mathematics, Meta Intelligence, Page 2

“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:

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 (17^th 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 17^th century classic Spanish play, Life is a Dream (Spanish: La 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 philosopher–physicist 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 Weyl–Hawking–Einstein 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.

Education and “The Three-Body Problem”

The brilliant math-watcher, Ian Stewart, says of this classic physics problem, the Three-Body Problem:

Newton’s Law of Gravity runs into problems with three bodies (earth, moon, sun, say).
In particular, the gravitational interaction of a mere three bodies, assumed to obey Newton’s inverse square law of gravity, stumped the mathematical world for centuries.
It still does, if what you want is a nice formula for the orbits of those bodies. In fact, we now know that three-body dynamics is chaotic–so irregular that is has elements of randomness.
There is no tidy geometric characterization of three-body orbits, not even a formula in coordinate geometry.
Until the late nineteenth century, very little was known about the motion of three celestial bodies, even if one of them were so tiny that its mass could be ignored.
(Visions of Infinity: The Great Mathematical Problems, Ian Stewart, Basic Books, 2014, page 136)

Henri Poincaré, the great mathematician, wrestled with this with tremendous intricacy and ingenuity all his life:

Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as “The Last Universalist,” since he excelled in all fields of the discipline as it existed during his lifetime.

Born: April 29, 1854, Nancy, France
Died: July 17, 1912, Paris, France.

We now think of applying in an evocative and not a rigorous mathematical way, the unexpected difficulties of the three-body problem to the n-body (i.e., more than three) problems of sociology or economics or history itself, and sense that social life is always multifactorial and not readily pin-downable, since “everything is causing everything else” and extracting mono-causal explanations must be elusive for all the planetary and Poincaré reasons and beyond.

This suggests to the student that novels are one attempt to say something about n-body human “orbits” based on “n-body” stances and “circumstances” with large amounts of randomness governing the untidy mess that dominates human affairs.

Words are deployed in novels and not numbers as in physics, but the “recalcitrance” of the world, social and physical, remains permanent.

Education and meta-intelligence would be more complete by seeing how the world, as someone put it, “won’t meet us halfway.” Remember Ian Stewart’s warning above:

“There is no tidy geometric characterization of three-body orbits…” and you sense that this must apply to human affairs even more deeply.

Early View Alert: Water Resources Research

from the American Geophysical Union’s journals:

Research Articles

Modeling the Snow Depth Variability w ith a High-Resolution Lidar Data Set and Nonlinear Terrain Dependency

by T. Skaugen & K. Melvold

Summary: Using airborne laser, 400 million snow depth measurements at Hardangervidda in Southern Norway have been collected. The amount of data has made in-depth studies of the spatial distribution of snow and its interaction with the terrain and vegetation possible. We find that the terrain variability, expressed by the square slope, the average amount of snow, and whether the terrain is vegetated or not, largely explains the variation of snow depth. With this information it is possible to develop equations that predict snow depth variability that can be used in environmental models, which again are used for important tasks such as flood forecasting and hydropower planning. One major advantage is that these equations can be determined from the data that are, in principle, available everywhere, provided there exists a detailed digital model of the terrain.

[Archived PDF article]

Phosphorus Transport in Intensively Managed Watersheds

by Christine L. Dolph, Evelyn Boardman, Mohammad Danesh-Yazdi, Jacques C. Finlay, Amy T. Hansen, Anna C. Baker & Brent Dalzell

Abstract: When phosphorus from farm fertilizer, eroded soil, and septic waste enters our water, it leads to problems like toxic algae blooms, fish kills, and contaminated drinking supplies. In this study, we examine how phosphorus travels through streams and rivers of farmed areas. In the past, soil lost from farm fields was considered the biggest contributor to phosphorus pollution in agricultural areas, but our study shows that phosphorus originating from fertilizer stores in the soil and from crop residue, as well as from soil eroded from sensitive ravines and bluffs, contributes strongly to the total amount of phosphorus pollution in agricultural rivers. We also found that most phosphorus leaves farmed watersheds during the very highest river flows. Increased frequency of large storms due to climate chaos will therefore likely worsen water quality in areas that are heavily loaded with phosphorus from farm fertilizers. Protecting water in agricultural watersheds will require knowledge of the local landscape along with strategies to address (1) drivers of climate chaos, (2) reduction in the highest river flows, and (3) ongoing inputs and legacy stores of phosphorus that are readily transported across land and water.

Will Forests Let Us Down?

Research Articles

Research Letters