Philosophical aspects of logic and set theory; truth status of mathematical axioms; Philosophy of Mathematics; philosophical aspects of mathematics in general; relation of mathematics to philosophy; etc. Consider also posting at http://philosophy.stackexchange.com/, where philosophy-of-mathematics ...

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18
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8answers
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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
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2answers
992 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 $\neg CH$ ...
32
votes
5answers
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 ...
7
votes
3answers
872 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
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1answer
531 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 ...
18
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7answers
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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 = ...
44
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9answers
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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
3answers
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
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2answers
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 ...
7
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5answers
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
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4answers
3k 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 ...
44
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2answers
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
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1answer
919 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 ...
1
vote
2answers
433 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
2answers
825 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
5answers
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 ...
14
votes
10answers
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
3answers
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 ...
24
votes
3answers
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. ...
8
votes
3answers
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
4answers
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 ...
32
votes
5answers
7k 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 ...
1
vote
4answers
485 views

Does the Golden Ratio Apply to Timing as Well? [closed]

I've seen the golden section applied to art, but does it apply to sound/timing as well? Just curious.
14
votes
2answers
932 views

synthetic differential geometry and other alternative theories

There are models of differential geometry in which the intermediate value theorem is not true but every function is smooth. In fact I have a book sitting on my desk called "Models for Smooth ...
4
votes
4answers
5k views

Badiou and Mathematics [closed]

Does anyone have an opinion on Alain Badiou's use of set theory? Is there anything interesting mathematically there? Also could anyone shed any light on the comment in the Wikipedia article link text ...
3
votes
2answers
481 views

Broken Symmetry

I have a tangled web of ideas about natural transformations, vector spaces, equivalence classes, local coordinates, etc. in my head that I'm trying to unravel. So here are some of the questions I ...
17
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8answers
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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 ...
10
votes
3answers
2k views

Math History Question about the exponential function

While tutoring a student recently, I have come across the situation of explain logarithms by first introducing functions of the form $$f(x)= a^x$$ where $a \ge 0,x\in \mathbb{R}$. My student then ...
11
votes
3answers
1k views

Is formal proof (formalized mathematics) interesting to practicing mathematicians? To educators? [closed]

Formalizing mathematical proofs so that they can be checked for correctness and manipulated by computer is a recurrent proposal, most notably stated in the QED manifesto (1994). The December 2008 ...
7
votes
2answers
2k views

How platonistic is your attitude towards mathematics? [closed]

A discussion in the n-category cafe about Manin's 'emotional Platonism' made me wonder how such a perception of mathematics is distributed among mathematicians and how that influences attitudes ...
-3
votes
2answers
458 views

what is logic without a proof system [closed]

Is there a proof for no proof ?
13
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7answers
2k views

Is no proof based on “tertium non datur” sufficient any more after Gödel?

There are many proofs based on a "tertium non datur"-approach (e.g. prove that there exist two irrational numbers a and b such that a^b is rational). But according to Gödel's First Incompleteness ...
60
votes
10answers
5k 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 ...