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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75 views

Can pure randomness be computed?

Algorithm for randomness usually use seed, and thus having an unique input it cannot be said to be completely random, so can pure randomness be theoretically computed?
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Is math arbitrary?

Math at its core begins with calling something true or false and following logic. WE for example call an odd number 2n+1, but what if we called an odd number 2n and flipped it for it to become an even ...
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What are some mathematical fields that can be useful to philosophers?

I am wondering if there's any field in mathematics that can help philosophers define things or help a philosopher make an argument for something. I am just wondering if there's any mathematics that ...
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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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Do picture proofs of the Pythagorean theorem make it empirical?

As I understand it, the Pythagorean Theorem, which defines the metric for Euclidean space, is said to be strictly mathematical in the sense that it is derived from a set of purely theoretical axioms (...
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Was Euclid's method of proof axiomatic?

Euclid's method of proof has often been described in textbooks as axiomatic, but was it really so? And if not, how else can Euclid's method be characterized?
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What is the connection between conscious mind and Gödel's incompleteness in a mathematical universe?

Assume that our universe is a mathematical one, similar to the one that Tegmark proposed (see here). In contrast to what I read there, let's assume that the axioms upon we build the universe are such ...
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8answers
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Do transfinite sets have practical applications?

This may not qualify as a philosophy question exactly, but I would argue that potential applications of pure mathematics are in the bounds of philosophical interest. Many innovations in pure ...
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102 views

Can math be done without syllogisms? [closed]

Question seems self explanatory. Is there anything in mathematics that can be stated to be true without using a logical syllogism? Had a discussion with somebody about this recently. Sorry if this is ...
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216 views

Did any philosopher make the claim that mathematics can be as illusory as visual information?

The Greeks postulated that the world we observe may be just an illusion and Kant based some of his philosophy on that very idea. From that idea, came the idea that mathematical truths are more certain ...
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1answer
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What are some philosophical arguments that explain why mathematics allow us to reach a greater truth than empirical evidences?

Is it really the case? Was there a proof of sort that shows mathematical facts are more certain than empirical facts? What are the arguments for and against that claim?
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The mereological account of sets

So it has come to my attention that David Lewis, David M. Armstrong and others tried a mereological account of sets. James Franklin states it as: Armstrong adopts David Lewis’s proposal that a ...
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What actually are meaningless symbols?

Some days ago our professor during the course of his lecture wrote the following definition of a polynomial. We say that an expression of the form a0 + a1x + a2x2 + ... + anxn is a polynomial of ...
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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 flat, how can the Earth be round? [closed]

Just another silly question that may deserve a wise answer.
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How much platonism do I need to handle the halting property?

I always considered myself as platonist (in contrast to formalist / finitist) but recently I realized (if this is actually true) that you need a bit of platonism to even make sense of questions like '...
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Number, Category and Set

Can it be said that a number is a category is a set? There is such a variety of ideas on numbers, categories and sets that probably anything one says about them will be controversial, but I was ...
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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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Why do the professional mathematicians believe blindly in so meaningless concepts as Infinity? [closed]

To refute such a concept as Infinity (or many infinities) in mathematics doesn't at all require all that big efforts mainly from its own definition in mathematics. To explain this very simple fiction ...
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1answer
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Are there clear counter-examples to this definition of mathematics?

Here I'll re-present the question about a definition of mathematics as being about deduction, that I've given in a prior posting, but here I'll further clarify that this might not be what is usually ...
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Infinity in modern integration theory

The Riemann integral itself doesn't work with infinity (±∞) as “endpoints”, you have to take a detour by calculating the integral for arbitrary endpoints ±z and then take the limit for z→∞, which ...
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To what extent can the invention of zero in India as a number be tied to Buddhist philosophy, if at all?

The Wikipedia entry on zero suggests that the ancient Greeks were unsure about the ontological status of zero. They asked themselves, 'How can nothing be something?' whereas in Buddhism, Sunyata or ...
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1answer
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Why must we choose an intuitionistic explanation over a paraconsistent one, given they are dual?

Given the anti-intuitive results of Quantum Mechanics, it is not surprising that Physicists would look for a deeper reason in the structure of the theory to explain what was then (and still is) ...
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How much this theory fulfills of criteria for a foundational theory of mathematics?

[EDIT] The criteria for a founding theory of mathematics, especially if it uses large cardinal axioms that I want to refer to are those of Harvey Friedman's 2000 criteria given in pages 5-6 of the ...
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Are there any strong reasons to still consider logical monism or a “One True Logic” in light of all the non-classical logics that have been developed?

I know there has always been some debate concerning whether or not a certain logical system (like classical logic) is the correct one, especially when it comes to propositional claims about the ...
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Are there counter-examples to this broad characterization of mathematics?

Mathematics can be broadly characterized as the study of non-trivial apriori symbolically displayed axiomatic systems. Or more elaborately the study of non trivial apriori implicit or explicit ...
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Distinguishing between procedure-like and collection-like mathematical objects

Is it useful/productive to draw a distinction between "active" things with "computational force" (procedure-like) and "passive" things without such force (collection-like)? Does this distinction have ...
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Kurt Gödel Quote: Why does learning what to disregard improve correct thinking?

What does Gödel mean by this quote? What for example should one disregard?
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Does Reflective Set Theory “RfST” fulfill the requirements of founding Category Theory and Mathematics?

On mathoverflow I've posed the question in the title in connection to Muller's 2001 criteria for a founding theory of mathematics, which largely raised in connection to Category theory [see here]. ...
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Was mathematics invented or discovered?

What would it mean to say that mathematics was invented and how would this be different from saying mathematics was discovered? Is this even a serious philosophical question or just a meaningless/...
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Similar to Douglas Adam's HGTTG, Is there any philosophy that views human society as a computation?

In Douglas Adam's Hitchhikers guide to the Galaxy, Earth is a supercomputer that is computing the the Ultimate question, whose answer is 42. I was wondering is Douglas Adams was inspired by any ...
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Does the snake bite its own tail: “Philosophy of philosophy”

I was just philosophizing about the philosophy of mathematics. Then at one point I philosophized: is there a philosophy of philosophy? Is that meta-philosophy, or is that just philosophy again? Can ...
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4answers
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Truth in Science vs. Truth in Math

Two scientists independently try to solve a problem to predict a certain phenomenon. The two scientists come up with different answers, but both of their solutions seem logical to each other. How do ...
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3answers
156 views

How can truth exist if every statement is ambiguous? [closed]

I have read online and personally believe that every statement has some degree of ambiguity to it. With this in mind, I was wondering how any propositions can be true. For example, I have heard some ...
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Modern Mathematical Objects and Empiricism

I don't have any formal knowledge in Philosophy. I am reading a book named 'Thinking About Mathematics' by S. Shapiro. In this book I have learnt about platonism and empiricism. Well, I think we, ...
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Is there a limit beyond which mathematics, if used correctly, cannot be applied to reality?

And if so, why, and which? Take this case: If I've two apples and I believe that with two more I'll have four apples, then I implicitly believe that summation applies to reality. Yet there are ...
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What is the empirical basis for justifying mathematics?

In the introduction of a very nice book by M. Giaquinto, called Visual Thinking in Mathematics, he investigates the conditions that give rise to mathematical knowledge - the following ideas are ...
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1answer
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What is Kant's view of a mathematical object?

I wonder what are mathematical objects - say, the number 1, a circle, etc. - for Kant? Do they have some kind of special status for him compared to ordinary (empirical) objects? Where exactly does he ...
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Are there any good philosophical arguments for or against Cantor's theorem, other than the ones that Cantor came up with?

I am looking for philosophical arguments for and against Cantor's theorem other than the ones Cantor came up with, if you know any, can you present them or a link to them? I post this in philosophy ...
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2answers
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Can results be predicted?

I wanted to know that, what can we assume as the result of some experiment which we have not conducted on the basis of mathematical proofs? I mean, in general, equations are created after analyzing ...
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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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How is ω-consistency different from ordinary consistency?

I've read Gödel's explanation and others but my understanding is unclear. Answers to the followup questions below would help: does ω-consistency have any relevance to methods or ideas not connected ...
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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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“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 philosophical ground for distinguishing logic and mathematics?

I was wondering why the field of mathematics and that of logic are perceived as two distinct fields. Although could be pleased with the intuition that logic is rather meta-mathematics, still would ...
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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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If our world is mathematical، Does not this increase the probability of being complex as well?

Tegmark's mathematical universe hypothesis, posits that reality is a mathematical structure. This mathematical nature of the universe, Tegmark argues, has important consequences for the way ...
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Do mathematicians use logic when adding things up?

I asked a similar question recently, and it was closed. So, bear with me. When two things are materially equivalent we don't add anything to work out how much we have of both together, right? If ...
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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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Is there a summary of Russell’s Principia Mathematica?

Perhaps better, is there an accessible version of the Principia? I am looking for a summary that would summarize and clarify Russell’s reasoning behind his famous conclusion that 1 + 1 = 2.