Battle of opinions about mathematical methodology.

in #mathematicslast year

That semester, I'm attending "Complex Analysis" course.
Unfortunately, because of fear of coronavirus, classes are remote.
My professor sent our group a photocopy of one page of book of William Fogg Osgood:

He wrote in 1936 there: "It is in no wise necessary to introduce the idea of a Riemann's surface. Analytic continuation along a path is all that is needed."
On the other side, later mathematicians like Arnold, disagree. MY teacher also say that Riemann surface is veyr important in XXI century.

This is not the only one abstract that is controversial.

You've probably heard about the concept of a graph - it is a set of vertices, some pairs of vertices are connected by some edges.
Planar graph is a graph that can be drawn on the plane in such a way, that not even two edges cross.
Colouring of a graph is a function from a set of vertices to a subset e. g. of natural numbers, such that two adjacent (connected by edge) vertices get different value (colour) in that function.
It is very easy to prove that every planar graph can be coloured by 6 colors. It is also easy to prove that every planar graph can be coloured by 5 colors.
It is unbelievably hard to prove that every planar graph can be coloured by 4 colors.


The proof uses computer and came in 1976. Authors - Kenneth Appel and Wolfgng Haken - broke the general case into thousands of smaller cases. And computer checked all of them. Proof is generally accepted, but noone really ever checked the code or the compiler which were used. Some mathematicians were sceptical. Later came other easier proofs, but all accepted proofs so far are using computers.

In 2000, CMA Institute chose 7 problems of Mathematics for XxI century. Solving any of them is paid 1000000 dollars.
So far, one was proved, Thurston's conjecure by Grigorij Perelman, in 2003. One the others is Riemann's Hypothesis.
But solving Riemann's hypothesis by numerical finding the nontrivial zero of Riemann's dzeta function does not give 1000000 dollars. Only "theoretical" proof guarantees that. But I'm not certain of even that!

In discrete mathematics, there was a disagreement about method called "generating functions".
Generating functions are formal power series (so they are functions, but they value is usually unimportant - only their "form").
They are used for example in combinatorics to count the combinatorial objects of a certain type - permutations, classes of graphs, partitions of a number, strings from some alphabet etc.
When counting simple objects, one can easily create a bijection between set of known number of elements, and the desired set. Therefore, we can count our class.
The harder objects, the harder this method is.
This is when generating functions come to action.
But this is "solving by trick". When I was a child and my grandmother was solving crosswords, she always tried to fill all words. Whereas I only wanted to find the smallest necessary words to fill "chosen" squares and obtain a password which is a solution of a crossword.
Generating function may be thought as me, whereas creating a bijection can be thought as filling whole crossword like my grandmother.

Here are two opinions about generating functions:


In 1968, Claude Berge (1926-2002) said:
"A property is understood better when one constructs a bijection than when
one calculates the coefficients of a polynomial whose variables have no
particular meaning. The method of generating functions, which has had a
devastating effects for a century, has fallen into obsolescence for this

This opinion is quite similar to the philosophy of Aristotle.
Similar opinion but not about GF was by Emmy Noether. Emmy Noether was against method of proving that a=b by proving that a<=b and a>=b. She said that one should understand the deeper nature why equality holds by proving it directly, without such two easier steps.
Interestingly, last semester I was attending course of Commutative Algebra. And one of Noether's theorem was proved the way she didnt like :)

The second opinion about generating functions (40 years later):


"Generating functions are the central object of the theory, rather than a
mere artifact to solve reccurences, as it is still often believed."
And this was said by Philippe Flajolet (1948-2011).
At that moment, it seems like generating functions did not die.
I think that all methods which give effort should be legal and respected.

Thanks for reading!