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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5
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3answers
699 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) $\underline{\quad \quad \quad \quad}$ ...
53
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15answers
4k views

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
252 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 $PRA+WF(\epsilon_0)$....
6
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1answer
645 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 ...
21
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10answers
4k views

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
147 views

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 ...
27
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18answers
6k views

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. ...
12
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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
116 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
754 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 ...
2
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0answers
182 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
251 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 ...
2
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1answer
178 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
254 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
284 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
89 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
114 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 ...
8
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0answers
463 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 ...
25
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8answers
3k 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 ...
114
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17answers
14k 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
697 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
490 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
380 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
643 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 ...
14
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0answers
541 views

Does inner model theory seek canonical models for large cardinals?

Like the author of this question, I have heard that a main goal of inner model theory is building canonical inner models for large cardinals. My questions are: (a) Is this accurate? (b) If so, in ...
17
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1answer
885 views

Axiom of Choice versus V=L in opposition to large cardinals

Consider the following two observations: The axiom $V=L$ is incompatible with large cardinal axioms that are somehow "too large", like measurable cardinals. The axiom of Choice is incompatible with ...
2
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0answers
288 views

Does this axiomatic system satisfy requirements for founding mathematics?

In this article, the author, F.A.Muller, suggests criteria for a founding theory of mathematics (pp:14-16). The author proposes $ARC$ Class Theory to embody these requirements. The motivation is ...
30
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3answers
2k views

Complex analytic vs algebraic geometry

This is more of a philosophical or historical question, and I can be totally wrong in what I am about to write next. It looks to me, that complex-analytic geometry has lost its relative positions ...
0
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1answer
456 views

“Mathematics is the science of the infinite” [closed]

The title is the first sentence of Hermann Weyl's 1930 essay, "Levels of Infinity." He focuses on "the distinction between actuality and potentiality, between Being and Possibility." He opines ...
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0answers
333 views

Theorems conditional on false conjectures

What is an example of a theorem that was conditional on a conjecture that later turned out to be false?
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4answers
2k views

Does Zorn's Lemma imply a physical prediction? [duplicate]

A friend of mine joked that Zorn's lemma must be true because it's used in functional analysis, which gives results about PDEs that are then used to make planes, and the planes fly. I'm not super ...
10
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1answer
843 views

Quantum functional analysis

Can one explain some philosophy behind "quantum functional analysis" (or "quantized functional analysis") which was initiated and developed by such researchers as: Ruan Z.-J., Pisier J., Effros E.G., ...
5
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3answers
421 views

Counting without one-to-one correspondence? [closed]

Ash and Gross in their wonderful book Fearless Symmetry found it worth mentioning (and thus suggesting) another way of counting for which "we do not even need to know how to count" (in the sense of ...
16
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0answers
1k views

What's the point of cubical type theory?

I have been following through the development of homotopy type theory since 2013 because I was really interested in the foundation of mathematics. The novel idea of combining programming with homotopy ...
0
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1answer
246 views

Criterion of completeness

Wittgenstein (PR 181) talks about a criterion of completeness for the irrationals. I am trying to understand what this might mean. Completeness of the reals, in the decimal number system, is the ...
58
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6answers
6k views

The logic of Buddha: a formal approach

Buddhist logic is a branch of Indian logic (see also Nyaya), one of the three original traditions of logic, alongside the Greek and the Chinese logic. It seems Buddha himself used some of the features ...
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1answer
279 views

What is the intuitive notion that ZF-Extensionality-Foundation+Collection can be said to capture? [closed]

This question has been moved to philosophy.stackexchange.com I'll try to abbreviate it here: the question asks about the "informal notion" that the fragment of $\text{ZFC}$ that is axiomatized by ...
15
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1answer
701 views

Does every model of ZF-foundation have an extension, with no new well-founded sets, where every set is bijective with a well-founded set?

This question follows up on an issue arising in Peter LeFanu Lumsdaine's nice question: Does foundation/regularity have any categorical/structural consequences, in ZF? Let me mention first that my ...
36
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11answers
9k views

Contemporary philosophy of mathematics

Starting to write an introduction to the philosophy of mathematics, I find tons of positions that are of historical interest. Which philosophical positions are explicitly considered these days, say in ...
34
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4answers
5k 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) ...
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2answers
386 views

What is against having distinct membership relations on sets in the Platonic realm?

This question is in connection with the question that I've asked at: Where do models of false theories exist? The answer to that question was that any consistent theory can have its primitives be re-...
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1answer
546 views

Where do models of false theories exist?

I have some difficulties in understanding the [uni]verse platonic view. How are we to understand the existence of a model of a false theory? what is the relationship of this model to the platonic ...
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1answer
277 views

Is it natural to hold that Ur-elements, small & big sets and proper classes exists? [closed]

The topic of this post was shifted to https://philosophy.stackexchange.com/questions/49504/is-it-natural-to-hold-that-big-sets-and-proper-classes-exist Since it was deemed to be a philosophical ...
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2answers
500 views

Reasoning Using Countable Subsets of Real Numbers

The purpose of my question is trying to understand whether, in some cases, we can achieve greater certainty of reasoning (say when dealing with statements about natural numbers, integers or rational ...
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0answers
287 views

The universe and multiverse views of set theory from the perspective of $ZFC^2$

(Note: the 'Second-order $ZFC$' ($ZFC^2$) I am talking about is the theory [in the second order language of set theory consisting of a single non-logical symbol $\in$ ] consisting of the axioms ...
12
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1answer
449 views

When we count the same set, must the number always be the same?

Return to Frege's question, What justifies arithmetic? And consider the ur-proposition that counting a finite set always produces the same number, and ask whether this has a logical justification, ...
53
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7answers
6k 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 ...
0
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1answer
524 views

Forcing the existence of a weakly inaccessible cardinal in some strong set theory

Does the fact that, assuming the consistency of $ZFC$, no proof that the consistency of "$ZFC$ implies the consistency of '$ZFC$ + There exists a weakly inaccessible cardinal'" can be formulated in $...

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