Questions tagged [philosophy]

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

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Difference between provability and truth of Goodstein's theorem

I have been thinking about the difference between provability and truth and think this example can illustrate what I have been wondering about: We know that Goodstein's theorem (G) is unprovable in ...
2k views

What kind of alternative mathematics systems exist? [closed]

What kind of alternative mathematics systems exist? What I mean is, mathematical systems that use a different sort of "basic premises" or e.g. logic(s) than the contemporary "mainstream" mathematics. ...
1k views

What exactly does tautology mean?

I'm struggling to understand the actual meaning of tautology. I know how to determine tautology with a truth table. But I don't understand what it actually means, so consider the following reasoning: ...
6k views

Does advanced math “power” more rudimentary math?

I came across this quote by Eric Weinstein that "when things got supposedly more advanced, they actually got simpler because mathematicians started revealing what was powering all the things that you ...
829 views

Why is negative minus negative not negative? Why is negative times positive not directionless?

What I understand is that the only difference between plus and minus is direction. I've never understood this: That +1 + +1 is ...
4k views

$e^{e^{e^{79}}}$ and ultrafinitism

I was reading the following article on Ultrafinitism, and it mentions that one of the reasons ultrafinitists believe that N is not infinite is because the floor of $e^{e^{e^{79}}}$ is not computable. ...
165 views

Why linear congruential generator is called random number generator?

As far my understanding linear congruential generator is used to produce pseudo random number. But the algorithm requires four parameters like, ...
83 views

Is it possible to construct a formal system such that all interesting statements from ZFC can be proven within the system?

I'm familiar with Godel's incompleteness theorem, which very basically states that there exists a statement that can be neither proved or disproved within a formal system powerful enough to include ...
789 views

Decidability and “truth value”

One can read in the Wikipedia page for "Gödel's incompleteness theorems": Undecidability of a statement in a particular deductive system does not, in and of itself, address the question of ...
178 views

Where do I go wrong with Presburger “multiplication”?

This is a speculation about the Presburger arithmetics, the Peano axioms with only addition added, vis-à-vis the same axioms with both addition and multiplication added. In the first case I understand ...
785 views

The boundaries of mathematics?

There are five possible answers to a multiple-choice question. Given that the student does not know the answer, what is the probability that the student chooses the first answer? That is not a well ...
399 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 ...
148 views

Should axioms be seen as “building blocks of definitions”?

This question is about the difference between a definition and an axiom. However, it does not address the following point: Whenever we define something, this is often written as a series of axioms. ...
114 views

Does Planck length contradict math? [closed]

I have a general question about math and infinity which really bothers me as a math student - can we actually divide every length by two? I would like to believe the answer is yes, because it ...
59k 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 ...
2k 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 ...
355 views

Is mathematics aprioristic? [closed]

Is mathematics aprioristic? I do not know. Some axioms of arithmetic and geometry arose clearly inspired by the observation of Nature. After that, those areas of mathematics were often developed with ...
777 views

Why did we settle for ZFC?

I understand the need for a solid foundation of mathematics. But it seems to me that we have definitely settled for ZFC and I would like to know if there are strong reasons for this or not. I am not ...
2k views

Are category-theory and set-theory on the equal foundational footing?

Set-theory is widely taken to be foundational to the rest of mathematics. So is category-theory. My question is: Are they two alternative, rival candidates for the role of a foundational theory of ...
If I have a line on an $XY$ grid: o---o---o---o---o a b c d e And I rotate it like so: ...