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.

Filter by
Sorted by
Tagged with
12
votes
1answer
596 views

What are the differences between Tarski's 1933 and 1956 truth definitions?

The paper "The Seven Virtues of Simple Type Theory" mentions that it uses the same trick (due to Tarski) to define the semantics that is also used by first-order logic. I interpreted this a reference ...
12
votes
1answer
858 views

What sources discuss Russell's response to Gödel's incompleteness theorems?

In his book My Philosophical Development Russell writes, In my introduction to the Tractatus, I suggested that, although in any given language there are things which that language cannot express, ...
11
votes
9answers
1k views

Is a proof still valid if only the author understands it?

Some time ago I was reading about the recent Shinichi Mochizuki's proof for the famous ABC conjecture. It's enormous and so incredibly difficult that at that time virtually nobody was able to ...
11
votes
3answers
8k views

What is the difference between an Ordinal number and a Cardinal number?

I'm trying to understand the real difference between an Ordinal and a Cardinal, especially in relation with transfinite cardinals. The stuff on Wiki is a bit too complicated. Can anyone make it simple ...
11
votes
8answers
2k views

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 ...
11
votes
3answers
1k views

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 ...
11
votes
1answer
320 views

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?" ...
11
votes
8answers
3k views

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 ...
11
votes
2answers
280 views

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 ...
10
votes
7answers
7k views

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 ...
10
votes
11answers
2k views

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 ...
10
votes
11answers
2k views

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 ...
10
votes
7answers
3k views

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 ...
10
votes
5answers
1k views

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 ...
10
votes
8answers
2k views

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 ...
10
votes
4answers
1k views

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 ...
10
votes
9answers
3k views

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 ...
10
votes
8answers
3k views

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 ...
10
votes
3answers
1k views

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 ...
10
votes
9answers
2k views

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 ...
10
votes
3answers
372 views

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 ...
10
votes
2answers
625 views

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 ...
10
votes
1answer
735 views

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 ...
10
votes
1answer
607 views

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 ...
10
votes
3answers
338 views

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 'being ...
9
votes
7answers
6k views

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 ...
9
votes
4answers
2k views

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 ...
9
votes
2answers
4k views

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.)
9
votes
3answers
975 views

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], ...
9
votes
8answers
782 views

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 - ...
9
votes
5answers
590 views

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, ...
9
votes
4answers
3k views

Can science work without mathematical formulations?

Ernst Mayr in his last book titled "What Makes Biology Unique?" argues that many of the theories in biology do not need any mathematical support. He says that much of biology is only conceptual and ...
9
votes
7answers
8k views

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?
9
votes
4answers
599 views

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 ...
9
votes
7answers
3k views

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 ...
9
votes
5answers
553 views

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 - ...
9
votes
5answers
621 views

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 ...
9
votes
3answers
1k views

“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 ...
9
votes
3answers
317 views

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 ...
9
votes
1answer
351 views

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 ...
9
votes
3answers
726 views

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 ...
9
votes
6answers
1k views

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 ...
9
votes
2answers
673 views

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 ...
9
votes
2answers
319 views

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 "...
8
votes
11answers
1k views

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 ...
8
votes
7answers
10k views

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 ...
8
votes
5answers
2k views

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 ...
8
votes
6answers
720 views

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.
8
votes
8answers
4k views

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 ...
8
votes
3answers
2k views

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 ...

1
2
3 4 5
19