# 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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### 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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### 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 ...
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 ...
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 ...
2k views

### 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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### 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 ...
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 ...
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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### 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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### 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. ...
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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### 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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### 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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### 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&...
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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### 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 ...
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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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 ...
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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### 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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### 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, ...
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 '...
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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### 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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### 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 ...
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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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