Playing the numbers game

Is dividing the number 1 the same as dividing one orange? Clearly not. For, whereas there is no minimal fraction, we do not know whether there is, or is not, a minimal part of an orange. Yet decisions in the real world are sometimes based too readily on maths. Take exam results. If Fred gets 80 per cent in physics and Jim 40 per cent, you can't conclude that Fred is twice as good as Jim at physics or 40 percentage points better. Yet the statistics remain and could be used unjustifiably for making decisions about the careers of Fred and Jim.

Putting it more philosophically, take 1+1 = 2. This does not mean: "Give me another and we'll have twice as much." The "ones" here live in the imaginary world of mathematics. They are the same as each other and do not refer to anything. In the real world, though, there are no ones or twos, just things.

Consider the number 3. If there are three marbles in a jar, you can find the marbles, but will search in vain for the "three". You can multiply numbers, but not marbles. You can define the number 3, but not three marbles. Even if you swap the marbles for measurements, the inexactitude of the real world, compared with the purity of the mathematical one, remains. It would be odd for me to claim that this piece of chocolate I was about to eat was exactly 3cm long.

One of the best illustrations of this distinction between maths and reality is to be found in the paradoxes of Zeno, the 5th-century BC Greek philosopher. The best known is the race between Achilles and a tortoise. To make it fair, the tortoise starts a few yards ahead. Contrary to some modern versions, all Zeno says is: "The pursuer will never overtake the pursued, for always when the pursuer comes to where the pursued was, the pursued will have moved further on." Modern mathematicians and physicists have mostly dismissed this paradox by invoking the concept of a limit in calculus. A few, however, to their credit, have said that this "solution" merely restates the assumptions Zeno was calling into question and have suggested connections with recent ideas about the fundamental nature of the physical world.

For the runners, it is the racetrack that matters. If they each have a mathematical line to race along, they will have the advantages of an infinitely divisible racetrack and the possibility of being level. The downside is that they will keep falling off, as mathematical lines do not exist. If, though, the track consists of sand, is there an identifiable point, or time, at which the two runners are exactly level, as far as we can tell? To be certain, we should need to take measurements as precisely as possible. Units of measurement, though, do not exist in the abstract. They are based on real things. Let us recklessly ignore the problems surrounding the concept of time and imagine that those ultimately precise units could be found, that is, ones based on the briefest event and the smallest thing. What would that mean? A nasty paradox, I'm afraid.

With one eye on the logicians, we can put it carefully like this: if there is an event than which none other is briefer and if some distance can be travelled during the time it takes, then that distance must be indivisible, since the event of travelling part of it would take less time and, if measurable, must be based on something than which there is nothing smaller.

If those conditions can be fulfilled, we can have our ultimately precise units T (time) and D (distance). This, though, would make motor racing rather dull, as there would then be only one speed in the universe, namely one D per T. How so? Well, you couldn't go at more than one D per T, as then you would cover one D in less than the shortest possible time, and you couldn't go at less than one D per T, as one D is the shortest possible distance. Perhaps the experimental approach is better, then. Give me that orange and let's start cutting.