FOM: Reductionism

[J P Mayberry]

	Robert Tragesser claims that "algebraic constructs" such as 
"groups, rings, modules, and fields, etc." (the list is Friedman's) are 
"materially/conceptually distinct from the set-theoretic organon 
(tools)" and that, moreover, "this distinction is more than just 
psychological". Now I think that Tragesser is just wrong about this: 
what notion of "group" do we have other than that of a structure 
satisfying the group axioms? What is that if not a "set-theoretic" 
construct? What methods of argument or definition are there in group 
theory that can't be - indeed, that aren't - justified on 
set- theoretical principles and without appealing to autonomous 
principles peculiar to group theory?
	In answer to Tragesser, Penelope Maddy has called attention to 
the analogy between the "foundational" roles of set theory in 
mathematics and of physics in natural science. Now no analogy is 
perfect, and the general point she is making is clear enough. But it 
seems to me that the analogy breaks down, and the way it breaks down 
goes right to the heart of the issue under discussion. Indeed, I think 
the breakdown is more illuminating than the analogy itself.
	That all of natural science rests on physics is something that 
has not been established *in detail*. We all believe that chemistry 
emerges out of quantum physics; but many of the details are hazy. The 
same goes for the emergence of biology out of chemistry, and, indeed, 
the details are much more hazy in that case. (What is the chemical 
explanation for embryonic development, for example? We have a picture 
in very broad brush strokes, but the details elude us. And, as everyone 
knows, the Good Lord is in the details.) When we come to psychology the 
"reduction" to physics is little more than a pius hope, though that 
hope may be useful in providing motivation for experimentalists. In 
short, the reduction of all natural science to physics must be 
classified as an Inspirational Ideal rather than as an accomplished 
fact.
 	In this respect, the contrast with the role of set theory in 
mathematics could scarcely be greater. For it is highly misleading to 
say that mathematics can be "reduced" to set theory On the contrary, 
*mainstream mathematics is constructed within set theory*. That is the 
way it is taught; that is the way it is practised. Set theory is built 
into our very conventions of exposition,. so "set theoretical 
foundations" just refers to how we *do* mathematics. When you think 
through what that means *in detail*, then all the talk about 
"alternative" foundations in category theory or topos theory seems a 
bit - indeed, more than a bit - unreal.

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J P Mayberry
J.P.Mayberry at bristol.ac.uk
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