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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.

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Are there concepts of mathematics today that aren't on a solid logical foundation?

In this answer, Arturo wrote: I would think like Sebastian, that most "working mathematicians" didn't worry too much about Russell's paradox; much like they didn't worry too much about the fact that Calculus was not, originally, on solid logical…
Red Banana
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Which concept is more primary in modern formal math: set, function or natural number?

My reasons to ask this question are the following: 1) formal definition of a function is "a subset of cartesian product of two sets" so it is defined with the set and natrual number as basic concepts. 2) in axiomatic set theory we need to be able to…
Igor
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What are the leading alternative foundations for mathematics?

I know that all of mathematics can be recast in terms of set theory. There are multiple choices for this set theory (some form of ZFC, NBG, NF, etc.), and so multiple possible set theoretic foundations for mathematics. I am told (though I know so…
Dennis
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Is there a simple formalism in which self-referential definitions can be made safely?

Consider the expression $g(f(x)).$ Its easy to draw this in a tree diagram; the bottom node is $x$, above that is $f$, and above that is $g$. Now consider the $(D(f))(x),$ for instance imagine that $D$ denotes differentiation. If you try to draw…
goblin GONE
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Definition Of Composition Of Functions

The standard definition of function composition is: Let $f : A \to B$, $g : B \to C$ then there is a composite function $g \circ f : A \to C$, given by $(g \circ f)(a) = g(f(a))$ with $a \in A$ Why could not be given a less restrictive definition…
Alberto Capitani
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What is an equation, set-theoretically?

In modern mathematics, everything is a set of some kind. For example, an ordered pair is a set, a binary operation is a more complicated set, a group is a yet more complicated set. But what about equations? How do you implement equations as a…
user107952
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What problems remain unanswered today regarding the nature of variables?

So far, to me, the greatest difficulty in studying philosophy is to recognize the importance of the problems: Exactly what make philosophers think these problems are worthy subject of study? Take Russell's The Principle of Mathematics §6 for…
George Chen
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In ETCS, how do we take the limit of an infinite diagram?

As everyone knows, the category $\mathrm{Set}$ is complete, or in other words all small limits exist. Now suppose we're using ETCS as a foundation. I think we can take finite limits without issue. But, how do we define the limit of a diagram with…
goblin GONE
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How does first order logic influence everyday mathematics?

I have read through a book about first order logic. It was interesting. However, when I read through undergraduate math texts it’s unclear about what system they are working in. They don’t specify anything at all about axioms or logical axioms. I’m…
Hasan Zaeem
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Axiom of Regularity and Self-Containment

I understand that the axiom of Regularity excludes $A=\{A\}$ but what about $A=\{A, \emptyset \}$?
Mario Pucci
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What is meant by a "Cantorian sense of a graph"?

I didn't get very far in this before I encountered this: $27 × 37 = 999$, then the comment This equality makes sense in the mainstream of mathematics by saying that the two sides denote the same integer and that × is a function in the Cantorian…
147pm
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What are the set-theoretic foundations of classical calculus?

I am wondering what set theoretical foundation is needed for the development of classical results of, say, calculus, such as taught in first years undergraduate courses. More concretely, I wonder if and where the full power of ZFC is needed, if…
Hanno
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Define numbers sets (e.g. $\mathbb{N,Z}$ etc.) as being subsets of next one...

Usually, numbers sets are constructed from $\mathbb N$ (with Peano's axioms) : then $\mathbb Z$ is viewed as a quotient set of $\mathbb N\times\mathbb N$, $\mathbb Q$ as a quotient set of $\mathbb Z \times \mathbb Z$, and $\mathbb R$ is defined with…
Célestin
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Shortcomings of "The Continuum" by Hermann Weyl?

I just read that Hermann Weyl used a certain type of type theory as the foundation of mathematics in his book "The Continuum" and he was able to derive calculus without the use of infinite sets. Naturally, I'm going to read it ASAP but I wanted to…
Praise Existence
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Is there a way to quantify distance between formal systems?

Suppose I and my twin embark on a project. I create a mathematical system, from scratch, based on the ZFC axioms. My twin, having read the HoTT book, decides to ground his system there. Does there exist a measure to gauge how related our two systems…
Kevin Binz
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