There are logical problems with practical induction, notorious since Hume. That we nonetheless freely (and even over-freely) constantly make use of it, might suggest that it reflects merely an innate disposition, reinforced because often successful (the world being, contingently, the way it is), thus logically on a par with the inveterate conviction of salmon that they need to swim upstream and spawn – which also usually works. So let me creep up on it genetically, via something at least similar to the notion of induction, which I did not always take for granted (hence its eventual attainment may count, albeit marginally, as an Idea).
For Parents Night, our third-grade teacher put samples on the blackboard of multiplication problems mastered by her class. Everyone in the class could multiply a one-digit number by a one-digit number, she explained; eighty percent could multiply a two-digit number by a two-digit number, and one pupil (unnamed, but in fact your correspondant) could multiply a three-digit number by a three-digit number. Not unnaturally proud of this accomplishment – which already was my first glimmering of an idea not widespread in the lowest-common-denominator society of primary school, viz. that individual abilities might differ – I jauntily asked my father, How many digits can *you* multiply?
The response I expected was something like “five” (he being an adult), or even “seven” (he being a scientist), or maybe even “ten” (what a special Dad!). Instead he looked puzzled, even embarrassed. It was not, he said, a well-defined question. Multiplying numbers of *any* length was just a small homogeneous step up from multiplying slightly shorter numbers. In practice you might become confused, and have to carefully write things down and check your work (indeed there *is* an individually varying limit as to how formidable a pair of numbers one can multiply in one’s *head*, but that was not the ability under discussion); yet in principle there was no limit to it, just as there wasn’t any highest number one could count to: if you make it as far as n, you can get to n + 1.
Now, he didn’t explain it quite as clearly as that, I imagine, but whatever it was he did say triggered a flash of insight in my small head. From that moment on I had a completely different attitude to arithmetic as taught, to school as conducted, even a different attitude towards knowledge and education in general. The teacher, from being an oracle, shrank to a small frail figure indeed. And the problems on the blackboard were no longer confined to their chalky two dimensions, but bored outward, into the void.
So far, there’s been no demonstration of an Idea beyond what we were born with, merely the application of same to phenomena initially conceived to be beyond its reach. That is, the child began by conceiving the multiplication of two-digit numbers and the multiplication of four-digit numbers to be as distinct as bipeds and quadrupeds, and we do not conceive six-legged insects and eight-legged spiders and octopuses to be some sort of inductive continuation of these. But by the time we have arrived at mathematical induction, there does seem to be something new under the sun. Given (by demonstration or observation) the truth of F(0), and proving that F(k) implies F(k+1), we suddenly, by these two little finite exercises, have access to an infinitity of truths, down to uncharted reaches of the sample space.
And lest you consider that elementary anecdote a mere trifle of childhood, the smooth and indefinite extensibility of multiplication being obvious to any mature mind,
consider this, from John von Neumann:
In an analogy machine [what we now call an analogical computer, when we call it anything at all, since these have largely fallen by the wayside -- ed.], a precision of 1 in 10^3 is easy to achieve; 1 in 10^4 somewhat difficult; ... 1 in 10^6 impossible …
In a digital machine, the above precisions mean merely that one builds the machine to 3,4,5, 6 decimal places … The transition .. gets actually easier …
--- quoted in James R. Newman, ed. World of Mathematics (1956), p. 2077
Compare further this parable: