Mathematics would seem to be the field with the most obviously objective standards of truth: A statement is accepted as true mathematically if it is proven clearly and rationally from given more-or-less-universal axioms; it is false if there is some counter-example; i.e. some object, structure, etc, which contradicts the statement. Basically everything a mathematician does (as I understand it) can be described as accomplishing one of these two tasks - proving a statement or finding a counter-example to it. And so we have two nice, clear-cut boxes: the Logically Provable, and the Provably False. At least, this is where we begin.
In Gödel's proof of incompleteness, he showed that for any interesting logical system (here interesting just means sufficiently complex; one example is Peano arithmetic) there are statements which are true for all objects in the system which are not provable. In other words, there are statements which are consistent with all other provable statements in the system, and which give us accurate information about objects in the system, but which cannot be proven from the system's axioms. I don't have the requisite background to really explain this in rigorous detail, but it throws out the two clean boxes of Provable and False. Mathematicians don't worry about this much (most of us don't work in foundational mathematics), but yeah. This is crazy.
This leaves someone like myself, whose work as a mathematician informs his perspective on life (for better or for worse, haha), with the interesting proposition that not all truths in life are rationally provable. Perhaps there are truths which I will never be able to show follow logically from necessary axioms, truths which have to be explored through art or literature or history. In these disciplines, in the humanities, I imagine there are ways to reach those missing truths about this system which we call reality, but we will have to believe them based on something besides logic. For they will, simply put, not be provable using logic's tools.
I realize these are somewhat scattered thoughts, but I think they reflect one aspect of the perspective that mathematics has to offer on the question of subjectivity and objectivity. There are certainly other aspects, and I think this is a fascinating question, but I haven't the time to think them out now. Thanks for reading this!
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Huh, we both wrote about the incompleteness theorems.
ReplyDeleteI've been working on understanding these quite a bit recently, even though they may only tangentially relate to my studies in experimental quantum optics. They are some of my favorite results.