**80**

votes

**23**answers

13k views

### Has philosophy ever clarified mathematics?

I've recently been reading some standard textbooks on the philosophy of mathematics, and I've become quite frustrated that (surely due to my own limitations) I don't seem to be gleaning any ...

**78**

votes

**16**answers

20k views

### What if Current Foundations of Mathematics are Inconsistent? [closed]

The title of the question is also the title of a talk by Vladimir Voevodsky, available here.
Had this kind of opinion been expressed before?
EDIT. Thanks to all answerers, commentators, voters, ...

**74**

votes

**11**answers

6k views

### Why do Groups and Abelian Groups feel so different?

Groups are naturally "the symmetries of an object". To me, the group axioms are just a way of codifying what the symmetries of an object can be so we can study it abstractly.
However, this heuristic ...

**67**

votes

**18**answers

9k views

### Can a mathematical definition be wrong?

This question originates from a bit of history. In the first paper on quantum Turing machines, the authors left a key uniformity condition out of their definition. Three mathematicians subsequently ...

**63**

votes

**39**answers

12k views

### nontrivial theorems with trivial proofs

A while back I saw posted on someone's office door a statement attributed to some famous person, saying that it is an instance of the callousness of youth to think that a theorem is trivial because ...

**59**

votes

**4**answers

8k views

### Nelson's program to show inconsistency of ZF

At the end of the paper Division by three by Peter G. Doyle and John H. Conway, the authors say:
Not that we believe there really are any such things as infinite sets, or that the Zermelo-Fraenkel ...

**56**

votes

**9**answers

7k views

### Is there any formal foundation to ultrafinitism?

Ultrafinitism is (I believe) a philosophy of mathematics that is not only constructive, but does not admit the existence of arbitrarily large natural numbers. According to wikipedia, it has been ...

**55**

votes

**13**answers

12k views

### Logic in mathematics and philosophy

What are the relations between logic as an area of (modern) philosophy and mathematical logic.
The world "modern" refers to 20th century and later, and I am curious mainly about the second half of ...

**55**

votes

**2**answers

16k views

### Is the analysis as taught in universities in fact the analysis of definable numbers?

Ten years ago when I studied in the university I had no idea about definable numbers, but I came to this concept myself. My thoughts were as follows:
All numbers are divided into two classes: those ...

**53**

votes

**3**answers

4k views

### What was Hilbert's view of Gödel's Incompleteness Theorems?

According to Solomon Feferman, in his slide presentation "Three Problems for Mathematics", Hilbert wrote (in regards to Gödel's second incompleteness theorem):
...the end goal [is] to establish as ...

**49**

votes

**9**answers

6k views

### When have we lost a body of mathematics because errors were found?

The history of mathematics over the last 200 years has many occasions when the fundamental assumptions of an area have been shown to be flawed, or even wrong. Yet I cannot think of any examples where, ...

**48**

votes

**17**answers

10k views

### Is rigour just a ritual that most mathematicians wish to get rid of if they could?

"No". That was my answer till this afternoon! "Mathematics without proofs isn't really mathematics at all" probably was my longer answer. Yet, I am a mathematics educator who was one of the panelists ...

**48**

votes

**7**answers

2k views

### How closed-form conjectures are made?

Recently I posted a conjecture at Math.SE:
$$\int_0^\infty\ln\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx\stackrel{?}{=}\frac{\pi}{2}(\mu^2-\nu^2),$$
where $J_\mu(x)$ and $Y_\mu(x)$ ...

**46**

votes

**9**answers

5k views

### How do they verify a verifier of formalized proofs?

In an unrelated thread Sam Nead intrigued me by mentioning a formalized proof of the Jordan curve theorem. I then found that there are at least two, made on two different systems. This is quite an ...

**45**

votes

**7**answers

5k views

### Why hasn't mereology succeeded as an alternative to set theory?

I have recently run into this wikipedia article on mereology. I was surprised I had never heard of it before and indeed it seems to be seldom mentioned in the mathematical literature. Unlike set ...

**43**

votes

**2**answers

6k views

### Lawvere's “Some thoughts on the future of category theory.”

In Lecture Notes in Mathematics 1488, Lawvere writes the introduction to the Proceedings for a 1990 conference in Como.
In this article, Lawvere, the inventor of Toposes and Algebraic Theories, ...

**42**

votes

**15**answers

6k views

### Abstract Thought vs Calculation

Jeremy Avigad and Erich Reck in their remarkable historical paper "Clarifying the nature of the infinite: the development of metamathematics and proof theory" claim that one of the factors of becoming ...

**41**

votes

**5**answers

3k views

### How to resolve a disagreement about a mathematical proof?

I am having a problem which should not exist. I am reading what I believe to be an important paper by a person - let me call him/her $A$ - whom I believe to be a serious and talented mathematician. A ...

**39**

votes

**9**answers

4k views

### Examples in mirror symmetry that can be understood.

It seems to me, that a typical science often has simple and important examples whose formulation can be understood (or at least there are some outcomes that can be understood). So if we consider ...

**37**

votes

**7**answers

7k views

### Arguments against large cardinals

I started to learn about large cardinals a while ago, and I read that the existence, and even the consistency of the existence of an inaccessible cardinal, i.e. a limit cardinal which is additionally ...

**34**

votes

**10**answers

3k views

### Believing the Conjectures

In Believing the axioms (I and II), Penelope Maddy proposes five "rules of thumb" that she then uses to justify large cardinal axioms in set theory. These extrinsic rules are modeled after the ...

**33**

votes

**4**answers

4k views

### Did Euler prove theorems by example?

In his 2014 book, Giovanni Ferraro writes at beginning of chapter 1, section 1 on page 7:
Capitolo I
Esempi e metodi dimostrativi
Introduzione
In The Calculus as Algebraic ...

**32**

votes

**8**answers

5k views

### Uninteresting questions with interesting answers [closed]

What are best examples of questions in mathematics that are not interesting until one knows the answers, whose answers themselves are what is interesting?
The thing that prompts me to post this is ...

**32**

votes

**5**answers

4k views

### Why do categorical foundationalists want to escape set theory?

This is a question that I have seen asked passively in comments relating to the separation of category theory from set theory, but I haven't seen it addressed in full.
I know that it's possible to ...

**32**

votes

**5**answers

8k views

### Categorical foundations without set theory

Can there be a foundations of mathematics using only category theory, i.e. no set theory? More precisely, the definition of a category is a class/set of objects and a class/set of arrows, satisfying ...

**32**

votes

**5**answers

4k views

### Were Bourbaki committed to set-theoretical reductionism?

A set-theoretical reductionist holds that sets are the only abstract objects, and that (e.g.) numbers are identical to sets. (Which sets? A reductionist is a relativist if she is (e.g.) indifferent ...

**31**

votes

**2**answers

4k views

### Why is this new result such a big deal?

This popular article reports a recent result in reverse mathematics, showing that a certain theorem in Ramsey theory is provable from RCA$_0$, the base theory in SOSOA. Then there are a bunch of ...

**29**

votes

**7**answers

3k views

### Can a problem be simultaneously polynomial time and undecidable?

The Robertson-Seymour theorem on graph minors leads to some interesting conundrums.
The theorem states that any minor-closed class of graphs can be described by a finite number of excluded minors. As ...

**28**

votes

**7**answers

2k views

### Why should we believe in the axiom of regularity?

Today I started reading Maddy's Believing the axioms. As I knew beforehand, it includes some discussion of ZFC axioms. However, I really hoped for a more extensive discussion of axiom of foundation/...

**28**

votes

**3**answers

3k views

### The influence of string theory on mathematics for philosophers.

I've agreed, perhaps unwisely, to give a talk to Philosophers about string theory.
I'd like to give the philosophers an overview of the status and influence of string theory in physics, which I feel ...

**27**

votes

**8**answers

2k views

### Interpretation of the Second Incompleteness Theorem

For simplicity, let me pick a particular instance of G\"odel's Second Incompleteness
Theorem:
ZFC (Zermelo-Fraenkel Set Theory plus the Axiom of Choice, the usual foundation of mathematics) does ...

**26**

votes

**3**answers

4k views

### Category of categories as a foundation of mathematics

In
Lawvere, F. W., 1966, “The Category of
Categories as a Foundation for
Mathematics”, Proceedings of the
Conference on Categorical Algebra, La
Jolla, New York: Springer-Verlag,
1–21.
...

**25**

votes

**14**answers

4k views

### Essential reads in the philosophy of mathematics and set theory

I am graduate student and have a decent understanding of logic and set theory.
Recently I have got interested in the philosophy of mathematics and set theory. I have read a number of papers by ...

**24**

votes

**7**answers

4k views

### Excellent mathematical explanations

In the Stanford Encyclopedia of Philosophy there is an entry on mathematical explanation.
The basic philosophical question is: What makes a proof explanatory?
Two main "models" of mathematical ...

**23**

votes

**10**answers

4k views

### Physics and Church–Turing Thesis

Is there constructed some set of physical laws from which we can logically obtain that any function that can be implemented in some device is Turing computable?
EDIT
I believe that if we restrict ...

**23**

votes

**4**answers

2k views

### In what ways did Leibniz's philosophy foresee modern mathematics?

Leibniz was a noted polymath who was deeply interested in philosophy as well as mathematics, among other things. From my mathematical readings I have the impression that Leibniz's stature as a ...

**22**

votes

**7**answers

2k views

### Supervenience in mathematics

I'm not quite sure if this is the right place to ask, and if this is the right way to ask, but I dare.
In philosophy (of mind, e.g.) the concept of supervenience is used:
"Supervenience [is] used ...

**22**

votes

**3**answers

1k views

### Mathematicians with Aphantasia (Inability to Visualize Things in One's Mind)

Are there any mathematicians with aphantasia? If so, could they please elaborate upon what their experience with mathematics is like?
I realize that this question probably falls outside of the scope ...

**21**

votes

**9**answers

5k views

### Why are proofs so valuable, although we do not know that our axiom system is consistent? [closed]

As a person who has been spending significant time to learn mathematics, I have to admit that I sometimes find the fact uncovered by Godel very upsetting: we never can know that our axiom system is ...

**21**

votes

**6**answers

2k views

### Philosophical Question related to Largest Known Primes

The other day while discussing math, and primes specifically, the following question came to mind, and I figured I'd ask it here to see what people's opinions on it might be.
Main Question: ...

**20**

votes

**7**answers

5k views

### What is Realistic Mathematics?

This post is partially about opinions and partially about more precise mathematical questions. Most of this post is not as formal as a precise mathematical question. However, I hope that most readers ...

**20**

votes

**5**answers

2k views

### What's special about the Simplex category?

I have been wondering lately what makes simplicial sets 'tick'.
Edited
The category $\Delta$can be viewed as the category of standard $n$-simplices and order preserving simplicial maps. The goal of ...

**20**

votes

**2**answers

2k views

### Euler's mathematics in terms of modern theories?

Some aspects of Euler's work were formalized in terms of modern infinitesimal theories by Laugwitz, McKinzie, Tuckey, and others. Referring to the latter, G. Ferraro claims that "one can see in ...

**19**

votes

**18**answers

8k views

### What are some applications of other fields to mathematics?

It is commonplace to consider applications of mathematics to other fields, especially the exact sciences. But what I would like to know about is the converse topic, namely:
What are some ...

**19**

votes

**13**answers

2k views

### An example of a proof that is explanatory but not beautiful? (or vice versa)

This question has a philosophical bent, but hopefully it will evoke straightforward, mathematical answers that would be appropriate for this list (like my earlier question about beautiful proofs ...

**19**

votes

**2**answers

2k views

### Should there be a true model of set theory?

As I understand it, there is a program in set theory to produce an ultimate, canonical model of set theory which, among other things, positively answers the Continuum Hypothesis and various questions ...

**18**

votes

**4**answers

2k views

### Are proper classes objects?

Many of us presume that mathematics studies objects. In agreement with this, set theorists often say that they study the well founded hereditarily extensional objects generated ex nihilo by the "...

**18**

votes

**8**answers

4k views

### To what extent is it true that “number theory = mathematics”? [closed]

In a thought-provoking answer to this MO question, Kevin Buzzard
and several commentators have described a multitude of ways in which
number theory is related to other parts of mathematics. It seems ...

**18**

votes

**7**answers

1k views

### Extensional theorems mostly used intensionally

Some theorems are stated and proved extensionally, but in practice are almost always used intensionally. Let me give an example to make this clear -- integration by parts:
$$ \int_a^b f(x)g'(x)ds = \...

**17**

votes

**8**answers

1k views

### The Importance of ZF

It seems as though many consider ZF to be the foundational set of axioms for all of mathematics (or at least, a crucial part of the foundations); when a theorem is found to be independent of ZF, it's ...