Axioms Fragment

1.130-32     G-c.1893-1

  130. The science which, next after logic, may be expected
to throw the most light upon philosophy, is mathematics. It
is historical fact, I believe, that it was the mathematicians
Thales, Pythagoras, and Plato who created metaphysics, and
that metaphysics has always been the ape of mathematics.
Seeing how the propositions of geometry flowed
demonstratively from a few postulates, men got the notion that the same
must be true in philosophy. But of late mathematicians have
fully agreed that the axioms of geometry (as they are wrongly
called) are not by any means evidently true. Euclid, be it

   * Metaphysics, bk. I, 982b-3a.
   ** See vol. 5, bk. II, ch. 7 and bk. III, chs. 2 and 3.
   *** Unpaginated fragment, c. 1893.
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observed, never pretended they were evident; he does not
reckon them among his {...}, or things everybody
knows, * but among the {...}, postulates, or things the
author must beg you to admit, because he is unable to prove
them. At any rate, it is now agreed that there is no reason
whatever to think the sum of the three angles of a triangle
precisely equal to 180 degrees. It is generally admitted that
the evidence is that the departure from 180 degrees ( if there
is any) will be greater the larger the triangle, and in the case of
a triangle having for its base the diameter of the earth's orbit
and for its apex the furthest star, the sum hardly can differ,
according to observation, so much as 0.1". It is probable the
discrepancy is far less. Nevertheless, there is an infinite
number of different possible values, of which precisely 180 degrees
is only one; so that the probability is as 1 to {.} or 0 to 1, that
the value is just 180 degrees. In other words, it seems for the
present impossible to suppose the postulates of geometry
precisely true. The matter is reduced to one of evidence; and as
absolute precision [is] beyond the reach of direct observation,
so it can never be rendered probable by evidence, which is
indirect observation.
  131. Thus, the postulates of geometry must go into the
number of things approximately true. It may be thousands of
years before men find out whether the sum of the three angles
of a triangle is greater or less than 180 degrees; but the
presumption is, it is one or the other.
  132. Now what is metaphysics, which has always formed
itself after the model of mathematics, to say to this state of
things? The mathematical axioms being discredited, are the
metaphysical ones to remain unquestioned? I trow not. There
is one proposition, now held to be very certain, though denied
throughout antiquity, namely that every event is precisely
determined by general laws, which evidently never can be
rendered probable by observation, and which, if admitted, must,
therefore, stand as self-evident. This is a metaphysical
postulate closely analogous to the postulates of geometry. Its fate is

  * Except the proposition that two lines cannot enclose a space, though only
one of the three best rnanuscripts places even this in the list. But what Euclid
meant was that two straight lines can have but one intersection, which is
evident.
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sealed. The geometrical axioms being exploded, this is for the
future untenable. Whenever we attempt to verify a physical
law, we find discrepancies between observation and theory,
which we rightly set down as errors of observation. But now it
appears we have no reason to deny that there are similar,
though no doubt far smaller, discrepancies between the law
and the real facts. As Lucretius says,* the atoms swerve from
the paths to which the laws of mechanics would confine them.
I do not now inquire whether there is or not any positive
evidence that this is so. What I am at present urging is that this
arbitrariness is a conception occurring in logic, encouraged by
mathematics, and ought to be regarded as a possible material
to be used in the construction of a philosophical theory, should
we find that it would suit the facts. We observe that
phenomena approach very closely to satisfying general laws; but
we have not the smallest reason for supposing that,they satisfy
them precisely.