1 An absent family of ideas .................................... 1
Although dicing is one of the oldest of human pastimes,
there is no known mathematics of randomness until the
Renaissance. None of the explanations of this fact is
compelling.
2 Duality ..................................................... 11
Probability, as we now conceive it, came into being about
1660. It was essentially dual, on the one hand having to
do with degrees of belief, on the other, with devices
tending to produce stable long-run frequencies.
3 Opinion ..................................................... 18
In the Renaissance, what was then called 'probability' was
an attribute of opinion, in contrast to knowledge, which
could only be obtained by demonstration. A probable
opinion was not one supported by evidence, but one which
was approved by some authority, or by the testimony of
respected judges.
4 Evidence .................................................... 31
Until the end of the Renaissance, one of our concepts of
evidence was lacking: that by which one thing can
indicate, contingently, the state of something else.
Demonstration, versimilitude and testimony were all
familiar concepts, but not this further idea of the
inductive evidence of things.
5 Signs ....................................................... 39
Probability is a child of the low sciences, such as
alchemy or medicine, which had to deal in opinion,
whereas the high sciences, such as astronomy or mechanics,
aimed at demonstrable knowledge. A chief concept of the
low sciences was, that of the sign, here described in some
detail. Observation of signs was conceived of as reading
testimony. Signs were more or less reliable. Thus on the
one hand a sign made an opinion probable (in the old
sense of Chapter 3) because it was furnished by the best
testimony of all. On the other hand, signs could be
assessed by the frequency with which they spoke truly.
At the end of the Renaissance, the sign was transformed
into the concept of evidence described in Chapter
4. This new kind of evidence conferred probability on
propositions, namely made them worthy of approval. But
it did so in virtue of the frequency with which it made
correct predictions. This transformation from sign into
evidence is the key to the emergence of a concept of
probability that is dual in the sense of Chapter 2.
6 The first calculations ...................................... 49
Some isolated calculations on chances, made before 1660,
are briefly described.
7 The Roannez circle (1654) ................................... 57
Some problems solved by Pascal set probability rolling.
From here until Chapter 17 Leibniz is used as a witness
to the early days of probability theory.
8 The great decision (1658?) .................................. 63
'Pascal's wager' for acting as if one believed in God is
the first well-understood contribution to decision theory.
9 The art of thinking (1662) .................................. 73
Something actually called 'probability' is first measured
in the Port Royal Logic, which is also one of the first
works to distinguish evidence, in the sense of Chapter
4, from testimony. The new awareness of probability,
evidence and conventional (as opposed to natural) sign,
is illustrated by work of Wilkins, first in 1640, before
the emergence of probability, and then in 1668, after the
emergence.
10 Probability and the law (1665) .............................. 85
While young and ignorant of the Paris developments
Leibniz proposed to measure degrees of proof and right in
law on a scale between 0 and 1, subject to a crude
calculation of what he called 'probability'.
11 Expectation (1657) .......................................... 92
Huygens wrote the first printed textbook of probability
using expectation as the central concept. His justification
of this concept is still of interest.
12 Political arithmetic (1662) ................................ 102
Graunt drew the first detailed statistical inferences
from the bills of mortality for the city of London,
and Petty urged the need for a central statistical
office.
13 Annuities (1671) ........................................... 111
Hudde and de Witt used Dutch annuity records to infer
a mortality curve on which to work out the fair price
for an annuity.
14 Equipossibility (1678) ..................................... 122
The definition of probability as a ratio among 'equally
possible cases' originates with Leibniz. The definition,
unintelligible to us, was natural at the time, for
possibility was either de re (about things) or de dicto
(about propositions). Probability was likewise either
about things, in the frequency sense, or about
propositions, in the epistemic sense. Thus the duality
of probability was preserved by the duality of possibility.
15 Inductive logic ............................................ 134
Leibniz anticipated Carnap's notion of inductive logic.
He could do so because of the central place occupied by
the concept of possibility in his scheme of metaphysics.
Within that scheme, a global system of inductive logic
makes more sense than Carnap's does in our modern metaphysics.
16 The art of conjecturing (1692[?] published 1713) ........... 143
The emergence of probability is completed with Jacques
Bernoulli's book, which both undertakes a self-conscious
analysis of the concept of probability, and proves the
first limit theorem.
17 The first limit theorem .................................... 154
The possible interpretations of Bernoulli's theorem are
described.
18 Design ..................................................... 166
The English conception of probability in the early
eighteenth century, guided by the Newtonian philosophy
espoused by members of the Royal Society, interprets the
stability of stochastic processes proven by the limit
theorems as evidence of divine design.
19 Induction (1737) ........................................... 176
Hume's sceptical problem of induction could not have arisen
much before 1660, for there was no concept of inductive
evidence in terms of which to raise it. Why did it have
to wait until 1737? So long as it was still believed that
demonstrative knowledge was possible, a knowledge in which
causes were proved from first principles, then Hume's
argument could always be stopped. It was necessary that
the distinction between opinion and knowledge should become
a matter of degree. That means that high and low science
had to collapse into one another. This had been an ongoing
process throughout the seventeenth century. It was
formalized by Berkeley who said that all causes were
merely signs. Causes had been the prerogative of high
science, and signs the tool of the low. Berkeley
identified them and Hume thereby became possible.
Bibliography .................................................. 186
Index ......................................................... 203
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