Is God a Mathematician_ - Mario Livio [69]
Figure 37
Figure 38
As I have noted several times already, probability and statistics become meaningful when one deals with a large number of events—never individual events. This cardinal realization, known as the law of large numbers, is due to Jakob Bernoulli, who formulated it as a theorem in his book Ars Conjectandi (The Art of Conjecturing; figure 38 shows the frontispiece). In simple terms, the theorem states that if the probability of an event’s occurrence is p, then p is the most probable proportion of the event’s occurrences to the total number of trials. In addition, as the number of trials approaches infinity, the proportion of successes becomes p with certainty. Here is how Bernoulli introduced the law of large numbers in Ars Conjectandi: “What is still to be investigated is whether by increasing the number of observations we thereby also keep increasing the probability that the recorded proportion of favorable to unfavorable instances will approach the true ratio, so that this probability will finally exceed any desired degree of certainty.” He then proceeded to explain the concept with a specific example:
We have a jar containing 3000 small white pebbles and 2000 black ones, and we wish to determine empirically the ratio of white pebbles to the black—something we do not know—by drawing one pebble after another out of the jar, and recording how often a white pebble is drawn and how often a black. (I remind you that an important requirement of this process is that you put back each pebble, after noting the color, before drawing the next one, so that the number of pebbles in the urn remains constant.) Now we ask, is it possible by indefinitely extending the trials to make it 10, 100, 1000, etc., times more probable (and ultimately “morally certain”) that the ratio of the number of drawings of a white pebble to the number of drawings of a black pebble will take on the same value (3:2) as the actual ratio of white to black pebbles in the urn, than that the ratio of the drawings will take on a different value? If the answer is no, then I admit that we are likely to fail in the attempt to ascertain the number of instances of each case (i.e., the number of white and of black pebbles) by observation. But if it is true that we can finally attain moral certainty by this method [and Jakob Bernoulli proves this to be the case in the following chapter of Ars Conjectandi]…then we can determine the number of instances a posteriori with almost as great accuracy as if they were known to us a priori.
Bernoulli devoted twenty years to the perfection of this theorem, which has since become one of the central pillars of statistics. He concluded with his belief in the ultimate existence of governing laws, even in those instances that appear to be a matter of chance:
If all events from now through eternity were continually observed (whereby probability would ultimately become certainty), it would be found that everything in the world occurs for definite reasons and in definite conformity with law, and that hence we are constrained, even for things that may seem quite accidental, to assume a certain necessity and, as it were, fatefulness. For all I know that is what Plato had in mind when, in the doctrine of the universal cycle, he maintained that after the passage of countless centuries everything would return to its original state.
The upshot of this tale of the science of uncertainty is very simple: Mathematics is applicable in some ways even in the less “scientific” areas of our lives—including those that appear to be governed by pure chance. So in attempting to explain the “unreasonable effectiveness” of mathematics we cannot limit our discussion only to the laws of physics. Rather, we will eventually have to somehow figure out what it is that makes mathematics so omnipresent.
The incredible powers of mathematics were not lost on the famous playwright and essayist George Bernard Shaw (1856–1950). Definitely not known for his mathematical talents, Shaw once wrote an insightful article about statistics and probability