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Posts Tagged ‘Hermann Weyl

A few chaotic glitches in sciences and philosophy?

Note: Re-edit of “Ironing out a few chaotic glitches; (Dec. 5, 2009)”

This Covid-19 pandemics has forced upon me to repost this old article.

Philosophers have been babbling for many thousand years whether the universe is chaotic or very structured so that rational and logical thinking can untangle its laws and comprehend nature’s behaviors and phenomena.

Plato wrote that the world is comprehensible.  The world looked like a structured work of art built on mathematical logical precision. Why?

Plato was fond of symmetry, geometry, numbers, and he was impressed by the ordered tonality of musical cord instruments.

Leibnitz in the 18th century explained “In what manner God created the universe it must be in the most regular and ordered structure”.

Leibnitz claimed that “God selected the simplest in hypotheses that generated the richest varieties of phenomena.

A strong impetus that the universe is comprehensible started with the “positivist philosophers and scientists” of the 20th century who were convinced that the laws of nature can be discovered by rational mind.

Einstein followed suit and wrote “God does not play dice.  To rationally comprehend a phenomenon we must reduce, by a logical process, the propositions (or axioms) to apparently known evidence that reason cannot touch.”

The pronouncement of Einstein “The eternally incomprehensible universe is its comprehensibility” can be interpreted in many ways.

The first interpretation is “what is most incomprehensible in the universe is that it can be comprehensible but we must refrain from revoking its sacral complexity and uncertainty”.

The second interpretation is “If we are still thinking that the universe is not comprehensible then may be it is so, as much as we want to think that we may understand it; thus, the universe will remain incomprehensible (and we should not prematurely declare the “end of science”).

The mathematician Hermann Weyl developed the notion: “The assertion that nature is regulated by strict laws is void, unless we affirm that it is related by simple mathematical laws.  The more we delve in the reduction process to the bare fundamental propositions the more facts are explained with exactitude.”

It is this philosophy of an ordered and symmetrical world that drove Mendeleyev to classifying the chemical elements; Murray Gell-Mann used “group theory” to predict the existence of quarks.

A few scientists went even further; they claimed that the universe evolved in such a way to permit the emergence of the rational thinking man.

Scientists enunciated many principles such as:

“The principle of least time” that Fermat used to deduce the laws of refraction and reflection of light;

Richard Feynman discoursed on the “principle of least actions”;

We have the “principle of least energy consumed”, the “principle of computational equivalence”, the “principle of entropy” or the level of uncertainty in a chaotic environment.

Stephen Hawking popularized the idea of the “Theory of Everything TOE” a theory based on a few simple and non redundant rules that govern the universe.

Stephen Wolfram thinks that the TOE can be found by a thorough systematic computer search: The universe complexity is finite and the most seemingly complex phenomena (for example cognitive functions) emerge from simple rules.

Before we offer the opposite view that universe is intrinsically chaotic let us define what is a theory.

Gregory Chaitin explained that “a theory is a computer program designed to account for observed facts by computation”.  (Warning to all mathematicians!  If you want your theory to be published by peer reviewers then you might have to attach an “elegant” or the shortest computer program in bits that describes your theory)

Kurt Gödel and Alain Turing demonstrated what is called “incompletude” in mathematics or the ultimate uncertainty of mathematical foundations.  There are innumerable “true” propositions or conjectures that can never be demonstrated.

For example, it is impossible to account for the results of elementary arithmetic such as addition or multiplication by the deductive processes of its basic axioms.  Thus, many more axioms and unresolved conjectures have to be added in order to explain correctly many mathematical results.

Turing demonstrated mathematically that there is no algorithm that can “know” if a program will ever stop or not.  The consequence in mathematics is this: No set of axioms will ever permit to deduce if a program will ever stop or not. Actually, there exist many numbers that cannot be computed.  There are mathematical facts that are logically irreducible and incomprehensive.

Quantum mechanics proclaimed that, on the micro level, the universe is chaotic: there is impossibility of simultaneously locating a particle, its direction, and determining its velocity.  We are computing probabilities of occurrences.

John von Neumann wrote: “Theoretical physics does not explain natural phenomena: it classifies phenomena and tries to link or relate the classes.”

Acquiring knowledge was intuitively understood as a tool to improving human dignity by increasing quality of life. Thus, erasing as many dangerous superstitions that bogged down spiritual and moral life of man.

Ironically, the trend captured a negative life of its own in the last century.  The subconscious goal for learning was  meant to frustrate fanatic religiosity that proclaimed that God is the sole creator and controller of our life, its quality, and its destiny.

With our gained power in knowledge we may thus destroy our survival by our own volition: We can commit earth suicide regardless of what God wishes.

So far, we have been extremely successful beyond all expectations.  We can destroy all living creatures and plants by activating a single H-Bomb or whether we act now or desist from finding resolution to the predicaments of climate changes.

I have impressions.

First, what the mathematicians and scientists are doing is not discovering the truth or the real processes but to condense complexity into simple propositions so that an individual may think that he is able to comprehend the complexities of the world.

Second, nature is complex; man is more complex; social interactions are far more complex.

No mathematical equations or simple laws will ever help an individual to comprehend the thousands of interactions among the thousands of variability.

Third, we need to focus on the rare events. It has been proven that the rare events (for example, occurrences at the tails of probability functions) are the most catastrophic simply because very few are the researchers interested in investigating them: scientists are cozy with those well structured behaviors that answer collective behaviors.

Fourth impression is that I am a genius without realizing it.  Unfortunately, Kurt Gödel is the prime killjoy; he would have mock me on the ground that he mathematically demonstrated that any sentence I write is a lie.  How would I dare write anything?

Efficiency has limits within cultural bias; (Dec. 10, 2009)

Sciences that progressed so far have relied on mathematicians: many mathematical theories have proven to be efficacious in predicting, classifying, and explaining phenomena.

In general, fields of sciences that failed to interest mathematicians stagnated or were shelved for periods; maybe with the exception of psychology.

People wonder how a set of abstract symbols that are linked by precise game rules (called formal language) ends up predicting and explaining “reality” in many cases.

Biology has received recently a new invigorating shot: a few mathematicians got interested working for example on patterns of butterfly wings and mammalian furs using partial derivatives, but nothing of real value is expected to further interest in biology.

Economy, mainly for market equilibrium, applied methods adapted to dynamic systems, games, and topology. Computer sciences is catching up some interest.

Significant mathematics” or those theories that offer classes of invariant relative to operations, transformations, and relationship almost always find applications in the real world: they generate new methods and tools such as theories of group and functions of a complex variable.

For example, the theory of knot was connected to many applied domains because of its rich manipulation of “mathematical objects” (such as numbers, functions, or structures) that remain invariant when the knot is deformed.

What is the main activity of a modern mathematician?

First of all, they do systematic organization of classes of “mathematical objects” that are equivalent to transformations. For example, surfaces to a homeomorphisms or plastic transformation and invariant in deterministic transformations.

There are several philosophical groups within mathematicians

1. The Pythagorean mathematicians admit that natural numbers are the foundations of the material reality that is represented in geometric figures and forms. Their modern counterparts affirm that real physical structure (particles, fields, and space-time…) is identically mathematical. Math is the expression of reality and its symbolic language describes reality. 

2. The “empirical mathematicians” construct models of empirical (experimental) results. They know in advance that their theories are linked to real phenomena.

3. The “Platonist mathematicians” conceive the universe of their ideas and concepts as independent of the world of phenomena. At best, the sensed world is but a pale reflection of their ideas. Their ideas were not invented but are as real though not directly sensed or perceived. Thus, a priori harmony between the sensed world and their world of ideas is their guiding rod in discovering significant theories.

4. There is a newer group of mathematicians who are not worried to getting “dirty” by experimenting (analyses methods), crunching numbers, and adapting to new tools such as computer and performing surgery on geometric forms.

This new brand of mathematicians do not care to be limited within the “Greek” cultural bias of doing mathematics: they are ready to try the Babylonian and Egyptian cultural way of doing math by computation, pacing lands, and experimenting with various branches in mathematics (for example, Pelerman who proved the conjecture of Poincaré with “unorthodox” techniques and Gromov who gave geometry a new life and believe computer to be a great tool for theories that do not involve probability).

Explaining phenomena leads to generalization (reducing a diversity of phenomena, even in disparate fields of sciences, to a few fundamental principles).  Mathematics extend new concepts or strategies to resolving difficult problems that require collaboration of various branches in the discipline.

For example, the theory elaborated by Hermann Weyl in 1918 to unifying gravity and electromagnetism led to the theory of “jauge” (which is the cornerstone theory for quantum mechanics), though the initial theory failed to predict experimental results.

The cord and non-commutative geometry theories generated new horizons even before they verified empirical results. 

Axioms and propositions used in different branches of mathematics can be combined to developing new concepts of sets, numbers, or spaces.

Historically, mathematics was never “empirically neutral”: theories required significant work of translation and adaptation of the theories so that formal descriptions of phenomena are validated.

Thus, mathematical formalism was acquired by bits and pieces from the empirical world.  For example, the theory of general relativity was effective because it relied on the formal description of the invariant tensor calculus combined with the fundamental equation that is related to Poisson’s equations in classical potential

The same process of adaptation was applied to quantum mechanics that relied on algebra of operators combined with Hilbert’s theory of space and then the atomic spectrum.

In order to comprehend the efficiency of mathematics, it is important to master the production of mental representations such as ideas, concepts, images, analogies, and metaphors that are susceptible to lending rich invariant.

Thus, the discovery of the empirical world is done both ways:

First, the learning processes of the senses and

Second, the acquisition processes of mathematical modeling.

Mathematical activities are extensions to our perception power and written in a symbolic formal language.

Natural sciences grabbed the interest of mathematicians because they managed to extract invariant in natural phenomena. 

So far, mathematicians are wary to look into the invariant of the much complex human and social sciences. Maybe if they try to find analogies of invariant among the natural and human worlds then a great first incentive would enrich new theories applicable to fields of vast variability.

It appears that “significant mathematics” basically decodes how the brain perceives invariant in what the senses transmit to it as signals of the “real world”. For example, Stanislas Dehaene opened the way to comprehending how elementary mathematical capacity are generated from neuronal substrate.

I conjecture that, since individual experiences are what generate intuitive concepts, analogies, and various perspectives to viewing and solving problems, most of the useful mathematical theories were essentially founded on the vision and auditory perceptions. 

New breakthrough in significant theories will emerge when mathematicians start mining the processes of the brain of the other senses (they are far more than the regular six senses). Obviously, the interested mathematician must have witnessed his developed senses and experimented with them as hobbies to work on decoding their valuable processes in order to construct this seemingly “coherent world”.

A wide range of interesting discoveries on human capabilities can be uncovered if mathematicians dare direct scientists and experimenters to additional varieties of parameters and variables in cognitive processes and brain perception that their theories predict.

Mathematicians have to expand their horizon: the cultural bias to what is Greek has its limits.  It is time to take serious attempts at number crunching, complex computations, complex set of equations, and adapting to newer available tools.

Note: I am inclined to associate algebra to the deductive processes and generalization on the macro-level, while viewing analytic solutions in the realm of inferring by manipulating and controlling possibilities (singularities); it is sort of experimenting with rare events.


adonis49

adonis49

adonis49

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