Questions tagged [mathematical-philosophy]

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 is one of the most popular tags.

Filter by
Sorted by
Tagged with
40 votes
4 answers
7k 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 Analysis, Craig Fraser, ...
7 votes
1 answer
312 views

Literature about formalization of "natural reasoning" in mathematical logic

In "Logic of sheaves of structures", X. Caicedo justifies the logic he introduces stating (more or less) that assertions about a point should really be understood as assertions about a ...
42 votes
9 answers
5k views

The sets in mathematical logic

It is well known that intuitive set theory (or naive set theory) is characterized by having paradoxes, e.g. Russell's paradox, Cantor's paradox, etc. To avoid these and any other discovered or ...
11 votes
7 answers
6k views

Where can you find Grothendieck's "Récoltes et Semailles"?

Where can you find Grothendieck's "Récoltes et Semailles"? Is it available anywhere?
-1 votes
0 answers
79 views

Minimal metatheory for Gödel's first incompleteness theorem [duplicate]

When studying the complete proof of Gödel's first incompleteness theorem, I started to harbor a metamathematical worry of circularity. For reference, I am studying the proof in A Concise Introduction ...
58 votes
7 answers
8k views

In what respect are univalent foundations "better" than set theory?

It was an ambitious project of Vladimir Voevodsky's to provide new foundations for mathematics with univalent foundations (UF) to eventually replace set theory (ST). Part of what makes ST so appealing ...
32 votes
11 answers
10k 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 (...
69 votes
5 answers
9k 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 ...
7 votes
2 answers
968 views

Explaining the consistency of PRA and ZF from predicative foundations

Recently I got interested in predicative foundations, mostly because of Laura Crosilla's work and because Agda employs a predicative type theory. From the point of view of a predicative foundation to ...
-2 votes
1 answer
195 views

Is there any case of remormalization in which we have to solve it by ways in two different systems? [closed]

In renormalization of physics, $$\sum_{j=1}^{\infty}j=-\frac{1}{12}$$ We may obtain the result in two ways: first we may redifine the sum so we have used two system of math with different definition ...
13 votes
6 answers
4k views

Non-constructive proofs vs. efficient algorithms

My question concerns what is meant by "nonconstructive", and whether it has ever been defined in terms of computational complexity. The wikipedia article on constructive proof begins, "a constructive ...
150 votes
45 answers
29k 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 ...
14 votes
5 answers
1k views

In what sense does the sentence $\operatorname{con}(\mathsf{PA})$ "say" that $\mathsf{PA}$ is consistent?

It seems common amongst logicians to think of "truth" as being relative to a particular structure. Consider, for instance, the first-order theory of groups. The sentence $\forall x\forall y(...
6 votes
0 answers
322 views

An imaginary disaster scenario - second order arithmetic is inconsistent

I think my question is a natural follow up of What would be some major consequences of the inconsistency of ZFC? Regarding the later question, I agree with the commentaries that probably an ...
8 votes
1 answer
594 views

Concept of bedrock and mantle in the multiverse view in the philosophy of mathematics

To be clear, I am not a mathematics educated student and I can not follow the details of the technicality of the forcing extension, but I feel that I have a good understanding of the big picture (of ...
31 votes
8 answers
9k views

Why is integer factoring hard while determining whether an integer is prime easy?

In 2002, the discovery of the AKS algorithm proved that it is possible to determine whether an integer is prime in polynomial time deterministically. However, it is still not known whether there is an ...
31 votes
3 answers
4k views

How much should the average mathematician know about foundations?

How much should an average mathematician not working in an area like logic, set theory, or foundations know about the foundations of mathematics? The thread Why should we believe in the axiom of ...
6 votes
5 answers
1k views

How to tell a paradox from a "paradox"?

Russell's paradox showed that naive set theory leads to a contradiction. This was something that was taken seriously and caused a lot of work. Now, Banach–Tarski paradox is arises from a result that a ...
13 votes
1 answer
2k views

Mathematical fictionalism

Have there been any successful mathematicians that also happen to be mathematical fictionalists? Let's say success is defined by at least one article published in a non-pay journal. I ask because ...
16 votes
1 answer
481 views

A textbook on foundations of geometry in spirit of Tarski

I am interested in a textbook for studying (and teaching) foundations of geometry in the spirit of Tarski. I know that there is a rather old German book [W. Schwabhäuser, W. Szmielew, A. Tarski, ...
32 votes
3 answers
5k 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 ...
38 votes
4 answers
6k views

On critical reviews of Hawking's lecture "Gödel and the end of the universe"

The search for a neat Theory of Everything (ToE) which unifies the entire set of fundamental forces of the universe (as well as the rules which govern dark energy, dark matter and anti-matter realms) ...
24 votes
12 answers
4k views

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 ...
6 votes
4 answers
9k 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 ...
49 votes
30 answers
7k views

Taking a theorem as a definition and proving the original definition as a theorem

Gian-Carlo Rota's famous 1991 essay, "The pernicious influence of mathematics upon philosophy" contains the following passage: Perform the following thought experiment. Suppose that you are ...
8 votes
5 answers
2k 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?
21 votes
10 answers
2k views

Comparative analysis of history of mathematics

I am a bit scared about writing this question because I am unsure if it is appropriate. However, here it is. Is there anything written about the history of mathematics from a comparative or (post)...
18 votes
4 answers
4k 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 ...
1 vote
1 answer
344 views

How are Koepke's ordinal computability and E-recursion related?

In Koepke's paper, "Turing Computations On Ordinals", one has the following (well-known) result: A set $x$ is ordinal computable from a finite set of ordinal parameters if and only if it is ...
8 votes
2 answers
884 views

Large cardinal near inconsistencies

I am looking for examples of results about large cardinals, large cardinal axioms, or other objects of high (or seemingly high) consistency strength that are almost inconsistencies. I am looking for ...
0 votes
1 answer
283 views

Nancy Cartwright's dichotomy

Nancy Cartwright introduced an interesting distinction with regard to modeling of physical phenomena. According to Cartwright, a mathematical theory is not applied directly to such phenomena. Rather, ...
54 votes
7 answers
9k views

Are there any fields of academic mathematics whose epistemic status as math is controversial within the academic community?

String theory (and related areas of purely theoretical quantum gravity, like loop quantum gravity) has a unique position within the academic physics community. Many academic physicists don't really ...
6 votes
9 answers
7k views

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 ...
52 votes
2 answers
3k views

Silver's approach to the inconsistency of $\mathrm{ZFC}$

As all probably know, Jack Silver passed away about one month ago. The announcement released, with delay, by European Set Theory Society includes a quote by Solovay about his belief on inconsistency ...
78 votes
21 answers
17k 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 ...
25 votes
6 answers
3k views

Is there a metamathematical $V$?

As with many of you, I've been following Peter Scholze's recent question about universes with great interest. In ring theory, we don't often have to deal with proper classes, but they occasionally ...
11 votes
1 answer
870 views

Causality, if any, in mathematics itself

Mathematicians often express comments like "X is true because Y and Z are true". One's sense of mathematical causation is also a major part of mathematical intuition. But causality per se is ...
10 votes
8 answers
5k 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 ...
5 votes
1 answer
466 views

Extensions of the Ackermann interpretation to nonstandard theories of arithmetic

In their paper, " On Interpretations of Arithmetic and Set Theory" (Notre Dame Journal of Formal Logic, Vol. 8, No. 4 (2007), pp. 497-510) in section 7, "Fragments of Arithmetic and Set ...
10 votes
4 answers
2k views

The name for an assumption made for the sake of contradiction

What is the name (or adjective) for an assumption made for the sake of contradiction? To be clear, I'm in search of an expression in the form "a(n) $\underline{\quad \quad \quad \quad}$ ...
65 votes
9 answers
13k views

Axiom of choice, Banach-Tarski and reality

The following is not a proper mathematical question but more of a metamathematical one. I hope it is nonetheless appropriate for this site. One of the non-obvious consequences of the axiom of choice ...
10 votes
0 answers
407 views

Theories of truth

Not knowing much about logic, I thought that in mathematics saying that a (closed) sentence $\varphi$ in a (formal) theory $T$ is "true" amounted to one of the following notions: Syntactic ...
40 votes
5 answers
6k 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 ...
6 votes
2 answers
504 views

Is there a source linking Robinson's work in wing theory with his theory of infinitesimals?

Abraham Robinson worked in applied mathematics for several decades. MathSciNet lists 12 articles by Robinson in wing theory. His production included the book Robinson, A.; Laurmann, J. A. Wing theory....
8 votes
0 answers
366 views

What is the relationship (if any) between constructivism, finitism and predicativism?

The terms “constructivism”, “finitism” and “predicativism” refer to ideas / currents in the philosophy of mathematics (or loosely defined conditions on a system of logic) that I think I understand ...
74 votes
11 answers
11k 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 ...
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 ...
5 votes
2 answers
485 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 ...
7 votes
3 answers
3k views

Are the categories of sets, abelian groups, and commutative rings unique?

Are the categories of sets, abelian groups, and commutative rings unique? Independence results like the independence of the generalized continuum hypothesis, the Whitehead problem, and the global ...
3 votes
1 answer
121 views

Equivalences between statements of (seemingly) different order

In Steve Simpson's excellent monograph SOSOA, we find Theorem X.4.4 which contains an equivalence (over RCA$_0^*$) between the following statements: The induction axiom for $\Sigma_1^0$-formulas (...

1
2 3 4 5
8