Cogency v2 n2

Wittgenstein and the Logic of Deep Disagreement / D. M. GODDEN & W. H. BRENNER
Conversion to a new concept-formation is not something arbitrary, if
that implies “pointless”; nor is it irrational, if that implies inappropriately
motivated. This might be illustrated by John Wisdom’s story in Lectures on
the Foundations of Mathematics of how his tutor persuaded him that 3 x 0
equals 0. It struck the young pupil as more “logical” to say that it equals 3.
His tutor persuaded him otherwise, not by intimidation (pressing his authority
as teacher), but by way of an argument by analogy:
Three multiplied by three = three threes (3 x 3 = 3 + 3 + 3),
Three multiplied by two = two threes (3 x 2 = 3 + 3),
Three multiplied by one = one three (3 x 1 = 3),
Therefore, by analogy,
Three multiplied by zero = zero threes (3 x 0 = 0).
The young Wisdom had an argument too: that if you multiply 3 x’s by 0,
that would be equivalent to not multiplying them at all (“multiplying them
by nothing”)–not a bad argument, abstractly considered! He was led to abandon
it by being given a perspicuous representation of the math he was being
taught, so he could understand how – not “3 x 0 =3” – but “3 x 3 = 0” fits
into the system he was being taught. Had he not been persuaded but persisted
in going his own way, his elders might have been forced to conclude
that he was unteachable when it comes to arithmetic.
2
The Pythagoreans were brought up with an arithmetic in which the only
numbers were integers and fractions of integers. Imagine the controversy
that must have arisen when one member of the brotherhood pointed out
that the hypotenuse of the 1-1 Right Triangle is neither an integer nor a
fraction of integers. The controversy needn’t have consisted in one party
offering non-rational inducement to the other; it consisted, one might imagine,
in pointing out analogies and disanalogies between established numbers
and these new candidates for the title, and in ‘weighing’ the analogies
and disanalogies in the light of the place of numbers in their home context
of measurement and calculation. Now, of course, we include “irrationals”
among the ranks of numbers with no trace of the aversion and hesitation
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