Probability: Degree of
(un)certainty expressed by a number between 0 and 1 inclusive and
attributed to a proposition or its
meaning, with 0 meaning that the
proposition is false, and 1 meaning that the proposition is
The serious mathematical study of the idea of probability was started in
the 17th Century, by Pascal and Fermat, and a little later by
Bernouilli, who produced one of its first treatises, the "Ars
Conjectandi", that also stands at the beginning of the mathematical
study of statistics. There were some forerunners, like Cardano and De
Witt, who were respectively concerned with games of chance and the
profitability of insurance, but it remains a somewhat curious fact that
although the concept of chance is
very old, since the ancient Greeks already knew it and wondered about
it, the mathematics of chance is relatively new.
After Pascal, Fermat and Bernouilli, quite a lot was done in probability
by mathematicians - De Moivre, Laplace, Gauss, Poisson, Boole, Pearson,
to name some - but the whole subject only got a sound axiomatic
foundation by A. Kolmogorov in the 1930ies, who proposed the following
axioms for it.
Let s and t be arbitrary formulas, and let "p(.)" be a function that is
read "the probability of":
K1. 0 <= p(s) <= 1.
K2. If s is logically true, then p(s)=1.
K3. If s logically implies t, then p(s) <= p(t).
K4. If s and t are logically inconsistent, then p(s or t)=p(s)+p(t).
These axioms allow the derivation of all theorems that were commonly
accepted by mathematicians for finite probabilities.
Since then, the mathematics of probability is a branch of what is known
as measure theory, which is concerned with sums and integrals,
and which has a fine abstract introduction by P. Halmos: "Measure
Also, Kolmogorov's axiom 4 is supplemented or strengthened in such
measure theoretical treatments by an axiom that allows the formation of
infinite sums, which is required for many applications and theorems.
Even so, the philosophical foundations of probability are far less
certain than the by now widely received mathematics of it, and there are
quite a few distinct interpretations of what probability and/or chance
really are, which differ from the assumption that in the end they are
wholly subjective or a mere measure of human ignorance to the assumption
that there really is chance and indeterminism in nature, and probability
theory represents it, to various positions inbetween.
What increased these problems in the 20th Century were the arisal of
indeterminism in physics (quantum mechanics) and the arisal of social
statistics and the testing of hypotheses, both of which have a fairly
firm mathematical foundation in measure theory, but a far less firm
foundation when one inquires into their possible factual truth or
There are very many texts about probability and its interpretations. If
we suppose for the moment that the mathematics of it is widely accepted
and well expounded by measure-theoretical mathematical texts, or
textbooks that are derived from this treatment, there still remain many
problems with what it all might mean. Seven able and interesting
W. Stegmüller: Probleme und Resultaten
der analytischen und Wissenschaftsphilosophie.
E. Adams: A Primer of Probability Logic.
C. Howson & P. Urbach: Scientific Reasoning - The Bayesian
R. Weatherford: Philosophical Foundations of Probability Theory.
A.A. Sheshnikov: Problems in Probability Theory, Mathematical.
Statistics and Theory of Random Functions.
H. Freudenthal: Waarschijnlijkheidstheorie.
T. Fine: Theories of probability.