Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
NCTM or The Math Forum.



Cantor's absurdity, once again, why not?
Posted:
Mar 14, 2013 5:47 AM


In a previous aritcle a while back, "marcus_b" wrote:
** start quote ** In math, the great pons asinorum now is Cantor's diagonal proof. There seem to be scads of people out there who just cannot quite get it and who yearn to achieve their rightful 15 minutes of fame by trying to shoot it down, thus thrusting themselves ahead of Cantor in the pantheon of mathematical geniuses. Can you shed some light on the motivation here? ** end quote **
The "cranks" who insist on pointing out the absurdity of Cantor's argument are puzzled that mathematicians "cannot quite get it".
Cantor's argument relies on accepting the notion of an actual infinite, and that's problematic. It is reasonable to accept the assertion that the purpose of mathematics is to provide a rigorous language and conceptual framework for reasoning quantitatively about realworld phenomena. It's a good bet that mathematicians before the time of Cantor would have heartily endorsed that view. And the actual infinite is not needed for the mathematics or the real world, as has been pointed out by many top notch mathematicians over the last century, from Poincare' to Feferman:
"The actual infinite is not required for the mathematics of the physical world" (Soloman Fefermanm, in an article titled "Is Cantor Necessary?")
That article also contains the following quote:
"I am convinced that the platonism which underlies Cantorian set theory is utterly unsatisfactory as a philosophy of our subject, despite the apparent coherence of current settheoretical conceptions and methods ... platonism is the medieval metaphysics of mathematics; surely we can do better" (Feferman)
Some of the "cranks" are saying that yes indeed, we can do better. We can do better by rejecting platonism and accepting falsifiability as part of the underlying philosophy of mathematics.
Falsifiability is a cornerstone of scientific reasoning. When we reason about the real world, we apply falsifiability, even if sometimes we do it only subconsciously. Falsifiability is part of the natural logic we are born with. It can be formalized in such a way that it can serve as a cornerstone of mathematical reasoning, and if we don't accept it as part of the underlying logic of our mathematics, then our mathematics is deficient as a language for reasoning about the real world.
Platonism, on the other hand, is a perversion of natural logic. It would be insane to apply platonism to reasoning about the real world. Falsifiability and platonism are not compatible. Falsifiability would exclude Cantor from mathematics. Just as the scientists use falsifiability as a criterion to distinguish real science from crackpot science, it can serve as a criterion to distinguish real mathematics from crackpot mathematics.
Cantorian mathematics is crackpot mathematics.



