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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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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101 views

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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255 views

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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78 views

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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156 views

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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186 views

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 ...
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What makes generalization of a mathematical notion correct?

As far as I know, the notion of open set has been formulated after the notion of open interval in the real line. Was the goal of generalization to allow the definition to work in higher dimensions? ...
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Should I trust mathematics?

First of all I'm not an expert in this field, please correct me if I'm lacking relevant knowledge here. A few hundreds years ago mathematics was largery based on intuition. People realised we need to ...
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On a question of likeness as elaborated by Parmenides

(This question is a tribute to Grothendieck died recently, and whose obituary is here he was amongst the first to theorise on higher morphisms - likenesses). Socrates in a dialectical debate with ...
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considering the line and circle as not just a contrary, but as a extremes on a continuum

Question: In Greek philosophy, it is generally taken that the line and the circle form a contrary. For example in Aristoteles Physics generally takes that motion can be formed out of this contrary, ...
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Is knowledge a form of energy? [closed]

To better describe my question, do the following experiment: Calculate x=12+26+67+71 Now you might have spent some time in getting the answer. You burnt sugar, you used up energy to get the ...
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Mathematical Platonism vs Platonic Platonism

According to the summary of Platonism (ie the Forms) by Aristotles Metaphysics: Besides sensible things, and the Forms, there are mathematical objects; of the first (the sensible) they share in ...
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Is the inconsistent (or paraconsistent) line a possibility?

According to the SEP: Another place to find applications of inconsistency in analysis is topology, where one readily observes the practice of cutting and pasting spaces being described as ...
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488 views

Is mathematics as pure as originally thought?

It is said that theorems in mathematics cannot be proved or disproved by experimentation. I assert that if a statement is decidable, then it can be proved or disproved in a finite amount of time by a ...
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264 views

What is the number 2?

My friend told me that he took a course in the philosophy of mathematics and said that they defined the number 2 to be "the set of all sets with two elements." I may be remembering wrong, but this is ...
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If a cone is divided in a plane parallel to its base, are the surfaces produced by the cut the same or different in size?

Democritus of Abdera, the ancient philosopher of Greece, reasoned over two thousand years ago: If a cut were made through a cone parallel to its base, how should we conceive of the two opposing ...
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Can we add Pi to Pi [closed]

Can we say we can add Pi plus Pi? (from http://deepturtel.blogspot.in/2015/01/pi-plus-pi.html): Are the rules of addition in Maths defined for such a process? Or do we always add only approximations ...
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Mathematical versus philosophical reasoning (and the mathematics of philosophical arguments)?

What is the difference between mathematical reasoning and philosophical reasoning and why isn't philosophy just considered to be a branch of mathematics? Is any study not a branch of mathematics ...
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How historically grounded is the standard narrative of the Irrationals in Antiquity?

Its commonly said that the Pythagoreans were unbalanced by the discovery of the irrationals; since their philosophy was predicated on ratios; ratios of two finite numbers. Still, it is natural to ...
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Input/output in 'mathematical' programming languages [closed]

More than once I have observed this: A person describes a functional programming language (as opposed to a programming language that makes heavy use of interspersed states), that person will say it is ...
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Creativity in mathematics [duplicate]

Hello can anyone especially e.g. an actual mathematician, from the horses mouth, explain what role this has in mathematics, both through the whole course of learning skills and mathematics, and ...
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Is mathematical platonism compatible with Platonism?

When calling themselves "Platonists" mathematicians usually mean that they feel they discover ideal facts that eternally exist in some way. My question is if this sentiment is consistent with Plato's ...
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What does Deleuze mean by “the world can be regarded as a 'remainder'”?

Deleuze in Difference & Repetition writes in the chapter named as the Assymetrical synthesis of the sensible: It is therefore true that God makes the world by calculating, but his calculations ...
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What is the intersection of Physics & Philosophy?

(I've already asked this question on Meta, but as one answer (by Joseph Weissman) pointed out this is already a philosophical question; so I thought it worth asking here). I've asked a number of ...
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Dual of identity relation?

Does anyone have any intuitions about what the dual of the identity relation might be? I.e. is there a 'natural' concept expressed by a statement such as 'it is not the case that a is not identical to ...
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Countable additivity

Graham Oppy writes in »Philosophical Perspectives on Infinity«: Without countable additivity, it seems – for example – that we must lose the result that an arithmetic sum of an infinite series is ...
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Why does Aristotle suggest One is not a number?

Parmenides showed Nothing is not the same as Zero; the second is a number, and the first is not, in more than one sense; it also differs from the Buddhist notion of Sunyatta, which is nothing in a ...
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What is one divided by zero?

The Pythagoreans were (probably apocryphally) disturbed by the discovery of irrational magnitudes; its useful pointed out here, that irrational means, in one sense, and not now generally alluded, ...
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Mathematical reductionism: senseless

Does it make any sense to ask if logic can be reduced to math? Truth be told I have no idea what the inverse logical reduction could look like. Naturally I'm familiar with a kind of reductionism, in ...
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Should the easiness with which math is applied to the world be a surprise?

I study physics at an undergraduate level. Since early on, I've was a person who thought math was 'logical' and as such, its applications to the world aren't really a surprise since math is so ...
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259 views

What purpose do mathematics and philosophy serve epistemologically (compared to sciences)?

Kant did not consider them sciences, but meta-disciplines that study a priori conditions of doing science. Indeed, both mathematics and philosophy permeate all empirical sciences to varying degrees, ...
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Are there any naturally occuring non-embedded manifolds?

Mathematicians are always insisting that manifolds need not be embedded, and by Occams Razor, are best thought without the surrounding ambient space. For example, the surface of a table is a 2d ...
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Do mathematician always agree at the end?

I know it's a off beat question but I thought philosophical answer would be better. I've been trying to study some different sciences in my life, ranging from biology to mathematics, and if I try to ...
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Do Mobius Strips have a front and back?

Mobius strips, are generally held to have only one side - if one marks a place on the strip and on what looks like the other side then a pencil can draw a line between these two points without ever ...