Questions tagged [foundations]

This tag is for questions about the foundations of mathematics, and the formalization of mathematical concepts in foundational theories (e.g. set theory, category theory, and type theory).

Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathematics. In this latter sense, the distinction between foundations of mathematics and philosophy of mathematics turns out to be quite vague.

Foundations of mathematics can be conceived as the study of the basic mathematical concepts (set, function, geometrical figure, number, etc.) and how they form hierarchies of more complex structures and concepts, especially the fundamentally important structures that form the language of mathematics (formulas, theories and their models giving a meaning to formulas, definitions, proofs, algorithms, etc.) also called metamathematical concepts, with an eye to the philosophical aspects and the unity of mathematics.

The search for foundations of mathematics is a central question of the philosophy of mathematics; the abstract nature of mathematical objects presents special philosophical challenges.

1131 questions
24 answers

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 through the process of deduction. As such, it would…
  • 2,851
  • 4
  • 21
  • 35
19 answers

Mathematical ideas that took long to define rigorously

It often happens in mathematics that the answer to a problem is "known" long before anybody knows how to prove it. (Some examples of contemporary interest are among the Millennium Prize problems: E.g. Yang-Mills existence is widely believed to be…
  • 14,399
  • 4
  • 29
  • 70
23 answers

Is math built on assumptions?

I just came across this statement when I was lecturing a student on math and strictly speaking I used: Assuming that the value of $x$ equals , ... One of my students just rose and asked me: Why do we assume so much in math? Is math…
Anz Joy
  • 1,410
  • 3
  • 12
  • 16
10 answers

Why can't you add apples and oranges, but you can multiply and divide them?

What is the algebraic difference between arithmetic operations, that prevents entities with different units from being summed or subtracted, but allows them to be multiplied or divided? This looks more like a question for Physics, but lengths and…
  • 1,153
  • 1
  • 9
  • 10
10 answers

Is mathematics just a bunch of nested empty sets?

When in high school I used to see mathematical objects as ideal objects whose existence is independent of us. But when I learned set theory, I discovered that all mathematical objects I was studying were sets, for example: $ 0 = \emptyset $ What…
Vinicius L. Deloi
  • 1,227
  • 1
  • 8
  • 12
8 answers

Does mathematics become circular at the bottom? What is at the bottom of mathematics?

I am trying to understand what mathematics is really built up of. I thought mathematical logic was the foundation of everything. But from reading a book in mathematical logic, they use "="(equals-sign), functions and relations. Now is the "=" taken…
  • 9,418
  • 5
  • 33
  • 96
9 answers

Why is the construction of the real numbers important?

There are a lot of books, specially in Real Analysis and set theory, which define the real numbers by Cauchy sequences or Dedekind cuts. So my question is why don't we simply define the Real numbers as a complete ordered field? What's the importance…
  • 22,250
  • 21
  • 85
  • 225
9 answers

Does mathematics require axioms?

I just read this whole article: which is also discussed over here: Infinite sets don't exist!? However, the paragraph which I found most interesting is not really discussed there. I think…
  • 12,734
  • 12
  • 60
  • 110
11 answers

Infinite sets don't exist!?

Has anyone read this article? 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 arguments, but with my limited knowledge of axiomatic set…
Nicolas Bourbaki
  • 1,636
  • 1
  • 11
  • 16
6 answers

Is it possible to formulate category theory without set theory?

I have never understood why set theory has so many detractors, or what is gained by avoiding its use. It is well known that the naive concept of a set as a collection of objects leads to logical paradoxes (when dealing with infinite sets) that can…
Matt Calhoun
  • 4,194
  • 1
  • 28
  • 52
13 answers

What is the definition of a set?

From what I have been told, everything in mathematics has a definition and everything is based on the rules of logic. For example, whether or not $0^0$ is $1$ is a simple matter of definition. My question is what the definition of a set is? I have…
John Doe
  • 2,807
  • 5
  • 35
  • 71
6 answers

Foundation for analysis without axiom of choice?

Let's say I consider the Banach–Tarski paradox unacceptable, meaning that I would rather do all my mathematics without using the axiom of choice. As my foundation, I would presumably have to use ZF, ZF plus other axioms, or an approach in which sets…
3 answers

Why is it worth spending time on type theory?

Looking around there are three candidates for "foundations of mathematics": set theory category theory type theory There is a seminal paper relating these three topics: From Sets to Types to Categories to Sets by Steve Awodey But at this forum…
Hans-Peter Stricker
  • 17,273
  • 7
  • 54
  • 119
9 answers

Where to begin with foundations of mathematics

I would like to know more about the foundations of mathematics, but I can't really figure out where it all starts. If I look in a book on axiomatic set theory, then it seems to be assumed that one already have learned about languages. If I look in a…
  • 41,457
  • 11
  • 67
  • 130
7 answers

How can someone reject a math result if everything has to be proved?

I'm reading a book on axiomatic set theory, classic Set Theory: For Guided Independent Study, and at the beginning of chapter 4 it says: So far in this book we have given the impression that sets are needed to help explain the important number…
2 3
75 76