Questions for Ludwig Boltzmann: Entropy and Methodology

Boltzmann’s significant and sometimes counterintuitive advancements in spacetime physics and philosophy have been well documented by Lawrence Sklar in Space, Time, and Spacetime and Philosophy and Spacetime Physics, and by other writers as well. In Vorlesungen über Gastheorie, Boltzmann himself sets his ideas forth.

The reader will be aware that Boltzmann argues that the universe is mostly in a state of equilibrium, with only rare instances of imbalance:

Man kann sich die Welt als ein mechanisches System von einer enorm großen Anzahl von Bestandteilen und von enorm langer Dauer denken, so dass die Dimensionen unseres Fixsternhimmels winzig gegen die Ausdehnung des Universums und Zeiten, die wir Äonen nennen, winzig gegen dessen Dauer sind. Es müssen dann im Universum, das sonst überall im Wärmegleichgewichte, also todt ist, hier und da solche verhältnismässig kleine Bezirke von der Ausdehnung unseres Sternenraumes (nennen wir sie Einzelwelten) vorkommen, die während der verhältnismässig kurzen Zeit von Äonen erheblich vom Wärmegleichgewichte abweichen, und zwar ebenso häufig solche, in denen die Zustandswahrscheinlichkeit gerade zu- als abnimmt.

Even more significant is his claim that in these rare pockets of disequilibrium, change is as likely to increase entropy as decrease it. This amounts to saying that time is as likely to move forward as backward, because, for Boltzmann, the motion toward entropy is time. For him, motion toward entropy does not happen within time, but rather that motion is time. The source of our observable universe is an anomaly, development away from equilibrium: our observable universe is a pocket within a larger universe.

Describing Boltzmann’s views, Craig Callender writes:

Boltzmann, however, explained this low-entropy condition by treating the observable universe as a natural statistical fluctuation away from equilibrium in a vastly larger universe.

Surveying Boltzmann’s conclusions, then, many questions might arise. Among them are these three:

First, Boltzmann’s hypotheses about the universe are based on, or prompted by, his investigation in gas theory. To which extent is it a valid methodological move, to apply the principles of gas theory to the entire universe? Aside from the observation that the universe includes solids and liquids, and is not composed of gas alone, it seems in other ways, too, that it is a great leap to assume that the principles which govern and explain the behavior of a gas inside a glass container sitting on Boltzmann’s laboratory table are sufficient to explain and govern the entire universe. Can Boltzmann validly generalize to the entire cosmos from a sealed beaker filled with nitrogen, oxygen, carbon dioxide?

Second, to which extent is Boltzmann’s assumption justified, that the universe is largely, almost entirely, in a state of equilibrium? And that any disequilibrium is a small, relatively microscopic, anomalous pocket within this larger universe? The glass container, filled with a gas, on Boltzmann’s laboratory table may well be largely, even entirely, in a state of equilibrium, but to assume the same of the entire universe seems again like an unwarranted generalization. Unlike the glass container in the laboratory, the universe contains near-perfect vacuums between galaxies, contrasted with the most dense possible compressions of matter elsewhere in the same universe. It contains the coldest possible and the hottest possible points. It seems that these would be pieces of evidence that perhaps disequilibrium is more pervasive in the universe than Boltzmann seems to indicate.

Third, do Boltzmann’s results contain a hidden assumption or requirement that there be some type of “meta-time” within which time operates? In order for Boltzmann to indicate that there would simultaneously be pockets within the larger universe, and that time in these different pockets could be going in different directions, there needs to be a larger meta-time in order for these to be happening “simultaneously.”

In addition to the above three questions, there are doubtless many more questions which can, and should, be posed both in order to understand and in order to evaluate Boltzmann’s work.

Faraday’s Method: Electromagnetism as Visual Intuitive Phenomena

In America — and perhaps in other parts of the world as well — secondary and postsecondary educational institutions have worked to create in the minds of students an almost instinctive or reflexive connection between mathematics and the natural sciences. Those who work in schools will automatically say “science” when prompted with the words “math and … ”

Yet the marriage between math and science may be merely a flirtation or friendship. The bond and similarity between the two is not as strong as is commonly supposed.

The thinkers who began the enterprises of modern mathematics and modern science saw them as two very different activities. The home of a priori rational certainty was found in mathematics, while approximation and tentativeness lived in the sciences.

The quadratic formula is the same in textbooks printed today or two hundred years ago — and will be the same centuries into the future. By contrast, the atomic weight of various isotopes of lead or gold, when calculated out to large numbers of significant figures, might be revised or refined over the years as successive editions of textbooks are printed for chemistry and physics classes.

There is a danger in overemphasizing the importance of mathematics in the natural sciences — and here one can also mean the observational sciences and empirical sciences. The non-mathematical aspects of scientific activity risk being ignored: the intuitive aspects.

There is an objection, of course, to say that seemingly non-mathematical properties like color and shape can in fact be reduced to mathematics: color is a wavelength which can be represented by a number; shape can be captured in an algebraic equation.

To this objection, one might respond by positing that a color is more than the number of its wavelength and a shape is more than its corresponding algebraic equation. As the human mind seeks correlations and systemic connections, a color or a shape functions differently, and is treated by the mind differently, than a number or an equation. It is in this seeking that scientific discoveries are made and insights gained.

Two scientists who were champions of mathematics — James Clerk Maxwell and Ludwig Boltzmann — nonetheless praised the intuitive and conceptual work of Michael Faraday. Faraday’s discoveries and descriptions of electromagnetic phenomena were shockingly free of mathematics, as historian Alan Hirschfeld writes:

Maxwell, the consummate mathematician, nonetheless understood the power of mathematics to mislead when not anchored in experiment or observation. In Faraday’s Researches, he encountered science in its purest form, “untainted” by mathematical manipulation. Here, he decided, would be the entry point for his own investigations into electricity and magnetism. In a later reflection, Maxwell sounds almost relieved that Faraday had stuck to his particular brand of investigation, thereby blazing a trail that Maxwell himself could follow.

Boltzmann explains one of Faraday’s many intuitive, i.e. non-mathematical, breakthroughs:

While the older system had held the centers of force to be the only realities, and the forces themselves to be mathematical conceptions, Faraday saw distinctly the continuous working of the forces from point to point in the intermediate space. The potential, which had hitherto been only a formula for lightening the work of calculation, was for him the bond really existing in space, the cause of the action of force.

Boltzmann uses the word ‘saw’ in the text above, emphasizing Faraday’s visual technique. Faraday’s work with magnetic fields was largely the work of observing patterns, e.g., the movement of iron filings. “By the light of his own clear conceptions,” Boltzmann writes, Faraday made “such great discoveries.” Boltzmann also explains Faraday’s effect on Maxwell:

Maxwell also, when he undertook the mathematical treatment of Faraday’s ideas, was from the very outset impelled by their influence into a new path.

Maxwell himself praises Faraday’s non-mathematical approach:

It was perhaps for the advantage of science that Faraday, though thoroughly conscious of the fundamental forms of space, time, and force, was not a professed mathematician. He was not tempted to enter into the many interesting researches in pure mathematics which his discoveries would have suggested if they had been exhibited in a mathematical form, and he did not feel called upon either to force his results into a shape acceptable to the mathematical taste of the time, or to express them in a form which mathematicians might attack. He was thus left at leisure to do his proper work, to coordinate his ideas with his facts, and to express them in natural, untechnical language.

Faraday was talented at drawing diagrams and sketches of magnetic fields and various configurations which were part of his experiments. The etymology of the word ‘intuition’ arises from verbs meaning to ‘see’ or to ‘look’ and Immanuel Kant chose the word ‘intuition’ — or rather, the German equivalent, Anschauung — which might prompt the reader to investigate any similarities between Kant’s thought and Faraday’s thought.

In any case, Faraday filled notebooks with drawings and illustrations, and it was in them and through them that he founded the modern science of electromagnetism and made his many significant discoveries: it was in and through the images and illustrations, not by means of equations and formulas.

Aside from Faraday’s visual method, there was a second feature of his thought which may have shaped his investigations and conclusions. His entire adult life was spent working in an organization known as the “Sandemanian” or “Glasite” church, as Alan Hirschfeld writes:

The Sandemanian church continued to hold a central place in Faraday’s life. He attended services and ritual feats, enjoyed the sense of community and, with rare exception, clung to its precepts.

The worldview of this faith shaped his exploration of electromagnetism. In the same way that a sort of spiritual humility caused Augustine to recognize the limits of human reason and caused Francis Bacon to formulate the sources of experimental error, so also Faraday was motivated to caution in his hypothesizing and to thoroughness in observation and experimentation.

Faraday understood the laws and principles of electromagnetism as being the products of God’s thought. He wrote: “for the book of nature, which we have to read is written by the finger of God.” Faraday understood lawlike phenomena, and nature’s laws, to be deliberately and rationally planned: “God has been pleased to work in his material creation by laws.”

There is a connection between Faraday’s intuitive approach to electromagnetism and his understanding of God. Verbs of sight were both metaphor and literally truth for Faraday in his investigations of electromagnetism: he was “looking” into God’s work and “saw” the rationality of it.