Joined: 09 Mar 2007
Location: Wembley, London, UK
|Posted: Sun Jan 14, 2018 1:46 pm Post subject: A New Look At Plato's Arithmetic
Plato makes this observation in his Parmenides (154b-d):
“If one thing is actually older than another it cannot be becoming older still, nor be becoming younger still, by any more than their original differences in age; for if equals are added to unequals the difference that results, in time or any other magnitude, will always be the same … [But] if an equal time is added to a greater time and to a less the greater will exceed the less by a smaller part.”
This is how David Fowler the Maths Historian introduces it in his Mathematics of Plato’s Academy (p.42 ed.1)
There’s no need to fuss over the way this is put. It’s meaning is clear: I am 7 years older than my brother. When I was 14, he was half my age, when I was twenty one, he was two thirds my age. But when I reached 70 he was nine tenths my age and seemingly catching me up. But of course he never will, try as he might, for there will always be seven years difference between us, even after we have both long turned up our toes.
Fowler goes on to apply this proposition according to his own preoccupation; but, as usually is the case, whatever Plato says can be applied to many preoccupations. I, for example, would like to apply it to other observations from the Dialogues in an attempt to discover what is behind Plato's and Socrates' thoughts on mathematics. We might look at David Fowler’s ideas at a much later stage.
The first thing to say is that we are talking of Equality here. This is of perennial interest to humankind because it is so closely tied to the concept of Justice. Young children soon learn to say “that’s not fair” and they should be listened to carefully to find out exactly what they are saying. Is it the seed form of “that’s unjust” or merely “gimme!”? At some stage the two kind of merge together and so deep is the fusion that only a deeper reasoning can unknot it. It must be obvious to all that hours of arguing from one’s own viewpoint without even glancing at the opposing view is eventually futile. I have experienced this many times on this very forum, and it can be quite chastening when you realise it is you who is the impediment - it is you who is being unreasonable.
Equality is the foundation of Justice. But it has to be shown clearly what this means otherwise it remains a cliché, worse still, a mere assumption.
In order to do this I need to pull a few key passages from the Dialogues and hopefully ‘a picture will emerge’.
Equality is not an assumption. Actually it is a Form, an eidos (a ‘look’ or ‘glance’ at the eternal – or perhaps it is that the eternal ‘glances’ at us, which actually makes more sense).
We look at Phaedo 74a where Socrates is trying to convey to his friends the meaning of anamnesis – remembrance. He is explaining to Simmias that remembrance can be triggered both by things which are similar and dissimilar.
“Consider whether this is the case: We say that there is something that is equal. I do not mean a stick equal to a stick or a stone equal to a stone, but something beyond all these, the Equal itself. Shall we say that this exists or not?”
The term “the Equal itself” - auto ton ison – is sometimes translated as “equal in the abstract” (see for instance Harold North Fowler’s Loeb edition). I cannot express what a terrible translation this is, completely skewing Plato’s intentions. It is Aristotle who thinks the transcendental is abstracted from sense data. Socrates has already made it clear that he isn’t talking of sticks and stones and anything that can be seen, touched, heard or tasted. Quite the contrary, it is these things when in relation to one another acquire a faint echo of already existing Form, Isos. Fowler is trying to force Plato’s flowers into an Aristotelian vase and they don't fit.
I need no more from this passage at present, but it has already shown us that the equality we search for in Justice, is equality itself. Another way of putting it is that the Forms of Justice and Equality participate or share in one another.
This though is only the beginning of the journey. Plato has much more to say, and it is for our keen pleasure that he sows his words alternatively and craftily into his longer texts and in a way that is often tangential to the accepted theme he appears to discuss.