on the lighter side of things .. this is by far one of the more interesting abstracts that i have read, in recent memory:
so i have no rebuttals, only rejoinders:
In other words, in mathematics the creation of monsters is recognized as part of the working practice of a creative mathematician. Why is this the case?
how else can you test how "reasonable" a theory is?
it's not like we mathematicians are tethered to reality. it's similar to having philosophical arguments, to the point where words break down and definitions become crucial.
put another way, empiricists have the luxury of running experiments; one prepares the setup, let the particles run, and records the data. theorists, on the other hand, are afforded no such tests. there is no difference between the design of a "thought experiment" and its implementation, so these degenerate examples (or "monsters" if you will) are our version of 'control experiments.'
Why perhaps is this so legitimate in mathematics whereas it may seem more transgressive in physics and its sibling empirical sciences?
because, unlike scientists, we can do very well without loyalties to reality.
i have met some brilliant physicists, but at the end of the day, they must cede to physical law .. or the current fashionable speculation of what should be the correct law (e.g. string theory).
up to human frailties, pure mathematicians are under no such obligation. i would even say that computer scientists (not engineers) are more free than physicists.
I suggest that rather than deductively erecting propositional lattices atop a fixed set of axioms and definitions, mathematicians practice the formation, deformation, and reformation of conceptual patterns.
that would be in keeping with what i've observed about research and researchers. we draw from examples, wonder if the phenomena we observe is truly general, or whether the situation can take on a degenerate turn.
other than logicians and their ilk, i know very few mathematicians who start with axioms and start "computing" with them, seeing what comes out.
I also suggest that mathematicians can exteriorize their practice in mathematics as technology.
i have heard the word "technology" thrown around, quite often.
usually it refers to new insights that allow us a better understanding of a previously not-well-understood phenomena, or a newly-tested set of "tools" that allow us to make many more (or simply more precise) conclusions that were previously impossible.
so yes: it sounds to me like the conceptual version of technology.
The example that I will trace is the formation of geometric measure theory in the late 1960's through the 1980's, in which several mysterious pathologies in the classical notion of area were resolved by a monstrous re-interpretation of surface using notions from real analysis (abstract measure theory).
hold on: what are you calling "monstrous" ..? (-:
Geοmetric Measure Theοry As a Monster Mash (link)it's interesting how well this guy understands mathematicians, which is something different from understanding mathematics. this is not to say that he cannot reconstruct the proof of the deformation theorem, but it sounds like he's thought a lot about how mathematicians think.
Sha Χin Weι
Abstract. I argue that mathematicians crucially make use of monsters -- entities and concepts that break form and expectation, and hybridize across alien (unrelated) theories. But I will also argue that (1) mathematicians do this consciously as a research practice, and (2) this is considered a perfectly legitimate, if not generic process. In other words, in mathematics the creation of monsters is recognized as part of the working practice of a creative mathematician.
Why is this the case? Why perhaps is this so legitimate in mathematics whereas it may seem more transgressive in physics and its sibling empirical sciences? I suggest that rather than deductively erecting propositional lattices atop a fixed set of axioms and definitions, mathematicians practice the formation, deformation, and reformation of conceptual patterns.
I also suggest that mathematicians can exteriorize their practice in mathematics as technology. The example that I will trace is the formation of geometric measure theory in the late 1960's through the 1980's, in which several mysterious pathologies in the classical notion of area were resolved by a monstrous re-interpretation of surface using notions from real analysis (abstract measure theory).
so i have no rebuttals, only rejoinders:
In other words, in mathematics the creation of monsters is recognized as part of the working practice of a creative mathematician. Why is this the case?
how else can you test how "reasonable" a theory is?
it's not like we mathematicians are tethered to reality. it's similar to having philosophical arguments, to the point where words break down and definitions become crucial.
put another way, empiricists have the luxury of running experiments; one prepares the setup, let the particles run, and records the data. theorists, on the other hand, are afforded no such tests. there is no difference between the design of a "thought experiment" and its implementation, so these degenerate examples (or "monsters" if you will) are our version of 'control experiments.'
Why perhaps is this so legitimate in mathematics whereas it may seem more transgressive in physics and its sibling empirical sciences?
because, unlike scientists, we can do very well without loyalties to reality.
i have met some brilliant physicists, but at the end of the day, they must cede to physical law .. or the current fashionable speculation of what should be the correct law (e.g. string theory).
up to human frailties, pure mathematicians are under no such obligation. i would even say that computer scientists (not engineers) are more free than physicists.
I suggest that rather than deductively erecting propositional lattices atop a fixed set of axioms and definitions, mathematicians practice the formation, deformation, and reformation of conceptual patterns.
that would be in keeping with what i've observed about research and researchers. we draw from examples, wonder if the phenomena we observe is truly general, or whether the situation can take on a degenerate turn.
other than logicians and their ilk, i know very few mathematicians who start with axioms and start "computing" with them, seeing what comes out.
I also suggest that mathematicians can exteriorize their practice in mathematics as technology.
i have heard the word "technology" thrown around, quite often.
usually it refers to new insights that allow us a better understanding of a previously not-well-understood phenomena, or a newly-tested set of "tools" that allow us to make many more (or simply more precise) conclusions that were previously impossible.
so yes: it sounds to me like the conceptual version of technology.
The example that I will trace is the formation of geometric measure theory in the late 1960's through the 1980's, in which several mysterious pathologies in the classical notion of area were resolved by a monstrous re-interpretation of surface using notions from real analysis (abstract measure theory).
hold on: what are you calling "monstrous" ..? (-:
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