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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Infinity recursive simulation on a turing machine?

Do you think, that it would be possible to run a infinite and recursive simulation of the universe on an turing machine?
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Prove a logical formula is equivalent to the contradiction if and only if the set it describes is empty

Let ψ be a well-formed-formula (wff). Prove that (ψ ≡ ⊥) ⇔ {x:ψ(x)}=Ø that is, the formula ψ is a contradiction if and only if the set it describes has no members. Note This question is not about ...
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Calculus for Higher Order Logic

Famously, there is no complete calculus for higher order logic that is effective. Therefore, given the rules and axioms of a effective calculus for higher order logic, we must find formulas that are ...
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Zeno and the denial of plurality

Zeno is well-known as the storyteller of Achilles and the Tortoise and how the tortoise never catches Achilles; which is against our experience; the question of how to square these two notions ...
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Are there more numbers than numerals?

If the universe is finite and numbers are infinite, would that mean that there are more numbers than numerals to name them?
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Is Brouwer's notion of time that of a continuum?

Brouwer formulates his foundational philosophy using a single a priori notion, that of time; expressing the kernel of his idea thus : (neo)intuitionism considers the falling apart of moments of ...
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Lambda calculus and its philosophy [closed]

I am aware that λ-calculus is more expressive than FOL. Speaking philosophically, what are lambda abstracts good for?
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Best philosophy of mathematics (text)book?

What work (or works) are best for beginning an independent study of the philosophy of mathematics?
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Mathematical intuitionism, possibly

Mathematical intuitionism asserts as a vital principle that it's not enough to assert that something exists, it must be constructed - that is shown. What does this mean for the truth value for a ...
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What is the Philosophy of Mathematics?

Basically, what exactly is Philosophy of Mathematics? Why not simply call it mathematics? At what point does a mathematician practice the philosophy of mathematics?
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Are there any strong reasons to still consider logical monism or a “One True Logic” in light of all the non-classical logics that have been developed?

I know there has always been some debate concerning whether or not a certain logical system (like classical logic) is the correct one, especially when it comes to propositional claims about the ...
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Formalism: philosophy of mathematics

Is it a contradiction, if I am a formalist but I think that mathematical objects are created by human mind ? In other words: Am I allowed to be a formalist and to believe that ...
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What form of formalist am I? [duplicate]

I am an formalist in the sense that I think that mathematics is just manipulation of symbols. But I think that this manipulation is motivated by the phantasy of humans: mathematical objects are for me ...
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Law of excluded middle: mathematical philosophy

Is it a contradiction, if mathematical objects are for me "mental constructions" -- like in intuitionism but I accept classical mathematics and the Law oft excluded middle ?
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Law of excluded middle in intuitionistic formalism

Does intuitionistic formalism accept the law of exluded middle?
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Mathematical philosophy: intuitionistic formalism

What is intuitionistic formalism? Tarski called himself an intuitionistic formalist.
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What qualifies something as “math”?

What classifies something as math? Is "math" simply performing operations with a certain set of axioms in mind? Is "math" anything that involves numbers? What about mathematical logic? Google ...
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Looking for references for some remark of Quine's

I'm looking for a comment I think I remember Quine having made. He's talking about our understanding of proofs. I think he says something along the following lines... If you understand many different ...
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Question about Imre Hermann's book Parallelismes

Apparently, in his book Parallelismes, Imre Hermann discusses Hilbert, Brouwer en Russell from the viewpoint of psycho-pathology. Does anyone know whether the entire book is about ...
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How does infinity of numbers and of space work?

Is infinity just continuous generation of numbers, or can space be actually infinite? If it is finite can we see it expand if we went to the edge? When I say "I am counting to infinity" does it mean ...
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Is a proof still valid if many people say that is true? [duplicate]

A proof is some explanation to convincing others that a statement is true (or false in case of a counterexample). As Yuri Manin once wrote: "A proof becomes a proof only after the social act of ...
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Definition of “proof”

I read a book of logical puzzles once and there was a little story in it (fictional) about some boy who received an F on his geometry test because his professor said his proof that all angles of a ...
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How to define a number [closed]

What is the positive real number (say less than one) that is not a rational nor an irrational number? I have encountered a mathematical problem that confused me about the definition of real ...
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How should one interpret modern mathematics if one doesn't believe in infinity?

I am an ultrafinitist. http://en.wikipedia.org/wiki/Ultrafinitism I don't believe there is a such thing as infinity. To me, it is obvious that there has to be a largest number; I just don't know what ...
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Introduction Modal Logic with emphasis on metaphysics - free will particularly - and also mathematical logic

I'm reading Timothy O'Connor's 'Persons & Causes The metaphysics of free will' and the first chapter briefly goes into (though without introduction) the application of modal logic and ...
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What are the properties of Mathematical Objects?

I have been thinking a lot about how one knows when an observation contains mathematical elements. Many years ago when I was in school, I found that there was often little time taken out to discuss ...
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Nature of Persons and Mathematics

First, let me define 'Person' as anything complex enough to perceive truths about reality in the level that we do. Now, i'm gonna provide two mutually exclusive options about the nature of persons 1 ...
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Does anyone know pi? Do computers know pi? [closed]

I'm sure that the former's been covered in the literature, what about the latter? Does anyone know the value of pi? Can computers? Allow me to explain... Contrary to popular belief and what ...
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How does Russell's argument 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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Is logic built on assumptions?

I'm sorry if this sounds like a stupid question, but how can we know that our logical approach to ideas is not in itself based on assumptions. For example, how can we know that the workings of the ...
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Mathematical theorems and science

We know mathematics is applied constantly in science to solve problems, even if sometimes (e.g: modern physics) there is not always a intuition of why such theorems can be applied (since there isn't ...
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Philosophical interpretation of computability of a finite math problem

There is an interesting debate in the area of Enumerative Combinatorics, a branch of Mathematics. Several mathematician are having a somewhat tongue-in-check debate whether a certain (very large and ...
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Does Math, or analogically Language really have any impact on our “Thoughts”?

Here I see many say, language has an important impact on our thoughts. But according to this question, Foucault in the preface to The Order of Things wrote how he 'laughed out loud' when he ...
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In what way can deductive arguments be held to increase knowledge? [closed]

By definition, the conclusion of any deductive argument follows directly from the premises. For example, consider the following famous syllogism: Premise 1 - All men are mortal. Premise 2 - ...
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Type theory and metaphor

In my experience, textbooks and introductory material on type theory (or constructive logic systems) are remarkably devoid of metaphor. I never found any introductory text in those fields that ...
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Does the mind work with unconscious mathematical underpinnings?

That is, is our mind geared to use math without our knowledge? Reading academic writings from the biological and social sciences, I observed patterns of methods widely used in maths/statistics such ...
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Mathematical Inventions [closed]

I have been shown some interesting results recently, and i want to see whether it is possible for me to change my point of view regarding the ontology of mathematics ( which now sees mathematical ...
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Can there be ugliness in the world of a Mathematical God? [closed]

Let us assume that Mathematics is infinite, represents the multiverse and beyond, and is the deterministic cause of everything we know to exist (the Big Bang and our universe, the formation of the ...
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How does GEB support AI?

Typically, Godel's Incompleteness theorems have been used to argue against the possibility that the human mind is essentially equivalent to a formal system. However, in Daniel Dennett's book "Darwin's ...
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Let A be a finite non-empty set and S a finite symbol set. Show that there are only finitely many S-structures with A as the domain [closed]

Let A be a finite non-empty set and S a finite symbol set. Show that there are only finitely many S-structures with A as the domain Let k be the number of elements in A, for all constant symbols c ...
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How can teleological explanations not fit with modern science?

Source: Prof Michael Sandel, Justice: ..., Episode 09: "ARGUING AFFIRMATIVE ACTION 52:21: We grew up and and we’re talked out of this way thinking about the world. 52:30: But here's a question: ...
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Can a point of space be identified?

Consider a single particle in empty space; by what argument can we say that it always occupies the same position or place? Space itself has no identifying mark or label being everywhere the same. ...
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How can you translate the mathematical statement 5=5 into a second order symbolic logic statement?

Hi I'm trying to describe the statement 5=5 using symbolic logic. I initially attempted to describe this into English as: There is such a thing x and such a thing y, such that x is equivalent to ...
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What are the historic stances on the epistemological status of mathematics?

I know that Plato and Kant thought it was synthetic a priori (although Plato would not have phrased it in that way). What other major thinkers have weighed in on this issue, on both sides of both the ...
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What did Poincaré mean when he said “Mathematicians do not deal in objects, but in relations among objects”?

The mathematician Henri Poincaré had said, "Mathematicians do not deal in objects, but in relations among objects; they are free to replace some object by others so long as the relations remain ...
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Is this enough to conclude that G is false?

Let us assume a logical system with only 3 axioms (laws of thought): Axiom 1: A statement that is true will remain true till a change is made in the system. Axiom 2: A statement may not be true and ...
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Is graph theory a good model for Seven Bridges of Koenigsberg?

I've asked this question on MathSE, but apparently people over there don't like philosophy. Seven Bridges of Koenigsberg is the problem whose solution (by Euler) gave a rise to graph theory and ...
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What is the minimum number of axioms required for a system of axioms?

What is the minimum number of axioms you need, apart from definitions and usage of the notation, such that you have a system that does not contradict itself? I would just think that the answer is ...
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Is there a poset based/category theoretical definition of God?

As a current atheist, who was a former theist, I feel that God is not a logically incoherent concept. However, many definitions of God, especially those that involve omnipotence, omniscience etc. are ...
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The belief that everything is expressible in mathematical terms?

For want of a better word, mathematicism will be defined as the belief that everything is expressible in mathematical terms. I'm not sure if this is a position that anyone affirms, as my thoughts on ...