The Race-horse and the Tortoise

Mossy O'Connor, my teacher, was usually full of sense, but one day he said something that was total nonsense.



"Supposing," he said, "a Race-horse and a Tortoise set off at full gallop from the same starting post and kept going at their best pace for ever! Then," he said, "according to the mathematicians, they would both reach infinity!"

"That," he said, "is the mystery of Infinity."

"That," I whispered to Dessie Breen, who sat beside me, "is nonsense."

"Shut up," said Dessie.

Dessie had a different intelligence to me. I could beat him at X's and O's or Draughts, but, whenever the keyhole boys played Pontoon, (for marbles, of course), Dessie was king. He had an ability to seize an advantage. When he made a plaster leprechaun, the local grocer bought it off him for a couple of pence. When I made one, the grocer said, "I already have one of those."

Dessie was not interested in abstract thought. He was not interested in whether Mossy's statement was nonsense, or even in hearing my argument. If the experts said the Tortoise and the Horse would both reach Infinity at the same time, he was happy with that.

But, wait a minute. Did you not hear what the teacher said? He said that a Tortoise, struggling to make one mile an hour, in a race with a Race-horse coasting along at forty miles an hour, after the longest possible track, despite being left far behind for almost the whole race, will nevertheless arrive at the finishing line at exactly the same time. Is this not clearly impossible nonsense?

I suppose I could have put my hand up and voiced my objection openly, but I did not have the courage, and my palms were red enough already.

This nonsense was not Mossy's fault, in a sense. He was only a national school teacher, and this fallacy was promulgated by high-brow masters of mathematics. However, he should have had the sense to add, "it looks like nonsense to me."

Supposing he actually said this to the Grand Professors of Math, they would say to him, "It is counter-intuitive, but we have proven it to be true."

Mossy could then retort, "Your 'proof' leads to an absurd conclusion; therefore it must be based on a false assumption."

The false assumption the mathematicians made, in this case, was that Infinity is Actual.

Later in life I would find that this question had been settled a long time ago by Aristotle, who said, "Actually, there is no such thing as Infinity," (though this is usually quoted as, "There is no such thing as Actual Infinity").

The argument of the Great Mathematicians that leads to the conclusion that the Race-horse and the Tortoise reach the same Infinity is based on the incorrect assumption that there is an Actual Infinity to reach.

The counting numbers, (1, 2, 3 ... ) go on forever, (indicated by the three dots), so we say it is an infinite series, or infinite Set. No matter how many stars or grains of sand you count, you can always add one (or a billion) more. You never reach Infinity or the end of the count. Even if you counted to the last star, you could always turn around and continue the count on the way back and then repeat this over and over again. You never reach Infinity, because there is no such thing, just going on and on.

In sympathy for the poor old Tortoise, I wrote a poem about it (when, of course, I was still in Mossy O'Connor's class in primary school):

On and on and on I go
Through hail and rain and sleet and snow;
On and on and on forever,
Never ceasing ever.

Thrush upon the thorny tree,
Turn your gaze not after me;
Upon the stony road don't look
That, with weary feet, I took.

On and on and on forever,
Stopping, resting, ceasing never.
On and on and on I go,
Never ceasing ever.

Perhaps we should start this counting business with Zero, (0, 1, 2 ... ), since, if you are going to count your marbles, or the number of people going into a football match, first you will empty the jar into which you will be tossing the counted marbles, or you will set the style-clock to zero. But the ancient Greeks and Romans did not have a symbol for Zero, so started their count at One.

Now, you could depict the counting series as (0, 1, 2, 3 ... ∞), meaning "zero, one, two, three and so on up to Infinity," but, if so, you must bear in mind that there is no number "Infinity" and all this actually means is "and so on without end." The Infinity symbol here is only a Notional number, a symbol to indicate that the sequence goes on forever. It does not represent an Actual number. Unfortunately, some great thinkers actually took this as "Actual Infinity," an ultimate value, and drew all kinds of daft conclusions, including the absurdity that the Race-horse and the Tortoise will reach the same Infinity. ("If you keep going forever, you will reach Infinity,  and so will the Tortoise and the Race-horse." Nonsense, you will never reach any place unless you stop, and then you are where you are, a place like any other).

Modern mathematicians now have the task of unravelling this fallacy from all the complicated formulas which were built up entangled with this erroneous premise.

To the best of my knowledge, however, they have not yet apologised or confessed to the people that, indeed, the Tortoise never, in fact, catches up with the Race-horse.



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