This site documents the Divine Mind:
the Universe that enables us to create ourselves
page 5: God's ideas, cybernetics and singularity
Table of contents
5.1: Plato's forms
5.2: A history of creation
5.3: Form and mathematics
5.4: The nineteenth century exploration of formalism
5.5: Cantor: the cardinal of the continuum
5.6: Gödel and uncertainty; Turing and incomputability
5.7: Chaitin, cybernetics and requisite variety
5.8: Uncertainty and Probability
5.1: Plato's forms
The traditional story of creation assumes that God had a plan in mind for the world they were about to create. This plan is known as ideas. Plato first used this term for his invisible forms which serve both as patterns for the structure of the world and the source of our knowledge of the world. Theory of Forms - Wikipedia, Aquinas, Summa I, 15, 1: Are there ideas in God?
Here, from a modern point of view, we run into a serious difficulty. We now understand information to be physical, represented by physical symbols, and there are no such physical symbols in the traditional God. God is absolutely simple, without any internal structure. We cannot imagine how such a God can carry information, so it is difficult to understand how there can be ideas in this God. Insofar as we consider the initial singularity to be absolutely structureless we cannot see how it can contain information either. Rolf Landauer (1999): Information is a Physical Entity, Aquinas, Summa, I, 3, 7: Is God altogether simple?
At the same time, we want to see the Universe as divine and we can see that it is full of physical symbols, like ourselves, all the other things that we see around us and this text. If we wish to understand how the Universe emerges within the initial singularity, we have to understand how the initial singularity, which is all that there is in the beginning, can create all the information we see around us. This leads us to the idea that the immensely complex structure of the Universe may be equivalent to the ideas in God, that is the Universe is the mind of God.
This raises our principal question: how does structure come to be from a structureless beginning? Since we are trying to understand the Universe as a mental creation, we must try to find an explanation in cognitive terms. An analogous problem exists in our own mind: where do our ideas come from? In our conscious lives, we find that some of our ideas arise from sensation and some just seem to come from nowhere. At one moment we are completely puzzled. Moments later we have an idea which solves the puzzle with insight and understanding. We talk about working things out, but on reflection there is usually no clear link between the puzzle and the solution. This site records a slow accumulation of insights over 60 years. Bernard Lonergan (1992): Insight: A Study of Human Understanding
Darwin's theory of evolution shows how new species, that is new physical symbols, come to be. Darwin's theory does not start at the beginning, however. It assumes the existence of living creatures and the environment in which they live. Given this starting point, Darwin explains how new species arise through variation and selection.
We must therefore go deeper, since we assume that the initial singularity is identical to the traditional God, we must ask how ideas come to be. In the traditional God, the answer is simple: God's ideas are in fact identical to the essence of God, eternal. The traditional God did not have to spend time making plans for the new Universe as an architect might plan a new building. Once the plan for the Universe is complete, the omnipotent power of God brings it to be. In the traditional story this took six days. In modern science, creation has taken about 14 billion years.
The traditional God is said to be eternal, existing outside time, so the idea of God taking time to plan does not fit the traditional picture. Since the ideas in the eternal divinity are part of the essence of God, they too are eternal. From our point of view, existing in time, time itself is a part of creation. It did not exist until God created the Universe.
5.2: A history of creation
We work our way through these problems step by step in the next few pages. Beginning on page 6 with the ideas developed by Darwin, we explore how random processes create variation and natural selection filters the variants to select the forms of life that survive in their environment. This process is not a matter of foresight or thought. It is simply a matter of fact that some variants live long enough to reproduce their kind.
We then turn on page 7 to explore the processes in our brains that generate new ideas, which often appear to come from nowhere. Then on page 8 we look at the Christian doctrine of the Trinity which explains how God the Father, through reflecting upon themself, gives rise to the Son. The Father and the Son, communicating through love, create the Spirit.
We finally make this analogy concrete by identifying the initial singularity as pure action, the definition of God that Aquinas drew from Aristotle. From the conclusion drawn from the theory of the Trinity, that the characteristic action of God is to create Gods, we trace a plausible route to the emergence of the Universe within the initial singularity. Where the Trinity stops at three divine persons, we assume that there is no limit to the emergence of new gods. As the ancient scientist Thales (may have) said, the world is full of Gods. Thales of Miletus - Wikipedia
The key theological problem with the Trinity is to understand how the three divine persons can be both completely divine and yet be distinct from one another without sacrificing the unity of the divinity. The same problem arises with the distinction of all the elements that go to form the present Universe. The answer we will find lies in quantum mechanics, the theory of communication between discrete persons or sources.
5.3: Form and mathematics
The steps outlined above move toward to conclusion we seek, but it remains to be shown that this plan will succeed. This is a role for logic and mathematics. The the most powerful feature of these sciences is formalism, which gains its power by abstracting from concrete reality and distracting detail. Mathematicians, often talk about ideals like infinity which cannot be physically realized. Every pile of beans is finite. We can count them one by one, and although it may take a long time, we know that we will eventually come to the last bean.
Plato was among the first to take the idea of disembodied form seriously. It has remained central to theology ever since and became central to mathematics under the influence of Cantor and Hilbert. It is the foundation of the common idea that we have an immaterial spiritual soul which, because it has no material parts which can come apart, will live for eternity. Hilbert's program - Wikipedia
Aristotle paved the way to understanding change by bringing Plato's forms down to Earth to be embodied in matter, the theory we call hylomorphism. This hypothesis was designed to explain change among material things even though the forms remain immutable and eternal. When Aristotle came to discussing human sensation and understanding however, he saw matter as an impediment. Senses function by receiving the form of the entity sensed. Their material construction constrains the ability of the senses to receive forms, but he felt that this difficulty is overcome to come extent by the organic structure of the sense organs. Hylomorphism - Wikipedia, Aristotle (436a et sqq): Parva Naturalia; De Sensu et Sensibili
When he came to intellectual knowledge, however, Aristotle seems to have felt that the presence of matter in the intellect was too restrictive. The intellect must therefore be separated from matter. This idea became central to human psychology and became argument for our possession of a spiritual and immortal soul. Hendrik Lorenz: Ancient Theories of Soul, Christopher Shields (Stanford Encyclopedia of Philosophy b): The Active Mind of De Anima III 5
Aquinas, following Aristotle, associated intellect with immateriality. He argued that since God is maximally immaterial, they are maximally knowledgeable, that is omniscient. Aquinas, Summa: I, 14, 1: Is there knowledge in God?
Since the development of computation and information theory, however, the idea that immaterial forms have independent existence has fallen out of favour. Instead we understand that information is carried by marks, like the printed symbols that constitute this page. Information stored in computers, discs and solid state memories is written into billions of memory locations, each of which has a specific address and can represent one bit of information. We have found that matter is enormously complex, right down to the level of atoms and fundamental particles, and so capable of representing huge amounts of information. Current technology hardly touches the surface of the ability of particulate matter to store information.
5.4 The nineteenth century exploration of formalism
Although we no longer think that disembodied forms have independent existence, the idea found a new life in the study of infinity and continuity. Disembodied forms may have theoretical reality in the minds of people like Plato and mathematicians. The study of infinity was motivated in the nineteenth century by the need to put the ideas of differential and integral calculus on a firm logical footing and solve Zeno's paradoxes. Calculus depends heavily on the notions of limits and continuity. Since the development of the Pythagorean theorem, it became clear that many geometric objects, like the diagonal of a unit square, cannot be represented by rational numbers. This led to the development of mathematical analysis whose principal subjects are the real numbers and complex numbers constructed from real numbers. The purpose of the real numbers is to provide a name for every point in a geometric continuum. Square root of 2 - Wikipedia, Zeno's paradoxes - Wikipedia, Real number - Wikipedia, Differentiable function - Wikipedia, Mathematical analysis - Wikipedia
5.5: Cantor: the cardinal of the continuum
George Cantor worked in a mathematical milieu where it is believed meaningful to create a continuum from closely spaced discrete points. He wanted to define a numeral to represent the cardinal of the continuum, ie the number of points required to make a continuous line. In mathematics, syntax enables us to construct numerals to represent any cardinal number by forming ordered sets of digits, as in decimal numbers. A countably infinite string of decimal digits is considered adequate to represent any element of the set of real numbers. Toward the end of the nineteenth century, George Cantor set out to use syntactic methods to represent the cardinal of the continuum and realised that ordered sets of symbols could be used to represent anything representable well beyond the limit of real numbers and into the domain of infinite multi-dimensional structures. Cantor (1897, 1955); Contributions to the Founding of the Theory of Transfinite Numbers, Cantor's theorem - Wikipedia
Cantor was a very religious man, and felt that his theory of transfinite numbers brought him close to the mind of God. He sought support for his ideas in discussion with theologians. He knew that Aquinas had argued against the existence of actually infinite multitudes in the created world but he considered that transfinite ideas could exist in the divine intellect:
. . . in the transfinite a vastly greater abundance of forms and of species numerorum is available, and in a certain sense stored up, than there is in the correspondingly small field of the unbounded infinite. Consequently these transfinite species were at the disposal of the intention of the Creator and his absolutely inestimable will power just as were the finite numbers.Aquinas, Summa, I, 7, 4: Can any infinite multitude exist?, Michael Hallett (1984): Cantorian Set Theory and Limitation of Size, page 23
It obviously makes no sense to create a continuum out of a set of discrete and isolated points so Cantor's search for the cardinal of the continuum ultimately failed. Nevertheless he made great contributions to mathematics by making infinite objects finite and tractable by putting them in boxes or sets. The core of Cantor's discovery is the idea of one-to-one correspondence between infinite sets by matching their elements pairwise so long as they can be uniquely identified. His set theory remains one of the foundations of modern mathematics. Paul Cohen (1980): Set Theory and the Continuum Hypothesis
Cantor's work led to renewed interest in formalism. Some theologians felt that Cantor's work verged on pantheism, but he was defended by Hilbert, who introduced an explicitly formalist approach to mathematics and declared Aus dem Paradies, das Cantor uns geschaffen, soll uns niemand vertreiben können. (From the paradise, that Cantor created for us, no-one shall be able to expel us.) Formalism (mathematics) - Wikipedia, Cantor's paradise - Wikipedia, Joseph Dauben (1990): Georg Cantor: His Mathematics and Philosophy of the Infinite, page 144 sqq.
If we are going to establish that the Universe is divine, it it necessary to unite physics and theology. At present physics is still bedevilled with problems arising from the attempts to unite quantum mechanics and the special theory of relativity. Attempts to quantize the general theory of relativity have produced many ideas, but progress is very slow. The principal source of these problems is the belief that this union implies that the foundations of the Universe envisaged by quantum field theory are infinite. It has therefore been necessary to develop a procedure known as renormalization to eliminate these infinities and make the theory realistically respectable. Renormalization - Wikipedia, Michio Kaku (1998): Introduction to Superstrings and M-Theory
The resulting theory is a bit of a mess. Meinard Kuhlman, writes:
In conclusion one has to recall that one reason why the ontological interpretation of QFT is so difficult is the fact that it is exceptionally unclear which parts of the formalism should be taken to represent anything physical in the first place. And it looks as if that problem will persist for quite some time.
Hopefully the discussion of the creation of the Universe that begins on page 9: The active creation of Hilbert space will provide some ideas to establish a fruitful union of quantum physics and evidence based theology and eliminate the misuse of infinity and other unphysical mathematical and theological ideals in our descriptions of the Universe we inhabit. Meinard Kuhlmann (Stanford Encyclopedia of Philosophy): Quantum Field Theory
5.6: Gödel and uncertainty; Turing and incomputability
Cantor's introduction of set theory not only clarified the foundations of mathematics, but also introduced paradoxes that led to careful reevaluation of mathematical proofs. Hilbert believed that a proper application of formalist methods would eventually solve all these problems. This was not to be.
The search for the roots of mathematics emphasized the importance of logic and gradually brought symbolic logic and mathematics closer together. Whitehead and Russell set out to use logic to develop mathematics and built on the work of logicians to write Principia Mathematica. Their idea was to deduce all the fundamental propositions of logic and mathematics from a small number of logical premisses and primitive ideas, and so to prove that mathematics is a development of logic. Whitehead & Russell (1910, 1962); Principia Mathematica
One of the most significant consequences of Whitehead and Russell's work was to enable Kurt Gödel and Alan Turing to discover snakes in Hilbert's paradise.
Hilbert defined what he considered to be the ideal features of formal mathematics. It should be consistent, complete and computable.
Consistency means that no argument could arrive at the conclusion P ≡ not-P. Completeness means that any properly formed mathematical statement can be proved either true or false. Computable means that there exists definite algorithms capable of proving consistency and completeness.
Kurt Gödel upset the first of Hilbert's expectations by establishing that consistent mathematics is incomplete. Gödel's incompleteness theorems - Wikipedia
Soon afterwords Alan Turing established the existence of incomputable functions. Alan Turing (1936): On Computable Numbers, with an application to the Entscheidungsproblem
Thomas Aquinas established a limit to God's omnipotence: God cannot make a contradiction or inconsistency exist, eg that Socrates should be simultaneously sitting and standing. From this it has traditionally been concluded that God has absolute deterministic control over everything. The theorems of Gödel and Turing show, however, that from a formal point of view, absolute consistency does not always imply absolute control. Even in the mind of a God totally constrained by consistency, there may still be room for incompleteness and incomputability. Aquinas, Summa I, 25, 3: Is God omnipotent?
One of the problems that led to the study of the limits to proof was Cantor's paradox. Cantor thought that there might be an absolute maximum transfinite number that could represent God. This cannot be true, however, if Cantor's proof that every transfinite number is followed by a greater transfinite number is true. Cantor's paradox - Wikipedia
5.7: Chaitin, cybernetics and requisite variety
Apart from the theory of information and communication, there is a second modern development about which the ancients knew very little even though it governed their efforts at controlling the world: cybernetics. One of its founders, Norbert Wiener, defined it as the science of control and communication in the animal and the machine. Norbert Wiener (1996): Cybernetics or Control and Communication in the Animal and the Machine, Cybernetics - Wikipedia
Following Galileo, mathematical modelling has become a primary tool of modern physics. Mathematics has progressed well beyond what was available to Galileo. Aquinas believed that an omniscient and omnipotent God has total deterministic control of every event in the world. Gödel found that logically consistent formal systems are not completely determined and Chaitin interpreted Gödel's work to be an expression of the limits to control known as the cybernetic principle of requisite variety: One system can control another only if it has equal or greater entropy than the system to be controlled. This principle suggests that a completely structureless initial singularity has no power to control anything, even itself. Insofar as the initial singularity acts, its action must be act at random. Only later, as the Universe grows in structure and entropy does the possibility of control emerge, but it can very rarely be perfect. The entropy of the future is almost always greater than the entropy of the past. Aquinas, Summa, I, 22, 3: Does God have immediate providence over everything?, Gregory J. Chaitin (1982): Gödel's Theorem and Information, W Ross Ashby (1964): An Introduction to Cybernetics
This same principle invalidates the idea that the traditional God would have planned the Universe from the beginning. One of the first attributes of God that Aquinas studied was their simplicity. According to Aquinas (and long mystical history) God is absolutely simple. Since there are no structures or marks in this God, they cannot store or process information in the modern sense and information is necessary for control.
5.8: Uncertainty and Probability
Probability is a dynamic concept which characterizes properties of sequences of events. One of the simplest such sequences is repeated tosses of a coin. If this is done fairly, we expect a true coin to land heads up half the time, at least in the long run.
Probability is characteristic of a situation where there is little or no control. If we find that a tossed coin is consistently showing heads, we suspect trickery. It is not easy to "load" a coin because it is so simple, but dice can be weighed so that they show one face more frequently that the expected probability of 1/6 and more complex systems like roulette wheels open themselves to more complete control. The probabilistic nature of quantum observations shows there is less deterministic control in nature than Laplace and Einstein expected. Laplace's demon - Wikipedia
While we seek the ideal probabilities in games of chance, there are other situations where we demand complete control in order to reduce as far as possible the probability of adverse events like plane crashes and lung cancer.
Things go wrong when we lose control, but it turns out that absolute control is the enemy of creativity. The inability of the initial singularity to control itself opened the way for the creation of new structures in the Universe, making emergence and evolution possible.
As we shall see, the divine initial singularity gradually builds up its structure, becoming more and more complex by the process of trial and selection we call evolution. Our Universe was not planned beforehand, therefore, but has evolved through its own unlimited random activity. Some products of this activity do not last, other are consistent and durable, and these are the ones selected to exist for longer or shorter periods, giving us the more or less permanent features of the world we see.
(Revised Sunday 4 August 2024)
Copyright:
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Notes and references
Further reading
Books
Ashby (1964), W Ross, An Introduction to Cybernetics, Methuen 1956, 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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Cantor (1897, 1955), Georg, Contributions to the Founding of the Theory of Transfinite Numbers (Translated, with Introduction and Notes by Philip E B Jourdain), Dover 1895, 1897, 1955 Jacket: 'One of the greatest mathematical classics of all time, this work established a new field of mathematics which was to be of incalculable importance in topology, number theory, analysis, theory of functions, etc, as well as the entire field of modern logic.'
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Cohen (1980), Paul J, Set Theory and the Continuum Hypothesis, Benjamin/Cummings 1966-1980 Preface: 'The notes that follow are based on a course given at Harvard University, Spring 1965. The main objective was to give the proof of the independence of the continuum hypothesis [from the Zermelo-Fraenkel axioms for set theory with the axiom of choice included]. To keep the course as self contained as possible we included background materials in logic and axiomatic set theory as well as an account of Gödel's proof of the consistency of the continuum hypothesis. . . .'
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Dauben (1990), Joseph Warren, Georg Cantor: His Mathematics and Philosophy of the Infinite, Princeton University Press 1990 Jacket: 'One of the greatest revolutions in mathematics occurred when Georg Cantor (1843-1918) promulgated his theory of transfinite sets. . . . Set theory has been widely adopted in mathematics and philosophy, but the controversy surrounding it at the turn of the century remains of great interest. Cantor's own faith in his theory was partly theological. His religious beliefs led him to expect paradox in any concept of the infinite, and he always retained his belief in the utter veracity of transfinite set theory. Later in his life, he was troubled by attacks of severe depression. Dauben shows that these played an integral part in his understanding and defense of set theory.'
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Hallett (1984), Michael, Cantorian Set Theory and Limitation of Size, Oxford UP 1984 Jacket: 'This book will be of use to a wide audience, from beginning students of set theory (who can gain from it a sense of how the subject reached its present form), to mathematical set theorists (who will find an expert guide to the early literature), and for anyone concerned with the philosophy of mathematics (who will be interested by the extensive and perceptive discussion of the set concept).' Daniel Isaacson.
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Kaku (1998), Michio, Introduction to Superstrings and M-Theory (Graduate Texts in Contemporary Physics), Springer 1998 ' Called by some "the theory of everything," superstrings may solve a problem which has eluded physicists for the past 50 years -- the final unification of the two great theories of the twentieth century, general relativity and quantum field theory. This is a course-tested comprehensive introductory graduate text on superstrings which stresses the most current areas of interest, not covered in other presentation, including: string field theory, multi loops, Teichmueller spaces, conformal field theory, and four-dimensional strings. The book begins with a simple discussion of point particle theory, and uses the Feynman path integral technique to unify the presentation of superstrings. Prerequisites are an acquaintance with quantum mechanics and relativity. This second edition has been revised and updated throughout.'
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Lonergan (1992), 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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Whitehead (1910, 1962), Alfred North, and Bertrand Arthur Russell, Principia Mathematica (Cambridge Mathematical Library), Cambridge University Press 1910, 1962 The great three-volume Principia Mathematica is deservedly the most famous work ever written on the foundations of mathematics. Its aim is to deduce all the fundamental propositions of logic and mathematics from a small number of logical premisses and primitive ideas, and so to prove that mathematics is a development of logic. Not long after it was published, Gödel showed that the project could not completely succeed, but that in any system, such as arithmetic, there were true propositions that could not be proved.
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Wiener (1996), Norbert, Cybernetics or Control and Communication in the Animal and the Machine, MIT Press 1996 The classic founding text of cybernetics.
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Links
Alan Turing (1936), On Computable Numbers, with an application to the Entscheidungsproblem, 'The "computable" numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by some finite means. Although the subject of this paper is ostensibly the computable numbers, it is almost equally easy to define and investigate computable functions of an integral variable of a real or computable variable, computable predicates and so forth. . . . ' back |
Aquinas, Summa I, 15, 1, Are there ideas in God?, ' . . . in other agents (the form of the thing to be made pre-exists) according to intelligible being, as in those that act by the intellect; and thus the likeness of a house pre-exists in the mind of the builder. And this may be called the idea of the house, since the builder intends to build his house like to the form conceived in his mind. As then the world was not made by chance, but by God acting by His intellect, as will appear later, there must exist in the divine mind a form to the likeness of which the world was made. And in this the notion of an idea consists.' back |
Aquinas, Summa I, 25, 3, Is God omnipotent?, '. . . God is called omnipotent because He can do all things that are possible absolutely; which is the second way of saying a thing is possible. For a thing is said to be possible or impossible absolutely, according to the relation in which the very terms stand to one another, possible if the predicate is not incompatible with the subject, as that Socrates sits; and absolutely impossible when the predicate is altogether incompatible with the subject, as, for instance, that a man is a donkey.' back |
Aquinas, Summa, I, 22, 3, Does God have immediate providence over everything?, ' I answer that, Two things belong to providence—namely, the type of the order of things foreordained towards an end; and the execution of this order, which is called government. As regards the first of these, God has immediate providence over everything, because He has in His intellect the types of everything, even the smallest; and whatsoever causes He assigns to certain effects, He gives them the power to produce those effects. Whence it must be that He has beforehand the type of those effects in His mind. As to the second, there are certain intermediaries of God's providence; for He governs things inferior by superior, not on account of any defect in His power, but by reason of the abundance of His goodness; so that the dignity of causality is imparted even to creatures.' back |
Aquinas, Summa, I, 3, 7, Is God altogether simple?, 'I answer that, The absolute simplicity of God may be shown in many ways. First, from the previous articles of this question. For there is neither composition of quantitative parts in God, since He is not a body; nor composition of matter and form; nor does His nature differ from His "suppositum"; nor His essence from His existence; neither is there in Him composition of genus and difference, nor of subject and accident. Therefore, it is clear that God is nowise composite, but is altogether simple. . . . ' back |
Aquinas, Summa, I, 7, 4, Can any infinite multitude exist?, ' . . . multitude in nature is created; and everything created is comprehended under some clear intention of the Creator; for no agent acts aimlessly. Hence everything created must be comprehended in a certain number. Therefore it is impossible for an actually infinite multitude to exist, even accidentally.' back |
Aquinas, Summa: I, 14, 1, Is there knowledge in God?, ' I answer that, In God there exists the most perfect knowledge. . . . it is clear that the immateriality of a thing is the reason why it is cognitive; and according to the mode of immateriality is the mode of knowledge. Hence it is said in De Anima ii that plants do not know, because they are wholly material. But sense is cognitive because it can receive images free from matter, and the intellect is still further cognitive, because it is more separated from matter and unmixed, as said in De Anima iii. Since therefore God is in the highest degree of immateriality as stated above (Question 7, Article 1), it follows that He occupies the highest place in knowledge.' back |
Aristotle (436a et sqq), Parva Naturalia; De Sensu et Sensibili, ' Having now definitely considered the soul, by itself, and its several faculties, we must next make a survey of animals and all living things, in order to ascertain what functions are peculiar, and what functions are common, to them.' back |
Cantor's paradise - Wikipedia, Cantor's paradise - Wikipedia, the free encyclopedia, 'Cantor's paradise is an expression used by David Hilbert . . . in describing set theory and infinite cardinal numbers developed by Georg Cantor. The context of Hilbert's comment was his opposition to what he saw as L. E. J. Brouwer's reductive attempts to circumscribe what kind of mathematics is acceptable; see Brouwer–Hilbert controversy.' back |
Cantor's paradox - Wikipedia, Cantor's paradox - Wikipedia, the free encyclopedia, 'In set theory, Cantor's paradox is derivable from the theorem that there is no greatest cardinal number, so that the collection of "infinite sizes" is itself infinite. The difficulty is handled in axiomatic set theory by declaring that this collection is not a set but a proper class; in von Neumann–Bernays–Gödel set theory it follows from this and the axiom of limitation of size that this proper class must be in bijection with the class of all sets. Thus, not only are there infinitely many infinities, but this infinity is larger than any of the infinities it enumerates.' back |
Cantor's theorem - Wikipedia, Cantor's theorem - Wikipedia, the free encyclopedia, ' In mathematical set theory, Cantor's theorem is a fundamental result which states that, for any set A , the set of all subsets of A, the power set of A, has a strictly greater cardinality than A itself. For finite sets, Cantor's theorem can be seen to be true by simple enumeration of the number of subsets. Counting the empty set as a subset, a set with n elements has a total of n 2 subsets, and the theorem holds because n2 > n for all non-negative integers. Much more significant is Cantor's discovery of an argument that is applicable to any set, and shows that the theorem holds for infinite sets also.' back |
Christopher Shields (Stanford Encyclopedia of Philosophy b), The Active Mind of De Anima III 5 , ' After characterizingnous the mind (nous) and its activities in De Animaiii 4, Aristotle takes a surprising turn. In De Anima iii 5, he introduces an obscure and hotly disputed subject: the active mind or active intellect (nous poiêtikos). Controversy surrounds almost every aspect of De Anima iii 5, not least because in it Aristotle characterizes the active mind—a topic mentioned nowhere else in his entire corpus—as ‘separate and unaffected and unmixed, being in its essence actuality’ (chôristos kai apathês kai amigês, tê ousia energeia; DA iii 5, 430a17–18) and then also as ‘deathless and everlasting’ (athanaton kai aidion; DA iii 5, 430a23). This comes as no small surprise to readers of De Anima, because Aristotle had earlier in the same work treated the mind (nous) as but one faculty (dunamis) of the soul (psuchê), and he had contended that the soul as a whole is not separable from the body (DA ii 1, 413a3–5). back |
Cybernetics - Wikipedia, Cybernetics - Wikipedia, the free encyclopedia, ' Cybernetics is a transdisciplinary approach for exploring regulatory systems, their structures, constraints, and possibilities. Cybernetics is relevant to the study of systems, such as mechanical, physical, biological, cognitive, and social systems. Cybernetics is applicable when a system being analyzed is involved in a closed signaling loop; that is, where action by the system generates some change in its environment and that change is reflected in that system in some manner (feedback) that triggers a system change, originally referred to as a "circular causal" relationship.' back |
Differentiable function - Wikipedia, Differentiable function - Wikipedia, the free encyclopedia, 'In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain. As a result, the graph of a differentiable function must have a (non-vertical) tangent line at each point in its domain, be relatively smooth, and cannot contain any breaks, bends, or cusps.' back |
Formalism (mathematics) - Wikipedia, Formalism (mathematics) - Wikipedia, the free encyclopedia, ' In foundations of mathematics, philosophy of mathematics, and philosophy of logic, formalism is a theory that holds that statements of mathematics and logic can be thought of as statements about the consequences of certain string manipulation rules. For example, Euclidean geometry can be seen as a game whose play consists in moving around certain strings of symbols called axioms according to a set of rules called "rules of inference" to generate new strings. In playing this game one can "prove" that the Pythagorean theorem is valid because the string representing the Pythagorean theorem can be constructed using only the stated rules.' back |
Gödel's incompleteness theorems - Wikipedia, Gödel's incompleteness theorems - Wikipedia, the free encyclopedia, ' Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The two results are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible, giving a negative answer to Hilbert's second problem. The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (i.e., any sort of algorithm) is capable of proving all truths about the relations of the natural numbers (arithmetic). For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that such a system cannot demonstrate its own consistency.' back |
Gregory J. Chaitin (1982), Gödel's Theorem and Information, 'Abstract: Gödel's theorem may be demonstrated using arguments having an information-theoretic flavor. In such an approach it is possible to argue that if a theorem contains more information than a given set of axioms, then it is impossible for the theorem to be derived from the axioms. In contrast with the traditional proof based on the paradox of the liar, this new viewpoint suggests that the incompleteness phenomenon discovered by Gödel is natural and widespread rather than pathological and unusual.' |
Hendrik Lorenz (Stanford Encyclopedia of Philosophy), Ancient Theories of Soul, ' Ancient philosophical theories of soul are in many respects sensitive to ways of speaking and thinking about the soul [psuchê] that are not specifically philosophical or theoretical. We therefore begin with what the word ‘soul’ meant to speakers of Classical Greek, and what it would have been natural to think about and associate with the soul. We then turn to various Presocratic thinkers, and to the philosophical theories that are our primary concern, those of Plato (first in the Phaedo, then in the Republic), Aristotle (in the De Anima or On the Soul), Epicurus, and the Stoics.' back |
Hilbert's program - Wikipedia, Hilbert's program - Wikipedia, the free encyclopedia, ' In mathematics, Hilbert's program, formulated by German mathematician David Hilbert, was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathematics were found to suffer from paradoxes and inconsistencies. As a solution, Hilbert proposed to ground all existing theories to a finite, complete set of axioms, and provide a proof that these axioms were consistent. Hilbert proposed that the consistency of more complicated systems, such as real analysis, could be proven in terms of simpler systems. Ultimately, the consistency of all of mathematics could be reduced to basic arithmetic.' back |
Hylomorphism - Wikipedia, Hylomorphism - Wikipedia, the free encyclopedia, 'Hylomorphism (Greek ὑλο- hylo-, "wood, matter" + -morphism < Greek μορφή, morphē, "form") is a philosophical theory developed by Aristotle, which analyzes substance into matter and form. Substances are conceived of as compounds of form and matter.' back |
Laplace's demon - Wikipedia, Laplace's demon - Wikipedia, the free encyclopedia, ' We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.' A Philosophical Essay on Probabilities, Essai philosophique dur les probabilites introduction to the second edition of Theorie analytique des probabilites based on a lecture given in 1794. back |
Mathematical analysis - Wikipedia, Mathematical analysis - Wikipedia, the free encyclopedia, 'Analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, sequences, series, and analytic functions. These theories are usually studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space).' back |
Meinard Kuhlmann (Stanford Encyclopedia of Philosophy), Quantum Field Theory, ' Quantum Field Theory (QFT) is the mathematical and conceptual framework for contemporary elementary particle physics. In a rather informal sense QFT is the extension of quantum mechanics (QM), dealing with particles, over to fields, i.e. systems with an infinite number of degrees of freedom. (See the entry on quantum mechanics.) In the last few years QFT has become a more widely discussed topic in philosophy of science, with questions ranging from methodology and semantics to ontology. QFT taken seriously in its metaphysical implications seems to give a picture of the world which is at variance with central classical conceptions of particles and fields, and even with some features of QM.' back |
Probability - Wikipedia, Probability - Wikipedia, the free encyclopedia, 'Probability is the likelihood or chance that something is the case or will happen. Theoretical Probability is used extensively in areas such as finance, statistics, gambling, mathematics, science and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems.' back |
Real number - Wikipedia, Real number - Wikipedia, the free encyclopedia, 'In mathematics, a real number is a value that represents a quantity along a continuous line. . . . The discovery of a suitably rigorous definition of the real numbers – indeed, the realization that a better definition was needed – was one of the most important developments of 19th century mathematics. The currently standard axiomatic definition is that real numbers form the unique Archimedean complete totally ordered field (R ; + ; · ; <), up to an isomorphism,' back |
Renormalization - Wikipedia, Renormalization - Wikipedia, the free encyclopedia, ' Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering values of quantities to compensate for effects of their self-interactions. But even if it were the case that no infinities arose in loop diagrams in quantum field theory, it could be shown that renormalization of mass and fields appearing in the original Lagrangian is necessary.' back |
Rolf Landauer (1999), Information is a Physical Entity, 'Abstract: This paper, associated with a broader conference talk on the fundamental physical limits of information handling, emphasizes the aspects still least appreciated. Information is not an abstract entity but exists only through a physical representation, thus tying it to all the restrictions and possibilities of our real physical universe. The mathematician's vision of an unlimited sequence of totally reliable operations is unlikely to be implementable in this real universe. Speculative remarks about the possible impact of that on the ultimate nature of the laws of physics are included.' back |
Square root of 2 - Wikipedia, Square root of 2 - Wikipedia, the free encyclopedia, 'Geometrically the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational.' back |
Thales of Miletus - Wikipedia, Thales of Miletus - Wikipedia, the free encyclopedia, ' Thales of Miletus (Greek: Θαλῆς (ὁ Μιλήσιος), c. 624/623 – c. 548/545 BC) was a Greek mathematician, astronomer and pre-Socratic philosopher from Miletus in Ionia, Asia Minor. He was one of the Seven Sages of Greece. Many, most notably Aristotle, regarded him as the first philosopher in the Greek tradition, and he is otherwise historically recognized as the first individual in Western civilization known to have entertained and engaged in scientific philosophy. |
Theory of Forms - Wikipedia, Theory of Forms - Wikipedia, the free encyclopedia, 'Plato's theory of Forms or theory of Ideas asserts that non-material abstract (but substantial) forms (or ideas), and not the material world of change known to us through sensation, possess the highest and most fundamental kind of reality. When used in this sense, the word form or idea is often capitalized. Plato speaks of these entities only through the characters (primarily Socrates) of his dialogues who sometimes suggest that these Forms are the only true objects of study that can provide us with genuine knowledge; thus even apart from the very controversial status of the theory, Plato's own views are much in doubt. Plato spoke of Forms in formulating a possible solution to the problem of universals.' back |
Zeno's paradoxes - Wikipedia, Zeno's paradoxes - Wikipedia, the free encyclopedia, 'Zeno's paradoxes are a set of problems generally thought to have been devised by Zeno of Elea to support Parmenides's doctrine that "all is one" and that, contrary to the evidence of our senses, the belief in plurality and change is mistaken, and in particular that motion is nothing but an illusion.' back |