Questions involving philosophy of mathematics

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2
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2answers
231 views

Is Physics really a rigorous subject? [on hold]

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 ...
66
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
235 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
57 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
25 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
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1answer
30 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
560 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
43 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
910 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
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0answers
30 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
365 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
189 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 ...
16
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?
92
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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"?
4
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0answers
68 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
133 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 ...
6
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4answers
164 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 ...
2
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1answer
70 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
73 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
77 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
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1answer
125 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
348 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
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1answer
169 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
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4answers
112 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
49 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
65 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 ...
5
votes
0answers
84 views

Are there examples of mathematical problems proven by abduction?

Proof by deduction is a simple principal. For example: All humans are mortal, and Bill is a human; Therefore, Bill is mortal. However, proof by abduction is a bit different. A famous example: ...
3
votes
2answers
257 views

Books on the philosophy of mathematics and logic

Here is a list of some books on the philosophy of Mathematics and logic founded in an article about this matter. I would like to buy one or maybe two of these or any other suggested books. I would be ...
1
vote
2answers
76 views

How can we explain the discrepancy between $\rightarrow$ (IF-THEN) and $\setminus$ (A-BUT-NOT-B)?

Let $\mathbb{B} = \{0,1\}$ denote the Boolean domain, ordered in the usual way. Then $\mathbb{B}$ is a lattice. It has a join operation $\vee$ that coincides with "OR," a meet operation $\wedge$ that ...
3
votes
2answers
68 views

Formal theories dealing with non-commutattive and non-transitive notion of equality

This question is inspired by a philosophical discussion which I don't want to bother you with. As far as I know when we use (or define) the statement "$x$ is equal to $y$" in logic and ordinary ...
0
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1answer
15 views

How could we define the existence of an object/element in the Euclidean space?

Let X be an object/element, What does it mean when I say "X is an object in the Euclidean space"? in other words, What differs an existed object from an unexisted one in the Euclidean space?
0
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1answer
43 views

Points and real intervals [closed]

The sorites paradox goes like this: Start with a heap of sand. Remove a grain of sand and you still have a heap; remove another, and another, and another, and you'll still have a heap. Eventually, ...
9
votes
1answer
64 views

Can the set of computable numbers be used as a theoretical basis for calculus?

I recall from my Real Analysis course that the rational numbers $\mathbb{Q}$ are not suitable for doing calculus, and I believe the reason was that $\mathbb{Q}$ does not possess the least-upper-bound ...
0
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0answers
95 views

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 $x$ ...
4
votes
2answers
177 views

Are “proofs” that are contingent upon physical reality valid?

Consider the following statement: Let $P$ be any polygon and let $A$ be a point inside of $P$. Then there exists at least one side of $P$ such that the perpendicular from $A$ to said side touches ...
1
vote
1answer
99 views

Does Mathematics exists apart from the mathematician? [closed]

Does Mathematics exists apart from the mathematician? Explain yourself. Mathematics seems to be a projection of the mind. But from where the mind originates? Can the source of the mind be known or you ...
154
votes
23answers
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 ...
2
votes
2answers
325 views

What is the “shape” of numbers in Number Theory?

While reading popular science book Fermat's Last Theorem I was amazed to find out that in number theory interesting things happen even at very large scales. For instance the Graham's number was named ...
0
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2answers
91 views

Does axiom of foundation/regularity protect against Russell-like paradoxes?

In ZF set theory the axiom of regularity (also called axiom of foundation) says that: In all nonempty sets x there is an element y such that x∩y=∅ As I been told that the intention of the axiom ...
2
votes
2answers
92 views

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 ...
1
vote
2answers
383 views

Is there such a thing as the Fundamental Theorem of ZFC?

I was initially meaning to ask to about a fundamental theorem of mathematics, but the word "mathematics" is very vague. So, my question is the title. I want to know if there is a theorem of ZFC that, ...
0
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2answers
137 views

Does infinity have a limit?? [closed]

Infinity being the far extent that the numerical system can stretch,can we say that infinity is actually a limit or infiity has another limit?
1
vote
3answers
168 views

When and where the concept of valid logic formula was defined?

I was stimulated by a recent question about Gödel Completeness Theorem. All my citations are from Jean van Heijenoort (editor) From Frege to Gödel A Source Book in Mathematical Logic (1967). Gödel's ...
2
votes
1answer
88 views

Building math theory on absurd axioms - reducing math to logic

I know similar questions have been asked and i know my terminology might be wrong but I am trying to come to an answer to whether math can be derived from logic. Wikipedia defines logic as use and ...
12
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7answers
558 views

Is it possible to alternate the law of mathematics?

I am freelance writer. Recently I have been planning a science fiction - just planning, nothing solid yet - and I was wondering would it be possible for some other universes that have different set of ...
4
votes
3answers
192 views

Do the Kolmogorov's axioms permit speaking of frequencies of occurence in any meaningful sense?

It is frequently stated (in textbooks, on Wikipedia) that the "Law of large numbers" in mathematical probability theory is a statement about relative frequencies of occurrence of an event in a finite ...
-2
votes
2answers
190 views

Is Infinity Needed in Maths? Does Infinity Actually Exist? [closed]

I'm asking this question as I have been having an on going online debate with a friend of mine. I claimed that Infinity does in fact exist in Maths and in Reality, as there's a whole plethora of ...