Questions tagged [philosophy-of-mathematics]

Philosophy of mathematics asks questions about mathematical theories and practices. It can include questions about the nature or reality of numbers, the ground and limits of formal systems and the nature of the different mathematical disciplines.

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79 views

What makes a statement mathematical?

What is the formal definition of a mathematical statement? We can all agree that the statement "Humans are apes" is not a mathematical statement, and the statement "4 is a prime number&...
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Axiom of Choice: correspondence or derivability?

I'd like to ask about a specific impression that I have about issues concerning the Axiom of Choice. It seems to me that either one claims that the axiom is an obvious fact about the modelled concept (...
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Summary of philosophical positions on how belief revision proceeds in mathematics?

Since mathematicians have embraced classical logic, as e.g. MacFarlane points out in his 2021 intro book to philosophical logic (§ 7.4), one needs to distinguish between [meta-]reasoning and argument/...
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How to evaluate Stephen wolfram’s book “A new kind of science” from a philosophy of science perspective? [closed]

1- Is Stephen wolfram’s new kind of science a kind of reductionism? If it is, is the criticism against the reductionism still work on it? 2- What’s the relationship between the new kind of science and ...
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127 views

Can truths about the natural numbers vary across possible worlds?

The truths of logic are the same in all possible worlds. However, what about truths about natural numbers? Like, for instance, is there a world where there are only finitely many primes, or a world ...
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Technique that “de-trivialising” contradiction (systems)?

Gödel proved that some systems cannot prove their own consistency. As I see, what Gödel proved is no other than that mathematics is freedom, the adventure of a free mind (I.e. not afraid of being ...
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Can the Dirac belt trick (among others!) prove that mathematics is real?

First, I link the following video: https://www.youtube.com/watch?v=Vfh21o-JW9Q It demonstrates the 'Dirac belt trick', which was created by (I believe, Dirac) to demonstrate the calculus of spinors. ...
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Do constructivists (or intuitionists) reject real numbers, except the computable ones?

SEP has a bunch of pages on what (various flavors) of intuitionists or constructivists seem to accept as a model theory or as a set theory (they actually seem to diverge on the latter, in the sense of ...
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Philosophy of Math that talks about group theory, (or other stuff that's math but not numbers or geometry)?

I have just realized today that anytime I've read something about the philosophy of mathematics, the focus is on numbers, figuring out what numbers are, whether they're real, the relationship of ...
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Are theorems of math theorems even before they are proven?

I considered asking this in the math SE, but I decided this was a better option. I got into an argument with a math professor who claimed that Fermat's Last Theorem was a theorem only after it was ...
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How can you make sense of “equinumerosity” in Hume's Principle in a logicist approach to math, without first having functions defined?

I'm pretty sure that I am misunderstanding something here, but I'm not sure what. How can you make sense of "equinumerosity" in Hume's Principle in a logicist approach to math, without first ...
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Why can anything be discovered in mathematics at all?

Imagine a Perfect Mathematician that has superhuman abilities -- if you give him or her a formal foundational system for mathematics like ZFC with all the underlying logical machinery, he or she is ...
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Can we ignore intent if we recognize the extent? [closed]

I am not sure whether the formulation of my question is adequate, but I hope I'll make it clear enough. As a mathematics student, I have come to notice that some mathematical concepts are not defined ...
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What is implications of well ordering theorem regarding order in nature?

I have recently come across well-ordering theorem. And I found that well-ordering theorem is equivalent to axiom of choice. And as far I know, axiom of choice is what we understand as free will, that ...
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281 views

What is a counter argument for the proposition that reaching the truth involves abandoning language and other intellectual instruments?

I have a linguist friend of mine who proposes that one should abandon all labels and paradigms to reach the ultimate truth, as they are deceptive. He proposes that you should strip all intellectual ...
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247 views

Is a complete mathematical description of reality possible?

There are definitely states of systems(like mind) which are not quantifiable. For mathematics to work in principle, we need states which are quantifiable or measurable. So, does this go to show that ...
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What should a person interested in the philosophy of mathematics know?

What philosophy should a person interested in the philosophy of mathematics know, at a minimum? Having delved into the subject, it looks like there are things you need to know.
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Did Zeno of Sidon really write that any geometrical system must have some unstated assumptions?

According to this site Zeno of Sidon argued that even if we admit the fundamental principles of geometry, the deductions from them cannot be proved without the admission of something else as well ...
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113 views

Probability vs Possiblity vs gambling knowledge gap for a beginner

Probability is a difficult subject for me to grasp. I watch many religious vs atheist vs philosopher debates on YouTube where probability is often brought up, and because of my poor understanding I ...
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206 views

Is it possible for there to exist a geometrically perfect square?

The corners of a geometrically perfect square should have no width. But if they have no width they don't exist. Therefore the corners must have a width. If they have a width they can be looked at as ...
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99 views

Is Fourier transform a human made tool or an act of nature? [duplicate]

I am a PhD students in physics, and my father is a Math researcher. One time, I asked him "Doesn't the fact that we can use math to explain things that happen in front of us, tell us that math is ...
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371 views

Why is the definition of the real numbers not contradictory? [closed]

I understand that a set whose members can, in principle, be enumerated (by having a formula) can be considered as a well-defined set. Therefore, set of all even numbers, multiples of 3, and so on ...
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Inductive argument for Con(ZFC)

If you ask a mathematician, particularly a set theorist, about whether ZFC is consistent, they will answer that we can't know for sure because of Gödel's theorems. If you ask what evidence at all is ...
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Is paraconsistent logic used in other areas of mathematics other than discrete mathematics and in other areas such as physics and philosophy?

Is paraconsistent logic used in other areas of mathematics other than discrete mathematics and in other areas such as physics and philosophy? I heard that paraconsistent logic is an area of discrete ...
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61 views

Is Constructivism (philosophy of mathematics) against classical logic?

Is Constructivism (philosophy of mathematics) against classical logic? I might be wrong, but mathematics' main branch of logic is based on classical logic, and I was wondering if Constructivism was ...
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109 views

Do mathematical entities look like anything?

My view is that mathematical entities are not physical or visual objects, so they do not look like anything. Is this view correct? I would love to know whether there are philosophers who claim ...
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Does panpsychism imply mathematical entities are conscious?

Does panpsychism claim that even mathematical entities, like numbers and functions and sets, are conscious entities? Or is it restricted to physical objects?
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225 views

Are some mathematical truths contingent on the laws of physics?

Are there at least some mathematical truths that would have been different had the laws of physics been different? Probably most mathematical truths would not change, but are there some that would? Or ...
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183 views

Philosophy of mathematics that is not logic

Is there any readings on philosophy of mathematics that does not fall into studies of logical foundations of mathematics such as "definition of 1", "incompleteness theorems", "...
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52 views

Is there more than one form of logic in mathematics?

Is there more than one form of logic in mathematics? I would be inclined that mathematics only cover one type of formal logic, but I would be interested to know if there are variants thereof or ...
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True Thinkables - with regards to the Identity Theory of Truth

What exactly is a 'true thinkable'? According to John McDowell, 'true thinkables' are identical with facts(1996:27-8,179-80). This seems, if i may, a bit truistic and am left with no concrete ...
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86 views

What is the current status of Foundation-of-Mathematics programmes?

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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42 views

Why did Voevodsky feel “objects of a category can never be equal”?

In a lecture that Voevodsky gave at the IAS on the different notions of equality. He specified how this came from people who worked with categories and their higher analogues. The main problem he ...
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Are truth values of all mathematical statements immutable?

Are there some mathematical statements whose truth values are not fixed, but can change? Probably something like 1+2=3 will always be true, but are there at least some mathematical statements whose ...
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Is there some non-classical logic where the van der Waerden theorem does not apply?

The van der Waerden theorem is a theorem in the branch of mathematics called Ramsey theory which states that for any given positive integers r and k, there is some number N such that if the integers {...
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189 views

Do sets and the empty set exist?

The original title of this question was supposed to be "Do sets exist?", but it was too short. In philosophy of mathematics we sometimes ask whether mathematical objects exist. I think this ...
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157 views

How do mathematical formalists account for unreasonable effectiveness of mathematics?

It will be agreed that mathematical formulae "work" in the sense that we have airplanes, bridges which have been built using mathematical concepts, and they workout in reality as expected. ...
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Axioms in philosophy

In mathematics one lays an initial sets of axioms and rules of inference, and builds a theory from there: does philosophy, nowadays, proceed like this? if yes, what are other currently "...
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Can we define the notion of an “omnipotent God” in terms of computational power?

A classic omnipotence paradox asks, "can an omnipotent God create a stone so heavy that He cannot lift it?" The problem here is that we take omnipotence to mean "capable of anything ...
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Does Gödel believe in the existence of his rotating universe?

I am wondering whether Gödel believe ain the existence of his rotating universe since he is a mathematical Platonist. I am also wondering in what entities believe mathematical platonists. For example: ...
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Explanation of Littlewood's profound philosophical joke

In "A mathematician's Miscellany", By J. E. Littlewood, I found this piece of conversation between Littlewood and Wittgenstein: "Schoolmaster: Suppose x is the number of sheep in this ...
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What is the connection between Lawvere and Cantor?

Lawvere wrote in a couple papers that Cantors word “menge” which is usually understood as “set” is actually a cohesive type. And the “kardinale” is the abstraction from this by getting rid of the ...
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Philosophy of science: Determinism and indeterminism in statistical methods of science

A variable is modeled as a random variable in a statistical model, often without reference to the question of whether it is random in reality. For example, when the outcome of a coin flip is modeled ...
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107 views

Does the following argument about the ontological nature of math exhibit poor reasoning?

Argument P1: Mathematics is the substrate upon which all natural phenomena occur and necessarily governs phenomena in the physical world. P2: One can experience something that is not mathematically ...
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What is the difference between properties and sets?

Is there a difference between properties and sets? To me, it would seem that the property of being non-self-identical is the same thing as the empty set, and the property of being (identical to x OR ...
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271 views

What documented justification is there for using physics to describe the nature of reality?

One of the earmarks of empirical/materialistic research and documentation is its insistence on rejecting and dismissing any subject matter that it deems irrelevant. This has always appeared to me just ...
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What is a set? (Is it possible to define a set?)

I've recently been studying set theory from some introductory textbooks (like Steinhart's "More Precisely" or Open Logic Project's "Sets, Logic, Computation"). I'm interested in ...
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Why is the notion of domain of discourse at all necessary in predicate logic?

The notion of domain of discourse (also: domain of discourse, universe of discourse, universal set, or universe) is a fixture of mathematical logic which is sometimes claimed to be necessary to the ...
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What is the 'axiomatic' or epistemological foundation of Analytic philosophy, what is its practice and purpose?

In researching the origin and purpose of the Analytical Tradition in philosophy, all that appeared was that it traces its origin to the 'Tractatus' offshoots following Wittgenstein and Russell, and ...
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How can we overcome the challenge of the anti statistical philosopher?

Conventional statistical inference has been strongly challenged by the anti statistical philosopher who uses the following example: Imagine a man. Imagine that every time a man opens his front door ...

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