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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Are there kinds of arithmetic that are decidable despite the Gödel theorem?

A proposition p in a consistent formal language is decidable when we can assert either the truth of p or the truth of not p (But not both, for then it would be inconsistent and we have already said ...
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Hilbert's Sixth Problem: Is Kolmogorov's solution the last word?

The demand for axiomatization of probability was put forward by Hilbert at the very beginning of the past century: it was the sixth problem in his famous twenty three problems he deemed of high ...
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Can mathematical sentences in different theories be identified?

My question motivated by a part of this page from Saul Kripke's book Naming and Necessity, which is also viewable on google books. In the middle of the page he say something, which seems unnatural to ...
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What is the role of the a priori nature of time in intuitionism?

According to Brouwer, intuitionists abandoned Kant's apriority of space but adhered to the idea that time was a priori. This Intuitionism considers the "falling apart of moments of life into two ...
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What would be gained philosophically if logicism succeeds?

Any logic similar to predicate logic (which Frege and Russell used?) is already a pretty complicated system. Its axioms don't seem to have something to do with analyticity as conceived by Kant (“...
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Is Aristotle's resolution of Zeno's paradoxes vindicated by motion in the intuitionistic continuum?

In Physics VIII.8, Aristotle refers to his usual resolution of Zeno's paradox of motion: We should make the same response to anyone who uses Zeno's argument to ask whether it is always necessary to ...
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Is mathematical platonism compatible with Platonism?

When calling themselves "Platonists" mathematicians usually mean that they feel they discover ideal facts that eternally exist in some way. My question is if this sentiment is consistent with Plato's ...
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What did Poincaré mean by intuition of pure number?

To what does Poincaré refer in his article Intuition and Logic in mathematics when he speaks about the intuition of pure number? He refers also to two other forms of intuition, besides the "intuition ...
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In Gödels Incompleteness theorem what is the notion of truth?

The entry on Gödels Incompletenss theorem in Wikipedia says: Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for ...
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Does Fodor present any argument for his use of computable methods in his view of the mind?

In defense of his Language of Thought Hypothesis (SEP article), Jerry Fodor argues that Thought is recursively compositional in just the same way that Language is. When we understand a sentence, we ...
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Difference between math and physics in terms of describing the/a universe?

"The difference between math and physics is that physics describes our universe, while math describes any potential universe" This was one of my math professor's arguments in trying to convince me to ...
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In what sense are proofs just arguments that convince us, not arguments that establish truth?

In mathematics and logic, it seems that once a proof of some theorem is discovered, then it is taken to be "absolute truth" within the axiomatic system from which it was derived. My question is: are ...
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Has there been a Cartesian revolution in mathematics?

In his book "Méthodes modernes en géométrie", Jean Fresnel wrote: il ne faut pas se faire d'illusions, Descartes résout des problème de géométrie, non parce qu'il a de la méthode, mais parce qu'...
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What did Poincaré mean when he said “Mathematicians do not deal in objects, but in relations among objects”?

The mathematician Henri Poincaré had said, "Mathematicians do not deal in objects, but in relations among objects; they are free to replace some object by others so long as the relations remain ...
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If the Platonic world exists how would we know?

If we assume existence of a non-material world of ideas that mathematics describes there are some questions that a Platonist has to address. 1) How is the ideal world related to the real one, where ...
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What are the arguments for and against “one true arithmetic”?

This question was born out of a discussion Is the real number structure unique? on Math SE, but since it is more philosophical than mathematical I decided to ask here. From Gödel completeness and ...
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Was Plato an idealist or a realist?

I'm a bit confused. As far as I know realism is opposed to idealism. But we can say that Plato was an idealist when speaking about Plato's forms. We also talk about Plato's realism in the philosophy ...
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What is the number 2?

My friend told me that he took a course in the philosophy of mathematics and said that they defined the number 2 to be "the set of all sets with two elements." I may be remembering wrong, but this is ...
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Why don't we have consensus in more complicated areas of logic?

When I once realised I don't really understand how and why proof by contradiction works, I started reading about it. And apparently I wasn't the only one who felt there's something wrong about it - ...
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Is it possible to generate logically valid sentences made up of “atomic contradictions”?

Is it possible to generate sentences that are made up of "atomic contradictions", but which remain logically valid as a whole? By "atomic contradictions", I mean atomic propositions that are not ...
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Does Tegmark's Mathematical Universe hypothesis allow existence of alternative mathematics?

Tegmark's mathematical multiverse hypothesis assumes that all mathematical structures exist as universes But do you know whether his hypothesis also allows/accept universes described by other types ...
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Are there any philosophical writings/works about Fourier transforms?

I am interested in reading about possible meaning and implications of the time-frequency duality in Fourier transforms and the uncertainty principle (not necessarily the usually discussed Heisenberg ...
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Is the axiom of infinity truly an axiom?

I hope I can communicate my concerns effectively, so I can reach an understanding about a topic that I've been reflecting and researching intensely on for a few days. I am thinking about actually ...
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A (possible) puzzle regarding John Lane Bell's “Abstract Sets”

John Lane Bell, is his paper "Abstract and Variable Sets in Category Theory" (go to Bell's Homepage to download it), defines an abstract set as follows: "An abstract set is then an image of pure ...
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What are the philosophical consequences of employing computers to do science and mathematics?

In recent years the steadily increasing computing capacity of computers has led to a lot of new areas in science. In most cases the computer is used to process huge sets of data which cannot possibly ...
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Are there any known deficits of “relevant logic”?

The principle of explosion is the law of classical logic and similar systems of logic, according to which any statement can be proven from a contradiction. Some early formal systems like Frege's ...
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Do mathematician always agree at the end?

I know it's a off beat question but I thought philosophical answer would be better. I've been trying to study some different sciences in my life, ranging from biology to mathematics, and if I try to ...
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Difference between logic and mathematics

I’ve read the article in the SEP about the philosophy of mathematics. I believe I follow most of it. However, I am a bit puzzled by something that may be due to some basic misunderstanding on my part....
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What does mathematical constructivism gain us philosophically?

Constructivists restrict the kind of entities they are willing to let into the mathematical domain; thus, e.g., Leopold Kronecker did not accept transcendental numbers as well as other entities (see ...
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Reading for philosophy of statistics / statistical inference

Had a quick look around and although there are some questions on statistics there is not one that asks this specific question. I am about to return to university to study a masters in a statistical ...
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Mathematical philosophy: intuitionistic formalism

What is intuitionistic formalism? Tarski called himself an intuitionistic formalist.
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Is there any connection between Structuralism and Category Theory?

Having only the a very cursory knowledge of Structuralism, there does appear to be some points of coincidence: Structuralism: Individual elements of culture must be placed within a System/Structure. ...
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Why is 2 considered a prime number? [closed]

Since there are no integer numbers between two and one, how can two be divisible by a number other than itself and one? Perhaps the definition of prime numbers is irrational and wrong, at least for ...
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How can probability statements be falsified?

Have studied recently some about philosophical views of probability and ran into an interesting problem put forward by Popper: According to Popper, probability statements are not strictly ...
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Is there an absolute infinite, mathematically?

After Cantor, mathematicians realised that infinities can be graded by size (cardinalities). But just as we view the naive infinite as bigger than any finite number, is there an 'infinite' greater ...
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Is there a way to avoid Gödel's incompleteness affecting mathematics as a whole?

I have been thinking about Gödel's incompleteness theorems and their ramifications for the whole of mathematics. In this question I assume some fixed formal system F expressive enough for the ...
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Could philosophy be top-down?

Could it be that, in the way that mathematics is based on set theory (at least the standard one) or another framework and is built bottom-up from that, philosophy starts from relationships between ...
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How long is the standard meter?

In the Philosophical Investigations §50, Wittgenstein writes: There is one thing of which one can say neither that it is one metre long, nor that it is not one metre long, and that is the ...
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Why is intuitionistic negation nonconstructive?

Can someone simply describe why intuitionistic negation is not constructive and why intuitionistic proof is constructive? in intuitionistic logic the notion of falsity has a 'subordinate' status, i....
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How does Popper overcome this critique?

Suppose P and Q are falsifiable theories (in the Popperian sense). Then it seems to me that 'P and Q' is a falsifiable theory (we can refute it by refuting A, or by refuting B), and so too is 'P or Q' ...
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Alternatives to Axiomatic Method

In his article The Pernicious Influence of Mathematics upon Philosophy (see Chapter 12 of this book) Rota says (my emphasis), The axiomatic method of mathematics is one of the great achievements ...
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A question regarding the similarity of relations from Russell's Introduction to Mathematical Philsophy

I do not understand the basis of one of Russell's claims at the end of the chapter 'Similarity of Relations' in his Introduction to Mathematical Philosophy. I have taken an excerpt and emboldened the ...
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Does Quine's dissolution of the Analytic/Synthetic distinction challenge mathematical realism?

I was surprised to learn that Quine is a mathematical realist (See this interview for example). I always assumed that his "Two Dogmas of Empiricism" and specifically his dissolution of the Analytic/...
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What's a name for the impossibility of identity?

It appears to me that no two things can ever be identical, yet the notion that they can has been deployed rather without pause about a billion times in theoretical literature in philosophy and ...
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352 views

Was Kant an Intuitionist about mathematical objects?

In regards to the ontology of mathematics, as far as I can understand, Kant believed that Mathematical objects existed only as features of our perception that influenced how we viewed things-in-...
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What does Russell mean by “term” in Principles of Mathematics?

Bertrand Russell in Principles of Mathematics defines a term as "Whatever may be an object of thought, or may occur in any true or false proposition or can be counted as one." Can someone elaborate on ...
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How does actual infinity (of numbers or space) work?

Is infinity just continuous generation of numbers, or can space be actually infinite? If it is finite can we see it expand if we went to the edge? When I say "I am counting to infinity" does it mean ...
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What is the difference between mathematical reasoning and philosophical reasoning?

Please see question in title. Why isn't philosophy considered to be a branch of mathematics? Is study of anything not a branch of mathematics, vague and imprecise?
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What's so bad about giving up the Axiom of Choice?

The Axiom of Choice (AoC) in set theory famously gives rise to controversial and counterintuitive theorems. (Examples: Banach-Tarski paradox and existence of non-measurable sets.) I'm aware of some ...
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Can you list examples of problems that can not be solved within a formal system but human beings have solved through construction or creativity?

This kind of problem is mentioned in a book I have read, but the book did not give a concrete example. If any such problem existed, this might help me understand human creativity. I think it would ...