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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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Can a different universe be built with three dimensions?

Can you theoretically create a universe that will have the same dimensions but will look different? For example, a paper that has a limit, or another similar dimension can it be different?
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133 views

What's so bad about giving up the Axiom of Choice?

The Axiom of Choice (AoC) in set theory famously gives rise to controversial and counterintuitive theorems. (Examples: Banach-Tarski paradox and existence of non-measurable sets.) I'm aware of some ...
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85 views

Examples of theories that assume the existence of an “External Reality”?

In this paper written by physicist Max Tegmark (https://arxiv.org/pdf/0704.0646.pdf) it talks about "External Reality Hypothesis". Specifically, he says: Although many physicists subscribe to the ...
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Is a “truly infinite reality” possible? [on hold]

A "truly infinite reality" must follow these guidelines: While a "truly infinite reality" normally allows the "re-occurrence" of most entities in either exact or self-similar form (like for example ...
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Does Tegmark's hypothesis include dynamical mathematical structures?

Tegmark's hypothesis is the idea that mathematical structures are physical and thus have physical existence (https://en.wikipedia.org/wiki/Mathematical_universe_hypothesis) Zuse's thesis says that ...
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110 views

Relation of Mathematical Propositions to Natural Language

Treating Natural Language as a language game, what role does it play in our understanding of mathematics? Does natural language provide meaning to mathematics? Does a proof of a conjecture, say FLT,...
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369 views

Deductively sound formal proofs of mathematical logic? [closed]

When we specify that True(x) is the subset of the conventional formal proofs of mathematical logic having true premises then True(x) is always defined and never undefinable. When true is defined as ...
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25 views

Help wanted - need descriptor for a partcular type/form of argument

I am writing a paper on cognition, and to simplify my discussion I need an adjective or descriptor for particular category of argument as follows: I am arguing for the necessity of a construct with a ...
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3answers
121 views

Why is 2 + 2 = 4? [closed]

It is clear that 2 + 2 = 4. It is also clear that applying the successor function on 1 yields the next number, i.e. 2, and this operation can be repeated infinitely. This method can be used to verify ...
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77 views

Syntactic VS Semantic Provability

Consider a new Conjecture C. The task is to determine whether the conjecture is true or false. Now let us suppose, after working very hard, we are finally able to establish the truth or falsehood of C....
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2answers
84 views

What's the meaning of this quote of Pythagoras on the good and bad principle?

Simone de Beauvoir attributed the following quote on the good and bad principles to Pythagoras in The Second Sex, page 114 : There is a good principle that created order, light, and man and a bad ...
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Discovery VS Invention in Mathematics [duplicate]

Asking specifically in the context of philosophy of Mathematics, on what basis do we classify or should classify a new expression as a Creation of Mind (Invention) OR a Discovery?
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Why is ZFC not as susceptible to Gödel's incompleteness as was the Principia Mathematica?

So, from what little I have read (such as this answer), it appears to be that one reason why the program of Logicism, as laid out in the Principia Mathematica, failed was that its goals (of finding a ...
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3answers
105 views

Probability calculus and Quantum Mechanics [closed]

I am not an expert and probably this question highlights this. Anyway, is the probability calculus used in Quantum Mechanics? Does the concept of probability adopted in Quantum Mechanics satisfy the ...
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1answer
325 views

Was Kant an Intuitionist about mathematical objects?

In regards to the ontology of mathematics, as far as I can understand, Kant believed that Mathematical objects existed only as features of our perception that influenced how we viewed things-in-...
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677 views

What are the discoveries that have been possible with the rejection of positivism?

I am wondering if the rejection of the positivism movement in philosophy lead to any major discoveries in mathematics and natural sciences? I am thinking it might have been able to contribute to those ...
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The nature of Nominalist Formalism

In this entry in the stanford encyclopedia of philosophy, it is stated that the theory of nominalist formalism deals with the metatheory problem of formalism as follows: Commendably, Goodman and ...
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1answer
252 views

Is there any physics-model version of Tegmark's hypothesis?

Tegmark's mathematical universe hypothesis is very interesting (https://en.m.wikipedia.org/wiki/Mathematical_universe_hypothesis) but it has virtually no support among physicists because it is too ...
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What does this Jacques Hadamard quote mean?

What does this Jacques Hadamard quote mean? The shortest path between two truths in the real domain passes through the complex domain. Is this a philosophical statement? what is its mathematical ...
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2answers
80 views

Must the physical phenomenon of the universe be differentiable?

The use of Calculus for the analysis of real-world phenomenon depends entirely on our universe not only being continuous, but being differentiable. By "real-world phenomenon" I mean things like the ...
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6answers
7k views

Falsification in Math vs Science

In the beginning it was thought that the statement 1+1=0 is false, and necessarily so. However, with the birth of modular arithmetic, it was found that indeed, 1+1 does indeed equal to 0 (in the mod 2 ...
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What was Wittgenstein's argument against Cantor's transfinite numbers and where did he make his objection?

G. E. M. Anscombe had this to say about propositions in Wittgenstein's Tractatus: (page 137) It seems likely enough, indeed, that Wittgenstein objected to Cantor's result even at this date, and ...
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1answer
200 views

Gödel's Results and Philosophy of Mathematics [closed]

Gödel's results essentially conclude that there are True but Unprovable statements in arithmetic. My thoughts are as follows: Axioms form the foundation of mathematics -because we need to assume ...
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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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2answers
198 views

Does philosophy of mathematics affect mathematical research?

I am interested in a special case of the general question about whether the philosophy of X has an effect on the research or practice of X. My special interest is in the area of mathematics. I am a ...
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1answer
168 views

Where to start with the philosophy of mathematics?

This may be a duplicate question. What is the best way to get started with the philosophy of mathematics? Given that I know (from university) the basics that are discussed (Set theory, Russell's ...
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1answer
196 views

Are axioms in mathematics comparable to hypotheses in experimental sciences?

Remark: my question deals more particularly with the axioms of set theory, arithmetic, probability theory, etc. I think the status of the axioms in geometry is clearer. The French fictitious ...
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1answer
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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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1answer
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Could generalization of scientific theories be possible by just adding an ad hoc hypothesis?

In a seventeenth century world the Newtonian model did mostly very well to describe how gravity works in the universe and did well with most empirical evidence of that time. Of course now we know that ...
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4answers
173 views

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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437 views

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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2answers
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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1answer
315 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
212 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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3answers
214 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
166 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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1answer
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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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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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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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98 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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1answer
164 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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164 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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184 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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222 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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4answers
181 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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1answer
201 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?" ...