Mathematics and the Laws of Nature

In his essay The Unreasonable Effectiveness of Mathematics in the Natural Sciences, Eugene Wigner says, “The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.” But in reality, it can be proved that a physical world — a world which has an order of place, with one part beside another, and an order of time, with one thing before another — must of necessity either follow mathematical natural laws, or it must be more or less intentionally designed in order to avoid this.

For example, suppose we attempt to determine how long it takes a ball to fall a certain distance. We do not need any particularly exact method to measure distances; for example, we could be measuring a fall of ten feet, taking foot in the presumably original sense of “the length of an adult human foot,” despite the noisiness of this measure. Nor do we need any particularly exact method to measure time; we could for example measure time in blinks. Something took 10 blinks if it took so long that I blinked 10 times before it was over. This would be even noisier than measuring in feet. But the point is that it does not matter how exact or inexact the measures are. If we have a world with place and time in it, we can find ways to make such measurements, even if they are inexact ones. Nor again do we need a way to get an extremely precise measure in blinks or in feet or in whatever of the physical quantity we are measuring; it is enough if we get a best estimate.

Now suppose we repeatedly measure, in some such way, how long it takes for a ball to fall a certain distance. After we have made many measurements, we can add them together and divide by the total number of measurements, getting an average amount of time for the fall. The question that arises is this: as we increase the number of measurements indefinitely, will that average converge to a finite value? or will it diverge to infinity or go back and forth infinitely many times?

Evidently it will not diverge to infinity. It is difficult to see any reason in principle why it could not go back and forth infinitely many times, for example the average fall time might tend toward 1/4 of a blink for a long time, then start tending toward 1/5 of a blink for a long time, and then go back to 1/4, and so on. But we should notice the kind of pattern that is necessary in order for this to happen. Suppose the average is 1/4 of a blink after 100 measurements. In order to get the average to 1/5, there must be a great many measurements 1/5 or below, or at least many measurements which are very much below 1/5. And the more measurements we have taken to get the average, the more such especially low measures are needed. So if we are at an average of 1/4 of a blink after 1,000,000 measurements, this average will be very stable, and it will require an extremely long series, more or less continuous, of especially low measurements in order to get the average down to 1/5 again. And the length of the “especially low” or “especially high” series which is needed to move the average will be increasing each time we want to move it again. In other words, in order to get the average to go back and forth infinitely many times, we need to have a rather pathological series of measurements, namely one that looks like it was designed intentionally to prevent the series from converging to an average value.

Thus the “natural” result, when things are not designed to prevent convergence to an average, is that such measures of distance and time and basically anything else we might think of measuring, like “how much food does an adult eat in a year”, will always converge to an average value as we increase the number of measurements indefinitely. Given this result it follows that it is possible to express the behavior of the physical world using mathematical laws.

Several things however do not necessarily follow from this:

It does not follow that such laws cannot have “exceptions”, since they are only statistical laws from the beginning, and thus are only expected to work approximately. So it is not possible to rule out miracles in the way supposed by David Hume.

It also does not follow that such laws have to be particularly simple. A simpler law will be more likely than a more complex one, for the reasons given in a previous post, but theoretically the laws governing a falling body could have 500 variables, which would be simpler than ones having 50,000 variables. In practice however this does not tend to be the case, or at least we can find extremely good approximate laws with very few variables. It may simply be the case that in order to have a world with animals in it, the world needs to be fairly predictable to them, and this may require that fairly simple laws work at least as a good approximation. But a mathematical demonstration of this would be extremely difficult, if it turns out to be possible at all.

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