Physics Lournal

Powered by 🌱Roam Garden

On The Tension Between Mathematics and Physics

Metadata

Author:: Miklós Rédei

Tags:: Philosophy of Science, Philosophy of Physics, Philosophy Of Mathematics

Abstract:

Due to the deep interrelation of Mathematics and Physics, there is inherent tension between the two fields, perceived by mathematicians, physicists and mathematical physicists. Also, an attempt is made to explain why mathematical rigor is typically unwelcome in Physics.

**1. The Supermarket Picture: The Relation of Physics

and Mathematics, and Tension of Type I:**

According to what is perhaps the standard picture of this relation, physics is a science because it is systematically mathematical, which has two major implications:

(a.) Physics focuses on precise measurement, aimed at specifying values of defined physical quantities: this is the work of quantitative experimental physics, and ensures the descriptive accuracy of physics.

(b.) Physics lays out mathematical models of physical phenomena, making explicit the working relationships between the measured quantities, a.k.a general quantitative physical laws.

Both descriptive accuracy and predictive success should be appreciated with some qualifiers: Descriptive accuracy is imperfect, and defined quantities are not purely empirical (sometimes based on theoretical observations), natural laws may reference entities that are not directly observable.

While the standard picture as characterized by (a) and (b) is exemplary of the role of mathematics in physics, it lacks nuance, as it is based on the "supermarket picture", of the relation between the two: positing mathematics as a supermarket, and physics as a customer.

That is to say, it's assumed that when a physicist is in need of some concept, or structure, to develop a model of physical phenomena or processes, they go to the "mathematical supermarket", browse the aisles and shelves, retrieves the necessary products, does the necessary tweaking, and then has the necessary tool for developing the model.

This assumption, is non-trivial, so much so that it doesn't hold often: mathematical concepts are often not readily available when the physicists/physics need them.

Additionally, developing the necessary mathematics may take a non-negligible amount of time: because physics can not wait for the concepts to be defined rigorously, Physics will necessarily be imprecise, and inconsistent during this time.

2. Examples of Tension of Type I

2.1: The Missing Derivative in Newton's Mechanics.

Newton needed a rigorous concept of the derivative to fully formulate the second law of motion, but such a concept was not made mathematically precise, until the 19th century, via the work of Cauchy and Weierstrass. This forced Newton to make due with what was available, using inconsistent logic involving infinitesimals.

More precisely, he used both algebraic and geometric methods in his calculations: The algebraic work was built upon the notion of the infinitesimal, and the geometric reasoning used "ultimate ratios of vanishing quantities". The algebraic work was published after the Principia, which is largely geometric.

In Newton's time, issues with both perspectives were raised.

Berkely observed in the Analyst that the limits of vanishing quantities in the Principia are just as opaque as the infinitesimals, as the ultimate ratio of these quantities is effectively 0/00/0.

Concerning the logic of infinitesimals, he writes:

"For when it is said, let the increments vanish, i.e. let the increments be nothing, or let there be no increments, the former supposition that the increments were something, or that there were increments, is destroyed, and yet a consequence of that supposition, i.e. an expression got by virtue thereof, is retained. Which, by the foregoing lemma, is a false way of reasoning. [...]

The foregoing Lemma being:

"If with a view to demonstrate any proposition, a certain point is supposed, by virtue of which certain other points are attained; and such supposed point be it self after- wards destroyed or rejected by a contrary supposition; in that case, all the other points attained thereby, and consequent thereupon, must also be destroyed and rejected, so as from thence forward to be no more supposed or applied in the demonstration.’ This is so plain as to need no proof."

While it is canon within the history of mathematics that Newton didn't have a consistent definition of the concept of a limit, and that only since the work of Cauchy has this notion been available, this view is contested by some individuals, however it is incontestable that the rigorous concept of the derivate was not available in the supermarket when it was required by Classical Mechanics.

2.2: The Missing Spectral Theory of Self-Adjoint Operators in Quantum Mechanics.

[Hilbert Space] was established systematically by von Neumann, summarized in his book. The book was based on three papers, published in 1927. Of these three papers, the first introduced the notion of abstract Hilbert space, and presented the "eigenvalue problem", of self-adjoint operators having a continuous part in their spectrum, in a mathematically rigorous form, without making use of Dirac's delta function. The complete, analysis of the spectral theorem was worked out in a following paper in 1930.

These papers, with their mathematical precision, had been preceded by a paper by Hilbert, Neumann and Nordheim.

This latter paper was the first attempt at axiomatizing quantum mechanics in standard Hilbert Space, but it is mentioned by the authors that the ideas presented are not defined rigorously:

From a mathematical point of view the method of calculation is rather unsatisfactory because one is never certain whether the operations involved are really admissible. For this reason we do not detail them further. But we hope to return to these issues on another occasion.

This was due to an assumption that the possible values of observable physical quantities are elements of the spectrum of the operators representing the observables.

2.3 Missing Ergodic Theory In Classical Statistical Mechanics before von Neumann.

The concept of ergodicity stretches back to Boltzmann in the 19th century. The ergodicity of a dynamical system that describes time evolution of a system of moving particles according to classical laws of motion should have ensured that the time averages of macroscopic objects is equivalent to the phase averages of them, but making this mathematically precise turned out to be very tricky.

3. Tension of Type I Drives Mathematical Development

4. Absence of Rigor in Mathematics?

5. Mathematical Precision isn't welcome in Physics

6. Why Physicists do not appreciate Mathematical Precision

This may contrast excellently (or relate excellently) to Dirac's The Relation between Mathematics and Physics.

Referenced in

On The Tension Between Mathematics and Physics