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 does Aristotle suggest One is not a number?

Parmenides showed Nothing is not the same as Zero; the second is a number, and the first is not, in more than one sense; it also differs from the Buddhist notion of Sunyatta, which is nothing in a ...
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What is one divided by zero?

The Pythagoreans were (probably apocryphally) disturbed by the discovery of irrational magnitudes; its useful pointed out here, that irrational means, in one sense, and not now generally alluded, ...
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Mathematical reductionism: senseless

Does it make any sense to ask if logic can be reduced to math? Truth be told I have no idea what the inverse logical reduction could look like. Naturally I'm familiar with a kind of reductionism, in ...
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Should the easiness with which math is applied to the world be a surprise?

I study physics at an undergraduate level. Since early on, I've was a person who thought math was 'logical' and as such, its applications to the world aren't really a surprise since math is so ...
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What purpose do mathematics and philosophy serve epistemologically (compared to sciences)?

Kant did not consider them sciences, but meta-disciplines that study a priori conditions of doing science. Indeed, both mathematics and philosophy permeate all empirical sciences to varying degrees, ...
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Are there any naturally occuring non-embedded manifolds?

Mathematicians are always insisting that manifolds need not be embedded, and by Occams Razor, are best thought without the surrounding ambient space. For example, the surface of a table is a 2d ...
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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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316 views

Do Mobius Strips have a front and back?

Mobius strips, are generally held to have only one side - if one marks a place on the strip and on what looks like the other side then a pencil can draw a line between these two points without ever ...
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When is Mathematics not about counting?

A comment on an answer I posted asserted that "Mathematics is NOT always about counting". My thoughts were that if there's a unit (inches / milligrams / light years etc), then someting is being ...
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Where does Quine say logic should have no ontological presuppositions?

I'm sure I recall Quine saying in various places that one distinction between logic and set theory is that logic should have no ontological presuppositions (or, at most should presuppose some thing ...
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What is mathematics? [duplicate]

Is mathematical practice: an act of discovery of eternal objects and ideas independent of human existence; an intuition-free game in which symbols are manipulated according to a fixed sets of rules; ...
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What proves the law of non-contradiction true?

How has man determined that "The Law Of Non-contradiction" is true? The first time I heard it, if I understand it correctly, I proved it false. Do I understand correctly, that a matter for ...
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Jorge Luis Borges suggests that using a lottery is an “intensification of chance.” Does this make sense?

By intensification of chance, Borges adds that a lottery brings "a periodic infusion of chaos into the cosmos." To me, the idea that chance can be "intensified" seems strange. However, I'm also not ...
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Are there any other branch of mathematical philosophy?

Are there any other branch of mathematical philosophy? I am refer to the mathematical logic, are there any other branch of mathematical philosophy? For example, philosophical graph theory would be ...
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Does phenomenology determine ontology?

There were many historical instances where phenomena could be explained by seemingly incompatible theories, Copernican and Ptolemaic systems, corpuscular and wave theories of light, interpretations of ...
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Are ordinal or cardinal infinities theories for real?

There are a number of notions of infinity in mathematics that are respectable. One of the first is 'the point at infinity' to the line or plane; but one can argue that this is a spurious infinity as ...
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Does mathematics apply to physics in one way or multiple ways?

Does mathematics apply to physics in one way or multiple ways? What do Aristotle and Plato think? It would seem that Aristotle thinks mathematics can be applied to physics in one way only because, ...
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470 views

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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Are there two different mathematics in philosophy?

I was looking at arguments about mathematics being a science (or not), here for example, but it seems that these arguments are more about some metaphysical idea of mathematics rather than the subject ...
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Existence vs tertium non datur

This is my first question here, sorry if it turns out to be a duplicate. Mathematical constructivism states that contradicting the non-existence of something won't imply its existence. Does it mean ...
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Can mathematics be sublime?

The Sublime is sometime used as a synonym for subtle or sophisticated but with aesthetic overtones. In the original setting for this aesthetic notion it was a combination of Beauty & Greatness, ...
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Sources for the aesthetics of mathematics

Many mathematicians often equate mathematics to art and find a deep beauty in its method, results, and ideas. The classic example of this romanticism is captured by G.H. Hardy's A Mathematician's ...
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Is it possible to assign an objective profundity to theorems in a formal system?

To be concrete and specific, let us say we are working in Peano Arithmetic. Is it possible to assign a partial order among theorems of Peano Arithmetic that agrees with our vague intuitive notion of ...
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Formal systems and interpretations for numbers

In my book (Hodel's Intro To Mathematical Logic), we are given several examples of formalized mathematical theories such as group theory, Peano arithmetic, etc. But I've had this ongoing confusion: ...
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Can anyone explain the very beginning of The Analysis of Matter to me?

Can anyone explain the very beginning of The Analysis of Matter to me? What exactly is it that he is saying is an aesthetic choice with respect to physics? I just opened up the book and can't get ...
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142 views

What is an infintely small unit (of time)

Is there a philosophy of the infinitely small? Does anyone apply it to qualitative experience, and ask if that is divided up into instants? It seems to me that the infinitely small could not be like ...
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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" ...