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 is zero different from nothing? [on hold]

How is zero different from nothing? Should I travel beyond Earth's atmosphere I do not travel into zero but nothingness. What is the difference?
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Can all mathematical reasoning be translated into traditional logic?

Can all mathematical reasoning be translated into traditional (Aristotelian, syllogistic) logic? It would seem not ∵ one cannot syllogistically establish the validity of the reasoning in the ...
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2answers
96 views

What does the term “mathematical logic” mean?

What is "mathematical logic"? Is it the logic of mathematical reasoning, or is it the claim that mathematics and logic are identical? Also, is "quantificational logic" a particular type of ...
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Nominalist views and contradictions

Given a flavor of nominalism which denies that simple sentences and existential quantifiers referring to mathematical objects are literally true (pretense theory, fictionalism, figuralism, etc.), ...
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101 views

Does Quine's dissolution of the Analytic/Synthetic distinction challenge mathematical realism?

I was surprised to learn that Quine is a mathematical realist (See this interview for example). I always assumed that his "Two Dogmas of Empiricism" and specifically his dissolution of the ...
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What questions or areas in the foundations of mathematics remain active research fields?

By foundations of mathematics I am referring to the mathematical, logical, and philosophical foundations of the subject. I'm interested in seeing which of these have active research going on within ...
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298 views

What does mathematical constructivism gain philosophically?

Constructivists restrict the kind of entities they are willing to let into the mathematical domain; thus, e.g., Leopold Kronecker did not accept transcendental numbers as well as other entities (see ...
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What motivated Gödel to arithmetize syntax?

What were the benefits of arithmetizing syntax for Gödel? What did the arithmetization of syntax allow for Gödel that was otherwise not possible?
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1answer
126 views

What was Leopold Kronecker's philosophical view which guided his mathematics?

Leopold Kronecker (1823-1891) rejected Georg Cantor's (1845-1918) transfinite numbers and sets; he also rejected infinitesimals as well as other mathematical entities - such as transcendental numbers. ...
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What is the philosophical ground for distinguishing logic and mathematics?

I was wondering why the field of mathematics and that of logic are perceived as two distinct fields. Although could be pleased with the intuition that logic is rather meta-mathematics, still would ...
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67 views

For a mathematical realist, is there a distinction between real mathematical objects and constructed mathematical objects?

Mathematical realists believe that mathematical entities exit independently of human minds. Mathematical objects have an objective independent existence, and they are discovered by mathematicians, not ...
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What are computable numbers, and what is their philosophical significance?

What are Computable Numbers? Is computability (or non-computability) some sort of technology-dependent characteristic of numbers (via e.g. Turing Machines)? What are the philosophical implications or ...
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Are normal numbers evidence true randomness exists?

Is the existence of normal numbers evidence true randomness exists, and in every possible world at that? Another mind blowing fact is that most of real numbers are normal, so the normal numbers is not ...
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How should one properly characterize mathematical conclusions?

I am a mathematics graduate student, not a philosophy student, so please bear with me. However, I am interested in investigating what exactly it is that I spend the majority of my week doing! As ...
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Is there a way to avoid Gödel incompleteness affecting mathematics as a whole?

I have been thinking about Gödel's incompleteness theorems and their ramifications for the whole of mathematics. In this question I assume some fixed formal system F expressive enough for the ...
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80 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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87 views

No absolutely consistent foundation for math: logically possible?

Is it logically possible that there really is no absolutely consistent formal system, already discovered or yet undiscovered, that can serve as a foundation for mathematics?
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Are “mathematically possible universes” the same as “logically possible universes”?

I recently watched this interesting interview with physicist Paul Davies. https://www.youtube.com/watch?v=vqZN_LGYHJc In the first couple of minutes he outlines some of the problems with a multiverse ...
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555 views

What was Cantor's philosophical reason for accepting the infinite but rejecting the infinitesimal?

I have begun inquiring recently into mathematical aspects of Georg Cantor's theory of transfinite numbers and sets, which he developed between the years of 1874 and 1897. Throughout his theory, Cantor ...
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1answer
206 views

Reference request + What philosophies are there as to ontology of numbers?

I have interest in ontology of numbers. I know that the two main schools of thought regarding the metaphysical status of numbers are mathematical Platonism and mathematical Nominalism. But I wonder - ...
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Phenomenology of abstraction

I'm looking for philosophical articles / books that try to describe the process of human abstraction, and what it actually consists of, from a first person perspective. Examples of the type of ...
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106 views

Are all the consequences of a science in the science's principles?

"Chaotic" differential equations are very simple principles compared to the more complex consequences of them. For example, the equations modeling the motion of a double-pendulum, ,are relatively ...
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63 views

Logic of inductive inference will free statisticans - why?

I'm doing my best try to understand this excerpt of Efron's article (1998) on Fisher: Fisher believed that there must exist a logic of inductive inference that would yield a correct answer to ...
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2answers
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What is the connection between conscious mind and Gödel's incompleteness in a mathematical universe?

Assume that our universe is a mathematical one, similar to the one that Tegmark proposed (see here). In contrast to what I read there, let's assume that the axioms upon we build the universe are such ...
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Was Kant incorrect to assert all maths as 'a priori'?

Preface: Kant's assertion is rebutted by Prof David Joyce who references non-Euclidean geometry and by the last sentence on Sparknotes which states that 'empirical geometry is synthetic, but it is ...
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If mathematics can predict how nature is what does is say about nature it self?

Using mathematics a lot of 'laws' of nature are discovered, like the Higgsboson gravitational waves Diracs anti-matter etc. But if something like mathematics can predict how nature or reality 'should ...
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1answer
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What is the notion of a proof of a proposition for Martin-Löf?

The notion of proof of a proposition is of the most fundamental notions in Martin-Löf's work on philosophical logic, since it is conceptually prior to the notion of truth - cf "Truth of a proposition, ...
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What is Kant's view of a mathematical object?

I wonder what are mathematical objects - say, the number 1, a circle etc. - for Kant? Do they have some kind of special status for him compared to ordinary (empirical) objects? Where exactly he talks ...
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Impossible triangles, in what sense do they exist?

In what sense do impossible triangles and their properties exist, if they do at all? There is a rule for right triangles which states that the altitude of the hypotenuse can't be greater than half of ...
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Is it appropriate to say that mathematical knowledge has intrinsic value for mathematicians?

I have been toying with the idea of value for a while and came up with this question a few days ago. I think that it is impossible to say that mathematical knowledge has intrinsic value for ...
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176 views

Must mathematical definitions be formal? If so, why?

Recently on Math SE, I offered the following definition of a "pre-function": A pre-function f from X to Y will be defined as a "function" which in general is not well-defined. This was rejected ...
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1answer
106 views

Existence of mathematical objects: how?

In mathematical philosophy, one asks the question "do mathematical objects really exist"? This is then followed by "yes" or "no" answers, but does the question even make sense? Is it even meaningful ...
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Is Tarski's theory of truth compatible with intuitionism?

Intuitionistically, truth is identified with provability: A is true means that it is possible to prove A. In his essay "Intuitionistic logic a philosophical challenge, Logic and Philoshophy" (1980) ...
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Are the assertions “2 + 2 equals 4” and “2 + 2 is 4” identical [closed]

I asked the same question at math.stackexchange but I thought that the answers here could be quite different. I hope I am not breaking protocol/etiquette by doing that, if I am then I apologize. If ...
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Existence of numbers, were they invented or discovered? [duplicate]

Fire is a good example of matter that human beings discovered; Fire has been a part of nature even before human beings found it and at some points and we have used it ever since we discovered the ...
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Hilbert's Sixth Problem: Is Kolmogorov's solution the last word?

The demand for axiomatization of probability was put forward by Hilbert at the very beginning of the past century: it was the sixth problem in his famous twenty three problems he deemed of high ...
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What is the real meaning of a “well-defined” notion? [closed]

I'm not talking about pure-mathematical functions and such, but rather more philosophical and seemingly less formal ideas, such as consciousness.
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Is number π empirical or a priori?

I used the example of π, but this applies to other transcendental numbers as well, such as e Kant classified statements into 4 epistemic categories based on two criteria: The Analytic/Synthetic ...
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What is(are) the importance(s) of formal reasoning

In Mathematics, we as an undergraduate are exposed for the first time (at least for me it was the case) to 'rigor'. For example, in Real Analysis classes we often use logical quantifiers in our ...
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290 views

How can probability statements be falsified?

Have studied recently some about philosophical views of probability and ran into an interesting problem put forward by Popper: According to Popper, probability statements are not strictly ...
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Literature on Mathematical Universe Hypothesis beyond Tegmark?

I'm looking for further reading about the Mathematical Universe Hypothesis. I'm intrigued and would like to know to which degree this has been discussed. I'm aware and have been looking at Tegmark's ...
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Size of a point [closed]

I know this may sound too simple or maybe too absurd to discuss, but I am having a hard time visualizing a point in space! In Euclid's Elements a 'Point' is defined as Something which has no part. ...
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1answer
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How do mathematical Platonists think about formal systems?

I read that platonists believe there are abstract mathematical objects. How do they think about formal systems? For example, Do they discriminate the set of natural numbers N on ZF and that on ZFC? ...
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207 views

Is logic subjective?

If logic is constructed from axioms, and axioms are depended on observation which in term could be subjective, does this means that logic could be limited to our observation, and not really absolute ...
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What actually are meaningless symbols?

Some days ago our professor during the course of his lecture wrote the following definition of a polynomial. We say that an expression of the form a0 + a1x + a2x2 + ... + anxn is a polynomial of ...
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How does Bernoulli's theorem make an inference from chance to frequencies and not vice versa?

I am currently working on an assignment of mine about Frequentism and Bayesianism. In my studies I've found this source: ...
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What is a straight line?

I am not a philosopher; I am an engineer with a reasonable grasp of mathematics. This question has been bothering me for a long time, and I have asked a variation of it to a mathematical community. ...
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1answer
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How would an interpretivist justify using game theory?

It is often said that interpretivist theory does not, in principle, reject quantitative methods. That said, how could an interpretivist defend using a quantitative method (e.g., statistical model; ...
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How can rational choice theory be explanatory?

In his work, John Harsanyi appears to have taken issue with classical social theorists' account of social phenomena. For example, he criticized Max Weber's typological approach on the grounds, "If we ...
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What does Russell mean by “term” in Principles of Mathematics?

Bertrand Russell in Principles of Mathematics defines a term as "Whatever may be an object of thought, or may occur in any true or false proposition or can be counted as one." Can someone elaborate on ...