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.

Boltzmann and the Direction of Time: Newton, Leibniz, and the Move toward Equilibrium

Historically, thought about time, like thought about space, emerged in the modern era in terms of a debate between Newtonians and Leibnizians. Those who followed the thought of Isaac Newton, who did his writing in the late 1600s and early 1700s, saw time as an absolute: as an independently existing framework within which objects existed and events happened.

Those who followed Gottfried Leibniz, who lived at approximately the same time, saw time as a relative measurement between events. If there were no events, there would be no time, because time is simply the separation between events.

Both views found ways to conceptualize the directionality of time. The Newtonians saw time as a one-way street: events happen at points on the timeline, and an observer is moving along the timeline in one direction only, from one event to the next.

The Leibnizians denied an independent reality to the timeline, saying rather that a later event and a prior event have a relationship to each other which is not symmetrical or reciprocal: an analogy to the parent-child relationship reveals that the parent and the child are real, while the concept of parenthood is a merely relative abstraction from the two real things. So it is also, Leibniz would suggest, with time.

When the question about the possibility of bidirectional time is raised — when one asks about time moving backwards — a challenge arises both for Newton and for Leibniz.

Attempts to explain the directionality of time often incorporate the concept of causality or the second law of thermodynamics or both.

In an intuitive and naive sense, it seems obvious that later events cannot cause prior events. This is an instinctive argument for the directionality of time.

The second law of thermodynamics is subject to many different phrasings, but a simplistic version says that entropy never decreases and that systems always tend toward maximum entropy. The directionality of time, then, is marked out as entropy increases, or at least fails to decrease.

Ludwig Boltzmann was a physicist and philosopher in Vienna. He did his work in the late 1800s and early 1900s. Much of his work dealt with the physical chemistry of gasses. In particular, he refined the mathematical formulation of Brownian motion and how gasses move toward entropy, equilibrium, and homogeneity.

Along the way, Boltzmann obtained some results which are perhaps counterintuitive and which challenge the common understanding of the directionality of time.

Boltzmann discovered that a gas, if contained in a finite space which changes neither in shape nor in total volume, and if in a state of equilibrium, will spontaneously develop local regions of disequilibrium. This seems like a violation of the intuitive understanding of the second law of thermodynamics.

Further, Boltzmann came to reject a simplistic version of Newtonian time, in which it would be said that systems move toward entropy over time. He came instead to view the movement toward entropy as time. On Boltzmann’s view, then, it would be said that the movement of systems toward entropy is time: time is the increase of entropy.

Combining these two ideas, Boltzmann concluded that there are instances in which time runs backward: times when pockets of disequilibrium develop in a system which has already obtained maximum entropy. As author Martin Gardner writes:

The most popular way to give an operational meaning to “backward time” was by imagining a world in which shuffling processes went backward, from disorder to order. Ludwig Boltzmann, the 19th-century Austrian physicist who was one of the founders of statistical thermodynamics, realized that after the molecules of a gas in a closed, isolated container have reached a state of thermal equilibrium — that is, are moving in complete disorder with maximum entropy — there will always be little pockets forming here and there where entropy is momentarily decreasing. These would be balanced by other regions where entropy is increasing; the overall entropy remains relatively stable, with only minor up-and-down fluctuations.

As counterintuitive as these results are, Boltzmann went even further. If that these principles hold for a gas in an unchanging container — imagine a corked test tube in a chemistry laboratory — then these principles will also hold true for the universe as a whole.

On the grand cosmic scale, Boltzmann hypothesizes, there might be regions within the universe in which time is running backward.

If one is to speak of time running backward, then it must be decided whether this will be explained in Newtonian terms, Leibnizian terms, or Boltzmann’s terms. In intuitive Newtonian terms, some sense can be made of time as an existing framework in which it might happen that entropy would decrease instead of increase: but if time is independent of the events which happen in it, then this decrease in entropy would not qualify as a reversal of time’s direction, even if it is a violation of the laws of thermodynamics.

In Leibnizian terms, one event succeeding another, or one state succeeding another, shapes the direction of time, and so likewise this decrease in entropy would not be a reversal of time.

On Boltzmann’s own terms, in which time is the movement toward entropy, this can be seen as a reversal of time. Yet it should be asked: does Boltzmann need to assume a larger Newtonian framework of independent time, in order to determine that time in the smaller region is running backward?

To complicate matters further, Boltzmann implies that it would be possible to have in the universe regions, some of which are moving toward equilibrium, and some of which are moving away from it. Here one wants to add the phrase: “at the same time.” But if those competing regions within the universe define their time as Boltzmann suggests, i.e., by the movement toward equilibrium, how then could it be said that these regions have time moving in opposite directions, unless there were a larger framework, a meta-time, of Newtonian nature, against which the direction of time in the smaller regions could be measured?

The question is: Does Boltzmann need a Newtonian meta-time to make his view of time succeed?

Martin Gardner continues:

Boltzmann imagined a cosmos of vast size, perhaps infinite in space and time, the overall entropy of which is at a maximum but which contains pockets where for the moment entropy is decreasing. (A “pocket” could include billions of galaxies and the “moment” could be billions of years.) Perhaps our flyspeck portion of the infinite sea of space-time is one in which such a fluctuation has occurred. At some time in the past, perhaps at the time of the “big bang,” entropy happened to decrease; now it is increasing. In the eternal and infinite flux a bit of order happened to put in its appearance; now that order is disappearing again, and so our arrow of time runs in the familiar direction of increasing entropy. Are there other regions of space-time, Boltzmann asked, in which the arrow of entropy points the other way? If so, would it be correct to say that time in such a region was moving backward, or should one simply say that entropy was decreasing as the region continued to move forward in time?

Boltzmann further concludes that in regions, or in the universe at large, in which equilibrium as been achieved, i.e., in which entropy is at its maximum, there is no time, or in Boltzmann’s own words, it is “dead.”

Can it make sense to speak of time running backwards, or in the case of an achieved equilibrium, of time stopping, unless there is some meta-time, some perspective from a higher level, from which it could be observed that time was so behaving? Martin Gardner asks:

If things come to a standstill in time and “then” reverse, what does the word “then” mean? It has meaning only if we assume a more fundamental kind of time that continues to move forward, altogether independent of how things in the universe move. Relative to this meta-time — the time of the hypothetical observer who has slipped unnoticed into the picture — the cosmos is indeed running backward. But if there is no meta-time — no observer who can stand outside the entire cosmos and watch it reverse — it is hard to understand what sense can be given to the statement that the cosmos “stops” and “then” starts moving backward.

There is no doubt that Boltzmann was an exceptionally brilliant thinker. Yet there are some difficult questions for him to answer.

Did he over-rely on the analogy to gasses? What might be provable or observable about a corked test tube filled with air might not apply to the universe as a whole. What justifies the transference? If principles have been understood from gasses in a finite container of unchanging shape and size, why would these principles apply to the universe as a whole?

Is Boltzmann justified in asserting that the universe as a whole is in a state of equilibrium? He makes the assertion that “the universe” is “everywhere in thermal equilibrium and therefore dead,” with the exception of small regions which “depart from thermal equilibrium” for a “relatively short time.”

Yet the universe as it is known demonstrates sharp distinctions between vacuums and dense astronomical bodies. It displays, not chaotic Brownian motion, but predictable Newtonian and Keplerian orbits. Observations, whether by optical telescope or by radar telescope or by space travel, do not reveal a homogenized universe.

The reader will want to consult Boltzmann’s Vorlesungen über Gastheorie, Band II, Kapitel 90.

Boltzmann made remarkable discoveries and had brilliant insights. Yet many questions about the direction of time remain to be answered.

Reductionism Then and Now: Pre-Socratic Physics

When the Milesian philosophers of Ionia began their reductionist project, they seem to have equated a universal systematic underlying principle for all reality with a principle for all matter. Perhaps they either didn’t distinguish matter from energy, or didn’t care about energy, or weren’t aware of energy.

As with all investigations of pre-Socratic thinkers, conjectures will remain tentative, due to sketchy textual sources.

Perhaps they saw energy in terms of powers which objects have, making the objects — i.e., matter — primary, and energy secondary.

When the Milesians sought a unifying principle, they looked to matter instead of to energy — what is the common principle behind all matter? — although they were aware of forces, as Leonard Susskind writes:

Of all the forces of nature, only three were known to the ancients — electric, magnetic, and gravitational. Thales of Miletos (600 BC) was said to have moved feathers with amber that had been rubbed with cat fur. At about the same time he mentioned loadstone, a naturally occurring magnetic material. Aristotle, who was probably late on the scene, had a theory of gravity, even if it was completely wrong. These three were the only forces that were known until the 1930s.

The history of physics changed direction at some point placing more emphasis on energy as an independent topic, rather than energy as merely a property of, or an ancillary to, matter.

So it is that post-Socratic and post-Newtonian physics seeks a Grand Unified Theory (GUT), not uniting all matter, but uniting all known forces. Physics as a discipline decided that it had worked about the basic principle of matter — all atoms are composed of electrons, neutrons, and protons, etc. — and turned to energy.

The ancients knew of the three main forces because they were clearly observable, as Leonard Susskind notes:

What makes these easily observed forces special is that they are long-range. Long-range forces fade slowly with distance and can be seen between objects when they are well separated.

While physicists search for a GUT, they focus primarily on electro-magnetic and nuclear forces. A further step would make a truly universal system by including gravity. This is called a ‘Theory of Everything’ (TOE).

While GUT and TOE remain speculative and controversial, they are also a continuation of the Milesian reductionist project, with a shift toward energy and away from matter.

While gravity is the most easily observable force, and therefore the first object of philosophical speculation, it is also the weakest force. This is counterintuitive to the extent that it is everywhere visible and to the extent that someone who’s had a brick dropped on his toes will not consider the force to be empirically weak.

Yet gravity is considered to be a weak force because for any one unit of matter, the measured force is small relative, e.g., to magnetic forces. Gravity’s force seems strong, exerting hundreds of pounds of force on each human being, because the earth’s mass is so large. By contrast, a magnet of much smaller mass than the earth would be able to exert an equal or greater force.

Gravitational force is by the far the most obvious of the three, but surprisingly it is much weaker than electromagnetic force. The reason is interesting and worth a short digression. It goes back to Newton’s universal law of gravitational attractions. Everything attracts everything else.

In the search for universal principles — whether GUT or TOE or the Milesian reductionist project — language is strained to capture the concepts. Is this the systemic principle that’s “behind” or “underneath” all reality? The prepositions betray language’s difficulties in capturing the idea: most prepositions are spatial, and yet the quest here is not for a primarily spatial relationship.

In sum, while the Milesians made significant progress toward a unifying principle which underlies all reality, their search seems to have been skewed toward matter at the expense of energy.

Reductionism and the Milesian Philosophers

Sorting through the ideas of Thales, Anaximander, and Anaximenes, the modern reader can be forgiven for finding some of what they thought to be odd. But with sympathetic reading, their proposals can be understood in ways which do, after all, make some sense.

The common thread connecting these three philosophers from Miletus is a project now called ‘reductionism.’

Looking at the variety in the world around them, these men asked whether there was some unifying reality which produced all of it, constituted all of it, and made it all intelligible. For these three thinkers, this was primarily on the level of physical objects.

What do a flower, a rock, a cloud, and the planet Jupiter all have in common? They are all composed of matter. But matter manifests itself in these divergent — very divergent — ways. What makes all these different things fall into the same category?

Phrasing the questions in a twenty-first century way, one might ask, what properties does all matter share? Certainly, a flower growing in a garden and the planet Jupiter seem to have very little in common. If they are both made of matter, then it is necessary to more closely understand what matter is.

The three Milesian philosophers were looking for a universal and ubiquitous principle — the basis of all matter — which would be the source and substance for everything. In this way, they are not so different from modern physicists.

The modern answer to the Milesian question might be: “Everything is made of protons, electrons, and neutrons.”

Seen in this way, the suggestion that everything is foundationally composed of water, or air, or some indeterminate stuff which has the capability of becoming water or air or fire or dirt, is not so odd. Interpreted with charity, these suggestions make sense, even if they’re not quite correct.

Water is composed of hydrogen and oxygen. Oxygen is the most common element in the earth’s crust, and hydrogen is the most common element in the universe. These two substances are everywhere in the environments which human beings inhabit. The rocks, plants, animals, and other objects encountered in daily life on earth are full of these two substances.

The choice of air as a potential primordial source for everything likewise has some reasonable aspects. The air on Earth is approximately 21% oxygen, and as noted above, oxygen is ubiquitous. Earth’s air is often laden with water, whether as clouds or as invisible vapor, which therefore includes hydrogen. Additionally, air is often filled with dust, which could be fine particulates of silicon, iron, or anything else.

The hypothesis of some primary substance called the ‘indeterminate’ — the ‘boundless’ or the ‘unlimited’ in various attempts to translate Anaximander’s Greek into English — resembles the concept of an undifferentiated stem cell, which can become any of many different types of cell, and resembles the concept of basic particles in physics, which can form atoms of any type.

The reductionist project of Thales, Anaximander, and Anaximenes is, then, not as odd as it seems, and has a significant similarity to aspects of modern physics.

Anaximander: Order Out of Chaos

The career of Anaximander was both destructive and constructive. He formulated objects to the views of Thales, and then assembled lines of reasoning to support his own views.

Against Thales, he argued that finding the systemic principle of the universe in any one element was too limited, too specific. For one particular element to be the source and foundational principle for the universe seemed impossible to Anaximander, because that one element would be locked into the narrowness of not being any of the other elements.

So Anaximander proposes a view that there is some indeterminate stuff that is not any one element, but contains the potential to give rise to each and all of them, as Donald Palmer writes:

For Anaximander, the ultimate stuff behind the four elements could not itself be one of the elements. It would have to be an unobservable, unspecific, indeterminate something-or-other, which he called the Boundless, or the Unlimited (apeiron in Greek). It would have to be boundless, unlimited, and unspecific because anything specific is opposed to all the other specific things in existence. (Water is not fire, which in turn is not air, and air is not earth [not dirt and rock]). Yet the Boundless is opposed to nothing, because everything is in it.

Anaximander’s language is vague, but the modern reader can consider concrete examples from the twenty-first century. In complex organisms, a stem cell is indeterminate, or undetermined — it can eventually become one of a long list of different and mutually exclusive types of cells.

Likewise, in the plasmatic chaos in the center of the star, subatomic particles are freely existing which will later constitute definite, but distinct, types of atoms: hydrogen, helium, lithium, beryllium, boron, carbon, etc.

So Anaximander’s proposal for a limitless boundless something as a foundational principle for the universe isn’t too bad, as Donald Palmer explains:

Anaximander seems to have imagined the Boundless as originally moving effortlessly in a great cosmic vortex that was interrupted by some disaster (a Big Bang?), and that disaster caused opposites — dry and wet, cold and hot — to separate off from the vortex and to appear to us not only as qualities but as the four basic elements: earth, water, air, and fire.

In his analysis of Thales, Anaximander presented an early version of the concept of entropy: the idea that, without some force to the contrary, the universe tended toward some homogeneous equilibrium.

It is also possible to read into Anaximander an early version of the hypothesis that life is the principle which opposes entropy: action which moves the universe away from entropy is life. The biological process might be the one force in the universe which moves things toward more order, and which makes more complex structures.

Rebellion among Philosophers: Anaximander Questions Thales

Given that Thales is widely considered to be the world’s first philosopher, it was left to the world’s second philosopher to be the world’s first intellectual rebel. It was a historical inevitability that, Thales having expressed some views, someone else would later express different views.

The task of the reader is to compare both sets of ideas. This task will require thought about evenhanded and fair readings of the two competitors. A stalemate or a tie is a perfectly acceptable outcome — as is the production of a third alternative arising from the comparison. The task is more about the thinking process and less about the outcome of a final judgment.

Every disagreement also involves a certain amount of agreement. While no two philosophers agree on everything, it is also true that no two philosophers disagree on everything. In the case of Thales and his successors, Donald Palmer points out that:

Several generations of Thales’s followers agreed with his primary insight — that the plurality of kinds of things in the world must be reducible to one category — but none of them seems to have accepted his formula that everything is water.

The second philosopher has two tasks: First, he must first produce reasons or evidence which support his disagreement with the first philosopher. Second, he must produce reasons or evidence which support his innovation, his new idea which is proposed as a replacement for the first philosopher’s idea.

So who was the world’s second philosopher?

Thales taught his philosophy, whether in a formal academic setting, or merely by example, we do not know. “His student Anaximander” lived from around 610 B.C. to around 546 B.C., and was “also from the city of Miletus, said that if all things were water, then long ago everything would have returned to water.”

Anaximander was in some sense a “student” of Thales, whether through formal education or through merely being exposed, firsthand or secondhand, to the ideas of Thales. It is not certain whether or not the two men ever met in person, although it is very likely, given that they lived in the same city at roughly the same time.

In any case, Anaximander argues against Thales — in the philosophical sense of ‘argumentation’ which means a calm presentation of a line of reasoning, not an emotional quarrel — by a technique which amounts to saying, “If what you say is true … ”

This technique is called reductio ad absurdum or simply reductio for short. The writer grants his opponent’s view, and then shows that this view entails something clearly illogical or false.

Anaximander’s first argument amounts to this: If it were true that everything is essentially water, then by now, everything would have returned to the simple state of water, and there would be nothing in the universe besides water.

In a slightly different argument, Anaximander points out that fire is the opposite of water, and asks how it would be possible for water to produce fire. He asks it as a rhetorical question, i.e., he expects no answer, because he thinks that the answer is obvious to everyone — water can’t possibly produce fire.

The reader will recall that the “principle of charity” is necessary here. One could quickly retort that, from the viewpoint of modern chemistry, water is hydrogen and oxygen, and can easily give rise to fire. But such a retort would miss the point.

Donald Palmer gives a more detailed account of Anaximander’s line of reasoning:

Anaximander asked how water could become its deadly enemy, fire — how a quality could give rise to its opposite. That is, if observable objects were really just water in various states of agitation — as are ice and steam — then eventually all things would have settled back into their primordial liquid state. Aristotle paraphrases him this way: If ultimate reality “were something specific like water, the other elements would be annihilated by it. For the different elements have contrariety with one another … If one of them were unlimited the others would have ceased to exist by now.” (Notice that if this view can be accurately attributed to Anaximander, then he subscribed to an early view of the principle of entropy, according to which all things have a tendency to seek a state of equilibrium.)

The modern reader might be tempted to agree with half of what Anaximander asserts here, and disagree with the other half.

One the one hand, the tendency of physical systems to move toward a state of equilibrium, which can be variously characterized as chaos or homogeneity, is a valuable insight. Anaximander might be credited with anticipating the famous second law of thermodynamics, and is a forerunner of thinkers like Rudolf Clausius, Max Planck, and Ludwig Boltzmann.

On the other hand, he blithely assumes that something can’t become its opposite, or that a quality can’t give rise to its opposite. After Anaximander, some later philosophers — among them, G.W.F. Hegel — will assert the view that a thing or a quality will always give rise to its opposite.

In any case, the world’s first philosophical revolution — Anaximander’s opposition to Thales — is groundbreaking and worth studying.