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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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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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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What makes a math problem “difficult”?

Someone shared with me a video of Fermat's last theorem: http://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem Obviously it's been solved now: is the answer in an information rich vocabulary, is that ...
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Circular nature of the cosmos. (π) [closed]

I've been pondering the irrational number that is pi and how it relates to the infinity of the universe. We often see many cycles in nature, current scientific theory states that the universe began as ...
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Universal quantifier in Russell's Theory of descriptions - Who is the UNIVERSE?

In Russell's 1905 paper "on denoting" in which he introduces his theory of descriptions, he uses his method of analyzing propositions that include denoting phrases (descriptive ones), by rewriting ...
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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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Where is philosophy today [closed]

Basically I'd like to know which are the main philosophical schools today (meaning maybe the last 5-10 or even 15-20 years), where they come from and, if there is a reasonable answer, where are they ...
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Why do we have a problem about understanding the concept of the “empty set”?

   The title seems quite bit more generalized, but I'm saying about the philosophers and mathematicians in the past who discussed about the concept of nothing, or the empty set. I'm ...
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Is Mathematics considered a science?

Science, generally is analyzing information gathered from observing phenomena, and coming up with theories to try and explain the phenomena. Then, attempting to predict a new phenomenon before it ...
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Is mathematics as a concept value-laden?

Joan Robinson in Economic Philosophy writes: Perhaps Gunnar Myrdal is too sweeping when he says (speaking as an economist) that 'our very concepts are value-loaded' and 'cannot be defined except ...
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Generalizing mathematical concepts

It's sometimes useful and interesting to generalize mathematical concepts - e.g., turning the familiar notion of number of things in a given set into the concept of cardinality of a set, etc. Lately I ...
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What would be the philosophical implications of a solution to the P versus NP problem? [closed]

The relation between the complexity classes P and NP is studied in computational complexity theory, the part of the theory of computation dealing with the resources required during computation to ...
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What would be the philosophical implications of a solution to the Riemann hypothesis?

In mathematics, the Riemann hypothesis, proposed by Bernhard Riemann (1859), is a conjecture that the nontrivial zeros of the Riemann zeta function all have real part 1/2. The name is also used for ...
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Does a thing being subtracted from ever disappear completely?

Suppose I take any finite length and subtract half of it continuously. So the size of the remainder, after each subtraction, is equal to its original length multiplied by one half taken to the nth ...
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Are mathematical statements necessary truths?

I apologize if a similar question has been asked here, but I haven't found it. Are mathematical statements necessary truths? By 'mathematical statements', I mean both mathematical axioms as well as ...
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Are the truths of euclidean plane geometry contingent truths?

Existence of non-euclidean geometries does not seem to imply an affirmative answer to that question. It might be possible that such geometries are formal constructs to abstract our primitive and ...
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Can mathematics be political?

The Liverpool Tate have an exhibition currently running that discusses the dialectic between politics and art - the situationists, Bertolt Brecht and earlier the patronage of the court. Mathematics, ...
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Can Euclid's Elements be used to rigorously prove 2+2=4?

It is possible to formally prove that 2+2=4 using Peano's five axioms for the natural numbers and elementary set theory (actually a long and tedious process). Is it possible to prove it based on the ...
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Is it an evolution or design that makes Math in Music so beautiful to our human ears? [closed]

Did humans 'evolve' to find beauty in Math? This means that they must of had some advantage (with good music taste) to pass on genetic code. According to ...
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Philosophical presuppositions and theoretical foundations of mathematics curriculum [closed]

Philosophical presuppositions and theoretical foundations of mathematics curriculum. The purpose of this essay is to explore the philosophical pre-suppositions and the theoretical foundations of a ...
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What is it that transforms mathematical equations into physical reality? [closed]

It's clear that mathematics is the language of physics . Mathematics is a tool for constructing many many different realities . Imagine that god decided to create the universe . He used mathematics to ...
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could an algorithm that writes algorithms be written? [closed]

Could an algorithm that writes algorithms be written? Wikipedia says: An algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Starting ...
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Actual infinite vs. Potential infinite [closed]

I'm looking for bibliography about the problem of Actual infinite vs. Potential infinite. I would appreciate information about papers or books treating this problem deeply, philosophical and ...
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Is calculus approximate?

We know that calculus does not take real values and takes variables like infinity and approaching zero. So can we conclude that even calculus cannot define reality and is different from the "real" ...
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What do we mean by the symbolic representation of nothing?

The word 'nothing' symbolically represents nothing-in-itself. But how can we refer to something that by definition is not there? To make this clearer: the word 'horse' refers to an actual living ...
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Can an infinite be undifferentiated?

Starting in amthematics: The infinite in mathematics must be differentiated: we have the sequence - 0,1,2,3...; where each number is distinct. The same goes for infinite ordinals and cardinals. ...
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Is the number of universes finite, countably infinite or uncountably infinite (and what size of uncountable if so)?

Assuming that the alternative universe theory is correct, how many alternate universes are there? From my understanding an alternate universe "pops up" whenever a particle goes from being in an ...
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Understanding philosophy from a mathematicians perspective

(I'm new, so please downvote, re-tag, flag, close or delete the question as appropriate. I'd appreciate a comment about why it was done.) I was reading this, and came across the paragraph Working ...
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Can definition be existence (in mathematics)?

The set omega, as the comment in this question points out, can be defined as the smallest set that is closed under succession and includes the empty set. This is enough to define it uniquely, but to ...
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Opposite Universe

Let us assume we live in a shared Universe which can be fully described mathematically, and that all mathematical variables have an opposite. 2 is opposite to -2, and true is opposite to false. When I ...
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Is Horse a Concept?

Frege famously said horse is not a concept (it is an object). When we consider the sentence 'Socrates is a philosopher', 'Socrates' is an object and 'philosopher' is a concept, and there is a copula, ...
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Epistemology of infinite sets

I'm trying to understand the difference between sets from an epistemological point of view. Let S be a finite set of cardinality = n, I find intuitive that I can hold epistemically the set S without ...
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Why does one worry about the existence of a number but not of a dog?

One can ask whether the number one exists, and there are a range of answers. In particular, Platonism holds that this number does exist in some abstract world. Now observe that number and one are ...
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What did Hardy mean by “ugly mathematics”?

The following quote is attributed to G.H. Hardy, a British mathematician: The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas like the colours or the words, ...
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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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difference between conception and theory

What is the difference between a conception of something and a theory of something? Is the conception more extensive in content or less? Example: iterative conception of sets and ZFC
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Hilbert Grand Hotel vs. Cantor: Can we postpone the solution into infinity?

We can accomodate infinite amount of people into an Hotel with infinite rooms. Next, we can add extra infinitude of people. (See Hilbert's paradox of the Grand Hotel.) Obviously, the trick is just to ...
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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, ...
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Illogical part of universe

This question is based on many assumptions: That since universe could not appear out of itself and could not be created by someone there has to be a other part of 'universe' (I am not sure if ...
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How would anyone know if they saw the equation of everything?

Given this answer by user34445 to the question: Final theory in Physics: a mathematical existence proof? ...
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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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What kind of math curriculum would one need to go through to understand Lacanian topology and knot theory?

All the latest psychoanalytic theory has been pursuing a highly mathematized trajectory which has left me in the dust as a philosophy/psychology student (4 years of liberal arts college with no math ...
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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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Is there provably a largest cardinal consistent with ZF

In the integers:1,2,3... we can say that there isn't a largest number; but it turns out with the advent of set theory we can posit a largest number omega that doesn't lie within the integers and yet ...
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What is the difference between a statement and a proposition?

I'm doing a MOOC on mathematical philosophy and the lecturer drew a distinction between a proposition and a statement. This is very puzzling to me. My background is in math and I regard those two ...
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On the Axiom of Infinity

Russell, in Principia Mathematica, says the following of his Axiom of Infinity: "The axiom of infinity will be true in some possible worlds and false in others" He is notoriously sheepish about ...
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What could we call as mathematical intelligence?

A professor once told me, for example, that the act of counting is widely regarded as a first sign of mathematical intelligence. Is it though?, can the act of counting (or performing simple arithmetic ...
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Is math independent of our sensory experience?

I've been asking myself this and other questions in the field of philosophy of mathematics. Could we, if we were isolated from any kind of sensory experience, be able to learn mathematics? This ...
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Badiou, Deicide & the return of the Transcendent

Badious philosophy is predicated on Set Theory, in its incarnation as the materialist set theory ZFC. He calls mathematics the very site of ontology. Nietszche famously declared the death of God (in ...
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What do we mean by the term “Number of things”?

I am reading the book "The Number-System of Algebra (2nd edition)." I have some problems with the the first article: "Number". The author has confined the concept of number of things to the groups ...