Questions involving philosophy of mathematics

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6
votes
2answers
421 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, ...
9
votes
1answer
177 views

Is there an online Q/A forum for the philosophy of mathematical practice?

Certain philosophers of mathematics are interested in aspects of the philosophy of mathematical practice. Mathematicians, perhaps, would be interested in philosophy that may affect their day to day ...
13
votes
4answers
690 views

How do mathematicians think about the existence of numbers?

Question: How do mathematicians think about the existence of numbers? And how did Newton, Euler, and other famous mathematicians thought about this concept? I know that existence of numbers is a ...
5
votes
4answers
342 views

Logic as subset of mathematics and mathematics as subset of logic

Is logic a subset of mathematics or is mathematics a subset of logic? I have heard the former view, but is there any argument for the latter?
2
votes
2answers
184 views

What does Russell mean when he defines the “Posterity… with respect to the immediate predecessor”?

The the Introduction to Mathematical Philosophy, Russell defines the "posterity" of a given number with respect to the relation "immediate predecessor" as all those terms that belong to every ...
8
votes
4answers
621 views

What would happen if ZFC were found to be inconsistent?

If, one fine day, someone found a contradiction in ZFC (or even ZF), what implications would such an event have for mathematicians? Is there currently any backup axiomatic system on par with ZFC that ...
51
votes
4answers
6k 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 ...
5
votes
2answers
121 views

On the existence of number systems, and the extent to which we can extend them

The more I think about math, the less I realize I know. Learning about complex numbers has called me to re-evaluate how I think of negative numbers, or even natural numbers. I have to say the ...
8
votes
4answers
429 views

What is straight line?

I have found the definition of line in metric space. It is general but has two problems. Considering about $\mathbb R^2$ equipped rectilinear distance, every line by this definition contains a ...
-2
votes
2answers
127 views

Is math the measurement of motion? [closed]

I know that math can be used to measure motion, like where something will end up over a period of time, but is math itself 'in' motion. Take a look at how you use math: You scribble, type, draw ...
5
votes
6answers
2k views

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 ...
1
vote
1answer
79 views

A Simpler Characterization of Inductive Definitions?

While reading appendix A of John Harrison's "Handbook of Practical Logic and Automated Reasoning" a somewhat advanced theorem is appealed to as a prerequisite for characterizing when an inductive ...
4
votes
2answers
269 views

What is the role of Topology in mathematics?

What is the role of Topology in Mathematics? Is it like Logic that you need in every areas of math?
2
votes
2answers
255 views

What is the use of such concepts as potential infinity and actual infinity?

I'm aware of such mathematical concepts as and potential infinity and actual infinity. But I do not understand how those concepts are being used. Are there any applications to such concepts? Are there ...
2
votes
3answers
128 views

Polynomials as concrete structures

Motivation The structuralist point of view on mathematical objects has two aspects: On the one side, a mathematical object is seen as a concrete structure of abstract dots, e.g. a graph. On the ...
5
votes
3answers
1k views

are there non-standard models of arithmetic in second order arithmetic?

non-standard models of arithmetic in second order arithmetic? Background: According to Godel's theorem, if we have, in a given consistent system S, a non-provable wff. A, then we can extend the ...
6
votes
4answers
287 views

Is probability objective?

As we know, probability is a measure of events. However, is it an objectively attribute of events, or just an illusion in ones' mind? For example, suppose that there is an empty black box with an ...
47
votes
10answers
12k views

How is a system of axioms different from a system of beliefs?

Other ways to put it: Is there any faith required in the adoption of a system of axioms? How is a given system of axioms accepted or rejected if not based on blind faith?
12
votes
1answer
506 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 ...
9
votes
4answers
488 views

What are the reasons for not supporting constructive mathematics

It is obvious that in constructive mathematics, you cannot use the law of excluded middle. What else would be the reasons for not adopting constructive stance in mathematics?
4
votes
2answers
269 views

Did Structuralism influence the formulation of Category Theory?

Having only the a very cursory knowledge of Structuralism ( it's a movement generally held to have originated in linguistics, then moving on to philosophy & literature), there does appear to be ...
6
votes
6answers
2k views

Michael Spivak in “Calculus” asserts that $\sqrt2$ cannot be proven to exist, and that such a proof is impossible. What does he mean by “exist”?

Michael Spivak in "Calculus" asserts that $\sqrt2$ cannot be proven to exist, and that such a proof is impossible. What does he mean by "exist"? How are you to prove that any number "exists"? Why ...
8
votes
6answers
1k views

Why accept the axiom of infinity?

According to my readings, Russell showed that a principle Frege used to reduce Peano arithmetic to logic lead to a contradiction. So, Russell tried to reduce mathematics to logic a different way but ...
5
votes
1answer
543 views

How do mathematical objects relate to the real world? (a little philosophy)

I am just going to give an example of what I mean using Skolem's Paradox. I don't want to get into Skolem;s Paradox itself or its "resolution." Skolem's showed that in first-order formulations of ...
25
votes
2answers
661 views

A few questions about intuitionistic mathematics

I have to write a paper on Intuitionism for my Philosophy of Science class and I'm struggling with a few concepts I have encountered in my self-study. The (intuitive) characterization of valid ...
4
votes
4answers
303 views

Are there any situations where you can only memorize rather than understand?

I realize that you should understand theorems, equations etc. rather than just memorizing them, but are there any circumstances where memorizing in necessary? (I have always considered math a logical ...
5
votes
4answers
2k views

Is McGee's counterexample to Modus Ponens accepted by the mathematical community?

In the mid 1980's Vann McGee proposed a counterexample to Modus Ponens: (a) If a Republicans will win the election, then if Reagan will not win, Anderson will win. (b) A Republican will win the ...
1
vote
1answer
247 views

Equivalence of sequences and subsets of natural numbers

For me, facts like the independence of the continuum hypotheses from ZFC cast a doubt on the "law of the excluded middle". (In this context, the doubt is that there might be no "final set theory" such ...
21
votes
3answers
1k views

What did Gauss think about infinity?

I have someone who is begging for a conversation with me about infinity. He thinks that Cantor got it wrong, and suggested to me that Gauss did not really believe in infinity, and would not have ...
3
votes
1answer
319 views

Probability and Axiom of Choice

I'm not a logician, so I apologize if what follows translates to nonsense. I would like to try to define a different theory of random choice. I hesitate to call it probability theory because I do not ...
4
votes
3answers
290 views

How can any statements be proven undecidable?

As I understand it, undecidability means that there exists no proofs or contradictions of a statement. So if you've proved $X$ is undecidable then there are no contradictions to $X$, so $X$ always ...
13
votes
6answers
2k views

What philosophical consequence of Goedel's incompleteness theorems?

I want to write a philosophical essay centered about Goedel's incompleteness theorem. However I cannot find any real philosophical consequences that I can write more than half a page about. I read the ...
49
votes
6answers
4k views

In what sense are math axioms true?

Say I am explaining to a kid, $A +B$ is the same as $B+A$ for natural numbers. The kid asks: why? Well, it's an axiom. It's called commutativity (which is not even true for most groups). How do I ...
2
votes
1answer
288 views

Russell Paradox and set theories

The Russell paradox arise in the Cantor set theory, but it can be avoided in the $ZF$ and in $NGB$ axiomatic set theory. Are there other axiomatic set theories in which this paradox can be avoided? ...
6
votes
1answer
148 views

Hidden structures

There is a lot of talk about "hidden structures" in the realm of mathematics: hidden structures in the ZFC system, hidden structures in the natural number system, and so on. Saunders Mac Lane ...
5
votes
0answers
312 views

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 ...
6
votes
2answers
363 views

Are there any non-self-referential statements that cannot be assigned a truth value?

Statements like A) A is false. or B1) B2 is true. B2) B1 is false. cannot be assigned a truth-value due to their ...
3
votes
0answers
183 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". ...
2
votes
1answer
87 views

Reference request for Intuitionism

I need to write an essay on Intuitionism for my Philosophy of Science class, and I'm looking for books which cover the following topics: Brouwer's Intuitionism, from both a philosophical and ...
3
votes
5answers
315 views

Essays on the real line?

Are there any essays on real numbers (in general?). Specifically I want to learn more about: The history of (the system of) numbers; their philosophical significance through history; any good ...
7
votes
2answers
146 views

Literature on general paradox?

I suppose this one teeters on the edge of un-mathematical, but here it goes... I've been on something of a logic binge lately and have (surprise, surprise!) especially been interested in the results ...
5
votes
4answers
418 views

From continuity to differentiability and analyticity- what's next?

Continuity is an intuitive concept. I will not dwell on the precise definitions of continuity and the rest here. Note that differentiability is a more restrictive condition than continuity, while ...
9
votes
5answers
959 views

The status of high school geometry

Okay, so we've all seen Euclidean geometry in primary and high school. Back then, I really thought of points as indivisible entities in space and lines as 'breadthless lengths'. As far as I could ...
7
votes
4answers
837 views

Is the set of all mathematical truths countable or uncountable?

Is the set of all theorems countable or uncountable? Maybe its a stupid question. I just wanted to know. I am led to think that since, we use a finite set of symbols and English letters, the set of ...
3
votes
0answers
398 views

Mac Lane and Eilenberg's motivations for category theory

I'm looking to understand the conceptual process that brought Eilenberg and Mac Lane in developing the basic concepts of category theory. I quote Mac Lane's book "Category theory for working ...
8
votes
2answers
887 views

Definition of “non-constructive proof”

I was wondering if it is possible to define exactly what a non-constructive (nc) proof is. I have often seen the concept associated with the use of principles such as the axiom of choice or the law of ...
12
votes
3answers
361 views

Is there any difference between a math invention and a math discovery? [closed]

From wikipekia: The calculus controversy was an argument between 17th-century mathematicians Isaac Newton and Gottfried Leibniz (begun or fomented in part by their disciples and associates – ...
1
vote
4answers
465 views

Does $3+2=5$ have a non-physical interpretation? [closed]

Normally we consider simple arithmetic to be related to the world of objects. So the sum $3+2=5$ means $3$ three apples and $2$ apples gives $5$ apples. But is there an alternative interpretation ...
1
vote
3answers
310 views

What is the difference between a parametric equation and a mathematic law?

First of all sorry for my English, I'm not used to communicate with this language. I want to ask something about a thing that I discovered while studying physics (AKA applied mathematics). There is ...
6
votes
6answers
949 views

Inherently discrete concepts

Are there any concepts which are naturally defined only for the integers and so far has resisted any attempts at extension to other fields such as rationals or reals? Does not meet criteria: ...