Godel, Cantor, Wiener and Schrodinger's Cat

Capitalist Pig Vs. Socialist Swine does a very good job of introducing his/her readers to Godel's Theorm. Where science and philosophy collide: Part I

Godel was the theoritical mathematician who at the ripe young age of 21 proved the mathmatical equivalent of the philosophical conundrum: Nothing is True everything is Permitted.

Gödel's Incompleteness Theorem
In 1931, the Czech-born mathematician Kurt Gödel demonstrated that within any given branch of mathematics, there would always be some propositions that couldn't be proven either true or false using the rules and axioms ... of that mathematical branch itself. You might be able to prove every conceivable statement about numbers within a system by going outside the system in order to come up with new rules and axioms, but by doing so you'll only create a larger system with its own unprovable statements. The implication is that all logical system of any complexity are, by definition, incomplete; each of them contains, at any given time, more true statements than it can possibly prove according to its own defining set of rules.

So I posted in this in his comments section:

Godels Theorm my gawd man what are you on.

And you explained it so well too, which means a). you are a math major b) a theoritical math major c) a physics major d) a theoritical physics major
e) Robert Anton Wilson

Didn't expcet that outta ya swine...Godel is as obscure as they get unless one is reading about Quantum physics and Dr. Schrodinger and his cat.....
While you are at it for math theorms that disprove the proof of math/physics
(which I maybe mistaken in my usage of the term) see The Mystery of the Aleph- Mathematics, the Kabbalh and the search for infinity. by Amir D. Aczel, Pocketbooks 2000.

Its about mathmetician George Cantor, and like Godel he went mad too. Mad I say Mad, bwahhhahaha

Cantors Theorm1

Cantor's theorem2

Georg Cantor's achievement in mathematics was outstanding. He revolutionized the foundation of mathematics with set theory. Set theory is now considered so fundamental that it seems to border on the obvious but at its introduction it was controversial and revolutionary. The controversial element centered around the problem of whether infinity was a potentiality or could be achieved. Before Cantor it was generally felt that infinity as an actuality did not make sense; one could only speak of a variable increasing without bound as that variable going to infinity. That is to say, it was felt that n -> ∞ makes sense but n = ∞ does not. Cantor not only found a way to make sense out an actual, as opposed to a potential, infinity but showed that there are different orders of infinity.

Biographies of Mathematicians-Georg Ferdinand Ludwig Philipp Cantor

GEORG CANTOR (1845-1918) AND THE "DEGREES OF THE INFINITE"

Opps Godel didn't go mad he went to Princeton.

Kurt Godel (1906-1978), elected to Academy membership in 1955, was noted for his contributions to the foundations of logic and mathematics. In a celebrated paper published in 1931, Godel first put forward what came to be known simply as "Godel's Theorem": In certain formal systems, there exist propositions that cannot be proved or disproved using the axioms of that system. With this theorem, Godel had effectively demonstrated that some mathematical propositions are undecidable. Godel's Theorem made a deep impact in the fields of mathematics and logic, and has been called the most significant mathematical truth of the 20th century. Godel was born in Brunn (now Brno), in what is now the Czech Republic. He studied physics in Vienna, and emigrated to the US in 1939, where he took a position at Princeton's Institute for Advanced Study. In addition to other honors, in 1975 he was awarded the National Medal of Science, the US government's highest scientific honor.

GODEL'S THEOREMS AND TRUTH

Godel's theorem
To better understand the impact which Godel's findings must have had on his peers, we should first describe the mathematical climate of the time.
In the nineteenth century it had been discovered, through the work of Riemann, Lobachevsky and others, that coherent models of geometry could be constructed in which Euclid's parallel postulate (that, given a line L and a point P in the plane, exactly one line exists which contains P and is parallel to L) did not hold. This, in itself, was a shock to many mathematicians: for millenia it had been assumed that Euclid's description of geometry, founded as it was on a "self-evident" and minimal set of axioms, was one of the firmest, most trustworthy branches of mathematical knowledge. The existence of non-Euclidean geometries not only challenged mathematicians' geometrical intuition, but also the Platonist view that mathematics consisted of discoveries about eternal, pure forms whose existence was objective and unquestionable. More "monstrosities" such as continuous functions which were nowhere differentiable soon appeared, further fueling the general loss of faith in geometry.

The modern development of the foundations of mathematics

in the light of philosophy

Kurt Gödel (1961)

Now it is a familiar fact, even a platitude, that the development of philosophy since the Renaissance has by and large gone from right to left - not in a straight line, but with reverses, yet still, on the whole. Particularly in physics, this development has reached a peak in our own time, in that, to a large extent, the possibility of knowledge of the objectivisable states of affairs is denied, and it is asserted that we must be content to predict results of observations. This is really the end of all theoretical science in the usual sense (although this predicting can be completely sufficient for practical purposes such as making television sets or atom bombs).

It would truly be a miracle if this (I would like to say rabid) development had not also begun to make itself felt in the conception of mathematics. Actually, mathematics, by its nature as an a priori science, always has, in and of itself, an inclination toward the right, and, for this reason, has long withstood the spirit of the time [Zeitgeist] that has ruled since the Renaissance; i.e., the empiricist theory of mathematics, such as the one set forth by Mill, did not find much support. Indeed, mathematics has evolved into ever higher abstractions, away from matter and to ever greater clarity in its foundations (e.g., by giving an exact foundation of the infinitesimal calculus and the complex numbers) - thus, away from scepticism.

And to think I actually understand this stuff, when I hated math in school. Thats because mathematics and physics are an integral part of philosophy. And what we learned in school was NOT. That's because they taught New Math that incomprehensible clap trap of the sixties that made understanding math as easy as reading Egyptian Hieroglyphics.

Then I read Euclid and Pythagoras, and viola, or was it Eureka, and I understood math as philosophical constructs not just numbers. That the key to understanding the universe was the Golden Section or Golden Ratio, that its construct is the Pentagram, the most ancient symbol of man in the universe, as illustrated by Leonardo Da Vinci's famous drawing of Man, and the funadmental principle underlying both Magick and Science, since they are related. (and you just knew I would get around to mentioning magick)

The Golden Section

The first mathematical occurrence of the golden section and it's associated figures is found in the school of thinkers founded by Pythagoras. The Pythagoreans, as they are known, adopted the pentagram as the symbol of health in their brotherhood, and it eventually came to be the distinguishing badge of their school. Unfortunately, little of their actual mathematics survives, but it is highly likely that they were the ones who derived the construction of the pentagon and decagon from the golden section.

The Golden ratio

Euclid, in The Elements, says that the line AB is divided in extreme and mean ratio by C if AB:AC = AC:CB. Although Euclid does not use the term, we shall call this the golden ratio. The definition appears in Book VI but there is a construction given in Book II, Theorem 11, concerning areas which is solved by dividing a line in the golden ratio. As well as constructions to divide a line in the golden ratio, Euclid gives applications such as the construction of a regular pentagon, an icosahedron and a dodecahedron. Here is how the golden ratio comes into the construction of a pentagon.


Children need to read and be taught mathematics based on these and other original texts, not New Math interpretations of the theorms. Had I, or any of us, been taught properly we would begin with studying Pythagoras, move to Euclid etc. Any Grade Five or Six student can understand these authors, after all they write clearly and explain their ideas without artithmical obfustication. Then we would understand that math is not just some set of numbers but is the rational description of the phyical world, including art, music,biology, etc.


Oh by the way the reason you can read this is because of Godel's Theorm of chance as it applies to computers.

"Thus chance has been admitted, not merely as a mathematical tool for physics, but as part of its warp and weft" Norbert Wiener

And because of Norbert Wiener's classic founding work cybernetics: Cybernetics or control and communication in the animal and the machine, MIT Press 1996

Weiner's work on cybernetics influenced the workers councils in Allende's Chile,to use computers to develop worker self management of industry, they were crucial to the rationalization of inputs and outputs!

Weiner stated the following in the 1950's and it still applies today
Our papers have been making a great deal of American "know-how" ever since we had the misfortune to discover the atomic bomb. There is one quality more important than "know-how" and we cannot accuse the United States of any undue amount of it. This is "know-what" by which we determine not only how to accomplish our purposes, but what our purposes are to be.

Norbert Wiener's personality was generous: "I want to be the master of nobody"
You knew I would get some sort of libertarian perspective into this article.

Since Leibniz there has perhaps been no man who has had a full command of all the intellectual activity of his day. Since that time, science has been increasingly the task of specialists, in fields which show a tendency to grow progressively narrower. A century ago there may have been no Leibniz, but there was a Gauss, a Faraday, and a Darwin. Today there are few scholars who can call themselves mathematicians or physicists or biologists without restriction.

A man may be a topologist or an acoustician or a coleopterist. He will be filled with the jargon of his field, and will know all its literature and all its ramifications, but, more frequently than not, he will regard the next subject as something belonging to his colleague three doors down the corridor, and will consider any interest in it on his own part as an unwarrantable breach of privacy.

- Wiener, Norbert; Cybernetics; 1948.

The Human Use of Human Beings- Norbert Wiener's Ideas at the Dawn of the Age of Computing
I have spoken of machines, but not only of machines having brains of brass and thews of iron. When human atoms are knit into an organization in which they are used, not in their full right as responsible human beings, but as cogs and levers and rods, it matters little that their raw material is flesh and blood. What is used as an element in a machine, is in fact an element in the machine. Whether we entrust our decisions to machines of metal, or to those machines of flesh and blood which are bureaus and vast laboratories and armies and corporations, we shall never receive the right answers to our questions unless we ask the right questions.

Read to succeed -- in math; study shows how reading skill shapes more than just reading

University at Buffalo psychologist identifies a connectivity fingerprint suggesting that the brain's reading network is at work across cognitive domains; 'reading affects everything,' he says

UNIVERSITY AT BUFFALO

Research News

 

BUFFALO, N.Y. - A University at Buffalo researcher's recent work on dyslexia has unexpectedly produced a startling discovery which clearly demonstrates how the cooperative areas of the brain responsible for reading skill are also at work during apparently unrelated activities, such as multiplication.

Though the division between literacy and math is commonly reflected in the division between the arts and sciences, the findings suggest that reading, writing and arithmetic, the foundational skills informally identified as the three Rs, might actually overlap in ways not previously imagined, let alone experimentally validated.

"These findings floored me," said Christopher McNorgan, PhD, the paper's author and an assistant professor in UB's Department of Psychology. "They elevate the value and importance of literacy by showing how reading proficiency reaches across domains, guiding how we approach other tasks and solve other problems.

"Reading is everything, and saying so is more than an inspirational slogan. It's now a definitive research conclusion."

And it's a conclusion that was not originally part of McNorgan's design. He planned to exclusively explore if it was possible to identify children with dyslexia on the basis of how the brain was wired for reading.

"It seemed plausible given the work I had recently finished, which identified a biomarker for ADHD," said McNorgan, an expert in neuroimaging and computational modeling.

Like that previous study, a novel deep learning approach that makes multiple simultaneous classifications is at the core of McNorgan's current paper, which appears in the journal Frontiers in Computational Neuroscience.

Deep learning networks are ideal for uncovering conditional, non-linear relationships.

Where linear relationships involve one variable directly influencing another, a non-linear relationship can be slippery because changes in one area do not necessarily proportionally influence another area. But what's challenging for traditional methods is easily handled through deep learning.

McNorgan identified dyslexia with 94% accuracy when he finished with his first data set, consisting of functional connectivity from 14 good readers and 14 poor readers engaged in a language task.

But he needed another data set to determine if his findings could be generalized. So McNorgan chose a math study, which relied on a mental multiplication task, and measured functional connectivity from the fMRI information in that second data set.

Functional connectivity, unlike what the name might imply, is a dynamic description of how the brain is virtually wired from moment to moment. Don't think in terms of the physical wires used in a network, but instead of how those wires are used throughout the day. When you're working, your laptop is sending a document to your printer. Later in the day, your laptop might be streaming a movie to your television. How those wires are used depends on whether you're working or relaxing. Functional connectivity changes according to the immediate task.

The brain dynamically rewires itself according to the task all the time. Imagine reading a list of restaurant specials while standing only a few steps away from the menu board nailed to the wall. The visual cortex is working whenever you're looking at something, but because you're reading, the visual cortex works with, or is wired to, at least for the moment, the auditory cortex.

Pointing to one of the items on the board, you accidentally knock it from the wall. When you reach out to catch it, your brain wiring changes. You're no longer reading, but trying to catch a falling object, and your visual cortex now works with the pre-motor cortex to guide your hand.

Different tasks, different wiring; or, as McNorgan explains, different functional networks.

In the two data sets McNorgan used, participants were engaged in different tasks: language and math. Yet in each case, the connectivity fingerprint was the same, and he was able to identify dyslexia with 94% accuracy whether testing against the reading group or the math group.

It was a whim, he said, to see how well his model distinguished good readers from poor readers - or from participants who weren't reading at all. Seeing the accuracy, and the similarity, changed the direction of the paper McNorgan intended.

Yes, he could identify dyslexia. But it became obvious that the brain's wiring for reading was also present for math.

Different task. Same functional networks.

"The brain should be dynamically wiring itself in a way that's specifically relevant to doing math because of the multiplication problem in the second data set, but there's clear evidence of the dynamic configuration of the reading network showing up in the math task," McNorgan says.

He says it's the sort of finding that strengthens the already strong case for supporting literacy.

"These results show that the way our brain is wired for reading is actually influencing how the brain functions for math," he said. "That says your reading skill is going to affect how you tackle problems in other domains, and helps us better understand children with learning difficulties in both reading and math."

As the line between cognitive domains becomes more blurred, McNorgan wonders what other domains the reading network is actually guiding.

"I've looked at two domains which couldn't be farther afield," he said. "If the brain is showing that its wiring for reading is showing up in mental multiplication, what else might it be contributing toward?"

That's an open question, for now, according to McNorgan.

"What I do know because of this research is that an educational emphasis on reading means much more than improving reading skill," he said. "These findings suggest that learning how to read shapes so much more."

What is Godel's Theorem?

January 25, 1999

Melvin Henriksen--Professor of Mathematics Emeritus at Harvey Mudd College--offers this explanation:

KURT GODEL achieved fame in 1931 with the publication of his Incompleteness Theorem.

Giving a mathematically precise statement of Godel's Incompleteness Theorem would only obscure its important intuitive content from almost anyone who is not a specialist in mathematical logic. So instead, I will rephrase and simplify it in the language of computers.

Imagine that we have access to a very powerful computer called Oracle. As do the computers with which we are familiar, Oracle asks that the user "inputs" instructions that follow precise rules and it supplies the "output" or answer in a way that also follows these rules. The same input will always produce the same output. The input and output are written as integers (or whole numbers) and Oracle performs only the usual operations of addition, subtraction, multiplication and division (when possible). Unlike ordinary computers, there are no concerns regarding efficiency or time. Oracle will carry out properly given instructions no matter how long it takes and it will stop only when they are executed--even if it takes more than a million year

Let's consider a simple example. Remember that a positive integer (let's call it N) that is bigger than 1 is called a prime number if it is not divisible by any positive integer besides 1 and N. How would you ask Oracle to decide if N is prime? Tell it to divide N by every integer between 1 and N-1 and to stop when the division comes out evenly or it reaches N-1. (Actually, you can stop if it reaches the square root of N. If there have been no even divisions of N at that point, then N is prime.)

What Godel's theorem says is that there are properly posed questions involving only the arithmetic of integers that Oracle cannot answer. In other words, there are statements that--although inputted properly--Oracle cannot evaluate to decide if they are true or false. Such assertions are called undecidable, and are very complicated. And if you were to bring one to Dr. Godel, he would explain to you that such assertions will always exist.

Even if you were given an "improved" model of Oracle, call it OracleT, in which a particular undecidable statement, UD, is decreed true, another undecidable statement would be generated to take its place. More puzzling yet, you might also be given another "improved" model of Oracle, call it OracleF, in which UD would be decreed false. Regardless, this model too would generate other undecidable statements, and might yield results that differed from OracleT's, but were equally valid.

Do you find this shocking and close to paradoxical? It was even more shocking to the mathematical world in 1931, when Godel unveiled his incompleteness theorem. Godel did not phrase his result in the language of computers. He worked in a definite logical system and mathematicians hoped that his result depended on the peculiarities of that system. But in the next decade or so, a number of mathematicians--including Stephen C. Kleene, Emil Post, J.B. Rosser and Alan Turing--showed that it did not.

Research on the consequences of this great theorem continues to this day. Anyone with Internet access using a search engine like Alta Vista can find several hundred articles of highly varying quality on Godel's Theorem. Among the best things to read, though, is Godel's Proof by Ernest Nagel and James R. Newman, published in 1958 and released in paperback by New York University Press in 1983.

LA REVUE GAUCHE - Left Comment: Godel, Cantor, Wiener and Schrodinger's Cat 

LA REVUE GAUCHE - Left Comment: Search results for MATH 

Mysticism and Mathematics

If physics is the empirical study of the world as it is then mathematics is the study of the mind of god, it is gnosis; an attempt to know the absolute. Thus the basis of mathematical reasoning is not reason nor reasonable but metaphysics and mystical.

As documented below,the crisis in mathematics at the turn of last century was a crisis between German rationalist philosophy and Jewish mysticism; the Kabbalah. The German and rationalist mathematical philosophers, including thier British scion's like Russel and Whitehead, were attempting to map the mind of god; the absolute, while the Jewish mystic mathematician Georg Cantor had gone mad, it aggravated his existing depression , when he discovered the mind of God. He looked upon the face of the abyss and the face of the abyss looked back at him.

Wanting to avoid the problem of the abyss, which is the mystical journey known as the Conversation with the Holy Guardian Angel, the German school attempted to refute Cantor's theorem. They did this by adopting a different metaphysics and mystical philosophy, one that was still rooted in the occult knowledge and traditions of the pre-enlightenment. An analysis that did not require an appreciation of the absolute, god, by expansive reasoning, by looking at the enormity of infinity, rather they looked inward. And in looking inward they had to answer one question, can the infinite be finite.Can it be formulated as a set of finite principles.

It took another mystic Luitzen Egbertus Jan Brouwer, to challenge Cantor. Brouwer was a Dutch mathematician and mystic, influenced by the works of Meister Eckhart as well as by the alchemistJakob Boehme. His mathematical premises were thus founded on the alienated sense of man separated from God, and most math commentators below miss this crucial point. They divorce his philosophical world view from his principa mathematica. At least one article below focuses upon the importance of his mystical thinking to his mathematical philosophy, which he published in a pamphlet in 1903.

Life, Art, and Mysticism

Luitzen Egbertus Jan Brouwer

Like the Master Ekhardt he believes that the search for the mathematical absolute; god, is best done through asceticism, seclusion of the mind.

Seclusion

by Johannes (Master) Eckhart

I have read many writings of both Pagan masters and the Prophets of the old and new Covenant, and have investigated seriously and with great zeal which would be the best and highest virtue by which Man could best become similar to God, and how he could resemble again the archetype such as he was in God when there was no difference between him and God until God made the creatures. If I go down to the bottom of all that is written as far as my reason with its testimony and its judgment can reach, I find nothing but mere seclusion of all that is created. In this sense our Lord says to Martha: "One thing is needed," this means: He who wants to be pure and untroubled has to have one thing, Seclusion.

His essay corresponds to the changing world view of Modernity that was occurring at that moment in history. Herman Hesse also reflected this change in thought that was occurring before and after WWI. and he too was influenced by Master
Eckhart.

Brouwer's paean to an ascetic mathematical gnosis of the mind of god is also reminiscent of his contemporary aesthete; the Russian composer Scriabin. Scriabin believed music, which is mathematical, and art the highest form of gnosis.

It was the artist, and the artist alone – not the scientist or politician – who could offer to mankind a form of gnosis achieved through the experience of ecstasy and
the act of creation that brought it about. And it was to this mission of artistic
creation that Scriabin was unyieldingly faithful despite all else.

Brouwer's influence on Godel would lead the two down separate gnostic paths of interpretation of principa mathematica. And yet both these paths reflect the dualistic nature of actual gnosticism, between the deniers of the world as it is and those who embrace the world as it is. Between the ascetic and libertine, the Cathars and the Adamites. Master Eckhart himself was a member of the heretical sect the proto-communist Brethren of the Free Spirit.

Mysticism and mathematics: Brouwer, Godel, and the common core thesis

David Hilbert opened ‘Axiomatic Thought’ with the observation that ‘the most important bearers of mathematical thought,’ for ‘the benefit of mathematicsitself have always [. . . ] cultivated the relations to the domains of physics and the [philosophical] theory of knowledge.’ We have in L.E.J. Brouwer and Kurt Godel two of those ‘most important bearers of mathematical thought’ who cultivated the relations to philosophy for the benefit of mathematics (though not only for that). And both went beyond philosophy, cultivating relations to mysticism for the benefit of mathematics (though not for that alone).

There is a basic conception of mysticism that is singularly relevant here.
(’Mysticism’ labels that.) That corresponds to a basic conception of philosophy
(’Philosophy’), also singularly relevant here. Both Mystic and Philosopher begin
in a condition of seriously unpleasant, existential unease, and aim at a condition
of abiding ease. For Mystic and Philosopher the way to that ease is through
being enlightened about the real and true good of all things. Thus Mysticism
and Philosophy are triply optimistic: there is a real, true good of all things,
the Philosopher and Mystic can become enlightened about it, and being thus
enlightened would give them ease.

That Enlightenment sought comes from some sort of cognitive or intelligent
engagement with what we will here call ‘the Good’. Some use ‘the Absolute’
when it seems important to emphasize that ‘the Good’ is unconditioned—there
is nothing behind it, nothing above it. Others use ‘the One’; still others, ‘God’.

It is natural to regard the Good as somehow mind-like, or like something (permanently) in mind. It should in either case be in some way homogeneous with, or in sympathy with, our minds, for the Good must attract and support the intelligent engagement of it by our minds. In that way it can enlighten us.

We have seen that both Godel and Brouwer were looking for mystical experiences,
in which an openness of the mind to the Absolute is operative. What
is disclosed in such experiences has the air of being something imparted to the
person. The imparting is preceded by a preparation or transformation of the
person. The self must be brought into a condition to receive, support, and appreciate what is to be disclosed. This preparation we see mentioned by both Brouwer (the abandonment of mathematics) and Godel (closing off the senses, etc.)

However, they made very different claims as to how what is disclosed in
such experience is related to mathematics. What strikes us is how the bond
between mathematics and mysticism is equally tight in Godel and Brouwer, but
that the signs are different so to speak. According to both, mathematics relates
individual thought to ultimate reality, but Godel thinks of a positive relation
and Brouwer of a negative one.

For Godel, doing mathematics is a way of accessing the Absolute. For
Brouwer, doing mathematics precisely prohibits access to the Absolute.
Put differently, according to Godel, mathematical experience reveals (part
of) Reality; according to Brouwer, mathematical experience conceals Reality.
A mystical disclosure in the relevant sense has about it the phenomenological
character of being a form of knowing or enlightened understanding; it discloses
the Good, the significant, the important, fundamental values.

We would like to end by making the following two remarks. First, of course one could, and usually does, engage in mathematics for its own sake, without any interest in relating it, be it positively or negatively, to mysticism. From Godel’s and Brouwer’s point of view, that would probably be not unlike the possibility to perform a hymn for its own sake, without any interest in the religious meaning it may have.

The second remark is related to the first. In spite of the incommensurability
of Brouwer’s and Godel’s positions, their respective motivations to take the
mystical turn may have much in common. Both were disgruntled with the
materialistic and formalistic philosophies prevalent at their times; both thought
that these philosophies could not do justice to the Good.

The Crisis in the Foundations of Mathematics

José Ferreirós
Draft, 26 July 2004

The foundational crisis is a well-known affair for almost all mathematicians. We
all know something about logicism, formalism, and intuitionism; about the hopes
to place mathematical theories beyond the shadow of any doubt; about the
impact of Gödel’s results upon our images of mathematical knowledge. But the
real outlines of the historical debate are not well known, and the subtler
philosophical issues at stake are often ignored. In the limited space available,
we shall essentially discuss the former, in the hopes that this will help bring the
main conceptual issues under sharper focus.

Usually, the crisis is understood as a relatively localized event, a heated
debate taking place in the 1920s between the partisans of “classical” (late 19th
century) mathematics, led by Hilbert, and their critics, led by Brouwer,
advocating strong revision of the received doctrines. There is however a second
sense and in my opinion a very important one, in which the “crisis” was a long
and rather global process, indistinguishable from the rise of modern
mathematics and the philosophical/methodological perplexities it created.

This is the standpoint from which the present account has been written.

Within this longer process one may isolate five noteworthy intervals:
1) around 1870, discussions about non-Euclidean geometries, function
theory, and the real numbers;

2) around 1885, fights in algebra, higher arithmetic, and set theory;
3) by 1904, debates on axiomatics and logic vs. intuition, the concept of the
continuum, and set theory;
4) around 1925, the crisis in the proper sense, transforming the main
previous views into detailed research projects;
5) in the 1930s, Gödel’s results and their aftermath.

Meanwhile, back in the 1900s, a young mathematician in the Netherlands
was beginning to find his way toward a philosophically coloured version of
constructivism. Egbertus Brouwer presented his strikingly peculiar (to some,
outrageous) metaphysical and ethical views in 1905, and started to elaborate a
corresponding foundation for mathematics in his thesis of 1907. His philosophy
of intuitionism derived from the old metaphysical view that individual
consciousness is the one and only source of knowledge.

Brouwer’s worldview was idealistic and tended to solipsism, he had an artistic temperament, his private life was eccentric; he despised the modern world, looking for the inner life of the self as the only way out (at least in principle, though not always in practice).

In the end though they never truly refuted Cantor, they merely built on his theorems. Brouwer created a topology of the mind of god while Godel proved that no set theories can be proven, which led to the Heisenburg uncertainty principle.

They all contributed to the ultimate alchemical paradox of modern physics that the observer influences what is observed, As Above-So Below, which we know today as quantum theory.

Modern mathematical philosophy is simply gnosis stripped of its religious iconography and poetry. In that it still remains kabbalistic ,as Cantor suggested, guided by mystics whose language is their own and who despite their philosophical differences remain of one mind. The purpose and outcome of their theories are an attempt to define and understand a pantheistic/monistic universe. Their failure to resolve the contradictions of their theories is their failure to embrace dialectics. Like their enlightenment counterparts, the Freemasons, they remain a secret society founded on mysticism.


------------------------------------APPENDIX-----------------------------------------------

THE METAPHYSICS OF MATHEMATICS

Historically, the starting point is Plato who proposed that mathematical reality consists of perfect forms independent of the physical world. This view of the subject matter of mathematics lies at one end of a spectrum of metaphysical views; towards the other end is the view is that the subject matter is a purely human artefact. Views towards the Platonic end are known as Platonist; towards the other end, anti- Platonist. That is a classification of metaphysical views. Epistemological views fall into two classes, roughly speaking mathematical truths are known (i) by reason, or (ii) by inference from the evidence of the senses supplemented by deduction. There are a few important epistemological views which fall into neither camp, notably those of Plato, Kant and Gödel.

Philosophy of Mathematics

Many philosophers have taken mathematics to be the paradigm of knowledge, and the reasoning employed in following mathematical proofs is often regarded as the epitome of rational thought. But mathematics is also a rich source of philosophical problems which have been at the centre of epistemology and metaphysics since the beginnings of Western philosophy; among the most important are the following:

1. Do numbers and other mathematical entities exist independently of human cognition?
2. If not then how do we explain the extraordinary applicability of mathematics to science and practical affairs? If so then what kind of things are they and how can we know about them?
3. What is the relationship between mathematics and logic?

The first question is a metaphysical question with close affinities to questions about the existence of other entities such as universals, properties and values. According to many philosophers, if such entities exist then they do so outside of space and time, and they lack causal powers; they are often termed abstract (as opposed to concrete) entities. If we accept the existence of abstract mathematical objects then an adequate epistemology of mathematics must explain how we can know about them. Of course, proofs seem to be our main source of justification for mathematical propositions but proofs depend on axioms and so the question of how we can know the truth of the axioms remains.

It is usually thought that mathematical truths are necessary truths; how then is it possible for finite, physical beings inhabiting a contingent world to have knowledge of such truths? Two broad views are possible: either mathematical truths are known by reason; or they are known by inference from sensory experience. The former rationalist view is adopted by Descartes and Leibniz who also thought that mathematical concepts are innate. Locke and Hume agreed that mathematical truths were known by reason but they thought all mathematical concepts were derived by abstraction from experience. Mill was a complete empiricist about mathematics and held both that mathematical concepts are derived from experience and also that mathematical truths like 2+2=4 are really inductive generalisations from experience. (N.B. Kant’s views on mathematics are complex and important; see Kant.)

The discovery in the mid-nineteenth century of non-Euclidean geometry meant that philosophers were forced to reassess the status of Euclidean geometry which had previously been regarded as the shinning example of certain knowledge of the world. Many took the existence of consistent non-Euclidean geometries to be a direct refutation of both Mill’s and Kant’s philosophies of mathematics. By the end of the nineteenth century Cantor had discovered various paradoxes in the theory of classes and there was something of a crisis in the foundations of mathematics. The early twentieth century saw great advances in mathematics and also in mathematical logic and the foundations of mathematics.

Most of the fundamental issues in the philosophy of mathematics are accessible to anyone who is familiar with geometry and arithmetic and who has had the experience of following a mathematical proof. However, some of the most important philosophical developments of the twentieth century were instigated by the profound developments that have taken place in mathematics and logic, and a proper appreciation of these issues is only available to someone who has an understanding of basic set theory and intermediate logic. To study philosophy of mathematics at an advanced level one ought really to have followed a course which includes proofs of Gödel’s incompleteness theorems.

Georg Cantor - Wikipedia, the free encyclopedia

Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845[1] – January 6, 1918) was a German mathematician. He is best known as the creator of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between sets, defined infinite and well-ordered sets, and proved that the real numbers are "more numerous" than the natural numbers. In fact, Cantor's theorem implies the existence of an "infinity of infinities". He defined the cardinal and ordinal numbers, and their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware.

Cantor's theory of transfinite numbers was originally regarded as so counter-intuitive—even shocking—that it encountered resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaréand later from Hermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections. Some Christian theologians (particularly neo-Scholastics) saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God on one occasion equating the theory of transfinite numbers with pantheism. The objections to his work were occasionally fierce: Poincaré referred to Cantor's ideas as a "grave disease" infecting the discipline of mathematics,and Kronecker's public opposition and personal attacks included describing Cantor as a "scientific charlatan", a "renegade" and a "corrupter of youth."Writing decades after Cantor's death, Wittgenstein lamented that mathematics is "ridden through and through with the pernicious idioms of set theory," which he dismissed as "utter nonsense" that is "laughable" and "wrong".Cantor's recurring bouts of depression from 1884 to the end of his life were once blamed on the hostile attitude of many of his contemporaries, but these episodes can now be seen as probable manifestations of a bipolar disorder.

The harsh criticism has been matched by international accolades. In 1904, the Royal Society of London awarded Cantor its Sylvester Medal, the highest honor it can confer. Cantor believed his theory of transfinite numbers had been communicated to him by God.David Hilbert defended it from its critics by famously declaring: "No one shall expel us from the Paradise that Cantor has created."

CANTOR'S PHILOSOPHICAL WRITING

Mathematics, in the development of its ideas, has only to take account of the immanent reality of its concepts and has absolutely no obligation to examine their transient reality.

… Mathematics is in its development entirely free and is only bound in the self-evident respect that its concepts must both be consistent with each other, and also stand in exact relationships, ordered by definitions, to those concepts which have previously been introduced and are already at hand and established. In particular, in the introduction of new numbers, it is only obligated to give definitions of them which will bestow such a determinacy and, in certain circumstances, such a relationship to the other numbers that they can in any given instance be precisely distinguished. As soon as a number satisfies all these conditions, it can and must be regarded in mathematics as existent and real.

"… the essence of mathematics lies entirely in its freedom".

Everything and More: A Compact History of Infinity

The best-selling author of Infinite Jest on the two-thousand-year-old quest to understand infinity. ONE OF THE OUTSTANDING VOICES of his generation. David Foster Wallace has won a large and devoted following for the intellectual ambition and bravura style of his fiction and essays. Now he brings his considerable talents to the history of one of math's most enduring puzzles: the seemingly paradoxical nature of infinity. Is infinity a valid mathematical property or a meaningless abstraction? The nineteenth-century mathematical genius Georg Cantor's answer to this question not only surprised him but also shook the very foundations upon which math had been built. Cantor's counterintuitive discovery of a progression of larger and larger infinities created controversy in his time and may have hastened his mental breakdown, but it also helped lead to the development of set theory, analytic philosophy, and even computer technology. Smart, challenging, and thoroughly rewarding, Wallace's tour de force brings immediate and highprofile recognition to the bizarre and fascinating world of higher mathematics.

THE MYSTERY OF THE ALEPH: MATHEMATICS, THE KABBALAH, AND THE SEARCH FOR INFINITY

by Amir Aczel Four Walls Eight Windows, New York, NY, 258 pp., 2000

Seeing a marked increase in the number of books on mathematics written for the general populace and published in the past few years has been nice, seeing so many of them take a historical view is even more exciting. The Mystery of the Aleph is a fine addition to this collection. Amir Aczel's topic is Georg Cantor and his discovery/invention of transfinite numbers. The book is a well-written nontechnical introduction to Cantor's life, set theory, transfinite numbers, the continuum hypothesis, and related mathematical and historical issues. While staying true to the mathematics Amir Aczel has written The Mystery of the Aleph with the attention to suspense and character development of a skilled story-teller.

The story begins with Cantor's death in a university mental clinic in 1918. Like a fine mystery writer Aczel draws us into the tale by concluding a short (9 pages) first chapter with the following:

"One fact is known about Georg Cantor's illness. His attacks of depression were associated with periods in which he was thinking about what is now known as 'Cantor's continuum hypothesis.' He was contemplating a single mathematical expression, an equation using the Hebrew letter aleph: 2... = ... . This equation is a statement about the nature of infinity. A century and a third after Cantor first wrote it down, the equation - along with its properties and implications - remains the most enduring mystery in mathematics. " (pp. 8-9)

From the mental clinic in Halle, The Mystery of the Aleph takes the reader back to the paradoxes of Zeno, to the Pythagoreans, and then to the Kabbalah, the Jewish system of secret mysticism, numerology, and meditations. Here Aczel introduces notion of the intense light of the infinity of God as a metaphor for the wonder of Cantor's infinities. The metaphor continues with good effect throughout the book. Although no actual clear connection between Cantor's work and the Kabbalah is established in The Mystery of the Aleph, the metaphorical connection is successful and contributes to the story.

Quickly the pace of the tale picks up and the reader is treated to wonderful discussions of Galileo's demonstration of the one-to-one correspondence between the natural numbers and the even natural numbers, Bolzano's pioneering work with infinite series, the mathematical hegemony of German universities in the late nineteenth century, and the powerful personalities of Weierstrass and Kronecker. Woven through it all we watch the development of Cantor as a mathematician, and the birth of modern set theory and transfinite numbers.

The Mystery of the Aleph then takes us to the questions of the foundations of mathematics that have haunted generations of mathematicians from Peano, Russell, Frege, Zermelo, Hilbert and Brouwer, to Godel, Turing, and Cohen. The story culminates in Cohen's proof of the independence of the continuum hypothesis from the axioms of Zermelo-Fraenkel set theory, Godel's incompleteness theorem, and Turing's argument for the undecidibility of the halting problem. Throughout this grand tour of the key issues of mathematics and infinity, The Mystery of the Aleph never lets us lose sight of the humanity (and the inevitable failures and successes that go with it) of these giants of mathematics. The book ends with a quote on a plaque in Halle commemorating Georg Cantor. It reads "The essence of mathematics lies in its freedom." (p. 228)

The Mystery of the Aleph is not a source of details on the mathematics of Cantor, Godel, and Cohen, but it is a wonderful source for a quick historical overview of the issues of infinity in modern mathematics, biographical information on Cantor and Godel, and a good introduction to the politics of mathematics in the nineteenth century. This book would be a valuable addition to a school library or a text to share with a student who has begun to wonder about infinity.

Reviewed by James V. Rauff Millikin University

Copyright Mathematics and Computer Education Spring 2001
Provided by ProQuest Information and Learning Company. All rights Reserved

Aleph

In gematria, aleph represents the number 1, and when used at the beginning of Hebrew years, it means 1000 (i.e. א'תשנ"ד in numbers would be the date 1754).

• Aleph number or cardinality, a measurement of mathematical sets
  • aleph null (), the cardinality of countable infinite sets
  • aleph one (), the cardinality of certain uncountably infinite sets (if the continuum hypothesis is true, then the set of all real numbers has this cardinality).
  • 
    
• In set theory, The Hebrew aleph glyph is used as the symbol to denote the aleph numbers, which represent the cardinality of infinite sets.

Aleph is the subject of a midrash which praises its humility in not demanding to start the Bible. (In Hebrew the Bible is begun with the second letter of the alphabet, Bet.) In this folktale, Aleph is rewarded by being allowed to start the Ten Commandments. (In Hebrew, the first word is 'Anokhi, which starts with an aleph.)

In the Sefer Yetzirah, The letter Aleph is King over Breath, Formed Air in the universe, Temperate in the Year, and the Chest in the soul.

Aleph is also the first letter of the Hebrew word emet, which means truth. In Jewish mythology it was the letter aleph that was carved into the head of the golem which ultimately gave it life.

Aleph also begins the three words that make up God's mystical name in Exodus, I Am That I Am, (in Hebrew, 'Ehye 'Asher 'Ehye), and aleph is an important part of mystical amulets and formulas.

Luitzen Egbertus Jan Brouwer

In 1905, at the age of 26, Brouwer expressed his philosophy of life in a short tract Life, Art and Mysticism described by Davis as "drenched in romantic pessimism" (Davis (2002), p. 94). Then Brouwer "embarked on a self-righteous campaign to reconstruct mathematical practice from the ground up so as to satisfy his philosophical convictions"; indeed his thesis advisor refused to accept his Chapter II " 'as it stands, ... all interwoven with some kind of pessimism and mystical attitude to life which is not mathematics, nor has anything to do with the foundations of mathematics' " (Davis, p. 94 quoting van Stigt, p. 41). Nevertheless, in 1908:

"... Brouwer, in a paper entitled "The untrustworthiness of the principles of logic", challenged the belief that the rules of the classical logic, which have come down to us essentially from Aristotle (384--322 B.C.) have an absolute validity, independent of the subject matter to which they are applied" (Kleene (1952), p. 46).

"After completing his dissertation [year?], Brouwer made a conscious decision to temporarily keep his contentious ideas under wraps and to concentrate on demonstrating his mathematical prowess" (Davis (2000), p. 95); by 1910 he had published a number of important papers, in particular the Fixed Point Theorem. Hilbert -- the formalist with whom the intuitionist Brouwer would ultimately spend years in conflict -- admired the young man and helped him receive a regular academic appointment (1912) at the University of Amsterdam (Davis, p. 96). It was then that "Brouwer felt free to return to his revolutionary project which he was now calling intuitionism " (ibid).

Kurt Gödel

Kurt Gödel (April 28, 1906 Brünn, Austria-Hungary (now Brno, Czech Republic) – January 14, 1978 Princeton, New Jersey) was an Austrian American mathematician and philosopher.

One of the most significant logicians of all time, Gödel's work has had immense impact upon scientific and philosophical thinking in the 20th century, a time when many, such as Bertrand Russell, A. N. Whitehead and David Hilbert, were attempting to use logic and set theory to understand the foundations of mathematics.

Gödel is best known for his two incompleteness theorems, published in 1931 when he was 25 years of age, one year after finishing his doctorate at the University of Vienna. The more famous incompleteness theorem states that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (Peano arithmetic), there are true propositions about the naturals that cannot be proved from the axioms. To prove this theorem, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers.

He also showed that the continuum hypothesis cannot be disproved from the accepted axioms of set theory, if those axioms are consistent. He made important contributions to proof theory by clarifying the connections between classical logic, intuitionistic logic, and modal logic.

Gödel's incompleteness theorems

From Wikipedia, the free encyclopedia

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In mathematical logic, Gödel's incompleteness theorems, proved by Kurt Gödel in 1931, are two theorems stating inherent limitations of all but the most trivial formal systems for arithmetic of mathematical interest.

The theorems are also of considerable importance to the philosophy of mathematics. They are widely regarded as showing that Hilbert's program to find a complete and consistent set of axioms for all of mathematics is impossible, thus giving a negative answer to Hilbert's second problem. Authors such as J. R. Lucas have argued that the theorems have implications in wider areas of philosophy and even cognitive science, but these claims are less generally accepted.

INCOMPLETENESS: THE PROOF AND PARADOX OF KURT GÖDEL

INCOMPLETENESS: THE PROOF AND PARADOX OF KURT GÖDEL by Rebecca Goldstein Atlas Books, 2005, 296 pp. ISBN: 0-393-05169-2

On page 253 of Incompleteness: The Proof and Paradox of Kurt Gödel, there is a photograph of Albert Einstein and Kurt Gödel walking together on the grounds of the Institute for Advanced Study (IAS). I doubt that any student of mathematics could fail to be moved by this photograph. What wonderful ideas are being exchanged? What new areas of mathematics, physics, logic, or philosophy were born in the conversations between these two giants of twentieth-century thought?

Incompleteness: The Proof and Paradox of Kurt Gödel provides many tantalizing glimpses at the life and work of Kurt Gödel. Rebecca Goldstein follows Gödel from his early days with the Vienna Circle to his last days at the IAS. Although there are several recent books about Gödel and/or his incompleteness theorems, Goldstein's stands out on three fronts.

First, Incompleteness is absolutely beautifully written. The style is conversational and the reader is carried along by the author's obvious joy in her subject matter. I read Incompleteness in three consecutive evenings. It was truly difficult to put down. Undergraduate students in mathematics, physics, or philosophy will find Incompleteness exciting. It will reaffirm their choice of study.

Second, Incompleteness is an excellent introduction to the personalities and philosophies of the iconic members of the Vienna Circle (Moritz Schlick, Rudolph Carnap, Otto Neurath, Hans Hahn, Herbert Feigl, Karl Menger, Kurt Godel) and celebrated visitors and participants (John von Neumann, Willard van Orman Quine, Carl Hempel, Alfred Tarski, and the very influential Ludwig Wittgenstein). The reader can't help but imagine the intense level of intellectual activity going on in a single location. Goldstein skillfully shows us how Gödel was influenced by and influenced the Circle, and contrasts his Platonism with Wittgenstein's philosophy of mathematics, formalism, and logical positivism. These passages are superb introductions to the state of the philosophy of science in the first half of the twentieth century.

Finally, Goldstein presents one of the best non-technical outlines of Gödel's proof of the incompleteness of arithmetic. Any mathematics teacher would do well to begin their students' understanding of Gödel's results with a reading from Incompleteness.

Incompleteness: The Proof and Paradox of Kurt Gödel is a delightful introduction to the life, work, and times of Kurt Gödel. Written in a captivating conversational style, true to its mathematical, philosophical, and historical content, and just plain fun to read, this book deserves a spot on the recommended reading list for undergraduates.

Reviewed by James V. Rauff

Millikin University

Copyright Mathematics and Computer Education Spring 2006
Provided by ProQuest Information and Learning Company. All rights Reserved

On Gödel's Philosophy of Mathematics

by Harold Ravitch, Ph.D.
Chairman, Department of Philosophy
Los Angeles Valley College

(i) In thinking that the paradoxes were devastating mathematics, various restrictions on the usual methods of mathematical reasoning were imposed.

(ii) No paradox has been discovered which Involves entities which are strictly speaking mathematical: the "set of all sets," the "greatest ordinal number," "sets which are elements of themselves," etc. are logical and epistemological entities which do not belong to classical mathematics proper.

(iii) The concepts of classical mathematics are meaningful, precise, and are capable of being understood because they meet standards of clarity and exactitude which are adequate for their purpose.

(iv) Hence, there is no justification for applying unnecessary restrictions to classical mathematics.

2.) The Vicious Circle Principle.

The search for a once-and-for-all solution to the paradoxes led Russell, Poincaré, and others to the observation that each of the paradoxes trades on a vicious circle in defining an entity which ultimately creates the paradox. Questions of circularity are as old as philosophy., but it was never realized how deeply they could permeate logic and mathematics. Indeed Gödel himself remarked that "any epistemological paradox" could have been employed to yield an undecidable statement of arithmetic. Of course many nontechnical works on logic warn us about circular definitions.

In axiomatic set theory, one of the legislative functions of the axioms is to prohibit the existence of sets which would cause trouble, and the various axiom systems can be classified according to the manner in which the paradoxes are blocked. If one however wishes to derive totally his mathematics from his logic, it is found that the process of Dedekind Cuts, the fundamental method of establishing the real number system, is badly in violation of the vicious circle principles.Hermann Weyl attempted a development of analysis in Das Kontinuum which adhered to the vicious circle principle, but he was unable to obtain the whole of classical analysis. Recent research has shown that more can be squeezed out of these restrictions than had been expected:

all mathematically interesting statements about the natural numbers, as well as many analytic statements, which have been obtained by impredicative methods can already be obtained by predicative ones.

We do not wish to quibble over the meaning of "mathematically interesting." However, "it is shown that the arithmetical statement expressing the consistency of predicative analysis is provable by impredicative means." Thus it can be proved conclusively that restricting mathematics to predicative methods does in fact eliminate a substantial portion of classical mathematics.

Gödel has offered a rather complex analysis of the vicious circle principle and its devastating effects on classical mathematics culminating in the conclusion that because it "destroys the derivation of mathematics from logic, effected by Dedekind and Frege, and a good deal of modern mathematics itself" he would "consider this rather as a proof that the vicious circle principle is false than that classical mathematics is false."

The vicious circle principle as usually stated is dissected by Gödel into four forms:

(1) No totality can contain members definable only in terms of this totality.

(2) No totality can contain members involving this totality.

(3) No totality can contain members presupposing this totality.

(4) Nothing defined in terms of a propositional function can be a possible argument of this function.

The core of Gödel's rejection of the vicious circle principle reduces to his rejection of the view that mathematical entities are "constructed by ourselves." We shall see that this argument hinges an the interpretation of 'construction', and on Gödel's faith in the consistency of the axioms of set theory underlying classical analysis.

Cantor Godel Brouwer Russell Frege Whitehead

A power point presentation of their contributions to Philisophica Mathematica.



See:

Godel, Cantor, Wiener and Schrodinger's Cat

Dialectics, Nature and Science

Kabbalistic Kommunism

For a Ruthless Criticism of Everything Existing

Goldilocks Enigma

9 Minute Nobel Prize

Is God A Cosmonaut

Cosmic Conundrum

My Favorite Muslim



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GODEL'S PSYCHOGEOGRAPHY

Stick to your lane: Hidden order in chaotic crowds

Mathematical research from the University of Bath in the UK brings new understanding of crowd formation and behaviour

Peer-Reviewed Publication

UNIVERSITY OF BATH

 

IMAGE: TILTED LANES CAPTURED IN A HUMAN-CROWD EXPERIMENT. THE LANES ARE FORMED BY TWO GROUPS OF PEOPLE MOVING IN OPPOSITE DIRECTIONS. THE INCLINATION RESULTS FROM A `PASS ON THE RIGHT’ TRAFFIC RULE. view more 

CREDIT: K. BACIK. B. BACIK, T. ROGERS

Have you ever wondered how pedestrians ‘know’ to fall into lanes when they are moving through a crowd, without the matter being discussed or even given conscious thought?

A new theory developed by mathematicians at the University of Bath in the UK led by Professor Tim Rogers explains this phenomenon, and is able to predict when lanes will be curved as well as straight. The theory can even describe the tilt of a wonky lane when people are in the habit of passing on one side rather than the other (for instance, in a situation where they are often reminded to ‘pass on the right’).

This mathematical analysis unifies conflicting viewpoints on the origin of lane formation, and it reveals a new class of structures that in daily life may go unnoticed. The discovery, reported this week (Friday, March 3) in the prestigious journal Science, constitutes a major advance in the interdisciplinary science of ‘active matter’ – the study of group behaviours in interacting populations ranging in scale from bacteria to herds of animals.

Tested in arenas

To test their theory, the researchers asked a group of volunteers to walk across an experimental arena that mimicked different layouts, with changes to entrance and exit gates.

One arena was set up in the style of King’s Cross Station in London. When the researchers looked at the video footage from the experiment, they observed mathematical patterns taking shape in real life.

Professor Rogers said: “At a glance, a crowd of pedestrians attempting to pass through two gates might seem disorderly but when you look more closely, you see the hidden structure. Depending on the layout of the space, you may observe either the classic straight lanes or more complex curved patterns such as ellipses, parabolas, and hyperbolas”.

Lane formation

The single-file processions formed at busy zebra crossings are only one example of lane formation, and this study is likely to have implications for a range of scientific disciplines, particularly in the fields of physics and biology. Similar structures can also be formed by inanimate molecules, such as charged particles or organelles in a cell.

Until now, scientists have given several different explanations for why human crowds and other active systems naturally self-organise into lanes, but none of these theories have been verified. The Bath team used a new analytical approach, inspired by Albert Einstein’s theory of Brownian motion, which makes predictions that can be tested.

Encouraged by the way their theory agreed with the numerical simulations of colliding particles, they then teamed up with Professor Bogdan Bacik – an experimentalist from the Academy of Physical Education in Katowice, Poland – and ran a series of experiments (such as the one modelled on King’s Cross) using human crowds.

Lead author Dr Karol Bacik said: “Lane formation doesn’t require conscious thought – the participants of the experiment were not aware that they had arranged themselves into well-defined mathematical curves.

“The order emerges spontaneously when two groups with different objectives cross paths in a crowded space and try to avoid crashing into each other. The cumulative effect of lots of individual decisions inadvertently results in lanes forming.”

The researchers also tested the effects of externally imposed traffic rules – namely, they instructed the participants to pass others on the right. In agreement with the theoretical prediction, adding this rule changed the lane structure.

“When pedestrians have a preference for right turns, the lanes end up tilting and this introduces frustration that slows people down,” said Dr Bacik.

“What we’ve developed is a neat mathematical theory that forecasts the propensity for lane formation in any given system,” said Professor Rogers, adding: “We now know that much more structure exists than previously thought.”

 

Parabolic lane formation captured in a human crowd experiment. The red group crosses the experimental arena ‘south to north’, and the blue group targets a narrow gate on the side. In agreement with the theory, the crowd spontaneously self-organizes into lanes shaped as (confocal) parabolas.

CREDIT

Credit: K. Bacik. B. Bacik, T. Rogers

Pedestrians finding order in a [VIDEO] 

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JOURNAL

Science

DOI

10.1126/science.add8091 

METHOD OF RESEARCH

Computational simulation/modeling

SUBJECT OF RESEARCH

People

ARTICLE TITLE

Lane nucleation in complex active flows

ARTICLE PUBLICATION DATE

3-Mar-2023

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