Post
by **Wyman** » Sun Feb 26, 2017 3:24 pm

Supervenience Part I

The second chapter (more or less) of Chalmers' book lays out the idea of supervenience, for use in his later analysis of the relationship between consciousness and physical systems.

Supervenience purports to describe a relationship between two 'systems.' I want to first give an example of what I think is the - or 'a' - model of what philosophers base such ideas upon. The example I give is of number theory, but geometry could do as well. Chalmers uses mostly examples from the physical sciences such as the relation between physics and biology, but such examples already introduce many ambiguities that take away from elucidation of the concept.

Here is a short, user-friendly demonstration of how number theory 'logically supervenes upon' set theory. No mathematical skills are necessary and no technicalities are utilized (unlike certain philosophers who go on and on with useless technical language obscuring the forest for the trees):

First, number theory can be constructed fairly simply from ordered pairs. Everyone knows what an ordered pair is - you used them extensively in geometry class to denote points on the Cartesian plane - (a,b). But ordered pairs do not have to be thought of as points, of course (and here is where number theory meets analytic geometry, or geometry meets algebra, but that is another story).

From the ordered pair, it is easy to define ordered triples, quadruples, and all the way up.

From ordered sequences, it is easy to define relations. Relations are sets of ordered pairs. We define certain characteristics of relations, such as symmetry, reflexivity, and transitivity (and their negations).

- Relations are of the form 'aRb' and we all know relations from grade school math - a=b, a>b, etc.. As an example, 'equality' is symmetrical,

reflexive and transitive, since a=a (reflexive), if a=b then b=a (symmetric) and if a=b and b=c then a=c (transitive). To see how relations are

just sets of ordered pairs, think of the natural numbers (1,2,3...). The '>' relation is just the set of all ordered pairs (x,y) such that x is greater

than y.

From relations, we have the tools necessary to create the axioms of number theory (by which I mean the natural numbers - from them we can build and get to rationals and irrationals). For instance, the 'basic' axiomatic relation used in number theory is just the 'successor' relation.

Now, back to set theory. An ordered pair is defined as (a,b) = {{a},{a,b}}. Don't ask why or try to understand it - all it means is that for every situation in which (a,b) is used (the ordered pair), we could replace it by the set-theoretical equivalent, {{a},{a,b}} - i.e. it just 'comes out right.' Any attempt to use the set theory notation beyond simple ordered pairs - to triples and beyond and relations - leads to pages of chicken-scratch, with brackets upon brackets upon brackets such that no one can keep it straight (only computers). But we know, in theory, that all ordered pairs can be so translated, and that is the important part.

The only other important ingredient are the numbers themselves. These can be defined in terms of sets. Very smart mathematicians such as Von Neumann and Cantor came up with satisfactory set-theoretic definitions of natural numbers. What they boil down to, simplistically, is that the number 'one' is the set of all sets with one element, 'two' is the set of all sets with two elements, etc.. If you wonder how that is not circular (defining 'one' in terms of sets with 'one' element), that is where the magic of the smart mathematicians comes in and you'll just have to take my word for it (it is not that difficult, just not necessary to understand what I am explaining).

Do not let any math nerds tell you it is 'more complicated than that.' It is, slightly, but the complications are not that complicated and if you understand what I've laid out, you understand how number theory is definable in terms of set theory. Nothing in number theory can fail to be expressible in set-theoretic terms. Therefore, any 'fact' of number theory supervenes upon set theory. That is, any 'fact,' such as '7 + 5 = 12' is in a way 'determined' by set theory - if the fact is proved in set theory, it cannot fail to be proved in number theory.

The analogue to what Chalmers discusses would be something like: any 'fact' (true proposition) of biology can be described, theoretically, in terms of physics (quarks and such). So any fact of biology cannot fail to be a fact of physics - stipulating that there are such things as 'truths' of biology and 'truths' of physics.

Chalmers wants to see whether the 'facts' of consciousness supervene upon the 'facts' of physics. If they do not, he says that that shows that consciousness cannot be described within a physical theory.