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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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What theorems are most important for the foundation of mathematics?

What are the mathematical theorems which are considered as the most important for the mathematics themselves? By importance I mean foundational to mathematics as a whole or foundational to a good ...
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95 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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276 views

What is the relevance of applicability to the natural sciences in pure mathematics?

I think I am coming to a good, new understanding of the relationship of pure mathematics to the natural sciences. A major concern of mine is just how reliable is rigorous (characteristically "pure") ...
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4answers
196 views

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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203 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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160 views

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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Believing in the axiom of Power Set

I am struggling to find a philosophical reason for believing in the axiom of power set, and I was hoping you can give me some justifications. I am not looking for answers of the form "it's convenient ...
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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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91 views

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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202 views

Are the “laws” of deductive logic empirically verifiable?

"Is Logic Empirical?" strongly suggests a question that I would like very much to get a handle on. That phrase is a title of an article by Hilary Putnam, and, according to synopses/reviews, the ...
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3answers
91 views

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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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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153 views

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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160 views

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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180 views

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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213 views

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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173 views

Is there an alternative to cardinalities that makes proper subsets smaller than their sets?

Cantor defined an infinite set as a set whose subset can be placed in a one-to-one correspondence with its subset. That is, take the set of all natural numbers: {0, 1, 2, 3, 4,...}. From that set, you ...
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143 views

Is there a natural example of a non-self-referential semantic paradox in philosophy?

A commonly studied paradox is the liar's paradox. The liar's paradox is to determine whether "this statement is false". The usual resolution is to state this the sentence is not actually a statement ...
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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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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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2answers
140 views

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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0answers
200 views

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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62 views

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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106 views

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
152 views

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
137 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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312 views

Mathematics and disagreements

I was just pondering as a mathematics major, is there a particular instance where a mathematician's work doe NOT require agreements among peer scholars of mathematics to determine the quality of the ...
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223 views

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
71 views

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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48 views

Does Tegmark himself include paraconsistent mathematical structures in his mathematical multiverse hypothesis?

Tegmark postulates in his hypothesis that all possible mathematical structures would exist. But does he include also possible mathematical structures described by other types of logic like ...
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1answer
100 views

Is mathematics something real or just an abstraction we created?

Is mathematics something real like something so correlated in our universe which would be different in other theoretically universes or is it just an abstract universe-independent layer/framework we ...
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2answers
102 views

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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642 views

What are the philosophical implications of using inconsistent mathematics?

I have stumbled across the idea of 'inconsistent mathematics' and could not fully understand: why mathematicians would prefer at times to work with inconsistent systems (from which I assume everything ...
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1answer
218 views

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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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.
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Are there infinitely many “unique” mathematical concepts? [closed]

The difficulty in formulating my question lies in defining what I mean by "unique." What I mean by "uniqueness": For example, the concepts that 1 + 1 = 2 5 + 2 = 7 6 x 3 = 18 6 - 9 = -3 etc. only ...
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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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Can infinity be made finite in certain conditions?

In mathematics there are not only infinitely big numbers, but also infinitely small numbers. One can consider arbitrarily small numbers that can exist only in the mathematical world. For example, ten ...
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287 views

Does Tegmark's Mathematical Universe hypothesis allow existence of alternative mathematics?

Tegmark's mathematical multiverse hypothesis assumes that all mathematical structures exist as universes But do you know whether his hypothesis also allows/accept universes described by other types ...
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Mathematical structuralism and Saussure

Is mathematical structuralism related with structuralism that arose from Saussurean linguistics?
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74 views

Is everything in the universe made of 0s and 1s? [closed]

Qubits are the quantum counterpart of the bits used in traditional computing. While traditional bits represent data as 0s or 1s, qubits are distinguished by what's known as superposition, or the ...
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Is there any correlation between numbers and sensory experience?

Numbers exist, that is clear to me, but is there any logical correlation between numbers and sensory experience? This question came while I was reading Einstein's comments on Bertrand Russell's theory ...
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Can a reason cost 5 dollars?

Imagine a school where no one can wear a red hat. John goes to school with a red hat costing 5 dollars. Someone says John's red hat "is" the reason why he can't get into the school. What is the ...
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2answers
86 views

What is the principle of underdetermination?

While studying I read about the principle of underdetermination of scientific theories. I made some researches online but I am more confused than before. I read about the Quine-Duhem holistic thesis, ...
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119 views

Mathematical Consensus

Can anyone give me a reason why mathematics may require consensus to determine the quality of knowledge from the general mathematical community? Also what would be the counter to such a claim, as in ...
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2answers
72 views

What is frequentism?

I am studying for an exam and I ran into frequentism. Honestly, I don't understand anything about that. Is frequentism related to probability only? Why are probabilities understood as frequencies? I ...
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What are the implications of relativity of Mathematics?

I came across the idea that the same statement can be true in one model while not true in another, while both models are being consistent. An example is "Is the sum of the angles of a triangle equal ...
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Is the commutative property of multiplication intuitive to everyone at first glance? [closed]

a × b = b × a Most people speak about this as if it is common sense. For example, let's say 265 * 7948 = 7948 * 265. I can't get my head around when arbitrarily putting this adding 7948,...
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Does an infinitesimal instant of time have zero duration? [closed]

Is there a philosophical argument supporting the hypothesis that an infinitesimal instant of time has zero duration? The reference to infinitesimal includes the modern presentation of it in non-...
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5answers
229 views

Can anything be less than one?

Zero itself seems to be an absurd number because if there is really zero of something, then nobody has ever sensed it. But even with temperatures, we don’t really have negative and positive ...