How Dedekind Screwed Up a Hundred Years of Mathematics

2005-12-10T20:29:00Z

I while back I was telling someone how I found it hard to believe there was a debate in the early 20th century over constructivism. If it were not for the existence of computers, I would never be a constructivist. In the early 20th century there basically no computers, so why would anyone defend the constructivist position?

My friend suggested that for most of the 3600 year history of mathematics, mathematics was done constructively. He suggested that it wasn’t until the mid 19th century did people do any effectively non-constructive reasoning.

I found this very reasonable. For number theory, all Π2 statements provable in PA are also provable constructively. This probably encompasses a large part of the work in number theory. The rest of the theory may very well have been constructively valid as well. For analysis, well, if you believe non-continuous functions aren’t really functions, then you have practically taken a constructivist’s position.

For more information he suggested I have a look at The Introduction of Non-Recursive Methods Into Mathematics by G. Metakides and A. Nerode. They give a little history lesson in their paper. Kummer and Kronecker developed an algorithmic theory of prime ideals in 1859. His theory of ideal primes was wholly algorithmic, and the criteria for two such formal linear combinations to give the same prime were computational. But Dedekind wasn’t happy with the equivalence relations in the theory and he created the set-theoretic formulation that we suffer with today. Although the set-theoretical formula is elegant, there are no equivalence relations, the result is a less computation theory.

Meanwhile people were struggling with real numbers and analysis. In 1817 Bolzano had identified what we call the Cauchy completeness property of the real numbers, but he still didn’t have a definition of what the real numbers were. In the early 1860s Weierstrass gave a definition of real numbers similar to the modern definition of Cauchy sequences with an equivalence relation. Once again, Dedekind wasn’t happy with this, and in 1872, he proposed his own notion of real numbers based on set-theory by Dedekind cuts of the rational numbers. Again Dedekind’s theory works to eliminate the computational content of mathematics.

In light of this history, a debate in the early 20th century seems more reasonable. Unfortunately, the wrong side won, but I don’t think the war is over.

An interesting question arises: What was the first accepted non-constructive theorem? Using Cauchy completeness, Bolzano proved the intermediate value theorem. This could be the first accepted non-constructive theorem. Exercise (easier), verify that Bolzano statement was a non-constructive version of the intermediate value theorem. Exercise (harder), check to see if there are any previous constructively invalid theorems.

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