Songs as Another Kind of Parallel University

Meta Intelligence is a heterodox view of education where formal education (courses, diplomas, universities, fields) are incomplete and limited without adding informal education which is part of your life such as movies, songs, conversations and images (paintings, posters, etc). Your “lifeworld” (Edmund Husserl’s apt coinage) fuses all the kinds of education where the word education means thought-provoking and illuminating. Even personal experience counts such as illnesses or bad marriages! Only via this Meta Intelligence will you achieve a glimpsed “holism.” (Meta Intelligence is that meta-field outside fields, borders and boundaries.)

Take songs.

Think back to Jim Morrison’s classic tune, “Riders on the Storm” which begins:

“Riders on the storm
Riders on the storm
Into this house, we’re born
Into this world, we’re thrown
Like a dog without a bone
An actor out on loan
Riders on the storm”

This song (by the Doors), expresses in a simple way Heidegger’s notion of human existence as partly governed by “Geworfenheit” which derives from “werfen,” to throw. “Geworfenheit” means “thrownness.” Jim Morrison and his band the Doors are songphilosophers without (probably) being Heidegger’s acolytes. Max Weber, one of the fathers of modern sociology, uses the word “disenchantment” to describe the modern world, “Entzauberung” in German, where “zauber” means “magicality” and “ent” means “removal of,” and “ung” means “condition of being.” The magic here does not mean something like a card trick but rather sacred mysteries, perhaps like the feeling a medieval European felt on entering a cathedral.

Enchantment in the West survived in our notions of romantic love and was associated with the songs and outlook of the medieval troubadours. Such romantic enchantment which is fading from our culture in favor of sex is still celebrated in the classic Rogers and Hammerstein song, “Some Enchanted Evening” from the forties musical and fifties movie, South Pacific.

The song lyrics give you the philosophy of romantic love as the last stand of enchantment:

“Some enchanted evening, you may see a stranger,
You may see a stranger across a crowded room,
And somehow you know, you know even then,
That somehow you’ll see here again and again.
Some enchanted evening, someone may be laughing,
You may hear her laughing across a crowded room,
And night after night, as strange as it seems,
The sound of her laughter will sing in your dreams.

“Who can explain it, who can tell you why?
Fools give you reasons, wise men never try.

“Some enchanted evening, when you find your true love,
When you hear her call you across a crowded room,
Then fly to her side and make her your own,
Or all through your life you may dream all alone.

“Once you have found her, never let her go,
Once you have found her, never let her go.”

Notice that “chant” is a component of enchantment.

One could say that conventional enchantment has been transferred to the world of science and mathematics where a deep beauty is intuited. Professor Frank Wilczek of MIT (Nobel Prize) wrote several books on this intersection of science and the quest for beauty whereas Sabine Hossenfelder of Germany has argued, per contra, that this will be a “bum steer.”

You should sense that like movies, songs give you a “side window” or back door into thinking and knowledge, which should be center stage and not depreciated.

Mathematics and the World: London Mathematical Laboratory

Stability of Heteroclinic Cycles in Rings of Coupled Oscillators

[from the London Mathematical Laboratory]

Complex networks of interconnected physical systems arise in many areas of mathematics, science and engineering. Many such systems exhibit heteroclinic cyclesdynamical trajectories that show a roughly periodic behavior, with non-convergent time averages. In these systems, average quantities fluctuate continuously, although the fluctuations slow down as the dynamics repeatedly and systematically approach a set of fixed points. Despite this general understanding, key open questions remain concerning the existence and stability of such cycles in general dynamical networks.

In a new paper [archived PDF], LML Fellow Claire Postlethwaite and Rob Sturman of the University of Leeds investigate a family of coupled map lattices defined on ring networks and establish stability properties of the possible families of heteroclinic cycles. To begin, they first consider a simple system of N coupled systems, each system based on the logistic map, and coupling between systems determined by a parameter γ. If γ = 0, each node independently follows logistic map dynamics, showing stable periodic cycles or chaotic behavior. The authors design the coupling between systems to have a general inhibitory effect, driving the dynamics toward zero. Intuitively, this should encourage oscillatory behavior, as nodes can alternately be active (take a non-zero value), and hence inhibit those nodes to which it is connected to, decay, when other nodes in turn inhibit them; and finally grow again to an active state as the nodes inhibiting them decay in turn. In the simple case of N = 3, for example, this dynamics leads to a trajectory which cycles between three fixed points.

The authors then extend earlier work to consider larger networks of coupled systems as described by a directed graph, describing how to find the fixed points and heteroclinic connections for such a system. In general, they show, this procedure results in highly complex and difficult to analyze heteroclinic network. Simplifying to the special case of N-node directed graphs with one-way nearest neighbor coupling, they successfully derive results for the dynamic stability of subcycles within this network, establishing that only one of the subcycles can ever be stable.

Overall, this work demonstrates that heteroclinic networks can typically arise in the phase space dynamics of certain types of symmetric graphs with inhibitory coupling. Moreover, it establishes that at most one of the subcycles can be stable (and hence observable in simulations) for an open set of parameters. Interestingly, Postlethwaite and Sturman find that the dynamics associated with such cycles are not ergodic, so that long-term averages do not converge. In particular, averaged observed quantities such as Lyapunov exponents are ill-defined, and will oscillate at a progressively slower rate.

In addition, the authors also address the more general question of whether or not a stable heteroclinic cycle is likely to be found in the corresponding phase space dynamics of a randomly generated physical network of nodes. In preliminary investigations using randomly generated Erdős–Rényi graphs, they find that the probability of existence of heteroclinic cycles increases both as the number of nodes in the physical network increases, and also as the density of edges in the physical network decreases. However, even in cases where the probability of existence of heteroclinic cycles is high, there is also a high chance of the existence of a stable fixed point in the phase space. From this they conclude that the question of the stability of the heteroclinic cycle is important in determining whether or not the heteroclinic cycle, and associated slowing down of trajectories, will be observed in the phase space associated with a randomly generated graph.

The paper is available as a pre-print here [archived PDF].

Education and Word and Number Hidden Vagueness

These mini-essays help students of any age to re-understand education in a deeper and more connected way.

They look for “circum-spective” intelligence. (Not in the sense of prudential or cautious but in the sense of “around-looking.”)

One of the things to begin to see is that explaining things in schools is misleading “ab initio” (i.e., from the beginning).

Let’s do an example:

In basic algebra, you’re asked: what happens to (x2 – 1)/(x – 1) as x “goes to” (i.e., becomes) 1.

If you look at the numerator (thing on top), x2 is also 1 (since 1 times 1 is 1) and (1 – 1) is zero. The denominator is also (1 – 1) and zero.

Thus you get 0 divided by 0.

You’re then told that’s a no-no and that’s because zeros and infinities lead to all kinds of arithmetic “bad behavior” or singularities.

You’re then supposed to see that x2 – 1 can be re-written as (x – 1)(x + 1) and since “like cancels like,” you cancel the x – 1 is the numerator and denominator and “get rid” of it.

This leaves simply x + 1. So, as x goes to 1, x + 1 goes to 2 and you have a “legitimate” answer and have bypassed the impasse of 0 acting badly (i.e., zero divided by zero).

If you re-understand all this more slowly you’ll see that there are endless potential confusions:

For example: you cannot say that (x2 – 1)/(x – 1) = x + 1 since looking at the two sides of the equal sign shows different expressions which are not equal.

They’re also not really equivalent.

You could say that coming up with x + 1 is a workaround or a “reduced form” or a “downstream rewrite” of (x2 – 1)/(x – 1).

This reminds us of the endless confusions in high school science: if you combine hydrogen gas (H2) with oxygen gas (O2) you don’t get water (H2O). Water is the result of a chemical reaction giving you a compound.

A mixture is not a compound. Chemistry is based on this distinction.

Math and science for that matter, are based on taking a formula or expression (like the one we saw above) and “de-cluttering” it or “shaking loose” a variant form which is not identical and not the same but functionally equivalent in a restricted way.

A lot of students who fail to follow high school or college science sense these and other “language and number” problems of hidden vagueness.
School courses punish students who “muse” to themselves about hidden vagueness. This behavior is pre-defined as “bad woolgathering” but we turn this upside down and claim it is potentially “good woolgathering” and might lead to enchantment which then underlies progress in getting past one’s fear of something like math or science or anything else.

One is surrounded by this layer of reality on all sides, what Wittgenstein calls “philosophy problems which are really language games.”

Think of daily life: you say to someone: “you can count one me.” You mean trust, rely on, depend on, where count on is a “set phrase.” (The origin of the phrase and how it became a set phrase is probably unknowable and lost in the mists of time.)

“You can count on me” does not mean you can stand on me and then count something…one, two, three.

In other words in all kinds of language (English, say, or math as a language) one is constantly “skating over” such logic-and-nuance-and-meaning issues.

The genius Kurt Gödel (Einstein’s walk around buddy at Princeton) saw this in a deep way and said that it’s deeply surprising that languages work at all (spoken, written or mathematical) since the bilateral sharing of these ambiguities would seem deadly to any clarity at all and communication itself would seem a rather unlikely outcome.

You could also say that drama giants of the twentieth century like Pinter, Ionesco and Beckett, intuit these difficulties which then underlie their plays.

All of this together gives you a more “composite” “circum-spective” view of what is really happening in knowledge acquisition.

Knot Theory and the Strangeness of Reality

The subfield of “knot theory” in math as a kind of geometry of “twistiness” gives us a deep “meta-intelligent” signal or lesson.

Meta-intelligent means “perspective-challenging” with or without full details of any subfield itself.

Consider this overview or comment on “knot theory” now:

“In mathematical knot theory, you throw everything out that’s related to mechanics,” Dunkel (MIT math professor) says. “You don’t care about whether you have a stiff versus soft fiber—it’s the same knot from a mathematician’s point of view. But we wanted to see if we could add something to the mathematical modeling of knots that accounts for their mechanical properties, to be able to say why one knot is stronger than another.”

But you immediately think: in the real world knots are not only twisted up in mathematically definable ways but are in fact actual shoelaces, neckties, ropes, etc, that have chemical and molecular properties before you describe their twist-and-tighten or slide-and-grip “shapes.”

Which is the real: the math or the “ropiness” of the ropes or the “laciness” of the laces?

The relationship between things and numbers is elusive.

Mathematicians have long been intrigued by knots, so much so that physical knots have inspired an entire subfield of topology known as knot theory—the study of theoretical knots whose ends, unlike actual knots, are joined to form a continuous pattern.

In knot theory, mathematicians seek to describe a knot in mathematical terms, along with all the ways that it can be twisted or deformed while still retaining its topology, or general geometry.

MIT mathematicians and engineers have developed a mathematical model that predicts how stable a knot is, based on several key properties, including the number of crossings involved and the direction in which the rope segments twist as the knot is pulled tight.

“These subtle differences between knots critically determine whether a knot is strong or not,” says Jörn Dunkel, associate professor of mathematics at MIT. “With this model, you should be able to look at two knots that are almost identical, and be able to say which is the better one.”

“Empirical knowledge refined over centuries has crystallized out what the best knots are,” adds Mathias Kolle, the Rockwell International Career Development Associate Professor at MIT. “And now the model shows why.”

As per usual in science, one is dazzled by the ingenuity of the quest and the formulations but puzzled by the larger implications since we can never decide whether math “made” us or we “made” (i.e., invented) math.

“Pre-Understanding” as a Pillar of Better Education

One pillar of our education enhancement effort is the concept of “pre-understanding” which argues that there usually is a step that has been skipped in education and that is the overview or guidance or “lay of the land” step that comes before courses become efficacious. To tackle a 900-page text-book seems soul-crushing in the absence of “pre-understanding” (i.e., where are we and why are we doing this) other than the coercive power of schools (grades, scholarships, recommendations, grad school admissions, etc.)?

A person senses (not incorrectly) that economics as a field of study seems tedious and solipsistic (i.e., “talking to itself” and not to the student).

Can we give students a “pre-understanding” that opens a backdoor or side window into the field, where such doors and windows were never seen or noticed?

A person is trying to decide what airline they should use in flying from Boston to Nepal.

Immediate concerns are of course price, flexibility of ticket, safety reputation of different airlines, schedules, weather forecasts, routes, etc.

A person might argue: Flight A stops in Tokyo and I can make use of that because my friend who lives in the area will put me up for a weekend, whereby we can do the town and sights, talk about old times, re-connect, etc. There’s also some other task or chore there I could do and so the Tokyo interruption is to my liking. There’s some risks associated with this (i.e., my fiancée who’s traveling with me might find it boring). I’m not sure (uncertainty).

Now suppose somebody tells you that such “decision theory” is at the heart of economics and involves four dimensions:

  1. Costs.
  2. Benefits.
  3. Risks.
  4. Uncertainties.

Whether you know it or not, you are optimizing some things (usefulness and pleasure of travel) and minimizing other things (time in the air, costs, safety risks, comfort, etc.).

You don’t realize that you’re making subtle decisional calculations where risks and uncertainties that cannot be quantified, are somehow being weighted and weighed and quantified by you, implicitly and the decision calculus is quite complicated.

Suppose you were now given to understand that economics is about economizing (i.e., budgeting your costs, benefits, risks and uncertainties, some of which are qualitative and subjective) but you find a way to assign some kind of numbers and weighting factors (i.e., importance to you) in your actual but more likely, intuitive calculations.

Goaded and prompted by this “pre-understanding” you might then pick up a standard guide to actual cost-benefit analysis (such as Mishan’s classic book) and go through this previously unseen “door” into the field without being crushed by the feeling that it’s all so tiresome in its appearance.

Similarly, if you take a math concept like the square root of minus one, think of it as an imaginary “unicorn” of the mind, then how is it that it appears constantly in all science and math such as Euler’s equation, Schrödinger’s equation, electrical engineering textbooks, etc.

How can something so elusive be so useful?

This “pre-understanding” quest or detour or episode could give you, the student, a deep nudge through a hidden window or door into “math world.”
Without this “trampoline of pre-understanding,” an “ocean of math intricacy” seems to loom before you.

Education and “Chaos”: The Example of Climate Change

Students will have heard on read descriptions of “chaos theory” which try to capture the phenomenon that a small change “here” or now might involve a mega-change somewhere else or later on or both. In other words, tremendous turbulence could arise from overlooked minutiae in some other region or domain. Chaos here does not mean lawless…it means lawful but in surprising ways, like a pendulum swinging from another pendulum where the laws of pendular motion are still in effect but the motions are “jumpy.”

This can be described as follows:

Chaos theory is a branch of mathematics focusing on the study of chaos—states of dynamical systems whose apparently-random states of disorder and irregularities are often governed by deterministic laws that are highly sensitive to initial conditions. Chaos theory is an interdisciplinary theory stating that, within the apparent randomness of chaotic complex systems, there are underlying patterns, constant feedback loops, repetition, self-similarity, fractals, and self-organization.

The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state (meaning that there is sensitive dependence on initial conditions). A metaphor for this behavior is that a butterfly flapping its wings in China can cause a hurricane in Texas.

Blaise Pascal (17th century) gives us the example of “Cleopatra’s nose.” Had her nose been longer, Pascal muses, she would presumably have not been so beautiful and this could have altered romantic entanglements and the behavior of rival Roman generals and world history might have moved along different pathways completely (recall Caesar and Cleopatra, the play).

All of this “strange science” applies to climate change.

In the Winter 2019/20 issue of Options, from the International institute for Applied Systems Analysis (IIASA, Austria headquarters), there’s a short piece that shows you how climate change has such “chaos-type” features which could “turbo-charge” changes already expected:

Will Forests Let Us Down?

Current climate models assume that forests will continue to remove greenhouse gases from the atmosphere at their current rate.

A study by an international team including researchers from IIASA, however, indicates that this uptake capacity could be strongly limited by soil phosphorous availability. If this scenario proves true, the Earth’s climate would heat up much faster than previously assumed.

(Options, Winter 2019/20 issue, IIASA, page 5, “News in Brief”)

Students should glimpse something here that points to a “deep structure.”
Climate scientists and climate modelers at this time are trying to re-examine and re-jigger predictions to include overlooked details that could add “chaotic dynamics” to the predictions. Knowledge itself is evolving and if you add knowledge changes and revisions to model ones, you have to conclude that even with this fantastic level of human ingenuity and scientific intricacy, we “see the world through a glass, darkly” because the facts, models, chaos math, overviews, are themselves in “interactive flux.”

Two Kinds of Extra Understanding: Pre and Post

We argue here in this proposal for an educational remedy that two dimensions of understanding must be added to “retro-fit” education.

In the first addition, call it pre-understanding, a student is given an overview not only of the field but of his or her life as well as the “techno-commercial” environment that characterizes the globe.

Pre-understanding includes such “overall cautions” offered to you by Calderón de la Barca’s 17th century classic Spanish play, Life is a Dream (SpanishLa vida es sueño). A student would perhaps ask: “what would it be like if I faced this “dreamlike quality” of life, as shown by the Spanish play, and suddenly realized that a life of “perfect myopia” is not what I want.

Hannah Arendt warns similarly of a life “like a leaf in the whirlwind of time.”

Again, I, the student ask: do I want such a Hannah Arendt-type leaf-in-the-whirlwind-like life, buried further under Calderón de la Barca’s “dream state”?

But that’s not all: while I’m learning about these “life dangers,” all around me from my block to the whole world, humanity does its “techno-commerce” via container ships and robots, hundreds of millions of vehicles and smartphones, multilateral exchange rates, and tariff policies. Real understanding has one eye on the personal and the other on the impersonal and not one or the other.

All of these personal and impersonal layers of the full truth must be faced and followed, “en face,” as they say in French (i.e., “without blinking”).

Call all this pre-understanding which includes of course a sense of how my “field” or major or concentration fits into the “architecture of knowledge” and not in isolation without connections or a “ramification structure.”

Post-understanding comes from the other end: my lifelong effort, after just about all that I learned about the six wives of King Henry VIII and the “mean value theorem”/Rolle’s theorem in freshman math, have been completely forgotten and have utterly evaporated in my mind, to re-understand my life and times and book-learning.

Pre-and post-understanding together allows the Wittgenstein phenomenon of “light falls gradually over the whole.”

Without these deeper dimensions of educational remedy, the student as a person would mostly stumble from “pillar to post” with “perfect myopia.” Education mostly adds to all the “fragmentariness” of the modern world and is in that sense, incomplete or even disorienting.

Education in this deep sense is supposed to be the antidote to this overall sense of modern “shapelessness,” to use Kierkegaard’s term.

Meaningfulness versus Informativeness

The Decoding Reality book is a classic contemporary analysis of the foundations of physics and the implications for the human world. The scientists don’t see that physics and science are the infrastructure on which the human “quest for meaning” takes place. Ortega (Ortega y Gasset, died in 1955) tells us that a person is “a point of view directed at the universe.” This level of meaning cannot be reduced to bits or qubits or electrons since man is a “linguistic creature” who invents fictional stories to explain “things” that are not things.

The following dialog between Paul Davies (the outstanding science writer) and Vlatko Vedral (the distinguished physicist) gropes along on these issues: the difference between science as one kind of story and the human interpretation of life and self expressed in “tales” and parables, fictions and beliefs:

Davies: “When humans communicate, a certain quantity of information passes between them. But that information differs from the bits (or qubits) physicists normally consider, inasmuch as it possesses meaning. We may be able to quantify the information exchanged, but meaning is a qualitative property—a value—and therefore hard, maybe impossible, to capture mathematically. Nevertheless the concept of meaning obviously has, well… meaning. Will we ever have a credible physical theory of ‘meaningful information,’ or is ‘meaning’ simply outside the scope of physical science?”

Vedral: “This is a really difficult one. The success of Shannon’s formulation of ‘information’ lies precisely in the fact that he stripped it of all “meaning” and reduced it only to the notion of probability. Once we are able to estimate the probability for something to occur, we can immediately talk about its information content. But this sole dependence on probability could also be thought of as the main limitation of Shannon’s information theory (as you imply in your question). One could, for instance, argue that the DNA has the same information content inside as well as outside of a biological cell. However, it is really only when it has access to the cell’s machinery that it starts to serve its main biological purpose (i.e., it starts to make sense). Expressing this in your own words, the DNA has a meaning only within the context of a biological cell. The meaning of meaning is therefore obviously important. Though there has been some work on the theory of meaning, I have not really seen anything convincing yet. Intuitively we need some kind of a ‘relative information’ concept, information that is not only dependent on the probability, but also on its context, but I am afraid that we still do not have this.”

For a physicist, all the world is information. The universe and its workings are the ebb and flow of information. We are all transient patterns of information, passing on the recipe for our basic forms to future generations using a four-letter digital code called DNA.

See Decoding Reality.

In this engaging and mind-stretching account, Vlatko Vedral considers some of the deepest questions about the universe and considers the implications of interpreting it in terms of information. He explains the nature of information, the idea of entropy, and the roots of this thinking in thermodynamics. He describes the bizarre effects of quantum behavior—effects such as “entanglement,” which Einstein called “spooky action at a distance” and explores cutting edge work on the harnessing quantum effects in hyper-fast quantum computers, and how recent evidence suggests that the weirdness of the quantum world, once thought limited to the tiniest scales, may reach into the macro world.

Vedral finishes by considering the answer to the ultimate question: Where did all of the information in the universe come from? The answers he considers are exhilarating, drawing upon the work of distinguished physicist John Wheeler. The ideas challenge our concept of the nature of particles, of time, of determinism, and of reality itself.

Science is an “ontic” quest. Human life is an “ontological” quest. They are a “twisted pair” where each strand must be seen clearly and not confused. The content of your telephone conversation with your friend, say. is not reducible to the workings of a phone or the subtle electrical engineering and physics involved. A musical symphony is not just “an acoustical blast.”

The “meaning of meaning” is evocative and not logically expressible. There’s a “spooky action at a distance” between these levels of meaning versus information but they are different “realms” or “domains.”

Education and the “Knowability” Problem

There was a wonderful PBS Nature episode in 2006 called “The Queen of Trees” [full video, YouTube] which went into details about the survival strategy and rhythms and interactions with the environment of one tree in Africa and all the complexities this involves:

This Nature episode explores the evolution of a fig tree in Africa and its only pollinator, the fig wasp. This film takes us through a journey of intertwining relationships. It shows how the fig (queen) tree is life sustaining for an entire range of species, from plants, to insects, to other animals and even mammals. These other species are in turn life-sustaining to the fig tree itself. It could not survive without the interaction of all these different creatures and the various functions they perform. This is one of the single greatest documented (on video) examples of the wonders of our natural world; the intricacies involved for survival and ensuring the perpetual existence of species.

It shows us how fragile the balance is between survival and extinction.

One can begin to see that the tree/animal/bacteria/season/roots/climate interaction is highly complex and not quite fully understood to this day.

The fact that one tree yields new information every time we probe into it gives you a “meta” (i.e., meta-intelligent) clue that final theories of the cosmos and fully unified theories of physics will be elusive at best and unreachable at worst. If one can hardly pin down the workings of a single tree, does it sound plausible that “everything that is” from the electron to galaxy clusters to multiverses will be captured by an equation? The objective answer has to be: not particularly.

Think of the quest of the great unifiers like the great philosopherphysicist Hermann Weyl (died in 1955, like Einstein):

Since the 19th century, some physicists, notably Albert Einstein, have attempted to develop a single theoretical framework that can account for all the fundamental forces of nature–a unified field theory. Classical unified field theories are attempts to create a unified field theory based on classical physics. In particular, unification of gravitation and electromagnetism was actively pursued by several physicists and mathematicians in the years between the two World Wars. This work spurred the purely mathematical development of differential geometry.

Hermann Klaus Hugo Weyl (9 November, 1885 – 8 December, 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland and then Princeton, New Jersey, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.

His research has had major significance for theoretical physics as well as purely mathematical disciplines including number theory. He was one of the most influential mathematicians of the twentieth century, and an important member of the Institute for Advanced Study during its early years.

Weyl published technical and some general works on space, time, matter, philosophy, logic, symmetry and the history of mathematics. He was one of the first to conceive of combining general relativity with the laws of electromagnetism. While no mathematician of his generation aspired to the “universalism” of Henri Poincaré or Hilbert, Weyl came as close as anyone.

Weyl is quoted as saying:

“I am bold enough to believe that the whole of physical phenomena may be derived from one single universal world-law of the greatest mathematical simplicity.”

(The Trouble with Physics, Lee Smolin, Houghton Mifflin Co., 2006, page 46)

This reminds one of Stephen Hawking’s credo that he repeated often and without wavering, that the rational human mind would soon understand “the mind of God.”

This WeylHawkingEinstein program of “knowing the mind of God” via a world-equation seems both extremely charming and beautiful, as a human quest, but potentially mono-maniacal à la Captain Ahab in Moby-Dick. The reason that only Ishmael survives the sinking of the ship, the Pequod, is that he has become non-monomaniacal and accepts the variegatedness of the world and thus achieves a more moderate view of human existence and its limits. “The Whiteness of the Whale” chapter in the novel gives you Melville’s sense (from 1851) of the unknowability of some final world-reality or world-theory or world-equation.

Education and the Pursuit of Improved Overviews

Professor Sherman Stein was a prominent mathematician and popularizer, and his book, Mathematics: The Man-Made Universe, is a modern classic. The subtitle “The Man-made Universe” already tells you that you’re looking at a clear exposition of “humans made math” in contrast to the “mathematics fundamentalism,” à la Professor Max Tegmark of MIT, whose tone seems to say mathematics allowed for reality and us.

This is of course a perfect “argument without end.” This is the kind of argument that should help a student to rethink their assumptions and not obsess about some once-and-for-all final understanding which can become an “idée fixe” (i.e., fixed idea in French, indicating being overly rigid or stuck).

In the preface to Professor Stein’s mathematics survey classic, he writes:

“We all find ourselves in a world we never made. Though we become used to the kitchen sink, we do not understand the atoms that compose it. The kitchen sink, like all the objects surrounding us, is a convenient abstraction.

Mathematics, on the other hand is completely the work of man.

Each theorem, each proof, is the product of the human mind. In mathematics all the cards can be put on the table.

In this sense, mathematics is concrete whereas the world is abstract.”

(Sherman Stein, Mathematics The Man-Made Universe, Dover Publications, “Preface” Third Edition, page XIII, 1999)

Meta-intelligence tells you if views of what is real, what is concrete, what is abstract, what is man-made, what is mathematical, are so radically different depending on the interpreter or analyst, it makes prudent sense to keep various views in one’s mind and modify them or juggle them as you go along. Our ability as a species to nail down for eternity what the nature of mathematics, humans and kitchen sinks are and how they all interrelate, is elusive and tangled up in language, as Wittgenstein keeps saying.