# 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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### Gauss, Cantor, and infinite confusion

There is an interesting comment by Gauss on "infinite magnitude as a
complete thing" that has invited varying interpretations. In a
well-known passage, Gauss criticized the use of infinity ...

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### Christoph Benzmüller and Gödel's ontological proof?

Are there any notable mathematical or logical issues within Christoph Benzmüller and Bruno Woltzenlogel-Paleo formalized Gödel's ontological proof (pdf) that has been identified by the community?

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### Proving that ZF is Artemov-consistent

As discussed in another MO question, Sergei Artemov has proposed that the standard formalization Con(PA) of "PA is consistent" is flawed, and has proposed a different way to formalize "...

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### Why should I believe Martin's Maximum++?

$\sf MM^{++}$ is a 'good' set-theoretic axiom because it is 'Maximum'. Of course, bigger is better. But I'd like to know exactly how the argument works.
Let me be clear about the question posed:
What ...

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### Is there any correspondence between Gödel and Kreisel that supports Kreisel's observation that Gödel changed his mind about his 1938 set theory note?

At a conference in 1965 there were some interesting comments made by Kreisel and Mostowski asserting that Gödel later changed his mind regarding his1938 note on his set theory results (see Problems in ...

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### What governs our "perception?" about the platonic realm of sets?

Here, I want to delve into what do we exactly feel about what constitutes a platonic existence of a set? Or what makes us think or actually a kind of feel or sense the existence of a set in the ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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(...

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### 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 ...

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### 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, ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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)...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 (...

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### Examples of "proof by generalising" [duplicate]

In a previous post I asked (Which theorems have Pythagoras' Theorem as a special case?).
Are there any compelling examples where it is significantly "easier"/"simpler" to prove ...

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### 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, ...

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### Minor Lefschetz principle

I once read (I think) the following equivalent formulation of the Minor Lefschetz principle:
If an elementary sentence holds for one algebraically closed field,
then it holds for every algebraically ...

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### Do you know any deep paradoxes or controversial hypothesis in category theory similar to those we have in set theory?

There is a lot of non-obvious and controversial topics and questions in set theory. From its begining in the first half of 20th century it have generated many paradoxes. For example there are ...

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### Quantification over uncountable sets

If some statements below are too imprecise/peculiar, please note that this is mostly due to my own lack of knowledge/understanding. Nevertheless, I will try to phrase the actual question in a more ...

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### Formalisation of intuitive concepts in the language leading to mathematical progress

In his work, Albert Lautman thinks the genesis of some mathematical works as a dialectic that takes place between opposite notions, like between global and local. He argues that while those notions, ...

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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...