As promised, I would like to pick up where I left off some time ago, when I discussed the Butterfly Dream. I mentioned that, in my opinion, the Butterfly Dream describes the dual concept of existence vs non-existence. I opined that existence/non-existence is just another pair of Yin-Yang opposites that should not be treated differently than other opposites, such as hot/cold or day/night. Furthermore, I argued that we need to incorporate non-existing objects into our formal theories. After all, the Butterfly Dream teaches us that it is not possible to determine with certainty whether or not an object exists. However, modern mathematics does not address this intrinsic uncertainty between existence and non-existence at all.
While these considerations are valid from a philosophical point of view, they raise several questions:
- Where is the best place to introduce intrinsic uncertainty into mathematics?
- How can we incorporate intrinsic uncertainty into mathematics?
- What are the implications for the current state-of-the-art?
I addressed these questions in a recent article entitled “Computational Complexity on Signed Numbers,” which I presented in an earlier post on the P/NP problem. Let me elaborate more on these questions here and in the following posts.
As for the first question, all places where mathematics explicitly postulates the existence of “something” are good places to introduce intrinsic uncertainty. When mathematics resorts to postulating the existence of things, it usually means that there is no way to prove their existence with the axioms of mathematics alone. Introducing axioms that postulate existence is therefore an act of desperation more than anything else. Unfortunately, mathematicians have been too careless with axioms postulating existence. I think that intrinsic uncertainty is a fundamental principle. It should play a dominant role and form the basis of mathematics.
As for the second question, there are two ways to introduce intrinsic uncertainty into mathematics. One way is to introduce uncertainty explicitly. However, adding explicit uncertainty requires statistical models that are in themselves not uncertain. In fact, statistical models, such as the normal distribution, are well-defined and so is their behavior for large numbers. The other way to introduce uncertainty is to make uncertainty an intrinsic part of mathematics. For instance, removing an axiom that postulates the existence of an entity, a set for example, is a simple way to introduce intrinsic uncertainty into mathematics.
In my following posts, I will discuss this in more detail. I will show exactly where I think is the most appropriate place to introduce intrinsic uncertainty into today’s mathematical edifice.