The Archimedeans - Greek Metamathematics

Greek Metamathematics

BY N. A. ROUTLEDGE

Achilles and the Tortoise - a Consideration

How often does the deep simplicity and insight of country folk confound the sophistries of the over-educated! It was round about 450 B.C. that a self-taught philosopher pointed out four paradoxes to the frequenters of the academies in the little Greek colony of Elea. Since then everyone with the slightest interest in philosophy or mathematics has been troubled by them and has developed highly ingenious theories for resolving them. But ingenuity has not been accompanied by conclusiveness: indeed, the paradoxes "have probably occasioned more inconclusive disputation than any equal amount of disguised mathematics in history."

Three of Zeno's posers have been dealt with quite reasonably, but the other, concerning Achilles and the Tortoise, has never received a treatment that did not leave at the back of the mind a nasty feeling that the solver has been a little too clever. The argument is:

  Achilles runs ten times as fast as the Tortoise.
  He gives it a start of 100 yards.
  When he has run this the Tortoise is 10 yards ahead.
  When he has run this 10 yards the Tortoise is 1 yard ahead.
  When he has run this 1 yard the Tortoise is 1/10 yard ahead.
  Etc.

(a) Thus, according to this argument, Achilles never overtakes the Tortoise.

(b) Whereas, of course, we know that he does.

The usual way of treating this is to say that

and that Achilles overtakes the Tortoise after going yards, but one cannot sum an infinite series in a finite length of time. This sounds, on a first hearing, as if it disposes of the contradiction, but the more one looks at it the less it seems to do so, and if one tries to rewrite Zeno's proof, one sees that the remarks have no bearing on the problem at all.

The complete solution introduces metamathematics - arguments about arguments. Metamathematics has been developed almost entirely in the last 50 years or so, and has yielded many startling and important results. The most comforting is the solution of Hilbert's Entscheidungsproblem: it has been demonstrated that no machine can deal with all mathematics. The most tiresome results concern unsolvability: one takes a logical system and shows that a certain statement in it cannot be shown to be true and cannot be shown to be false. Zeno's paradox in its correct form is precisely of this kind.

To resolve the paradox we merely alter the statement (a) to:

Thus, no argument of this kind, however long we continue it, will ever lead us to the conclusion that Achilles overtakes the Tortoise.

Does this now conflict with the statement (b)? Only if one argues:

Since our argument cannot show that Achilles overtakes the Tortoise, it must be true that he does not.

We can only assert this if we are sure that every statement can be demonstrated true or else shown false by such arguments. But this is not so, for these arguments allow us to say nothing concerning, for example, where the Tortoise is when Achilles has gone 112 yards.

The paradox has thus vanished.

We can introduce a formal logical system:

The symbols used will be R, ( , ) , ; , 0. I shall use the abbreviations:

1 for 00

2 for 000

3 for 0000

...

n for n + 1 zeros in a row.

I have just one axiom: R(0; 1)
and just one rule of proof:

From R(n ; m) we may conclude R(n + 1 ; m + 1).

Then one may easily see that R(n ; 0) is not provable for any n. If we interpret R(n ; m) as meaning that:

At the nth stage of Zeno's argument Achilles is 1000/10^m yards behind the Tortoise except when m = 0, when Achilles is level with the Tortoise,

we see that this system formalises Zeno's method of argument.

The really thrilling thing is to see how near to discovering metamathematics the Greeks were, and it is amusing to speculate what the trend of history would have been had they done so.

Eureka, 13.

Reproduced from Eureka 27 pages 24-26.