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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Size of a point [on hold]

I know this may sound too simple or maybe too absurd to discuss, but I am having a hard time visualizing a point in space! In Euclid's Elements a 'Point' is defined as Something which has no part. ...
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How do mathematical Platonists think about formal systems?

I read that platonists believe there are abstract mathematical objects. How do they think about formal systems? For example, (1)Do they discriminate the set of natural numbers N on ZF and that on ...
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155 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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253 views

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 does Bernoulli's theorem make an inference from chance to frequencies and not vice versa?

I am currently working on an assignment of mine about Frequentism and Bayesianism. In my studies I've found this source: ...
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What is a straight line?

I am not a philosopher; I am an engineer with a reasonable grasp of mathematics. This question has been bothering me for a long time, and I have asked a variation of it to a mathematical community. ...
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1answer
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How would an interpretivist justify using game theory?

It is often said that interpretivist theory does not, in principle, reject quantitative methods. That said, how could an interpretivist defend using a quantitative method (e.g., statistical model; ...
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How can rational choice theory be explanatory?

In his work, John Harsanyi appears to have taken issue with classical social theorists' account of social phenomena. For example, he criticized Max Weber's typological approach on the grounds, "If we ...
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What does Russell mean by “term” in Principles of Mathematics?

Bertrand Russell in Principles of Mathematics defines a term as "Whatever may be an object of thought, or may occur in any true or false proposition or can be counted as one." Can someone elaborate on ...
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Does it make sense to say an expanding line segment is finite?

Whereby I assume a line segment of fixed length is uncontroversially finite. If we take a line segment that is increasing in length over time; does it really make sense to say the segment is finite ...
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What is the relation between proof and observation?

Recently in his 2015 Hirzebruch Lecture in Bonn, Arthur Jaffe re-amplified his famous perspective that finding proof in mathematics is analogous to making experimental observation in physics. In ...
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1answer
151 views

What are the major philosophical interpretations of probability?

Are there important philosophical interpretations of probability? What are the major "schools" or frameworks? What is their relation to formal systems of probability (for instance - the orthodox ...
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Does a proof necessarily entail an “explanation”?

While some mathematical proofs provide an explanation of why a theorem is true, this is not true of all proofs. Proof does not necessarily entail explanation. This recent post by Conifold highlights ...
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Why do people perceive the randomness of events so poorly?

People who are not trained in statistics and randomness (and even sometimes those who are) tend to draw horrible conclusions about whether an event is random or caused. Fundamentally my question is - ...
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Does one have to become a Platonist to refuse to be a Platonist?

I believe the answer is no, but Scott Aaronson on his blog just gave in interesting argument to the contrary. This is in connection with the now famous paper Undecidability of the Spectral Gap, and ...
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Is a “fair coin toss” a logical contradiction?

A previous question asked about the reality of the gambler's fallacy, in which logic appears to offend common sense. In light of the answers, I am now wondering about the other side of the coin, so to ...
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Why don't fair coin tosses “add up”? Or… is “gambler's fallacy” really valid?

I have always been perplexed by a seeming paradox in probability that I'm sure has some simple, well-known explanation. We say that a "fair coin" or whatever has "no memory." At each toss the odds ...
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4answers
226 views

Could philosophy be top-down?

Could it be that, in the way that mathematics is based on set theory (at least the standard one) or another framework and is built bottom-up from that, philosophy starts from relationships between ...
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Godel's Incompleteness Theorem [duplicate]

To What Extent Can Gödel's Incompleteness Theorem Be Applied To Real Life Explanations And Proof Of The Existence Of Things?
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What does Poincaré mean by intuition of pure number?

To what does Poincaré refer in his article Intuition and Logic in Mathematics when he speaks about the intuition of pure number? My answer is that he may refer to a sort of intuition related to the ...
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Can someone provide proper criticism of the author's view in this article? Since I am from east, I may be prejudiced towards his statements [closed]

Can someone provide proper criticism of the author's view in this article? Since I am from east, I may be prejudiced towards his statements ...
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How does pre-axiomatic geometry relate to axiomatic one?

If geometry is something abstract how did historical mathematicians discover theorems within the same framework of axioms, although they were unaware of the axioms? And what is the relation of the ...
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Is geometry mathematical or scientific?

Is Euclidean geometry a mathematical theory? Or is it a scientific theory? If taking it to be a mathematical theory would it be due to having alternative geometries? If so, is it in some way related ...
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1answer
81 views

Alan Turing on the philosophy of mathematics [closed]

What was Alan Turings opinion on the philosophy of mathematics? Was he a platonist? A formalist? If not: what else?
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Was Gödel already a Platonist when he discovered/proved his incompleteness theorems?

Since when was Gödel a platonist? Was he already a platonist when he discovered/proved his incompleteness theorems?
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What is the difference between identity and equivalence?

While the problems of identity seem to have more heft in philosophy, I am actually more interested in the meaning of "equivalence" as symbolized in the (=) sign and, in guilt by association,with ...
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1answer
91 views

Interpreting Leibnitz's “Law Of Continuity” [closed]

For my personal reason, I have to write a blog which tries to explain about the symbols of the differential as well as the integral. In that process, I hit upon the Leipnitz's "Law Of Continuity",, ...
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What is the difference between Intuitionism, Formalism, and Logicism?

What is the difference between Intuitionism, Formalism, and Logicism? Namely - on which issues do they disagree? And what is the relation of those schools of thought to Platonism, Nominalism, and ...
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How would Hume classify computer generated mathematical proofs?

Hume's fork divides knowledge of the world into: Analytic a priori: relations of ideas. Synthetic a posteriori: matters of fact, empirical statements about the world. How would Hume classify ...
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“Undefinable numbers” in real analysis?

What if any important results in real analysis make use of the notion of an "undefinable" real number? (Whatever "important" may mean to the reader.) Or is it used more in the philosophy of ...
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Does relating objects implies in the search of a common unity?

In mathematics, it seems that when we try to find relations about objects we are forced to set a unique object as a basis for the construction of each other object. For example: take one rectangle and ...
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Should I take Math GRE Subject Tests for applying to Philosophy Phd to study Philosophy of Math?

If I was interested in studying philosophy of another field (e.g. philosophy of math or physics), would philosophy phd programs I apply to be interested to see my scores on the gre subject test for ...
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3answers
103 views

Is mathematical Platonism geometric?

A fairly standard approach to mathematical Platonism might start with the observation that any circle actually drawn is not a true circle; and one can imagine or have in mind a perfect circle - a ...
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1answer
77 views

Why can't type theory be applied on personality types? [closed]

Given that type theory is a system for describing mathematics, and given that "Math can be applied everywhere", why is it that type theory can't seem to be applied to personality type theory? I've ...
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1answer
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Does the idea of experimental mathematics entail Platonism?

One form of Platonism entails the actual existence of mathematical objects. Godel and others were known to feel intuitively that they were working with and actually perceiving entities different from, ...
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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 ...
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Is infinite divisibility of Something the same concept as Nothing?

There must be some kind of proof for that. I have always be intrigued by the notion that if something is endlessly divisible then that would mean that it is nothing indeed. (An example? Matter which ...
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9answers
678 views

How can numbers be infinite? [closed]

Abstract or not, the number concept originates from the world, so it seems to me there are only as many numbers as things, and there comes a point where what we call numbers are only hollow ...
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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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How long is the standard meter?

In the Philosophical Investigations §50, Wittgenstein writes: There is one thing of which one can say neither that it is one metre long, nor that it is not one metre long, and that is the ...
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books or references on an introduction of mathematical philosophy/philosophy of mathematics

I am wanting to begin a self-study concerning mathematical philosophy, and I was wondering if anyone would suggest reading material of a modern introduction to mathematical philosophy. I stress on ...
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How does Chihara's account of Frege's objection to Hilbert's criterion of truth and existence makes sense?

I have recently read about the Frege-Hilbert controversy regarding the axioms of Euclidean Geometry. In Chihara's book it is written (under the subsection Hilbert's criterion of truth and existence) ...
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1answer
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Prove ∀w(∀v((v=w∧φ(v))⇔φ(w)))

In this math question of mine, an answer pointed me to this theorem: ∀w(∀v((v=w∧φ(v))⇔φ(w))) which in turn, the answerer stated, implies another theorem: ∃v(v=t∧φ(v))⇔φ(t) which was the ...
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0answers
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Where can I learn about the philosophy behind mathematical and logical proofs?

I'm looking for something that dives into the philosophical idea of a "proof," and explains how the subjects of mathematics and logic deal with it. Does anyone have any book or article recommendations ...
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7answers
577 views

Why might you not accept ¬(¬A) = A?

What motivates intuitionism's rejection of double negation: If A exists, then ¬(¬A) = A. I can't see what's wrong this statement or why someone would reject it.
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On what logic is all of classical mathematics true but undecidable statements are neither true nor false?

Not on classical logic obviously since it validates excluded middle, but less obviously not on intuitionistic logic either. Intuitionists identify truth and provability and discard excluded middle, ...
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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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2answers
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Subformulas of the WFF (∀x) ((∀y) ((x ∈ y) ∨ (y ∈ x )))

Consider the well-formed formula in set theory (∀x) ((∀y) ((x ∈ y) ∨ (y ∈ x ))). I believe there are 5 subformulas: (x ∈ y) (y ∈ x) ((x ∈ y)∨(y ∈ x)) (∀y) ((x ∈ y)∨(y ∈ x)) (∀x) ((∀y) ((x ∈ y)∨(y ∈ ...
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If the ZFC axioms cannot be proven consistent, how can we say for certain that any theorems in mathematics have been proven?

The ZFC axioms are the basis of modern mathematics. But Gödel's 2nd Theorem says that it is impossible to prove that these axioms are consistent. Hence, it is possible (if ZFC is inconsistent) that ...
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Is there something to theory as theory is practice? [closed]

Practice can be explained with theory, but what can theory be explained with? I would say that theory can only be affected directly by the reality itself. However neither one of us experiences the ...