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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Does ${\sf ZC + Universes + ZFC}^V$ meet Muller's criteria for a founding theory of Mathematics?

I was re-thinking Muller's criteria in Sets, Classes and Categories: page 14 for a theory that founds Mathematics. To him, it should be able to be a foundation of Category Theory. He lays down six ...
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-4 votes
1 answer
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Soft question around the equivalent forms of Riemann hypothesis: an overview of the role of these equivalent forms [closed]

Along decades, in the course of investigations around the Riemann hypothesis (I add [1] as general reference) professional mathematicians discovered/stated many much (several hundred of) equivalent ...
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Deontic logic and axiomatic pluralism

Background assumptions/definitions/questions (note that "S, A, B" stand for some sentence or other at issue per possible question; note also that these sentences can be imperatives, and in ...
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3 votes
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Does the following variant of common belief exist?

Let $A$ be a finite set of agents and $\mathtt{B}_a$ a modal operator where $\mathtt{B}_ap$ means agent $a$ believes proposition $p$. For now I don't assume any properties of $\mathtt{B}_a$, though ...
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The LNC as a mathematical theorem

One of the most intriguing things I've read about over the last few years is Diaconescu's theorem, which says that, in some forms of constructivist/intuitionistic set theory, even if the law of the ...
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-2 votes
1 answer
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Has there been any mathematical study of causality?

Causality seems to play an important role in physics. There also seems to be a close parallel between "$P$ causes $Q$" and "if $P$ then $Q$." Mathematical logic studies logical ...
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45 votes
7 answers
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Swimming against the tide in the past century: remarkable achievements that arose in contrast to the general view of mathematicians

I would like to ask a question inspired by the title of a book by Sir Roger Penrose ([1]). The germ of this is to ask about the role, if any, of the fashion in research of pure and applied mathematics....
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'Maximising interpretative power entails maximising consistency strength'?

I'm hoping there is a clear mathematical answer to this question (hence asking it here) rather than anything more exegetical (in which case it's presumably not appropriate for this site). In his paper ...
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3 votes
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What does it mean to solve an equation?

Assume that we want to find all integer (or rational) solutions to the polynomial Diophantine equation $$ P(x_1,\dots,x_n) = 0 $$ where $P$ is a polynomial with integer coefficients. Do we have a ...
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7 votes
2 answers
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Equivalences of $n$-categories

This question is an extension of my previous question last year (see [2020]) in which I asked about the (consensus of a) definition of a weak $n$-category. Here are some background: while strict $n$-...
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Are there mistakes in the proof of FLT?

This semester, I teach a graduate course in epistemology of mathematics and as a case study, I assigned students a discussion on the epistemological status of Fermat's Last Theorem according to ...
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Can finite sets be non-c.e. depending on how they are presented?

I ask the question because of the following statement found in Mark Burgin's paper, "Algorithmic complexity of recursive and inductive algorithms", Theoretical Computer Science 317 (2004) 31-...
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26 votes
7 answers
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Why is game theory formulated in terms of equilibrium instead of winning strategies?

Game theory, on the outset, seems to invite the questions, "what can I do to win" or "how do I beat my opponent?" So many people who are not familiar with game theory look to game ...
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11 votes
1 answer
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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 ...
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What is an oracle, really? [closed]

Regarding oracles, might this be a reasonable description of their inner workings (this from Hartley Rogers, Jr.'s text, Theory of Recursive Functions and Effective Computability)? Why should I ask ...
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7 votes
2 answers
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Analytic/synthetic distinction in mathematics besides geometry?

In a recent answer to an old MO question, I made a distinction between a "definition" of a mathematical object in the sense of axioms that characterize it, and a "definition" that ...
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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 ...
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To be is to be an element?

The distinction between sets and proper classes has interesting ontological consequences. One way to define a proper class, in the context of set theory, is to state that it is not an element, in a ...
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6 votes
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Does simple theory of types + ambiguity prove axiom of infinity?

Does simple theory of types + ambiguity prove axiom of infinity? The simple theory of types known as $\sf TST$ is a multi-sorted first order theory, syntactical restrictions include $\in$ being a ...
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51 votes
6 answers
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Siegel zeros and other "illusory worlds": building theories around hypotheses believed to be false

What are some examples of serious mathematical theory-building around hypotheses that are believed or known to be false? One interesting example, and the impetus for this question, is work in number ...
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Erotetic inference and extrinsic justification?

Gödel introduced his notion of what has come to be called extrinsic justification in the following terms: Furthermore, however, even disregarding the intrinsic necessity of some new axiom, and even ...
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-2 votes
1 answer
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Can mathematics help in defining free-will? [closed]

In the celebrated Free Will Theorem of Conway and Kochen it is made use of "free will" without giving a "mathematical definition" of it. The definition of the experimenter is the &...
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2 votes
2 answers
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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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1 vote
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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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33 votes
14 answers
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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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6 answers
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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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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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7 votes
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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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2 votes
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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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1 vote
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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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54 votes
9 answers
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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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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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22 votes
5 answers
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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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50 votes
5 answers
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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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1 vote
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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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10 votes
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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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19 votes
2 answers
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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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8 votes
3 answers
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Books on the 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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16 votes
2 answers
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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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9 votes
3 answers
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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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69 votes
22 answers
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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, ...
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6 votes
2 answers
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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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1 vote
1 answer
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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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6 votes
4 answers
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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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54 votes
15 answers
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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 ...
5 votes
1 answer
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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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7 votes
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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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23 votes
10 answers
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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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1 vote
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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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