The math-philosophy tag has no wiki summary.

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### Does there exist a non-trivial Ultrafinitist set theory?

Does there exist a set theory T-which has not yet been proved to be inconsistent-and in which
one can prove the existence of (1) the empty set (2) sets that are singletons and (3) sets which
have ...

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**12**answers

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### 2D Problems Which are Easier to Solve in 3D

It sometimes happens that 1D problems are easier to solve by somehow adding a dimension. For example, we convert linear differential equations for a real unknown to a complex unknown (to use complex ...

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**1**answer

604 views

### intensional equality in type theory

I want to know why we add an intensional equality in type theory to definitional equality ?
What is the aim with this intensional equality ?
thanks

**4**

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**0**answers

432 views

### sine and Archimedes' derivation of the area of the circle

The elementary "opposite over hypotenuse" definition of the sine function defines the sine of an angle, not a real number. As discussed in the article "A Circular Argument" [Fred Richman, The College ...

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**3**answers

872 views

### What is the status of irrational numbers within finitism/ultrafinitism?

According to constructivism a mathematical object to prove that it exists". There are several formulas to calculate pi, such as:
so I take it ...

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**6**answers

820 views

### Proof by `universal receiver'

Anyone following the news knows about the major breakthoughs that have taken place recently in $3$-manifold topology. These have come via a route whose big-picture I find to be conceptually ...

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**8**answers

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### ULTRAINFINITISM, or a step beyond the transfinite

Cantor has, in the immortal words of D. Hilbert, given all of us a paradise (or perhaps, I would rather say, a great vacation spot), the TRANSFINITE.
$\aleph_0, \aleph_1,\aleph_2\dots$
the lists ...

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vote

**0**answers

459 views

### Arguments against Reductio ad Absurdum [closed]

Could Reductio ad Absurdum not be consireded a valid proof method? Are there any compelling arguments against it, or at it's favor?
I feel like I am assuming some metamathematical hypothesis about my ...

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votes

**1**answer

232 views

### comprehension and ideal elements

A not uncommon thought in philosophy is that we should distinguish (in philosophy, anyway) between "sparse" ("real", "serious") and "abundant" ("ideal", "superficial") properties/classes and ...

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votes

**8**answers

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### Have we ever lost any mathematics?

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, ...

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vote

**1**answer

192 views

### Mathematical analysis of Lewisian concepts, esp. natural properties

David Lewis was one of the great philosophers of our time. He was a genuine philosopher, his focus was on theoretical metaphysics. And he had something to say about mathematics. His last book - he ...

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**14**answers

3k 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 ...

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**0**answers

821 views

### Which mathematical questions are objectively true or false? [closed]

Which parts of mathematics are objectively true or false like finite arithmetic and which are only true, false or undecidable relative to a particular axiom system such as Euclidean geometry?
...

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**2**answers

920 views

### Is Algebraic Geometry really natural? [closed]

Dear All!
I recently had a conversation with one mathematician who reckons that all sorts of combinatorial results are nothing compared to the things done in the algebraic geometry. As I do not have ...

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votes

**5**answers

1k views

### History of Logic Development

Where can I find a book which explains the development of modern logic, e.g. Tarski, Frege, Peano, up untill Wittgenstein, Russel?

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**0**answers

388 views

### a priori grounds of mathematics [closed]

Hi,
from the title of the question you may guess I'm new to the site, and I am. Even though I've read the FAQ and I know that this isn't the place for such open questions (I believe it is open, but ...

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votes

**3**answers

581 views

### Variable-centric logical foundation of calculus

Since calculus originated long before our modern function concept, much of our language of calculus still focuses on variables and their interrelationships rather than explicitly on functions. For ...

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**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 ...

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**0**answers

953 views

### Is there a finite-dimensional vector space whose dimension cannot be found? [closed]

Is there a finite-dimensional vector space whose dimension cannot be found? Assume, we have somehow constructed a vector space whose dimension is finite, but yet unknown. Is there always an algorithm ...

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**3**answers

1k views

### Unprovable sentence about integers

Is there any natural* statement S about the natural integers such that if PA contains no contradictions then neither PA+S nor PA+not S contains a contradiction?
If unknown, where can I read about the ...

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**0**answers

2k views

### Group Theory, Game Theory, a bit of Philosophy and a post in Tao's blog

I've decided to write this post after reading the incredibly beautiful and highly recomended post by Terence Tao ...

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votes

**5**answers

2k views

### A meta-mathematical question related to Hilbert tenth problem

I am reading Bjorn Poonen's very nice survey on Hilbert's Tenth problem
(http://www-math.mit.edu/~poonen/papers/uniform.pdf), and while I believe I understand the mathematics well, I have widespread ...

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votes

**2**answers

973 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 ...

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**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 ...

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votes

**1**answer

829 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 ...

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**5**answers

2k 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 ...

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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 ...

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votes

**1**answer

880 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 ...

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**1**answer

781 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 ...

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**1**answer

917 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 ...

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**5**answers

2k 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 ...

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**3**answers

920 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 ...

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**10**answers

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### 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 ...

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**1**answer

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### 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 ...

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951 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 ...

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**6**answers

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### 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: ...

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votes

**13**answers

10k 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 ...

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vote

**4**answers

733 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 ...

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votes

**6**answers

4k views

### Why hasn't mereology suceeded 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 ...

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votes

**1**answer

185 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 ...

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votes

**9**answers

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### 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 ...

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votes

**3**answers

947 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, ...

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**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 ...

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**10**answers

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### 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 ...

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**5**answers

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### 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 ...

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### 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 ...

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vote

**0**answers

439 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 ...

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votes

**2**answers

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

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

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**4**answers

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### 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.
...

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980 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 ...