Abbreviations

176 The shaping of necessity
diagram.) Now, this is equivalent to saying that the angles ΖΑΓ,
ΗΑΒ are both identical with the angle ΖΑΗ. ΑΓ and ΑΗ are the
same line, and ΑΒ and ΑΖ are the same line (this again can be
supported by some passages earlier in the text, but only with the
greatest difficulty – while of course this is immediately obvious in
the diagram). In this way, the claim can be propositionally deduced,
and perhaps some New Maths crank may still inflict such
proofs upon innocent children. But clearly the Greeks did not,
and the assertion was immediately supported by the diagram and
nothing else.
(f ) Finally, assertions may be intrinsically obvious. It might be thought
that Greek mathematics knows only a few of these, namely Euclid’s
axiomatic apparatus, but actually there are many intrinsically
obvious assertions which the Greeks do not formulate as axiomatic.
Take Euclid’s Elements iii.5, the clinching of the argument: 14
...� ΕΖ ëρα τ≥ ΕΗ �στ�ν �ση � �λáσσων τ≥ µείζονι· �περ
�στ�ν êδ�νατον
‘. . . so ΕΖ is equal to ΕΗ, the smaller to the larger; which is
impossible’.
The ‘which is impossible’ clause is a starting-point here, tantamount
to saying that the smaller cannot be equal to the larger. I do
not know of anything in Greek mathematics which legitimates this.
This is not meant as a criticism – I see, together with Euclid, that
the assumption is correct as far as he is concerned. 15 But it means
that the assumption is indeed a direct intuition.
Incidentally, how do we know, in this case, that ΕΖ is smaller
than ΕΗ? The answer is that we see this (fig. 5.5). This should
remind us that the distinction made between types of startingpoints
cuts across starting-points, not between them. Assertions
may be obvious through a variety of considerations.
To recapitulate, then, the sources of necessity identified above were:
(a) Explicit references
(b) The tool-box
14 176.15–16.
15 It ceases to be true as soon as set-theoretical discussions of infinity, so central to modern
mathematics, are started. The differences in cognitive styles between Greek and twentiethcentury
mathematics owe much to such real differences in mathematical content.