**5**

votes

**2**answers

1k views

### Is beauty at the high school level even possible? [closed]

This question is a follow up to 74841, and follows from a suggestion by Gian-Carlo Rota that beauty as judged by the educated public differs from that experienced by mathematicians (he gives Euclidean ...

**13**

votes

**5**answers

2k views

### What is the “reason” for modularity results?

The question is a little wishy-washy, but I take my cues from other popular questions that relate to the philosophy behind the mathematics as Why do Groups and Abelian Groups feel so different? .
I ...

**2**

votes

**1**answer

838 views

### Notion of Truth and Axioms

Hello,
The proofs in logic often use the notion of truth.
Can we ignore the notion of truth, if we add axioms to the Peano's axioms ?
Is it possible to prove Gödel's first incompleteness theorem ...

**40**

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 ...

**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 ...

**6**

votes

**1**answer

922 views

### How are mathematical objects defined from an ultrafinitist perspective?

I remember attending a lecture given by an ultrafinitist who denied that curves are a set of points, he would only say that any particular point may or not be on the curve. Similarly for algebraic or ...

**12**

votes

**1**answer

865 views

### Why should I believe the Singular Cardinal Hypothesis?

The Singular Cardinal Hypothesis (SCH) is the statement that $\kappa^{cf(\kappa)} = \kappa^+ \cdot 2^{cf(\kappa)}$ for every singular cardinal $\kappa$ (or various equivalent statements).
It is ...

**-1**

votes

**1**answer

971 views

### Is it possible to construct a finite mathematical universe? [duplicate]

Possible Duplicate:
Is there any formal foundation to ultrafinitism?
Very recently I have come across the skeptic opinions of a school of mathematicians(ultrafinitist) over a physically ...

**12**

votes

**5**answers

3k views

### What should philosophers know about math? [closed]

I know, I know... this is not a technical question. Nevertheless, I believe this is the right place to ask such question.
I am sure many of you read about philosophy, including philosophy of ...

**13**

votes

**3**answers

1k views

### Are there examples of nonconstructive metaproofs?

This came up in a question on the xkcd forums. Is it possible to have a nonconstructive metaproof, i.e. a proof that there exists a proof in some formal system which does not construct said proof? Are ...

**10**

votes

**10**answers

4k views

### Is PA consistent? do we know it?

1) (By Goedel's) One can not prove, in PA, a formula that can be interpreted to express the consistency of PA. (Hopefully I said it right. Specialists correct me, please).
2) There are proofs ...

**15**

votes

**1**answer

1k views

### Martin's “Philosophical Issues about the Hierarchy of Sets”

Some months ago (October 2010), in the context of the Workshop on Set Theory and the Philosophy of Mathematics, Professor Donald A. Martin gave a talk entitled "Philosophical issues about the ...

**7**

votes

**6**answers

987 views

### Seemingly emergent structures in mathematics

I rather suspect that this must have come up here on MO already, but my handful of searches didn't turn up the thread, so...
I'm curious about examples of mathematical structure that seems to arise ...

**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: ...

**55**

votes

**13**answers

11k 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 ...

**1**

vote

**4**answers

861 views

### Condition of possibility = Co-Implication

Sorry, but I do not know another place to post this question.
Condition of possibility is an important philosophical concept. Naively, this concept could be formally defined this way:
$q$ is a ...

**44**

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 ...

**3**

votes

**1**answer

197 views

### Class Separation, Oracles, Relativization

It is known, there exists oracles A, B s.t.:
$P^A = NP^A; P^B \neq NP^B$, showing that any proof of P vs NP must be non-relativizing.
Questions:
(1) Can we actually use Oracles to separate ...

**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 ...

**8**

votes

**3**answers

1k views

### Kunen's use of Countable Transitive Models

Hi,
I have a doubt concerning Kunen's exposition of forcing in his classical book (arguably $the$ book on forcing). When dealing with Countable Transitive Models to set up the forcing machinery, ...

**12**

votes

**5**answers

1k views

### Intended interpretations of set theories

In his Set Theory. An Introduction to Indepencence Proofs, Kunen develops $ZFC$ from a platonistic point of view because he believes that this is pedagogically easier. When he talks about the intended ...

**22**

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 ...

**13**

votes

**5**answers

2k views

### Consistency strength needed for applied mathematics

Given that we can never proof the consistency of a theory for the foundations of mathematics in a weaker system, one could seriously doubt whether any of the commonly used foundational frameworks (ZFC ...

**8**

votes

**5**answers

2k views

### Why is it OK to rely on the Fundamental Theorem of Arithmetic when using Gödel numbering?

Gödel's original proof of the First Incompleteness theorem relies on Gödel numbering.
Now, the use of Gödel numbering relies on the fact that the Fundamental Theorem of Arithmetic is true and thus the ...

**2**

votes

**0**answers

449 views

### Single logic foundation vs. multi-logic foundation

Dear all,
I have always wondered why I have never read anything about this topic. My question is, are there are any books or articles covering this subject?
With this topic I mean the philosophical ...

**4**

votes

**2**answers

4k views

### where can you find Grothendieck's “Recoltes et Semailles”?

Where can you find Grothendieck's "Recoltes et Semailles"?
Is it available anywhere?

**10**

votes

**4**answers

2k views

### Is finitism an extreme form of constructivism?

I hope this question is not too soft for MO.
The Wikipedia says about finitism that it is an extreme form of constructivism. See http://en.wikipedia.org/wiki/Finitism. I doubt that this is correct.
...

**11**

votes

**6**answers

1k views

### Reasons for the importance of planarity and colorability?

Could it have been foreseen that - exemplarily - planarity and colorability would turn out to be such important concepts in graph theory (there's almost no textbook on graphs without two chapters ...

**11**

votes

**2**answers

555 views

### Inconsistency and workaday independence.

Set-theoretic topologists, for example, encounter many propositions that turn out independent from set theory. Sometimes these results require novel forcing arguments, but often they simply rely on ...

**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 ...

**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 ...

**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 ...

**28**

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 ...

**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 ...

**14**

votes

**1**answer

734 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 ...

**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 ...

**54**

votes

**2**answers

15k 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 ...

**37**

votes

**7**answers

6k 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 ...

**13**

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 ...

**75**

votes

**16**answers

19k 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, ...

**17**

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

897 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 $F(W) \to F(U)$ ...

**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 ...

**57**

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 ...

**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 ...

**11**

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 ...

**9**

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

1k 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 ...