Adonis Diaries

Posts Tagged ‘John von Neumann

Incomplete: Simplify (Einstein, Godel, Turing, Chaitin…)

One thing we know is that life reinforces the hypothesis that the world is infinitely complex and most of its phenomena will remain incomprehensible, meaning unexplained.

For example, no theory of life evolution was able to predict the next phase in evolution and the route taken to the next phase. The reason we have difficulty discovering how living organism adapt to the environment to survive, in longer term.

We don’t know if laws in biology will exist in the same meaning of laws of physics or natural phenomena.

For example, is the universe simple or complex, finite or infinite?

The mathematician Chaitin answered: “This question will remain without any resolution, simply because we need an external observer outside our system of reference, preferably non-human, to corroborate our theoretical perception.”

(A few of my readers will say: “This smack of philosophy” and they hate philosophy or the rational logic deducted from reduced propositions that cannot rationally be proven)

So many scholars wanted to believe that “God does not play dice” (Einstein) or that chaos is within the predictive laws of God and nature (Leibniz), or that the universe can be explained by simple, restricted set of axioms, non-redundant rules (Stephen Hawking).

Modern mathematical theories and physical observations are demonstrating that most phenomena are basically behaving haphazardly.

For example, quantum physics reveals that hazard is the fundamental principle in the universe of the very tiny particles:  Individual behaviors of small particles in the atomic nucleus are unpredictable.  Thus, there is no way of measuring accurately speed, location, and direction of a particle simultaneously; all that physics can do is assigning probability numbers.

Apparently, hazard plays a role even in mathematics.

For example, many mathematical “true” statesmans cannot be demonstrated, they are logically irreducible and incomprehensible.

Mathematicians know that there exists an infinity of “twin” prime numbers (odd number followed by even number) but this knowledge cannot be proven mathematically.

Thus, many mathematicians would suggest to add these true “propositions” but non demonstrable theories to the basic set of axioms.

Axioms are a set of the bare minimum of “given propositions” that we think we know to be true, but the reason is unable to approach them adequately, using the logical processes.

Einstein said: “What is amazing is that the eternally incomprehensible in nature is comprehensible”; meaning that we always think that we can extend an explanation to a phenomenon without being able to proving its working behaviors.

Einstein wrote that to comprehend means to rationally explain by compressing the basic axioms so that our mind can understand the facts; even if we are never sure how the phenomenon behaves.

For example, Plato said that the universe is comprehensible simply because it looks structured by the beauty of geometric constructs, the regularity of the tonality in string instruments, and steady movement of planets…

Steven Weinberg admits that “If we manage to explain the universal phenomenon of nature it will not be feasible by just simple laws.” (I agree with Weinberg in that statement. Consequently, comprehension will be limited to the few scientists who can handle and visualize complex equations)

Many facts can be comprehended when they are explained by a restricted set of theoretical affirmations:  This is called the Occam Razor theory which says: “The best theory or explanation is the simplest.”

The mathematician Hermann Weyl explained: “We first need to confirm that nature is regulated by simple mathematical laws.  Then, the fundamental relationships become simpler the further we fine-tune the elements, and the better the explication of facts is more exact.”

So what is theory?

Informatics extended another perspective for defining theory: “a theory is a computer program designed to take account of observed facts by computation.  Thus, the program is designed to predict observations.  If we say that we comprehend a phenomenon then, we should be able to program its behavior.  The smaller the program (more elegant) the better the theory is comprehended.”

When we say “I can explain” we mean that “I compressed a complex phenomenon into simple programs that “I can comprehend”, that human mind can comprehend. 

Basically, explaining and comprehending is of an anthropic nature, within the dimension of human mental capabilities.

The father of information theory, John von Neumann wrote: “Theoretical physics mainly categorizes phenomena and tries to find links among the categories; it does not explain phenomena.

In 1931, mathematician Kurt Godel adopted a mental operation consisting of indexing lists of all kinds of assertions.

His formal mathematical method demonstrated that there are true propositions that cannot be demonstrated, called “logically incomplete problems

The significance of Godel’s theory is that it is impossible to account for elemental arithmetic operations (addition or multiplication) by reducing its results from a few basic axioms.  With any given set of logical rules, except for the most simple, there will always be statements that are undecidable, meaning that they cannot be proven or disproven due to the inevitable self-reference nature of any logical systems.

The theorem indicates that there is no grand mathematical system capable of proving or disproving all statements.

An undecidable statement can be thought of as a mathematical form of a statement like “What I just said is a lie”:  The statement makes reference to the language being used to describe it, it cannot be known whether the statement is true or not.

However, an undecidable statement does not need to be explicitly self-reference to be undecidable. The main conclusion of Gödel’s incompleteness theorems is that all logical systems will have statements that cannot be proven or disproven; therefore, all logical systems must be “incomplete.”

The philosophical implications of these theorems are widespread.

The set suggests that in physics, a “theory of everything” may be impossible, as no set of rules can explain every possible event or outcome. It also indicates that logically, “proof” is a weaker concept than “true”.

Such a concept is unsettling for scientists because it means there will always be things that, despite being true, cannot be proven to be true. Since this set of theorems also applies to computers, it also means that our own minds are incomplete and that there are some ideas we can never know, including whether our own minds are consistent (i.e. our reasoning contains no incorrect contradictions).

The second of Gödel’s incompleteness theorems states that no consistent system can prove its own consistency, meaning that no sane mind can prove its own sanity.

Also, since that same law states that any system able to prove its consistency to itself must be inconsistent, any mind that believes it can prove its own sanity is, therefore, insane.

Alan Turing used a deeper twist to Godel’s results.

In 1936, Turing indexed lists of programs designed to compute real numbers from zero to 1 (think probability real numbers).  Turing demonstrated mathematically that no infallible computational procedures (algorithms) exist that permit to decide whether a mathematical theorem is true or false.

In a sense, there can be no algorithm able to know if a computer program will even stop.

Consequently, no computer program can predict that another program will ever stop computing.  All that can be done is allocating a probability number that the program might stop.  Thus, you can play around with all kinds of axioms, but no sets can deduce that a program will end.  Turing proved the existence of non computable numbers.

Note 1: Chaitin considered the set of all possible programs; he played dice for each bit in the program (0 or 1, true or false) and allocated a probability number for each program that it might end.  The probability that a program will end in a finite number of steps is called Omega.  The succession of numbers comprising Omega are haphazard and thus, no simple set of axioms can deduce the exact number.  Thus, while Omega is defined mathematically, the succession of the numbers in Omega has absolutely no structure.  For example we can write algorithm to compute Pi but never for Omega.

Note 2:  Bertrand Russell (1872-1970) tried to rediscover the founding blocks of mathematics “the royal highway to truth”.  He was disappointed and wrote: “Mathematics is infected of non proven postulates and infested with cyclic definitions.  The beauty and the terror of mathematics is that a proof must be found; even if it proves that a theory cannot e be proven”

Note 3:  The French mathematician Poincaré got a prize for supposedly having discovered chaos.  The article was officially published when Poincaré realized that he made a serious error that disproved his original contention.  Poincaré had to pay for all the published articles and for destroying them.  A single copy was saved and found at the Mittag-Leffler Institute in Stockholm.

Ironing out a few chaotic glitches; (Dec. 5, 2009)

              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 found 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 natures 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 Herman 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; Murry Gell-Mann used “group theory” to predicting 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 Wolfran 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 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.

            My fourth impression is that I am a genius without realizing it.  Unfortunately Kurt Gödel is the prime kill joy; 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?




October 2020

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