Tag: mathematics

Economics-Watching: Second-Quarter GDP Growth Estimate Unchanged

[from the Federal Reserve Bank of Atlanta]

The growth rate of real gross domestic product (GDP) is a key indicator of economic activity, but the official estimate is released with a delay. The Federal Reserve Bank of Atlanta’s GDPNow forecasting model provides a “nowcast” of the official estimate prior to its release by estimating GDP growth using a methodology similar to the one used by the U.S. Bureau of Economic Analysis.

GDPNow is not an official forecast of the Atlanta Fed. Rather, it is best viewed as a running estimate of real GDP growth based on available economic data for the current measured quarter. There are no subjective adjustments made to GDPNow—the estimate is based solely on the mathematical results of the model.

Recent forecasts for the GDPNow model are available here [archived PDF]. More extensive numerical details—including underlying source data, forecasts, and model parameters—are available as a separate spreadsheet [archived XLSX]. You can also view an archive of recent commentaries from GDPNow estimates.

Please note that the Atlanta Fed no longer supports the GDPNow app. Download the EconomyNow app to get the latest GDP nowcast and more economic data.

Latest estimate: 2.4 percent — July 25, 2025

The GDPNow model estimate for real GDP growth (seasonally adjusted annual rate) in the second quarter of 2025 is 2.4 percent on July 25, unchanged from July 18 after rounding. The forecasts of the major GDP subcomponents were all unchanged or little changed from their July 18 values after this week’s releases from the U.S. Census Bureau and the National Association of Realtors.

The growth rate of real gross domestic product (GDP) measured by the U.S. Bureau of Economic Analysis (BEA) is a key metric of the pace of economic activity. It is one of the four variables included in the economic projections of Federal Reserve Board members and Bank presidents for every other Federal Open Market Committee (FOMC) meeting. As with many economic statistics, GDP estimates are released with a lag whose timing can be important for policymakers. In preparation for FOMC meetings, policymakers have the Fed Board staff projection of this “advance” estimate at their disposal. These projections—available through 2008 at the Philadelphia Fed’s Real Time Data Center—have generally been more accurate than forecasts from simple statistical models. As stated by economists Jon Faust and Jonathan H. Wright in a 2009 paper, “by mirroring key elements of the data construction machinery of the Bureau of Economic Analysis, the Fed staff forms a relatively precise estimate of what BEA will announce for the previous quarter’s GDP even before it is announced.”

The Atlanta Fed GDPNow model also mimics the methods used by the BEA to estimate real GDP growth. The GDPNow forecast is constructed by aggregating statistical model forecasts of 13 subcomponents that comprise GDP. Other private forecasters use similar approaches to “nowcast” GDP growth. However, these forecasts are not updated more than once a month or quarter, are not publicly available, or do not have forecasts of the subcomponents of GDP that add “color” to the top-line number. The Atlanta Fed GDPNow model fills these three voids.

The BEA’s advance estimates of the subcomponents of GDP use publicly released data from the U.S. Census Bureau, U.S. Bureau of Labor Statistics, and other sources. Much of this data is displayed in the BEA’s Key Source Data and Assumptions table that accompanies the “advance” GDP estimate. GDPNow relates these source data to their corresponding GDP subcomponents using a “bridge equation” approach similar to the one described in a Minneapolis Fed [archived PDF] study by Preston J. Miller and Daniel M. Chin. Whenever the monthly source data is not available, the missing values are forecasted using econometric techniques similar to those described in papers by James H. Stock and Mark W. Watson and Domenico Giannone, Lucrezia Reichlin, and David Small. A detailed description of the data sources and methods used in the GDPNow model is provided in an accompanying Atlanta Fed working paper [archived PDF].

As more monthly source data becomes available, the GDPNow forecast for a particular quarter evolves and generally becomes more accurate. That said, the forecasting error can still be substantial just prior to the “advance” GDP estimate release. It is important to emphasize that the Atlanta Fed GDPNow forecast is a model projection not subject to judgmental adjustments. It is not an official forecast of the Federal Reserve Bank of Atlanta, its president, the Federal Reserve System, or the FOMC.

Wrestling with History: Alexis de Tocqueville

Alexis de Tocqueville, a brilliant French historian, wrote Democracy in America. This book is a supreme example of U.S.-watching.

Another book of his, Recollections, shows him wrestling with history itself. If we remember that Clio is the muse of history, then we might say that Recollections is the chronicle of de Tocqueville’s encounter with her.

The question of human history and what de Tocqueville called “the world’s destiny” are described as follows:

l wrote histories without taking part in public affairs, and politicians whose only concern was to control events without a thought of describing them. And I have invariably noticed that the former see gen­eral causes everywhere, whereas the latter, spend­ing their lives amid the disconnected events of each day, freely attribute everything to particular incidents and think that all the little strings their hands are busy pulling daily are those that control the world’s destiny. Probably both of them are mistaken.

For my part I hate all those absolute systems that make all the events of history depend on great first causes linked together by the chain of fate and thus succeed, so to speak, in banishing men from the history of the human race. Their boasted breadth seems to me narrow, and their mathematical exactness false. I believe, pace the writers who find these sublime theories to feed their vanity and lighten their labours, that many important historical facts can be explained only by accidental circumstances, while many others are inexplicable. Finally, that chance, or rather the concatenation of secondary causes, which we call by that name because we can’t sort them all out, is a very important element in all that we see taking place in the world’s theatre. But I am firmly convinced that chance can do nothing unless the ground has been prepared in advance. Antecedent facts, the nature of institutions, turns of mind and the state of mores are the materials from which chance composes those impromptu events that surprise and terrify us.

Alexis de Tocqueville, Recollections, 1893, Anchor Books, page 78.

De Tocqueville warns us that the world’s destiny is always murky and what he calls a labyrinth and a whirlwind. He says:

Mentally I reviewed the history of our last sixty years and smiled bitterly to myself as I thought of the illusions cherished at the end of each phase of this long revolution; the theories feeding these illusions; our historians’ learned daydreams, and all the ingenious false systems by which men sought to explain a present still unclearly seen and to foresee the unseen future.

Recollections, page 83.

He continues:

Shall we reach, as other prophets as vain perhaps as their predecessors assure us, a more complete and profound social transformation than our fathers ever foresaw or desired, and which we ourselves cannot yet conceive; or may we not simply end up in that intermittent anarchy which is well known to be the chronic incurable disease of old peoples? I cannot tell, and do not know when this long voyage will end; I am tired of mistaking deceptive mists for the bank. And I often wonder whether that solid land we have sought for so long actually exists, and whether it is not our fate the rove the seas forever!

Recollections, pages 83-84.

And yet, with all that profound uncertainty, he offers a very sweeping interpretation of French history from the French Revolution (1789) to the French Revolution of 1848. The famous painting by Eugène Delacroix, Liberty Leading the People (French: La Liberté guidant le peuple), commemorating the July Revolution of 1830, falls in between.

Despite de Tocqueville’s warnings about the slipperiness of historical judgement, he arrives at an extremely precise interpretation of his own:

Seen as a whole from a distance, our history from 1789 to 1830 appears to be forty-one years of deadly struggle between the Ancien Régime with its traditions, memories, hopes and men (i.e. the aristocrats), and the new France led by the middle class. 1830 would seem to have ended the first period of our revolutions, or rather, of our revolution, for it was always one and the same, through its various fortunes and passions, whose beginning our fathers saw and whose end we shall in all probability not see. All that remained of the Ancien Régime was destroyed forever. In 1830 the triumph of the middle class was decisive and so complete that the narrow limits of the bourgeoisie encompassed all political powers, franchises, prerogatives, indeed the whole government, to the exclusion, in law, of all beneath it and, in fact, of all that had once been above it. Thus the bourgeoisie became not only the sole director of society, but also, one might say, its cultivator. It settled into every office, prodigiously increased the number of offices, and made a habit of living off the public Treasury almost as much as from its own industry.

Recollections, page 5.

Reviewing the first sentence from the quote above, one can see a deep characterization of an era, with the conclusion “in 1830 the triumph of the middle class was decisive…” Notice the profound paradox that on one hand de Tocqueville spoke of the elusiveness of history despite providing the definite description of this period. Contrast “seen as a whole from a distance” with one of the themes of his recollections, that it is not given to us to understand history.

Kierkegaard and Existence

There are various striking intuitions about human existence. For example, in his brilliant memoirs, Speak, Memory, Nabokov begins with the deep reflection where human existence is compared to a baby in a cradle, rocking, completely vulnerable and uncertain. All of this is bracketed by two episodes of infinite darkness. The first episode took place before you were born and the second takes place after you’re gone. Your existence is a temporary flame, like that of a lit match.

A MetaIntelligent comment on this would be that the profound ingenuity of the 19th century mathematicians analyzing the size and nature of infinity (e.g., Richard Dedekind or Georg Cantor) cannot in the last analysis wrestle down human existence into mathematics.

The modern progenitor of this kind of human existence-watching is the Danish genius Søren Kierkegaard. In one of his masterpieces, Concluding Unscientific Postscript to Philosophical Fragments (1846), he makes the claim that knowledge, theory, speculative thinking and infinity-watching à la Dedekind and Cantor, cannot possibly explain human existence, because it subsumes all of these.

In 2025, this would mean that the Kierkegaard sense of things would tell you that neuroscience can never really explain how existence is sensed by a living person.

Kierkegaard writes, “in my view the misfortune of the age was precisely that it had too much knowledge, had forgotten what existence means, and what inwardness signifies.” He continues, “for a knowledge-seeker, when he has finished studying China he can take up Persia; when he has studied French he can begin Italian; and then go on to astronomy, the veterinary sciences, and so forth, and always be sure of a reputation as a tremendous fellow.”

By way of contrast, “inwardness in love does not consist in consummating seven marriages with Danish maidens, then cutting loose on the French, the Italian, and so forth, but consists in loving one and the same woman, and yet being constantly renewed in the same love, making it always new in the luxuriant flowering of the mood.” (Concluding Unscientific Postscript to Philosophical Fragments, page 232.)

Kierkegaard’s kind of existence-watching can be understood as a turning-upside-down of the famous phrase from Descartes, “I think, therefore I am.” For Kierkegaard, “I am, therefore I think.” Notice that “I think” is an epistemological statement or knowledge-watching. “I am” is an ontological statement.

This existentialist tradition of putting ontology before epistemology finds its culmination in Heidegger. As he says in his opus, Being and Time (1927), “human being is ultimately the being for whom being itself is an issue.”

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 song–philosophers 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 cycles—dynamical 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 (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.