The History of Mathematics: A Brief Course

522 18. PROBABILITY AND STATISTICS

standard deviation. This work brought the normal distribution into a more or less

standard form, and it is now often referred to as the Gaussian distribution.

1.10. Philosophical issues. The notions of chance and necessity have always

played a large role in philosophical speculation; in fact, most books on logic are

kept in the philosophy sections of libraries. Many of the mathematicians who have

worked in this area have had a strong interest in philosophy and have speculated on

what probability means. In so doing, they have come up against the same difficulties

that confront natural philosophers when trying to explain how induction works.

Granted that like Pavlov's dogs and Skinner's pigeons (see Chapter 1), human

beings tend to form expectations based on frequent, but not necessarily invariable

conjunctions of events and seem to find it very difficult to suspend judgment and live

with no belief where there is no evidence,^13 can philosophy offer us any assurance

that proceeding by induction based on probability and statistics is any better than,

say, divination such as one finds in the / Ching? Are insurance companies acting

on pure faith when they offer to bet us that we will survive long enough to pay

them more money in premiums than they will pay out when we die? If probability

is a subjective matter, is subjectivity the same as arbitrariness?

What, then, is probability, when applied to the physical world? Is it merely

a matter of frequency of observation, and consequently objective? Or do human

beings have some innate faculty for assigning probabilities? For example, when we

toss a coin twice, there are four distinguishable outcomes: HH, HT, ÔÇ, TT. Are

these four equally likely? If one does not know the order of the tosses, only three

possibilities can be distinguished: two heads, two tails, and one of each. Should

those be regarded as equally likely, or should we imagine that we do know the

order and distinguish all four possibilities?^14 Philosophers still argue over such

matters. Simeon-Denis Poisson (1781-1840) seemed to be having it both ways in

his Recherches sur la probabilite des jugemens (Investigations into the Plausibility

of Inferences) when he wrote that

The probability of an event is the reason we have to believe that

it has taken place, or that it will take place.

and then immediately followed up with

The measure of the probability of an event is the ratio of the num-

ber of cases favorable to that event, to the total number of cases

favorable or contrary.

In the first statement, he appeared to be defining probability as a subjective

event, one's own personal reason, but then proceeded to make that reason an objec-

tive thing by assuming equal likelihood of all outcomes. Without some restriction

on the universe of discourse, these definitions are not very useful. We do not know,

for example, whether our automobile will start tomorrow morning or not, but if

(^13) In his Formal Logic, Augustus de Morgan imagined asking a person selected at random for an
opinion whether the volcanoes—he meant craters—on the unseen side of the moon were larger
than those on the side we can see. He concluded, "The odds are, that though he has never thought
of the question, he has a pretty stiff opinion in three seconds."
(^14) If the answer to that question seems intuitively obvious, please note that in more exotic appli-
cations of statistics, such as in quantum mechanics, either possibility can occur. Fermions have
wave functions that are antisymmetric, and they distinguish between HT and TH; bosons have
symmetric wave functions and do not distinguish them.