Questions tagged [philosophy]

Questions involving philosophy of mathematics. Please consider if Philosophy Stack Exchange is a better site to post your question.

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10
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1answer
261 views

How do we know we get the right answer?

The problem of ontology is one much discussed in mathematical philosophy with much categorization into different schools of thought, but the problem of epistemology seems to be less discussed; ...
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469 views

Infinitely many axioms of ZFC vs. finitely many axioms of NBG

It is known that ZFC needs infinitely many axioms, but NBG (Neuman-Bernays-Gödel set theory) is finitely axiomatizable (as first-order theories of course). But both theories agree completely on the ...
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543 views

Principle of mathematical induction

In his book “Introduction to Mathematical Philosophy” Bertrand Russell seems to reach the conclusion that mathematical induction is a definition and not a principle. In essence he states that ...
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90 views

Codifying ways to think and write about imprecise ideas?

This question will not be about affine spaces; rather those are mentioned here as one of many examples. A vector space has an underlying set and a field of scalars and an operation of linear ...
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153 views

Apparent Arbitrariness in Mathematics

Something about definitions in mathematics has always interested – confused? - me, I call it “arbitrariness in Mathematics” - it's a bad name, but I don't know a better one. Let me explain: 1st - ...
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140 views

Can one define informational content of a mathematical expression?

At least in physicist's thinking, information, vaguely, is something that allows one to select a subset from a set. Say, a system can be in states A and B, we have done a measurement on it (...
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375 views

How much are mathematics driven by applications?

At some point this provocative question came to my mind: Are mathematics mostly driven by applications? I am taking into account some of the comments made to my original question so I want to make ...
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91 views

Arithmetic systems without Induction

It's often said that AC is a controversial axiom and so often in my math classes when it is used a brief comment is made to the effect of "we can prove this without Zorn's Lemma but it's more work". ...
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515 views

Why do the axioms of equality suffice?

In this answer, Henning Makholm axiomatizes the notion of equality as follows: Reflexive axiom, Symmetry axiom and Transitive axiom: The properties we need are the pure equality axioms: $x=...
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237 views

Can the tehniques of higher level mathematics solve most of Olympiad level math problems through straighforward applications?

Working through many Olympiad math problems(pre-undergrad) I've found that simple applications of undergrad math will solve many of them. Does this trend go on? Can it be that Putnam problems are ...
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288 views

Why are divergent Fourier series all so 'HARD'?

I'm not sure if this question is appropriate or even making sense, but I still feel curious: why are every example of divergent Fourier series SO COMPLICATED? It usually takes pages to construct and ...
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looking for good book on the history of formalism

In 1868 Beltrami published a paper ""Saggio di interpretazione della geometria non-euclidea" that seems to have led to the formalist philosophy of mathematics. But what was written exactly what were ...
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90 views

Constructivism implied or not

Let me take up some details in the answer of another question. Submitted by user hyg17: Heading: All real numbers can be expressed as a limit of rational numbers? The question was: Let $C$ be a set ...
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251 views

Finitistic objections to the current mathematical model

I recently read this pdf: Warning Signs of a Possible Collapse of Contemporary Mathematics, and I'm having some trouble understanding the issues it raises. The author says that the consistency of ...
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1answer
84 views

Propositional Logic and Redundancy

There are Philosophical problems with the Material conditional. The Dutch philosopher Emanuel Rutten has written an article about it, titled: Dissolving the Scandal of Propositional Logic? From that ...
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1answer
94 views

Should “not P” be interpreted as “P implies a contradiction”?

From this answer, (...) "not P" should be interpreted as the assertion "P implies a contradiction". Is this the (only/widespread/mainly) accepted definition of ...
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94 views

Is Leibnizian calculus embeddable in first order logic?

We just published an article making what we feel is a plausible case in favor of an affirmative answer in Foundations of Science, see preprint here. The basic argument is that while such a requirement ...
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99 views

On the (Pre-)History of Sheaf Theory

In the wikipedia page on sheaf theory I found the following statement which somehow puzzled me: some of the facets of sheaf theory can also be traced back as far as Leibniz. Could anyone explain ...
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1answer
149 views

Sheaves in Philosophy

I once found a book on google.books. It was about the applications of sheave theory to philosophy or more general to social studies. I don't remember for sure. i just know it was not the book Sheaves ...
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259 views

Strange Consequences of Large Cardinals in Probability

Large cardinal axioms are very strong hypothesizes and as any other strong hypothesis they have many strange consequences in mathematics. On the other hand we know that if we bring even the least ...
3
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1answer
219 views

Errors of Euler interpretation?

To complement the recent post on Euler's errors, I would pose the following question: what common errors of Euler interpetation appear in the literature? What errors are attributed to Euler's work in ...
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85 views

Is there a link between level of abstraction and use of numbers?

One of my friend who stopped studying maths in high school told me once You study maths, can you help me fill my tax forms? In her mind, advancing in maths studies implied manipulating an ...
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302 views

Does the concept of predicativity need to be formalized to go beyond Feferman-Schutte ordinal?

Feferman-Schütte ordinal is sometimes said to be: ....first impredicative ordinal, though this is controversial, partly because there is no generally accepted precise definition of "predicative". ...
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1answer
116 views

What formal proof systems are capable of proving $\forall x \exists y x = y$ without needing to apply $\forall$-I to $\exists y x = y$?

I am interested in some philosophical questions that depend on whether the open formula $\exists y x = y$ is a logical truth. I'm making the assumption that some logical systems are intended, in the ...
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1answer
159 views

Is mathematics a syntax?

I have read that syntax is symbol and semantics is meaning those symbols convey. Is mathematics a syntax? Where is semantics in mathematics? What gives meaning to mathematics, to those symbols I ...
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93 views

Intuitionism and theoretical physics

In the book by Kleene "Introduction to Metamathematics" I have read that Poincare was intuitionist. Nevertheless, due to the fact that I am an undergraduate student in physics, I know that Poincare ...
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49 views

Non-set-theoretical foundations

Nowadays most ideas of foundations are based on some set theories. But are there some non-set-theoretical foundations, I mean are there some ideas of creating a theory which can foundate other math ...
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263 views

Can we hope for an elementary proof of a conjecture of Goldbach?

It is written on Wikipedia: "During the 20th century, the theorem of Hadamard and de la Vallée-Poussin also became known as the Prime Number Theorem. Several different proofs of it were found, ...
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97 views

Insight in the diagonal method as explained by Girard

I'm currently reading "The Blind Spot" by J.Y.Girard, and came across this passage about diagonalization: Is this way of describing diagonal arguments in general legitimate? If not is there another ...
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63 views

How to apply rigor to proof sketches

When doing analysis, I have difficulties to recite the proofs given in the lectures in oral examinations. I am able to do calculations with the theorems that I encountered. (Note that in central ...
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99 views

Not computable (but definable) real numbers, not related to the halting problem?

The existence of not computable real numbers has been eating at me for a long time. However, I have not seen any example of such a number not related to the halting problem. Could any such number ...
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144 views

Different ways to define Kripke structure

The Wikipedia page https://en.wikipedia.org/wiki/Kripke_structure_(model_checking)#Example has an example of a Kripke structure $M = (S,R,L)$; however, others define Kripke structures as $M = (S,R,V)$ ...
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1answer
90 views

Cauchy's real line and math philosophy till XIX

I have to write an essay concerning philosophy of mathematics until the end of $XIX$ century. I've heard that the reason why the Cauchy's theorem (if continuous functions $f_n \rightarrow f$ then $f$ ...
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60 views

How can we recognize if something is a number?

There are formal definitions of various types of numbers; natural numbers, real numbers, ordinal numbers, cardinals etc. And we all regard them as some type of number. Are there properties that are ...
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139 views

“Probable” truth in mathematics

This might be more of a philosophical question, but why in mathematics is the tendency to only accept formal proof as a means of finding out what's true? In the physical sciences there's no such ...
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105 views

Is there any solution to Frege's criticisms of Hilbert's Geometry without the application of Model Theory?

Recently I have come across the interesting debate of Frege and Hilbert regarding the Foundations of Geometry. It seems to me that the main concern of Frege was on the Logical Consistency of Hilbert's ...
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1answer
228 views

Does math have to be learned linearly?

I am asking because often times one doesn't know where to start in math. "Just learn what you need" is very vague and unspecific ... for example, assume I'm a beginner at Algebra and was considering 3-...
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1answer
227 views

Do circles exist

So I was wondering about circles today and if they really do exsist. If you graph a circle in function mode, your equation looks like$$y=\sqrt{1-x^2}$$ Now for simple purposes lets take a portion of ...
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97 views

What if segments are not infinitely divisible?

I almost got myself mixed up I a philosophical discussion again. Somebody was talking about the Planck time and length which are, according to him, the minimal possible time and distance, and how ...
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55 views

What is this process/action called in English?

it is a fairly generate question regarding a terminology. People without science or engineering discipline makes an unfounded claim X, but people with such discipline start with proven facts A, B, C, ...
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3answers
88 views

What is the real being(entity?) the word set or class denotes?

It is somewhat philosophical(at least to me). The question is as above. What is the 'substance' referred to as by the word set or class? Especially how is the thing called class defined? I cannot ...
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1answer
78 views

Can we see all mathematical concepts as (possibly uncountable-time) algorithms?

Note: I am interpreting “algorithms” broadly, to include algorithms that require an infinite and even uncountable number of computational steps. It seems to me that any definition can be seen as a ...
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144 views

What is a fair coin?

The title of this question is almost a retorical question. My point is that there is no way to define probability in a non circular manner. Let's say the probality of getting a tail when tossing a ...
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2answers
136 views

Is there any “good” definition for what constitutes “applied mathematics”?

Is there any "good" definition for what constitutes "applied mathematics"? Wikipedia lists stuff such as statistics, optimization. However, e.g these have certainly "pure mathematical" aspects to ...
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1answer
62 views

Assumptions necessary to justify the method of proof by contradiction.

It seems to me, these assumptions are necessary to demonstrate proof by contradiction: i) Every proposition must belong to $T$ or $F$. ii) No proposition belongs to both $T$ and $F$ iii) If having $...
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38 views

The use of square in maths

The variance of a random variable is defined as $E[(X-E[X])^2]$. In machine learning and linear regressions, loss is sometimes calculated with the squared error. In both cases, the main function of ...
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39 views

Increase in conditional probability for contradictory hypotheses in bayesian confirmation theory?

Although this question has a philosophical slant and my motivations for asking it are philosophical, I'm going to justify asking this in the mathematics stack exchange in two ways: 1) I've asked ...
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93 views

How can one think conceptually about Type Theory when one explains the differences between ZFC and Type Theory?

It's a big question, this I am quite aware of, so please excuse my little understanding on the subject but with reference to the following question, to which I would rather like to extend a little, ...
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87 views

What counts as a witness in constructive mathematics?

In order for a mathematical object to be accepted by a constructivist, a witness of a such object much be constructed. For example, we may let points be witnesses for natural numbers. For a real ...
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Game theoretic foundation of Kantian imperative like thinking

Let's say I'm involved in some project (e.g. humanity) in which I can invest some time, but I have no direct information about the amount of time spent by the other project members. Some common sense ...