The math-philosophy tag has no wiki summary.

**9**

votes

**5**answers

2k views

### Proper classes and their consequences

I have two main questions:
What is a proper class? I've read that it's collection of objects that's "too big" to be a set, but in what sense is such a collection "too big"? Since I'd like this post ...

**23**

votes

**6**answers

2k 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 ...

**14**

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 ...

**14**

votes

**1**answer

670 views

### How is Fredkin and Toffoli's Conservative Logic related to Linear Logic?

In the answers to this question, Timothy Gowers asks:
I've been interested in this question for some time. I haven't put any serious thought into it, so all I can offer is a further question ...

**49**

votes

**8**answers

6k 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 ...

**40**

votes

**2**answers

13k 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 ...

**31**

votes

**7**answers

5k 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 ...

**12**

votes

**3**answers

2k views

### What was Weierstrass's counterexample to the Dirichlet Principle?

Shortly after his work on the foundations of geometry David Hilbert turned his attention to finding a suitable statement of the Dirichlet principle, from which to prove the Riemann mapping theorem and ...

**15**

votes

**2**answers

3k views

### Question arising from Voevodsky's talk on inconsistency

This question arises from the talk by Voevodsky mentioned in
this recent MO question. On one of his slides, Voevodsky says that
a general formula even with one free variable describes a subset of ...

**74**

votes

**16**answers

17k 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, ...

**16**

votes

**2**answers

2k views

### Universe view vs. Multiverse view of Set Theory

Here I refer to Hamkins' slides:
http://lumiere.ens.fr/~dbonnay/files/talks/hamkins.pdf
particularly, to the "Universe view simulated inside Multiverse", p. 22.
My question is: is it very unsound ...

**12**

votes

**1**answer

772 views

### evil properties, higher category theory and well-chosen tensor products

Let's start with the following random example: If $F$ is a presheaf, then for every chain of open subsets $U \subseteq V \subseteq W$, the morphisms $F(W) \to F(V) \to F(U)$ and $F(W) \to F(U)$ ...

**20**

votes

**6**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 ...

**51**

votes

**3**answers

6k 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 ...

**17**

votes

**7**answers

4k 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 ...

**9**

votes

**2**answers

1k views

### Proving Independence of Axioms by Exhibiting Models Which Don't Satisfy Our Intuition

I recently saw the proof of the independence of ZF (with allowance for multiple empty sets) and AC. The proof constructed the model based on a set theory generated by infinitely many empty sets and ...

**7**

votes

**7**answers

2k views

### Meaning of Kronecker's comment to Lindemann

At the Mactutor history page, it is said that Kronecker remarked to Lindemann:
"What good your beautiful proof on [the transcendence of] π? Why investigate such problems, given that irrational ...

**8**

votes

**3**answers

2k views

### Use of Conjectures to Prove a Theorem

Name a theorem T that has a proof based upon the truth of a conjecture C, and also has another proof based upon the falsehood of the same conjecture C, but for longtime has no known direct proof that ...

**6**

votes

**2**answers

965 views

### When is a statement provable?

We all know that Gödel showed that there, in a formal system, are true statements that are non-provable (undecidable). In ZFC, there's Kaplansky's Conjecture, the Whitehead problem, etc.
We can also ...

**12**

votes

**3**answers

1k views

### Are there natural examples of mathematical statements which follow from consistency statements?

Motivation
One of the methods for strictly extending a theory $T$ (which is axiomatizable and consistent, and includes enough arithmetic) is adding the sentence expressing the consistency of $T$ ( ...

**63**

votes

**18**answers

8k 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 ...

**13**

votes

**3**answers

1k views

### Has there ever been a weaker Church-like thesis?

Background. The Church-Turing thesis, in one of its many equivalent formulations, states that the intuitively computable arithmetical functions are exactly those computed by Turing machines.
...

**30**

votes

**15**answers

5k 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 ...

**9**

votes

**6**answers

1k views

### Defining variable, symbol, indeterminate and parameter

Are there precise definitions for what a variable, a symbol, a name, an indeterminate, a meta-variable, and a parameter are?
In informal mathematics, they are used in a variety of ways, and often in ...

**19**

votes

**9**answers

4k 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 ...

**1**

vote

**2**answers

727 views

### Mathematics of the Anthropic Principle [closed]

A form of the anthropic principle is as follows: "We can observe the universe only because we can exist within it in some way such that we can observe it, and it exists such that we can observe it."
...

**45**

votes

**34**answers

10k 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 ...

**13**

votes

**2**answers

2k views

### Clarification of Gödel's second incompleteness theorem

I am sorry for the following question, because the actual answer to this question is in the beautiful works of Feferman and Jeroslow, but, unfortunately, I havn't any time to go into that specific ...

**4**

votes

**0**answers

612 views

### BGG category everywhere implies generalized Kazhdan-Lusztig formula?

Maybe this question is vague. I am not an expert on what I asked, if I made mistake, please point out.
BGG category was discoverd in Lie algebra setting. One has Verma module $M(\lambda)$, ...

**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 ...

**6**

votes

**1**answer

844 views

### Using the multiverse approach to decide the law of the exluded middle?

Recently, in response to deciding the Continuum Hypothesis $CH$, Hamkins and Gitman have proposed consider a multiverse of set-theoretic universes, some in which $CH$ is true, some in which $\neg CH$ ...

**29**

votes

**5**answers

3k 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 ...

**7**

votes

**3**answers

787 views

### What are trig classes like within a universe that's “noticeably” hyperbolic?

[I want to think that this question has an answer, but it may be more a "community wiki" discussion. Feel free to re-tag.]
What are trig classes like within a universe that's "noticeably"[*] ...

**0**

votes

**1**answer

527 views

### Formal definition of 'useful' ?

Has anyone worked out a formal, general-enough definition of what is 'useful', so that it could reflectively be used in mathematics? I am aware of the work in utility theory from economics (but ...

**16**

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 = ...

**43**

votes

**9**answers

4k 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 ...

**2**

votes

**3**answers

1k views

### The problem of infinity [closed]

Background and motivation
The following is copied from my blog since someone thought it was the clearest statement I had made regarding a problem I recently posed. On their advice, it is a community ...

**-3**

votes

**2**answers

1k views

### Finite versus infinite on non-Hausdorff topologies [closed]

Question: Does there exist some real-valued function $f(x)$ where $f: \mathbb{R} \to \mathbb{R}$, for which $\lim_{x \to \infty}$ converges on a non-Hausdorff topology but does not converge on a ...

**5**

votes

**5**answers

3k views

### Models of ZFC Set Theory - Getting Started

For just any first-order theory: What are the sets I am supposed/allowed to think of when thinking of models as sets (of something + additional structure)?
Provided:
I can think of models of any ...

**1**

vote

**4**answers

2k views

### Are all mathematical theorems necessarily true?

Define a formal tautology as a statement where by the nature of its atomic components there exists no truth-value assignment where it is not true. A contingent statement is a statement that is true by ...

**43**

votes

**2**answers

5k 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, ...

**5**

votes

**1**answer

882 views

### Co-Objects are better [closed]

This is a rather vague question, but perhaps we can talk about it.
There are two types of mathematical objects (which don't exclude each other):
A) There is a good description of morphisms defined ...

**0**

votes

**2**answers

410 views

### In what sense Fraissean view point shows Model Theory can be done without any formal syntax and deduction rule?

In this post I want to look at an issue I was in doubt when looking at the comment of F. G. Dorais in the post In model theory, does compactness easily imply completeness?
F. G. Dorais remark was:
...

**11**

votes

**2**answers

799 views

### Where are we working when we prove metamathematical theorems?

I am posting my comment from this question as a separate question, as was recommended to me.
(EDIT: I'm sorry if it ended up being too similar a question, I just wanted to phrase it in the ...

**32**

votes

**5**answers

3k 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 ...

**15**

votes

**10**answers

1k views

### Can you prove equivalence without being able to calculate it?

In mathematics we often seek to classify objects up to an equivalence relation, where two objects A and B are said to be equivalent if there exists a map $f:A\rightarrow B$ satisfying certain ...

**7**

votes

**3**answers

2k views

### randomness in nature [closed]

What is the explanation of the apparent randomness of high-level phenomena in nature?
For example the distribution of females vs. males in a population (I am referring to randomness in terms of the ...

**20**

votes

**3**answers

3k 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.
...

**8**

votes

**3**answers

1k views

### categorification of logic

has there be an effort to categorify first order logic? More particularly, structures in the sense of logic.
If so, then every structure of a first order theory is a category. so in particular, the ...

**7**

votes

**4**answers

1k views

### Alternative axiom to induction

Is anyone aware of alternative axioms to induction? To be precise, consider peano axioms without induction PA-. Is there any axiom/axiom schema that is equiconsistent to induction, assuming PA-? If ...