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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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3answers
2k views

Relation of Gödel's incompleteness theorems and Karl Popper falsification

Falsifiability is considered a positive (and often essential) quality of a hypothesis because it means that the hypothesis is testable by empirical experiment and thus conforms to the standards of ...
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2answers
57 views

Is it a logical fallacy? A question about majority opinion and samples

Say a community of X number of people is notified of a sudden policy change by higher authorities, and Y number of people from the community express their opinions for or against the policy change. ...
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8answers
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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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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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2answers
75 views

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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1answer
179 views

Ur-primitive notions of logic and mathematics and especially Ur-primitive notions of objective existence of abstract entities [closed]

I am very puzzled about how mathematics came to be as it is in its modern form, where it is extremely embellished to depend on formal rules of logical inference and on the notion of axiomatic ...
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5answers
408 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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15answers
12k 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? ...
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0answers
141 views

Is this a good answer about about the meaningfulness of platonism? [closed]

Is the following a good answer to the question(s) posed at the following site? Is Mathematical Platonism a meaningful thing? Namely, here is my proposed answer: @Ajax, I'm not sure this will help, ...
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1answer
144 views

The notion of knowledge for Kant and mathematical objects

As far as I understand the notion of knowledge in Kantian philosophy, we cannot speak of knowing something unless there is a relation between its concept and some object of intuition in experience. ...
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2answers
147 views

Max Tegmark's Mathematical Universe

Max Tegmark believes the universe to be a mathematical structure, and he further claims any mathematical structure with self-aware substructure will perceive itself in a physical world. What exactly ...
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4answers
121 views

Why there is geometry in nature? [closed]

We perceive our surrounding as a 3 dimensional world. Geometry is part of math which are a collection of abstract concepts that arose in our mind. Most of the things around us can be described in ...
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3answers
186 views

In mathematics , can't we say that the intuition of the original author of a true conjecture is the proof of it just indescribable on paper- pen

I think that when a person put on a true conjecture in mathematics (we have verified it to be true) ,proof must came in his mind in the form of what we called intuition. I mean if his saying is ...
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2answers
148 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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2answers
180 views

Is Mathematical Platonism a meaningful thing? [closed]

How can we be sure that asking If Mathematics is Platonic or If it is Our Own Construction is a meaningful thing to ask, and not just abuse of natural language? Can somebody provide a convincing ...
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1answer
139 views

Why should a system of set theory represent the following property of x: ((x = h) or (x = k))) using the set that Frege used?

I have been considering the possibility of an approach to set theory that represents the following property of the free variable x, in the terms of parameters h and k: ((x = h) or (x = k))) by means ...
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3answers
211 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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1answer
67 views

Are there any systems of mathematics that permit such a wide range of ways to formulate ideas

... that there is no algorithm for determining whether or not a given sequence of symbols is a wff ("well-formed formula"), but instead non-trivial proofs are required, so that some sequence of ...
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0answers
135 views

Is there an existing, published philosophy of mathematics that was designed to avow or declare all of the following four points of view?

Just as medicine is about human health, and astronomy is about the universe beyond the Earth's atmosphere, with the defining nature of medicine not white coats or stethoscopes, and the defining nature ...
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1answer
33 views

How could I possibly apply rule-utilitarianism in real life?

If I believe in rule-utilitarianism as the best moral system we have to judge what actions are right and wrong, how would I go about applying this system pratically in real life decisions? For ...
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0answers
64 views

What goes wrong if we explain Russell's paradox as resulting from an overly rigid link between property satisfaction and elementhood?

An excerpt from a question at Math SE (Bounty of 100 available for an answer): What goes wrong if we explain Russell's paradox as resulting from an overly rigid link between property satisfaction and ...
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1answer
316 views

What is going on with Russell's “beauty cold and austere” of mathematics?

I admit that this is an idle question, but I wondered why it is that mathematics appears "beautiful cold and austere" to those who are particularly gifted at it. The full quite from wikipedia on this ...
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1answer
192 views

Are there recent coherence theory of truth for mathematical truths?

Are there any recent works (papers, books, etc) in philosophy of mathematics where it is given an account of mathematical truth in terms of a coherence theory of mathematical truth? I am interested ...
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3answers
171 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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4answers
350 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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5answers
246 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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1answer
224 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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1answer
93 views

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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0answers
42 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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3answers
181 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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2answers
214 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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0answers
104 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
161 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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5answers
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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7answers
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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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2answers
65 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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3answers
133 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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2answers
123 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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314 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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2answers
631 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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1answer
279 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
37 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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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 ...
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3answers
114 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
333 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
192 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
554 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
187 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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2answers
268 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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3answers
306 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 ...