Saturday, August 25, 2012

Euclid's Common Notions

I was thinking about Euclid's common notions yesterday, and in particular about what we get if we focus only on them. The common notions, or axioms, if you will remember, are (in usual translation, going back to Heath):

(1) Things which are equal to the same thing are also equal to one another.
(2) If equals are added to equals, then the wholes are equal.
(3) If equals are subtracted from equals, then the remainders are equal.
(4) Things which coincide with one another equal one another.
(5) The whole is greater than the part.

What do we get if we look only at this? A very simple mereology, that is, a system of reasoning about parts and wholes. Axiom 5 tells us that a whole is distinguishable from its parts, and vice versa. The other four axioms tell us what we can conclude if parts or wholes are indistinguishable (indiscernible/equal/exactly similar) in some way. Axiom 1 tells us how things relate to each other when they are both indistinguishable from the same thing (the answer is: indistinguishably!); it's about equality with respect to equality. Axiom 2 is about equality with respect to addition; in other words, they tell us something we can conclude about the wholes when parts are joined to other parts. Axiom 3 is about equality with respect to subtraction; that is, they tell us something we can conclude about leftover parts when we remove parts from a whole.

Axiom 4 is interesting. The Greek means something like "Things that fit onto each other are alike/equal"; it tells us that if you can fit something (part or whole) onto something (part or whole) that they are indistinguishable. So this is about equality with respect to superposition, whether of parts or wholes. Axioms 2, 3, and 4, then, allow for three kinds of mereological operation: joining part to part in order to get equal wholes (addition), removing parts from wholes in order to get equal parts (subtraction), and fitting part or whole on top of part or whole in order to get equals (superposition).

It should be noted that in none of these cases do the parts and wholes actually have to be spatial parts and wholes. For instance, here is a pretty straightforward case of something that can be fit to the same mereological principles: propositions (wholes) and terms (parts). If intersubstitutable terms are predicated of intersubstitutable terms (addition), the propositions are intersubstitutable; if intersubstitutable terms are removed from intersubstitutable propositions, the leftover terms are intersubstitutable; substitution is 'coinciding' or 'fitting on top of'; propositions include more than their terms; and things intersubstitutable with the same thing are themselves intersubstitutable.

We get something geometrical in our usual sense only when we add definitions and postulates. The definitions give us kinds of objects in space, and properties of those objects; the postulates are guidelines for accurately constructing those same objects (they are things you are asked to do when drawing diagrams -- for instance, the first postulate literally asks you to bring your strokes directly from marker to marker, while the fourth asks that all your upright corners be alike). Any Euclid-style approach to geometry, then, has three elements: definitions of kinds of spatial objects and their properties; a method for constructing spatial objects; and a mereology, or system for reasoning in terms of parts and wholes. We can get variations by changing any of these. We could change the definitions; and in a sense a part of Newtonian physics is Euclid where our definitions are not of kinds of spatial objects but kinds of locomotion. We could change the methods of construction, and this is where we get all sorts of different geometries, ranging from Euclidian geometry with neusis to Riemann and others. And we could also change the mereology.


Thinking more about this, there are famously three Books of Euclid's Elements that are independent of any other book: Book I, Book V, and Book VII. These all share the same common notions (mereology) and postulates (method of construction), but they differ according to their definitions. The geometry of Book I is a comparative mereology of dimensional constructions -- dimensional here being what I meant above by 'spatial', and meaning 'having to do with length, breadth, etc.'. The definitions of Book V have to do with measured magnitudes (a magnitude is part of another when it measures the other); and those of Book VII are concerned with measured numbers (a number is part of another when it measures the other). So even within Euclidean geometry we see the way in which definitions establish the objects that are reasoned about with a simple mereology and a method of construction, in order to get a geometrical theory.

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