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Notes

[Notebook: Transfinite field theory DB 56]

[Sunday 23 May 2004 - Saturday 28 May 2004]

[page 84]

Sunday 23 May 2004

We take a much broader view than Lonergan of the 'integral heuristic structure of proportionate being'. Lonergan, pp 415 sqq. Lonergan constrains this structure using a theory of knowledge rooted in the experience of insight. We take the view that whatever metaphysics is, if it is to be of any value it must be written down somewhere, on paper, in metastable media or in the minds of the people. Further, since there is no a priori reason to put a size on proportionate being, we start with the biggest writing system imaginable, the Cantor Universe. Instead of using a theory of knowledge to constrain the term proportionate being, we use the theories of writing, known as logic and mathematics. In the formal and natural sciences

[page 85]

we are concerned to separate writings that make sense from those that do not. This occurs at two levels we call them the formal and the natural. So all writings that formally contradict themselves are rejected as possible true accounts of natural systems. [an article of faith is involved here, that the Universe is reasonable] Having passed the formal filter, writings can be passed through the natural filter to see if they correspond to anything actually happening in the world.

We identify the formal filter with metaphysics and the natural filter with the experience based (empirical) sciences.

Next we want to show that quantum mechanics is the local version of the output of the formal filter and quantum field theory the global version of this output which results when we mate quantum mechanics and special relativity.

Pais Inward page 244: 'The rejection of forces acting instantaneously at a distance [the most unrealistic feature of the Newtonian picture, possible because the Universe [was] a puppet of god] in favour of forces transmitted from point to neighbouring point by continuous fields [that determine the probability distribution of discrete events] meant the end of a purely mechanical picture of the physical world.' Pais

Now we have the beginning of a communication network, a field being a continuous communication network, constrained by various conserved flows - probability, action, energy, momentum.

Einstein: 'Since Maxwell's time, Physical Reality has been thought of as being represented by continuous fields [an approximation to discrete networks] . . . not capable of any mechanical interpretation. This change in the conception of Reality is the most profound and most fruitful that physics has experienced since the time of Newton. [Einstein in James Clerk Maxwell, CUP 1931].

[page 86]

Field is a thing with a personality, the dynamical interactions of the elements that it 'drives' (and in the case of gravitation) is driven by. Can we deduce the properties of the continuous fields by taking discrete networks to their infinitesimal limits? This idea underlies Feynman's path integral method. Feynman, Zee.

Monday 24 May 2004

Quantum field theory is the modern version of Aristotle's De Generatione et Corruptione. McKeon pp 470-531, Aristotle.

Why does interfacing special relativity and quantum mechanics introduce so much complexity? Should we kiss simple explanations goodbye, or is there a new simplicity waiting for someone with the right point of view?

What we seek to express in plain terms is 'how the world works' which we take to be equivalent to how god works, so that we are in a position to fit in and modify, to our best advantage, given the constraints of sustainability and justice, which we impose on our unconstrained power probably for reasons of symmetry (beauty, aesthetics). My worst pain is to see people destroying one another and their capital, houses, social structure, languages and so on. What we particularly need is a clear 'sociotype' to handed on from generation to generation through education of our children [maybe better psychotype, the network mind].

These notes mostly talk about what I would like to do without a lot of progress toward doing it. Faith, hope and charity are required here to continue the search so that even if I do not find what I am looking for, I might contribute to a climate in which others will continue the search.

Pais page 333 ''The quantum number [nk] of an oscillator is equal to the number of quanta with corresponding [nu].' Born, Heisenberg and Jordan. Let us have a look at this simple yet quite extraordinary remark which opens the door to quantum field theory.'

page 333: 'To distinct energy levels of the initial oscillator correspond (they propose) distinct numbers of photons. In the new interpretation a transition from one level to another must therefore mean that particles with energy h n are either

[page 90]

made or else disappear.' Photons are massless so E = mc2 does not apply, but the notion of creation and annihilation applies to massive particles as well, provided the energy can be obtained from somewhere or given to something. The creation of a photon involved mass loss in the atom and vice versa. How do we generalize these ideas to the creation and annihilation of people, civilizations, planets etc? Obviously by showing how large events are composed of small events. We see the Universe as a network of events based on countable, planck sized events and working up to the whole.

We conceive of the world as a set [or function] {structure, probability} [f(structure) = probability].

We can guarantee that the structure is somewhere in the TN, ie the 'wave function' of the Universe. The probability we calculate using the methods of quantum mechanics. The rule is that the probability of a certain structure is proportional to the absolute square of the wave function at that point.

The task of the prudent manager is to adjust the probability of events, eg to avoid lost time injury, gain in efficiency, etc. The quantity flowing through the transfinite network is probability, and all the nodes manipulate the probabilities of the links and vice versa [since probability is normalized to one, this flow is conserved, probability moving from one event (form]) to another].

Such a manager will always be testing to find and correct errors in the system to ensure its safe and efficient working. There comes time in the life of every plant, where maintenance and modification are no longer feasible, and the time comes for the plant to be annihilated and a new one created.

We give up the idea that everything has to be made of something (ie we can ignore what it is made of [Landauer] and simply concentrate on what it does (ie things which are in principle observable). This is the same as transfinite symmetry [symmetry with respect to complexity] : when we operate at a certain peer level, the levels below and above us are more or

[page 91]

less invisible. [so we do not have to know nuclear physics or politics to live].

Energy is symmetry with respect to time. Momentum is symmetry with respect to space. Is action symmetry with respect to complexity, ie one act is one act no matter how big it is, god or photon.

Pais page 350: Pais. On the early days of quantum field theory: Heisenberg: 'You know it was not the quantum mechanics. In quantum mechanics everything came out much simpler and much better than what I expected. Somehow, when you touched it and you had a disagreeable difficulty at the end you saw "Well, was it that simple?" Here in electrodynamics it does not become simple. Well, you can do the theory, but it will still never become simple.

Pais: 'So it is to this day and it will never be otherwise.'(?)

'It turned out on several occasions that the context of quantum field theory was much richer than expected - in some ways similar to what happened to the classical theory of general relativity.

page 331: 'There can be no question that a history of our penetration into the mysteries of matter must take cognizance of this inner life of quantum field theory.'

We construct an explanation of something by finding two different points of view that may be equated.

F = ma
Pump = machine for giving potential energy to a fluid.
etc.

So we seek the model that goes into the equation

model = f(world)

[page 92]

Born and Jordan: Born and Jordan.

B & J replace the functions of numbers used in classical mechanics with functions of matrices based on the arithmetic of matrices.

Such matrices must be infinite. May they also be transfinite, so that the outer shell of the Hamiltonian governs the vast complexity of the Cantor Universe. So we can imagine people (represented by suitably complex vectors and operators) are governed by the same form Hamiltonian as electrons and photons, something constructed from kinetic and potential energy.

Quantum mechanics elucidates the connection between structure and probability that controls the world. The constraint is that the sum of the probabilities of different competing structures is 1. This is a profound truth, and lies at the heart of Quantum mechanics as a metric science.

Now we come to life. The wave function determines both the nature and frequency of the outcomes, so if we want to influence the outcomes we must influence the wave functions. The order in my body is maintained by dissipation (Prigogine). I follow the arrow of time, being a heat engine of sorts. How do we build a heat engine on a quantum system, and other sorts of engine can we build, computing engines for instance.

A Carnot engine is reversible, which means that it conserves entropy. The entropy of the hot reservoir is passed conservatively to the cold reservoir and mechanical (zero entropy) energy is made available. In reverse, by absorbing mechanical energy, we have a refrigerator.

[page 93]

Quantum mechanics is also reversible = unitary. The common thread is the conservation of entropy H = sum over i pi log pi. Normally we think of i as a natural number, but what if we make it larger. Normally physics handles this transition from discrete to continuum by the change from sum to integral.

So how do living creatures manipulate the wave functions of wilderness in order to survive? The function of an enzyme is to increase the probability of a certain reaction by lowering its activation energy or by putting in energy in order to make the desired process work, ie active pumping of ions etc.

Tuesday 25 May 2004
Wednesday 26 May 2004

We think of physics as the shell or operating system of the metaphysics, including the hardware. We run on the physical hardware.

We can thus imagine the world as a layered network process analogous to the internet. This seems so clear, but I keep saying it to myself again. We may think of the metaphysics as being spatially inside the physics, while being logically outside it.

Cantor's theorem is complexity invariant, in that it works just as well for any aleph(n) --> aleph(n+1), n >= 0 transition. We would like to think that quantum field theory is also complexity invariant, so that it fits into the transfinite network naturally, and describes it subject to a) finite velocity of communication and countability

[page 94]

of ℵ0, giving us conservation of all countable entities, energy, momentum and action. In each case both the number of different states and the number of quanta occupying the state is countable. This constraint defines the 'ground state' of the transfinite oscillator.

Thursday 27 May 2004

I would like to understand the detail of quantum field theory but I am not well motivated because it seems far too complicated to be the description of something whose existence is unconstrained by historical detail. We see that formally, this condition can apply anywhere as long as there are enough markers (symbols) and weak enough constraints on the rearrangement of markers to enable exponential growth of change as generation follows generation. We might say the transfinite network is constrained by exhaustion, in that there is no way to increase the number of symbols that it generates.

We begin to understand the Cantor Universe by looking at features which are invariant with size (since it is too big to comprehend).

Life is perfect (for me) and yet I worry. Because life is not perfect everywhere, and there is a chance that the disease will spread to me and mine. This is the practical foundation of my worry and the other is aesthetic. I do not like to see suffering.

Kolmogorov: foundations of probability. Kolmogorov

page 1: Every axiomatic (abstract) theory, admits, as is well known, an unlimited number [of concrete interpretations besides those from which it was derived.]

page v: '. . . after Lesbegues publication [of his theories of measure

[page 95]

and integration] the analogies between the measure of a set and probability of an event and between integral of a function and mathematical expectation of a random variable became apparent. These analogies allowed a further extension; thus, for example, various properties of independent random variables were seen to be in complete analogy with the corresponding properties of orthogonal functions.

Kolmogorov page 1-2: '. . . if our aim is to achieve the utmost simplicity both in the system of axioms and the further development of the theory, then the postulational concept of a random event and its probability seem the most suitable.'

Friday 28 May 2004

We have equated metaphysics with both quantum mechanics and quantum field theory, but would a better choice be field theory in general conceived as network theory taken to the continuum limit. The best way to represent the world? Everything we see in the world is an event, a discrete entity with a beginning a middle and an end. Calculus tries to expose the middle by bringing the beginning and the end closer to one another, but this tends to squash the detailed process between points on a curve out of existence. This is natural at the Euclidian/Aristotelian level of geometric complexity. Now from Cantor . . .

The arithmetic shell: despite their great difference in internal complexity 1 (sheep) + 1 (sheep) = 2 (sheep) just as 1 + 1 = 2 holds for atoms, galaxies, in fact any conserved (additive) quantity.

The continuous paradigm could be holding us back. We observe countable events, and we can say that all our information comes in the form of counts taken at different points in spacetime.

[page 96]

We develop the explanation of these counts in terms of an underlying deterministic continuum called (for historical reasons) the wave equations. Waves are are handy because they have both discrete (wavelength, energy) and continuous features, so they are a natural bridge from the continuum. Going the other way, from discrete to continuous, we use the theory of computation. This theory (based on set theory) allows us to detect the boundaries between discrete and continuous and to constrain the functions in the continuum.

Our survival depends on our competence at reading and manipulating the world. We can encode our more predictable manipulations as algorithms for getting from one state to another, the state of independent ingredients, butter, flour, eggs, sugar etc to the state of a cake for instance. These manipulations are often very simple, and often at the high end of research and technology, quite complex and closely controlled in order to get the desired outcome, a state of the art x-ray telescope, say or a system to test the general theory of relativity in the vicinity of earth.

The general algorithm executer is the Turing machine.

Since Kant and perhaps beyond, the ding-an-sich has been considered to be beyond the human mind, but this may not be necessarily so, since of the Universe started from nothing we can expect some parts of it to be quite simple, dual in fact. The unitary simplicity of god is unanalyzable and so not much use to us except as an object of contemplation.

The connection between metaphysics and quantum field theory is via the network model which we map in passing onto function space, Hilbert space and the use [the model] to make a theory of knowledge, so bringing us back to Lonergan's standpoint.

Then we apply Lonergan's axiom as we know, so it is.

Saturday 29 May 2004

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Further reading

Books

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Aristotle, and H Rackham (translator), Aristotle, XIX, Nichomachean Ethics, Loeb Classical Library 1934 ' . . . This book opened my eyes to the true meaning of "Philosophy". The translation is in modern English, free from the back-to-front syntax of the Ancient Greek text (which makes it impossible to understand the meaning of a sentence until you reach the end of it!). The subject matter is "Ethics". However, a modern author may have called it something more akin to "The Meaning of Life" or "The Art of Living". Aristotle proceeds with simple and clear logic, to reveal the objective of human struggle in this life. He demonstrates a deep understanding of the Human Being, what we are and what we are not, what makes us act in one way or another and what makes us feel joy or distress. He addresses anxienties of the modern human, such as the question of nature or nurture, the moral action versus the practical, violence versus non-violence. His recommendations for living this life in a manner that is meaningfull and rewarding are profound yet simple.. . . ' Agis Liberakis 
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Ashby, W Ross, An Introduction to Cybernetics, Methuen 1964 'This book is intended to provide [an introduction to cybernetics]. It starts from common-place and well understood concepts, and proceeds step by step to show how these concepts can be made exact, and how they can be developed until they lead into such subjects as feedback, stability, regulation, ultrastability, information, coding, noise and other cybernetic topics' 
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Feferman, Anita Burdman, and Solomon Feferman, Alfred Tarski: Life and Logic, Cambridge University Press 2004 Review "A chain smoker, a heavy drinker, a frequent user of 'speed', a relentless womaniser, and a man of Napoleonic self-regard and worldly ambition. This is not how one pictures an eminent Professor of Logic. And yet, this is how the great logician, Alfred Tarski, emerges from this marvellous biography. The Fefermans, of course, are uniquely qualified to lead the reader through the intricacies of Tarski's work, which they do very engagingly and with great expository skill. Tarski's colourful personality is conveyed with prose that is economical, superbly readable and extremely vivid, and the whole book is a joy to read." Ray Monk, Professor of Philosophy, University of Southampton 
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Feynman, Richard P , and Albert P Hibbs, Quantum Mechanics and Path Integrals, McGraw Hill 1965 Preface: 'The fundamental physical and mathematical concepts which underlie the path integral approach were first developed by R P Feynman in the course of his graduate studies at Princeton, ... . These early inquiries were involved with the problem of the infinte self-energy of the electron. In working on that problem, a "least action" principle was discovered [which] could deal succesfully with the infinity arising in the application of classical electrodynamics.' As described in this book. Feynam, inspired by Dirac, went on the develop this insight into a fruitful source of solutions to many quantum mechanical problems.  
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Fowles, John, The Collector, Back Bay Books 1997 Amazon Product Description 'The Collector (1963) is disturbing, engrossing, unforgettable -- the story of an obsessive young man and the girl he kidnaps and holds prisoner in his cellar.' 
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Kolmogorov, A N , and Nathan Morrison (Translator) (With an added bibliography by A T Bharucha-Reid), Foundations of the Theory of Probability, Chelsea 1956 Preface: 'The purpose of this monograph is to give an axiomatic foundation for the theory of probability. ... This task would have been a rather hopeless one before the introduction of Lebesgue's theories of measure and integration. However, after Lebesgue's publication of his investigations, the analogies between measure of a set and mathematical expectation of a random variable became apparent. These analogies allowed of further extensions; thus, for example, various properties of independent random variables were seen to be incomplete analogy with the corresponding properties of orthogonal functions ... ' 
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Lonergan, Bernard J F, Insight : A Study of Human Understanding (Collected Works of Bernard Lonergan : Volume 3), University of Toronto Press 1992 '... Bernard Lonergan's masterwork. Its aim is nothing less than insight into insight itself, an understanding of understanding' 
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McKeon, Richard, and (editor), The Basic Works of Aristotle, Random 1941 Introduction: 'The influence of Aristotle, in the ... sense of initiating a tradition, has been continuous from his day to the present, for his philosophy contains the first statement, explicit or by opposition, of many of the technical distinctions, definitions, and convictions on which later science and philosophy have been based...' (xi) 
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Nixon, Richard Milhous, The Memoirs of Richard Nixon, Buccaneer Books 1994 Amazon editorial review: 'Former President Richard Nixon's bestselling autobiography is an intensely personal examination of his life, public career, and White House years. With startling candor, Nixon reveals his beliefs, doubts, and behind-the-scenes decisions, and sheds new light on his landmark diplomatic initiatives, political campaigns, and historic decision to resign from the presidency. Throughout his career, Richard Nixon made extensive notes about his ideas, conversations, activities, and meetings. During his presidency, from November 1971 until April 1973, and again in June and July 1974, he kept an almost daily diary of reflections, analyses, and perceptions. These notes and diary dictations, which are quoted throughout this book, provide a unique insight into the complexities of the modern presidency and the great issues of American policy and politics.' 
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Pais, Abraham, 'Subtle is the Lord...': The Science and Life of Albert Einstein, Oxford UP 1982 Jacket: In this ... major work Abraham Pais, himself an eminent physicist who worked alongside Einstein in the post-war years, traces the development of Einstein's entire ouvre. ... Running through the book is a completely non-scientific biography ... including many letters which appear in English for the first time, as well as other information not published before.' 
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Pais, Abraham, Inward Bound: Of Matter and Forces in the Physical World, Clarendon Press, Oxford University Press 1986 Preface: 'I will attempt to describe what has been discovered and understood about the constituents of matter, the laws to which they are subject and the forces that act on them [in the period 1895-1983]. . . . I will attempt to convey that these have been times of progress and stagnation, of order and chaos, of belief and incredulity, of the conventional and the bizarre; also of revolutionaries and conservatives, of science by individuals and by consortia, of little gadgets and big machines, and of modest funds and big moneys.' AP 
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Rawson, Philip, Tantra: Indian Cult of Ecstasy, Crescent 1988 Jacket: 'Suggesting as its final goal a vision of cosmic sexuality, Tantra embodies fundamental patterns of symbolic expression in a view of life which offers a uniquely successful antidote to the anxieties of our time. The act of creation is continuous: therefore sexual intercourse between human beings can be a microcosmic representation of the creative process -- a symbolic tribute to the great Goddess from whose womb, and through whose wisdom, all things in the Universe are manifested in Time.' 
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Tarski, Alfred, Cardinal Algebras, Oxford University Press 1949 'This book is an axiomatic investigation of the novel types of algebraic systems which arise from three sources: the arithmetic of cardinal numbers; the formal properties of the direct product decompositions of algebraic systems; the algebraic aspects of invariant measures, regarded as functions on a field of sets. ... The book is replete with novel algebraic notions; it is written in logical style; all theorems (important and unimportant) are explicitly stated, and the proofs are carefully cross-referenced.' Saunders MacLane 
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Tymoczko, Thomas, and (Editor), New Directions in the Philosophy of Mathematics: An Anthology, Princeton University Press 1998 Jacket: 'The traditional debate among philosophers of mathematics is whether there is an external mathematical reality, something out there to be discovered, or whether mathematics is the product of the human mind. ... By bringing together essays of leading philosophers, mathematicians, logicians and computer scientists, TT reveals an evolving effort to account for the nature of mathematics in relation to other hman activities.' 
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van der Waerden, B L, Sources of Quantum Mechanics, Dover Publications 1968 Amazon Book Description: 'Seventeen seminal papers, dating from the years 1917-26, in which the quantum theory as wenow know it was developed and formulated. Among the scientists represented: Einstein,Ehrenfest, Bohr, Born, Van Vleck, Heisenberg, Dirac, Pauli and Jordan. All 17 papers translatedinto English.' 
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Papers
Born, Max, Werner Heisenberg and Paul Jordan, "Zur Quantenmechanik II (On quantum mechanics II)", Zeitschrift fur Physiks, 35, , received November 16, 1925, page 557 - . Translation available in van der Waerden, B L, Sources of Quantum Mechanics, Dover Publications 1968 . back
Links
The Internet Classics Archive | On Generation and Corruption by Aristotle 'Written 350 B.C.E , Translated by H. H. Joachim. ... 'Our next task is to study coming-to-be and passing-away. We are to distinguish the causes, and to state thedefinitions, of these processes considered in general-as changes predicable uniformly of all the things that come-to-be and pass-away by nature. Further, we are to study growth and 'alteration'. We must inquire what each of them is; and whether 'alteration' is to be identified with coming-to-be, or whether to these different names there correspond two separate processes with distinct natures.'' back
Being There - Wikipedia Being There - Wikipedia, the free encyclopedia 'Being There is a 1979 American comedy-drama film directed by Hal Ashby, adapted from the 1971 novella written by Jerzy Kosiński. The film stars Peter Sellers, Shirley MacLaine, Melvyn Douglas, Jack Warden, Richard A. Dysart, and Richard Basehart. Douglas won the Academy Award for Best Actor in a Supporting Role and Sellers was nominated for Best Actor in a Leading Role. This was the last Peter Sellers film to be released while he was alive.' back
Dakini - Wikipedia Dakini - Wikipedia, the free encyclopedia 'A dakini (Sanskrit: डाकिनी ḍākinī; Tibetan: མཁའ་འགྲོ་མ་ khandroma, Wylie: mkha' 'gro ma, TP: kandroma; Chinese: 空行女) is a tantric deity described as a female embodiment of enlightened energy. In the Tibetan language, dakini is rendered khandroma which means 'she who traverses the sky' or 'she who moves in space'. Sometimes the term is translated poetically as 'sky dancer' or 'sky walker'.' back
Fluctuation dissipation theorem - Wikipedia Fluctuation dissipation theorem - Wikipedia, the free encyclopedia 'In statistical physics, the fluctuation dissipation theorem is a powerful tool for predicting the non-equilibrium behavior of a system — such as the irreversible dissipation of energy into heat — from its reversible fluctuations in thermal equilibrium. The fluctuation dissipation theorem applies both to classical and quantum mechanical systems. Although formulated originally by Nyquist in 1928, the fluctuation-dissipation theorem was first proved by Herbert B. Callen and Theodore A. Welton in 1951.

The fluctuation dissipation theorem relies on the assumption that the response of a system in thermodynamic equilibrium to a small applied force is the same as its response to a spontaneous fluctuation. Therefore, there is a direct relation between the fluctuation properties of the thermodynamic system and its linear response properties. Often the linear response takes the form of one or more exponential decays.' back

Piety - Wikipedia Piety - Wikipedia, the free encyclopedia 'The word piety comes from the Latin word pietas, the noun form of the adjective pius (which means "devout" or "good"). Pietas in traditional Latin usage expressed a complex, highly valued Roman virtue; a man with pietas respected his responsibilities to other people, gods and entities (such as the state), and understood his place in society with respect to others.' back
Pontifical Catholic University of Chile - Wikipedia Pontifical Catholic University of Chile - Wikipedia,.the free encyclopedia 'The Pontifical Catholic University of Chile (UC or PUC) (Spanish: Pontificia Universidad Católica de Chile) is one of the six Catholic Universities existing in the Chilean university system and one of the two Pontifical Universities in the country, along with the Pontifical Catholic University of Valparaíso. It is also one of Chile's oldest universities and one of the most recognized educational institutions in Latin America.[1] Since it is a Pontifical University, it has always had a strong and very close relationship with the Vatican. It was founded on June 21, 1888 through a decree issued by the Santiago Archbishop. Its first chancellor was Monsignor Joaquín Larraín Gandarillas, and at the very beginning, the university only taught two subjects, law and mathematics. It is part of the Universities of the Rectors' Council of Chilean Universities, and although it is not state-owned, a substantial part of its budget is given by state transferences under different concepts.' back
René Thom - Wikipedia Rene Thom - Wikipedia, the free encyclopedia René Frédéric Thom (September 2, 1923 – October 25, 2002) was a French mathematician. He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became world-famous among the wider academic community and the educated general public for one aspect of this latter interest, his work as founder of catastrophe theory (later developed by Erik Christopher Zeeman). He received the Fields Medal in 1958.' back
Rudolf Clausius - Wikipedia Rudolf Clausius - Wikipedia, the free encyclopedia 'Rudolf Julius Emanuel Clausius (Born Rudolf Gottlieb,[1] January 2, 1822 – August 24, 1888), was a German physicist and mathematician and is considered one of the central founders of the science of thermodynamics.[2] By his restatement of Sadi Carnot's principle known as the Carnot cycle, he put the theory of heat on a truer and sounder basis. His most important paper, On the mechanical theory of heat, published in 1850, first stated the basic ideas of the second law of thermodynamics. In 1865 he introduced the concept of entropy.' back

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