On Numbers: Large Numbers I

“The calculation of probabilities is of the utmost value, […] but in statistical inquiries there is need not so much of mathematical subtlety as of a precise statement of all the circumstances. The possible contingencies are too numerous to be covered by a finite number of experiments, and exact calculation is, therefore, out of the question. Although nature has her habits, due to the recurrence of causes, they are general, not invariable. Yet empirical calculation, although it is inexact, may be adequate in affairs of practice.” (Gottfried W Leibniz [letter to Bernoulli], 1703)

“Further, it cannot escape anyone that for judging in this way about any event at all, it is not enough to use one or two trials, but rather a great number of trials is required. And sometimes the stupidest man|by some instinct of nature per se and by no previous instruction (this is truly amazing)| knows for sure that the more observations of this sort that are taken, the less the danger will be of straying from the mark.” (Jacob Bernoulli, “The Art of Conjecturing”, 1713)

“If thus all events through all eternity could be repeated, by which we would go from probability to certainty, one would find that everything in the world happens from definite causes and according to definite rules, and that we would be forced to assume amongst the most apparently fortuitous things a certain necessity, or, so to say, FATE.” (Jacob Bernoulli, “The Art of Conjecturing”, 1713)

“And thus in all cases it will be found, that although Chance produces Irregularities, still the odds will be infinitely great that in the process of time, those Irregularities will bear no proportion to the recurrency of that Order which naturally results from ORIGINAL DESIGN.” (Abraham de Moivre, “The Doctrine of Chances”, 1718)

“Things of all kinds are subject to a universal law which may be called the law of large numbers. It consists in the fact that, if one observes very considerable numbers of events of the same nature, dependent on constant causes and causes which vary irregularly, sometimes in one direction, sometimes in the other, it is to say without their variation being progressive in any definite direction, one shall find, between these numbers, relations which are almost constant.” (Siméon-Denis Poisson, “Poisson’s Law of Large Numbers”, 1837)

“We know the laws of trial and error, of large numbers and probabilities. We know that these laws are part of the mathematical and mechanical fabric of the universe, and that they are also at play in biological processes. But, in the name of the experimental method and out of our poor knowledge, are we really entitled to claim that everything happens by chance, to the exclusion of all other possibilities?” (Albert Claude, [Nobel Prize Lecture], 1974)

“The trouble with integers is that we have examined only the small ones. Maybe all the exciting stuff happens at really big numbers, ones we can’t get our hand on or even begin to think about in any very definite way. So maybe all the action is really inaccessible and we’re just fiddling around. Our brains have evolved to get us out of the rain, find where the berries are, and keep us from getting killed. Our brains did not evolve to help us grasp really large numbers or to look at things in a hundred thousand dimensions.” (Paul Hauffman, “The Man Who Loves Only Numbers”, The Atlantic Magazine, Vol 260, No 5, 1987)

“The law of truly large numbers states: With a large enough sample, any outrageous thing is likely to happen.” (Frederick Mosteller, “Methods for Studying Coincidences Journal of the American Statistical Association, Volume 84, 1989)