Page 11 - GALIET THE BEAUTIFUL INNATE: Meno & Theatetus Plato IV

Galiet & Galiet
simply multiplying areas by two does not equate to dividing the length of a side by two. In the end, Socrates leads the slave to accept that:
4(A/2)=2A
1. by means of the quadruple square and the
diagonal, and
2. the line of the double square is the diagonal of
the simple square.
a. That is, x=(2)1/2l.
This example does not fully and successfully support Socrates’ argument for reminiscence and immortality of the soul. First, the truth of this assertion depends on a supposition accepted by Socrates and the servant without discussion: the existence of at least one square, that is, a plane figure of four equal sides and four 90o angles. Since a century and a half or so, it is possible to conceive of worlds where this supposition does not occur. This suggests various alternative interpretations to Plato’s thesis.
If it means that one knows a-priori that in the realm of the Forms, the square constructed over the diagonal of another is equal to its double, the thesis is false. If a realm of geometric ideas exists,9 it consists of many sub-worlds, the greater number of which might exclude squares. If it means that one knows of the above a-priori to be valid in this world, which imitates or shares in an ideal geometrical realm in which squares exist, the thesis is quite arguable. It can be sustained that the problem of the geometric structure of real space needs to be resolved empirically, experimentally and not a-priori, for it is a question of physics and not of mathematics. If Plato’s thesis means that one knows a-priori hypothetical
9 Plato distinguished between the Realm of Forms and the Realm of geometric entities. The latter is also eternal, immutable and purely intelligible, but it was distinguished from it because it included many objects of one same form, i.e. many circles, etc.
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