Marcus G. Daniels wrote: > If Math is a way to create nodes and augment a network of related formal > systems, it doesn't mean that these transactions are against the same > graph, or even that it is necessary to go to the first node of a graph > to understand why it is valid to add this or that node to a large > graph. If it were necessary, then I suppose Math would be `bigger' > than the total set of formal systems.
Well, the relationships between formal systems is probably not a clean graph where each node is completely distinct. It's probably more like a dynamically evolving sub-structure that generally supports many less dynamic superstructures. (I'm thinking of it in analogy with neutral networks, where the formal systems are at the phenomenal level and the alphabet, grammar rules, and axioms are at the genomic level.) We'd need some sort of measure of the difference between formal systems. To my very limited knowledge, we don't have that, in general. So, it's not obviously appropriate to think about the "size" of math in relation to that of formal systems graphically (or even discretely), as you've done. I also don't intend to say that math is "bigger" than formal systems, only that it is "more". I know I said "larger"... sorry for that. And I also don't think it's necessarily appropriate to speak of the relations as "transactions" ... at least not yet, not until we have some concept of the dynamism (if any) of the substrate on which math sets (which, in my more speculative hours, I tend to think emerges from the way our central nervous systems relate to the world out there). The existence proof I'm pointing out as an example of how math is more than formal systems (Tarski's indefinability or the GIT) merely shows that what we call math is not fully captured by formal systems (or _a_ formal system). Proofs of the GIT go a bit further in that they show mathematical techniques for showing that math is not fully captured by formal systems. But it's not a good enough measure to describe all the differences between elements of any set of formal systems. Besides, the fact that, when using math, we can swap out one referent (e.g. groups, sets, and topological spaces using the same commutative diagram) for another also demonstrates the point that math is more than formal systems, because formal systems are "rigid". So even _if_ we could think of this relational structure as a graph, the fact that the graph might change based on aspect without damaging the math involved makes it even clearer. Then again, I've got a head cold at the moment. [grin] So, what I think is clear might well be confused. -- glen e. p. ropella, 971-219-3846, http://tempusdictum.com ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org
