Tag Archives: incompleteness theorem

Five common realities – facts en logic 9


Carla, Man and Narrator are walking around the Piazza di Santa Croce.

“I will come back on the synthesis between the world of the Upanishads and the Mahābhārata that Narrator has highlighted. Do you realise the difference between the “Thusness”-aspect and the “Concourse  of Things”-aspect?”, asks Man to Carla and Narrator.

“I have understood your explanation of this difference, but I wonder if in reality there is a difference between “thusness” and “concourse of things”. It seems to me that the “concourse of things” is another way of looking at “thusness”, or do I overlook something?”, says Carla.

“Carla may be right”, says Narrator.

“Carla is right. In my explanation, I have underlined the difference between both aspects to create a look on “Thusness” in two different ways. In the introduction to the “Awakening of Faith” initially the “Thusness”-aspect is indicated by “Thusness in essence (in emptiness and form)”; and “Saṃsāra – or Concourse of Things” is described as “Thusness in manifestations and characteristics”. After that, in the introduction the concepts of “Thusness”-aspect and “Concourse of Things”-aspect are both used to clarify both manifestations of “Thusness”. After reading this introduction I have fully realised what is meant by “evam” as first word – and also as summary – of all the Buddhist Sutras; “evam” includes everything, nothing is excluded”, says Man.

“Also with “evam” I have my usual question about the definition of this fundamental principle. When “evam” is finite, then Gödel’s second incompleteness theorem is applicable on “evam” [1]. But in case “evam” is infinite and all-encompassing, then nothing may exist outside “evam” to prove it or discuss “evam”; in case of infinity and all-encompassiveness, “evam” is by definition complete, because outside “evam” nothing exists. I will leave this question for the time being, because I think the answer is located in the inconceivable”, says Carla.

“It is an introduction to the “Awakening of Faith” and not an introduction to the “Awakening of Science”. The question of “evam” is a religious question; a question to the origin where people can fall back on when they don’t know or they cannot know. I think Carla is right; the answer probably lies in the inconceivable”, says Narrator.

“As stepping stone to “God in search of Man” – the book title includes my first name –, I use the book “Ich und Du” (“I and You”) from 1923 by the religious philosopher Martin Buber [2] (honorary professor in Frankfurt am Main) who in 1938 had escaped from the other regime in Germany by fleeing to Jerusalem. “In the beginning is the relation [3]” according to Martin Buber. Man can only say “I” due to “you” (or “it”), the relationship with others (and things) is dialogical. “I” and “you” are not separate objects or things; there is no “I” without “you”, there exists only a reciprocal relationship to one another. By interpreting this religiously – “In every You, we call the infinite all-encompassing [4]” – the relation with God is dialogic: in the all-encompassiveness we cannot describe God, but we can only appeal; our live is an existential dialog with the infinite all-encompassing “You”. Science together with religion offers no doctrine according to Martin Buber, but wisdom.

feiten en logica 91[5]

I read an example of this wisdom founded in science and religion on the backside of the book “God in search of Man” where the words of Baäl Sjem are mentioned as guidance:

If a man has seen evil, may he make no fuss.

Let he be aware of his own evil, and work hard to avoid it.

Because what he has seen, is also inside him.

Within the framework of the “Awakening of Faith” the words by Baäl Shem are crystal clear; all the good and the evil is – just like you and me – included in “evam” or “Thusness”. All the good and the evil is within us. Would Martin Buber see good and evil as manifestations of “Ich und Du“, as dialogical relationship between me and God, or would he place good and evil into his second dialogical relationship “Ich und Es” (“I and It”) ? I do not know; I’ll leave this question for the time being until we will arrive at God in the shape of a human being during my introduction.

When reading the first chapters of “God in search of Man” – in Sanskrit “Man” means amongst others “to think/consider/observe” – I was struck by the similarities in structure with the “Awakening of Faith”. Abraham Joshua Heschel had chosen the following three ways for this quest for God:

  • God – for Abraham Joshua Heschel this is the unspeakable all-encompassing One from the “Awakening of Faith”.
  • Revelation (unveiling or disclosure)
  • Resonance (respons)

The last two ways for the quest for God show similarities with “evam” where the “revelation” looks like “Thusness in essence (in emptiness and form)” during the transition to “Thusness in manifestations and characteristics” that looks like “resonance”. Is this coincidence or is this a fundamental resemblance with the “Awakening of Faith” of man?”, says Man.

“I think there is a fundamental difference between your introduction to the “Awakening of Faith” on one hand, and the reciprocal relationship between I and you by Martin Buber and the quest of God to human beings on the other hand. In Hua-yen Buddhism there is in principle no other, because all appearances and illusions arise from and are interwoven with “One”. Martin Buber and Abraham Joshua Heschel search and/or experience a dialogue with an everlasting all-encompassing You: there is a certain separation between I and You. This is similar to one key question on our quest: “Are you and I connected or are we separated“. I don’t know the answer, but it seems necessary to investigate this question more in depth”, says Carla.

“Maybe both ways of seeing are two manifestations of one and the same within Indra’s Net. Shall we first visit the inside of the Basilica di Santa Croce. Inside is the crucifix that was seriously damaged during the flood in 1996. This may offer a transition to God in the shape of a human being in our world”, says Narrator.

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[1] See for a simplified explanation of the evidence of this second incompleteness theorem: Nārāyana, Narrator, “Carla Drift – An Outlier, A Biography”. Amsterdam: Omnia – Amsterdam Publisher, 2012, p. 154

[2] See also: http://nl.wikipedia.org/wiki/Martin_Buber

[3] Source: Buber, Martin, Ik en Jij. Utrecht: Erven J. Bijleveld, 2010 p. 24; see also the first sentence in the Gospel of John.

[4] Source: Buber, Martin, Ik en Jij. Utrecht: Erven J. Bijleveld, 2010 p. 110, 111

[5] Source image: http://en.wikipedia.org/wiki/Martin_Buber

[6] Source image: http://fr.wikipedia.org/wiki/Basilique_Santa_Croce_de_Florence

Five common realities – facts and logic 7


Carla, Man and Narrator have visited the inside of the Baptisterium San Giovanni and they are now standing outside for the closed Eastern door of the Baptisterium.

“This is according to Michelangelo the “Porta del Paradiso” or the gateway to paradise. On the top panel in the left door, “Adam and Eve in paradise”, “the fall” and “the expulsion from paradise” are shown. Paradise and the fall are beautiful metaphors for the human illusion of the paradisiacal possibility to an all-encompassing knowledge of the organised chaos. After Gödel had eaten from the apple of wisdom with the proof of the two incompleteness theorems – whereafter the illusion of omniscience of “facts and logic” was basically unreachable forever – the scientific world knew that humanity has forever no access to the paradise of omniscience.

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It is absolutely right that the “Porta del Paradiso” is closed and there is a fence before this gate”, says Carla.

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“I have read in my city guide that both doors are copies of the original doors that were severely damaged by the ravages of time and the flood of the river Arno [3] on November 6, 1966. Because of this flood, several panels were torn from the Porta del Paradiso [4]. I think the flood of the river Arno was also a manifestation of organized chaos”, says Man.

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“Certainly. The river has exceeded the banks to a nearly equal height in 1333 and in 1557 AD. Perhaps this is the reason that the living quarters in the Florentine palaces are located on the first floor.

[6]

OLYMPUS DIGITAL CAMERA[7]

This flood is an outstanding example of the organised chaos. Within reasonable certainty it can be stated, that once every few hundred years a similar flood can occur in Florence, like – within certain limits – all other water levels in the river can manifest themselves at any given time with a certain probability – this is the ordered nature of the organised chaos. But no one can predict on which day in the future a similar flood as in 1333, 1557 and 1966 AD will happen – this is the chaotic nature of the organised chaos”, says Carla.

“Good example. Yesterday I was looking for more information about Gödel. I read Gödel’s ontological proof of God. I have a copy of this proof for you”, says Narrator.

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“I can rarely accept a first definition without questioning. During my study at Delft University of Technology I had always asked my teachers where the first definition was based on. Initially teachers could give a limited answer to my question, but by asking further questions the answer always came down to the platitude: “We must start somewhere“. This answer remained unsatisfactory for me, I was actually always asking for an answer to “Why” while at best teachers could give an answer to “How – within a certain context”. Looking back, my change from my study of Applied Physics to the subject Humanities can be traced to the fact that Applied Physics has no answers available to why facts and logic manifest itself in a certain manner to us. Coming back to the first definition in Gödel’s ontological argument: why has God-like only positive properties and no negative characteristics or imaginary properties – as imaginary numbers in electrical engineering [9]? Based on the lack of an answer to my last question and based on Gödel’s first incompleteness theorem, Gödel’s ontological proof cannot contain a complete system – or a complete description of God-like. In the case Gödel’s ontological proof would contain an all-encompassing system – which is not the case in my opinion  – the consistency of the axioms cannot be proven from within its own system according to Gödel’s second incompleteness theorem. For completeness, I must say that my statements are based on scientific logic – and not on religious principles [10]. I think Gödel’s ontological proof is only a first exercise of Gödel in the field of religion and no more”, says Carla.

“I think you’re right”, says Narrator.

“Sounds convincing. During lunch I would like to hear your opinion about facts and logic of the look on God as described by Abraham Joshua Heschel in “God in Search of Man” [11] and the two aspects of “One” in the “Commentary on the Awakening of Faith” by Fa-Tsang [12], says Man.

“This fits nicely with Gödel’s ontological proof. My introduction to the mind of the warrior can wait for a while. Shall we look for a place for our lunch? ”, says Carla.

“I know a nice place in the Piazza di Santa Croche”, says Narrator.


[1] Source image: http://en.wikipedia.org/wiki/Florence_Baptistery

[2] Source image: http://nl.wikipedia.org/wiki/Baptisterium_(Florence)

[3] See also: http://en.wikipedia.org/wiki/1966_Flood_of_the_Arno_River and http://it.wikipedia.org/wiki/Alluvione_di_Firenze_del_4_novembre_1966

[4] Source image: http://nl.wikipedia.org/wiki/Baptisterium_(Florence)

[5] Source image:http://it.wikipedia.org/wiki/Alluvione_di_Firenze_del_4_novembre_1966

[6] Source image: http://de.wikipedia.org/wiki/%C3%9Cberschwemmung_in_Florenz_1966

[7] Source image: http://en.wikipedia.org/wiki/1966_Flood_of_the_Arno_River

[8] Source image/text: http://en.wikipedia.org/wiki/G%C3%B6del’s_ontological_proof

[9] Calculations of electronic circuits are considerably simplified by use of imaginary numbers with a real value equal to zero. See also: http://en.wikipedia.org/wiki/Imaginary_number

[10] See also the statement made by Prof. Dr. W. Luijpen on the scope of science in relation to religion referred to in the post “Five common realities – facts and logic 6”

[11] See also: Heschel, Abraham Joshua, God in Search of Man: A Philosophy of Judaism. New York: Farrar, Straus and Giroux, 1976 (Reprint – original published in 1955).

[12] See also: Vorenkamp, Dirck, An English Translation of Fa-Tsang’s Commentary on the Awakening of Faith. New York: The Edwin Mellen Press. 2004

Five common realities – facts and logic 6


The next morning Carla, Man and Narrator drink a cup of coffee on a terrace in front of the Baptisterium San Giovanni [1] opposite the Basilica di Santa Maria del Fiore in Florence.

801px-Baptistry_Florence_Apr_2008[2]

“This morning I have thought about your introduction to the development of science. It is an impressive summary in brevity and in depth. At one point I would wish to make an addition; in my opinion human science had started much earlier than the moment people started giving meaning and perceiving meaning within rituals to increase survival. I think human science had started at the first conscious creative act during giving meaning and taking meaning to sound, to feeling, during love and raising children. Do you agree?”, Narrator asks Carla.

“Fully. Besides I have also ignored the difference between atomism [3] – wherein all facts are based on the smallest possible basic elements, particularism [4] – wherein facts and logic serve to promote our own interests above (and, if necessary, at the expense of) the interests of others, pluralism [5] – wherein different systems of facts and logic co-exist within a certain balance, and holism [6] – wherein facts and logic form a coherent whole. I hope to be able to include particularism in an introduction on the mind-set of the warrior. Also, I have ignored the many wrong representations of facts and the fallacies to serve a certain interest. Shall I now proceed with the orderly chaos – both fragmented and universal?”, says Carla.

“I am curious about your remark “both fragmented and universal”. Last night I had read in a different context a chapter on the coherence of both subject”, says Man.

“Please add information where you see fit. Actually from the beginning of science, people have usually tried to create an order out of chaos by looking at reality as an ideal. Facts that didn’t fit within the ideal frame of mind – such as friction, air resistance and unwelcome religions and cultures – were neglected as being irrelevant, or were fiercely contested during religious wars. By the end of the third scientific revolution, scientists thought that only a few obstacles within the basic elements of scientific knowledge had to be overcome, such as knowledge about the transfer of the gravity and the nature of light, before the paradise of the omniscience could be accessed within which everything would be known and may be explained by the application of the basic elements. The East doors of the Baptistry – or Porta del Paradiso [7] – are a beautiful metaphor for this way of thinking.

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With knowledge of these panels and of the interrelationship between these panels, scientists had thought to obtain the doors the heavenly omniscience.

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The organised chaos – fragmented and universal – prevented access to the paradisiacal omniscience with unambiguous repeatable predictability of facts in our world.

With “organised chaos” I mean that within certain limits every possible fact has a certain chance to manifest itself at a certain moment. As example I take a grazing cow in a delimited field with just enough food for the cow: the cow will graze within certain limits (the field and the edge that the cow can just reach with her mouth); each clump of grass in the field has a certain chance to be eaten at a certain moment; a butterfly – flying-by – can change the grazing of the cow after which the cow wil use another grazing pattern; this other grazing pattern has no influence on overall grazing of the total field in the long run, but it makes a huge difference for the life of several clumps of grass in the proximity of the cow [10].

With “fragmented and universal” I try to explain that the fragmented organised chaos manifests itself within a particular environment – such as the delimited field for the cow and such as the grasshopper in a matchbox [11] – and that the universal organised chaos takes place within the overall universe. “Fragmented and universal” relate to each other as the clouds, waves and ocean to the universal organised interplay and chaos within the total universe. The clouds, waves and the ocean are manifestations of the universal organised chaos, such as the weather – in the short term with a good predictability up to 4 days, and with good predictability over the long term – is also a manifestation of the universal organised chaos in space.

feiten en logica 64[12]

With “unambiguous repeatable predictability of facts” I mean that the course of facts has the same outcome at identical starting positions. In everyday life, there is rarely an identical starting position, so a “uniform repeatable predictability of facts” arises rarely. Well there are many starting positions with similar characteristics: these situations often show a similar predictability of facts, but with minor differences in the starting positions the progress of the facts can show an organised chaotic behaviour in specific situations.

Under ideal conditions the constant of Heisenberg [13] limits accuracy of the determination of the starting position and of observations within the quantum mechanics [14].

In 1931 Gödel has published [15] the formal proof of the two incompleteness theorems [16]:

  • If a system – for example, a system (or grasshopper) in a matchbox – is consistent, then the system cannot be complete.
  • The consistency of the axioms cannot be proven from within the own system.

The combination of the “organized chaos”, the “limitation of observation within quantum mechanics” and “observations that – within the theory of relativity – depend on the way of observing” limit the “unambiguous repeatable predictability of facts”. The consistency of logic is seriously limited by the Gödel’s second incompleteness theorem.

By both restrictions, the ambition of the third revolution in science to unambiguously know and describe our world, was basically stalled. Part of the logic took distance from the organised chaos of daily life: this intuitionistic [17] logic only focused itself on symbol. Another part of the logic connected symbols with assumptions of reality as an extension of the intuitionistic logic – this extension was also called superintuitionistic logic. Classical logic was regarded in the superintuitionistic logic as the most strongly coherent system, which was seen as an intermediary – or intermediate [18] – between classical and intuitionistic logic. In a detour within another framework – symbols instead of rituals – the impact of the first revolution in the scientific development of mankind is demonstrated once again.

This is my introduction to the organised chaos; I hope you were able to follow me”, says Carla.

“Impressive in all respects. If I understand you correctly, then according to Gödel’s second incompleteness theorem the cohesion within a holistic system cannot be proven from its own system. This means that the cohesion of the “All-encompassing One” cannot be shown within itself”, says Man.

“That’s right within the conceptual framework of Gödel. I should add that Gödel proved his incompleteness theorems in a mathematical manner, using symbols that do not necessarily have an interpretation within our daily life. Some technical scientists only recognise symbols and ideal conditions as pure science. In the late 1970s, Prof. Dr. W. Luijpen – professor of philosophy of science at Delft University of Technology – had made the following statement during his lecturers: “Solely recognising symbols and ideal scientific conditions as the only scientific reality is a religious statement. Religion is not the area of expertise of technical scientists: this recognition is, until now, no scientific statement“. In line with Popper and Kuhn I do not rule out that outside mathematics – with its world of symbols – the two incompleteness theorem of Gödel may not be applicable under certain circumstances”, says Carla.

“Fascinating thoughts. Let us visit the Baptisterium San Giovanni. I suggest to come back to your introduction later on”, says Narrator.

“That’s good”, say Carla and Man.


[1] See also: http://en.wikipedia.org/wiki/Florence

[2] Source image: http://en.wikipedia.org/wiki/Florence_Baptistery

[3] See also: http://en.wikipedia.org/wiki/Atomism

[4] See also: http://en.wikipedia.org/wiki/Political_particularism

[5] See also: http://en.wikipedia.org/wiki/Cultural_pluralism

[6] See also: http://en.wikipedia.org/wiki/Holism

[7] See also: http://en.wikipedia.org/wiki/Florence_Baptistery

[8] Source image: http://nl.wikipedia.org/wiki/Baptisterium_(Florence)

[9] Source image: http://nl.wikipedia.org/wiki/Baptisterium_(Florence)

[10] Source metaphor: Stewart, Ian, Does God Play Dice? London: Penguin Books, 1992², p. 132

[11] See also: Nārāyana, Narrator, Carla Drift – An Outlier, A Biography. Omnia – Amsterdam Publisher, 2012, p. 15 en p. 151 – 156

[12] Source image: http://en.wikipedia.org/wiki/Ocean

[13] See also: http://en.wikipedia.org/wiki/Heisenberg_uncertainty_principle

[14] See also: http://en.wikipedia.org/wiki/Quantum_mechanics

[15] Zie ook: http://en.wikipedia.org/wiki/Kurt_G%C3%B6del

[16] Zie ook: Nārāyana, Narrator, Carla Drift – An Outlier, A Biography. Omnia – Amsterdam Publisher, 2012, p. 154

[17] Zie ook: http://en.wikipedia.org/wiki/Intuitionism and http://en.wikipedia.org/wiki/Intermediate_logic

[18] Zie ook: http://en.wikipedia.org/wiki/Intermediate_logic

Carla Drift – Looking back at my innocence


Before we started the quest for “Who are you”, I told Narrator in a few words about my youth – the years of my innocence.

“There was once a girl that was so clever that everywhere she was an outlier. She surpassed all the knowledge of her environment. This girl was so wise to show this special gift to nobody. Very soon she discovered that this gift completely confused her environment. Now and then she showed a glimpse whereof she thought.

In elementary school children learned to add, multiply and divide. This girl already calculated in the infinite or in the uncountable as she called it. Countable was all that fitted within a box of the “knowable”. Hereby she thought about the matchbox in which she formerly had caught a grasshopper.

[1]

When the class learned to count until ten, the content of the Matchbox was ten for her. For the class innumerable was at that time “ten plus one”. When the class learned to count until one hundred, from then on countable was one hundred; “hundred and one” was innumerable and so on as far as the classmates could count.

The countable and therefore the knowable grew along with the knowledge of the classroom and the innumerable became bigger and bigger. This girl learned that the countable – so the content of the matchbox “L” – changed along the changes of the environment. The uncountable was then still “L+1”. This girl started to add the countable, so when for the class L was equal to ten, the girl decided to place ten matchboxes in a row: for her “10 times L” was equal to one hundred; infinite was then ten matchboxes plus 1. She placed hundred matchboxes in a row and “hundred times 10” or “1000” was countable and infinite was “hundred boxes plus 1”. She did the same with boxes that were getting smaller like Russian dolls. Infinitesimal was one size smaller than the smallest knowable.

[2]

And zero was an empty table without any box or doll. She wrote this as “O”. This was very easy for her.

For simplicity, this girl decided to write the infinite as “L + 1”; This was equal to the largest box plus one or the greatest number of knowable boxes plus 1.

Now this girl was so far that she saw infinitely – or L + 1 – as a matchbox of all knowable plus one. She began in the first class of primary school to calculate with the infinite, which was also an outlier that fell outside the knowable. For infinite the same rules applied, but it the infinite was still outside the knowable of the others: in this way she remained in touch with arithmetic lessons of her classmates. The ordinary multiplication tables were applicable for the infinite and normal division rules applied to the division of the infinite – a piece of cake. Increases the knowable and the infinite is just slightly larger; decrease the smallest knowable and the infinitesimal small is just slightly smaller.

According to her the infinite or L + 1 was the evidence for the existence of God on the Catholic primary school. God could adopt all dimensions depending on the circumstances required, but God himself was larger than the knowable so he remained all encompassing. If the changes increased rapidly, God also increased quickly and vice versa. And because God was all encompassing or L + 1, God took the required form immediately. In this way the girl derived and integrated in the second class of elementary school. The most beautiful thing was that God was no foreigner, he was also an outlier just like her. God made woman and man (as knowable) like his image – also the outliers like her were created like his image. She made the knowable slightly larger because she was an outlier. Later she adjusted her view on God.

In the second class of elementary school she read in a book from the library – that was smuggled through her father – about primes. she decided to look at primes as matchboxes for calculation purposes. According to her new calculation method the core numbers were L, 2L, 3l, 5l, 7l, 11l, 13L, 17L, 19L and so on as primes. With these primes all known matchboxes could be created [3].

In the fourth grade of elementary school she saw in the library at the Department of mathematics a book on Gödel. In this book she read Gödel’s two incompleteness theorems [4]. She borrowed this book via her father. By naming L + 1 she already knew the first incompleteness theorem and with her new calculation method – whereby she used the core numbers L, 2 L, 3 l, 5 l, 7 l, 17 l, 11L, 13L, 19L according to the sequence of primes – she saw immediately the second incompleteness theorem; we can never prove the whole arithmetic L because there will be always a L + 1. This evidence is a piece of cake.

She purposely made a few mistakes in long divisions [5] in order to appear normal.

In the fifth and sixth class of primary school a new schoolmaster let her read the book “Cosmic View, The Universe in 40 Jumps” by Kees Boeke. With her father she studied astronomy and microscopy. She calculated the Kepler orbits on her own. In a course mechanics within theoretical physics [6] at the University of Technology in Delft, she saw these calculations again. One of the two authors was an outlier [7] in the field of mathematics and physics.”

[8]

After this brief description of my years of innocence in elementary school, Narrator and I decided to start the quest “Who are you” together. During the preparations we invited Man Leben – after the death of his second life companion – to go along. He accepted the invitation “With hope and consolation”.


[1] Source image: http://en.wikipedia.org/wiki/Match

[2] Source image: http://fr.wikipedia.org/wiki/Fichier:Floral_matryoshka_set_1.JPG

[3] See also: http://en.wikipedia.org/wiki/Prime_number

[4] See: http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

[5] See also: http://en.wikipedia.org/wiki/Long_division

[6] See: http://en.wikipedia.org/wiki/Course_of_Theoretical_Physics

[7] See also: http://en.wikipedia.org/wiki/Lev_Landau

[8] Source image: http://en.wikipedia.org/wiki/Course_of_Theoretical_Physics