Let’s first consider the correctness in mathematics. First, there is “axiom” in mathematics. I assume that the axiom is correct. And I think that what is guided logically by combination of axioms is “right”.
By using the metaphor of boxes used in (1.1.2) Modeling and abstraction, “knowledge that is certainly correct” exists as a foundation, and the knowledge properly stacked on the foundation is correct knowledge. This is the definition of “right” in mathematics.
As a reason to show that knowledge X is correct, suppose we use another knowledge Y. Until we show Y is correct, we can not saay X is correct. In this way of reasoning, we somewhere need to declare “this is correct” without any basis.
Among the axioms that are determined to be correct without any basis, what particularly calls debate will be “axiom of choice”. This is the axiom that you can create a new set by selecting an element from each set for any family of sets (*3) that do not have empty set (*2) as an element. For example, if there is a set {A, B, C}, a set of {D, E} and a set {F, G, H}, we can select one element from each and create a set {A, D, F}.
This looks like a natural look at first glance. However, the example given is a set of finite elements. Let’s consider a set of infinite number of elements. For example, the “sphere of radius 1” is “a set of infinite number of points whose distance from the center is 1 or less”. Banach and Tarski of mathematicians proved that if we admitting that the choice axiom is correct, this sphere can be divided into four, two spheres of radius 1 can be made by rotating and combining two by two . In other words, when you rotate and combine it, the volume doubles. This is contrary to intuition.
Some mathematicians think that choice axioms should not be accepted as it leads to counterintuitive results. On the other hand, there are mathematicians who think that they should accept the choice axiom. The majority is on the side those who think it is better to accept. Whether axioms are correct is a matter of opinion divided by people.
*1: This idea has a name: “foundationalism”. Foundationalism - Wikipedia *2: “empty set”: It is a set with no elements. *3: “family of sets”: A set whose elements are sets (a set of sets). *4: Banach-Tarski’s theorem. It is also called the Banach-Tarski paradox because of its counter-intuitive consequences. Banach–Tarski paradox - Wikipedia *5: Some readers may surprise that the majority of mathematician admit the counter-intuitive consequences. They think that the intuition is wrong. Human intuition is often wrong when infinity is involved. For example, intuition says that number of even numbers (2, 4, 6, …) is less than the number of natural numbers (1, 2, 3, …)for example. However, “even numbers are less than natural numbers” is wrong. Because for any natural number n, there is a corresponding even number 2n. There are the same number of even numbers and natural numbers. Of cource, if you have “natural numbers up to 10,000” and “even numbers up to 10,000”, the even numbers are less. But this is when the number of elements in the set is finite. Our intuition confuses infinity with large finite numbers.
This page is auto-translated from [/nishio/(7.1.1) “right” in Mathematics](https://scrapbox.io/nishio/(7.1.1) “right” in Mathematics) using DeepL. If you looks something interesting but the auto-translated English is not good enough to understand it, feel free to let me know at @nishio_en. I’m very happy to spread my thought to non-Japanese readers.