There are several level of mathematical foundation. Of course, the most basic one is to ignore foundations at all. This is the intuitive approach. Mathematics becomes a social thing, where a convincing proof depends on the knowledge and experience of the author and his audience. Results become a matter of discussion, just like in any other science. This includes discussion not only about the validness of the argument, but also about the validness of the hypothesis. Truth is then just a mirror of our changing ignorance. For many, math would be no longer math. For others, math becomes suddenly interesting. In fact, this philosophical idea of making math may reflect the real life math more than we like to admit. For me, this approach may be good for schools, but it is not sufficient on the university level.
The highest level of foundation is set theory. Its idea is that math is based on sound logical principles and axioms. Many mathematicians still believe that the axioms reflect reality in some way, despite the results that we owe to Gödel. In fact, there is no axiomatic system, which contains the complete truth, not even for such an elementary subject as integer arithmetic. By Gödels incompleteness theorem, any model of arithmetic will always satisfy a statement, which cannot be proved by the axioms. And, there are essentially different models of arithmetic. Even worse, we will never know if our axiomatic system is contradictory in itself or not.
To overcome this difficulty, mathematicians such as Brower have thought of intermediate levels of foundations. One is finite mathematics. In this mathematics, the square root of 2 exists only as some potential object. We must be very careful about the meaning of „potential“. E.g.:
- We can effecively compute a sequence of finitely expressible rational numbers, such that the square of these numbers approaches 2 as close as we like.
- We can effectively say, from which number on the distance becomes less than a specific epsilon.
- Each other effectively constructable sequence of numbers with the same property is close to the one sequence, we constructed.
- We can effectively estimate, how close the other sequence is.
By „effectively“ we mean, that there is an algorithm to compute the sequence, which could be executed on a computer, ending in a finite time.
To show the difference, this approach makes, let us prove the „existence“ of the square root of 2. Both ways, we can use bisection to compute a constructive sequence of intervals, such that
\(b_n-a_n \le 1/2^n, \quad a_n^2 < 2 < b_n^2 \)
In the school book approach, we derive the convergence of both interval ends to some common c. Then we use the continuity of the square function to derive that c is equal to the square root of 2.
In the finite approach, things are not so easy. We need to use the equi-continuity of the square function, which is a more useful concept anyway. It is easy to see that each polynomial is equi-continuous on each finite interval. In fact, the modulus of continuity is effectively computable. Here we have
\(|a_n^2-b_n^2| \le 2 |a_n-b_n| \)
It follows immediately, that the square of the interval ends converge to 2. We can also effectively estimate the speed of convergence. There is also the converse estimate, valid on [1,2],
\(|a_n-c_n| \le |a_n^2-c_n^2| \)
This proves that any other sequence, the square of which converges to 2, is close to our sequences.
As we see, thinking this way is a lot more difficult. No wonder, we do not teach that way. However, it might be worth the effort to think about such constructive approaches from time to time. My „intuitive“ feeling is, that they are closer to reality than the simpler approach we are used to.
Maybe I show you in another post how to do the same for a proof of the main theorem of Algebra saying that each polynomial factors in the complex plane.