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February 17th, 2006

To phil_of_science:

Let's liven up this journal!  A friend, who is also a self-proclaimed mathematician and logician, questioned the logical foundations of science with the following experience:

"...if you ask a student of physics or any sort of person who believes in the validity of science the question, "Will the laws of gravity continue to hold tomorrow?" you will receive one of two possible answers: "Yes, certainly." or "It seems very probable that they will." I will show that the first statement doesn't follow necessarily from assumptions and that <i>the second statement makes no sense at all</i>."

This later, italicized assertion is one that baffles me because I have never (to date) been a skeptic of probability.  Are there substantial reasons for questioning chance? regardless of context?  What I mean is that I would like an abstraction of probability without toying with temporal contexts (can one avoid the textbook example that varies time? ie. 'there is a chance A <i>will occur</i> so probability is a meaningful experience').  I think a reduction of probability to magic or deceptive anomalies would expose the argument as illogical or lead to determinism.

Next, my friend defines induction and disagrees with its conclusions:

"The foundational axiom of inductive reasoning--and all of science--is that if X(n) holds for n < p, then X(p) holds also. If it has always worked before, then it will work again.  I don't accept this. ... It works--and it works through science--because we modern humans believe in science."

First, I have never seen induction defined symbolically.  Does anyone have a good source for an induction theorem?  

Second, I applaud my friend's defense of, despite mathematics and logic, my very own epistemology in belief systems!  But, I feel that belief systems are used here to undermine science's vitality.  Is there a better defense of science than a dogmatic, "I believe in it" speech.  Isn't my use of a deductive theorem also dependent upon my belief in deduction? and mathematics?

A discussion begins...