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

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A philosophical question about an hypothetical theorem/equation of everything

Preamble I'm not a mathematician. I'm just curious. Please forgive my pseudo formalism. Please allow me, a non mathematician, to have just questions. Definition A mathematical theorem is a statement ...
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1answer
79 views

Can you choose -1 as the multiplicative unit? And what is a positive number?

If one starts with the cyclic group of integers and want to introduce multiplikation the ordinare choice of multiplicative identity is the generator 1. But since 1 and its inverse -1 is sort of the ...
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0answers
68 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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2answers
109 views

Higher-Order Logic in ordinary Mathematics?

Do we use the language of higher-order logic in ordinary mathematics? (If yes: Can you give some examples?) Or are we always working with first-order logic? Comment: Maybe you are going to say that ...
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1answer
106 views

Some Philosophical Questions About Mathematics and Logic [closed]

The following questions may seem very philosophical and I guess that you guys will tell me that this is not the right place for asking them. But for me it is important to get answers from ...
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0answers
26 views

Completeness property in signal analysis

Why completeness is an important property for signal analysis such as Fourier? What if we don't have a such a property?Many books discuss that the vector should not have a hole to complete.what is ...
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3answers
209 views

Logic & Reality [closed]

Maybe just a quick preface first before the question. I recently started a YouTube channel where I'm trying to clear up confusions I see on various (usually philosophical topics). In my 2nd video, the ...
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0answers
57 views

Question on a note by Kreisel

In Kreisel´s "Two notes on the foundation of set theory" he writes in a footnote that 2 is to be considered as measurable and omega is also measurable. Further more he goes on to say that an ...
2
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1answer
71 views

Meaning of “existence” for an uncomputable function related to the Halting Problem

Take the set of all Turing Machines $TM$, we can divide this set in two: $P$, the set of all Turing Machines that will halt if starting from an empty tape, and $Q$, its complement: the set of all ...
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0answers
58 views

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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2answers
77 views

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
801 views

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 "set"...
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1answer
124 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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3answers
356 views

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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16answers
6k views

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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2answers
108 views

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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2answers
116 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
147 views

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 ...
6
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1answer
204 views

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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5answers
284 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 ...
3
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1answer
97 views

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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1answer
22 views

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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2answers
88 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
79 views

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 ...
8
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1answer
289 views

When is a function 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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1answer
186 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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3answers
98 views

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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12answers
7k views

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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5answers
1k views

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 instance,...
3
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1answer
83 views

What exactly is wrong with this argument (Lucas-Penrose fallacy)

Argument "For every computer system, there is a sentence which is undecidable for the computer, but the human sees that it is true, therefore proving the sentence via some non-algorithmic method."
7
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1answer
188 views

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 $G=...
3
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2answers
125 views

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 others. ...
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0answers
39 views

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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2answers
395 views

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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5answers
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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 ...
5
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1answer
213 views

What are numbers? [duplicate]

The title is a bit of clickbait, but I think it's justified. How did I came to ask this question In programming, many programming languages have concepts of a hierarchy of numerical types. Often ...
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2answers
306 views

Motive for the definition of inner product

Mathematicians pride themselves on writing proofs of propositions in an elegant way, but frequently (maybe even usually?) neglect to formally write motivations of definitions with the same elegance, ...
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2answers
182 views

Why are Euclid axioms of geometry considered 'not sound'?

The five postulates (axioms) are: "To draw a straight line from any point to any point." "To produce [extend] a finite straight line continuously in a straight line." "To describe a circle with ...
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2answers
98 views

How to generate complicated looking identities such as $\sqrt [3] {2 + \sqrt 5} - \sqrt [3] {2 - \sqrt 5}=1$ easily?

How to generate complicated looking identities, or even more complicated looking identies such as $\sqrt [3] {2 + \sqrt 5} - \sqrt [3] {2 - \sqrt 5}=1$ easily? I saw the identity to be shown. What is ...
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1answer
201 views

Why a line is said to have infinite number of points? [duplicate]

Why a line is said to have infinite number of points? Is this so because a line is ever lasting or we can not count how many points does it have? Finite means: Having an end. Infinite means: No end! ...
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1answer
444 views

I need help understanding Frege's definition of number

I have really been trying to understand Frege's definition of a number or at least gain a strong intuition of it. However, my attempts have not been fruitful. If someone could help me it would be much ...
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2answers
70 views

When the probability model of an experiment is correct?

Suppose I wanted to tell what's the probability of event $A$: getting 2 tails in a row of 5 coin tosses. According to the classical definition of probability, the probability of this event is equal to ...
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2answers
150 views

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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0answers
73 views

curves in Poincare half space (3 dimensional hyperbolic geometry)

Okay maybe I am going a bit ahead of my self The Poincare half plane still has many mysteries for me But still I was puzzeling about the 3 dimensional variant of it. So lets assume an hyperbolic 3 ...
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1answer
142 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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4answers
214 views

Peano's Axioms: Mathematical Philosophy

In Peano Axioms, why is it necessary to define number and successor. Does not using them imply that we know what they mean? Or could they have just as easily been any two arbitrary terms which are not ...
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3answers
2k views

Avoiding proof by induction

Proofs that proceed by induction are almost always unsatisfying to me. They do not seem to deepen understanding, I would describe something that is true by induction as being "true by a technicality". ...
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3answers
263 views

Is there an area of study regarding why certain mathematical definitions are useful?

Often in my studies I'll come across an definition, which I understand, and but don't necessarily see why the particular definition was chosen to be studied. For example, the topological axioms (...
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2answers
113 views

Error in Introduction to Mathematical Philosophy

Is this an error in the text or am I reading incorrectly. What am I missing? Introduction to Mathematical Philosophy Page 18 Definition of Number “A relation is said to be “one-one” when, if $x$ has ...
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82 views

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 ...