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

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What is the physical significance of arithmetic operations?

Here is an example of what I mean by physical significance: When we use some geometric or trigonometric identity, let us say Pythagoras' theorem to calculate the length of the diagonal of a field, ...
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10answers
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How far can one get in analysis without leaving $\mathbb{Q}$?

Suppose you're trying to teach analysis to a stubborn algebraist who refuses to acknowledge the existence of any characteristic $0$ field other than $\mathbb{Q}$. How ugly are things going to get for ...
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Physical meaning of autonomous system of ODE. [on hold]

According to Wikipedia: Many laws in physics, where the independent variable is usually assumed to be time, are expressed as autonomous systems because it is assumed the laws of nature which hold ...
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Is it possible to distinguish rest and movement in hyperbolic universe?

Imagine a large body (for example, a planet) in 3D hyperbolic space. Now imagine the planet starts moving in a straight line at constant speed. In Euclidean space, all points would move along ...
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4answers
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As of August 2015, is the “set” of all gold medalists in the 2016 Olympics a set?

As of August 2015, is the "set" of all gold medalists in the 2016 Olympics a set? I think it is since the defining property is very clear. However, given any $x$, we do not know if $x$ is in this ...
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Is mathematics one big tautology?

Is mathematics one big tautology? Let me put the question in clearer terms: Mathematics is a deductive system: it works by starting with arbitrary axioms, and deriving therefrom "new" properties ...
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10answers
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Problems that are largely believed to be true, but are unresolved

Are there unsolved problems in math that are large believed to be true, but for reasons other then statistical justification? It seems that Goldbach should be true, but this is based on heuristic ...
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1answer
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Is a derivation a proof?

Is there a difference between "derivation" and "proof"? I imagine a derivation is a type of proof but that proofs are perhaps more general. Although then again, I suppose every proof should be ...
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35answers
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Do complex numbers really exist?

Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious ...
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Why do we not have to prove definitions?

I am a beginning level math student and I read recently (in a book written by a Ph. D in Mathematical Education) that mathematical definitions do not get "proven." As in they can't be proven. Why not? ...
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1answer
48 views

Proof of identity A = A or 1 = 1

Is 1 = 1 an assumption? I feel it's a very good assumption, but is there a proof for it? Imagine a world where people were contesting it, where equivalence wasn't a common sense concept. In reality no ...
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Is there a variational principle explaining the distribution of prime numbers?

Imagine the set of positive integers $n\geq 1$ as a system $n\Rightarrow n+1$, in this set for each $N>1$, with euclidean algorithm or using Erathostenes sieve, we can find all prime numbers $p\leq ...
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Plantinga's logical argument for mind-body dualism [closed]

Some may feel this is not appropriate for the mathematics stack exchange, but it is a question in logic, and I feel it is entirely a good fit. The following argument has been put forth by the ...
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2answers
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About the cardinality of natural numbers [Solved]

I had learned that the set is countable if and only if it is finite or countably infinite. We know well that the set $\mathbb{N}=\{1,2,3,4,\dots\}$ is an infinite set. In order to find out if the ...
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5answers
213 views

What exactly is real number?

This question may sound philosophy, but it has been bothering me for a very long time, therefore I have to ask it here. The story goes back when my first time reading Apostol's Calculus, then I had ...
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2answers
87 views

Can math be learned backwards? [closed]

In C++, we can reverse engineer and performance binary analysis to know exactly what a piece of binary will do, even without seeing the original source code. In math, can this be done? Basically, can ...
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1answer
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Math is no more valid than string theory or fan fiction? [closed]

I heard from a math expert that stated maths are not a single bit more accurate or valid than fan fiction or string theory. This was in a discussion regarding the philosophy of math and whether math ...
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1answer
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Can the generalized continuum hypothesis be disguised as a principle of logic?

A cool way to formulate the axiom of choice (AC) is: AC. For all sets $X$ and $Y$ and all predicates $P : X \times Y \rightarrow \rm\{True,False\}$, we have: $$(\forall x:X)(\exists y:Y)P(x,y) ...
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1answer
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Finding unique rules for a finite number of initial steps, using Information theory

Is there a unique way to determine which rule provides the sequence that matches a finite number of initial steps, choosing the rule that needs the least amount of information to be described? ...
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Complex analysis is more “real” than real analysis

In physics, in the past, complex numbers were used only to remember or simplify formulas and computations. But after the birth of quantum physics, they found that a thing as real as "matter" itself ...
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1answer
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Troubling questions about probability

Suppose we have some random phenomena. Is it true that any event concerning the phenomena has a fixed "correct" probability? That is, the correct probability is the relative number of occurrences of ...
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2answers
69 views

From the perspective of the multiverse theory, would maths “work the same” in every possible Universe?

I've had an interesting discussion with a friend recently and I was arguing that in every possible Universe, mathematics would always have to work the same, i.e. $1 + 1 = 2$ would have to be true for ...
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1answer
336 views

Exactly who popularized the modern definition of domain and codomain of functions?

In Whitehead and Russell's Principia, domain is the referents of relation; converse domain is the relata. Modern function in mathematics is just one special case of relation whose referent is unique ...
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How do we distinguish between characteristic 0 and characteristic p for very large p?

This is a somewhat soft question, apologies if it turns out to be trivial/nonsensical. Background: I was half-asleep one morning, not quite through my first cup of coffee, and thought about the ...
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1answer
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Proving a theorem of logic

At the moment I'm going through a book which treats logic in a very rigorous axiomatic way. But I just got stuck in this theorem that I can't seem to be able to solve (I'm still trying hard). The ...
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4answers
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How do I convince someone that $1+1=2$ may not necessarily be true?

Me and my friend were arguing over this "fact" that we all know and hold dear. However, I do know that $1+1=2$ is an axiom. That is why I beg to differ. Neither of us have the required mathematical ...
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When a function is a dimension?

The concept of dimension is used in many different contexts. Generally a dimension is a function that has as domain some family of sets ad has value on a set that, in the most common situations, is ...
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What is a negative number?

I'm trying to get to an abstract definition of a negative number that would fit in with the basic concept of addition/subtraction. There are questions here about multiplying and dividing negative ...
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1answer
174 views

Since arithmetic has a model (thus it is consistent) why care if consistency can't be proved?

Since arithmetic has a model, the numbers as we know them, it is consistent. Why do we care if consistency can't be proved within arithmetic? Do I miss something, ie in what we can consider a model?
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How to interpret material conditional and explain it to freshmen?

After studying mathematics for some time, I am still confused. The material conditional “$\rightarrow$” is a logical connective in classical logic. In mathematical texts one often encounters the ...
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3answers
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Numbers and reality [closed]

I have a question I don't really know how to formulate, so apologies for the cloudy mess. The topic is on the meaning of numbers, that is when I say 3, I am referring to, say, $3$ avocados, or $3$ ...
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0answers
279 views

Gödel's Completeness Theorem and logical consequence

At the end of a long process of "rumination" on "old" math log textbooks, I've found the "missing link" - from my personal point of view - between some issues I've raised in the previous months : (i) ...
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2answers
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What are the uses of cross-theoretic identifications within mathematics?

I've been thinking about the identification of objects from different mathematical theories. For example, when you do set theoretic constructions of the natural numbers and identify, e.g., 0 with the ...
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What do people mean by “finite”?

Many arguments about the foundations or philosophy of mathematics centre on the question of whether or not there exist objects or entities (such as certain sets) which are not "finite". (For ...
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1answer
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Presentation of a group question

So I know that given a presentation of a group $G$, one can derive from the relations of the group presentation any element in the group $G$ right. However, I do have some confusion. If we take ...
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1answer
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Do second-order categoricity proofs require a background concept of set?

In his article "The Set-Theoretic Multiverse", Joel David Hamkins (as part of his reply to Donald Martin's argument that the set-theoretic universe is unique, found in "Multiple Universes of Sets and ...
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Is a proof still valid if only the writer understands it?

Say that there is some conjecture that someone has just proved. Let's assume that this proof is correct--that it is based on deductive reasoning and reaches the desired conclusion. However, if ...
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Structuralist slogans

I am afraid to make a bad impression by misusing this forum but I am looking for as-many-as-possible mathematically inspired formulations and references to one (sometimes vague) idea. The idea is ...
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2answers
75 views

particular property and completeness?

I was puzzeling with the almost standard definition of completeness: In mathematical logic and metalogic, a formal system is called complete with respect to a particular property if every formula ...
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An adequate difference between $\forall x\in A:P(x)$ and $(\forall x)(x\in A\rightarrow P(x))$?

Ever since I was a young student I have felt doubts about the traditional $(\forall x)$-expression: starting a statement with such an irrational lack of focus doesn't seems reasonable! I mean, all $x$ ...
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Naturality in linear algebra

Question. How can we formalize these intuitions about predicates on matrices? Let $P$ denote a predicate on matrices, so that $P(A)$ is true for some choices of matrix $A$ and false for all ...
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Imagening the Thurston geometries

I can (more of less) imagine how it would look if space was Euclidean, spherical of hyperbolic. But there are 8 Thurston geometries see https://en.wikipedia.org/wiki/Geometrization_conjecture how ...
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Which mathematical ideas most influenced the way you think?

This is not a question about how you use a formula or mathematical method to solve quantitative problems - that is applied mathematics. Rather, I'd like to hear how deeper ideas gained through the ...
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Study of all published works of Bertrand Russell on foundations of mathematics: Please recommend his works.

Study of all published works of Bertrand Russell on foundations of mathematics: Please recommend his works. I think Bertrand Russell was a special mind and I set a goal for myself to study all his ...
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Why can't differentiability be generalized as nicely as continuity?

The question: Can we define differentiable functions between (some class of) sets, "without $\Bbb R$"* so that it Reduces to the traditional definition when desired? Has the same use in at least ...
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2answers
664 views

Who stole the axioms in Natural Deduction?

The study of Gentzen's sequent calculus give me the opportunity to propose some reflections about the concept of logical truth. I'll refer to the english edition of Gentzen's works : The collected ...
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2answers
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What is mathematical definition of a fluid?

I am searching the precise and mathematical definition of a fluid for a long time but I did not find it anywhere. What I mean by precise and mathematical can be understood by the following: There is ...
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Why is it important, that mathematics can be formalized in set theory?

Why is it important, that mathematics can be formalized in set theory? As one can read in the thread Are there areas of mathematics that cannot be formalized in set theory? Today known mathematical ...
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Infinite sets don't exist!?

Has anyone read this article? This accomplished mathematician gives his opinion on why he doesn't think infinite sets exist, and claims that axioms are nonsense. I don't disagree with his arguments, ...
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How come mathematics is applicable to the real world?

Often in mathematics one constructs a set of some sort, let's name it $A$. We've constructed it in an abstract way, so, a priori, structural aspects of $A$ are yet unknown to us, until we prove them. ...