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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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Some questions about the material conditional and entailment in intuitionist math

In an excellent answer to a question about the history of material implication, @Bumble notes: Unfortunately, the word ‘implies’ is ambiguous between these meanings. In particular, mathematicians are ...
mudskipper's user avatar
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What is something that math cannot be applied to and doesn't involve math?

I have been asked this question, yet I am unable to answer it. The issue with this question is that I have given all that I know, therefore I too am at a loss. What I do know is it has no concept of ...
Smartarse69 5000's user avatar
6 votes
5 answers
630 views

Is it fair to say truth is used more in logic than in math? If so, what are the reasons for doing so?

Right out of the gate in logic we see sentences (propositions) like "all men are mortal" and we say both that it is a true proposition (e.g. independent of being a premise) and that in a ...
J Kusin's user avatar
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3 votes
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Objection to indirect proof in Intuitionism

From my understanding, Brouwer's conception of intuitionism is that mathematical objects only exist in the mind once they have been constructed. And we can create constructions using computable ...
BENG's user avatar
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Has anyone ever studied which proof types are feasible for which theorems in mathematics? If not, why not?

For instance, when asked to prove that sqrt(2) is irrational, we go straight for the proof by contradiction where we assume it’s equal to a/b in lowest terms and end up with a and b not being in ...
asdf555's user avatar
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1 vote
1 answer
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Is maths and computation anything other than addition? [closed]

This question may have been best answered on the mathematics site or the computer science site but however I think there is an argument to it being a philosophical question aswell. Quantity is a ...
8Mad0Manc8's user avatar
1 vote
0 answers
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Intuitionist perspectives on Greek mathematics

My question pertains to how intuitionist perspectives on the philosophy of mathematics might apply to Greek geometry and number theory. It seems that the standard examples given to justify the ...
Menander I's user avatar
2 votes
1 answer
155 views

Can Internal Set Theory provide a complete system of arithmetic?

Internal Set Theory (IST) is a conservative extension of ZFC that adds three axioms that serve to define a predicate standard such that all numbers are either standard or not. There are finitely many ...
Bumble's user avatar
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Is infinity a number?

So I've been on a number of math fora, part of learning some calculus (not much of set theory, no). To my surprise I found what I would describe as strong resistance from some folks against (using) ...
Hudjefa's user avatar
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3 answers
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Mathematical Realism and 0=1

Mathematical Realism is the notion that mathematical truth exists, and is not subjective or merely a mental construction. Inspired by Noah Schweber’s recent post on Math Stack Exchange: https://math....
PW_246's user avatar
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2 votes
8 answers
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Could mathematical truths have been otherwise?

Could at least some mathematical truths have been otherwise? Or are all mathematical truths necessarily true? For example, is it the case that "there exists only one complete ordered field up to ...
user107952's user avatar
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What papers or books should I read in order?

I have been reading literature on modal set theory and am currently reading Putnam's "Mathematics Without Foundations," which is known for being one of the earliest presentations of this ...
유준상's user avatar
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1 answer
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Must constants denote exactly one thing? [closed]

Consider the algebraic equation C2 - 1 = 0 C is a constant. When we solve this equation we report it's solution as C = 1 or C = -1. But that can't be an exclusive or. Thus, we can say C = 1 and C = -...
lee pappas's user avatar
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3 votes
1 answer
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Why does this proof hold?

I'm currently reading Mathematics Without Numbers: Towards A Modal-Structural Interpretation by Geoffrey Hellman, and I'm on pages 26-27. It seems like Hellman is discussing opposition to viewing ...
유준상's user avatar
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2 answers
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Does a mathematical object that does not contradict itself have to exist?

I have recently finished the chapter on constructing the real numbers in my Analysis textbook (via Dedekind cuts). At first the natural numbers, then the whole numbers and the rational numbers were ...
user19213592's user avatar
5 votes
3 answers
1k views

Is intuitionistic mathematics situated in time?

Mathematics, or at least classical mathematics (that is, mathematics based on classical logic), is thought to be timeless. A theorem that is proven to be true in classical mathematics was true always ...
user107952's user avatar
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1 answer
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Logical Pluralism and Anti-Psychologism

Are there any forms of logical pluralism which reject psychologism? It seems that logical pluralism is predicated on the notion that the diversity of logic is attributed to the diversity of thought. ...
GhostRocket's user avatar
7 votes
6 answers
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Why do we have problem of concept of set?

I am reading George Boolos's "The Iterative Concept of Set," and in the first chapter of this paper, he criticizes Cantor's definition of a set as a whole or totality of objects, pointing ...
유준상's user avatar
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2 votes
1 answer
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How can we point to a specific element of an abstract mathematical space that has no distinctive elements with respect to the space's structure?

An example of such a space is a Euclidean affine space. Consider the statement "point O is the origin of the system". How could we clearly specify and convey what point O is supposed to be, ...
jvf's user avatar
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"Entails" in probability propositions: Why are these claims made by the author true?

I've just began reading "Probability: A Philosophical Introduction" by D.H. Mellor. In chapter 1, section 8, he states: ... the epistemic probability of a proposition conditional on ...
AmagicalFishy's user avatar
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4 answers
110 views

If a predicate doesn't determine a set, does that predicate even exist in the first place?

I thought of asking this in the Math Stack Exchange, but then I thought this stack exchange is better. Certain predicates define sets, such as "x is not equal to x". Other predicates do not, ...
user107952's user avatar
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How do you prove mathematical induction without the notion of a set?

EDIT - Peano's axioms for N can't be used to answer this question, because they assume induction. So what axioms can be used? I am thinking the following: P1. x ∈ N iff x=1 ∨ ∃y (x=y' ∧ y ∈ N) P2. 0'...
lee pappas's user avatar
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3 votes
1 answer
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Who was the first philosopher to describe approximation?

Who was the first philosopher to describe what we now call curve fitting or approximation? Pierre Duhem discusses this a bit in Aim & Structure of Physical Theory, pt. 2, ch. 3 "Mathematical ...
Geremia's user avatar
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2 votes
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Need help understanding how certain mathemetical statements across the landscape can seemingly contradict (e.g. Cantor-Hume vs Euclid)

Here are the main components to my understanding on this issue: Almost all of math can be given in a foundation of set theory Different math can seemingly contradict, e.g. in Euclidean geometry ...
J Kusin's user avatar
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4 votes
1 answer
87 views

Is all rigor mathematical in nature?

Is all rigor, mathematical rigor? Or are there other forms of rigor besides mathematical rigor? For example, is there such a thing as philosophical rigor, or scientific rigor? Personally, I think all ...
user107952's user avatar
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2 votes
6 answers
602 views

Is the B-theory of time only compatible with an infinitely renewing cyclical reality?

I'm not a mathematician and I may be misunderstanding some aspects of this concept. According to the B-theory of time, the flow of time is an illusion, and every point in time exists equally. If this ...
Blaxium's user avatar
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10 votes
13 answers
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Can LLMs have intention?

In many movies, you have seen an AI robot moving here and there, doing this and that with an intention. Is it possible that a generative AI-like language model (e.g., ChatGPT) could ever do that? ...
Shriman Keshri's user avatar
4 votes
3 answers
420 views

How do skeptics explain axioms not being arbitrary?

I get infinite regress but surely the axioms of ZFC or arithmetic were not so much chosen as discovered and intuited and thought about. They certainly didn't just grab whatever was around them and say ...
Ehudjd Ejeijr's user avatar
6 votes
5 answers
5k views

Is there a paradox in the proof of Godel's incompleteness theorem?

The Gödel Incompleteness Theorem was a major discovery in modern logic that has consistently attracted the attention of scientific and philosophical circles. However, since the Gödel Incompleteness ...
Zhang Hong's user avatar
1 vote
4 answers
204 views

How can objects be nonexisting?

A square circle. Obviously, this is contradictory, but i feel odd saying it doesnt exist as well. thats not the bestw ay to say it. but, then again, whatg do we even mean in mathematics or logic by ...
Lawrence Lee's user avatar
2 votes
1 answer
56 views

How do numbers and quantities relate in Aristotle's philosophy?

I'm trying to find out how numbers (and other mathematical objects) and quantities relate in Aristotles philosophy. For example if the distance between point A and point B is "287 miles", ...
r0k1m's user avatar
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5 votes
3 answers
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Idealization and Abstraction of Space

I found myself pondering about space and realized that aside from the general notion of dimension (that being the minimum number of coordinates needed to specify an entity in the space under ...
Max Maxman's user avatar
3 votes
1 answer
247 views

Frege's implication theory

In a comment in the answer to a question I previously posted, someone said MacColl asserted that the Boolean translation of "Every X is A" as XA=X does not work due to the "ivalid"...
user1274233's user avatar
4 votes
0 answers
99 views

What does it mean to say that two theorems (provable statements) are 'equivalent'?

sometimes one sees/reads assertions such as "[the bounded inverse theorem] is equivalent to both the open mapping theorem and the closed graph theorem", but taken formally and literally this ...
ac15's user avatar
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3 votes
0 answers
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Knowing-that-we-know in plenitudinous Platonism

SEP background: If every consistent mathematical theory is true of some universe of mathematical objects, then mathematical knowledge will, in some sense, be easy to obtain: provided that our ...
Kristian Berry's user avatar
6 votes
0 answers
60 views

Can Balaguer’s argument we don’t, and couldn’t, have any good argument for Platonism or ficitonalism in math extend to realism/antirealism in general?

Mark Balaguer is a philosopher who advances the position there is one form of mathematical Platonism, that every consistent mathematical object exists, and one form of anti Platonism, ficitonalism. ...
J Kusin's user avatar
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1 vote
0 answers
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Symbolic logic and that which cannot be commited to language? [closed]

Imagine I have a one to many mapping function. Whatever x is it maps it to all other elements except x. So if I have x=1 then it maps it to every number except 1. Now, I claim this is nothing but the &...
More Anonymous's user avatar
-3 votes
3 answers
265 views

A question about the Notion of Limits in Mathematics [closed]

When we say ( Lim{x → x0} f(x) = k ), are we implying exact equality or merely approaching?
HAMDI ABDERRAHMENE's user avatar
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1 answer
120 views

Is there a set theory which implies the interval [0, 1] but no more?

A deductive system (as a collection of judgments and rules of inference) can be used to describe something commonly called a “set theory”. We can imagine a priori there are certain properties we would ...
Julius Hamilton's user avatar
6 votes
9 answers
2k views

Is mathematical truth empirical?

I’m inclined to think that if someone says mathematical truth is empirical, it would trivialize the meaning of “empirical”. For example, to say that we do not know a priori the answer to certain math ...
Julius Hamilton's user avatar
5 votes
3 answers
211 views

Why Do Magnetic Field Lines Point Clockwise Around a Current? [closed]

ANSWER TO QUESTION Moving charges produce magnetic fields. A negative moving charge (i.e., an electron) produces a North Pole which points anticlockwise around a moving current. A positive moving ...
Teragreg's user avatar
2 votes
1 answer
54 views

Distinction between classical essential (primary) and non-essential (secondary) properties of matter vs. modern primary-secondary qualities?

Primary qualities according to modernity (Galileo, Descartes, Hobbes) are qualities that are quantitative/mathematical. Everything else cannot be reduced to mathematics—e.g., a sensible is a secondary ...
ashadow4u's user avatar
3 votes
7 answers
1k views

Can falsehood be measured? If so, would it be continuous or discrete?

This question occurred to me after reading the question "Does it make sense to say that one false scientific theory is closer to the truth than another?" From what I gathered, the only way ...
Hooman's user avatar
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3 votes
3 answers
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Why are pure powers of the empty set insufficient as a definition for ordinals?

I recently discovered a philosophical term that gives expression to a paradigm that had been circling in my head. G. E. Moore discussed the “paradox of analysis”, which is similar to what I think of ...
Julius Hamilton's user avatar
6 votes
4 answers
1k views

Nothingness: What philosophical concept relates to how the empty set is a subset of every set?

Harry Potter and the Deathly Hallows: “Where do vanished objects go?" "Into nonbeing, which is to say, everything," replied Professor McGonagall. "Nicely phrased," replied ...
BCLC's user avatar
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3 votes
4 answers
568 views

What are some critiques of my philosophy about approaching claims of truth using the scientific method? [closed]

mathematician here. I occasionally go down philosophy rabbit holes and end up in some dark mental states. It always stems from examining the foundations of mathematics. As a foreword, I am not making ...
Fraser Pye's user avatar
2 votes
3 answers
394 views

Are all validities isomorphic or equivalent to valid proofs?

Are all validities isomorphic or equivalent to valid proofs? I ask because of the following: Let P be the set of prime numbers ∃x∃y(y>x∧y∈P)→∀x∃y(y>x→y∈P) translates to the following proof: P1....
Lorenzo Gil Badiola's user avatar
0 votes
0 answers
94 views

On the Peano Axioms (Set Theory) [duplicate]

According to Wolfram Math World, the Peano Axioms are as follows: 0 is a number If x is a number, then the successor of x is a number 0 is not the successor of a number Two numbers of which the ...
Lorenzo Gil Badiola's user avatar
5 votes
2 answers
2k views

Can the laws of physics and the constants of nature exist in a fundamental sense without mathematical realism?

Can the laws of physics and fundamental constants of nature exist without fundamental mathematical constants, operators, and equations also existing? In other words, can there be fundamental physical ...
user avatar
1 vote
0 answers
61 views

Can every idea including mathematical ideas be reduced to a series of simpler idea without information loss?

Can every idea including mathematical ideas be reduced to a series of simpler idea without information loss? You would naturally think this is the case since most ideas could be explained using a ...
Sayaman's user avatar
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