contain enough sets to make the theory sensible.

Per the Wiki article: "The concept of a Grothendieck universe can also be defined in a topos." So if you're as as big a believer in topos theory as you say, you have a ready-made proof within topos theory of the consistency of ZFC.

Professional mathematicians, even category theorists -- especially category theorists -- understand that if you can't ground your categorical theory in set theory one way or another, you don't have a good theory.

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.

It's perfectly possible (and probable) that I wrote something wrong, but I would like to know what's the mistake that I made. — Mephist

I'm sorely behind in responding to my mentions but I am getting to this point soon. I will explain in detail why you are wrong in your response to my demonstration of the existence of the empty set.

Of course, the nature of the empty set is essential to understanding what a "set" is, and if a theory has contradictory premises, then I object to the theory in its entirety, it needs to be reformulated — Metaphysician Undercover

Take it up with Frege, Russell, Zermelo, von Neumann, and all the other brilliant 20th century set theorists including those working at the forefront of knowledge today such as Hamkins, Steele, Woodin, Shelah, and others. Your childish objection to modern math and science is noted. You don't have to repeat it. I heard you the first 20 times. Your nihilistic point does not gain sanity by repetition.

What I object to is the claim of "existence" for objects which have a contradictory description. This is not nihilistic, but a healthy skepticism. The attitude demonstrated by you, that we might assign "existence" arbitrarily is best described as delusional. — Metaphysician Undercover

I can't take that as much of a criticism, since by your own criterion you regard the entire community of working professional mathematicians as delusional, and perhaps the physicists too. You have put me into some great company, that, frankly, I hardly deserve.

I do agree that I have not provided a definition of mathematical existence that you would find satisfactory. I'm thinking the issue over but it's tricky. However you are someone who regards the simple adjunction to the rational numbers of a formal square root of two as completely beyond the pale. I confess that I'm at a loss to respond to such philosophical nihilism. The principles by which we accept the rational numbers are no different and no logically simpler than those by which we adjoin a formal square root of two.

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.

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.

Mephist seems to have no rebuttal to the arguments which demonstrate that the "empty set" is a contradictory concept, and unlike you, seems about ready to face the reality of this. — Metaphysician Undercover

When I challenged you on this point, you admitted that it's not only the empty set, but set theory in its entirety that you object to. Then you added that you reject modern physics as well. Nihilism. You are speaking nonsense. For @Mephist's part, he read a book on category theory but knows very little actual math. The fact that he's confused about the empty set, even when shown its existence proof from the axioms of set theory, supports that conclusion.

I didn't change idea: there is no contradiction in the axiomatic definition of sets given by ZFC, at least for what has been discovered until now. It has not even been proved that ZFC is not contradictory, however; but since nobody has found any contradiction in ZFC after 100 years of using it, I would guess that it is consistent.
By the way, dependent type theory - at least a subset of the version used in coq - has been proved to be consistent (but of course it is not complete - no way to avoid Godel's incompleteness theorem). — Mephist

Was that a yes or a no? Stop dancing. You're wrong on the facts, wrong on the math. Why are you trying to placate @Metaphysician Undercover's nutty ideas?

What do you think a set is, if not anything that obeys the rules of set theory?

Yea, this discussion is going in circles without any hope of a conclusion. I would like to finish discussing about empty sets! — Mephist

You retract or stand by your claim that there is no empty set?

This is a "set_or_nothing", not a "set" — Mephist

If it's not a set, which do you disagree with: The axiom schema of specification? Or that the natural numbers are a set?

Does the smiley mean that you don't actually believe what you wrote but that talking to @Metaphysician Undercover has caused you to lose your grip? What does the smiley mean? Why did you claim there is no empty set? If you so claim, what do you do with the brief existence proof I just gave?

When I put the same question to @Metaphysician Undercover, he admitted that it's not the empty set he objects to, but rather the entirety of set theory. That's a nihilistic position but at least it's a position. You have none that I can see.

Yes. — Zelebg

Are you keeping it secret?

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.

You are misinterpreting and I already explained your error. — Zelebg

Do you know how to implement 2 and 5? Yes or no?

No, I did not say that. I asked you what is the problem, then suggested you do not know what is the problem, and finally I said what is the real problem and how to solve it. Overall, nothing to do with 2 and 5, but 3 and 4. — Zelebg

Someone using your handle wrote:

↪fishfry
Because nobody knows how to implement 2 and 5, or even if it's possible to do so.

Not known is 'what' to implement, so 'how' is not even a question yet. But what I am suggesting here is both what and how to implement, and relevant part is hardware configuration, not software modules. — Zelebg

I ask again: How do you implement 2 and 5?

You think the problem is 2. and 5. and I say the so called 'hard problem of consciousness' is 3. and 4., which needs to be implemented with some kind of display / camera system, rather than by any kind of software algorithm. — Zelebg

You said you offered suggestions on how to implement 2 and 5. I didn't see them. Changing the subject doesn't answer my question. How do you implement 2 and 5?

Not known is 'what' to implement, so 'how' is not even a question yet. But what I am suggesting here is both what and how to implement, — Zelebg

Ok. How do you implement 2 and 5?

Let us know when you've figured out how to implement 2 and 5.

Why is that a problem? — Zelebg

Because nobody knows how to implement 2 and 5, or even if it's possible to do so.

Maybe you are right: sets cannot be empty. — Mephist

So what do you make of the set $\{x \in \mathbb N : x \neq x \}$?

You reject the axiom schema of specification? You don't think $\mathbb N$ is a set? I really want to hear this.

How did Archimedes calculate pi? — TheMadFool

I found this but I don't know anything about it beyond Googling around.

I mean, it's a given that they have a decimal expansion e.g
pi = 3.14159...I suspect this involves some kind of division e.g . pi = circumference/diameter and that's a fraction isn't it? — TheMadFool

$\pi$ is no longer defined geometrically. Here is the modern analytic definition.

First you establish the theory of convergent power series in one complex variable, analogously to how it's done in freshman calculus for a single real variable.

Then you define the function $\displaystyle \exp(z) = \sum_{n=0}^\infty \frac{z^n}{n!}$, which you can show converges for all complex numbers $z$.

You then define the functions

$\cos(z) = \frac{\exp(i z) + \exp(-i z)}{2}$

and

$\sin(z) = \frac{\exp(i z) - \exp(-i z)}{2 i}$

It's not hard to show that $exp$, $cos$, and $sin$ are the usual exponential, cosine, and sine functions of pre-calculus, extended to the complex domain. However no triangles or circles or geometric notions are involved at all. These definitions are purely analytic, meaning that they pertain only to the theory of complex analytic functions.

Then what is $\pi$? It's the smallest positive zero of the sine function. A glance at the unit circle shows that this property uniquely characterizes the number $\pi$ as we usually understand it.

This is how an analyst thinks about the number $\pi$. I mention this to give an example of how mathematicians go about "defining an irrational number" using the machinery of modern math.

https://en.wikipedia.org/wiki/Exponential_function#Complex_plane

In terms of calculating the digits of $\pi$, you could in fact use the power series for $exp$, the definition of $sin$, and a root-approximating algorithm to compute as many digits as you like. Of course I imagine calculator manufacturers use different algorithms but either way you can see that no triangles or circles are ever involved. It's all about convergent infinite power series.

A famous but inefficient algorithm for the digits of $\pi$ is based on the Leibniz formula

$\displaystyle \frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \dots = \sum_{n = 0}^\infty (-1)^n \frac{1}{2 n + 1}$

It's fun to write a little program and watch it converge.

I am not a mathematician, but my understanding is that this is exactly backwards. Set theory can be established within category theory, but category theory cannot be established within set theory. "Set" is one of the categories, but there are others that need not and do not conform to standard set theory. — aletheist

I'm not personally in accord with this point of view.

The relationships among category theory, set theory, various flavors of type theory, and other candidate foundations is not a simple matter to be summed up in a phrase.

I found an informative and insightful thread here.

Are category-theory and set-theory on the equal foundational footing?

The entire thread is well worth your time for anyone interested in contemporary foundations. In particular Derek Elkins's response is comprehensive and mind-expanding.

A few points. First, CT does not model ZFC. Rather, Elkins notes that "ETCS [Elementary theory of the category of sets] is equivalent to Bounded Zermelo set theory (BZ) which is weaker than ZFC."

Secondly, CT doesn't properly account for mathematical existence. This quote is a comment by Michael Greinecker:

"Set theory is full of axioms that guarantee that some things exist, which can be used to show that other things exist and finally that all the mathematical objects we want to exist do exist. Category theory doesn't really do that. You can formulate existence statements in categorical terms, but it is much less clear what kind of foundations category theory is meant to supply."

So it's not fair to say that you can "get set theory from category theory" or "do set theory within category theory." Those are facile statements. Facile means, "appearing neat and comprehensive only by ignoring the true complexities of an issue; superficial." That's apt.

On the other hand, can we do category within set theory? The conventional wisdom would be that classes are too big to be sets. The category of sets, for example is surely not a set. The category of Abelian groups is not a set. One way around this is to consider only "small" categories in which the objects and morphisms form sets. Another way is to say that we are agnostic as to whether a category contains "all" possible instances of an object type; but rather contains "enough" for any argument you need to make. I have seen this point of view expressed but don't have a reference at the moment.

Or we could just say that categories in general are proper classes, in the sense of "Predicate satisfiers that are too big to be sets." And then the argument is that since ZFC doesn't have proper classes, you can't do category theory within ZFC.

That is the argument. But what about a set theory like Morse-Kelly or Von Neuman-Bernays-Gödel set theory, two set theories that DO incorporate proper classes? Can you do category theory in those set theories? Good question.

This brings up a larger question: Which set theory, and which version of category theory? There are various flavors of each. The Stackexchange thread brings out this point in more detail. I truly hope people will read that page to get a sense of the many interrelated and nontrivial issues.

Set theory and category theory are not in a cage match to the death, as some seem to think. They're complementary ideas in a toolkit. It's like the programmers arguing over functional versus object-oriented. They're tools, not religions.

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.

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?

?? 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

1. Camera A: visual input extern -> feeds into 2.
2. Program A: subconsciousness & memory -> feeds into 3.
3. Display A: visual output inner -> feeds into 4.
4. Camera B: visual input inner -> feeds into 5.
5. Program B: consciousness & free will -> feeds into 6.& 2.
6. Speaker: audio output extern — Zelebg

Let us know when you've figured out how to implement 2 and 5.

That's not quite right. I learned how the game was played, then decided I didn't want to play it. The fact that it was a game, and the rules referred to nothing "real" probably made me think of it as a waste of time — Metaphysician Undercover

You reject all formal systems not based strictly on physical reality? Do you drive on the correct side of the road appropriate to your jurisdiction? Traffic laws are a made-up game too.

Again, nihilism. You reject games, you reject abstraction, you reject science. Fun to argue all day long on an Internet forum but I truly doubt you actually live this way. I bet you obey traffic laws even though they are not laws of nature.

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.

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.

If you are asking me to accept the precepts of the system without judging them, then you are being unreasonable. — Metaphysician Undercover

Were you like this when you learned to play chess? "This is the knight." "But no it's not REALLY a knight. Real knights don't make moves like that, they slay dragons and rescue damsels. I refuse to accept the rules of your game till you tell me what they mean outside of the game."