The Temptation of Ludwig Boltzmann

A True Story

by Roderick T. Long

[Written for Prof. David Layzer’s course “Science A-22: Chance, Necessity, and Order,” 17 January 1983 (Harvard, sophomore year, age 18); subsequently kindly submitted by Layzer for the James Bryant Conant Competition on Natural Science (for the best essay written in a science course), which it miraculously won – ultimately leading, by an odd sequence of events, to the paper’s being listed as an out-of-print book on Amazon. (Layzer was a cool dude; see more about him here.)

It always sounds impressive to say that I won a prize in natural science, given that it’s an area outside my specialisation; but that’s misleading, since it’s really a science-fiction story about the philosophical implications of a particular scientific theory, namely Boltzmann’s theory of probability. (Note that if that theory were true, this story would likewise be true; hence the subtitle.)

The quotation “I have found it necessary to deny knowledge, in order to make room for faith” is from Immanuel Kant’s Critique of Pure Reason. The quotation “A difference which makes no difference is no difference,” which I’d learned from James Blish’s Star Trek novel Spock Must Die!, is usually attributed either to William James or to C. S. Peirce, but I’ve never seen an exact citation. With regard to Boltzmann’s Latin quotation, not only does nihil not decline in classical Latin, but ex is elided before a consonant, so ex nihilo would be e nihil.]

Behold this moment! From this gateway, Moment, a long, eternal lane leads backward: behind us lies an eternity. Must not whatever can walk have walked on this lane before? Must not whatever can happen have happened, have been done, have passed by before? And if everything has been there before – what do you think, dwarf, of this moment? Must not this gateway too have been there before? And are not all things knotted together so firmly that this moment draws after it all that is to come? Therefore – itself too? For whatever can walk – in this long lane out there too, it must walk once more.

And this slow spider, which crawls in the moonlight, and this moonlight itself, and I and you in the gateway, whispering together, whispering of eternal things – must not all of us have been here before? And return and walk in that other lane, out there, before us, in this long dreadful lane – must we not eternally return?

– Friedrich Wilhelm Nietzsche
Also Sprach Zarathustra, III.ii.2


To the Analytic-Synthetic Dichotomy –
a bane to science, but a boon to séance.

*     *     *

On a winter evening in Österreich in 1872, Ludwig Boltzmann was calculating furiously. Before him, on the oakwood desk, complex mathematical formulæ filled sheafs [sic, for “sheaves”] of yellow paper; and the few remaining pristine sheets were even now succumbing to Boltzmann’s energetic pen. The characteristic scrawlings would soon illuminate the world as the statistical formulation of the Second Law of Thermodynamics.

The probability of any given outcome of a trial occurring in a series of trials, Boltzmann reasoned, increases as the number of trials increases. The probability of tossing a coin 3759 times and getting heads every time is 2-3759 – not a very likely outcome. But in a series of 23759 trials, one would logically expect a run of 3759 heads to occur exactly once. And in an infinite number of trials, the unlikely event would occur an infinite number of times.

Suddenly the frantic scratching of his pen came to a halt as the implications of this conclusion flooded in on him. Modern astronomy had proven conclusively that the universe was infinitely extended in space and time. If so, then every possible combination of molecules, no matter how unlikely, must have already occurred a infinite number of times, and must, even now, be occurring in an infinite number of places. But this conjecture rendered the very statistical laws he had worked out more or less useless; how could the Law of Entropy Production provide a basis for scientific predictions, if that law also predicted an infinite number of statistical fluctuations?

At that moment, a vague form began to shimmer into view on the opposite side of Boltzamnn’s desk. As it coalesced, Boltzmann recognised the figure, with a start. There, in his study, in the midst of an odd framework of gears and levers, stood Ludwig Boltzmann himself! The Doppelgänger wore Boltzmann’s suit, Boltzmann’s beard, and Boltzmann’s expression of complacent ascetism [sic, for “asceticism”]. The only variation from the original matrix was a pair of flight goggles which the visitor proceeded to remove.

“Who are you? What are you doing here?” Boltzmann demanded.

“I have been observing your progress,” the second Boltzmann replied, “and I am here to inform you that your surmise is correct. Anything that can happen, is happening somewhere; this very conversation is taking place in infinite variations across the universe. In some versions, the only difference is that my suit contains one more atom than that of my counterpart. In others, we are both gaseous beings, conversing by electrical flashes. I come from a probability-system very similar to this one; I am the Ludwig Boltzmann of this world, as you are of yours.”

This spectacular metaphysical confirmation of his theory left the first Boltzmann almost speechless. But another question remained: “How did you arrive ... in that incredible manner ....”

Boltzmann-2 indicated the mechanical assembly which Boltzmann-1 had noticed earlier. “Allow me to present my greatest invention: the Hypothetical Calculus. This device allows me to select the particular probability-system I wish to visit, and transports me there instantaneously.”

Now Boltzmann-1 felt he had a legitimate right to object. “What do you mean by ‘instantaneous’?” he asked. “As wide as my statistical theory is, it only permits events which are physically possible – and instantaneous travel is not one of them.”

Boltzmann-2 smiled indulgently. “I was using ‘instantaneous’ in the most relative sense,” he replied. “What actually occurs, I suspect, is that both my body and my vehicle are temporarily frozen into timelessness – arrested in time, I should say – and that I am then propelled toward my destination at sub-light speed. It may take millions of years for me to arrive, but, from my point of view, the trip passes in the blink of an eye.”

“But you can then never return to your home? For millions of years must have passed there as well!”

“Whenever I have returned to my point of origin, nothing whatever has changed; my entire world has been ‘frozen’ just as long as I have, waiting to be ‘thawed out’ on my return. Or perhaps it isn’t the same world at all – just an identical one, all of whose events occur a million years later. Consequently, everything turns out just as if my travel were really instantaneous. But enough explanations; allow me to give you a demonstration.”

“You want me to get in that thing and be frozen for a million years?”

“That’s just a way of putting it. Perhaps instantaneous travel is physically possible; what does it matter, so long as the result is the same? ‘A difference which makes no difference is no difference.’”

Boltzmann-1 could think of no reason to distrust someone who bore such a striking resemblance to himself; so he climbed into the futuristic contraption alongside Boltzmann-2. “But please,” he said, “be careful where you aim this thing! I wouldn’t want to wake up in the middle of the sun.”

“Or worse yet,” smiled Boltzmann-2, “in a probability-system where all your molecules suddenly flew apart at once. Don’t worry; I’ll select only a mildly weird world. In some parts of the universe, improbably events occur regularly and in an orderly way, so that a whole system of what you would call ‘magic’ exists. I’ve visited one world where, whenever anyone waved his hand, a platter of food appeared – that is, molecules spontaneously combined to form a platter of food. Such events are highly improbable, but this particular world was in a highly improbable state, owing to a highly inevitable statistical fluctuation; so its inhabitants accepted this common event as a natural law. I’ve visited worlds where reciting mysterious incantations would turn lead into gold – at least, the two events always accompany one another – and one rather embarrassing world where H2O was an aphrodisiac. The particular world I’ve selected to visit has its own intriguing examples of magic; there are photographs that move, and can be transmitted by wireless; there are ships which are heavier than air, but which nevertheless soar through the sky; there are very intense beams of light which can cut through steel; and bombs which can blow up an entire city and spread disease for miles around.”

“It sounds like a very unpleasant place,” remarked Boltzmann-1.“Are you sure it’s safe to visit?”

“It’s not much different from Earth; they even call it Earth. But see for yourself.” After manipulating several dials, Boltzmann-2 activated the Hypothetical Calculus; and infinite blackness descended.

The next sensation was a rush of air; Boltzmann-1 looked down, and, through the grillwork beneath his feet, he could see a city laid out at a giddy depth. “We’ve materialised, or whatever, in midair!” he shouted, to be head above the wind. “Won’t we fall?”

“I’ve selected the probability that we won’t,” Boltzmann-2 replied cheerily. And sure enough, the invention descended gracefully and majestically at the steady rate of four miles an hour. “We will land in the Boston Common.”

“Boston? Then we’re in the United Colonies of America?”

“Not exactly; in this world, the British lost the War of 1812. Did in mine, too. But that’s not the oddest thing about this world. You see, this Earth’s scientists would disagree with your assumption, for instance, that the universe is infinitely old. It so happens that all the light-phota that arrive on Earth have ‘improbably’ deviated from course and altered their wavelengths, thus giving astronomers a very distorted view of reality. They have inferred from this systematic misinformation that space is curved, that the universe is expanding, and that time started a mere few billion years ago!”

“But that’s ridiculous!” retorted Boltzmann-1. “haven’t their philosophers told them that nothing can rise from nothing? Nihil ex nihilo fit – or nihil ex nihil fit; I remember nihil never declines in Classical Latin but always declines in Mediæval latin – or is it the other way around?”

“If you’d read any Quine you’d know better than to ask such questions. But look! quite a crowd has gathered to greet us.”

“How is that?” asked Boltzmann-1. “Who knew we were coming?”

“Nobody knew; I can’t understand it. Ah! wait – watch these fellows closely.”

Boltzmann-1 did as his mentor instructed, and quickly noticed something very odd. Most of the people in the crowd were standing still; but those who were moving, were moving backwards!

“My mistake,” admitted Boltzmann-2. “I forgot to specify the direction of the arrow of time. This world is almost exactly like yours – but the chief difference is: the Second Law of Thermodynamics (which is only a statistical law, after all) runs backwards. Information-generating processes result in equilibrium, and entropy-generating processes result in order. Waterfalls run upwards, molecules rush from low pressure zones to high, and so forth. That’s why all these people are here; according to their time-sense, we’ve already been here for quite a while, and they’ve just gathered to see us off. I’ll feed the same information in, and switch the selector, so we’ll up on an Earth exactly like this one, but with the time-sign reversed.”

“But can you do that? Don’t we have to stay here a while, since, according to these people, we’ve ‘been’ here a while?”

“That’s the interpretation we’d use if we stayed, but since we’re going, we can just assume that their coming to meet us was simply another infinitely probable improbable event – that they were ‘fated’ to congregate here even if we had never arrived.”

“But how can you describe the same event by two radically different interpretations?” asked Boltzmann-1, his scientific sense offended.

Boltzmann-2 grinned. “In either case, the mathematical description of the event remains the same, and that’s all that matters. As the founder of our science said, ‘All things are numbers.’” The ground was now only a few yards away.

“I’m beginning to think the founder of our science was Herakleitos, not Pythagoras.”

“Nonsense! With my Hypothetical Calculus, I can step into the same river as many times as –”; Boltzmann-1 failed to catch the slight hesitation in his Doppelgänger’s voice. “As many times as I like!” Boltzmann-2 concluded firmly, and so saying, he plunged himself, his comrade, and his invention into the endless dark once more.

They re-emerged in Boston Common, this time minus the crowd. “The most interesting thing about this world,” Boltzmann-2 told Boltzmann-1, “is that here, too, a Ludwig Boltzmann discovered the law of infinite statistical fluctuations. But our twin’s theory was ‘disproven.’ Order on Earth couldn’t be ‘just a local fluctuation’ – astronomers ‘discovered’ that space was homogeneous as far out as anyone could go.”

“So did the astronomers on my world,” said Boltzmann-1, “but they haven’t looked far enough, as yet.”

“So they haven’t,” replied Boltzmann-2. But on this world, they can never look far enough. From this Earth, space really is homogeneous infinitely far out, in all directions!”

“But how is that possible? What happened to the universe we just came from? The one with the planet where lead turns into gold, and so forth?”

“Oh, it’s the same universe, alright. But that part of the universe is at, say, ½∞ from here, while this system’s homogeneity stretches out to only, say, ¼∞ – which takes eternity to get to, anyway.”

“Then you have lied to me,” Boltzmann-1 said suddenly.

“I? Lied?”

“Yes! If the distance between my Earth and this Earth is truly infinite – or some semi-infinite fraction thereof – then, just as these Earthmen can never leave their homogeneity, so we could never have gotten here. One can only cross an infinite distance after an infinite amount of time – that is to say, never! So this calculating machine of yours cannot possibly work by the freeze-and-thaw method you described earlier!”

Boltzmann-2 blushed. “I see I must tell you the truth. You did not ‘travel’ from your Earth to this one. You ceased to exist on your Earth – your molecules spontaneously disassociated – and other molecules came together, here, to form a new you. No ‘message’ was transmitted across an infinite distance – it was just another improbable coincidence which had to happen sub specie æternitatis.”

“You mean I’m not the same me who woke up this morning?” cried Boltzmann-1, horrified.

“There may not even have been a you this morning. Perhaps you and I were spontaneously assembled here, complete with the memory of having existed before, whether we did or not.”

“If I’d known I was committing theoretical suicide, I would never have stepped into this thing of yours.”

“I know. That’s why I didn’t tell you. But look at the advantages you gain from dying and being reborn all over the universe.” Boltzmann-2 flicked a switch, and the Hypothetical Calculus disappeared from the Boston Common – to the consternation of passersby.

“That darkness!” cried the Boltzmann-1 who began to be that moment, an infinite distance away. “That darkness that always accompanies the trips – I thought it was the ‘time-freezing.’ Now I realise it was death! You just killed me!”

“No,” replied Boltzmann-2. “A Boltzmann just like me killed a Boltzmann just like you – and himself, as well. And here we are: look!”

Boltzmann-1 looked. Never had he seen anything more lovely: a rainbow-hued landscape lay before him, with lacy, towering needle-mountains as delicate as spiderwebs or chains of butterflies. “Let there be the complete works of Euripides,” whispered Boltzmann-2 – and they were there. “Let there be wealth without measure! Let there be diamonds as large as cabbages!” he shouted – and they were there. “Let there be the most beautiful of all women!” – he sang – and she was there.

Boltzmann-2 turned to Boltzmann-1. “I find this probability-system a particularly congenial one,” he said. “Here, everything that one may imagine is at one’s command; anything may be synthesised out of the subatomic particles of the air. But if you don’t like it, you can always switch the dial – and specify another world which pleases you more. Is my Hypothetical Calculus not the greatest invention of all time? Is it not Paradise Regained?”

“I am almost convinced,” Boltzmann-1 answered solemnly. “But first, satisfy my scientific curiosity – how does the Calculus work? If everything in the universe is subject to the wild caprice of chance, how can you successfully direct your device to one probability-system or another?”

A shadow fell across Boltzmann-2’s face. “I don’t know.”

“You don’t!”

“No. I just happened to construct it one day – I was trying to build a turbo-generator – and it started depositing me here and there all over the cosmos. There’s no rational way to explain it – there’s no way the different probabilities could possible be specified as I specify them. The operation of the Hypothetical Calculus must be ascribed entirely to chance.”

“You mean it works according to no principle at all? Then how does it take you where you want to go?”

“It just happens to. It’s always worked so far. What more can I ask?”

“You trust your very existence to – a coincidence?”

“Everything is a coincidence. I thought you understood that.”

“Then this wonderful world where wishes come true – might reverse itself tomorrow, contrary to your specifications?”

“Of course.”

“Take me back home! Take me back at once – to the rational world of Galileo, of Kepler, of Newton!”

“Stay here, my brother,” Boltzmann-2 pleaded. “Let it grow on you. Certainty isn’t worth that much. You want security; I want to hunt unicorns! You want a world without risk. I want to rule the stars!”

Remembering the peculiar nature which this particular probability-world at last temporarily possessed, Boltzmann-1 said fiercely: “Let it be that you take me home – now.”

Against his will, Boltzmann-2 found his hands once more bringing down the utter night.

And once more they stood in Boltzmann-1’s study, on a winter evening in 1872, in Österreich.

“I am back in a world of predictability. And here I shall stay.”

“In the next century, even if everything proceeds ‘normally,’ a physicist named Heisenberg will come along and make hash of your predicable world.”

“Let him come. Quantum uncertainty is sedate compared to your realm of frightful flux.”

“I made a mistake when I didn’t specify your reaction. Next time I’ll pick a world whose Boltzmann agrees with me.”

“You have no way of knowing your device won’t drop you into a volcano next time. Somewhere there’s already a you whose Calculus has failed him.”

“I have no way of knowing I won’t suddenly drop into a volcano anyway, with or without the Calculus. So I use the Calculus.”

“How do you stand it – not knowing?”

Boltzmann-2 smiled, and quoted: “‘I have found it necessary to deny knowledge, in order to make room for faith.’”

“I prefer knowledge.”

“No, you don’t; or you wouldn’t flee from what I’ve shown you, back to your illusion of an ordered universe. You have seen the brilliance of what is possible – and have turned back to what is familiar.”

“Touché. Then you have our faith, and I have mine. But yours is in the wind, while mine is in the rock.”

“Your faith is no less precarious than mine – only less interesting. My Hypothetical Calculus predicts that, with a few minor exceptions – such as my presence here – your world will obey the normal laws for all eternity. But, as you point out, I have no way of knowing that the Calculus is right. A ‘normal’ course of affairs is much more probable than the alternative – but you cannot assume that you are living in one of the areas where only probable things happen. Any such assumption would be perfectly gratuitous. Your house is built on quicksand, my friend.”

“Yes – but it is my house,” replied Boltzmann-1.

Boltzmann-2 inclined his head in acknowledgment of his double’s decision. “To faith, then!” he laughed, with a dazzling grin, slipping his goggles down over his eyes.

“To faith,” Boltzmann-1 said softly as Boltzmann-2 and the Hypothetical Calculus dissolved into the interstices of the universe.

Boltzmann-1 sat down once more at his desk and examined the calculations he had left unfinished. Picking up his pen, he wrote a sentence which was to puzzle mathematicians for decades to come: “Thus, we see that the hypothesis of infinite statistical fluctuations is repugnant to the nature of probability; and from this it follows that the observable universe is a representative sample of the whole.”

Ludwig Boltzmann walked to the window and looked out at the moonlight on the surface of the lake. He wished Boltzmann-2 well; but he did not regret his own decision. He belonged here, in the world of cause and effect, of natural law, of scientific knowledge – a world fixed and immutable. Boltzmann walked to the flickering lamp and extinguished it.

Then, as he had done the night before, and every night before that, Ludwig Boltzmann turned himself into a rabbit and went to sleep.

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