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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How do we know if a mathematical proof is valid?

Georg Cantor has showed there are more real numbers than natural numbers in his diagonal argument. Assuming that two sets have the same size if we can make a pair up elements from set A with elements ...
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Why would Wittgenstein say we can't have a perfect language?

I have been reading Wittgenstein's Philosophical Investigations and my question is how does he come to realize that we can't have a perfect language. For instance I would say math is a perfect ...
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What are the truth-values of intuitionistic logic?

Classical propositional logic is bivalent, that is its set of truth-values has cardinality 2 (True & False). Intuitionistic logic drops the law of the excluded middle; does it have the same set of ...
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What was the impact of the discovery of non-euclidean geometry on Kantian thought?

This is mainly a historical question. In Gary Hatfields introduction to Kants Prologomena, he says: After the discovery of non-Euclidean geometry, Kant’s claims for the synthetic a priori status of ...
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How is 0 defined?

I know that the naturals are assumed by the axiom of infinity, but the relationship between them (eg 1+0=1), must be rule based or defined at the very least. Basically I want to know what makes 0 or ...
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Is Cantor's theorem based on a fallacy?

The Brazilian philosopher Olavo de Carvalho has written a philosophical “refutation” of Cantor’s theorem in his book “O Jardim das Aflições” (“The Garden of Afflictions”). Since the book has only been ...
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What makes something mathematics?

What classifies something as math? Is "math" simply performing operations with a certain set of axioms in mind? Is "math" anything that involves numbers? What about mathematical logic? Google ...
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Can mathematics be separated from the physical world?

I am a math enthusiast, with very little interest in physics. In fact, today I thought to myself how can I expel the physical world from mathematics completely. However, this has proved to more ...
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What are computable numbers, and what is their philosophical significance?

What are Computable Numbers? Is computability (or non-computability) some sort of technology-dependent characteristic of numbers (via e.g. Turing Machines)? What are the philosophical implications or ...
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How do we separate rules of logic from non-logical constraints?

I think that very often the idea of 'constraint' appears in mathematics. For example, when a triangle is considered, 3 points are constrained not to be co-linear, and then we try to discover the ...
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Validity of mathematical induction

Are there philosophical positions that reject the validity of mathematical proofs by induction? If so, what are the implications? I know that mathematical intuitionists reject the law of the excluded ...
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Why does statistics work?

Statistics deals with probability, where even an extremely unlikely event has some chance of happening. What if there's a series of these unlikely events going on for thousands of year. I mean it ...
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Was Bishop Berkeley part of the Enlightenment and if so - how did it fit his adherence to religion?

In his The Analyst Berkeley argued, among other things, that mathematicians must not "submit to Authority, take things upon Trust" and so expressed a view of the Enlightenment. This made me think: if ...
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Is a “fair coin toss” a logical contradiction?

A previous question asked about the reality of the gambler's fallacy, in which logic appears to offend common sense. In light of the answers, I am now wondering about the other side of the coin, so to ...
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Has there been any philosophical investigation into the role of aesthetics in mathematics?

There are many mathematicians who talk about the particular beauty of a subject. They may say a particular result is pretty. It may be beautiful. It seems to me play a fundamental role in the ...
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How does Russell's argument for identity refute that of Wittgenstein's?

In My Philosophical Development Russell wrote, I come next to what Wittgenstein had to say about identity, which has an importance that may not be obvious at once. To explain this theory, I must ...
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Is there a sheaf-theoretic description of para-consistent logics?

Paraconsistent logics drop the notion of global consistency, instead they have a notion of local consistency. In sheaf-theory, or categorical logic, as in topos theory, there is a notion of local ...
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Can one still derive paradoxes from the amended version of Naive Set theory given by Cantor in a letter to Dedekind?

Consider the following definition of set given by Cantor in a letter to Dedekind: If on the other hand the totality of the elements of a multiplicity can be thought of without contradiction as '...
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What does this Jacques Hadamard quote mean?

What does this Jacques Hadamard quote mean? The shortest path between two truths in the real domain passes through the complex domain. Is this a philosophical statement? what is its mathematical ...
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Is mathematics an art?

I'm thinking of art in the traditional sense as visual, musical or literary. Mathematics certainly requires technique, and hence one can say craftmanship. But whereas the production of an art (at ...
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What are the major criticisms of Alain Badiou's claim that mathematics is ontology?

Building on Was mathematics invented or discovered? I would like to know what the major criticisms are of Alain Badiou's claim that mathematics is "the very site of ontology" (in Being and Event.)
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Why isn't Cantor's diagonal argument just a paradox?

Cantor's diagonal argument concludes the cardinality of the power set of a countably infinite set is greater than that of the countably infinite set. In other words, the infiniteness of real numbers ...
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Why do people perceive the randomness of events so poorly?

People who are not trained in statistics and randomness (and even sometimes those who are) tend to draw horrible conclusions about whether an event is random or caused. Fundamentally my question is - ...
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Does Popper's theory of falsification apply to mathematics?

Mathematics is generally & popularly judged a science in the basic duality: science - humanities. As enemies and collaborationists. The border heavily & fiercely policed. However, it seems to ...
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Is there any justification for the existence of sets?

In this Reddit comment I was explaining how natural numbers could be built from the empty set: A standard set-theoretic way of defining the natural numbers[1] 1,2,3,... is based on the empty set, ...
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Is mathematics a language?

Galileo gave the metaphor that the natural world is written in the language of mathematics, but is mathematics even a language?
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“Continental” philosophers who have worked on the Philosophy of Mathematics?

Who are some philosophers that are generally placed in the continental tradition but who have done some work in the philosophy of mathematics. I know that Husserl has some great work in philosophy of ...
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What is the difference between depth and surface information?

I was looking for an answer to this question: Was Euclid's method of proof axiomatic? While doing so I ran across an abstract of Jaakko Hintikka for an article "What is the axiomatic method?" ...
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What are the historic stances on the epistemological status of mathematics?

I know that Plato and Kant thought it was synthetic a priori (although Plato would not have phrased it in that way). What other major thinkers have weighed in on this issue, on both sides of both the ...
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How does “higher-order logic” differ from “normal” (first order?) predicate logic?

How does “higher-order logic” differ from “normal” predicate logic? I assume the latter is consistently called ”first order logic”. So where are the differences between these? What kinds of statements ...
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Where did Gödel write that first-order logic is the “true” logic?

In "On How Logic Became First-Order" Matti Eklund writes (p. 2/148): It appears to be widely held today that arguments from Skolem and Kurt Gödel, both alleged proponents of the thesis that ...
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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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What are the properties of Mathematical Objects?

I have been thinking a lot about how one knows when an observation contains mathematical elements. Many years ago when I was in school, I found that there was often little time taken out to discuss ...
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A Question Regarding Russell's Paradox

Consider the 'set' behind Russell's Paradox: R = { x | x is a set and x ∉ x } in light of Cantor's definition of set ("aggregate"/Menge) in his CONTRIBUTIONS TO ...
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How should one interpret modern mathematics if one doesn't believe in infinity?

I am an ultrafinitist. http://en.wikipedia.org/wiki/Ultrafinitism I don't believe there is a such thing as infinity. To me, it is obvious that there has to be a largest number; I just don't know what ...
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If the universe is infinite, shouldn't I already have been contacted by a time and space travelling doppelgänger?

If the universe is infinite, by virtue of chance it means that every possible configuration of matter must exist somewhere (according to this documentary). Therefore, if the universe is infinite and ...
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Does mathematics always need axioms?

To do mathematics, one obviously needs definitions; but, do we always need axioms? I was thinking that a statement like For all prime numbers, there exists a strictly greater prime number. cannot ...
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Why might you not accept ¬(¬A) = A?

What motivates intuitionism's rejection of double negation: If A exists, then ¬(¬A) = A. I can't see what's wrong this statement or why someone would reject it.
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How does Frege's definition of number solve the Julius Caesar problem?

How does Frege's definition of number solve the Julius Caesar problem? Frege's definition of number in the end of Foundation is such: the number belonging to the concept F is the extension of the ...
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What does it mean for an axiom to be logical?

I have recently been hearing the phrase logical axiom being thrown around in reference to the philosophy of mathematics and I'm having a hard time understanding what one might mean when they are using ...
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Are there mathematical properties a mathematical object might have only contingently?

It is generally assumed that mathematics is necessary, such that any mathematical theorem is necessarily true. This can be read as a de dicto necessity such that for any mathematical proposition p, []...
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How do we know how to follow a rule?

This question seems to either be at the forefront or the background of countless philosophical enquiries. Much has been written on Wittgenstein's rule paradox (e.g. Kirke's Wittgenstein: On Rules and ...
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What are some arguments for the golden ratio making things more aesthetically pleasing?

What are some (not necessarily good) arguments that painters, architects, designers, musicians, etc. basing their work on the golden ratio φ makes their work more aesthetically pleasing? I think ...
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What is the difference between identity and equivalence?

While the problems of identity seem to have more heft in philosophy, I am actually more interested in the meaning of "equivalence" as symbolized in the (=) sign and, in guilt by association,with ...
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What is the philosophical problem with Skolem's Paradox?

I guess there are two questions here. QUESTION 1: Skolem's Paradox shows that countability is relative in first-order logic, but where is the relativity? In this first question, I will do the ...
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Is there a known limit to relationship between physics and mathematics?

I am much interested in discussions such as Wigner's "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". It's quite amazing that mathematics so well applies to our universe, and ...
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Why were Kant's categories used in the mathematical category theory?

I am curious exactly how mathematical categories were inspired by Kant's categories. The SEP article on category theory says: In order to give a general definition of the [natural transformation], ...
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What are the major philosophical interpretations of probability?

Are there important philosophical interpretations of probability? What are the major "schools" or frameworks? What is their relation to formal systems of probability (for instance - the orthodox ...
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What are some introductory books about the philosophy of mathematics?

What well-written introductory books are there about a mathematical point of view on the philosophy of mathematics and its different school of thought? By this I mean a book that has some ...
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Should I trust mathematics?

First of all I'm not an expert in this field, please correct me if I'm lacking relevant knowledge here. A few hundreds years ago mathematics was largery based on intuition. People realised we need to ...