Solitude_within said:
freejroll, what would an "anti-inductionist" have to say about mathematics? Would two plus two always equal four? Or would that too fail the induction test?
Also, how do we have knowledge of anything at all, since most of our information is gained by experiencing things in the past?
Thanks.
I'm not an anti-inductionist, so I'm not quite sure if I'm qualified to answer your question. Its just that there doesn't seem to be any good reason to accept induction as a good method of reasoning. This doesn't mean that I believe it is false. It could be a good method of reasoning. We just don't know how to
justify this. Which of course leads us to question why we accept it in the first place...
I think mathematics is primarily a
language that is deductive. I believe there are inductive principles in some mathematics, but your question seems to be primarily aimed at deduction. Though I could be wrong!
It is interesting to note that philosophy has pointed out many problems in mathematics, too. The problem of the one and the many plays into this very much. Depending on your philosophical stance on epistemology or ontology, some people might think that 2 + 2 does not equal 4. In fact, a monist, believing that all plurality is an illusion (and all is in fact one) might be forced to claim that 1 + 1 = 1. There are many questions about the validity of mathematics.
Sounds crazy, but there are actually very good arguments against mathematics in philosophy. Many people don't read about them because they don't go searching for them, and someone running around claiming that mathematics doesn't work and that 1 + 1 = 1, would generally be shut off by the public, and his arguments wouldn't be considered. The same goes for induction. Besides we seemed to have created a system for ourselves, and if we are to decide that it doesn't really work, or we have no good reason to believe in it, it could lead to chaos. It makes good thinking and conversation, though. I definately believe these problems are important. It's just that, well, regardless, it's kind of hard to debate them and if we did, it would be a million times harder to fix them.
For example, mathematics tells us that there are more irrational numbers than rational numbers (the diagonalization method proves this) or more real numbers than rational numbers (this proof is easy, since real encompasses rational and irrational). However, both classes continue to infinity, which seems to suggest that there are different levels of infinity. This doesn't seem to make much since, since infinity is... well infinity.. All of these questions have been debated for some time (and good answers have come out -- whether correct or incorrect). It would be a good topic for another thread for sure.
And for your last question, I don't know how we have knowledge of anything at all. Do we?