III
The systems we have discussed—logical, mathe-
matical, physical, metaphysical, ethical, or
legal—all
have in common a dogmatic character. Axioms were
supposed to compel assent through their inherent self-
evidence transmitted to later propositions by means
of demonstrations. In the modern period we see this
conception gradually
disintegrating, until we reach by
degrees our present conception of
axiomatics. This
conception was attained by a progressive dissociation
of the two hitherto intimately related components of
the idea of axiom
(self-evident and primary prop-
osition).
The transformation was accomplished in
two stages: the first, in the
seventeenth century in
connection with the advent of experimental
physics;
the second, the beginning of which can be dated in
the early
nineteenth century, with the construction of
non-Euclidean geometries.
Descartes still required that the principles of philos-
ophy (including natural philosophy or physics)
satisfy
two conditions at the same time: “... one, that
they
be so clear and so self-evident that the human mind
cannot doubt
their truth when it concentrates on judg-
ing
them; second, that the knowledge of other things
depends on the principles
which can be known without
these other things but not
conversely” (Letter, preface
to the French version of the Principles of Philosophy,
1647). Nevertheless, he
admitted, and practiced him-
self, when he
needed to, another mode of exposition,
although he regarded it as less
perfect than the deduc-
tive mode. This other
mode consisted in regarding basic
propositions (general principles) not as
principles of
demonstration but as requiring, on the contrary, proof
by the empirical verification of their consequences. In
this he was in
agreement, albeit accidentally, with the
practice of the new
seventeenth-century physicists
who, following Galileo and Torricelli, were
active
around him and Mersenne: Pascal, Roberval, Gassendi.
These
principles, that is to say, the propositions from
which deduction starts,
are only “suppositions” or
“hypotheses” in two senses of these words: premisses
and conjectures. For, as Pascal maintained, in physics
the experiments
furnish the true principles, that is, the
foundations of our knowledge. When in the order of
exposition,
though not in the order of discovery, the
order of the propositions is
always the same, viz., that
of a deductive synthesis, the meaning of the
proof is
reversed: instead of extending beyond the premisses
to the
consequences, the truth rebounds from the con-
sequences to support the premisses.
The operational distinction in Physics between the
propositions which it
states as its principles and those
that it invokes to establish them, is
nowhere better
illustrated than in the Newtonian theory of
gravitation,
expounded in Book III of the Principia;
it instigated
the battle between Cartesians and Newtonians that
lasted
for half a century. One of the chief reasons for
the antagonism of the
Cartesians was the idea of at-
traction or
action at a distance, which, far from being
a clear idea demanding assent
as self-evident, was on
the contrary, unintelligible to them. To which
the
Newtonians replied: clear or obscure, self-evident or
not, the
principle of gravitation is still a compelling
truth because experience
confirms it in very many
precise ways.
Thus classical science was faced with the institution
of a sharp separation
between the experimental method
of the physicists and the demonstrative
method of the
mathematicians. The result was the uncertain status
of
Mechanics, halfway between geometry and physics.
In the middle of the
eighteenth century, the Academy
of Berlin offered a prize for the best
answer to the
question whether the laws of nature are necessary or
contingent truths, that is to say, whether they are
directly or indirectly
purely rational statements or, on
the contrary, simply experimental
findings. D'Alem-
bert, a declared Newtonian,
replied in the way a
Cartesian would, and presented Dynamics as a demon-
strative science. And in the next
century, William
Whewell still wondered about the “paradox of
neces-
sary truths acquired by
experience,” which suggested
to him his theory about the
“progress of evidence.”
A nondogmatic use of deductive method was, in a
manner of speaking, quite
ancient. Without discussing
the accidental employment of this method by
the
mathematician in his indirect proofs, or by the dialec-
tician in his refutations through
reduction of his adver-
sary's arguments to
absurdity, we find in antiquity a
systematic use of the
hypothetico-deductive method
in expounding one of the sciences of nature
which had
already reached a high level of development. Alongside
of
physical astronomy there had actually developed,
after Plato, a so-called
formal or mathematical astron-
omy which was
connected especially with Ptolemy's
work, and which continued to thrive in
the Middle
Ages; it did not require that its principles should be
true
but only that they allow one to calculate exactly
the empirical data
(“to save the phenomena,”
σώζειν
τὰ
φαινόμενα).
This was the notion to which Osiander
also referred when, in his
introduction to Copernicus'
De revolutionibus orbium coelestium (1543), he
asked
the astronomer “to imagine and invent any hypotheses
whatsoever,” adding: “it is not necessary for these
hypotheses to be true or even probable, the only suffi-
cient condition is that they must lend themselves
to
a calculation which agrees with the observations.”
However, the new physics was just as opposed to these
hypotheses which are
neither true nor false, as it was
to dogmatic theses. If it no longer
posits its principles
as self-evident, it nevertheless proposes them as
truths.
As Newton expressly says in the first of his Rules for
Philosophy (Regulae
philosophandi) it does not suffice
that the causes invoked by the
scientist should be fit
to explain the phenomena, but they must be true
in
addition. This new way of employing the hypothetico-
deductive method would
discredit in time and relegate
to oblivion the ancient method of
“saving the phe-
nomena.”
To summarize, the deductive order, or axiomatic
order in the wide sense, may
function in three different
ways, depending on the identifiable nature of
its basic
propositions: (1) a categorical deduction which dem-
onstrates the truth of the consequences
by the truth
of the principles laid down dogmatically; (2) a hypo-
thetical deduction which proves a posteriori the truth
of its provisional hypotheses
by the truth of their con-
sequences;
finally, (3) a pure hypothetical deduction
whose principles are introduced
as fictions removed
from the domain of what is true or false, so that
truth
comes into play only on the level of logical conse-
quences. Whereas Mechanics tried for a while to per-
severe still in the first conception
(categorical deduc-
tion), classical physics
opted clearly for the second (a
posteriori
hypothetical inference), rejecting the two
past legacies of demonstrative
physics and fictive
physics. Even when Physics seeks to combine a body
of experimental laws into a unifying theory and even
when it reduces this
theory to a certain mathematical
structure, as did J. Lagrange in his Analytical Me-
chanics
(Mécanique analytique, 1788), Joseph
Fourier
in his thermodynamics (Théorie
analytique de la cha-
leur, 1822),
and J. C. Maxwell in his Treatise on Elec-
tricity and Magnetism (1873), it does not
postulate its
axioms either as necessary truths or as arbitrary
fictions,
but as a system of hypotheses; the truth of these hy-
potheses being tested finally by the
precise and un-
erring agreement of the many
consequences of the
hypotheses with the experiential data. It was only
around 1900 that this radically hypothetical conception
of Physics was
revived, on the one hand with the
critique of scientific dogmatism by H.
Poincaré, P.
Duhem, and H. Vaihinger's Philosophy of As If (Philos-
ophie des als ob, 1911); on the other
hand, and above
all, the hypothetical conception came to the fore when
the
logico-mathematical sciences were put into axiom-
atic form, in the strict and most rigorous sense which
the term
“axiomatic” has assumed today.
