I did once sit down and consider the difference between the induction of
mathematics and the induction of science, that I may discover why one is
reasonable and the other is not.
As my problem, I decided to induct on the coming and going of days, and
the fact that we observe the sun rises at the beginning of each day. We
may decide from this that since the sun has risen every day we have observed,
that the sun will rise at the beginning of every day. This seemed a clearly
incorrect reason, because of post hoc, but for another reason as well.
In mathematics, when we induct on some series in proving that it is
equivalent to some general formula, we have the luxury of the entire series
as a starting point. The formula then can be shown true because what the
formula states is in fact the same as what the series is. When we induct on
the sun rising every day, however, we do not have the complete set of days
to start with (only those in the past, and remembered) and so our induction
fails.
Considered another way, given the set "(5, 5, 5, 5, 5,...)", we add nothing
new to the meaning here by generalizing "All the numbers are five"). However,
given the incomplete set "(3, 3, 3, ", we cannot state that they are all three
(they may, in fact, all BE threes, but such a conclusion would be fallaciously
drawn given this partial set).