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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190
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29answers
29k views

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/...
56
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25answers
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Why is there something instead of nothing?

A simple but fundamental question. The "something" means the whole Universe (known and unknown), it could be represented as the reality version of the set of all sets, which is itself debated. It ...
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13answers
5k views

What are the necessary conditions for an action to be regarded as a free choice?

A common philosophical question revolves around the existence of free will, but what I've found is that these debates seem to gloss over the concept of "free will" itself, either taking it as a given ...
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1answer
475 views

Does Quine's dissolution of the Analytic/Synthetic distinction challenge mathematical realism?

I was surprised to learn that Quine is a mathematical realist (See this interview for example). I always assumed that his "Two Dogmas of Empiricism" and specifically his dissolution of the Analytic/...
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2answers
365 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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3answers
270 views

Have the exact sciences tried to obtain their legitimacy from “outside” the human being?

I'm not really specialized in the history of science. But it seems for me that as the time passed, the exact sciences tried to do that. For example: The second is measured in relation to the spinning ...
9
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3answers
699 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 ...
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2answers
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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 ...
6
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1answer
416 views

Was there a Kantian influence on Hilbert's formalist programme?

In this paper by Cassou-Nogues which is on an aspect of the mathematical philosophy of Cavailles he quotes the mathematician Hilbert (a colloborator of Einstein in Gottingen) ...We find ourselves ...
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6answers
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Is the use of inconsistent definitions a logical fallacy?

I am not asking for a defense of or pro/con of the existence of an omnipotent (or multiple omni-x) being, or for the existence of square-circles or any other similar thing. These arguments are well ...
14
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4answers
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Is geometry mathematical or empirical?

Is Euclidean geometry a mathematical theory, or is it a theory of empirical science? If taking it to be a mathematical theory would it be due to having alternative geometries? If so, is it in some ...
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11answers
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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 ...
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5answers
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How does actual infinity (of numbers or space) work?

Is infinity just continuous generation of numbers, or can space be actually infinite? If it is finite can we see it expand if we went to the edge? When I say "I am counting to infinity" does it mean ...
15
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7answers
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Is Logic Empirical?

We use the logical system that we know from observations (empirical data) holds true in the world we live in (please correct me if I am wrong). Hence the axioms of logic we choose are themselves ...
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6answers
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Why is Aristotle's objection not considered a resolution to Zeno's paradox?

It seems to me, perhaps naïvely, that Aristotle resolved Zenos' famous paradoxes well, when he said that, Time is not composed of indivisible nows any more than any other magnitude is composed of ...
10
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2answers
625 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 ...
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4answers
985 views

Is logic subjective?

If logic is constructed from axioms, and axioms are depended on observation which in term could be subjective, does this means that logic could be limited to our observation, and not really absolute ...
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1answer
235 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?" ...
24
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4answers
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What are the philosophical implications of category theory?

I have heard about topoi being the ideal entities to use for foundations of mathematics (since we are able to reasonably interpret our theories in them), so I imagine there might possibly be some ...
17
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13answers
10k views

Is mathematics truth?

In mathematics there are imaginary numbers which cannot be represented directly in reality (the physical world). For example, you can't have i apples where i = √-1 (square root of -1) Can we ...
16
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13answers
11k views

Are numbers real?

I am confused as to what numbers are. Numbers are defined to be what they are, so numbers aren't real? But numbers are found in nature, right? So if we invented them, how can they be found in nature? ...
21
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4answers
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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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14answers
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Is mathematics founded on beliefs and assumptions?

Note: I originally posted the question in meta.math.stackexchange.com but I reckon this would suit a more philosophical audience so I am posting it here. Background: I am a 28 year old ...
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7answers
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What are the foundations of philosophy?

I'm a student majoring in mathematics. I've taken a course in mathematical logic and a course in set theory. My problem is basically that I'm always finding philosophical concepts, for example syntax, ...
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11answers
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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 ...
14
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1answer
327 views

Are mathematical suppositions of physical theories determined uniquely according to Aristotle and Plato?

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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4answers
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What is mathematical existence?

When I make a claim in a proof that a mathematical entity exists, is this no more than saying that the theory I'm working within is consistent, and that all the steps upto that point in the proof are ...
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2answers
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What are the truth-values of intuitionistic logic?

Classical propositional logic is bivalent, that is its set of truth-values has cardinality 2 (True & False). Intuitionistic logic drops the law of the excluded middle; does it have the same set of ...
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4answers
287 views

Is there an alternative to Cantor's 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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3answers
574 views

What are the arguments for and against “one true arithmetic”?

This question was born out of a discussion Is the real number structure unique? on Math SE, but since it is more philosophical than mathematical I decided to ask here. From Gödel completeness and ...
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3answers
275 views

What philosophies does Wigner's “Unreasonable Effectiveness of Mathematics” threaten?

Wigner's paper "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" is a well-known paper in the community of the philosophy of mathematics. The overbearing question in his paper ...
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2answers
238 views

What does Wittgenstein mean when he says “there are no numbers in logic”?

From the Tractatus: 5.453 All numbers in logic stand in need of justification. Or rather, it must become evident that there are no numbers in logic. There are no pre-eminent numbers. What does ...
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3answers
324 views

Can all mathematical reasoning be translated into traditional logic?

Can all mathematical reasoning be translated into traditional (Aristotelian, syllogistic) logic? It would seem not ∵ one cannot syllogistically establish the validity of the reasoning in the ...
6
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1answer
564 views

Are analogies between ethics and mathematics philosophically coherent?

Analogies between ethics and mathematics are pretty common – probably because of their shared a priori nature. Philosophical laymen use them, like “Scott Alexander” (no, you don't need to know him), ...
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3answers
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Is First Order Logic (FOL) the only fundamental logic?

I'm far from being an expert in the field of mathematical logic, but I've been reading about the academic work invested in the foundations of mathematics, both in a historical and objetive sense; and ...
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4answers
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What are the philosophical implications of Gödel's First Incompleteness Theorem?

Gödel's First Incompleteness Theorem states Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any ...
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13answers
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Do numbers exist independently from observers?

Do numbers have an objective existence? If life had not evolved on planet earth would there be numbers or are numbers an invention of human minds? Are there any relevant works that discuss this? (I ...
44
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15answers
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Is Mathematics always correct?

It seems Mathematical theories/Laws/Formulas are the least questioned in all of the sciences. Is mathematics that good at being closest to the laws of universe, or is it just a logical tool of our own ...
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12answers
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What should philosophers know about math and natural sciences?

My question is whether a lack of knowledge about formal mathematics or theoretical science in general would have an impact on a philosopher's ability to think and make judgments. Why should a ...
34
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3answers
10k views

Did Russell understand Gödel's incompleteness theorems?

Russell was active in philosophy (although no longer in math) for many years after the Gödel's 1931 publication. Gödel's paper were not obscure, and Russell would have been aware of their effect on ...
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9answers
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Do the laws of logic exist independently of human or animal consciousness?

Are the laws of mathematics and logic, such as if a=b, and b=c, then a=c just constructs of the human mind, or does the universe hold an innate logical structure to it, which the physical part of the ...
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5answers
5k views

What is the difference between mathematical reasoning and philosophical reasoning?

Please see question in title. Why isn't philosophy considered to be a branch of mathematics? Is study of anything not a branch of mathematics, vague and imprecise?
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7answers
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Selection of logical connectives {¬,∨,∧,⇒,⇔} in set theory?

Nearly every treatment of set theory, whether Paul Halmos' Naive Set Theory, Herbert Enderton's Elements of Set Theory, Patrick Suppes' Axiomatic Set Theory, etc. introduce a common set of logical ...
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9answers
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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 ...
21
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5answers
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What was Cantor's philosophical reason for accepting the infinite but rejecting the infinitesimal?

I have begun inquiring recently into mathematical aspects of Georg Cantor's theory of transfinite numbers and sets, which he developed between the years of 1874 and 1897. Throughout his theory, Cantor ...
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8answers
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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 ...
13
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1answer
794 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, ...
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10answers
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Why is 2 considered a prime number? [closed]

Since there are no integer numbers between two and one, how can two be divisible by a number other than itself and one? Perhaps the definition of prime numbers is irrational and wrong, at least for ...
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4answers
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Cantor and infinities

I know we have accepted Cantor's ideas a long time ago and many mathematicians use sets and infinities without ever realizing that thinking about sets and infinities intuitively fails, because there ...
9
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3answers
305 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 ...