(3) Induction
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:
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