Abstract
If mathematics, unlike entomology, is unreasonably effective, it should be possible to say with, at least some precision, what it means for a mathematical object, structure, or theory to be applied to an object, structure or theory that is resolutely not mathematical. If it is possible to say as much, I have not found a way in which to say it. Mathematics is about mathematics; and so far as the Great Beyond is concerned, while it is obvious that mathematics is often applied, it is anything but obvious how this is done.
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Notes
- 1.
Parson's phrase, the “instance of a structure” is not entirely happy. Predicates have instances; properties are exemplified; structures just sit there.
- 2.
See [13 pp. 198–204] for interesting remarks.
- 3.
By Euclidean geometry, I mean any axiomatic version of geometry essentially equivalent to Hilbert's original system—the one offered in chapter 6 of [5] for example.
- 4.
Indeed, it is not clear at all that the surface of my desk is either a two- or a three-dimensional surface. If the curved sides of the top are counted as a part of the top of the desk, the surface is a three-dimensional manifold. What then of its rectangular shape? If the edges are excluded, where are the desk’s boundaries?
- 5.
In a well-known passage [12] Albert Einstein remarked that to the extent that the laws of mathematics are certain, they do not refer to reality; and to the extent that they refer to reality, they are not certain. I do not think Einstein right, but I wonder whether he appreciated the devastating consequences of his own argument?
- 6.
Quantum considerations, I would think, make it impossible to affirm any version of an Archimedian axiom for points on a physical line.
- 7.
For very interesting if inconclusive remarks, see the round-table discussion by a collection of Field medallists in [8 pp. 88–108], especially the comments of Alain Connes on p. 95.
- 8.
Defined as the ratio of two lengths, radians are in any case dimensionless units.
- 9.
See, for example, [7 pp. 56–58].
- 10.
Curiously enough, this is a point that Weyl himself appreciates [23 pp. 15–17].
- 11.
See Eilenberg [11], from a philosophical point of view, interest in semigroups is considerable. A finite state automata constitutes the simplest model of a physical process. Associated to any finite state automata is its transition semigroup. Semigroups thus appear as the most basic algebraic objects by which change may abstractly be represented. Any process over a finite interval can, of course, be modeled by a finite state automata; but physical laws require differential equations. Associated to differential equations are groups, not semigroups. This is a fact of some importance, and one that is largely mysterious.
- 12.
This familiar argument has more content than might be supposed. It is, of course, a fact that quantitative measurements are approximate; physical predicates are thus inexact. For reasons that are anything but clear, quantitative measurements do not figure in mathematics; mathematical predicates are thus exact. It follows that mathematical theories typically are unstable. If a figure D just misses being a triangle, no truth strictly about triangles applies to D. Mathematical theories are sensitive to their initial descriptions. This is not typically true of physical theories. To complicate matters still further, I might observe that no mathematical theory is capable fully of expressing the conditions governing the application of its predicates. It is thus not a theorem of Euclidean geometry that the sum of the angles of a triangle is precisely 180 degrees; ‘precisely’ is not a geometric term. For interesting remarks, see [18].
- 13.
- 14.
The notion of a solution to a differential equation is by no means free of difficulties. Consider a function f(x) = Ax, and consider, too, a tap of the sort that sends f to g(x) = Ax + μx, where μ is small. Do f and g represent two functions or only one?
- 15.
See also [4] for a very detailed treatment of similar themes.
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Berlinski, D. (2015). Mathematics and its applications. In: Davis, E., Davis, P. (eds) Mathematics, Substance and Surmise. Springer, Cham. https://doi.org/10.1007/978-3-319-21473-3_6
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