popper-logic-scientific-discovery

sequence, so that we can attribute to each a positive probability in such
a way that their sum converges and does not exceed unity. And it
means, further, that to laws which appear earlier in this sequence, a
greater probability must be attributed (in general) than to laws which
appear later in the sequence. We should therefore have to make sure
that the following important consistency condition is satisfied:
Our method of ordering the laws must never place a law before another one if it is
possible to prove that the probability of the latter is greater than that of the former.
Jeffreys and Wrinch had some intuitive reasons to believe that a
method of ordering the laws satisfying this consistency condition may
be found: they proposed to order the explanatory theories according to
their decreasing simplicity (‘simplicity postulate’), or according to
their increasing complexity, measuring complexity by the number of
the adjustable parameters of the law. But it can be shown (and it will be
shown in appendix *viii) that this method of ordering, or any other
possible method, violates the consistency condition.
Thus we obtain p(a, e) = 0 for all explanatory hypotheses, whatever
the data e may be; that is to say, we obtain (2), and thereby indirectly
(1).
(An interesting aspect of this last proof is that it is valid even in a
finite universe, provided our explanatory hypotheses are formulated in
a mathematical language which allows for an infinity of (mutually
exclusive) hypotheses. For example, we may construct the following
universe. 12 On a much extended chessboard, little discs or draught
pieces are placed by somebody according to the following rule: there is
a mathematically defined function, or curve, known to him but not to
us, and the discs may be placed only in squares which lie on the curve;
within the limits determined by this rule, they may be placed at random.
Our task is to observe the placing of the discs, and to find an
‘explanatory theory’, that is to say, the unknown mathematical curve, if
possible, or one very close to it. Clearly, there will be an infinity of
possible solutions any two of which are incompatible, although some
of them will be indistinguishable with respect to the discs placed on
the board. Any of these theories may, of course, be ‘refuted’ by discs
placed on the board after the theory was announced. Although the
12 A similar example is used in appendix *viii, text to note 2.
appendix *vii 385