"Mathematics - Application and Applicability", by Mark Steiner.
Included in The Nature of Nature: Examining the Role of Naturalism in Science (2011). Edited by Bruce L. Gordon and William A. Dembski. ISI Books: Wilmington, Deleware.
- Canonical applications (theories developed to describe an application) vs. non-canonical (applying mathematics in situations other than those that created them)
- Distinguishing applications of mathematics from mathematics itself
- Exploration of individual thinkers and their attempts to reconcile mathematics and the empirical world (including Gottlob Frege, Hartry Field, Eugene Wigner), with the group-theoretic leading the charge
"What are we doing when we 'apply' mathematics to Nature?"
"Other skeptics argue the reverse, that we should expect mathematics to be applicable because (unlike what [Eugene] Wigner asserts) the real and ultimate source of mathematical concepts is experience; thus, it is not at all unreasonable that mathematical concepts should return the favor."
"We may fundamentally misrepresent what we do not detect - for example, when we think that the four dimensions of space-time are continua (they may, for all we know, be lattices)."
"I am persuaded by John Locke's central insight that the world could turn out to be fundamentally unlike what it appears to be, and the framework of space and time is no exception".
Mark gives examples of canonical nonempirical applications, including an explanation of how the definition of of multiplication in terms of repeated addition is wrong. They present instead a set-theoretic presentation, that could be "'applied' to addition".
They then moves on to group theory, which they explain was not set up to describe symmetries in the world, but rather constructed to be applied internally to mathematics itself - specifically to algebra (by Galois). They claim that Galois, by studying groups of equations in order to learn more about the equations, "inaguarate[d] modern mathematics, in which the main applications of mathematics are to mathematics itself."
They move into an exploration of group theory in quantum mechanics - kicking diving into an exploration of electron spin. He tangentially references Heisenberg's 1932 assertion that the neutron and proton are actually tow different states of the same particle ("today called the 'nucleon'"), which I though was pretty cool. They move on from discussing electrons and nucleons to hadrons. Group theory is further supported in Murray Gell-Mann and Yuval Ne'eman's predication of a 10th hadron in the SU(3) family and Gell-Mann's theoretical identification of "quarks" stemming for a prediction of "existence of the [three pions] form the three-dimensional representation."
And yet the author cautions our reading of group theory's applications by saying, "the historical players would mostly have rejected this characterization." After centuries of math being used to derive quantitative results from phenomena we could also describe qualitatively, we now can see our "mathematics [giving] even the qualitative descriptions".
Gottlob Frege demonstrates that applications can involve concepts (i.e. two-ness) of things rather than the things themselves, thereby side stepping the need to reconcile mathematics and the empirical world. This, and the exploration of the group-theoretic (with specific phenomena being signified by the properties of their symmetry group), is the bridge between mathematics and empirical reality this articles is building.
Mark ends the article by throwing the ball back up in the air when he asks "whether Pythagoreanism is an acceptable doctrine".