More History of London

See Small Fragment of the History of The Corporation of The City of London:

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That sent me off thinking about triangles again, so I spent the whole evening talking to Google AI about Euclid and Aristotle. It told me Aristotle didn't refer to anything in Euclid's Elements because Aristotle died 20 years before Euclid wrote the Elements.  (See Andrei Rodin's letter in Support of Svetlana Mesyats --- I hope she didn't spend too much of her grant money on AI chatbots!). 

I made a video of the transcript but YouTube seems to have eaten it. Ah, no, it eventually showed up. I guess Google AI had to do character recognition to check that all this wasn't against community guidelines, ... Wow, that must have burned some GPU cycles, ...

This is to do with the comment I made on About Logic - Analytic and Synthetic Mathematics on the connection between Euclid's Proposition 32 in Book 1 and the Parallel Postulate.

Q. what is the final lemma in book 13 of euclid's elements?

A. 

Q. 

Q. I wonder whether this proposition or maybe proposition 18 is what Aristotle was referring to (Metaphysics, Book 9, Part 9) when he wrote "It is an activity also that geometrical constructions are discovered; for we find them by dividing. If the figures had been already divided, the constructions would have been obvious; but as it is they are present only potentially"? i.e. that the geometrical constructions in the Elements were all produced by analysis, starting with the whole three dimensional (steroidal) space.

A. 

Q. What does it mean to say that "actuality is prior to potentiality." Further on in Metaphysics Book 9 Part 9 Aristotle writes "Obviously, therefore, the potentially existing constructions are discovered by being brought to actuality; the reason is that the geometer’s thinking is an actuality; so that the potency proceeds from an actuality; and therefore it is by making constructions that people come to know them (though the single actuality is later in generation than the corresponding potency)." Clearly then "prior to" does not mean "earlier in generation" when referred to the single actuality of the geometer's thought.

A. 

Q. It never occurred to me that "actuality is prior to potentiality" is here being applied to the geometer. It is being applied to the thought of the geometer, who is potentially measuring a whole, namely the earth, that being the meaning of the term geometry. This is why I think the actuality of the thought of the geometer is the apprehension of the whole space of solid bodies, and that is why I think proposition 18 of Euclid's Elements is related to this. In Aristotle's Physics, Part 9 we read "Necessity in mathematics is in a way similar to necessity in things which come to be through the operation of nature. Since a straight line is what it is, it is necessary that the angles of a triangle should equal two right angles. But not conversely; though if the angles are not equal to two right angles, then the straight line is not what it is either." He goes on to say that it is not so in things brought to some end or for some purpose.

A. 

Q. It seems my use of the phrase "potentially measuring the earth" has been misunderstood, as has my use of the term apprehension when referred to a whole.

A. 

Q. My intuition about Proposition 10 (and Proposition 18) is that they give a way to construct an infinite number of geometrically consistent "meshes" which bear certain definite proportions (in the sense of book X) each to the others and which thereby demonstrate the necessity of all the prior propositions, starting from Proposition 1 of Book 1. In this structure I would expect Proposition 16 of book 10 to be pivotal, because of what Aristotle wrote about its unique place in Scientific Knowledge (I forget where he wrote it) as being proper to five distinct sciences. 

Thank you for this exchange, it was helpful. If someone at Google finds it useful and they make the world a better place then I am sure Aristotle will have been pleased to know!

A. 

See About Logic on Natural Language and the Meaning of σύνεσις. 

I think they really like getting free training data for their AI. I wonder whether it does any good though. I feel like a vampire has been sucking my blood all night. 

See Philip Wadler doing Stand-Up in Edinburgh. 

The last paragraph of Aristotle's Physics, Book II, Part 9: 

The necessary in nature, then, is plainly what we call by the name of matter, and the changes in it. Both causes must be stated by the physicist, but especially the end; for that is the cause of the matter, not vice versa; and the end is 'that for the sake of which', and the beginning starts from the definition or essence; as in artificial products, since a house is of such-and-such a kind, certain things must necessarily come to be or be there already, or since health is this, these things must necessarily come to be or be there already. Similarly if man is this, then these; if these, then those. Perhaps the necessary is present also in the definition. For if one defines the operation of sawing as being a certain kind of dividing, then this cannot come about unless the saw has teeth of a certain kind; and these cannot be unless it is of iron. For in the definition too there are some parts that are, as it were, its matter. 

See Tim Maudlin with an Interesting Idea About Relativity.

Norman Wildberger is using this stuff to write his book: 

See his Box Arithmetic Book playlist.

My comment:

4:12 It's nice to be able to look stuff up, but you also need to think about who controls these resources and who has access to them. I don't think we can just assume that these issues will be ironed out eventually. I am not happy with AI as it is presently for that reason. 

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