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.

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2answers
389 views

Are ITTM's necessary to compute Turing's “computable numbers” and what does that mean for ordinary recursion theory?

In his celebrated paper, "On Computable Numbers, With An Application To the Entscheidungsproblem", Turing defines a "computable number" as follows: The "computable" ...
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0answers
178 views

generate all possible theories compatible with axioms [migrated]

I am currently trying to learn about the fundations of mathematical logic, and the incompleteness theorem. I was curious to know if there's a way, given some given axioms, to analyze all the possible ...
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235 views

Is this theory using defined notions of classes, sets, and membership interpretable in ZFC?

The main difference with this formal theory is that it depends in an essential manner on a defined notion of class, set, and set membership $\in$, rather than the usual appraoch of leaving them ...
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15answers
4k views

Is pure mathematics useful outside of mathematics itself? [closed]

From time to time Mathoverflow allows soft questions because they are arguably best answered by active mathematicians and they can benefit other mathematicians/PhD students/math undergraduates. I ...
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5answers
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Can you do math without knowing how to count?

Are there mathematical theories that do not use intuitive integers? [That is, do not use integers to write statements.] Can you propose a theory that describes natural integers, without using ...
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1answer
441 views

Is a function needed here?

This question is related to my question Can we choose an element from a class?. However, I decided to create a separate question. Let $H$ be a complex Hilbert space and $H_1,\dotsc,H_n$ be closed ...
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0answers
264 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 ...
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1answer
386 views

Can we choose an element from a class?

Let $H$ be a complex Hilbert space and $H_1,...,H_n$ be closed subspaces of $H$. Set $H_0:=H_1\cap H_2\cap...\cap H_n$ and let $P_i$ be the orthogonal projection onto $H_i$, $i=0,1,2,...,n$. I study ...
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2answers
645 views

Is there an analogue of the Lost Melody Theorem in ordinary recursion theory and if not, why not?

In their arXiv preprint, "Infinite Time Turing Machines" (arXiv:math/9808093v1 [math.LO] 21 Aug 1998) Hamkins and Lewis state the Lost Melody Theorem for ITTM's as follows: Lost Melody ...
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9answers
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Proofs of theorems that proved more or deeper results than what was first supposed or stated as the corresponding theorem

Recently, I figured out that a colleague of mine has had published during recent years a proof of a theorem in which he was actually proving a deeper result which we both thought to be still open. ...
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2answers
953 views

Has there been any serious attempt at a “circular” foundation of mathematics?

As far as I know, there is no published attempt at a "circular" foundations of mathematics though I'ave seen it noted by many category theorists and logicians without in-depth analysis, e.g ...
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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 ...
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5answers
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Metamathematics of buts

Something I learned (probably in middle school) that always bothered me is that the truth value of "and" and "but" are basically the same. If you were going to assign a truth-...
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0answers
348 views

Is this a good way of conceptualising the current status of Foundation of Maths projects?

I have been reading 'A Very Short Introduction to Mathematics' by Timothy Gowers and at one point he mentions that most of the mathematical proofs can be finally resolved to a set of logical ...
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71 views

Tutte polynomial of a complete graph and full symmetric group

A complete graph has a perfect permutation symmetry. How such permutation symmetry is reflected in the corresponding Tutte polynomial? This question arises in the context of "Quantum Graphity&...
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1answer
543 views

Gödel on pure mathematics and medieval theology

I was watching this youtube video recently where Gregory Chaitin paraphrases something from one of Gödel's unpublished essays (apparently published now). It is at the 4:48 mark of the video Gregory ...
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2answers
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Can we take a supremum over all Hilbert spaces?

In my paper On the optimal error bound for the first step in the method of cyclic alternating projections, I defined functions $f_n:[0,1]\to\mathbb{R}$, $n\geqslant 2$, by $$ f_n(c)=\sup\{\|P_n\dotsm ...
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2answers
747 views

Books on relationship between the Socratic method and mathematics?

Apart from books on heuristics by George Polya. When trying to engage with and understand mathematical concepts and when applying abstract mathematical concepts to model "continuum" or real ...
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1answer
763 views

A “paradox” about the inner model problem

As stated in Woodin, Davis, and Rodriguez - The HOD dichotomy, a longstanding open problem in set theory is to construct a canonical inner model for supercompactness. In general there are various ...
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3answers
885 views

What does T+non-Cons(T) mean?

I am puzzled by the following question, which is about the philosophical meaning of some axiomatic system. Suppose that a formal axiomatic system (containing arithmetics) $T$ is consistent. Let a ...
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22answers
8k views

Small ideas that became big

I am looking for ideas that began as small and maybe naïve or weak in some obscure and not very known paper, school or book but at some point in history turned into big powerful tools in research ...
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2answers
612 views

Gödel's ontological proof & Benzmüller's work

For a decade or so, Christoph Benzmüller from Berlin has explored Gödel's ontological proof (and variants) of existence of God. He uses the proof assistant Isabelle/HOL. He recently posted a preprint, ...
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2answers
268 views

Searching for an early, highly theoretical, even philosophical, math paper on models or small-world networks

All I can remember is that it was very high-level / abstact and kind of philosophical, explaining (the discovery or interdependence of) small world networks. I assume that it was +50 years old and '...
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1answer
261 views

Reference request on Gentzen's proof of the consistency of PA

I've always been interested in having a good understanding of Gentzen's proof of the consistency of arithmetic. Being more precise, he showed that $PRA + WF(\epsilon_0) \vdash Con(PA)$. I am ...
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4answers
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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) $\underline{\quad \quad \quad \quad}$ ...
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15answers
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Request for examples: verifying vs understanding proofs

My colleague and I are researchers in philosophy of mathematical practice and are working on developing an account of mathematical understanding. We have often seen it remarked that there is an ...
6
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1answer
297 views

How can we know the well-foundedness of $\epsilon_0$?

I think the question can be quite philosophical, but I see that $WF(\epsilon_0)$ is widely accepted as one of the attributes of the natural numbers. Gentzen proved $Con(PA)$ with $PRA+WF(\epsilon_0)$....
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1answer
713 views

What types are to mathematical proofs as types à la Martin-Löf are to constructive proofs, and what's wrong with them?

The question is motivated by this surprising sentence from Freek Wiedijk's The QED Manifesto Revisited. I agree that the QED-like systems that exist today are not good enough to start developing ...
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10answers
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The meaning and purpose of "canonical''

This question is jointly formulated with Neil Barton. We want to know about the significance of canonicity in mathematics broadly. That is, both what it means in some detail, and why it is important....
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0answers
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Can “description” of models revive formalism?

A model of a theory is a structure (e.g. an interpretation) that satisfies the sentences of that theory. Wikipedia Let $A$ be a set of sentences in some language that has only one extra-logical ...
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20answers
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Modeling in pure math

We all know that models play a major role in scientific practice. (By "model" here I mean any of various kinds of entities that function as representations or descriptions of real-world phenomena. ...
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2answers
1k views

Origin of the noun “mathematician” [closed]

I have read that Pythagoras's fraternity had two kinds of members, the 'acousmaticians', who were allowed to attend the lectures, and the 'mathematicians', who had been initiated. Is this the origin ...
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0answers
127 views

Arithmetization of Syntax: Can any semantic be encoded as syntax?

It is my understanding that Gödel Encoding and "Arithmetization of Syntax" can be used to represent any logical system. This is exemplified by the encoding of a Universal Turing Machine. "According ...
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3answers
810 views

Regarding Gentzen's note regarding 'Godel-points' (i.e., “Where is the Godel-point hiding?”)

Consider the following note written by Gerhard Gentzen in early 1932, on the onset of his work on a consistency proof for arithmetic: The axioms of arithmetic are obviously correct, and the ...
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0answers
453 views

Compatible and incompatible sets [closed]

Definition of the compatibility relation I have defined a relation $\mathsf{C}$ for sets that captures (to some extent) the notion of compatibility. In order to do this, we need an operation $': \...
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1answer
257 views

Formalizing ontological optimism

Inform speaking ontological optimisms means that everything that possibly exists in the abstract reality actually exists. From this principle we (again informally) get the Axiom of infinity, the Power ...
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1answer
190 views

Overview of interpretations of classical probability

The Stanford Encyclopedia of Philosophy has a nice overview of numerous different interpretations of probability (classical as opposed to quantum) with an extensive bibliography. What books would ...
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0answers
373 views

Can third-order arithmetic prove the consistency of second-order arithmetic?

I'm trying to get a deeper understanding of Buss's version of Gödel's speedup proof. In short, if we assume that $Z_0$ is first-order arithmetic, $Z_1$ is second-order arithmetic, and so on, then for $...
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1answer
307 views

Proving independence with large cardinals?

Suppose I want to prove some statement S is independent of ZFC. Now instead of the usual approach of making models, I do the following: - Take two large cardinal axioms L1 and L2 - Prove that ZFC + L1 ...
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0answers
90 views

Is there a criterion for reducibility of equi-interpretable theories?

I want to coin a notion for reducibility of theories. Generally this goes like that: if we have two equi-interpretable theories $T;Q$ and it is harder to interpret $T$ in $Q$ than to interpret $Q$ in $...
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1answer
133 views

Generalized Fourier integral and steepest descent path, saddle point near the endpoints

I am looking forward to solving the integration in the following equation with the assumption that $ka$ is very large \begin{align} H = 2jka\int_{-\pi/2}^{\pi/2}\cos{(\varphi-\phi)}e^{jka[\cos{\...
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3answers
2k views

The Lucas argument vs the theorem-provers — who wins and why?

In his paper, "Minds, Machines and Gödel", J.R. Lucas writes the following: Gödel's theorem [First Incompleteness Theorem, that is—my comment] must apply to cybernetic machines, because it is of ...
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0answers
471 views

When is a paper finished? [closed]

How much truth is there to Chomsky's remark that "mathematicians stop working when things get too difficult"? https://youtu.be/atupfHizJxM?t=453 Is this true in your own work? How do you know you ...
28
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8answers
4k views

Why not adopt the constructibility axiom $V=L$?

Gödelian incompleteness seems to ruin the idea of mathematics offering absolute certainty and objectivity. But Gödel‘s proof gives examples of independent statements that are often remarked as having ...
116
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17answers
15k views

Pressure to defend the relevance of one's area of mathematics

I am a set theorist. Since I began to study this subject, I became increasingly aware of negative attitudes about it. These were expressed both from an internal and an external perspective. By the “...
14
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2answers
742 views

Set-theoretical foundations of Mathematics with only bounded quantifiers

It seems that outside of researchers in Mathematical Logic, mathematicians use almost exclusively bounded quantifiers instead of unbounded quantifiers. In fact, I haven't observed any other practice ...
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2answers
520 views

Constructivist defininition of linear subspaces of $\mathbb{Q}^n$?

Let me preface this by saying I'm not someone who has every studied mathematical logic or philosophy of math, so I may be mangling terminology here (and the title is a little tongue in cheek). I (and ...
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2answers
394 views

Is the notion of measurable cardinal definable from the perspective of set-theoretical potentialism?

Consider the definition of measurable cardinal (this definition was found in Neil Barton's paper, "Large cardinals and the iterative conception of set"): Definition 8. A cardinal $\kappa$ ...
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5answers
2k views

Is there any physical or computational justification for non-constructive axioms such as AC or excluded middle?

I became interested in mathematics after studying physics because I wanted to better understand the mathematical foundations of various physical theories I had studied such as quantum mechanics, ...
7
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1answer
689 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 ...

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