Stephen I. Ternyik:
The Epistemological Heritage of Erdös Pal: N is more than a Number.


The mathematical talent of Paul Erdös, in search for the hidden Proofs from The Big Book, teaches us about the soulful world journey of the scientific mind towards patterns of eternity. This magic of numbers reveals that N is more than a number and that one is the beginning of all human reasoning about natural or physical reality. Creative tools (numbers, letters) of the human spirit have shaped our develop-mental history and help us to detect the concealed vibrations and frequencies of the universe we live in via the perfection of cultural techniques.

Key terms: one; mathetics; proofs; The Big Book; eternal patterns; unified force.

Írta:   Stephen I. Ternyik

Magan valalkozo/tudos (1985 ota)

It is now well known that we live in the technological age of electronic social media, communication networks and automated information-processing; free informatics design, open source and knowledge working are definitely future trends of human productivity. Evolutionary and anthropological limits of humankind cannot be erased by these technical motions, but human communicative culture will be changed forever.

Concerning these future events, Paul Erdös/Erdös Pal (1913-1996) was a brilliant trendsetter or role model; this mathematical genius came out of a Hungarian group of scientific researchers (e.g. Polya G/1887-1985; Neumann Jv/1903-1957; Turan P/1910-1976) who were all ‘spiritual children’ of Fejer Lipot (1880-1959). Behind this kind of mathematical science worked the hidden paradigm that the blueprint of reality is concealed in numbers and can be revealed by the scientific soul/mind of the logical/empirical researcher.

At that time, a ‘perfected’ scientific infrastructure for free mathematical research did exist already in Hungary, for example regular Journals with price questions, nationwide contests and loose formation of study groups at the university level where mentors/topics could be chosen via voluntary choice. Of course, these were not yeshivas or laboratories, but communicative study partnerships (‘give me a study partner or death’, an old Hebrew saying).

The cosmopolitan spirit of Paul Erdös, dedicated to empirical mathematical ‘revelation’, made him a role model of the future scientist; his permanent research into the Proofs from The Book and deep need for humane communication are the motivational driving factors of his scientific life. Despite all horrific setbacks and experiences, he found a way to mystically survive by inquiring into universal unity (oneness) for aesthetic reasons and intellectual beauty, i.e. his mathematical mind tried to grasp the spiritual essence of the physical order of this world, we may call it ‘mathematical redemption’ or ‘intuitive reconciliation’ of the human mind.

For a deeply religious person of ethical monotheist faith this looks like a real and Ersatz human link towards the creative divine order of the eternal upper force; however, science is factually the psychological metamorphosis of religion and philosophy at another stage of human development, i.e. the human mind tries to better and more exactly understand the deeper layers (laws/principles) of the natural process order or: G-d has the Big Book, the beautiful proofs of mathematical theorems are listed here ( a saying attributed to Paul Erdös).

Among the  quantitative record of 1525 mathematical articles from Erdös are 511 research papers with direct co-authors; he visited these colleagues constantly throughout the world, lived out of a suitcase, had very few possessions, donated his scarce money to needy researchers and ‘turned coffee into theorems’ (A.Renyi). Erdös’ scientific network had no hierarchies, was not organized, of informal nature, and he was the central knot of all hubs and connections. It was surely not a random network, but a probability network of potential communicators where knowledge working was focused on creative problem-solving. Human and scientific networks of this kind are a social phenomenon and economically very efficient: A) the costs of knowledge ‘transactions’ are very low, and B) they work better without a formal structure, in legal and monetary terms. Motivation and creativity are more valued than mechanical efficiency, i.e. it is quite the opposite inter-action model of the current economization and commercialization of academia where intellectual property rights have become somehow paramount. Such an open source model is based on brilliant ideas (ideals), factual or empirical performance (experience) and real problem-solving (results); it resembles as communicative ensemble a Jazz band (create and reset) and not a labor hierarchy (command and control), i.e. human inter-action follows the quality principle of optional growth, in quantitative terms: the ethical imperative is to reduce complexity via small steps, circular feedbacks and process strategy, i.e. a variable dependency of independent elements via a specific temporal content is at work, e.g. only variety can create variety.

The Erdös collaboration technique of scientific communication is a prototype of Jewish epistemology or learning process how Jews gain knowledge; a much solidified body of knowledge and skills is slowly being extended by thematic human communication and permanent exchange. Not an academic homo clausus is the ideal of knowing, but a serious knowledge worker who socially co-operates, shares and cares about the scientific results of the research discipline; professional and biographical background are minors and the topic of investigation becomes the major priority. In this case, the magic of numbers (not Hebrew letters which are actually also numbers, according to the religious science of Kabballah) in space-time and time-space drives the permanent and perennial re-search for the eternal patterns of universal force as a temporal moment of being here in this world. From this point of view, mathematical genealogy is a scientific approach to decipher the historical journeys of the human mind via the discovery route of the natural working order, i.e. it is mathetics, the science of learning. In addition, Erdös was very good at didactics and promoted a lot of mathematical talents (e.g. B.Bollobas, J.Kruskal, L.Lovasz, and T.Tao).

Travelling with a Hungarian passport, being officially affiliated with the Hebrew University of Jerusalem and the Hungarian Academy of Sciences, researching at several prominent American and British universities, and visiting scientific conferences globally ( in 25 countries), Paul Erdös was a true world citizen, incorporating humane science and peace in times of the cold war.

Approaching a first conclusion:

The essentially Jewish epistemology of Erdös made him an original cosmopolitan knowledge worker in the research field of mathematical science who was well ahead of today’s open source, social media networks and communication culture. His life-long re-search into the concealed number magic of Proofs from the Big Book reveals a real and Ersatz human link to the eternal upper and unified force of all proofs.

We could stop here and point to reference resources, conferences, scientific problems and research methodologies of modern contemporary mathematics as related to Paul Erdös, i.e. the pure intellectual legacy of a continuous searcher for elegant proof methods. Of course, we have also to do some homework and to understand the underlying math and the abstract scientific value of the Erdös methodical problem-solving strategy which was a social product of really applied creativity and human ingenuity. In other words, the mathematical world that average inhabitants of this planet normally experience is an earthly mixture of daily accounting techniques, general school memories and continuous number applications; this was definitely not the Erdös world. However, infinite calculus and continuous analysis, basic techno-logical tools of engineering, computation and other applied natural sciences, were also not his ballpark or -if you want- cognitive playground.

The discrete or finite world that Paul Erdös mathematically explored (inhabited) was the scientific foundation of the digital age and the majority of research contributions, concerning sets and graphs, were basic inquiries into the computability of theorems via intuitive proofs. As a scientific consequence, this intuitive work developed fundamental computational patterns of human exact thought for the technical age of automated information-processing or informatics as we know it today. So, this is not the formal mathematics of the modern technician, engineer or applied scientist, but the profound exploration of the deeper layers of structural and functional reality patterns of this physical world, working by energy and matter in discrete terms.

The research into N is definitely more than a scientific exploration of number symbols; it is the quest for discrete patterns of energy and the information flow. As a result of this spiritual way of thinking, mathematics is a science of the human mind as related to rationality; rationality is consequently about efficient human reasoning while morality is about ethical human action, both systems of reasoning fall under the categorical dichotomy of false or right. The globalization of mathematical thought (since about 1900 )of which Paul Erdös was a vital and pioneering part, has to be better and deeper understood, and not only for the explanation of the many successful applications in modern (quantum) physics. It was a decisive historical process and the most important single elements (sparks) for grasping modern complexity methodically were such spirits or human minds like Erdös who was dynamically inter-connecting to any possible willing source of mathematical inter-action, i.e. the mutual continuity and dependency of this spiritual work in math is an original chain of human ideas, dating 4000 years back to the Abrahamic times of Sumer and Mesopotamia, and it was translated via discrete mathematical thought into computational modernity.

As we want to deeper and better understand the epistemological and ontological heritage of Paul Erdös, concerning Hungarian Jewry, scientific ingenuity and  cosmopolitan world culture, it is crucial to delve into the biographical and professional details of the Erdös cosmos; this will be on -one side- social psychology in-depth ( as related to critical life events and inter-action patterns) and on the -other side- a fresh look into mathematics as globalized discipline (after having coped successfully with a basic cognitive crisis, in terms of scientific and technological history, i.e. formalism vs. intuitivism).

If both of your parents are mathematicians, physicists and teachers, the probability of getting professionally also involved in one of the above mentioned vocational directions is well above 80%; however, this is no proof for mathematical talent and a matter of circumstances: Intellectual ‘capital’ can get lost as life goes on and many talented people walked into lethal detours. The deeper look into the biographical process of Paul Erdös reveals a very protective behavior from side of his parents, an early detected strong mathematical talent and an upbringing in a rational culture of free thought; his intuitive scientific formation was via home-schooling, KöMal (Közepiskolai Matematikai es Fizikai Lapok/math and physics sheets for young students) and he received his PhD fast track (1930-1934) from Budapest University (today ELTE). The magic numbers of the discrete world captured his soul/mind at the developmental stage of early youth, saved surely his life and built the rational ‘rock of life’. Manchester, Princeton and Jerusalem were important locations of his voyage, but Budapest remained his spiritual root center of cosmopolitan gravity.

We can detect a strong interest in politics (to avoid politics), a cordial sociability (human connectedness), and celibacy for scientific reasons, skeptical humor and a permanent ambition to creative mathematical problem-solving until his physical death in Warszaw at a scientific conference. Intuitive rationality may be the best words to describe the overall attitude or life style of Paul Erdös; intuitive rationality is a continuous combination of internal meditation and external concentration, but in this case, not focused on holy texts or divine hermeneutics. The Erdös reverence was done by deep immersion into the discrete and combinatory mystical magic of proofs; it resembles the kabalistic interpretation of the physical world via mathematical and numerological means. This constant re-search into eternal patterns is ultimately very similar to religious activity, i.e. knowing and experiencing more about the concealed harmony of universal force; the intuitivism (‘fruit’) of deeds (proofs) counts more than the formality (‘shell’) of creed (theory). Here lies the inter-section between mathematics and theology or religion and science; formality and intuition are complementary forces of human inquiry and philosophically intersect where standard knowledge meets learning by faith, i.e. formal religion did interfere with Erdös’ faith in intuitive proofs and his own life experiences did rationally not confirm the formal statements of any official faith. Consequently, he concluded that hidden proofs from The Big Book can be re-searched via intuitive rationality; this lead to an ‘irrational’ merger of a wanderlust life style and a communicative scientific method, in terms of practicalities  and coping with every-day tasks. Also the scientific genius is rooted to the earth and the fruits of his work remain earthly products, even though they are pointing to patterns of eternity and higher force.

Approaching a second conclusion:

Hungarian Jewry and scientists have played a central and vital role in the globalization of the mathematical discipline in the 20th century and its physical applications in natural science, e.g. computers; networking capacity, economic generosity and mathematical creativity have made of Paul Erdös a role model for the cosmopolitan scientific researcher. Knowledge networking, sociability and humane creativity are surely the key traits of future scientists and advancing globalization humanely will be driven by this intellectual epistemology. Erdös anticipated this trend of the next scientific stage of human co-operation by almost 100 years via a creative technique of survivalist rationality and morality.

The epistemological heritage and intellectual legacy of Paul Erdös was molded by three distinct cultural factors: A) Mathematical problem-solving; B) Hungarian reflective mind; C) Jewish learning techniques. Scientific vagabondery has reinforced this molding of intellectual cultures into a cosmopolitan epistemological framework of case-oriented (proofs) and inter-personal research communication (networking). Erdös was not very much interested in mathematical theory-building; he stayed in the formal and intuitive system of traditional mathematical thought, developing it further by a discrete combination of puzzling and communicating via a world wide web of connections.

The theoretical statement of mathematician L. Kronecker (1823-1891) that ‘G-d made the integers, all else is the work of man’ bears fundamental importance for understanding the psycho-logical consequences of mathematical (human) and natural science. As it has been explained many times, the lack of communication between   the natural and human sciences, including its applied inter-sections like math, is a cultural tragedy (CP. Snow, 1959); the mal-applications and mal-interpretations of math and physics in the social sciences (e.g. global financial economics) are one part of this cultural dilemma, i.e. the two mathematical approaches of theoretical precision and proof practice are not reconciled. Letters and numbers are cognitive tools to better and deeper understand the construction principles/laws of the natural world order, but it is not wise to attribute divinity to cultural tools and techniques of the human mind. A clear physical distinction between the human mind and eternal patterns does exist, concerning life-time priorities, meaning and free choice. In any case, Erdös believed in the discovery of the hidden proofs from The Big Book; this is truly a real and Ersatz human link to the above mentioned statement.

Paul Erdös was bad at recognizing faces and names, but when he came to know the telephone number of a colleague then he immediately could connect name and research topic. This goes to show that his memory of contacts was based on numerical processing even in inter-personal relationships which illustrates the intellectual and epistemological merger of human life-style and science; the limits of the mathematical method as ‘linguistic tool’ are set by the human mind itself. Erdös was fascinated by the creativity and beauty of mathematical proofs and this kept him breathing for 83 years; you can find his grave at Kozma Street Jewish Cemetery, Budapest (plot:17A-6-29).

Approaching a summary and third conclusion:

The life-span of Paul Erdös covers the end of the Austro-Hungarian Empire (of which his father Lajos was a soldier and prisoner of war in Russia/1914-1920), WW1 and WW2, the Shoah, the Israeli war of Independence, the Iron Curtain/ Cold War and the fall of the Berlin wall, and …more. This is quite a lot for a mortal human being and can only be done by a survival strategy. Consequently, Erdös’ brain/mind was trained to decipher the mathematical structures of this world by numbers, re-searching into the hidden proofs from The Big Book which G-d has concealed for humankind to discover. His open source approach was more than futuristic, pointing to a new kind of scientific research co-operation, based on a deep knowledge base, social connectivity and physical relocation; he was primarily a problem-solver and not a theory-builder. Mother Anna was the most important woman in his life and she nurtured also his mathematical talent by educating him to a high level by private means. The mystery and magic of numbers, the deep immersion into theoretical proofs and intuitive rationality were his cultural tools of human survival and meaning; it is also reported that he enjoyed dining out in a good restaurant. The intellectual heritage of Erdös’ epistemology is to search for meaning not only in the purity of theoretical proofs, but to socially connect/communicate in an open source manner about scientific problems and to investigate as creatively as possible. A humanization of the scientific discourse is a rational and moral imperative in our times of economic over-competition, knowledge over-specialization and interpersonal alienation. N is definitely more than a number and Aleph is definitely more than a letter; beyond being cultural tools of human survivalist rationality and morality, they serve a higher purpose, i.e. the detection of eternal patters, unified force and spiritual unity. Zero and One are the most important numbers while Aleph and Beth are the most important letters;  the true purpose of science is most probably to provide a vehicle for the perception and observation of G-dliness in  the worldly order of natural processes/systems. Science is not the view from above, but the empirical view from below. When empirical science will become true science then will theoretical practice reveal the unity of this world as the simple unity of the eternal upper force. Science is a relative reference system of the absolute, if it is performed in a balanced manner of rationality and morality; this is also the solution of the paradox that Erdös Pal sought: G-d made the integers, all else is made by man like him.


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