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calculations in question, but such as they would be if they adapted themselves exactly to the
augmentations or diminutions of what might be called the mathematical reasons for believing (les
raisons mathématiques de croire). One must avoid regarding these reasons for belief as intrinsic
characteristics of things, thereby restoring objectivity to probability. These reasons are themselves
subjective and consist in our knowledge, not of the causes of an expected and unknown events but of
the limits of the field beyond which we are sure that these causes will not operate, and of the division
of this field into two unequal portions, one called favorable odds, the other unfavorable odds, whose
inequality can be calculated. I do not know by what combination of physical, physiological, and
psychological causes the hand of a child draws one number and not another at a lottery; but I know
(negative certainty) that the number which comes up will be between 1 and 100 and not above, since
there are only 100 tickets, and further (positive certainty) I know that I have 10 tickets and that
consequently there are 90 I do not have. Here the hypothesis consists of considering these two
certainties as the partial equivalent of the knowledge of the causes, which I cannot have.
Once this hypothesis is accepted, everything follows easily, and it appears fairly natural to think
that the degree of belief of a man invincibly ignorant of the real causes must be proportional to the
mathematical value of the reasons as I have defined them. From this point of view, the calculation of
credibility, that is, what may be affirmed and denied, would be the very kind of algebraic logic that
modern logicians have dreamed of; the symmetrical counterpart of this science would be the
utilitarian doctrine of Bentham, the ethics of a plus b which would be called the calculation of
positive and negative desirabilities. But for the mathematician as for the utilitarian the difficulty is
justifying the duty they impose on me to believe or desire more or less, or otherwise than I do in fact
believe and desire. And as far as the mathematicians are concerned, why should I accept the
debatable hypothesis which serves as the basis for their elaborate scaffolding of formulas? Now in
reality these mathematical reasons for belief which I was discussing are to belief what, according to
the psychophysicists, the degree of outside excitation, such as luminous intensity, is to the degree of
the impression of light. Not that it is appropriate to extend to this new case the famous logarithm of
sensations. But according to whether faith is influenced by desire or by repulsion it should be noted
that the increases proceed more rapidly or more slowly than the parallel increases of mathematical
probability. The inhabitants of a town of 10,000 are frightened when 10 cases of cholera break out. If
the next day there are 20 cases, their alarm will more than double, whereas if there were 20 cases
from the beginning, the initial alarm would not have been noticeably different. If I have 10 lottery
tickets and take 10 more, will my hopes of winning double? Not at all, though certain personalities
predisposed to fancy and inclined to hope rather than to fear may be exceptions on this point.
Then note that in the same man and concerning the same type of event, increases of faith (foi), after
being more rapid than parallel increases of probability, can become slower, or vice versa. In general
when belief has nearly attained its maximum state (which we call certitude), its rate of increase slows
down appreciably. In the sciences one may note the singular resistance to the definitive establishment
of a theory which was already recognized as almost proved at a time when it could explain only half
as many facts. Each day new facts are found to support the transformist or atomistic hypotheses, but
the faith of their proponents is not thereby increased nearly as much as it was at first by the discovery
of much less convincing facts. With simple indications Newton almost acknowledged his conjecture
as a law. Since then multiple astronomical observations have increased the proofs of his theory a
hundred times over, but the scholars’ faith in it could not become one hundred times greater. Even
though certitude is not essentially different from the other degrees of belief and is simply one of the
extremes of their series, the passage from belief to certainty we know to be a kind of change of state,