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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221 views

Hilbert's formalism and game formalism differences and similarities

I've recently encountered differences between Hilbert's formalism and game formalism. They seem pretty much similar in my eyes. I wish to understand in what way does Hilbert’s formalism resemble game ...
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Infinity - Sizes vs Types

Suppose there is a line, infinitely long in both directions. Make arbitrarily "uniform" cuts or "integers". Obviously there are infinitely many of these. And there are arbitrary "lengths" BETWEEN ...
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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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371 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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39 views

Was Russell's purpose in Principia Mathematica to formalize all the possible constructions of logic?

In writing Principia Mathematica, was one Russell's purposes to formally describe all the possible variations of logical concepts and reasoning that can be used in mathematics? For example, suppose ...
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29 views

Would it create difficulties if a proposed model theory had two or more distinct representations for the same predicate? [closed]

If there would be difficulties using such a model theory, then what is an example of a difficulty that would arise? The reason that I ask is that if no difficulties arise, or if the difficulties can ...
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127 views

Are there systems of set theory structured to allow various ideas to be formulated, instead of structured to deduce foregone conclusions? [closed]

When the conclusions are prejudged to be true, and premises are invented to fulfill the criterion of reaching those foregone conclusions, the concepts may be distorted, and there may be a risk of ...
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In classical model theory, how does one represent the one-place predicate R(x) where for all x, R(x) iff (x∉x)? [closed]

If we try to represent the predicate by means of the set of values that satisfy it, then of course we run into Russell's paradox. Now, in ZF, we could simply use the whole domain of the theory, but ...
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186 views

Did early Wittgenstein view mathematics as “sense-less” or “non-sensical”?

G. E. M Anscombe makes the following distinction between Wittgenstein's use of sense-less (sinnlos) and nonsense (unsinnig): (page 163) We must distinguish in the theory of the Tractatus between ...
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169 views

Although Russell's paradox has the virtue of simplicity, is it a distraction from other paradoxes of naive set theory?

Given that Russell's paradox exhibits a contradiction in naive set theory, the interpretation of the binary relation "∈" called "membership" (where the expression "x ∈ m" is pronounced as "x is an ...
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123 views

To connect or to disconnect mathematics and platonism?

How [do philosophers] strongly support or refute the view that: mathematics is a bag of tricks for real-world problem solving; undecidable statements are an irrelevant and harmless side-effect of an ...
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205 views

What philosophical axes did 19th century mathematicians have to grind?

Tim Button's presentation of set theory motivates the subject by providing a history of 19th century mathematics where the notion of limit allowed definitions of the derivative and continuity. These ...
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100 views

Justification for applied mathematics

I'm familiar with some philosophy of mathematics and what does mathematical knowledge mean (Plato, Kant, Frege, Brouwer etc.). And as mathematicians, we establish a set of axioms and work up from ...
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1answer
159 views

Did physicist Eugene Wigner think that every mathematical structure existed as an isolated universe?

I have read that Eugene Paul Wigner thought that all mathematical structures had physical existence. Does that mean that he believed in a multiverse containing all mathematical structures as separate ...
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579 views

A Question Regarding Russell's Paradox

Consider the 'set' behind Russell's Paradox: R = { x | x is a set and x ∉ x } in light of Cantor's definition of set ("aggregate"/Menge) in his CONTRIBUTIONS TO ...
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Why isn't Cantor's diagonal argument just a paradox?

Cantor's diagonal argument concludes the cardinality of the power set of a countably infinite set is greater than that of the countably infinite set. In other words, the infiniteness of real numbers ...
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63 views

How are geometry and space related? [closed]

How are geometry and space related? I am asking in what type of relationship are "space" and "geometry". I can think of the relationship "necessity", but it's a very general relationship. I can't ...
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128 views

Why is it argued that an argument has one and only one conclusion?

Why can't an argument have more than just one conclusion? If we assume some premises and we assume them to be true, then by some inference rules we are sometimes able to deduce more than just one true ...
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212 views

Is mathematics a mental idea?

Is mathematics a mental idea? According to this answer, a mental idea cannot exist without a mind. If mathematics is a mental idea, what does this imply about the laws of physics which can be ...
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2answers
136 views

Poincare says we are born geometric or arithmetic thinkers. Which was Grothendieck and why?

Poincare proclaims that the mathematical continuum originates from the sensible intuition and that intuition by pure number or logic alone could not have given us this notion. Source for the claim: ...
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116 views

How much math must tenured full philosophy professors know?

I'm talking math not logic. I'm referring to full tenured Professors of Philosophy at world famous universities like Oxbridge, Ivy League, Stanford, or MIT. Please be specific and type the math course ...
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309 views

What is model theory?

I have never been able to understand any need or even any benefit of model theory. Both Rudolf Caranp and Richard Montague showed how to encode semantics directly in the syntax. Can you help me ...
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624 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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278 views

Can all formal systems be generalized as specified relations between finite strings? [closed]

https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathematics) In the philosophy of mathematics, formalism is the view that holds that statements of mathematics and logic can be considered to be ...
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1answer
35 views

Kant's Notion of Synthetic A Prioiri as Logical Entailment

Is there something wrong about interpreting Kant's notion of synthetic a priori statements to be logical entailments? I understand, I think, that Kant didn't want to say such statements (e.g math ...
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4answers
336 views

can we reason about logic?

People who study mathematical logic make arguments about logic itself. So it seems that people take for granted an "intuitive logic" (otherwise, how would they form arguments?). So the observation is ...
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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 ...
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111 views

Are there natural language examples of n-ary relations greater than 3?

Binary relations are obvious, and I see the need to have 3-ary relations such as "being in between things". Are there natural language examples of relations greater 3? Edit: without combining binary ...
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1answer
330 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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5answers
187 views

A comprehensive introduction to relationship between math and experience

I am a mathematician with interest in physics and pure logic and exists one problem: the connection between math and physics. Math concerned on pure universal truths and physics concerned on ...
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7answers
532 views

Do whole numbers other than zero actually exist?

Think about counting up: you start from 0. There are many decimals in between 0 and 1, actually, an infinite amount of decimals are there. So in the same way that there is no last number there is no ...
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1answer
185 views

Poignancy because of Gödel's theorems - why?

Why do Gödel's incompleteness theorems make mathematicians so sad? There are complete, decidable and consistent fragments of mathematics like the arithmetic of real numbers, complex numbers, ...
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264 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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293 views

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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1answer
73 views

what is the ontology-ideology distinction in phil of math

Quine proposed a distinction between ontology - the doctrine of what there is - and ideology - the complex terms and predicates expressible in one's theoretical language (though in his 1971 paper he ...
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7answers
268 views

Why does mathematics work in the physical sciences?

Why does mathematics work in the physical sciences? I looked at reddit, and they said that it's not surprising it does, just because that's what it's there for. But there's definitely a question ...
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1answer
106 views

Why is Math not Logic? [duplicate]

So I've heard, "Math is not logic," because logic has no notion of order. However, consider the following argument: There once was a man on a mountaintop. He came down, murdered a villager's cat, and ...
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3answers
605 views

Is the real number line actually real when we construct it?

Intuitionism is akin to constructivism in mathematics but not quite the same from what I can tell. In the usual treatment of the real line, the additional numbers are found between the rationals by ...
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Does Popper's theory of falsification apply to mathematics?

Mathematics is generally & popularly judged a science in the basic duality: science - humanities. As enemies and collaborationists. The border heavily & fiercely policed. However, it seems to ...
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Minimalist philosophical assumptions to reason, deductively/inductively

I've been thinking a lot about foundations of science recently and I was wondering. Are there existing books/essays/works, or were there attempts that try to achieve the following: With a ...
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4answers
163 views

Mathematical proof of a philosophical theory

Can I prove a philosophical theory mathematically? If yes? How? For example, can the theory of materialism be proved mathematically?
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189 views

If we assume logic is correct, does it imply that our consciousness proccesses real information?

[major edits] Even if our consciousness is an illusion (even in the sense Denett suggests), the mere fact we see some information flowing across the universe means there is at least something that ...
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106 views

How can you determine if a hypothesis (mathematical logic ones) is falsifiable enough to be “good”?

We had a group discussion and the prof gave us the following question and left. The problem is that I hardly understand the question. How can you determine if a hypothesis (in particular, ...
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What are numbers? [closed]

What are numbers? Does the number two exist? If so, how?
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5answers
291 views

Can a solid theory ever exist without any axioms?

In math, numbers and addition are logically defined by Zermelo Set Theory, a small group of axioms upon which everything else can be built. Could it be possible to have a working theory, (in any field ...
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79 views

Help with an existential natural deduction proof

From the assumption ∃x∃y R(x, y) I need to derive the conclusion ∃y∃x R(x, y) From the comments: I tried to use Existential Elimination but I can't figure out how to do it properly.
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566 views

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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Why do we need geometry for pure math?

Karl Weierstrass had a very interesting critique of Riemann's work. Supporters of Riemann, claim that a pure logician would never have been able to see the things that the "geometric imagination" of ...
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Mechanics of Perception

How is perception formed? By perception I mean 'thought' or 'idea' of the World. What I see by itself does not contribute anything to thought. Only an acknowledgement can contribute to structuring of ...
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Question on Hypothetico-Deductive Method

I had another quiz related Hypothetico-Deductive (HD) Method. I couldn't answer this because the way it was posed is so baffling to me. I am so sorry to ask all the basic questions (I think all ...