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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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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 ...
105
votes
16answers
12k 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 “...
13
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2answers
646 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 ...
8
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2answers
425 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
356 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 $\kappa$ is ...
7
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5answers
1k 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
votes
1answer
554 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
votes
0answers
448 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 ...
16
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1answer
811 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
271 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 ...
23
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2answers
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 ...
-1
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1answer
408 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 ...
5
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0answers
313 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?
22
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4answers
1k 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
votes
1answer
654 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
votes
3answers
417 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 ...
8
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0answers
655 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
231 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 ...
50
votes
5answers
5k 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 ...
-3
votes
1answer
266 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
votes
1answer
627 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
8k 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. My question to the research community in mathematics is which positions are ...
33
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3answers
4k views

On Critical Reviews of Hawking's “Gödel and the End of the Universe” Lecture

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) ...
1
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2answers
375 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-...
2
votes
1answer
525 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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votes
1answer
263 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 ...
6
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2answers
435 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 ...
1
vote
0answers
260 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 ...
11
votes
1answer
403 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, ...
50
votes
7answers
5k 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
votes
1answer
457 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 $...
14
votes
3answers
984 views

History of the abstract method in mathematics

Recently I have "finished" a 13-year on and off research on the history of the mathematical notion of equivalence. At the end of which, I learned that we owe the nowadays rather elementary process of "...
70
votes
4answers
17k views

The enigmatic complexity of number theory

One of the most salient aspects of the discipline of number theory is that from a very small number of definitions, entities and axioms one is led to an extraordinary wealth and diversity of theorems, ...
13
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2answers
2k views

(Fictive) story of a time where people reasoned only up to isomorphism

I seem to remember reading once a story that some mathematician had written to justify the use of categories, or isomorphisms or equivalences, or something like that. The story goes something like ...
5
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0answers
621 views

Why are real-valued measurable cardinals never explicitly mentioned in Gödel's “What is Cantor's Continuum Problem”?

It is a matter of mathematical folklore that Gödel "entertained the idea of so called stronger axioms of infinity deciding $CH$...." (this quote from Radek Honzik's paper, "Large cardinals and the ...
37
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5answers
4k views

How to improve writing mathematics?

My first language is not English. How can I improve my mathematical writing. I feel like the only things I can write down are numbers and equations. Is there any good suggestion for improving writing, ...
26
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5answers
4k views

Why aren't functions used predominantly as a model for mathematics instead of set theory etc.? [closed]

If definitions themselves are informally just maps from words to collections of other words. Then in order for one to define anything, they must inherently already have a notion of a function. I mean ...
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2answers
692 views

Can only the constructible sets be proven to exist in $ZF$ without benefit of extra assumptions? [closed]

I am interested in asking the following question: What sets can be proven to exist in $ZF$ without the benefit of extra assumptions? (Thanks to Toshiyasu Arai for inspiring me to ask this variation ...
41
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1answer
2k views

Hilbert's alleged proof of the Continuum Hypothesis in “On the Infinite”

As is known, Hilbert attempted a proof sketch of the Continuum Hypothesis in the latter part of his paper, "On the Infinite". It is also known that it is false. Has there ever been a published ...
4
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0answers
183 views

Have chromatic techniques actually been used to compute more stable homotopy groups of spheres?

I have heard the perspective that chromatic homotopy theory is meant to break apart the stable homotopy groups of spheres into manageable pieces, and that this perspective has led to various insights. ...
1
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0answers
96 views

Can $GPK^{+}_{\infty}$ + $AC_{WF}$ prove "$ZFC$ proves that the class of ordinals is not weakly compact for definable classes?

Consider the topological set theory $GPK^{+}_{\infty}$ +$AC_{WF}$, where $GPK^{+}_{\infty}$ is defined as follows (this from Olivier Esser's papers, "An Interpretation of the Zermelo-Fraenkel Set ...
1
vote
0answers
76 views

Second-order characterizability and forcing

In their paper, "On Second-Order Characterizability", Hyttinen, Kangas, and V$\ddot a$$\ddot a$n$\ddot a$nen define that notion as follows: Let us call a structure $\mathfrak U$ second order ...
9
votes
1answer
522 views

How are material set theory and structural set theory related from the point of view of category theory?

In his comments to both cody and Nik Weaver regarding his answer to user7280899's mathoverflow question "What kind of foundation are mathematicians using when proving metatheorems?", Mike Shulman ...
33
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1answer
1k views

Silver's approach to the inconsistency of $ZFC$

As all probably know, Jack Silver passed away about one month ago. The announcement released, with delay, by European Set Theory Society includes a quote by Solovay about his belief on inconsistency ...
49
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10answers
7k views

Axiom of choice, Banach-Tarski and reality

The following is not a proper mathematical question but more of a metamathematical one. I hope it is nonetheless appropriate for this site. One of the non-obvious consequences of the axiom of choice ...
3
votes
1answer
273 views

Extensions of the Ackermann interpretation to nonstandard theories of arithmetic

In their paper, " On Interpretations of Arithmetic and Set Theory" (Notre Dame Journal of Formal Logic, Vol. 8, No. 4 (2007), pp. 497-510) in section 7, "Fragments of Arithmetic and Set Theory", Kaye ...
5
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1answer
146 views

Is $PRA$ + $TI({\epsilon_0})$ mutually interpretable with some theory in the language of set theory?

As is well known, the following theory is equiconsistent with $PA$: $ZFC$ with the axiom of infinity replaced by its negation. Since this theory is equiconsistent with $PA$, it would seem ...
2
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1answer
301 views

Is the statement “All numbers are counting numbers” independent of $PA$?

In his paper, "Completed versus Incomplete Infinity in Arithmetic" (look under "www.math.princeton.edu/$\sim$nelson/papers.html" under the subheading "Infinity"), the late Edward Nelson defines the ...
19
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3answers
1k views

Why would the category of sets be intuitionistic?

This question is probably really naive. And, I hope the title doesn't come off as too combative. I think that topoi of $\mathbf{Set}$-valued sheaves provide an excellent motivation for higher-order ...
10
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2answers
348 views

Why is the notion of algorithm a primitive one in Brouwer's intuitionism?

I've seen several times people mentioning that the notion of an algorithm / a computation is taken as a primitive notion in L. E. J. Brouwer's intuitionism. For instance, in Varieties of Constructive ...