This article is the appendix to my new ebook, Scientific Natural Philosophy, published by Bentham Science Publishers.
There is much wisdom in what this author’s best academic friend, Professor V. Lakshmikantham, says on the subject of research: do it! Some researchers rely on the serendipity of science, others on sharpening their tools and reading the work of others leaving no time to think and ending up nowhere. Perhaps, this author’s experience can offer some insights.
His academic career was interrupted 15 years straight during the political turmoil in the Philippines capped by the imposition of martial law by President Ferdinand Marcos, 1972 – 1986. The situation was the least this researcher needed at the time especially when he found himself not really in immediate danger but in a wide field in the general direction of the business end of the barrel of the gun. He left for the United States in 1971 and found a new imperative: build a support movement for the end of martial rule during which time he shelved his mathematical career completely.
When people’s power ousted President Marcos in 1986 the author returned to the Philippines but faced a daunting challenge: he found himself at ground level while his international colleagues were up on the fifteenth floor of the Tower of Academe. Retracing their steps would have been impossible and historical precedence was not in his favor either. The closest to his experience was Bertrand Russell’s when he was removed from his post as professor of mathematics at Cambridge University, England, for his pacifist activity during the First World War. When he got his post back 26 years later through the effort of British mathematician G. H. Hardy, he went around mathematics and remained a philosopher.
Two years after his return to the Philippines the author was groping in the dark for a couple of years although he got results that would prove important later – the generalized integral and derivative. He knew that proving a theorem here and there would not help. The 15-year handicap needed something spectacular to overcome.
Then he posed the right question: why are there long-standing unsolved problems in mathematics and physics? The answer: the underlying fields are inadequate. The most famous problem in mathematics then was the 355-year-old Fermat’s conjecture (popularly known as Fermat’s last theorem or FLT) and in physics, the 200-year old gravitational n-body problem posed by Simon Marquis de Laplace at the turn of the 18th Century. He made a critique of the underlying fields of FLT – foundations, number theory and the real number system – and opened a can of worms that we saw earlier the most devastating of which being the inconsistency of the field axioms of the real number system that David Hilbert wanted to avoid. The inconsistency was first shown by Felix Brouwer by way of a counterexample to the trichotomy axiom [5] and, later, by this author with his own version of the counterexample to it [21,78]. The remedy was a simple construction of the decimals, using only three axioms, into the contradiction-free new real number system [21] that has, at the same time, countably infinite counterexamples to prove Fermat’s conjecture false. It retains the valid and interesting properties of the real numbers and has new elements that qualitatively and quantitatively model some physical concepts such as the superstring.
The work was not over yet; no journal would touch the author’s work with a ten–foot pole. Then he scoured mathematical landscape around the world to find the most influential mathematician in the field who can help. He found one: Professor V. Lakshmikantham, Editor-in-Chief of many scientific journals and founder of the broad and only rapidly expanding mathematical field today: nonlinear analysis. He was going to address the founding session of the International Conference on Dynamic Systems and Applications in Atlanta, 1995, organized by Professor M. Sambandham, Editor of the journal that published [21] and [44]. It was there where the author presented his introductory paper on the generalized integral and derivative, FLT, and the gravitational n-body problem, in the paper “Probabilistic mathematics and applications to dynamic systems including Fermat's last theorem” which was published in the proceedings [14]. Most of all, however, he got the lead to the World Congress of Nonlinear Analysts in Athens, 1996, sponsored by the International Federation of Nonlinear Analysts of which Professor Lakshmikantham was the President. He submitted his solution of the gravitational n-body problem which was accepted for presentation at the Congress. It was at this Congress where, through the initiative Mrs. Escultura, Professor Lakshmikantham got a copy of his book, Diophantus: Introduction to Mathematical Philosophy (With probabilistic solution of Fermat’s last theorem) [13], which he liked. He published the full paper, Exact solutions of Fermat’s equation (A definitive resolution of Fermat’s last theorem), in the journal, Nonlinear Studies, 1998, official publication of the International Federation of Nonlinear Analysts (now shared with the GVP – Lakshmikantham Institute for Advanced Studies, GVP College of Engineering, J. Nehru Technical University, India). One of the major results of the paper, characterization of undecidable propositions, gave the clue to the inadequacy of the present methodology of physics – mathematical modeling (now quantitative modeling). The introduction of qualitative mathematics and modeling as remedy was the crucial factor in the solution of the gravitational n-body problem that required the discovery of the superstring and the initial 11 natural laws needed for the solution that provided the foundations of GUT.
The passage of time has a way of distorting one’s memory. In this author’s earlier papers on GUT it is mentioned that qualitative modeling was introduced in 1997 to solve the gravitational n-body problem. In fact, it was introduced and was the main contribution of the author in his Ph. D. thesis [46], The Trajectories, Reachable Set, Minimal Level and Chain of Trajectories of a Control System, and in 1991 he published the book, Introduction to Qualitative Control Theory [22]. It was applied to physics for the first time in 1997 [41].
Perhaps success in research has something to do with one’s academic training. This author’s Ph. D. advisor, Professor L. C. Young of the University of Wisconsin lectured only on really new topics and did not give any examination. Instead, he threw this author into the lion’s den, so to speak, to lecture at his faculty seminars (which was always well-attended) before mathematicians a number of whom really famous, some visiting professors. He discussed most of the hand written manuscripts of Young’s book, Lectures on the Calculus of Variations and Optimal Control Theory [117] and recent papers by famous mathematicians then like L. Cesari and Peter Lax and reviewed mathematical papers submitted for publication (of course, Young signed the evaluation in each case this author being unknown). Young trained only 13 Ph.D.s who made a mark in diverse fields that fulfilled his vision: a good teacher is one who trains his pupil to beat him somewhere [118].
In his recommendation for the author’s appointment at the University of Illinois Chicago which he got hold of many years later by virtue of the Freedom of Information Act, Professor Young wrote:
Dr. Edgar Escultura originally joined a remarkable group that I was fortunate enough to have working with me from about 1963 to 1970, a group consisting of post-doctoral visitors as well as really promising graduate students. I must admit that I had at first some doubts as to Escultura’s suitability. In such a group, the standards are tough: only really good students as a rule will join it – students with the drive and confidence to tackle what is reputed to be hard. Much was expected, among other things, of the Seminar presentations: a tradition of excellence had established itself. Escultura surprised me by being able to fit into it. Indeed, on the strength of his Seminar presentations, I have no doubt that he can be an excellent teacher. My second surprise came when in due course Escultura went on to write a fine thesis under me. It was much more than a nice piece of work for solving some problem: there really were concepts of his own devising and the more I think of them, the more I feel that these concepts may prove important by opening up new methods in a number of fields far removed from that he was tackling. In this respect, the period of enforced interruption that he has experienced may be all to the good; it has allowed problems in a number of areas to mature to a stage at which they are ripe for his ideas. At any rate, I have no doubt that he will make significant contributions. I can recommend him in the highest terms.
The letter was quite providential. Unfortunately, he was unable to confirm it; although he reached the ripe old age of 95 and passed away on Christmas Eve, 2000, the author lost contact with him since 1993.
This author believes that everyone is endowed with the same potential. How much of that potential is realized depends on how he develops his thought through formal training, experience and his own initiatives and creativity. He agrees with Mahatma Gandhi that life is an experiment (the author extends this view to the development of biological species [31] where every species experiments with various anatomical and physiological innovations) and if one succeeds in an endeavor anyone else will. (Taken from “An Autobiography or The Story of My Experiments with Truth” by M. K. Gandhi, Navajivan Publishing House, Ahmedabad-14, 1927; translated from the Gujarati by Mahadev, 209 ed.)
The author stands on the shoulders of others. At the bottom of the article, Abstracts and Summary of Publications, on his websites, http://users.tpg.com.au/pidro
the author acknowledges his indebtedness to them:
I owe much of my achievements to others: R. A. Favila and M. Sambandham opened the doors to mathematics and science; L. C. Young inspired and trained me well and V. Lakshmikantham discovered and walked me over to the frontiers of science and mathematics.
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