> When you say 'induction', do you mean our way of constructing
> inductive proofs? I do think that the Greeks could and may have
> produced such kinds of arguments, but not one around this passage
> in The Theaetetus. What would they have been proving here? You
> need some simpler and clearer case than that to get things across,
> say something like the result on side-and-diameter numbers, that
>
> D^2 - 2 S^2 = 2 s^2 - d^2 = 1 .
I meant "induction" not in the sense of "mathematical induction", but in
the sense which philosophers usually use the term, which one can call
reasoning non-deductively, or (as some interpret it) probabilistically.
Another frequent charactization of philosophical induction (as I will call
it) is the inference of general conclusions from particular instances.
(This doesn't necessarily involve a passage from finite to infinite -- one
may simply not have examined all of a finite set of instances.)
To take an old example, one may be willing to conclude from the fact that
the sun has arisen with respect to the earth (using Ptolemaic coordinates)
every day since time immemorial, that it will also rise tomorrow, as a
"general truth". But then maybe an asteroid will knock our Earth out of
its orbit and out of our solar system, and it will then no longer be the
case that the sun will rise again tomorrow with respect to the earth. Or,
one might argue, with less force, that our universe is expanding, and
therefore will continue to do so forever. But then reasonable cosmological
theories have been proposed in which this isn't so. Or it may be a simple
matter of arguing that if Joe Smith always stops for too many beers on his
way home from work, he will continue to do so, barring accidents, death,
membership in Alcoholics Anonymous, etc. As you may know, the so-called
"problem of induction" is a famous one among philosophers, and has been
much discussed since antiquity. It was so done in Aristotle, and, I think,
has adumbrations in Plato's *Theaetetus*.
This is, I take it, quite different from mathematical induction, in which
one accepts some sort of induction axiom, and concludes upon successful
application of it that something or the other is true for all integers in
some infinite set of them.
If you ask any philosopher acquainted sufficiently with the works of Plato,
he or she would be very surprised to hear that the *Theaetetus* was
considered to be about mathematics. As to why Plato was concerned to give
historical details about incommensurability, that was, I think, Plato's way
when writing his dialogues. He gives historical details about many things
throughout his writings, for example about what previous so-called
pre-Socratic philosophers thought about many matters besides mathematics.
> Finally, why is Root 17 notorious? Is there any other reason outside
> this dialogue and perhaps those who comment on it?
>
Notorious because so many historians of mathematics have tried to explain
on mathematical grounds why Theaetetus (or Theodorus, or Plato) stopped at
17, and didn't continue to, say, 18 and 19, and so on. They assumed
usually that there was some procedure that worked up to 17, but failed
thereafter, or at least sometimes failed. I once, nearly 40 years ago, had
quite a collection of explanations by various people, especially German
historians of mathematics of the later 19th and earlier 20th centuries,
some of which I found downright humorous as far as historical credibility
was concerned, however admirable they were as pure mathematics. If my
memory is right, you can, for example, find one such explanation in van der
Waerden's *Science Awakening* (he was Dutch, of course).
I tend to the belief that this was a place where Plato, in his sly way,
using his version of Socrates as a mouthpiece, was giving an example
concerning the philosophical problem of induction.
Gordon Fisher gfisher@shentel.net