The limits of knowledge

An illustration of M4, the sponge after four iterations of the construction process

We will never know everything but you already know that, right? Georg Cantor proved this to be so with infinities within infinites using his Canter Sets to make it happen. The Menger Sponge, seen above, is a 3D representation of the set seen below. 

Zoom in Cantor set. Each point in the set is represented here by a vertical line.

The Set ... Using the idea of self-similar transformations

Cantor's set pointing to infinities residing in infinities within finite space shows, indirectly, the inherent limits of knowledge.

Why this exercise in contemplating this fascinating issue? Well, the limits of knowledge remains forever universal simply because we cannot ascertain the complete totality of a given event let alone properly describing the initial condition of said event in question as stated by chaos theory. 

An animation of a double-rod pendulum at an intermediate energy showing chaotic behavior. Starting the pendulum from a slightly different initial condition would result in a vastly different trajectory. The double-rod pendulum is one of the simplest dynamical systems with chaotic solutions.

For deterministic systems, even if one could not know exactly what would happen in practice (because of inexact or otherwise incomplete knowledge of earlier states), well-formed probabilistic estimates of the most likely outcomes might still be possible. Thus, even if we cannot know whether a particular flipped coin will end up heads or tails, we know that its end state will surely be either heads or tails and even what the likely distribution of heads and tails will be over any repeated set of tosses.

Implicit in that probabilistic analysis, however, is an assumption that all the possible outcomes can be stated in advance. Without foreknowledge of this sample space, one lacks the basis for compiling the appropriate statistics. Systems for which the sample space cannot be prestated seem to preclude the formation of meaningful probability statements.

As described in the main text, we believe that economic systems fall into this category. One cannot anticipate all possible future innovations by simply listing all possible combinations of prior ones: it is impossible to know what features of those earlier goods might be useful in the future because new goods and services can invent completely novel uses of old ones (and sometimes eliminate old ones).

Full knowledge of the movements of a  two body system works but not for three.

In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's law of universal gravitation.[1] The three-body problem is a special case of the n-body problem. Unlike two-body problems, no general closed-form solution exists,[1] as the resulting dynamical system is chaotic for most initial conditions, and numerical methods are generally required.

End result, the limits of knowledge remains forever absolute so ... as per the Tao, assume nothing. :)

The limits of knowledge/rev X

Tommy Parker for Quanta Magazine

Recursion, feedback loops and the journey to forever resonates when viewing this terrific illustration showing just how wondrous reality truly is. At the same time, while looking at the image, the limits of knowledge comes to the fore as one can never know everything about any given event thanks to Heisenberg's uncertainty principle, Godel's incompleteness theorems and Quantum Mechanics because there is no certitude, ever. 

This sentence is false.

Proof of the limits of knowledge writ large applies because if the sentence is true, how can it be false at the same time?

Uncertainty also applies in classic space as one cannot ascertain the momentum of a cue ball and its position at the same time. You can get very very close but not perfect.

As for certitude, go to a parking lot that's full. You know a space will open but not when.

David Hilbert (left) wanted to put mathematics on firmer foundations. Kurt Gödel (center) and Alan Turing later showed that Hilbert’s dream was impossible.

University of Göttingen (left); Kurt Gödel Papers, the Shelby White and Leon Levy Archives Center, Institute for Advanced Study; GL Archive/Alamy Stock Photo

Researchers worried about the limits of formal mathematical reasoning long before the P versus NP problem was first articulated — indeed, long before the beginning of modern computer science. In 1921, struggling with the same question that would grab Carmosino’s attention nearly a century later, the mathematician David Hilbert proposed a research program for grounding mathematics in absolute certainty. He hoped to start from a few simple assumptions, called axioms, and derive a unified mathematical theory that met three key criteria.

Hilbert’s first condition, consistency, was the essential requirement that mathematics be free of contradictions: If two contradictory statements could be proved starting from the same axioms, the whole theory would be unsalvageable. But a theory could be free of contradiction and still limited in its reach. That was the motivation for Hilbert’s second condition, completeness: the requirement that all mathematical statements be either provably true or provably false. His third criterion, decidability, demanded an unambiguous mechanical procedure for determining whether any mathematical statement was true or false. Addressing a conference in 1930, Hilbert declared: “Our slogan shall be ‘We must know, we will know.’”

Just a year later, Gödel delivered the first blow to Hilbert’s dream. He proved that a self-defeating statement like “this statement is unprovable” could be derived from any appropriate set of axioms. If such a statement is indeed unprovable, the theory is incomplete, but if it’s provable, the theory is inconsistent — an even worse outcome. In the same paper, Gödel also proved that no mathematical theory could ever prove its own consistency.

The limits to knowledge rules ... yet again.

Then in 1936, a 23-year-old graduate student named Alan Turing rephrased Hilbert’s decidability condition in the then-unfamiliar language of computation and dealt it a fatal blow. Turing formulated a mathematical model — now known as the Turing machine — that could represent all possible algorithms, and showed that if Hilbert’s procedure existed, it would fit within this model. He then used self-referential methods like Gödel’s to prove the existence of undecidable statements — or, equivalently, uncomputable problems that no algorithm could solve.

How to negotiate reality ...

Diagram of the OODA loop

Situational awareness applies as does Destruction and Creation.

Observe, Orient, Decide, Act,

Observe, Orient, Decide, Act, a set of actions defined by John Boyd's OODA loop. shows, in depth, the steps every organism takes in order to survive also applies to better understanding how the limits of knowledge, driven by uncertainty, incompleteness and entropy, depict a reality we will never fully understand.  To see why, the animation depicting one example of chaos beckons. :) 

An animation of a double-rod pendulum at an intermediate energy showing chaotic behavior.
Starting the pendulum from a slightly different initial condition would result in a vastly different trajectory. The double-rod pendulum is one of the simplest dynamical systems with chaotic solutions.

Interesting enough, OODA, when examined closely, relates to the act of creation without question.

Cause I'm a million miles away

And at the same time, I'm right here in your picture frame. Jimi Hendrix | Voodoo Chile

1. The Act of Creation - Arthur Koestler: The creative act is the joining together of two or more seemingly dissimilar ideas that generate a heretofore unexpected result. i.e. Chemistry - H20 consists of 2 Hydrogen + 1 Oxygen atoms, both explosive yet, when combined, becomes water. 
2. The joke: Henny Youngman - Take my wife, Please. :)
3. Creativity starts with curiosity, the start point to the asking of a question:
   Why is this the way it is?
4. Question everything - Einstein
5. Creation & destruction is part of the creative process as everything has a cost thanks to the two laws of thermodynamics.
6. The ability to see relationships is key as everything is connected because reality is a Quantum space and entanglement stitches it together.
7. The creative act is always visual. Think Stevie Wonder and Songs in the Keys of life.
8. Play is indispensable. What if ... is the real deal here. A sense of wonderment is key.
9. Creativity takes courage as new ideas disrupt the "natural" order of things.
10. Creativity is problem-solving. It matters not the discipline used save that said discipline shapes how the creative act is produced.
11. The act itself is ineffable and cannot be explained. Like quantum, creativity just happens. It cannot be forced, only acted upon when the idea shows up. The finished act can be explained after the fact without question.
12. Practice makes better. Beginners mind and intuition rule.
13. Attention to details.
14. Passion is a prerequisite to create anything of value.
15. The creative act is selfish. One does not create for others, only for oneself.
16. The size of the problem solved dictates the significance of the creative act. i.e. Mixing anchovies with pineapples on a pizza is a creative act. The size or importance of the problem solved, not so much while Einstein's Theory of relativity proving that Newton was wrong at universal scale, is.
17. Creativity, like tech and existence, has no morality. It simply is.
18. Stay hungry, stay foolish - Stewart Brand/Whole Earth Catalog. :)
19. AI is but a tool, the issue now becomes, does the tool become the master? A good question to ask is it not?
20. Being first to come up with a new idea is essential. Darwin & Wallace relates as Wallace was two weeks later than Darwin in explaining why evolution matters.
21. The Act of Creation, whenever possible, runs counter to the limits of knowledge as needs warrant.

Circularity

 Characteristics of great art

1. Clarity.
2. Simplicity. It's easy to be complex, simplicity is hard. Einstein: Keep it simple, not simpler.
3. Good artists create, great artists steal. - Picasso
4. Great art always involves the process of discovery.
5. Timeless. Bach, like fine wine, never ages.
6. No spin. 
7. Details, details, details.
8. As stated before, great art applies to all disciplines.
9. Connections to differing disciplines apply.
10. Less is more. Dizzy Gillespie
11. Great artists are never afraid to make mistakes.
12. Doggedness. It takes thousands of hours to become good at something. 

TANSTAAFL: a plan for a new economic world order
by Pierre Dos Utt (1949)

Connections example ...

Kanji characters for shodō (書道)

Kind of Blue is considered to be the greatest jazz album of all time even though
Oliver Nelson's The Blues and the Abstract Truth is right up there on the pantheon of the greatest of all time IMHO. This is not about KOB per se but rather about Bill Evans' liner notes and why his writing is just as profound as the album itself because it's all about improv and it's role in the creative process.

To whit.

There is a Japanese visual art in which the artist is forced to be spontaneous. He must paint on a thin stretched parchment with a special brush and black water paint in such a way that an unnatural or interrupted stroke will destroy the line or break through the parchment. Erasures or changes are impossible. These artists must practice a particular discipline, that of allowing the idea to express itself in communication with their hands in such a direct way that deliberation cannot interfere.

The resulting pictures lack the complex composition and textures of ordinary painting, but it is said that those who see well find something captured that escapes explanation.

This conviction that direct deed is the most meaningful reflections, I believe, has prompted the evolution of the extremely severe and unique disciplines of the jazz or improvising musician.

Details example ... 

There was a problem with the orbit of Mercury. Seems it was off by 1/100th of a degree thanks to Newtonian mechanics, an error Einstein knew had to be corrected if we are to better understand how the universe works at deep level. To that end, General Relativity and the notion of space time came to be or in John Wheeler's words, Spacetime tells matter how to move; matter tells spacetime how to curve.

A notion of Einstein's Special Relativity ... Vue style :)

Remember, walking is controlled falling. 

There is no off ramp :)

Mirror Beach

S = entropy

k_{b} = Boltzmann constant

\ln = natural logarithm

\Omega = number of microscopic configurations

Being rather limited in math to say the least, this "simple" equation explains in a way even I can understand as entropy is the driving engine of the 2nd law of thermodynamics. When paired with the first, the two define reality as being an environment sustained by the ever moving essence of energy.

Entropy is a measure of disorder. It corresponds with how many possible microscopic configurations can underlie an overall state.

Heat flowing from hot water to cold water

The First Law: Energy can neither be created or destroyed, only transformed

The Second Law: Isolated systems spontaneously evolve towards thermal equilibrium 

—the state of maximum entropy/disorder.

The rabbit hole beckons. :)

Exactly 200 years ago, a French engineer introduced an idea that would quantify the universe’s inexorable slide into decay. But entropy, as it’s currently understood, is less a fact about the world than a reflection of our growing ignorance. Embracing that truth is leading to a rethink of everything from rational decision-making to the limits of machines.

Sadi Carnot,  aged 17 in the uniform of a cadet of the École polytechnique.
Portrait by Louis-Léopold Boilly (1813).

Factoid: Reading through Carnot’s book a few decades later, in 1865, the German physicist Rudolf Clausius coined a term for the proportion of energy that’s locked up in futility. He called it “entropy,” after the Greek word for transformation. He then laid out what became known as the second law of thermodynamics: “The entropy of the universe tends to a maximum.”

But it's more than that ...

Rudolf Clausius (left) originated the insight that entropy tends to increase. Ludwig Boltzmann rooted this rise in statistical mechanics. Theo Schafgans (left); Creative Commons

Physicists of the era erroneously believed that heat was a fluid (called “caloric”). Over the following decades, they realized heat was rather a byproduct of individual molecules bumping around. This shift in perspective allowed the Austrian physicist Ludwig Boltzmann to reframe and sharpen the idea of entropy using probabilities.

Heisenberg's Uncertainty Principle & Entropy, a match make in heaven

Claude Shannon

Entropy experienced a rebirth during World War II. Claude Shannon, an American mathematician, was working to encrypt communication channels, including the one that connected Franklin D. Roosevelt and Winston Churchill. That experience led him to think deeply about the fundamentals of communication over the following years. Shannon sought to measure the amount of information contained in a message. He did so in a roundabout way, by treating knowledge as a reduction in uncertainty.

In essence, entropy's an information construct as per John Wheeler's From It to Bit

From John Wheeler (1983), Law Without Law. In: Quantum theory and measurement.

"It from bit symbolizes the idea that every item of the physical world has at bottom — at a very deep bottom, in most instances — an immaterial source and explanation; that what we call reality arises in the last analysis from the posing of yes-no questions and the registering of equipment-evoked responses; in short, that all things physical are information-theoretic in origin and this is a participatory universe. 

The limits of knowledge rules

An illustration of M4, the sponge after four iterations of the construction process

We will never know everything but you already know that, right? Georg Cantor proved this to be so with infinities residing within infinities using his theory of transfinite numbers to make it happen. The Menger Sponge, seen above, is a 3D representation of Cantor's theory.

In closing ...

Low

High

In closing ...

“Nothing in life is certain except death, taxes and the second law of thermodynamics,” wrote(opens a new tab) Seth Lloyd, a physicist at the Massachusetts Institute of Technology. There’s no sidestepping this directive. The growth of entropy is deeply entwined with our most basic experiences, accounting for why time runs forward and why the world appears deterministic rather than quantum mechanically uncertain.

But despite its fundamental importance, entropy is perhaps the most divisive concept in physics. “Entropy has always been a problem,” Lloyd told me. The confusion stems in part from the way the term gets tossed and twisted between disciplines — it has similar but distinct meanings in everything from physics to information theory to ecology. But it’s also because truly wrapping one’s head around entropy requires taking some deeply uncomfortable philosophical leaps.

Everything's connected and entangled, this is why we rubes have a consistent view of a "uncertain" reality we will never fully understand. Awesome to a fault without question if you're one willing to question. :)

And so it goes ... K. Vonnegut

Plato's cave

 

Plato's allegory of the cave by Jan Saenredam, according to Cornelis van Haarlem, 1604, Albertina, Vienna

BRT has written copiously about The Limits of Knowledge as quantum reality works in ways beyond the kin of man. From Plato to Einstein, the quest to learn how existence works never ends as nature's the ultimate trickster, forever keeping the true essence of herself forever hidden from our prying eyes.

More than a half century ago, Richard Feynman advised us to accept that nature makes no sense. “Do not keep saying to yourself … ‘But how can [nature] be like that?’” Feynman warns in The Character of Physical Law, “because you will get ‘down the drain,’ into a blind alley from which nobody has yet escaped. Nobody knows how it can be like that.” Most physicists have followed Feynman’s advice. Ignoring the oddness of quantum mechanics, they simply apply it to accomplish various tasks, such as predicting new particles or building more powerful computers.

Today, there are more interpretations than ever, but they deepen rather than dispel the mystery at the heart of things. The more I dwell on puzzles such as superposition, entanglement and the measurement problem, the more I identify with the piranha. I’m blindly thrashing about for insights, epiphanies, revelations. Every now and then I think I’ve grasped some slippery truth, but my satisfaction is always fleeting. Sooner or later, I end up crashing into an invisible barrier. I don’t really know where I am or what’s going on. I’m in the dark.

Plato's Cave, ...

In the allegory, Socrates describes a group of people who have lived chained to the wall of a cave all their lives, facing a blank wall. The people watch shadows projected on the wall from objects passing in front of a fire behind them and give names to these shadows. The shadows are the prisoners' reality, but are not accurate representations of the real world. Three higher levels exist: the natural sciences; mathematics, geometry, and deductive logic; and the theory of forms.

Socrates explains how the philosopher is like a prisoner who is freed from the cave and comes to understand that the shadows on the wall are actually not reality at all. A philosopher aims to understand and perceive the higher levels of reality. However, the other inmates of the cave do not even desire to leave their prison, for they know no better life.[1]

Allegory of the cave. From top to bottom: The sun ("good" ideas) Natural things (ideas) Reflections of natural things (mathematical objects) Fire (light) Artificial objects (creatures and objects) Shadows of artificial objects, allegory (image, analogy of the sun and of the divided line)

The limits of knowledge indeed ...

The Limits to Growth ...

Illustration by Maria Contreras

Prior to the rise of oil, the standard of living for the great majority of people on planet earth was marginal at best. As per Thomas Hobbes ... In such condition there is no place for industry, because the fruit thereof is uncertain: and consequently no culture of the earth; no navigation, nor use of the commodities that may be imported by sea; no commodious building; no instruments of moving and removing such things as require much force; no knowledge of the face of the earth; no account of time; no arts; no letters; no society; and which is worst of all, continual fear, and danger of violent death; and the life of man, solitary, poor, nasty, brutish, and short’ 

Frontispiece of Leviathan by Abraham Bosse, with input from Hobbes

With advent of oil comes abundance ...

John Maynard Keynes once observed that dating from “say, to two thousand years before Christ—down to the beginning of the 18th century, there was no very great change in the standard of life of the average man living in the civilised centres of the earth. Ups and downs certainly. Visitations of plague, famine, and war. Golden intervals. But no progressive, violent change.” At best, he calculated, the average standard of living had no more than doubled in the previous four millennia, essentially because, when that epoch began, we already knew about fire, banking, the sail, the plow, mathematics; we learned little new that would have accelerated economic growth; and throughout that stretch the planet mostly ran on the muscles of people and animals, supplemented by the power of wind and water. Then, in the eighteenth and nineteenth century, we started to harness the combustion of coal, gas, and oil, and everything changed. That’s because a barrel of oil contains 5.8 million British thermal units’ worth of energy. Nate Hagens, the director of the Institute for the Study of Energy & Our Future, ran the numbers: “One barrel of oil has the same amount of energy of up to 25,000 hours of hard human labor, which is 12.5 years of work. At $20 per hour, this is $500,000 of labor per barrel.” A barrel of oil costs about seventy dollars at this week’s market price.

To call that energy revolution liberating hardly suffices. Suddenly, people could easily venture beyond their villages, or build dwellings large enough to afford some privacy, or stay up all night if they wanted to read. After four thousand years of economic stasis, we were suddenly in a world where the average standard of living doubled in a matter of decades, and then doubled again and again and again and again. And we liked it so much that it became the raison d’être of our political life. In the United States, the per-capita G.N.P. grew twenty-four per cent between 1947 and 1960, when Jack Kennedy, campaigning for President, pointed out that Russia’s growth rate was “three times as fast,” a gap he tried to narrow while in office. Between 1961 and 1965, the G.N.P. grew at a rate greater than five per cent a year, and the percentage of Americans living in poverty dropped by nearly half by the end of the decade. If there was anything Americans agreed on, it was that they wanted more, please. In the 1996 campaign, for instance, the Republican Vice-Presidential nominee, Jack Kemp, demanded that we double the rate of growth, while Bill Clinton’s Secretary of the Treasury, Larry Summers, said that we “cannot and will not accept any ‘speed limit’ on American economic growth. It is the task of economic policy to grow the economy.”

and cost ...

But a critique of growth was emerging in the postwar years as well, most concisely in a 1972 report commissioned by the Club of Rome titled “The Limits to Growth.” A team of M.I.T. economists used computer models (then something of a novelty) to show that, if we kept growing at the then-current rate, the planet could expect ecological collapse sometime toward the middle of the twenty-first century. That prediction turns out to have been spot-on: a report published in Nature on the last day of May concluded that we have already exceeded seven of eight “safe and just Earth system boundaries” that it studied—from groundwater supplies and fertilizer overuse to temperature. “We are moving in the wrong direction on basically all of these,” Johan Rockström, the paper’s lead author and the director of the Potsdam Institute for Climate Impact Research, told reporters. 

Click here to get Limits to Growth in PDF.

Somewhat right ... Dall'e 2 experiments ... :)

The double slit experiment/rev 1

Plato's cave/rev XX

The single slit ...

When measurement occurs

Superposition/rev X

Strange Attractors/Chaos/rev XXX

The limits of knowledge

Channeling Francis Bacon

Channeling Francis Bacon/Rev II

Dall'e-2 has growing pains, particularly when trying to accurately describe quantum physics in the guise of the 1805 Thomas Young Double Slit Experiment. Plato's cave is my favorite as an incorrect but fanciful expression of the most beautiful experiment in science as it resonates, in indirect fashion, to Plato's attempt to describe the true essence of existence. The fourth picture shows the act of "measuring" a given subatomic particle as said entity exists both as a wave and a particle at the same time until measured, an act that not only collapses the entity in question into a particle but also shows where the most likely location of the particle might be. :) 

The 5th shows superposition when a sub-atomic entity is not measured. 

The 6th reflects chaos theory and its notion of a strange attractor. 

The 7th depicts the limits of knowledge as man will never know the full extent of reality.

The last two pix were intended as "A watercolor painting in the style of Andrew Wyeth of an annoyed snow leopard in a spaceship" but Dall'e 2 decided a Francis Bacon take would be better. :)

As reference, here's an accurate pix of Young's wonderful experiment.

Joonasl/Wikimedia Commons/CC BY 3.0

The single slit image by yours truly's spot on IMHO. :)

Limits of knowledge rev XX

Blue indigo

Night for Day

Open window

Stone wall

Solitude

 Emergence

All these pix show aspects of the great invisible whether it be Night for Day or the infrared driven images of Solitude and Blue Indigo as our eyes cannot see reality in a light spectrum outside of the tiny subset of what we see without the aid of tech. 

The visible light spectrum is the segment of the electromagnetic spectrum that the human eye can view. More simply, this range of wavelengths is called visible light. Typically, the human eye can detect wavelengths from 380 to 700 nanometers.

As for the stone wall, this image indirectly shows the passage of time with the oldest at the bottom and the latest at the top. As for Open Window & Emergence, darkness invokes mystery even in the bright of day.

To yours truly, the limits of knowledge resides everywhere, even in the commonplace as depicted in the pictures seen above. :)