Glen - > Yes, I can see that. The answer(s) to the question, though, is still > interesting. I tried to allude to it by offering the 3 alternatives: chain, > tree, mesh from the foundations up to "sophisticated" conclusions. It seems > to me that if one reacts to something like TheoremDep with a sense of > "concatenation", then that implies they think about this sort of reasoning as > a chain or, at least, a tree. Reading ahead, I want to respond to this with anticipation toward what you write in the next paragraph. When I first saw the TheoremDep, I thought of a tree which is pretty close to a DAG because of the obvious constraint that we don't want circular dependencies/logic in our proofs. I didn't think of concatenation as much as composition. In principle, one could unroll or serialize all the dependencies of a theorem and include them in the proof. I remember when I took a (belated?) course in the History of Mathematics near the end of my BS in math. I think I took it as credit toward a history course, though it was at least 2nd year math in some of it's content. One interesting point was the clay-tablet records from Mesopotamia where algebraic solutions of (all?) quadratics and many cubics were enumerated, but redundantly across different contexts, expressed as "story problems". It may not have been as bad as offering the same solution to determining crop yields for wheat and rye in separate story-problems, but less immediate abstractions/analogies would have been redundantly "solved". > For context, I have the same problem with what we've (I've?) come to call > "physics-based biological modelers". Their focus is on physically realistic > simulation engines, upon which they implement chemically realistic reactions, > upon which they implement biologically realistic mechanisms. I think there is good motivation for believing that Chemistry "emerges" from physics and Biology "emerges from" Chemistry and Physics. At the same time, the point of emergent phenomena is that the governing "forces" between entities are NOT directly or obviously based in the next lower level of abstraction. So, while Metabolic Networks *DO* operate on Chemistry (and maybe some Physics like diffusion?), the governing logic IS relational, as you point out. Higher levels of abstraction like protein expression networks and (as my last example on Gene Ontologies represent) higher levels of abstraction have their own logic which "emerges" from the lower levels. > I (being the soft-X type of person I am), tend to focus on *relational* > modeling, which can be usefully applied at any layer of the heterarchy. I think that is the *point* (if we need to invoke a Final Cause here) of these "layers" of emergent functionality. To support more and more complex types of relations (not just linear, heirarchical, etc.)... I haven't thought it through well, but I'm guessing that (for example), metabolic networks, or gene expression networks have a qualitative higher complexity than *most* physics phenomena (though Feynman diagrams describe "relations" between subatomic particles, and there MAY be known/interesting/relevant examples with similar complexity?) > Or, at least, I prefer a "middle out" method, starting at the layer where > you understand/care the most. The physics-based people tend to think in > terms of chains and trees, whereas the relational people tend to think in > terms of networks. I call the phyics-based people "Grand Unified Modelers" > because they're seeking the One True Model. ... very Monistic. I do believe that there is a conceit of Physics that if there is a place for a Monistic GUT it would be in Physics, but I think there is room for the same in Chemistry, Biology, Sociology, Economics, etc... within the domain circumscribed by the "floor and ceiling" of emergence(s)? >> The old fascination with why mathematics (as a pure abstraction) seems to >> line up so well with physical reality comes up again. I think there is >> some kind of anthropic principle involved? Do you have any parallax on >> this? > I tend to believe it boils down to *attention*, intentionality, purposeful > behavior. It strikes me that math can unambiguously describe *anything*, > real or fantasy. And where non-ambiguity is required, math will be > successful. I think this is accurate, maybe tautological... > Bands like Tool argue against me in some sense, because they use a lot of > math in the creative composition of their tunes and lyrics. But, then again, > they're nerds and have high expectations for how tightly a performance hangs > together ... which is, again, a requirement for non-ambiguity. I think the point in "interesting" endeavors is to provide enough familiar structure to be "familiar" and then add to (abberate?) it enough to be "surprising". > My guess is math would be less successful if we tried to use it to, say, > foster creativity in children ... at least for now. I would claim that more than a little of my own personal creativity was based IN mathematics as a child/adolescent. It was the abstract language of math that allowed me to see (and manipulate?) patterns across more disparate domains than "natural language" allowed. It wasn't the lack of ambiguity (because my clumsy application re-indroduced ambiguity) in Math that drew me, but the ease and expressiveness of abstraction. > The author of TheoremDep cites Metacademy (here's a good example: > https://metacademy.org/graphs/concepts/incompleteness_of_set_theory#focus=mh2y5zs9&mode=explore). > That strikes me as a system amenable to edge weight adjustment. E.g. How > important is it to understand Russell's Paradox, really, for understanding > how ZF fits into this "lineage of reasoning"? I am working on a project involving the quantification and visualization of uncertainty. In that it is implicit that we want to propogate uncertainty through many different processes or models or scenarios. It seems somewhat to be a parallel to what you describe here around MetAcademy. > the kind of graph I'm imagining, it may be better to imagine the lemmas as > edges and the nodes as something else (conceptions of reality, maybe?). > Lemmas would, then, be transitions from one state to another.
I do see the (potential) value of considering this "dual" of the dependency graph... a Lemma as you say, can be considered a transition from one state of knowledge or conviction or proof to another state. - Steve > ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com archives back to 2003: http://friam.471366.n2.nabble.com/ FRIAM-COMIC http://friam-comic.blogspot.com/ by Dr. Strangelove
