Questions involving philosophy of mathematics

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14
votes
4answers
1k views

Why do we need to learn Set Theory?

I was planning to write some article for the Mathematics magazine of our college and it occurred to me that it will be a good idea to write about the impact and importance of Set Theory. I plan ...
8
votes
4answers
202 views

The nature of infinities

I have been thinking about the nature of infinity lately. I have no experience with higher mathematics or theorems regarding infinity, so please forgive me if my ideas on this topic are extremely ...
4
votes
6answers
409 views

Question on induction technique

When one uses induction (say on $n$) to prove something, does it mean the proof holds for all finite values of $n$ or does it always hold when even $n$ takes $\pm\infty$?
1
vote
8answers
590 views

Are the real numbers really uncountable?

Consider the following statement Every real number must have a definition in order to be discussed. What this statement doesn't specify is how that loose-specific that definition is. Some examples ...
7
votes
5answers
156 views

Why do we first introduce the open set definition for continuity instead of the neighborhood definition?

After (nearly) completing my course in topology, something weird just stuck out to me which I hadn't considered before. When first discussing continuity, we often use the following definition: Let ...
92
votes
17answers
12k views

Is 10 closer to infinity than 1?

This may be considered a philosophy but is the number "10" closer to infinity than the number "1"?
82
votes
9answers
4k views

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 ...
0
votes
0answers
15 views

Request for good resources on 'history of infinity' topics [migrated]

Im writing/starting with my bachelor thesis, the subject is about "infinity": what the hell is it, why do we accept it, but most of all my goal is to give an overview of the history of the ...
2
votes
1answer
33 views

Properties of the simplest object in n-dimension

In my boredom, I was thinking about why the simplest 3d object (i.e. the one with the least faces, sides, vertices) was the tetrahedron. After it made sense to me, I realized some cool stuff which was ...
7
votes
2answers
504 views

Could computers someday discover theorems or find demonstrations?

Cloud computing and quantum computers bring computers to what seems like a limitless calculation power? If one sees all the mathematical operations and theorems as a toolset that a computer can use, ...
2
votes
2answers
344 views

Is Physics really a rigorous subject? [closed]

Though I can't give a precise definition of the term rigor (or better to say mathematical rigor) but intuitively in case of mathematics one may note that when we say that 'the proof is rigorous' we ...
4
votes
2answers
73 views

Why do we focus so much in math on functions (as a subclass of relations)?

Why is it that math so focuses on the subclass of relations known as functions? I.e. why is it so useful for us in nearly all branches of mathematics to focus on relations which are left-total and ...
156
votes
24answers
12k views

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 ...
1
vote
2answers
51 views

Are there undecidable problems for which a solution has been found?

I mean are there examples of problems that have been proven to be undecidable, in the sense that it would not be possible to devise a deterministic computer program that outputs a solution for an ...
0
votes
3answers
35 views
4
votes
6answers
107 views

the purpose of induction

After getting an answer (in a comment) from peter for this question I have a follow up question. If, in all horses are the same color problem for example, we need to use reason, reason which is ...
129
votes
24answers
10k views

Can a coin with an unknown bias be treated as fair?

This morning, I wanted to flip a coin to make a decision but no coins were in reach. There was however an SD card on my desk: Given that I don't know the bias of this SD card, would flipping it be ...
282
votes
35answers
30k views

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 ...
3
votes
0answers
64 views

Goldbach's conjecture can't be proved to be undecidable?

Conjectures concerning natural numbers which could be settled by a counterexample can, as far as I understand, not be proved to be undecidable without being proved not having a counterexample at the ...
1
vote
3answers
158 views

How much of Mathematics is limited by our writing? [closed]

I'm sorry if this question is too vague or otherwise a stupid question. Suppose the mathematicians in some alien civilisation similar to ours sculpted their Mathematics in three dimensions (or ...
1
vote
0answers
45 views

Is there a name for this constant? (0.0100011011…)

It's the simplest number I could think of that contains any finite binary code in its digits: $$\begin{align} c &= 0.0100011011000001010011100101110111...\\ &= ...
41
votes
9answers
3k views

Does mathematics require axioms?

I just read this whole article: http://web.maths.unsw.edu.au/~norman/papers/SetTheory.pdf which is also discussed over here: Infinite sets don't exist!? However, the paragraph which I found most ...
8
votes
4answers
2k views

how do we assume there is infinity?

Definition of infinite: A set is infinite iff it is equivalent to one of its proper subsets. We know that our universe doesn't contain infinite number of elements (including subatomic particles), so ...
13
votes
7answers
2k views

Why do statements which appear elementary have complicated proofs?

The motivation for this question is : Rationals of the form $\frac{p}{q}$ where $p,q$ are primes in $[a,b]$ and some other problems in Mathematics which looks as if they are elementary but their ...
68
votes
4answers
7k views

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 ...
7
votes
2answers
240 views

New Axioms of Infinity

Axiom of Infinity says there is an inductive set (i.e. a set which includes $\emptyset$ and is closed under successor operator). Formally: $Inf:\exists x~(\emptyset\in x~\wedge~\forall y\in ...
5
votes
0answers
66 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 - ...
1
vote
1answer
28 views

maximum number of possible rules of a sequence?

Read this http://www.mathsisfun.com/algebra/sequences-finding-rule.html and also http://en.wikipedia.org/wiki/The_Oxford_Murders_%28film%29 where the scene about the murder note left behind contains a ...
-1
votes
1answer
36 views

A Question Regarding Representing $\mathscr P$($\omega$) as a Digraph and CH

It is well known that one can represent sets as digraphs. What is the proper digraph representation of $\mathscr P$($\omega$)? I ask this because $\mathscr P$($\omega$) is $\Pi_1$ in the Levy ...
13
votes
2answers
577 views

Is First Order Logic (FOL) the only fundamental logic?

I'm far from being an expert in the field of mathematical logic, but I've been reading about the academic work invested in the foundations of mathematics, both in a historical and objetive sense; and ...
2
votes
1answer
169 views

What's with conditionals in mathematical logic?

Having a bit of difficulty understanding the conditional ($\rightarrow$) in mathematical logic. I read up on the already-existing questions and it did help me understand it better (the 'promise' ...
1
vote
1answer
46 views

Impredicativity and set theory

I have thought about an example in set theory, but I don't know if its legal to do it, maybe someone can help. Let $\emptyset$ be given and let $A$ be a non empty set. Let us create the subset $X = ...
14
votes
1answer
928 views

What is the “opposite” of the Axiom of Choice?

One might think that, trivially, the "opposite" of AC is $\neg$AC. However, thinking about it differently, I'm not sure this is intuitively the case. AC says that every set has a choice function. ...
3
votes
0answers
32 views

Philosophical implications of P vs NP proof?

Wikipedia article on P vs NP says that "a proof either way would have profound implications for ... Philosophy" without providing further details. So I was wondering what could be the philosophical ...
2
votes
5answers
382 views

What is maths? “Maths is the study of ______”? [closed]

I can fill in the blank by just listing the different fields of maths but my goal is to define all of mathematics. An answer that I would've accepted a few years ago is "Maths is the study of ...
7
votes
3answers
213 views

Why is the Power Set Operation Inherently Vague?

It is a somewhat common view among mathematicians/philosophers (who have an opinion on the subject) that the power set operation is inherently vague. They go on to say that its inherent vagueness is ...
17
votes
7answers
2k views

Why do we stop at exponentiation stage in arithmetic of natural numbers?

In natural numbers the unary successor operator $S$ is the most natural function which maps each number to the next one. Furthermore we may consider the binary relation $+$ as an iteration of $S$. ...
25
votes
11answers
5k views

Good books on Philosophy of Mathematics

Where can I learn more about the implications, meta discussions, history and the foundations of mathematics? Is Russell's Introduction to Mathematical Philosophy a good start?
4
votes
0answers
70 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 ...
37
votes
8answers
5k views

Infinite sets don't exist!?

Has anyone read this article? Set theory 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 ...
5
votes
1answer
137 views

What goes wrong in the following argument that our conception of “set” is inconsistent?

This question might sound facetious, but it is a genuine question which I am very much interested in. I apologize in advance if it is too conceptual or philosophical, but I'm optimistic that I might ...
2
votes
1answer
73 views

Can arithmetic truths fix the truth value of the Continuum Hypothesis?

Many logicians and philosophers believe that all sentences expressible in the language of Peano Arithmetic have determinate truth-values, even though no nice formal system can capture all of these ...
2
votes
1answer
75 views

Why the dual of some results are true while others are false?

In mathematics, many results have their "dual" versions. In many cases, if a result is true, then its dual is true as well. However, there are some examples while the dual of a true statement is ...
3
votes
1answer
81 views

Is the first-order incompleteness of a theory (like arithmetics, set theory or logic itself) avoidable in a second or higher-order axiomatizations?

Can we avoid the first-order incompleteness of a theory (like arithmetics or set theory) in a second-order theory which contains the previous? How does it depend on the chosen semantics or models? If ...
0
votes
1answer
127 views

Is Bell's Notion of “Abstract Set” Flawed?

Consider the following definition of "abstract set" given by John L. Bell (who wrote the book "Set Theory: Boolean-valued Models and Independence Proofs") from his preprint "Abstract and Variable ...
5
votes
3answers
359 views

Good Sources for Lecture Movies in Set Theory, Logic and Philosophy of Maths

Of course as any other researcher I'm not able to attend any scientific event in my research area. But it is always interesting and useful to watch the lecture movies of these events. I will ...
0
votes
1answer
171 views

Hand-incalculable Problems

Let's define a "hand-incalculable problem" as a mathematical problem that can not be solved by available human calculation power (using only writing materials and utensils) at a specific date and ...
0
votes
4answers
116 views

what is $e$ really? what is its meaning? [duplicate]

I don't get it how we came up with $e$ and how can nature use this number so much! that is what I have been told and I only know that $e$ is a specific constant like $\pi$! I understand that $\pi$ ...
2
votes
2answers
50 views

Undecidability and truth

Are there undecidable problems for which a single truth exists? For example, the question about parallels is not decidable from Euclid axioms. But multiple answers are valid and give different kinds ...
1
vote
1answer
68 views

Can a proof be too long? [closed]

Suppose that there is an omnipotent oracle saying that there is a proof for Riemann hypothesis but its proof is so long the universum would collapse before mankind will understand the proof. Would ...