Author: Richard Melson

Chance and Necessity

If we free-associate the word “chance” you may arrive at “I’ll take my chances” before potentially arriving at Thomas Bayes’ “Doctrine of Chances”. Jacques Monod was an early explorer of the collision of probability and chance in biology.

THE PRINCE OF CHANCE

Jacques Monod grew up just down the coast from Monte Carlo in Cannes, France, another town famous for its casinos and, later, its film festival. Graced with movie star looks—one prominent French journalist described him as a “prince” who resembled Hollywood icon Henry Fonda, as well as considerable musical talent, and an exceptional intellect, Monod struggled to decide on a career path through his twenties. After distinguishing himself in the French Resistance, Monod rose to fame not as an actor or musician, but as a brilliant biologist. He shared the 1965 Nobel Prize in Physiology or Medicine for seminal discoveries about how genes work.

A pioneer in the field of molecular biology, Monod was privy to the blizzard of discoveries in the 1950s and early 1960s about the molecules that determined the characteristics of living things—what Monod and others dubbed “the secrets of life.” He kept close company with a relatively small international community of leading researchers. For example, when James Watson and Francis Crick cracked the structure of DNA (deoxyribonucleic acid) in 1953, Monod was one of the first with whom Watson shared the breakthrough.

But as a Frenchman steeped in his culture’s deep philosophical traditions, Monod was interested in science for more than just science’s sake. After the war, Monod befriended France’s leading philosopher-writer Albert Camus, and the two men pondered questions of human existence in Left Bank cafés. Monod felt that the public misunderstood the principal purpose of science as being the creation of technology. Rather, Monod believed technology was merely a by-product. He said, “the most important results of science have been to change the relationship of man to the universe, or the way he sees himself in the universe”—a relationship of equally intense interest to his friend Camus.

Monod thought that there were profound philosophical implications of the new molecular biology, particularly in the realm of heredity, which had gone largely unnoted in the broader culture. Several years after his Nobel Prize and Camus’ untimely death, he decided to write a book to try to bring the meaning of modern biology to laypersons.

“[T]he ‘secret of life’…has been laid bare,” he wrote.

“This, a considerable event, ought certainly to make itself strongly felt in contemporary thinking.”

Monod used several chapters to describe the insights that had very recently emerged from the study of DNA and the deciphering of the genetic code. He understood this knowledge would be unfamiliar to most readers, so he included an appendix with chemical structures of proteins and nucleic acids, and a primer on how the genetic code worked.

In a matter-of-fact style, he explained genetic mutations as accidental alterations—substitutions, additions, deletions, or rearrangements—in the text of DNA, in the sequence of the long strings of chemical bases (ACTTGATAA, etc.) that make up genes.

Then, almost without warning, he turned to the broader implications of how mutations arise in DNA. It is worth quoting him at length for after 111 pages of background, he delivered one of the most powerful ideas in five centuries of science (all italics are original):

“We call these events accidental; we say they are random occurrences. And since they constitute the only possible source of modifications in the genetic text, itself the sole repository of the organism’s hereditary structure, it necessarily follows that chance alone is at the source of every innovation, of all creation in the biosphere.

“Pure chance, absolutely free but blind, at the very root of the stupendous edifice of evolution: this central concept of modern biology is no longer one among other possible or even conceivable hypotheses. It is today the sole conceivable hypothesis, the only one that squares with observed and tested fact. And nothing warrants the supposition—or the hope—that on this score our position is likely ever to be revised.

“There is no scientific concept, in any of the sciences, more destructive of anthropocentrism than this one.”

In essence, heretofore obscure discoveries in biochemistry and genetics (largely studied at that time in simple bacteria) had upended two millennia of philosophy and religion that put humans at the center or apex of creation. “Man was the product of an incalculable number of fortuitous events,” Monod wrote. “The result of a huge Monte Carlo game, where our number eventually did come out, when it might not well have appeared.”

Le Hasard et la nécessité (Chance and Necessity) appeared in France in October 1970. It was a fairly technical book with several chapters on philosophy and genetics, and those appendices full of chemical diagrams. A first-time author, Monod did not know what reactions to expect.

The merde hit the fan.

The book received dozens of reviews across France and quickly became a bestseller—second only to the French translation of Erich Segal’s Love Story (this was France after all. After it was translated into English, reviews and interviews with Monod were featured in several of the most prominent British and American newspapers and magazines.

Many commentators immediately recognized the threat chance posed to traditional ideas of humanity’s origins and purpose. To Arthur Peacocke, a British biochemist turned prominent theologian, Monod had put forth “one of the strongest and most influential attacks of the century on theism.” A flurry of articles and books appeared with titles such as Anti-Chance: A Reply to Monod’s Chance and Necessity, Beyond Chance and Necessity, and God, Chance, and Necessity. Monod was invited to debate philosophers and theologians both in France and abroad, on television, radio, and in print.

American Calvinist theologian and pastor R.C. Sproul summed up the high stakes posed by chance in the first page of his book Not A Chance:

“It is not necessary for chance to rule in order to supplant God. Indeed, chance requires little authority at all if it is to depose God; all it needs to do the job is to exist. The mere existence of chance is enough to rip God from his cosmic throne. Chance does not need to rule; it does not need to be sovereign. If it exists as a mere impotent, humble servant, it leaves God not only out of date, but out of a job.” More than two hundred pages later, Sproul concluded:

“Chance as a real force is a myth. It has no basis in reality and no place in scientific inquiry. For science and philosophy to continue to advance in knowledge, chance must be demythologized once and for all.”

Sproul and other critics argued that what scientists perceived as chance merely reflected a lack of knowledge of true causes. Perhaps that was the expression of hope to which Monod alluded—the hope that as scientists learned more, our position on the role of chance would somehow be revised.

A SECOND CHANCE

The ensuing fifty years have not played out as either Monod or his detractors hoped. The Frenchman thought that the new insights from molecular biology should be a turning point for modern society-away from traditional beliefs about causes in the natural world toward one that embraced randomness and our chance existence.

Ha! Fat chance. The excitement and fuss stirred by Chance and Necessity simmered down, and Monod passed away a few years later. Surveys reveal that the majority of Americans, for example, continue to believe that everything on earth happens for His reasons.

But Monod’s critics should take no comfort. The province of chance in the biosphere and human life has been revised, although not at all in the scope or direction that they hoped.

The domain of chance has expanded into realms neither Monod nor anyone else imagined.

As we have learned much more about the history and workings of the planet, we have been startled to discover how the course of life has been buffeted by a variety of cosmological and geological accidents—without which we would not be here. As we have explored human history, we have seen how pandemics, droughts, and other civilization-changing episodes have been triggered by random, singular events in nature that easily might not have happened. And as we have probed human biology and the factors that impact our individual lives, we have caught chance red-handed, reigning over the often-thin line between life and death.

This book tells the stories that Monod could not—of astonishing discoveries from the planetary to the molecular scale, from great upheavals across the globe to the machinery of chance that operates within every cell of every creature, including ourselves. And while these discoveries vaporize the comforts of anthropocentrism, the story of chance, I hope you will come to agree, is much more than highfalutin philosophy or the refutation of theologians’ wishful thinking.

I hope that you are awed—awed by the power and the drama of asteroids slamming into the planet, of continents colliding, and of the rapid rising and falling of ice and oceans; awed by the realization that we live on (and are at the mercy of a planet that is far more unstable than our short lives perceive; awed by the knowledge of how random chance is at the source of all of the beautiful and wondrous creatures with whom we share the planet; awed by the unique invisible accidents that made each one of us; and awed by the fact that we humans, recent descendants of bands of hunter-gatherers who persevered through a period of exceptional chaos, have in just the last fifty years or so, figured all of this out!

My goal here is to be comprehensible without being comprehensive. It is almost trivial to claim that the world is the way it is or that we are here because of a long chain of chance, albeit fortunate, events. The explanatory power I seek comes from specificity. It is essential to unpack some of those events to appreciate how they shape the direction of life. The layout of the book follows a simple three-part logic. I’ll begin with inanimate, external chance events that have shaped the conditions for life (Part One, “Stuff Happens”), and then turn to the internal random mechanism within every creature that generates the adaptations to those conditions (Part Two, “A World of Mistakes”). Then, I bring the story to the personal level (Part Three, “23 and You”) and how chance impacts our natural lives, as well as our deaths. Our chance-driven existence shatters long-held beliefs about humanity’s place and raises challenging questions about the meaning and purpose of our lives. In the Afterword, I’ll offer some possible replies with the help of some special guests.

This is a relatively small book for a really big idea. Science has given us a handful of really big ideas over the centuries, but they have been received in funny ways. Darwin had a huge idea that was very simple to understand, and even though the evidence is massive and everywhere, many refuse to believe it. Einstein had a brand new idea, and even though few understand it or the evidence for it, most everyone seems to believe it. Monod had a great idea, but these days most people (other than scholars) have not heard of it, or of him.

My greatest hope, then, is that this short book might be chance’s second chance.

Sean B. Carroll, A Series of Fortunate Events: Chance and the Making of the Planet, Life, and You, Princeton University Press, 2020, pp. 6-12.

Modern biology as described by Monod pushes the linkage of probability, randomness and chance to the center. At the time, this was a very radical and profound way of looking at biology. If we combine Monod’s biology with theoretical physics like Lawrence Krauss’ A Universe from Nothing: Why There Is Something Rather than Nothing, we realize that accidentality is the bedrock on which our knowledge of the universe is built.

Modernity and its Nightmares

Scholars console themselves with the notion that calamitous acts of death and destruction such as The Holocaust, the Bengal famine, and the atomic bombings of Hiroshima and Nagasaki are freak, singular aberrations. Psychologically, it’s reassuring to pigeonhole these as exotic historical events for academic study and push them away.

In 1989, Zygmunt Bauman published Modernity and the Holocaust, in which he argues the opposite: that these events are intrinsically linked to modernity. A symptom of the nightmare of modernity can be seen in Shlomo Sternberg’s Dynamical Systems, where he includes a photo of Felix Hausdorff, with a caption explaining the latter’s suicide (to avoid an extermination camp) amid his brilliant mathematical analysis.

At the end of World War II, Allied forces agents detained ten leading German scientists who were thought to have worked on Nazi Germany’s nuclear program. When one of these men, the Nobel Prize-winning chemist Otto Hahn, heard about the atomic bombings of Hiroshima and Nagasaki, he is said to have had a nervous breakdown, feeling guilt for his discovery and its use in this tragedy.

The Harvard Nobel Prize-winning economist Amartya Sen focused on the Bengal famine. In Poverty and Famines: An Essay on Entitlement and Deprivation, he explained that the Bengal famine was not the result of food shortages due to flooding, locusts or crop failures, but rather resulted from income shortages and lack of political clout.

The rise of bureaucracy, entwined with modernity, has removed any sense of culpability, as technological breakthroughs can be used to cause horrific events. You may have seen the 2023 film Oppenheimer, which depicts the American theoretical physicist’s guilt over developing atomic weapons amid Congressional hearings. In this framework, Fritz Haber was instrumental in developing the large-scale synthesis of fertilizers and explosives and is considered the “father of chemical warfare” in contrast to the advances he gave us in agriculture.

To quote Stephen Dedalus from James Joyce’s Ulysses, “History is a nightmare from which I am trying to awake.”

Can a Movie Excite an Unexpected Intellectual Adventure?

Consider the political thriller The Eagle Has Landed. There’s a scene where Col. Radl discusses Jung’s concept of synchronicity. For example, a man checking out drops a coin near the register and searches for it momentarily before giving up. Years later, he is at the same store and finds a similar coin. He rationalizes this with a mundane story, not some cosmic notion of fairness.

…Jung was working, in collaboration with Pauli, on his treatise on Synchronicity: An Acausal Connecting Principle, which was published together with Pauli’s essay on Kepler in one volume. This was evidently meant as a symbolic act: one of the greatest physicists of the century joining forces with one of its greatest psychologists. The result was a stimulating exercise in unorthodox speculation, but at the same time sadly disappointing. It did not amount to a theory in the proper sense, but rather to a universal schema, both very bold and very vague.

Jung’s treatise hinges on his concept of “Synchronicity”. He defines it as “the simultaneous occurrence of two meaningfully but not causally connected events”;1 or alternatively as “a coincidence in time of two or more causally unrelated events which have the same or similar meaning2…equal in rank to causality as a principle of explanation”.3 This is an almost verbatim repetition of Kammerer’s definition of “Seriality” as “a recurrence of the same or similar things or events in time or space”—events which, as far as can be ascertained; “are not connected by the same acting cause”. The main difference appears to be that Kammerer emphasises serial happenings in time (though, of course, he includes contemporaneous coincidences in space), whereas Jung’s concept of synchronicity seems to refer only to simultaneous events—although he includes precognitive dreams which occurred sometimes several days before the events. He tried to get around the time paradox by saying that the unconscious mind functions outside of the physical framework of space-time; thus precognitive experiences are “evidently not synchronous but are synchronistic since they are experienced as psychic images in the present as though the objective event already existed”.4 One wonders why Jung created these unnecessary complications by coining a term which implies simultaneity, and then explaining that it does not mean what it means. But this kind of obscurity combined with verbosity runs through much of Jung’s writing.

Although Kammerer’s “Seriality” and Jung’s “Synchronicity” are as similar as a pair of gloves, each fits one hand only. Kammerer confined himself to analogies in naive physical terms, rejecting ESP and mentalistic explanations. Jung went to the opposite extreme and tried to explain all phenomena which could not be accounted for in terms of physical causality, as manifestations of the unconscious mind: “Synchronicity is a phenomenon that seems to be primarily connected with psychic conditions, that is to say with processes in the unconscious.”5 Its deepest strata, according to Jungian terminology, are formed by the “collective unconscious”, potentially shared by all members of the race. The “decisive factors” in the collective unconscious are the archetypes which “constitute its structure”.6 They are, as it were, the distilled memories of the human species, but cannot be represented in verbal terms, only in elusive symbols, shared by all mythologies. They also provide “patterns of behaviour”7 for all human beings in archetypal situations—confrontations with death, danger, love, conflict, etc. In such situations the unconscious archetypes invade consciousness, carrying strong emotions and—owing perhaps to the archetype’s indifference to physical space and time—facilitate the occurrence of “synchronistic” events. The appearance of the scarab while the patient was telling her archetypal dream is considered by Jung as an illustration of this nexus. The same applies to the detonations in Freud’s bookcase during Jung’s visit, indicating the explosive nature of their father-son relationship: “Meaningful coincidences which are to be distinguished from meaningless chance-groupings—therefore seem to rest on an archetypal foundation. At least all the cases in my experience and there is a large number of them—show this characteristic?”8

Elsewhere in the essay he writes:

Synchronistic events rest on the simultaneous occurrence of two different psychic states. One of them is the normal, probable state (i.e., the one that is causally explicable), and the other, the critical experience, is the one that cannot be derived causally from the first. In the case of sudden death, the critical experience cannot be recognised immediately as “extra-sensory perception” but can only be verified as such afterwards. …In all these cases, whether it is a question of spatial or of temporal ESP, we find a simultaneity of the normal or ordinary state with another state or experience which is not causally derivable from it, and whose objective existence can only be verified afterwards. …An unexpected [mental] content which is directly or indirectly connected with some objective external event coincides with the ordinary psychic state: this is what I call synchronicity.”9

  1. Jung (1960), p. 441 [ref]
  2. Jung (1960), p.551 [ref]
  3. Jung (1960), p. 435 [ref]
  4. Jung (1960), p. 445 [ref]
  5. Jung (1960), p. 511 [ref]
  6. Jung (1960), p. 436 [ref]
  7. Jung (1960), p. 438 [ref]
  8. Jung (1960), p. 440 [ref]
  9. Jung (1960), pp. 444-5 [ref]
Arthur Koestler, The Roots of Coincidence, Picador, 1974 (first published by Hutchinson & Co. Ltd., 1972), pgs. 94-97.

Can a movie trigger an unexpected intellectual adventure?

Usually, when considering this question, you might see a film on a specific subject, then perhaps read an article or try to find a book about it.

In his essay on Synchronicity (Jung’s term for meaningful coincidences of events separated in space and/or in time,1 Jung examined some of the beliefs surrounding apparently related incidents which seem to have no causal connection. These incidents could be, for example, the coinciding of a patient’s dream with an actual event corresponding to its occurring at the same time some distance away, it could be ESP phenomena—response to some event that does not become known through any sense; it could be a horoscope reading which corresponds to the observed character of the individual or his self-image, or an astrological prediction which seems to be borne out in subsequent events. The possibility of finding meaning in these correspondences had tantalized Jung for many years. The beginning of his serious study goes back to the days when Albert Einstein was developing his first theory of relativity. During this time he was a guest on several occasions for dinner in Jung’s home. In a letter on Einstein and synchronicity, Jung wrote: “It was Einstein who first started me off thinking about a possible relativity of time as well as space, and their psychic synchronicity.”2

  1. In The Structure and Dynamics of the Psyche, C. W. 8, pp. 417-531. [ref]
  2. Letter to Dr. Selig dated 25 February, 1953, in Spring, 1971, p. 127. [ref]
June Singer, Boundaries of the Soul: The Practice of Jung’s Psychology, Anchor Books edition, 1973, p. 398.

The Eagle Has Landed begins with newsreel footage depicting Mussolini’s rescue by the Germans during the Gran Sasso raid. This probably inspires the plot to abduct Churchill. Come to the present. Did the operation to capture Venezuelan president Nicolás Maduro and his wife, Cilia Flores, stem from this, or is there no causal connection? All of this is the intellectual adventure of Jung’s synchronicity.

We Are Finite. Does This Affect What We Can Know?

§2. What Can We Know?

The theme of this section is that We Are Finite: while our knowledge of the number 3 is intimately associated with experiences of some kind or other of triples of one or another sort, surely nothing like that is the case for our knowledge of the number ω or, for that matter, the number 10101010.

To avoid misunderstanding, let me emphasize that the issue being addressed is not “How can we have knowledge of mathematical objects despite their abstractness?” but rather “How can we have knowledge of infinite mathematical objects despite their huge size and consequent remoteness from experience?” Here is what I mean by remoteness: Not only do we not have experience of infinite mathematical objects, but we do not have any experience of anything suitably like them. The number 2 bears some relation to pairs of objects. The points and lines of Euclidean geometry bear some relation to pencil points and lines. But there does not seem to be anything suitable to play any analogous role for infinite mathematical objects.

Of course infinite mathematical objects are abstract. The issue of remoteness is an addition to that of abstractness. Both are apparent epistemological difficulties caused by the distance of mathematical objects from experience.

But if we divide the problem of the abstract into two parts, the problem of the finite and the problem of the infinite, it becomes clear that the two have quite distinct features.

The problem of the abstract, in Paul Benacerraf’s words, is this [Ben73, p. 409]: “the concept of mathematical truth, as explicated, must fit into an over-all account of knowledge in a way that makes it intelligible how we have the mathematical knowledge that we have. An acceptable semantics for mathematics must fit an acceptable epistemology?” For Benacerraf, an acceptable semantics is a Platonist one, and so his problem is that of the difficulties involved in giving an acceptable account of knowledge of abstract objects. Benacerraf concentrated on a weaker version of the problem: not on accounting for the mathematical knowledge that we have but on accounting for how we can have any mathematical knowledge whatever. “The minimal requirement, then, is that a satisfactory account of mathematical truth must be consistent with the possibility that some such truths be knowable” [Ben73, p. 409]. Philosophers have tended to concentrate on the minimal requirement and to offer up solutions to the problem of the finite to solve it.

Even those skeptical about the existence of any abstract mathematical objects at all want to endorse the truism ‘2 + 2 = 4’ in some way or other that acknowledges that it is better than ‘2 + 2 = 5’ even though they may not grant that it is true. Even without an abstract number 2, one is still faced with explaining the general fact that the members of two nonoverlapping pairs form a quadruple.

Whatever one’s views about mathematical objects, it is necessary to make sense of our counting, computing, and bookkeeping activities. Skepticism about small finite mathematical objects—in particular small natural numbers is just not doubt about the acceptability in some form or other of many of the putative facts about them. Moreover, there are many stories one could tell about a source for genuine knowledge concerning some finite mathematical objects, including, as the most trivial special case, knowledge of their existence. Take, for example, small natural numbers. Various explanations of our knowledge about them might invoke the experience of time, the experience of bunches of physical objects or of patterns exhibited by them, or the sequencing of words in sentences. Other explanations might rely on the exigencies of the construction of theories of the physical world. We are faced with too many ways of accounting for our knowledge of small finite mathematical objects, not too few. For example, Parsons [Par80] showed how to account for such knowledge on the basis of our linguistic capacities, while Maddy [Mad90] showed how to do it on the basis of experiences with medium-sized physical objects.

Benacerraf’s minimal requirement can surely be met with an acceptable solution to the problem of the finite—though I make no claim to know what the actual solution is. That is a question whose answer involves detailed psychological information about how people typically actually acquire knowledge of small finite mathematical objects, and we do not yet have sufficiently detailed information to answer it. Acquisition of the number concept does, however, involve both linguistic components—learning to count aloud—and experience of medium-sized physical objects—counting them using the spoken number sequence. It is therefore likely that the actual solution involves components of both the one proposed by Parsons and the one proposed by Maddy—and probably other components as well.

In sharp contrast to the situation about ‘2 + 2 = 4’, many of those who are skeptical about the existence of infinite combinatorial collections would want to doubt or deny the Axiom of Choice—not only its truth, but its acceptability in any form whatever. General facts about the infinite are not robust in the same way that the facts of counting, computing, and bookkeeping are. Moreover, it is not at all clear what we can fall back on as a source of mathematical knowledge concerning the infinite—what can play the role that bunches and sequences of moments, objects, or words seem so well suited to play for small finite mathematical objects. It is that lack that raises the problem posed by the remoteness of the infinite: it seems that we cannot have grounds to know what we find we actually do know about the infinite.

In Chapter VIII I shall show that as a matter of fact the combinatorial infinite is not remote—it has pretty much the same kinds of ties to experience as do small natural numbers. (That may be a bit misleading see Chapter VIII for a more careful formulation.) That solves the problem of the remoteness of the infinite philosophical problems concerning infinite mathematical objects become just like the familiar ones concerning finite mathematical objects. That is important because the problems concerning finite mathematical objects are not skeptical ones—the genuine doubts about the acceptability of our theory of the infinite are refuted. It also provides the essential missing ingredient for an explanation of the grounds on which mathematicians are entitled to make claims of self-evidence. But before presenting the solution, it is necessary to become clearer on the nature of the problem.

The two Benacerrafian problems—of the finite and the infinite—are both important, and every adequate philosophy of mathematics must be compatible with solutions to them. Nonetheless, the problem of the infinite deserves special emphasis because it is in danger of being lost as the result of the huge amount of attention being devoted to the Benacerrafian problem of the abstract in its simplified guise as the problem of the finite. That loss would be most unfortunate. The problem concerning the infinite was a primary concern of the philosophy of mathematics for many years—as may be seen in the work of Brouwer and that of several philosophers discussed by Benacerraf, such as Hilbert, Gödel, and Quine.

Shaughan Lavine, Understanding the Infinite, Harvard University Press, 1994, pgs. 162-165.

You do not doubt that ‘7 – 7 = 0’, but you cannot say ‘∞ – ∞ = 0’. In math lingo, the latter is indeterminate. Does this mean we are incapable of understanding the indeterminate, or is this just the nature of the universe? Consider Eugene Wigner’s lecture, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”. He noted, “It is important to point out that the mathematical formulation of the physicist’s often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena.”

Still, complete understanding often seems to elude us.

“Fog Everywhere” Continued

[read the previous essay on this topic]

Consider mathematics and the aspect of fog. Nietzsche argued that the world is not knowable to us as a whole. This includes trying to express it with numbers. A recent example is Max Tegmark’s assertion that the universe is not something explained by mathematics; rather, it is itself mathematics. To contrast, Karl Jaspers, summarizing Nietzsche, wrote:

We cannot say what the world as a whole is. It is false to change all processes into a familiar world of our own, and then say: “All is will (everything wills); all is pleasure or pain (everything endures); all is motion (everything flows); all is tone (everything sounds); all is spirit (everything thinks); all is number (everything reckons).” Nietzsche warns us against all notions of the whole: “Let us guard against thinking that the world is a living being…or that the universe is a machine…Let us guard against saying that there are laws of nature…Let us guard against thinking that the world eternally creates novelties.” All these “shadows of God” darken actuality. We are within the world, and the whole of the world is, as a whole, not accessible to us.

Karl Jaspers, Nietzsche: An Introduction to the Understanding of His Philosophical Activity, Henry Regnery Company, 1966, p. 293

Notice in the paragraph quoted above that Nietzsche specifically excludes reducing everything to numbers, in opposition to Tegmark and the simulation hypothesis.

Robert Kanigel describes Srinivasa Ramanujan as “the man who knew infinity” in his biography of the same title, but mathematicians disagree on the ontological status and utility of infinity. The discussion of applying Ramanujan’s sum to arrive at -1/12 is very intricate and complex.

To make things even more puzzling, math students are confronted with the Gaussian integral, with its positive and negative infinities.

You can gain additional insight into infinity with:

Cauchy and Weierstrass had eliminated infinitely small and infinitely great numbers from analysis and replaced them by limits. But the theory of limits that thereby became so central required a clearer theory of the real line, that is to say, a theory of the irrational numbers. And that theory promptly reintroduced the infinite into analysis. The old infinity of infinitesimal and infinite numbers was simply replaced by the new infinity of infinitely large collections.12

In 1831 Carl Friedrich Gauss said [Kli72, p. 994], “I protest against the use of an infinite quantity as an actual entity; this is never allowed in mathematics. The infinite is only a manner of speaking, in which one properly speaks of limits to which certain ratios can come as near as desired, while others are permitted to increase without bound.” But only 52 years later, we find this in Cantor’s Grundlagen [Can76, p. 75]: “The idea of considering the infinitely large not only in the form of the unlimitedly increasing magnitude and in the closely related form of convergent infinite series…but to also fix it mathematically by numbers in the definite form of the completed infinite was logically forced upon me, almost against my will since it was contrary to traditions which I had come to cherish in the course of many years of scientific effort and investigations.”

12 See Russell’s Principles of Mathematics [Rus03, p. 304] for a related sentiment.

Shaughan Lavine, Understanding the Infinite, Harvard University Press, 1994, pgs. 38-39

In 1961, the great Russian physicist George Gamow’s popular math and physics overview was republished to great acclaim among science and math aficionados. Intriguingly titled One, Two, Three…Infinity, the book gives you a hint that the relationship of numbers and integers to the concept of infinity is still intriguing even after the intricate analyses of Cauchy, Weierstrass et al.

Let us conclude by considering Zeno’s dichotomy paradox of motion, in which “That which is in locomotion must arrive at the half-way stage before it arrives at the goal.” As the goal is divided into halves, one must complete an infinite number of tasks, which Zeno maintains is an impossibility.

[read the previous essay on this topic]

“Fog everywhere” and Other Confusions

Charles Dickens gives us wonderful sociopolitical insight in his serial Bleak House, with his imagery of impenetrable fog in the second paragraph of chapter 1:

Fog everywhere. Fog up the river, where it flows among green aits and meadows; fog down the river, where it rolls deified among the tiers of shipping and the waterside pollutions of a great (and dirty) city. Fog on the Essex marshes, fog on the Kentish heights. Fog creeping into the cabooses of collier–brigs; fog lying out on the yards and hovering in the rigging of great ships; fog drooping on the gunwales of barges and small boats. Fog in the eyes and throats of ancient Greenwich pensioners, wheezing by the firesides of their wards; fog in the stem and bowl of the afternoon pipe of the wrathful skipper, down in his close cabin; fog cruelly pinching the toes and fingers of his shivering little ’prentice boy on deck. Chance people on the bridges peeping over the parapets into a nether sky of fog, with fog all round them, as if they were up in a balloon and hanging in the misty clouds.

Charles Dickens, Bleak House. Bradbury & Evans, 1852-1853.

Think of the Trump–Epstein cover-up and the machinations of the U.S. government to attempt to conceal everything in a similar fog. We discussed another dimension of our ignorance with Friedrich Nietzsche’s assertion that we “are unknown to ourselves”.

Let’s consider a third level of our confusion and how we attempt to extricate ourselves by expressing the world around us through mathematics. For example, the square root of -1 is i. ii is approximately 0.208. To a student encountering this concept for the first time, it can be inscrutable that an imaginary number to the power of itself results in a real number. To quote Wikipedia:

In electrical engineering, signal processing, and similar fields, signals that vary periodically over time are often described as a combination of sinusoidal functions (see Fourier analysis), and these are more conveniently expressed as the sum of exponential functions with imaginary exponents, using Euler’s formula.

Wikipedia, Euler’s formula [with added links]

The esteemed physicist Roger Penrose has said on multiple occasions that he believes the realm of imaginary numbers or complex analysis will be more informative in physics than real numbers. (See his classic book, The Road to Reality: A Complete Guide to the Laws of the Universe.)

Let’s conclude with our inevitable mortality as living beings as another source of perplexity and confusion. As we grow older, half of our mind is fixated on the enjoyment we get from life, while the other half is focused on the anxiety that it is not forever. We attempt to sidestep this anxiety by clinging to the escapist thought that “Besides, it’s always the others who die.”

Is Reality Fundamentally Analyzable or Opaque?

Imagine your childhood. You’re looking out a window in the early afternoon. You see the clouds in the sky, the passers-by and traffic down the street. You have no trouble differentiating a police car from an ambulance. A summary of this can be expressed in the philosophical quote, “your version of the world shows up for you, like a friend at the bus stop.”

All of this suffers from an invasion from two sides: clarity and opacity. Consider the last sentence of the preface from Frederich Nietzsche’s On the Genealogy of Morality (below):

Preface

I

We are unknown to ourselves, we knowers, we ourselves, to ourselves, and there is a good reason for this. We have never looked for ourselves, — so how are we ever supposed to find ourselves? How right is the saying: ‘Where your treasure is, there will your heart be also’;1 our treasure is where the hives of our knowledge are. As born winged-insects and intellectual honey-gatherers we are constantly making for them, concerned at heart with only one thing — to ‘bring something home’. As far as the rest of life is concerned, the so-called ‘experiences’, — who of us ever has enough seriousness for them? or enough time? I fear we have never really been ‘with it’ in such matters: our heart is simply not in it — and not even our ear! On the contrary, like somebody divinely absent-minded and sunk in his own thoughts who, the twelve strokes of midday having just boomed into his ears, wakes with a start and wonders ‘What hour struck?’, sometimes we, too, afterwards rub our ears and ask, astonished, taken aback, ‘What did we actually experience then?’ or even, ‘Who are we, in fact?’ and afterwards, as I said, we count all twelve reverberating strokes of our experience, of our life, of our being — oh! and lose count … We remain strange to ourselves out of necessity, we do not understand ourselves, we must confusedly mistake who we are, the motto2 ‘everyone is furthest from himself’ applies to us for ever, — we are not ‘knowers’ when it comes to ourselves…

  1. Gospel according to Matthew 6.21.
  2. ‘Jeder ist sich selbst der Fernste’ is a reversal of the common German saying, ‘Jeder ist sich selbst der Nächste’ ‘Everyone is closest to himself’ i.e. ‘Charity begins at home’, cf. also Terence, Andria IV. 1.12.
Nietzsche, Frederich, On the Genealogy of Morality, Translated by Carol Diethe, Cambridge University Press, 1994, pp. 3-4.

Go back to our initial example of the childhood window. On one hand, our version of the world shows up for us. On the other, since “we are not ‘knowers’ when it comes to ourselves”, we show up as an opaque version of ourselves. As he states at the beginning of the preface, we are unknown to ourselves. If asked, you would most likely not remember the point at which you could differentiate the police car from an ambulance. Imagine meeting a friend during that same period, and instead of thinking of the enjoyable conversation you will have, but the concern that all of these things are impermanent.

The Danish thinker Søren Kierkegaard teaches us that the reason we are opaque to ourselves is that the self is a synthesis waiting to be made. This synthesis would connect the momentary pleasures of time with friends and the impermanence of these things.

Consider the opacity built into science. Science claims that ultimately, there is a logical, mathematical way of understanding what you see while looking out the window. Contrast this with the Dutch historian Pieter Geyl’s assertion that history is an “argument without end.” It could be that scientific inquiry is also a quest without end.

What is the great takeaway from all of this? On Kierkegaard’s side, we face the opacity wall and on the materialist side, we still have the opacity of what we have yet to understand. With each discovery, we continue to see more that we do not yet understand.

Songs and Tweaking Our Understanding, Part 2

Let’s look at three songs: “Cabaret” (from the musical of the same title), “Is That All There Is?” and “More Than a Woman” by the Bee Gees.

In “Cabaret” the character Sally Bowles sings, “Start by admitting from cradle to tomb / It isn’t that long a stay / Life is a cabaret, old chum…” She asserts that life is so short that one shouldn’t take it seriously and should be treated as boisterous fun.

Earlier, she sang:

I used to have this girlfriend known as Elsie
With whom I shared four sordid rooms in Chelsea
She wasn’t what you’d call a blushing flower
As a matter of fact, she rented by the hour

The day she died the neighbors came to snicker:
“Well, that’s what comes from too much pills and liquor”
But when I saw her laid out like a queen
She was the happiest corpse I’d ever seen

Does the idea of a happy corpse ring true? As a corollary of this, is it plausible that a smile survives hours after death?

While “Cabaret” compares life to the titular cabaret, there are many other songs with simplified comparisons of life, none of which cover the full breadth of human experience. The French philosopher Maurice Merleau-Ponty wrote that the world is inexhaustible, and to extend this, so too is life.

In “Is That All There Is?” each question has several levels. When confronted with the fire that claims her childhood home, the disappointment of the circus and finally the loss of her love, the song echoes “Cabaret” with the chorus:

Is that all there is?
If that’s all there is my friends, then let’s keep dancing
Let’s break out the booze and have a ball

Much like the way “Cabaret” repeats “Life is a cabaret, old chum / Come to the cabaret”.

In “More Than a Woman” by the Bee Gees, they sing, “We can take forever, just a minute at a time”. Can we visualize this in the concrete terms of the song? In Western tradition, we often associate the search for forever with young love. What does it mean to fuse the momentary and the forever?

The Bee Gees song gives us an entry to the changing sense of time experienced by a living person, as contrasted to the objective time of a fossil carbon-dated by a paleontologist.

Songs and Tweaking Our Understanding

In any public place, you’re likely to be surrounded by faces reflecting an unshakeable puzzlement of people who have sleepwalked through life. Songs capture this perplexity. Take “Hier encore” by Charles Aznavour, which has subsequently been translated into a myriad of languages and covered by many artists.

Yesterday, when I was young
The taste of life was sweet as rain upon my tongue
I teased at life as if it were a foolish game
The way the evening breeze may tease a candle flame
The thousand dreams I dreamed, the splendid things I planned
I always built, alas, on weak and shifting sand
I lived by night and shunned the naked light of day
And only now I see how the years ran away

Yesterday, when I was young
So many drinking songs were waiting to be sung
So many wayward pleasures lay in store for me
And so much pain my dazzled eyes refused to see
I ran so fast that time and youth at last ran out
I never stopped to think what life was all about
And every conversation I can now recall
Concerned itself with me, me and nothing else at all

Yesterday, the moon was blue
And every crazy day brought something new to do
I used my magic age as if it were a wand
And never saw the waste and emptiness beyond
The game of love I played with arrogance and pride
And every flame I lit too quickly, quickly died
The friends I made all seemed somehow to drift away
And only I am left on stage to end the play

There are so many songs in me that won’t be sung
I feel the bitter taste of tears upon my tongue
The time has come for me to pay for yesterday
When I was young
Young, young…

Notice that “Yesterday, when I was young” is a poetic conceit discussing his youth. Orbiting this thought is a song by Crosby, Stills & Nash, the first of which is “Wasted on the Way”, regretting the singers’ wasted lives. Another is “We May Never Pass This Way (Again)” by Seals & Crofts. These songs tweak our understanding by reminding us in a painful, yet melodious way, that we are all swept along by life’s pressures and events.

Imagine that one is aware of this trend early enough to avoid it and attempts to sidestep this fate. However, there’s a danger on the other side of mindless snarling rebellion, in the vein of Billy Idol’s “Rebel Yell”.

Finally, there are musical styles such as Portuguese fado, characterized by mournful tunes and lyrics that straddle the loss of one’s true love and the loss of Portuguese influence in the world, such as Latin America supplanting Portugal as the center of Portuguese culture. In all of these, the “once upon a time” is at the level of youth, one’s romance and eventually life, which “never comes again”.

Perhaps the most sobering line in popular music is “all we are is dust in the wind” (from “Dust in the Wind” by Kansas). One might benefit from a moment of quiet reflection on this lyric.

Is There a Scheme of Things Underlying Everything?

In The Thibaults (the novel sequence for which Roger Martin du Gard won the 1937 Nobel Prize in Literature), a fundamental motif is the question of whether the universe is coherent. Roughly speaking, there are three competing schools of coherence—science, religion and art. Antoine Thibault discusses with the Abbé (French title for abbot):

Antoine did not seem to hear him. “Just think,” he exclaimed, “what it means to a youngster, when he’s turned loose, by gradual stages, on mathematics, physics, chemistry! Suddenly he discovers that he has all space, the universe, for his playground. And after that, religion strikes him as not only cramped, but false, illogical. Untrustworthy.”

Roger Martin du Gard, The Thibaults, translated by Philip Thody & Ellen Kennedy, Bantam Modern Classic Edition, Viking Press, 1968. page 762.

The climax of this debate appears when Antoine says:

“…I talked just now about Universal Order and a Scheme of Things; but that was merely to talk like everyone else. Actually it seems to me that we’ve as many reasons to question the existence of a Scheme of Things as to take it for granted. From his actual viewpoint the human animal I am observes an immense tangle of conflicting forces. But do these forces obey a universal law outside themselves, distinct from them? Or do they, rather, obey—so to speak—internal laws, each atom being a law unto itself, that compels it to work out a kind of ‘personal’ destiny? I see these forces obeying laws which do not control them from outside, but join up with them, which do nothing more than in some way stimulate them.…And anyhow, what a jumble it is, the course of natural phenomena! I’d just as soon believe that causes spring from each other ad infinitum, each cause being the effect of another cause, and each effect the cause of other effects. Why should one want to assume at all costs a Scheme of Things?…”

Roger Martin du Gard, The Thibaults, translated by Philip Thody & Ellen Kennedy, Bantam Modern Classic Edition, Viking Press, 1968. page 768.

The topic of an underlying scheme of things is close to the central question of Western civilization. In Plato’s Republic, we have the allegory of the dark cave occupied by humanity. It looks at shadows dancing on the wall, projected by a fire. Liberating humanity requires leaving the cave and climbing to the surface of the earth, glimpsing the sun for the first time. From here, with the help of philosophy, humanity flies off and encounters the Logos and the Eidos. Mathematical truth crowns this journey.

Aristotle, Plato’s star pupil and later rival, takes this quest and focuses on the biological. The Athenian tradition invents theory, a pillar of Western tradition, culminating in modern science. Jerusalem’s competing tradition, the Judeo-Christian worldview, derives its scheme of things from divinity before biology and mathematics. Thus, the novel’s debate is ultimately the struggle between Athens and Jerusalem.