2018-02-12 18:07:13 UTC
-- Plato attributes this to Socrates
It's been over a quarter of a century now that I've been promoting an idea that I think is very important. It's revolutionary. Here's the idea:
We (mathematicians) can and should view mathematics as the language of science. Especially, we should use that as a guiding principle when reasoning about the foundations of mathematics.
Modern pure mathematics is deficient as a language of science. In particular, the subject we call the foundations of mathematics is missing two key ideas:
1) Falsifiability is the cornerstone of scientific reasoning, and it can and should be formalized and integrated into the logic of mathematics.
2) Reasoning under uncertainty. Scientific reasoning always takes uncertainty into consideration. So reasoning under uncertainty should be viewed as the foundational logic of mathematics.
The reason mathematicians rejects those two ideas is, evidently, because they cling to Cantor's theory of infinity, which is incompatible with those ideas.
When I studying mathematics in graduate school, I wanted to work in the field of the foundations of mathematics, and the idea I wanted to work on was to build a foundation for mathematics that is built upon those two ideas--falsifiability, and reasoning under uncertainty. My claim is that such a foundation for mathematics would serve as a foundation for artificial intelligence, and would be a major advance in that field.
I was surprised by the mathematicians' reaction to what I was proposing. Especially on the internet! Besides suggesting the idea is a crank idea, they sometimes would accuse me of believing I am channeling God, or other silly things like that.
It only recently occured to me that I should respond by pointing out exactly where my ideas are coming from. They come from self knowledge. That is, as intelligent beings, we can make progess in understanding ourselves. We can understand how we reason about what we call reality. And so my radical claim is that mathematics should be consistent with our understanding of ourselves, and especially, how we reason about reality. And Cantor's theory of infinity fails that criterion.