appolose said:
Sqrl said:
appolose said:
Sqrl said:
So you have a problem with anything proved using mathematical induction, or its basic concept at least, as a basis?
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No; math is (I think) a system of definition and logic (on the other hand, what was Russell trying to prove?).
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The concept is used outside of math as well, but math is the best way to explain it:
The basic premise of mathematical induction is that if you prove, or in this case accept, the base case as true you don't have to prove every case. Instead of trying to prove every related case you simply prove that, in general terms, the next case is also true.
A good example is helpful, but math is the best way to express it:
Problem: Prove that the sum of the first 'n' odd numbers is equal to n squared.
Let S(n) = the sum of the first n odd numbers greater than 0.
We need to show that S(n) = 1 + 3 + … + (2n – 1) = n2
Base Case (n = 1): S(1) = 1 = 12
The result holds for n = 1.
Induction Hypothesis: Assume that S(k.) = k2
We must show that S(k+1) = (k + 1)2
S(k+1) = 1 + 3 + … + (2k - 1) + (2(k + 1) - 1)
= S(k.) + 2(k + 1) - 1 (by definition of S(n))
= k2 + 2(k + 1) - 1 (by the induction hypothesis)
= k2 + 2k + 1
= (k + 1)(k + 1) (factoring)
= (k + 1)2
Therefore, we can conclude that since S(1) = 1 and that S(k.) logical implies S(k+1) then S(n) is
equal to the sum of the first n odd numbers for all n > 0.
The idea is that you can prove an infinite number of cases by proving a basic case and then proving a generalized case. I think you can do something similar in the example I quoted as well. Once you have decided to accept sensory data, even if reluctantly, it logically follows (logic being a result of our experience with our sensory input) that you can expound on this. Even to the point of proposing other parts of reality.
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You cannot expound upon sense data: we have no idea what it could represent (if anything) (or, so that's been my contention). Yes, we can accept sense data, but not any judgements upon it (that is, deciding what, if anything is responible for sense data).
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Do you see the part that Sqrl said that you really disagree with? IMO, this is it:
(logic being a result of our experience with our sensory input)
In my opinion, logic, language, the whole kit-and-caboodle (whatever the hell that means) is "a result of our experience with our sensory input." Which is why I find your skepticism to be self-contradictory.
And also, that's why I think it really does matter where you think your logic and language come from. If not from sensory input, then what?
