GÖDEL’S REFUTATION OF THE MECHANICAL MODEL OF EXPLANATION
(Demise of Positivistic Model in Mathematics)
Gödel’s Challenge to Naturalistic, Mechanistic Model in the Post-Modern Culture
PROGRESS OF SCIENCE AND THE FOUNDATIONS OF MATHEMATICS:
FROM EUCLID TO GÖDEL
Is Mathematics autonomous? That assumption is suspect after Gödel. I argued in the Newtonian section of this thesis that he actually employed two methods in developing his scientific method: Hypothesis fingo, i.e., experimentalism and Mathematics. But how and why does Mathematics enable science to progress? Why, if it is, is Mathematics the language of nature, history and society? What is the relationship of theory commensurability or incommensurability to the translation capability of mathematics? How is Mathematics related to the problem of observation language and presuppositions? Is all observation theory-laden (cf. Hanson, et. al.)?
The developments in Mathematics over the past three centuries move from the certitude expressed by Descartes-- I have resolved to quit only abstract geometry, that is to say, the consideration of questions that serve only to exercise the mind, and this, in order to study another kind of geometry, which has for its object the explanation of the phenomena of nature to these words by Professor John C. Slater, until recently at the Massachusetts Institute of Technology:
The physicist finds very little help from the mathematician. For every mathematician like von Neumann who realizes these problems and contributes practically to them, there are twenty who have no interest in them, who either work in fields of remote interest to physics, or who stress the older and more familiar parts of mathematical physics. Is it any wonder that in such a situation the physicist, looking at the mathematicians, feels that they have strayed from the path which has led to the past greatness of mathematics, and feels that they will not regain this path until they again resolutely enter the main current of progress of mathematical physics, the current which in the past has led to the most fruitful development of mathematics. . . . That, the physicist firmly feels, is the only path through which the mathematician of the present can achieve greatness.
HISTORIOGRAPHY OF PHYSICAL SCIENCE
First draft of chapter in my forthcoming work. God. Creation. Rationality and Progress of Science with Special Attention to T. Kuhn's Theory of Scientific Development.
MATHEMATICAL ROOTS OF TWENTIETH CENTURY PHILOSOPHY:
THE KANTIAN CAMEL OF CONSTRUCTIVISM:
MATHEMATICS AND MATHEMATICIANS TRAJECTORIES OF INFLUENCE
The Second Coming expresses our present parallelizing pluralism,— Turning and turning in the widening gyre The falcon cannot hear the falconer; Things fall apart, the center cannot hold; Mere anarchy is loosed upon the world . . .Keats
The leading philosophers, at the turn of the century, were: 1) Bradley and Bosanquet in England; 2) Bergson in France; and 3) James and Royce in this country. But by now the results of the Kantian paradigm are everywhere apparent. Indeed, all the disagreements, from major engagements to minor skirmishes, have been conducted within, around, or over what may be called the Kantian paradigm.
When Kant's Critique of Pure Reason was published in 1781 the dominant philosophical school was a form of metaphysical and epistemological dualism, which entails two types of entities in the universe: 1) minds, and 2) material objects. i.e. mind knows objects (and other minds - see A. Plantinga's Other Minds, Cornell University Press) by means of mental states, variously called: 1) ideas, 2) representations, 3) impressions, 4) phantasms (see my syllabus, Epistemology for changing views Subject or Object) that are caused by these objects and that resemble them. Despite crucial differences on many vital points, the Lockeian empiricists and the Cartesian rationalists agreed that the mind is directly acquainted only with its own states, that is, its ideas are its only means of access to the outside, i.e., objective world.
Hume clearly pointed out that if the mind knows only its own states, its own states are all that it knows. But Hume did not argue that only Hume exists; on the contrary, he believed in the existence of other minds and of an external world (see H. H. Price, Hume on the External World). Yet, he maintained that these shared beliefs were incapable of proof. They are merely the expressions of "a blind and powerful instinct of nature." It is not reason or logic, but only custom, that is, "the great guide of human life" (see my forthcoming Rationality and Historiography of Scientific Progress with special attention to thesis of T. Kuhn concerning "Paradigmatic Revolutions" for critique).
It was at this juncture where the entire post-Renaissance philosophical paradigm threatened to collapse in solipsism and scepticism, that Kant came on the intellectual/cultural scene. Since Hume had demonstrated the breakdown of the hypothesis that truth consists in the minds being in agreement with objects, Kant proposed to try the opposite hypothesis that truth consists in the agreement of objects with minds. That is, he proposed to abandon the old view that the mind passively records what is "out there" and to try instead the hypothesis that it selects and structures what is out there. This hypothesis entailed that the mind contains selecting and organizing principles and that it is possible to learn what those principles are. If we can ascertain this, as Kant assumed he could, it follows that an absolutely certain knowledge of nature can be had—not an absolutely certain knowledge of particular facts out of the basic structure of nature as far as we can experience it.
For the basic structure of nature so considered is a product of the mind's activity, not something independent of that activity. (Cf. Kant assumed that he was explaining Newtonian philosophy of science—contra Hume's similar assumption; so Whitehead, Hartshorne, et al., assume that their process philosophies articulate the Einsteinian paradigm.) Any experience which depends on empirical observation is merely probable? but the basic law that every event has a cause, on which the whole procedure of physics rests, is a priori, i.e., certain, because the human mind structures its experience in a cause-effect way. While the range of assured knowledge is limited in scope, Kant believed that the mind's organizing and synthesizing activities were sufficient to justify and warrant the fundamental principles of Newtonian physics, above all the principle of the uniformity of nature as to which Hume had maintained that our belief rests on mere blind instinct rather than on logic or evidence (see Jones, Brehier, Copleston, and my Research Bibliography in Philosophy).
Kant described his hypothesis about the knower and his relation to the objects of his knowledge as a 'Copernican-like revolution' - (see Hanson's Critique of N. K. Smith's translation and commentary). Just as Copernicus had shifted the frame of reference from the earth to the sun, so Kant shifted the frame of reference from objects to the mind. What Copernicus brought about admittedly was an enormous shift in perspective, with momentous consequences? in calling his own hypothesis 'Copernican,' Kant was claiming that it was an equally revolutionary shift in perspective, In this estimate he was correct, but paradoxically, his hypothesis had an almost directly opposite effect. Whereas Copernicus' astronomical hypothesis had demoted the earth (and with it, man) from the center to the periphery, Kant's epistemology hypothesis brought man, the knower, back to the center. Now the 'primary imagination' (Coleridge) allowed the poet to exalt his role. No longer merely a pleasing imitator of nature, man is a creative god in his own right.
The historic developments from Kant to Hegel (see my Hegel, Marx, and Liberation Theologies?
and Philosophy of Religion - syllabi) towards the revival of realism (G. E. Moore's "The Refutation of Idealism," 1903; and Meinong/Brentano -theory of the nature of consciousness) made a radical departure from the entire idealistic, constructivist paradigm possible - see Ewings, History of Idealism; ed. Realistic Foundations of Phenomenology. This radical subjectivism of idealism was challenged by Moore's analysis of 'mental' and 'physical.' Eddington's 'two tables problem,' i.e., the table of physics and the table of common sense? If the former is real, must not the latter be an illusion? This was one of the questions to Russell addressed himself (see Levi, chp. 9 and 11, for analysis of Russell and Wittgenstein, et al.). A brief statement concerning the developments in nineteenth century theories of logic and numbers are in order before addressing the 'Received View' of Logical Positivism and its collapse and consequences. (See my syllabi Historiography of the Physical Sciences; and Historiography of Theories of Logic.)
Kantian Constructivism and Nineteenth Century Mathematics
From Newton to Positivism the paradigm of Truth was mathematics, specifically Euclidean Geometry. Few would dispute that mathematics has influenced the development of both Western philosophy and theology (see Tillich's acknowledgement in his Systematic Theology, I, p. 107). Mathematics has had considerable influence on contemporary theology, not only through Husserl and his existentialist followers, but also through two other major philosophical traditions, that which is generally known as Linguistic Analysis, and the Process Philosophy of Whitehead. Mathematics has an influence on theology - comparable to that which it occupied through Greek philosophy and the rationalistic philosophy of Descartes and Leibniz on Medieval and modern theology respectively, for both content and structure. Number in Pythagorean philosophy and geometric figure in Cartesian philosophy, egs., were considered to be primary content of the respective philosophies (see - Kirk/Raven, The Presocratic Philosophers, Cambridge, 1962, p. 216). Martin Heidegger employs non-mathematical insights of Kierkegaard and Nietzsche in a methodological framework determined by the early mathematic philosophy of E. Husserl. It is not that the content of mathematics is utilized in existential philosophy but that distinctions in contemporary mathematics provide the context for existential content to be cast in its present form.
The Euclidean Paradigm: Classical Paradigm of Proof and Truth
The formulation of the axiomatic method and the related discovery of general mathematics by Greek mathematicians is closely associated with the development of Greek philosophy (see G. J. Allman, Greek Geometry from Thales to Euclid (Dublin University Press, 1889); and T. L. Heath, The Thirteen Books of Euclid's Elements (3 vols., 2nd ed., New York: Dover, 1956, pb.). Plato's mathematical presuppositions have had a great influence on both the philosophy and theology of Western culture.
The Elements of Euclid afford us the best source for an examination of the mathematics, and the presuppositions associated with it, that influenced Plato and — his student Aristotle. Euclid's fundamental presuppositions of Elements, I will call the Euclidean paradigm.
The major characteristic of the Euclidean paradigm is the emphasis on geometrical symbolism. The Elements is concerned as much with number theory as it is with geometry, but both are described with geometric variables (common sense, symbolism, language and logic). This fact, perhaps, expresses the Greek mathematician/ philosopher's contempt for the common logistica which "does not consider number in the true sense" (see Scholium to Plato's Channinides; Ivor Thomas, Selections Illustrating the History of Greek Mathematics, 2 vols. (Cambridge: Harvard Univ. Press, 1951), I, p. 17.), perhaps motivated by a much deeper necessity. In order to describe the general emphasis of Euclid on geometrical symbolism, we go directly to an example. This e.g. is chosen as one which represents both relations in numbers and relations in geometric figures. Why did the Greeks choose a geometric symbolism instead of another numerical one which was readily available to them? Proposition 2 of Book II of the Elements is: "If a straight line be cut at random, the rectangle contained by the whole and both of the segments is equal to the square on the whole" (T. L. Heath, The Thirteen Books of Euclid's Elements, 3 vols. (Dover, pb.), I, p. 376).
A C B
D F E
This proposition is equivalent to the statement that for any given line segment AB and any arbitrary point C on AB, the area of ADFC and CFEB is equal to that of ADEB, where ADEB is the square of AB, and CF is drawn parallel to either AD or BE. This proposition is obvious by inspection (note the place of 'intuition' - 'seeing' in Husserl, Whitehead and Wittgenstein, et al., in each case the intuition-seeing paradigm derives for a mathematical paradigm). One notes that the whole is made up of the two parts, and that is sufficient 'to see' that the proposition is true.
This proposition can be translated into number theory using algebraic variables with no difficulty. Noting that the length of a line segment corresponds to a number and that a rectangle represents the product of two numbers whose "sides are the numbers which have multiplied one another" (Elements, Book VII, Definition. 17), the algebraic representation comes out: (a+b) (a+b) = a(a+b) + b(a+b) for any real numbers a and b. In this algebraic statement, the variables a and b have the same function as Euclid's line segment variables. We have introduced four new symbols, two for operations in arithmetic, that of addition and multiplication, one for equality, and a .symbol for grouping. Each of these symbols, with the possible exception of a multiplicative sign, was available to the classical Greek mathematicians. (See Howard Eves, An Introduction to the History of Mathematics (N.Y.: Rinehart and Co., Inc., 1959), p. 40.) One of the earliest written mathematical documents, the Rhin papyrus (ca. 1650 B.C.) has symbols for plus and minus, one for equality, and even a symbol for the unknown. The question must be asked - "Why did the Greeks choose geometric symbolism in preference to the common symbolism involved in arithmetic operations?" The reason for the use of geometric symbols instead of arithmetic ones may be obvious enough. It is that one may 'see and understand 'relationships'" more easily when they are presented in pictorial form (cf. Wittgenstein's Picture Theory of Meaning and observational emphasis in Learning/Educational Theory) than when they are presented in other symbolism. It is intuitively clear that the proposition is correct because we can visually see that the two parts of the square make up the whole, whereas the algebraic identity (a+b)(a+b) = a(a+b) + b(a+b) is by no means obvious on inspection.
A second crucial reason why the Greeks may have chosen geometric over algebraic symbolism is that they had no symbol for an irrational number. The scandal that the discovery of irrational numbers presented to the Pythagorean brotherhood may be the first, e.g., of religious faith being undermined by reason (cf. Kuhn's Paradigmatic Revolution anomalies and the Received View). What may have been only a convenience for the representation of rational numbers became a necessity for the representation of irrational ones.
Thus, the axiomatic method of Euclid comes to us in geometric form. In this form it has conditioned the thinking and intuition of mathematician/philosopher' alike for over two thousand years.
Geometrical symbolism carries along with it the intuitions associated with spatial relationships. Indeed, geometry has been defined traditionally as the science of space (cf. Newtonian 'absolute space' vs. Einsteinian 'relativized space' and its implications for logic of induction and any empirically based theory of science. See my forthcoming Historiography of Theories of Rationality and Progress of Scientific Knowledge). A mathematics that is characterized exclusively in geometrical symbolism as in The Elements allows one to assume a relationship between mathematic form and the human subject comparable to the observed between space and the human subject. Just as the human subject is perceived to be necessarily spatial, i.e., subject to spatial relationships, so is he presumed to be subject to mathematical relationships in a similar way. Implicit in this understanding of the nature of mathematics is the belief that mathematics is a description of 'necessary relationships' to which each existent entity must conform. Little distinction is made between the ontological position of the human subject- and that of other entities. Each has its relationship to mathematics comparable to the universal relationship that all things have as spatial (see section in my syllabus - Historiography of Physical Sciences).
Mathematics becomes a description of an overarching reality “that includes the mathematician as a part of this reality. (See E. Nagel, The Structure of Science, chp. 8 "Space and Geometry," pp. 203ff., and chp. 9 "Geometry and Physics," pp. 234ff.).
The second fundamental characteristic of the Euclidean paradigm is the emphasis upon and use of the axiomatic method. This method of the Elements has four different types of logical constructs: 1) Definitions, 2) postulates, 3) common notions, and 4) propositions. The definitions describe or point to the basic things which the propositions are about; a) points, b) lines, c) figures, d) numbers, etc. They refer extra-system to entities that are the object of examination. Postulates are regarded as permissible constructions but have a logical status with the common notions. Both 'common notions' and postulates may be called axioms. The axioms are basic assumptions, or that which is perfectly obvious, about the things that the definitions describe. Propositions are statements about the things the definitions describe which' are proved by means of the axioms.
The definitions in the Elements occupy a position in relationship to axioms, propositions, etc., that has been thought to be natural and appropriate for two thousand years, but which was seen to be in error by formalistic developments in the nineteenth century (see my syllabus - Historiography of Logical Theory; and especially W. and M. Kneale, The Development of Logic; Bochenski, A History of Formal Logic; G. Frege, The Foundations of Arithmetic; and Gödel’s Theorems). The definitions come first in the Elements, before the postulates, the common notions, and the propositions. They attempt to describe and characterize the nature of things outside the system about which the axioms and propositions speak. They present, the nature of the subject matter of the Elements, points, triangles, squares, etc. They denote the objects to be examined.
After the more rigid formalization of the axiomatic method had occurred in the latter part of the nineteenth century, definitions were found to be in Russell's terminology, "theoretically superfluous." In the new axiomatic formal systems, definitions refer only infra-system in distinction to the extra-system reference of definitions in the Elements. In a modem axiomatic system, there is no necessary reference of definitions to anything outside the system. This development will "lead" to the loss of the concepts of universal and necessary truths essential for the Christian/scientific paradigm of truth claims. The Euclidean descriptive use of definitions enhanced the objective status of transcendent mathematical objects. The very form of the axiomatic method suggested that mathematical relationships are "out there" and objective to the observer. This confirmed the feeling of the general "thereness" that geometrical figures appear to have as they are ingressed in phenomenal objects.
Both the use of geometrical symbolism and axiomatic method provide a natural 'unity' for mathematics. There is a multiplicity of different mathematical relationships in the Elements, and each one of them is formulated in geometrical symbolism. The unity of the multiplicity of relationships is found in the unity of an abstracted space. Just as all spatial relationships are in the one universal space, so are all mathematical relationships (i.e., geometrical ones) in an “abstract” mathematical space. One aspect of the unity of Euclidean mathematics is seen, using contemporary terminology, in terms of its interpretation in space.
The other aspect of the unity of Euclidean mathematics comes from the use of the axiomatical method (see Bochenski's Contemporary Philosophical Methods, chp. On 'Axiomatic Method.'). The whole of Euclidean mathematics is deduced from the firm foundation of a few unquestioned axioms and intuitively clear definitions. Mathematics is seen to be unified in one axiomatic system. The singular axiomatic system of Euclid must be contrasted with multiplicity of different axiomatic systems that have come into being since the nineteenth century and the problem this has posed for the nature of the unity of mathematics, and ultimately the unity of science movement (see my syllabus - Historiography of the Physical Sciences and the Crisis in the Received View, i.e., Positivism).
One of the fundamental presuppositions in any mathematical philosophy is the assumed relationship between the constructive ability of the mathematician to create mathematical relationships and the "objective status" of these relationships prior to or apart from human construction of them. (This development has staggering implications for certain forms of Presuppositionalism, Evidentialism in Christian Eristics, assumptions regarding "proof" derived from consensus, coherence and consistency, and the "objectivity of the scientific enterprise which employs the language of nature - mathematics to explain scientific Truth/Knowledge claims.) There is no question in the philosophy of Plato and Aristotle that the mathematical objects exist apart and prior to man's discovery of them. A mathematician does not bring into existence a figure by constructing it. The mathematically constructed objects are ontologically real in themselves and do not depend on the mathematician for their existence (contra Locke - who affirmed that a natural number is constituted by the activity of the mind, or with that of the later Wittgenstein who states that the mathematician is an inventor and not a discoverer of mathematical truth). Compare this with the Elements in which, care is taken to insure that the definitions are not merely verbal but do actually refer to "existent" mathematical figures. (See Aristotle, Posterior Analytics 92b, 12; and T. L. Heath, (trans.), Euclid's Elements, I, 142.)
The Euclidean paradigm has had a powerful impact on Christian theology through the a priori method, e.g. Augustinian/Reformed theological paradigm; also existence proofs as determinative of the form of the proofs of God via Plato/Aristotle/Stoics/ Anselm, et al. It is imperative that we take note of the fact that Christian theology has been structured in terms of the Euclidean paradigm and that paradigmatic revolutions in mathematics have precipitated radical changes in the structure of religious content.
Perceptive Paradigm Shift; From Euclidean to Non-Euclidean Space
Both Archimedes and Diophantes, however, complemented the Euclidean paradigm in the classical Greek period, but it was not until Descartes that principles inherent in their mathematics actually challenged the presuppositions of the Euclidean paradigm. In his analytic geometry, Descartes established a correspondence between algebraic equations in two variables and plane geometric curves. The correspondence between curves and equations, established by Descartes, was made possible by his utilization of “Cartesian coordinates”, two lines at some fixed angle so that one point, e.g., (x,y), could represent two lengths, x and y, on the respective lines. Since Greek mathematics had represented the product of two lines as an area, and the product of three lines as a volume, and laid no adequate representation for 4th and higher powers of quantities, it was necessary for Descartes to develop a method for the representation of a product as the length of a line segment and not of an area of volume. By means of his technique, a quantity to any power could be represented as a length of line segment. Thus, Descartes was able to establish a one - one correspondence between the elements of arithmetic, numbers, and the elements of geometry, line segments,- so that the addition and multiplication of line segments give line segments just as addition and multiplication of numbers give numbers.
There is a tension between alternative possibilities in Descartes' mathematical work: (a) the resolution of geometry into algebra, or (b) the resolution of algebra into geometry. Descartes sought a mathematics that transcended both the disciplines of algebra and geometry, i.e., 'Universal Mathematics' (mathesis universalis). That Descartes did not translate geometry into that higher method, but instead chose to translate algebra into geometry was significant for his own philosophy and for that which followed him. Descartes chose to translate algebra into geometry for two reasons: (1) geometry possessed for him an intuitive clarity that algebra did not, and (2) he believed that geometrical figure alone "gives us a means of constructing the images of all objects whatsoever" (Descartes, Rules for the Direction of the Mind, Haldane/Ross (eds.)(New York: Dover, 1955), I, p. 63). Mathematics became for him geometrical, and all physical phenomena, everything that is extended, became for him mathematical. Descartes had the vision that the whole universe, except mind, could be readily formulated into mathematical laws pictured geometrically. According to him God had brought it about that geometrical figures are immutable and eternal and are conformable with the real nature of things (Descartes' Reply to Objections V, Haldane, II, p. 226).
Descartes' discovery re-emphasized the geometric aspects of the Euclidean paradigm and the position that true mathematics is essentially a mathesis universalis, i.e., an over-arching strata of relationships to which geometric intuition was reinforced by Descartes' mathematical successes is indicated in a comment by Leibniz where he found it necessary to call attention to the fact that Descartes' algebra was not necessarily about lines (Leibniz, The Art of Discovery, ed. P. P. Wiener. Leibniz Selections (New York: Scribner's, 1951), p. 57.
The aspect of Descartes' philosophy which fundamentally contradicts the Euclidean paradigm is his position of founding philosophy (and mathematics) on the awareness of the existence of the ego (this is an example of a Kuhnian paradigmatic revolution) . That the clear and distinct ideas of the axioms of mathematics are. secondary to the clear and distinct idea of our own existence is seen in the fact that we "may . . .doubt of the demonstration of mathematics" (Descartes, Principle V, Haldane I, p. 220), but "we cannot doubt of our existence without existing while we doubt," (Principle VII, I, p. 221) In the Euclidean paradigm, "thinghood existence" and "human existence" are comparable ontological entities because both are interpretations of an over-arching geometrical system. Descartes asserted a fundamental dualism where extended substances are of an ontologically different sort than thinking substances . The human ego may radically transcend both mathematics and the world. (Cf. Augustinian affirmation of transcendence of the human ego via Intuitive Subjectivity.)
The introduction of a concept of the transcendence of the human ego over mathematics provided for Descartes a problem which he was unable to solve.
Descartes' Two Loyalties
His two loyalties were (1) mathesis universalis which was able to order and describe all of man's concerns, and (2) a transcendent ego which could in no way be described by mathematics. His "fundamental problem" was, how can the ego be radically transcendent over mathematics and yet be adequately characterized by mathematics? .This unsolved problem enters the modem intellectual arena with critical force in The First Critique of I. Kant. The problem and its solution is tied closely to developments in nineteenth-twentieth centuries' mathematics. Kant's answer was vitiated by developments within the nineteenth century. The problem is significant for the developments in Phenomenology, Existentialism, and especially the philosophies of Whitehead (as interpreter of Einsteinian paradigmatic revolution), Husserl, and Wittgenstein, whose efforts at solution provide the aspect of their philosophies which radically effect the development of new theological paradigms.
The Fundamental Problem; Transcendence - Object or Ego?
Historically, Leibniz, Pascal, and Locke expressed important views on the fundamental problem. Their gradual awareness of the transcendence of the human subject over mathematics precipitated their concern for the problem.
Blaise Pascal is the single philosopher between Descartes and Kant who took seriously both the radical transcendence of the mathematician over mathematical systems and the necessity of his conforming to mathematical relationships. In his De lespirit geometrique, Pascal affirms that he can "demonstrate truths already found ... in such a way that the proof will be invincible" (see B. Pascal, De lespirit geometrique, 2 vols. (Paris: 1860), II, p. 335). Radical shifts occurred in Pascal's Fen sees, where he acknowledges Montaigne's influence on himself with the words, "it is not in Montaigne, but in myself, that I find all that I see in him," he then affirms, not the scepticism of Montaigne, but the transcendence of roan over all mathematical and technical reason. According to him in the Pensees, mathematical and technical reason is not suited to the study of man. The study of man has its own Reason, and this may or nay not have any affinity with technical reason. "A hundred contradictions might be true" (see Pensees, pp. 20, 55, 95-6, 91). "The order of thought is to begin with self" (ibid., p. 55). Pascal affirmed that Descartes' heinous error is the beginning with self for the purpose of reaffirming 'technical reason' to which man is understood as necessarily subject.
Reason vs. Reason
The degree of transcendence of Reason over technical, i.e., mathematical, reason is such that mathematical reason is called in question. Pascal expresses a fundamental scepticism about technical reason when he asserts that "It may be that there are true demonstrations; but this is not certain" (ibid., p. 55). "We have an incapacity of proof" (ibid., p. 127). Yet existential Reason, by the Grace of God, leads to truth. "We have an idea of truth, invincible to all scepticism" (ibid). In following Reason, however, we should not exclude (technical, i.e., mathematical) reason. "Two extremes: to exclude reason, to admit reason only" (ibid., p. 90).
Pascal broke with the Euclidean paradigm in two fundamental ways: (1) he de-emphasized geometric figure as the appropriate mathematical symbolism, and (2) he reformulated the understanding of the nature of definition. Pascal found within arrays of numbers a pattern which he did not interpret geometrically, egs., his arithmetic triangle represents a pattern that is applicable-to-combinations, egs., coefficients of binomial expansion, and certain mathematical structures of probability. (See H. W. Turnbull, The Great Mathematicians (New York: New York Univ. Press, 1961), p. 89.)
A definition, to Pascal, consists of "the arbitrary application of a name to things which are clearly designated by terms perfectly known" (Geometrique, II, p. 336) . The utility of a definition "is to elucidate and abbreviate discourse, in expressing by a single name . . . what could otherwise be only expressed by several terms" (ibid.). For Pascal, definitions refer only intra-system; that which refers extra-system are the undefined or primitive terms.
The utilization of literal symbolism which does not necessarily point to geometric figure and a clear understanding of the nature of definition, as Pascal had, tends to subvert Euclidean assumptions. The literal symbolism becomes more an end in itself, and is' that which is constructed and manipulated by the mathematician. The mathematician gives the literal symbolism or undefined terms meaning, if it has any. (Cf. Camap, the Vienna Circle and their 'Received View.') The mathematician is perceived in a different relationship to mathematics than in the Euclidean paradigm. He is more creator than discoverer, more transcendent to the mathematical system than conformable to it. Pascal gives no consistent answer to the 'fundamental problem'; this is, perhaps, due to his conviction that such an attempt would subject the reasons of the heart (Reason) to technical reason, i.e., mathematics. Pascal's answer to the fundamental problem entails the Christian faith. .Contemporary existentialism, e.g., that of Heidegger or Sartre, differs from that of Pascal by being interpreted through the consistent and unified philosophical system of E. Husserl where an attempted solution to the fundamental problem is explicitly given. Perhaps it is no accident that the "founder of modern algebra" (Pascal) developed an existential stance that is compatible with the existentialism that developed after modem algebra came into its own late in the nineteenth century.
From Locke's Constructivism to Kant's Synthetic A Priori
Prior to further radical development in mathematics, Locke's description of the practical constructive ability of the mind to formulate mathematical ideas found its way into the synthetic part of Kant's synthetic a priori and remains visible in contemporary mathematics in the emphasis of the Intuitionists on constructive methods. Kant's assertion that natural members may represent relations more adequately than geometrical symbolism was vindicated by the Berlin school in the nineteenth century. Locke was also opposed to any conception of an innate a priori. He recognized that many of the claims for a priori innate ideas were based on the assumed authority of the axiomatic method. Both the axiomatic method and the privileged use of geometrical symbolism fell before his criticism. They are "archetypes of the mind's own making, not intended to be the copies of anything," and are put together "without considering any connexion they have in nature" (J. Locke, An Essay Concerning Human Understanding, 2 vols. (New York: Dover, 1959), II, p. 230).
Though accepting the place of mathematical ideas as 'internal,' and though depreciating geometry as fundamental mathematical discipline, Locke did accept the idea of extension as a primary quality of things, and thereby, had a natural way of understanding, in some manner, the nonnativeness of geometry over external things. Berkeley brought extension "inside" claiming that it, like color, is a secondary quality.(see Burt's Metaphysical Foundations of Modern Science concerning- - -primary/secondary qualities) . With the transference of the relationship of cause and effect from outside things to inside ideas by Hume, the process was complete. The necessary regularity of the objective world as described by a normative mathematics was seen to be untenable. The human mind was seen to create its mathematical ideas to handle other ideas about primary sense data. The Euclidean paradigm was rejected in toto. The human subject was seen to have a natural transcendence over all mathematics.
The Enlightenment paradigm of autonomous man was now complete. The Enlightenment (seventeenth/eighteenth centuries) united a vastly ambitious program, a program of secularism, humanity, -cosmopolitanism, and freedom, above all, freedom in its many forms - freedom from arbitrary power, freedom of speech, freedom of trade, freedom to realize one's talents, freedom of aesthetic response, freedom, in a word, of moral man to make his own way in the world. In 1784, when the Enlightenment had done most of its work, Kant defined it as man's emergence from his self-imposed tutelage, and offered as its motto - Sapere aude - "Dare to Know"; take the risk of discovery, exercise the right of unfettered criticism, accept the loneliness of autonomy (Kant, What is Enlightenment?; see my syllabi - The Enlightenment; and The Making of the Contemporary Mind for analysis of differences between a Paradigmatic Revolution and an Ideological Shift - see Popper).
Kant on the Axiomatic Method and Geometric Symbolism
Kant sought to uphold an a priori necessary mathematics to which the world necessarily conforms while accepting the basic criticism by Hume of this position. He attempted to maintain the Euclidean paradigm while allowing an adequate understanding of the transcendence of the human subject over mathematics. Though, he does not allow the axiomatic method a place in the formulation of a total philosophy; this is metaphysics. The legitimacy of axiomatic method is limited to geometry. The axioms of geometry have apodictic certainty. The things which Kant 'looked at' in order to build his philosophy on anschauung were geometrical figures. The common logistica - e.g. 7 + 5 = 12, had formed the basis for algebraic generalizations for centuries, but in the time of Kant, it was still the habit to translate numerical relationships into geometrical symbolism in order to acquire an "intuitive" foundation for these relationships. Kant's acceptance of the Euclidean paradigm affirms that mathematics is pure a priori. Our perception of anything is perception in terms of certain mathematical relationships (cf. Theory-laden observation language -see my syllabus - Historiography of Physical Sciences). Here we note the Kantian acceptance of the Newtonian philosophy of science/mathematics, i.e., that mathematics has the power to describe the world in a comprehensive and necessary manner. Kant sought to construct a philosophy which would justify Newtonian mechanics and its explanatory power.
Yet, Kant accepts the transcendence of mathematician over mathematics in the sense that the mathematician constructs mathematical judgments. According to Kant, all mathematical judgments are synthetical by utilizing Locke's emphasis on the constructability of natural numbers. The concepts of geometry are also constructed. That a straight line is the shortest path between two points is a synthetical proposition because it joins the concept of straightness (non-quantity) with quantitative concept "shortest."
Science, Common Sense and Truth
One of Kant's most critical problems is to show how mathematics can be both a priori and synthetic. He maintained that there are two categories, space and time, "existing in" consciousness to which all sense perception must conform. These categories are not available for observation. The category of space, eg., cannot be examined directly, only spatial things can be seen. But geometrical figures are representations of spatial things corresponding to the category of upace which presents itself to "intuition," which is pure and independent of perception of spatial things themselves. Judgments of geometry are put together synthetically with an appropriate spatial figure in mind. Even the consciousness of oneself "is with all our internal perception?, empirical only, and always transient" and therefore subject to the category of time (Kant, Critique of Pure Reason, "Transcendental analytic," I, 2,3,4).
There is, however, a "pure, original, and unchangeable self consciousness" to which the categories are logically related and which "precedes all data of consciousness" (ibid.). This "transcendental unity of self - consciousness" belongs to me (the noumenal self) and thus I form the basis of its existence. Kant attempted to fuse both the rationalist and empiricist traditions, but what he missed from the Euclidean paradigm is that an account of the necessary laws of the activity of consciousness can never describe adequately the essential character of an object of consciousness. Mathematical objects, in some way, stand over against the activity of consciousness. An epistemological examination of the knowledge of the nature of number or of a geometric figure is not -the same thing as an account of the activity of counting or constructing figures by synthetic a priori operations. A number may have some relationship to structures of consciousness, but it must be distinguished from activities determined by this structure. The recapture of this aspect of the Euclidean paradigm, i.e., the objectivity of mathematical relationships, which was lost in Kantian philosophy, and the emphasis on a radical subjectivity for greater than that of Kant's, characterize a position that we shall call "mathematical existentialism," which has greatly effected contemporary philosophy/theology.
From Crisis to Collapse; Breakdown of the Euclidean Paradigm
The transition from the Newtonian/Euclidean paradigm led the way for solid accomplishment in the new mathematics.
Newtonian Paradigm and Newtonianism
One of the greats in this paradigmatic shift was Joseph Louis Lagrange. Newton was to him "the man of genius par excellence" (E. T. Bell, Men of Mathematics (New York: Simon and Schuster, 1937), p. 170).
Lagrang-e's masterpiece, Mecanique analytique, was published in Paris in 1788. It was thoroughly in the Newtonian tradition, yet he broke with the Euclidean paradigm by presenting his comprehensive and rigorous mathematical system of mechanics without the use of geometric symbolism. By contrast, in Newton's Principia every mathematical relationship is conceived and proved by recourse to geometric figure. Lagrange substituted for Newton's geometrical presentation one which consisted of equations of variables and numbers. Lagrange succeeded in presenting a mathematical mechanics without the use of geometric figure or a geometrical understanding of the infinitesimal. His success did not invalidate- the use of geometric intuition in analysis. This was to come later in the work of Weierstrass.
At the turn of the century he was engaged in a work to develop the calculus without any concept of infinitesimal and without Newton's concept of a limit (see his Lessons on the Calculus of Functions, 1801).
Demise of the Self-Evident
Augustine Louis Cauchy (1789-1857) introduced a rigor in mathematics that separated the eighteenth and nineteenth centuries' analysis. He shifted the focus of attention from geometric figures to that of the properties of the real numbers. As a result, he eliminated "Motion" from the calculus (Newton had introduced moving points into the set of Cartesian coordinates and defined the derivative (his fluxion) at the rate of change of a moving point (his fluent). Cauchy made it possible, in principle, to dismiss all such intuitions of movement. Cauchy's convergence conditions are not based on these finite choices by the mathematician but on the choices involving "for all," i.e., "for all positive integral values of p," etc., and, hence, do not devolve from any specific activity of the mathematician but depend on the logical nature of the definition.
The final blow to geometric intuition came at the hands of Weierstrass (1815-1897). He produced an example that showed a direct contradiction in intuition " involving the classical Newtonian concept. He demonstrated the existence of a continuous curve which at no point possesses a derivative. If we. maintain a geometric intuition which includes movement in the Cartesian coordinates, we have to consider a curve being generated where at no time is there a ratio between the velocities of the y and x movement respectively. We came to the contradictory conclusion that a curve is being generated by a moving point which at no time can move. Or alternately we must conceive of a point moving which at no time has a '-'•' definite velocity. These e.g. point to the fact that no geometrical intuition is adequate to describe what is actually the case mathematically when the curve is presented in terms of limits involving sets of real numbers handled in terms of formal logic. The new formal devices allowed the presentation of a curve that could not be pictured geometrically. The astonishing thing about the curve is that, though incapable of geometrical representation, it was found to be descriptive of scientific phenomena - namely that of Brownian Movement. Thus, not only was the assumed adequacy of geometric figure to present mathematics challenged by the e.g., but also challenged was the assumption that physical phenomena are more naturally susceptible of geometrical characterization than by other abstract and formal means.
With the breakdown of the Euclidean paradigmatic intuition, the natural candidate for replacement of geometric figure was the mathematical content which had been in competition with geometric symbolism throughout the Greek era and eighteenth/nineteenth centuries, namely, ordinary numbers. A crucial reason for the abandonment of geometric intuition and figure was its inadequacy to represent known (formal) mathematical relationships without paradox.
Geometrical Figure/Natural Number
The fundamental shift in intuition from geometric figure to that of the natural numbers further upset the Euclidean paradigm concerning the relationship of the mathematician's constructive ability to that of the objectivity of mathematical relations. In the Euclidean paradigm mathematical relationships were considered to be objective and "out there." A natural number, however, may be thought of as create by the act of counting. As Dedekind (of Berlin school) affirmed, "numbers are the free creations of the human mind" (Richard Dedekind, Essays on the Theory of Numbers, tr. W. W. Beman (New York: Dover, pb., 1963), p. 31). Thus the transcendence of the mathematician over mathematics seems to be heightened by this understanding. Mathematics thus appears to be merely a radically subjective creation of the human intellect (contra Husserl's Logische).
The greatest nineteenth century mathematician was Johann Friederich Carl Gauss. Ha questioned the uniqueness of the Euclidean paradigm at the age of sixteen. He developed a rigorous formalism which was extended by the works of Boole, Hilbert, Schroder, Peano, Frege, et al.
As number replaced the Euclidean paradigm, it, too, was later found to be inadequate as a content for the whole of mathematics, and there occurred a shift to logic (Principia Mathematica) as the primary content. The increasing formation of the mathematics of Principia Mathematic disallowed intuitive logic as the fundamental content of mathematics. As the question of which mathematical content, i.e., geometrical figures, numbers, etc., is most appropriate for the ground of mathematics has been historically significant. The contemporary concern is whether or not there is any content at all that is sufficient to ground the whole of mathematics. The end results of the new mathematics is that the Euclidean paradigm has been completely dislodged. No longer are there assumed mathematical objects which one seeks to pattern by schematic mathematical devices. No longer is there an assumed unity to mathematics. There are a multiplicity of different mathematical systems and a multiplicity of different mathematical contents; therefore, mathematics is no longer a universal paradigm of proof/demonstration and Truth. Gödel’s incompleteness theorem has shown the impossibility of formulating one all-inclusive mathematical axiomatic system, thus the demise of the Unity of Science Movement and the 'Received View' of the nature of scientific knowledge and its progress.
The axiomatic method today is a rigorous formalized system having no necessary relationship to intuitive content. Definitions in this system no longer point to mathematical objects; they are simply shorthand devices for handling other symbols. In short, the Euclidean paradigm has been weighed and found wanting. But new paradigms have arisen to compete for our allegiance.
New Mathematical Paradigm
In The Foundations of Mathematical Logic (New York: McGraw Hill, 1963), Rp- 8-16, Haskeil Curry claims that there are two main types of opinion in regard to the nature of contemporary mathematics: (1) The constensive, and (2) the formal. He coined the word "constensive" as a translation for the German word inhaltlich. In the constensive viewpoint, mathematics is assumed to have a definite subject matter or content. The objects with which mathematical statements deal are understood to exist in some sense and the statements are considered true as they conform with the facts of this existence. According to the formal point of view, mathematics is characteristic "more by its method than its subject matter" (ibid.). its objects are normally unspecified, but if they are specified, their exact nature is such that their characterization is irrelevant as far as mathematics is concerned.
Curry divides contemporary constensivism into two groups: (1) Platonism? and (2) critical constensivism. In both the fundamental problem is - "How can the mathematician be understood to be both transcendent to mathematical objects and necessarily conformable to them? (Cf. Same issue with Historicism and Historicity of all Reality in Social Theory.) Kant answers the above question. But since his constructivism developments have emphasized both, the transcendence of the mathematician over the mathematical form and an emphasis on the objective and normative status of mathematical objects. New formulations of the relationship of the mathematician to mathematical form have and continue to influence contemporary philosophical/theological presuppositions.
Three Philosophies of Mathematics
Three influences are: (1) Husserlian phenomenology/existentialism, (2) A. N. Whitehead's process philosophy, and (3) L. Wittgenstein's philosophy of mathematics/ language. Each of these three men were mathematicians. Each was involved in the 'fundamental problem' as it was precipitated by formalistic development in mathematics; each constructed his solution to the 'fundamental problem' with regard to the relationship of mathematicians to mathematical form; sach also continues to influence contemporary philosophico-theologico articulations. G. C. Henry (in his Ph.D. thesis) suggests that the common core of the influence of the above trinity is their mathematico-existentialism.
The fundamental starting point of all forms of existentialism (see my exposition in syllabus) is the position that existence precedes essence. Classical Western thought has presupposed that what is given to man in his experience is an ordered set of qualities (relations, etc.). These qualities as structured in certain ways make up all the objects of human knowledge. A critical question for an existentialist/non-existentialist distinction is whether "existence" is itself one of the qualities (egs. forms, categories, space, time) along with the others. Hegel taught that 'existence' is one of the qualities of reality, not prior to existence, as in Aquinas, et al. Hegel's paradigm opens Western thought up to: (1) Sociology of Knowledge, (2) Cultural Relativism, (3) Structuralism, (4) Contextualism, etc. Kant's transcendent became an immanent transcendent in Hegel's phenomenology. The rational development of scientific progress is impossible, i.e., irrational given Hegel's paradigm (see my forthcoming Historiography of Rationality and Theories of Scientific Progress with Special Attention to T. Kuhn's Concept of Paradigmatic Revolution) . The term "existentialism" has meaning, however, only if that which is understood to precede essence is first of all human existence.
The mathematical philosophies of Husserl, Whitehead and Wittgenstein share three characteristics in common: (1) Mathematico-existentialism is the viewpoint that mathematics is considered to be grounded in human existence. The primary "where" of mathematics is human creativity. (2) Mathematico-existentialism is the full recognition of mathematical relationships as objects of some sect. Though mathematical relationships be grounded in human existence, their objectivity, and apartness from individual human subjectivity, cannot be rationally denied. (C£. objectivity of mathematical relationship - Kant/Idealistic tradition/Dewey's Instrumentalism/ Bridgeman's Operationalism/Wittgenstein's Language Game, etc.) In any form of pragmatism the truth of mathematical instruments, however, is affirmed dependent on their success in use, and not due to their own objectively existing status. (3) Mathematico-existentialism necessarily entails that human existence be found in the world as the ground of any mathematical relationships that exist in the world. Mathematical relationships as objective to the mathematician are to be found in the world.
The Great Demise: Two Cultural Absolutes
In the-process of-twentieth -century scientifico-philosophico development the Two Cultural Paradigms - Christianity and Science - have been reduced to ‘Language Games’, and neither are or can be necessarily universally True. As men continue to create mathematical paradigms/ we exist in a narcissistic, pluralistic, fragmented world. This is our greatest challenge as believers in Jesus Christ.
MATHEMATICAL ORIGINS OF MAJOR MOVEMENTS
Edmund Husserl has had an enormous influence on contemporary philosophy/theology. Though not a Roman Catholic himself, his thought had a most immediate effect on those Catholics who were his followers, and has finally found its way into more official and normative Catholic theology. Through his students Heidegger and Sartre have coma aspects of existentialism as it is applicable to contemporary theology. The biblical theology of R. Bultmann, e.g., reflects the fundamental presuppositions of Heidegger. Finally, through the clarification of the nature and meaning of phenomenology, Husserl has helped bring into existence a method for the examination of World Cultures and World Religions which is presently called Phenomenology of Religion.
I. Mathematician Turned Philosopher: Husserl
A. Studied mathematics, physics, astronomy, and philosophy
B. Three semesters at Leipzig (1876-1878)
C. Transferred to Berlin in 1878 - to study under the most productive mathematicians of the period - Kummer, Kronecker, and Weierstrass.
D. Kronecker influenced Husserl to consider philosophy of mathematics, i.e., The Foundations of Mathematics. Husserl was critical of Kronecker’s 'finite procedures' and 'infinite sets.'
E. March, 1881, left Berlin, went to Vienna to complete doctoral dissertation under Leo Konigsberger, entitled Beitrage zur Theorie der Variationsrechnung. Received doctorate in mathematics - 1883.
F. After 1883 Husserl pursued philosophy. Returned to Vienna to study with Franz Brentano (see A. D. Osborn - in bibliography).
G. F. Brentano (ex-Roman Catholic priest with medieval tendency) was an anti-Kantian, pro-English empiricist (esp. Mill). Kant was inadequate because of disproportionate emphasis on original intuition at the expense of adequate scientific rigor. Brentano's position was - The true method of philosophy is none other than that of natural science, eg., Vienna Circle - 30 years later. Attempted to broaden empiricism via new type of experience - ideale Anschauung. Described via Descriptive Psychology.
H. Brentano - sent Husserl to study with Carl Stumpf at Halle, at which he did inaugural dissertation - Uber den Begriff der Zahl: psychologische Analyse.
I. Husserl's early Constensive Viewpoint determined (?) nature/content of Begriff der Zahl and Philosophic der Arithmetik. Accepted primacy of natural numbers - from Kronecker/ Weierstrass. Natural numbers may be considered to be determined by the acts of counting. "Every philosophy of mathematics must accordingly start with the analysis of the concept of number. Such analysis is the aim of this work. The means which it employs belong to psychology as they must if such a study is to achieve, sure results." (Quote from Osborn, p. 33 - from Begriff der Zahl.)
J. Husserl later realized error of his constensive position. His new method -of seeing and examining - he called phenomenology (see H. Spiegelberg's History of Phenomenological Movement, 2 vols., Leiden - indispensable!). The - controversy -which led to his development was the Cardinal-Ordinal-Controversy concerning priority. Tension between .'subjective construction' of these numbers and their 'objective existence.'
K. Gottlob Frege - a constensive Platonist - saw tension in Husserl between 'objects of mathematics' and psychological clarification of the concepts. Frege complained that "Everything is transformed into something subjective" (Beach/Black, Phil. Writings of G. Frege (Oxford; Blackwell, 1952), p. 79).
L. Ultimate issue - Subjectivism/Objectivism. "The psychological questions which are connected with the analysis of the concepts of multiplicity, unity, and number as far as they are given to us really and not through indirect symbolism" (Phil. of Math., pp. VII-VIII).
"Consider the symbolic ideas of multiplicity and number and attempts to show how the fact that we are almost entirely confined to symbolic concepts of number determines the meaning and object of the arithmetic of numbers" (ibid.). Cf. Relationship of 'objects' to 'formal procedures.'
"The obvious possibility of generalizations or inflections of formal arithmetic, . . . Naturally, from here I had to advance to the more fundamental questions as to the nature of the form of knowledge as distinct from the matter of knowledge, and as to the meaning of the difference between formal (pure) and material determinations, truths, laws" (Logische Untersuchungen, p. VI, p. 55 - now available in E. T.).
M. Husserl's critique of Ernst Schroder's Vordesungen uber die Algebra der Logik (3 vols., Leipzig, 1890-1905), is also applicable to Russell's/ Whitehead's Principia Mathematica. In this work there is a presupposition that the symbolic presentation actually refers to a universal logic from which the whole of mathematics may be deduced. The breakdown of the thesis of Principia Mathematica due to awareness that the system refers only to a very small and technical section of logic is also confirmation of Husserl's criticism of Schroder's logic. There is a great difference, according to Husserl, between a universal logic and a symbolic system. The logic of a formal logical system is not investigated by a formal logical system. Husserl's passion - 'Go to the logical objects themselves.' Husserl moves towards objectivity and the Euclidean paradigm. See Husserl's Ideas -concerning nature of "objectivity in general." The whole realm of formal ontology, which is 'pure logic' and includes the mathesis universalis, is the science of 'object in general.'
N. Nature of Intentionality/Noetic/Noematic.
0. Transcendental Subjectivity/Transcendental Ego is what is primordially most real.
P. Human Existence: Husserl/Heidegger/Sartre. J. P. Sartre, "La Transcendence de L’Ego, Equisse d'une description phenomenologique," Recherches Philosophiques, VI, 1936/1937.
Q. Problem of Unity of Consciousness (Sartre, I‘Etre et Ie Neant).
"All the results of phenomenology begin to crumble if the I is not, by the same title as the world, a relative existent: that is to say, an object for consciousness" (Sartre, ibid.).
R. Husserl thought that Heidegger was still bogged down in "psychologism" (Being and Time; Richardson's classic exposition - Through Phenomenology to Thought).
S. Phenomenology/Realism (Ego out of consciousness)/Idealism (Ego in Consciousness/ Relativism (Lost transcendence of Ego)
T. Husserl's -influence on Roman Catholic thought/comparative religion for Wach, et al. "norms and values were to be explained historically, psychologically, and sociologically" (J. Wach, Comparative Study of Religions (New York: Columbia Univ. Press, 1958), pp. 1-6); R. Otto, Idea of Holy, 1917; M. Muller, Comparative-Mythology, 1856; Sacred Books of the East, 1897; Positivistic period in late 1890's; 1890's revolt against Positivism; Kierkegaard-objectivity without involvement is powerless; E. Troeltsch, "historical flux cannot yield norms for faith/action." Cf. Hermeneutical paradigm shift. Goal of objectivity without Involvement was found to be illegitimate.
U. Wach states that phenomenology, along with neo-Kantianism (both French-Boutroux Circle; and German Positivism, e.g. Cassirer) and the philosophy of Bergson made possible the advent of the third era. He lists Otto, Scheller, van der Leeuw as important. (Wach, ibid., p. 5). He was student of Husserl at Freiburg - doctoral dissertation - "Grundzuge einer Phaenomenologie des Erlsungsgedankens.")
V. Protestant Theology - Heidegger (Marburg 1924/25) /Tillich (Marburg 1924/25)/ Bultmann (Marburg) (see Kegley/Bretall, eds. The Theology of P. Tillich (New York: Macmillan, 1952), p. 14).
II. Process Philosopher; Einstein and Whitehead (Whitehead's philosophy seeks to interpret Einstein's Scientific Paradigm)
A. Whitehead, like Husserl, was a mathematician.
B. Whitshead and C. Hartshorne
C. Nels Ferre/H. R. Niebuhr
D. Breakdown of Logical Positivism after WWII - ca. 1950 (my Historiography of Physical Sciences).
E. Compare - Whiteheadian Philosophy Language Analysis and Existentialism.
F. Whiteheadian Renaissance: John Cobb, Schubert Ogden, Frederick Ferre (see my syllabus. Historiography of Biological Theories).
G. Whitehead's A Treatise on Universal Algebra, 1898 (Hafner, 1960), two years before Husserl's Logische Untersuchungen.
H. Whitehead grounds unity in generalized geometry; and develops position from Treatise to that Principia Mathematicia where, as with Husserl, he seeks the unity of mathematics in logic. In Euclidean paradigm - 'space and geometric representation' has been understood to form the ground/unity of mathematics, whether in terms of geometric constructions of Euclid or Kantian categories.
I. Whitehead's dependence on Georg Riemann's Uber die Hypothesen welche der Geometrie zu Grunde liegen (1854); compare with Whitehead's An Introduction to Mathematics (London: Oxford Univ. Press, 1948). Heart of issue - is how we explain the fact that we do perceive 'the material world' if only Russell's 'between points of space' and 'instants of time' (see Russell's Critique of Leibniz in A Critical Exposition of the Philosophy of Leibniz, London: Allen/Unwin, 1958). Perception is capable of 'seeing' 'beyond' space/time or else there is neither 'universe' nor macro or micro worlds. In addition to Treatise and mathematical concepts, he published "The Axioms of Projective Geometry" - 1906; "The Axioms of Descriptive Geometry" - 1907; and An Introduction to Mathematics - 1911.
J. Decade - 1900-1910 - Russell/Whitehead - Principia Mathematica (Vol. I, 1910: other two in 1912/1913).
K. Influence of Peano/Gödel.
L. "Mathematics is the science concerned with the logical deduction of consequences from the general premises of all reasoning" (“Mathematics” Encyclopaedia Britannica, 11 ed., vol. XVII, p. 880); Whitehead's summary of Principia in Ency. Brit., 14 ed., XV, 1929).
M. Wittgenstein (student of Russell) - Tractatus became standard work of movement called Logical Positivism and had characterized the nature of mathematical propositions as "tautological."
N. Gödel’s 'Incompleteness Theorem' strikes at the root of the fundamental motivation of Principia Mathematica, the desire to reconstensivize mathematics into a unified whole. Gödel’s theorem proves that such a goal is. impossible to attain.
0. 'Relations' to WH. were 'logical notions.' An ‘eternal object’ in WH's Process and Reality is a potential relationship of actual entities.
P. WH's Influence on Theology, egs. Lionel Thomton, Henry Nelson Wieroan, Williaa. Temple, C. Hartshome, Cobb, Ogden, et al. Wh's principle of Organism is impersonal. To mistake process (or organism) with God or to call it Personality, is an example of the error of "misplaced concreteness."
III. Ludwig Wittgenstein and Logical Positivism.
A. Wittgenstein, unlike Husserl and Whitehead, was not a professional mathematician. His concern was foundations of mathematics, ground of all knowledge and communication, namely logic.
B. His philosophy of mathematics was first set forth in Tractatus Logico-Philosophicus (London: 1961); later extended in his Remarks on the Foundations of Mathematics (Oxford: E.T. 1956).
C. At one time it was thought Tractatus solved inherent instability in Principia and also all philosophical problems.
D. Centrality of mathematics in his Philosophical Investigation.
E. Wittgenstein and Logical Positivists: Vienna Circle: 1) Moritz Schlick, 2) Fr. Waismann, 3) Neurath, 4) Zilsel, 5) Feigel, 6) Juhos, 7) Neider, 8) Carnap, 9) Kraft.
Mathematicians: 1) Hahn, 2) Menger, 3) Radakovic, and 4) Gödel.
F. American Parallel Movement: 1) Morris, 2) Langford, 3) Lewis, 4) Bridgman, 5) Kagsl, 6) Reichenbach, and 7) R. V. Mises; England - 1) Russell, 2) Popper, 3) Ayer.
G. Two basic sources of Logical Positivism/Empiricism are: 1) New Symbolic Logic of Principia Mathematica and 2) Empiricism. Differs from old empiricism of Mill and Spencer in that it does not require Mathematics to be empirically verified. Mathematics is independent of experience; and gives no knowledge of reality. The propositions of mathematics are not synthetic but analytic, i.e., tautologous. Pure rationalism which can say everything about mathematics through the new formalism, can say nothing about the experienced world. All knowledge of matters of fact, all scientific knowledge, etc., must come from empirical sources.
H. Logical Positivists recognize radical transcendence of mathematical form over world of phenomena but see no ontological (Platonic) reality in these forms. Cf. Relationships of transcendent forms and the empirical world with human logic and language as it characterizes and utilizes the world. Unity of L. P. movement based in agreement/disagreement with Tractatus. Wittgenstein's critique of his Tractatus in his Philosophical Investigations also destroys credibility of Logical Positivism per se.
I. Wittgenstein's solution - 'The Language Game' - Logic and mathematics are exclusively a creature of man’s invention. Language Game is a form of “social activity where different 'players' have different parts” (Philosophical Investigations, p. 39). Ordinary language is not one unified interrelated homogenous whole, but a vast motley of different language games.
J. What and where of mathematics in Philosophical Investigators and Remarks on the Foundations of Mathematics, Wittgenstein locates mathematics in ordinary language. Technical language (mathematics) are a part of ordinary language. Both are to be understood in terms of Language Games, e.g. intuitive crises and contradictions in mathematics over ambiguity of words or symbols.
K. Wittgenstein's Philosophy of Science - cf. Statements about relations. A great part of scientific theory concerns statements about functions. How are functions used in the Language Game? (Blue and Brown Books, p. 95.) Cf. Language Games, Paradigm Shifts and new discovery? Wittgenstein says that the "Mathematician is an inventor, not a discoverer" (Remarks, p. 167) .
L. Nature of ‘Seeing’ - second part of Philosophical Investigations, section XI. Compare with Kuhn's Paradigm, Incommensurability thesis, etc.
M. Wittgenstein and Contemporary Theology.
1. Tractatus, Verification thesis, and hostility to theology, ethics, and aesthetics.
2. Philosophical Investigation - and concern for analysis of Religious Discourse. Influences on Ayer's Language, Truth, and Logic.
3. John Wisdom (1920-1944 - parable of gardener)/John Hick/William Zuurderg (New Essays in Philosophical Theology).
4. Frederick Ferre/John Hutchison/Paul van Buren (Death of God - period 1960's).
5. Is theism meaningful? (John Wisdom, “God”, Logic and Language, ed. Flew, Oxford, 1951, pp. 187-206); Philosophy and PsychoAnalysis, Oxford, 1957, esp. pp. 248-282.
6. Wittgenstein's Philosophical Investigation and use of picture puzzles -II, XI - to analyze the place of 'interpretation' in 'seeing,' eg. Observation as Theory Laden - see my forthcoming work concerning Paradigms, Theories, Models, World-Views, and Meaning.
7. Meaning of 'Seeing' - Before any analysis of seeing occurs, one must actually "see" in the context of some language game. Cf. objectivity and Tacit Knowledge of Polanyi.
Clearly, the Trajectories of Influence move from philosophies of mathematics to major contemporary theological voices. Philosophy carried on in the twentieth century was placed in our trajectory by Kant/Hegel/Marx/Husserl/Whitehead/Wittgenstein. Now, in our narcissistic, pluralistic age, there is a renewed clamor for Order. What is the nature of and where is this order to be located and grounded and verified is our challenge as Christians in The Decade of the new Millineum. Maranatha!
Boole, G. The Mathematical Analysis of Logic. Cambridge: 1847.
_______. An Investigation of the Laws of Thought. -New York: Dover, 1854 ed.
Cantor, G. Contributions to the Founding of the Theory of Transfinite Numbers. New York; Dover, 1915.
Cowan, J. L. "Wittgenstein's Philosophy of Logic." The Philosophical Review, LXX (July, 1961); 362-75.
Dantzig, T. Number, the Language of Science. Garden City: Doubleday, 1954.
Frege, G. The Foundations of Arithmetic. New York: Harper Torch, 1960.
Gödel, K. On Formally Undecidable Prepositions of ‘Principia Mathematica’ and Related Systems. Edinburgh: Oliver Boyd, 1962.
Nagel, E., and Newman, J. R. Gödel’s Proof. New York: New York Univ. Press, 1958.
Quine, W. Van Onnan. Methods of Logic. New York; Henry Holt and Co., 1950.
Russell, B., and Whitehead, A. N. Principia. Paperback ed.
Husserl, E. Logische Untersuchungen. Halle: Vol. I, 1900; Vol. II, 1901.
_______. Ideas. Paperback.
Lauer, Q. The Triumph of Subjectivity. New York: Fordham Univ. Press, 1958.
Osborn, A. D, The Philosophy of Edmund Husserl in Its Development from His Mathematical Interests to His First Conception of Phenomenology in Logical Investigation. New York: International Press, 1934.
Whitehead, A. N. Process and Reality. New York: 1960.
_______. Science and the Modern World. New York: 1960.
Wittgenstein, L. Philosophical Investigations. New York: Macmillan, 1953.
_______. Remarks on the Foundations of Mathematics. Oxford: Blackwell, 1956.
_______. Tractatus Logico-Philosophicus. London: Routledge and Kegan Pub., 1961.
(see also my Research Bibliography in Philosophy; and the Bibliography on Wittgenstein.)
MATHEMATICAL FOUNDATIONS OF 20TH CENTURY PHILOSOPHY:
PRIOR TO GÖDEL - 1932
It has been claimed that there are two models of truth: (1) Empiricism Model, which yields results that are never more than probable, and (2) Rationalism Model, which yields true results but provides no new information that is not contained in its a prioris. Since one does not know any a priori transcendent truths and transcendence is not empirically observable, transcendent truth claims do not express anything. The only alternative for any transcendence claims is then mysticism. Such a line of reasoning has a number of flaws, but an important contradiction to it can be seen in Gödel’s Theorem which demonstrates that the human mind can transcend the purely formalized logic to which arithmetic has been reduced. The historical background of this proof will be given, followed by a discussion of the proof itself, and then the consequences of the proof on mathematics and truth claims.
Greek geometry under Euclid is an important background to what will later be discussed. Geometry was reduced to a complete axiomization in which all new theorems could be obtained by logical operations upon the initial axioms. The axiomization of geometry was considered the model of science at its best which should be emulated in all fields at all levels. This remained an ideal towards which little was realized until the nineteenth century.
Gottlob Frege sought to systematize arithmetic by showing that all arithmetic notation can be reduced to purely logical ideas. All axioms in mathematics would be deduced from a small number of basic propositions. If this project was to be carried out, arithmetic would need a consistency proof. Frege thought he had established one until Bertrand Russell discovered a paradox in it. A consistent system is one in which every formula is not derivable from the axioms of the system. In an inconsistent system, both a formula and its negation may both be derived from its axioms. There is therefore no distinction between valid and invalid formulae. Since anything would be possible in such a system it could not form an adequate basis for science or any other field where the law of contradiction applies. The discovery of paradoxes such as Russell's put pressure on those who hoped to demonstrate the consistency of arithmetic. Added pressure came from another direction. Euclidean geometry had not been demonstrated consistent, but it appeared to describe the world in which people lived consistently. With the advent of systems founded on different axioms such as non-Euclidean geometry, it was no longer obvious that the system spoken of described what was obviously true. If one is going to use a mathematical system that appears to be false (by "common sense" standards), it will require stronger logical support to gain acceptance.
In 1910, with the publication of Principia Mathematica. Whitehead and Russell seemed to have accomplished the reduction of mathematics to logic. Mathematics was presented as a deductive system with a limited number of axioms, theorems followed from these by rules of inference, and deductions were reflected in a chain of formulae, or a calculus. The following challenge was put forth by Russell: “If there are still those who do not admit the identity of logic and mathematics, we may challenge them to indicate at what point, in the successive definitions and deductions of Principia Mathematica, they consider that logic ends and mathematics begins.” The claim is a bold one, and if true would be the complete accomplishment of Frege's original design. Furthermore, if mathematics is reducible to logic, the consistency of mathematics is reduced to that of the consistency of logic. That this did not prove to be the case will be seen later.
Basic to an understanding of Gödel is a knowledge of the work of David Hilbert. Hilbert, like Russell and Whitehead, identified symbolic logic and number theory. He inverted the idea of the calculus in Principia Mathematica to prove deductive theorems in an arithmetic system rather than solving arithmetic problems by deductive reasoning. He did this by making the rules of symbolic manipulation in a calculus represent the logical rules of inference, the initial formulae of the calculus correspond to the axioms of the logical system, and the proof that a formula occurs in the sequence of formulae of a calculus yields a proof that a proposition can be deduced from the axioms of a deductive system. The symbols in the calculus have only those values assigned to them by definition in the axioms of the system. Theorems and postulates are "strings" of meaningless marks which are constructed according to the rules for combining the elementary signs of the system. The idea is to eliminate the possibility of any concealed meanings that might bring unexpected results. In a completely formalized system it was hoped that unexpected results would be prevented by the construction of the system. Hilbert then originated the idea of metamathematics to describe statements about the calculus. A mathematical statement is a statement made in terms such as 1+1=2 while a metamathematical statement is a statement about the terms and symbols of the system such as "1+1=2 is an arithmatic proposition," Mathematics is a formal calculus, while metamathematics might apply to the description of that formal calculus. He then suggested the idea of an absolute consistency proof which would demonstrate that two contradictory formulae could not be derived from the axioms of the same system. Such a proof would have to make use of only the axioms of the formalized system and would be finitistic, that is, it could not make reference to an infinite number of structural properties of formulae or to an infinite number of operations of formulae. An absolute consistency proof is contrasted with a relative one in that it is not built on a model of another system involving other axioms. A relative proof might claim that the system of whole numbers is consistent if Euclidean geometry is. The problem with such a proof is that the model itself may contain concealed contradictions. Through the formalization of mathematics based on logic presented in Principia Mathematica Hilbert hoped to establish an absolute consistency proof for mathematics.
To form a formalized system as that pictured by Hilbert or actually formulated by Whitehead and Russell requires a basic system of axioms. Giuseppe Peano formed a set of such axioms which were to be incorporated into Principia Mathematica. These axioms form a monomorphic set, that is, they form a set of axioms which can not meaningfully be enlarged by new axioms expressed in the same symbolism. Additional axioms in such a system will always be consequences of the formulae already stated as axioms. A little-mentioned predecessor to Gödel’s work was that of Julius Dedekind. He proposed the idea of a "Bild" which is the image of a system S as mapped in the system S1. A rule or function is created which automatically maps functions from one system into the other. Furthermore, the map contains a mapping of the mapping function. The mapping function is a metamathematical function with regard to at least one of the two systems. A system that is capable of mapping statements about itself is usually one open to paradoxes.
A further background to Gödel’s theorem is the area of paradoxes. Russell's and Richard's paradoxes are among the better known. The former is formulated as x r if x x and then by substituting r into the equations: r r if r r. In English the interpretation would be "x is- an element of the set r if it does not contain itself". The second sentence asks whether r can be an element of itself: "r is an element of r if r is not an element of r." The paradox results from the use of self-reference by asking whether r can apply to itself. Richard's paradox makes a similar move in that it has an ambiguous distinction between statements about the system and statements within the system. If these formulations could be made without confusing metamathematical and mathematical statements, the result would be either an undecidable proposition or a contradiction. That is essentially what Gödel did.
The first step in the explanation of this theorem is connected with Hilbert's idea of a formal calculus that represents metamathematical propositions within the functions of a mathematical system. The axioms of the system are those, of Peano added to the logic of Principia Mathematica. The signs of the system are the natural numbers although they are used to represent symbols, functions, and functions about functions. The basic signs of the system are represented by the first seven odd numbers, variables by prime numbers greater than thirteen raised to the power of the variable, sentential variables are squares of primes greater than ten, and predicate variables are represented by cubes of primes greater than ten. All formulae of the system are recursively defined, that is, they are recursively defined by two earlier functions of the system or they can be obtained by substitution from an earlier function. The first four propositions of the system are based on the definition of a recursive system and on the logical relationships of negation, equivalence, and "or." Since functions are represented by numbers within this arithmatic system, metamathematical propositions can be represented as functions dealing with the number representing a function. What can or can not be done in the calculus also can or can not be done in the whole number arithmatic that the calculus represents. Gödel’s calculus has been described as co-ordinate metamathematics to compare its mapping of metamathematical propositions in mathematics to the Cartesian mapping of algebraic propositions in geometry. Since primes are the only numbers used and every number has a unique factorization, it is always possible to discover what function is represented by a particular number; More needs to be said about the idea of recursiveness. Applied to numbers it means that each number in a sequence can be specified by knowing the first number and a rule which specifies the (k+l)th number in terms of the kth number and or k. Applied to functions recursiveness means that it is the final term in a finite term in a finite series of functions in which each function is defined by a rule involving two earlier functions. The advantage of this definition is that an infinite sequence can be decided without working out its totality, but simply by analyzing its rule of construction.
While the essential proposition in Gödel’s proof in the sixth, which states the existence of an undecidable proposition, the fifth needs to be stated because its application to two functions is the basis for the sixth, The fifth proposition insures the recursive nature of the system. If a number series R is recursive, it can be proved (if true) by constructing a finite series of prepositional statements with the axioms, and if it is false its negation could be so constructed. The proposition stated is: “To every recursive relation R(x1....xn ) there corresponds an n-place
relation-sign r (with free variable u1...un) such .......
that for every n-tuple of numbers (x1....xn ):
the following hold: R(x1....xn ) Bew Sb u1...un
R(x1....xn ) Bew Neg Sb u1...un
The first statement corresponds to the idea that if the function is true it is provable by a series of substitutions of the number-sign .(for which Z is the abbreviation) for the free variable u. The second statement corresponds to the proof of its negay tion if it is not true. The bar on top (note: the bar is underneath to accommodate this paper’s type face only) of R in the second sentence means negation. The "Bew" stands for "is provable" the Sb for "substitution" and the Neg for "negation". Connected with this proposition is stated the definition of - consistency in a class of formulae c. It amounts to the statement that an individual substitution involving the class -sign a can not be a consequence of c at the same time as the negation of the generalization of a. This means that a function can not be true for individual values that are substituted into an equation at the same time as the negation of the generalization of the function. The function can not be proved false for every value yet true for some individual values. Generalization asserts that the class of numbers represented is the universal class, that every number is a member of the class denoted by a particular sign.
Next Gödel states his key proposition; Proposition VI: “To every -consistency recursive class c of formulae there correspond recursive class-signs r, such that neither v Gen r nor Neg (v Gen r) belongs to Flg(c) (where v is the free variable of r).” A class-sign is a formula of Principia Mathematica with one free variable which is of the type of the natural numbers (remembering that natural numbers can represent functions, variables, or sequences of functions). "F1g" means "is a consequence of". The proposition says that there can be stated formulae which are not consequences of the system and that their negations are not either. According to proposition five it would seem that any recursive formula would be a consequence of the basic axioms of the system, and that if it were not, its negation would be, The relation Q is then defined which states that Q(x,y ) is equivalent to:
xB Sb y Z(y).
The relation is based on earlier definitions Gödel has already stated, all of which are recursive, and therefore define a recursive function. The sole symbol that has not been previously introduced is "B" which means "proof." The relation states that there is not a proof of the function at a particular value of x. Since the relation is recursive, it is subject to the specifications of the two formulae stated in Proposition V, This operation is carried out using the free variables 17 and 19 and using q as a relation-sign to yield:
xB Sb y Z(y)
Bew Sb q Z(x)
for the first formulation and
xB Sb y Z(y)
Bew Neg Sb q Z(x) Z(y)
for the second.
To simplify these equations, the class-sign p is introduced which is defined as "17 Gen q" and has the free variable 19. Second, p’s relation to the above is defined by the recursive class-sign r which is identified as:
Sb q Z(p).
From these two relations the following are derived:
Sb p 19 = 17 Gen r and Sb q 17 19 = Sb r 17.
Z(x) Z(y) Z(x)
These values can be substituted directly into the two equations that were obtained from defining Q in terms of Proposition V to obtain the following:
x B (17 Gen r) Bew Sb r 17
and xB (17 Gen r)
Bew Neg Sb r 17.
The two statements amount to saying that if the function is performed its implications will include the provability of the negation of a specified substitution and the provability of that specified substitution. If 17 Gen r is provable, then so are the two opposite statements it implies. If two opposite statements are possible in the system it meets the standards of being an inconsistent system. Since the system was defined as being a consistent one the statement must be unprovable, At this point it might be pointed out that the process started by applying a proposition that stated provability to a relation that stated the lack of proof for a particular function, Through the use of substitution and generalization the function is manipulated to require a number for the substitution in its proof that would be the number of a negated provability statement. In essence the statement says: "I am not provable in this system (c)." Since if the system is consistent, the statement can not be proved, by informal logic what it asserts is true. Since it is true and since no false formula can be proven, if the system is consistent the negation of the statement is also unprovable.
It may be obvious that there are comparisons between Gödel’s theorem and the "Liar Paradox" which asserts: "What I am now saying is false." The differences keep this from being a paradox. Truth and falsehood are not properties defined on the formal system, provability and its negation are. Provability is not an empty statement such as asserting one's own falsehood, but it is an examinable property that can be examined without reference to meaning. It should be remembered that one property of the system was that the marks in the calculus should be uninterpreted, or meaningless, Concepts like truth and falsehood are -meaning concepts that are unallowable in a formalized system.
The error of other mathematical paradoxes is also avoided. It is not a metamathematical statement that is being considered within the system. It is the representation of a metamathematical statement of a proposition's unprovability by a mathematical formulation within the system. However, since the calculus does represent the arithmatic system of Principia Mathematica, undecidability in ;the calculus demonstrates undecidability for the latter system. In other words, one can state true but unprovable within the system statements in mathematics. The ability to demonstrate such a statement was one purpose of mapping on the natural number system in the first place. The system can work in reverse as well. One knows the statement; "this statement is unprovable in this system" is a true one by metamathematical reasoning, but it is still a true statement expressed within the mathematical system. The consequences of all this for arithmetic are rather important.
Proof or Consequences
The first point this theorem obviously makes is that there are undecidable formulae in the formalization of arithmatic and therefore the basic axioms of the system must be incomplete. It might be suggested that the system could be made complete by the addition of further axioms, such as making the unprovable formula an axiom of the system. One could always state a new theorem that stated that it was improvable on the newly formulated formal system. One purpose of postulating the system of formulae c was so that systems based on Principia Mathematica but including additional axioms could also be covered. One can readily see that for any formal system a similar theorem could be constructed that would be unprovable within the system and yet true. One might suggest adding the idea that the true formulae of elementary arithmetic form a productive set. This would eliminate Gödel’s theorem from the system, possibly. First of all, this is a metamathematical proposition. "Productive" seems to be a term of meaning, which is not likely to be mapable within the axioms of a system. Second, "productive" is a relative term. Many decisions in mathematics rest on a decision as to whether the information necessary for a proof is at hand. In such a situation, the inability to prove a formula would be productive information. The choice as to what results are not productive is at best going to be random. It would seem that Hilbert's goal of the complete formalization of mathematics based on logic has been demonstrated impossible. The basic axioms might be reconstructed as a polymorphic set which would be inconsistent at points and allow for contradictions within the system. If the system of axioms were weakened be dropping the possibility of multiplication within the system and making addition the only basic function, the system can be shown to be decidable for all formulae. By the standards of Principia Mathematica, a system lacking multiplication is a defective system, it is not comprehensive enough to carry out the functions of natural numbers. The other direction would be to strengthen the system by the addition of axioms. Various means of doing this have already been suggested and seem to be unhelpful. The conclusion is that it is impossible to develop an axiomization of elementary number theory which is both consistent and complete.
Gödel’s theorem is not a unique oddity. The same idea has been proved in three different ways by Gödel, Rosser, and Kleene. Furthermore, it has been seen that similar theorems could be formulated in any formalized system, Gödel’s theorem is not the only one that can not be proved within even the system of Principia Mathematica. Since the consistency of the system is what prevents his theorem from being provable, a relation can be stated that Gödel’s theorem implies the consistency of the system. Since Gödel’s theorem is not provable within the system, by the rule of detachment, the consistency of arithmetic is not provable within the system. If there is no way of testing the decidability of a formula within a system, that system is said to lack a decision process. Since it has no mechanical test for all formulae, elementary number theory lacks a decision procedure. For practical purposes, this means that there could be false statements within the system, but being unprovable, they might go undetected.
It would seem that the formalist approach itself is in error. The intuitive side of mathematics can not be eliminated, nor can it be formalized into a completely logical system. In answer to the challenge quoted from Bertrand Russell earlier as to where logic ended and mathematics begins, an answer can be posed. Logic ends with quantification theory which includes only the function of addition and mathematics starts with the inclusion of multiplication.
A consistency proof for natural numbers has been posited, but it involves extending the system of natural numbers into transfinite numbers. Transfinite ordinals are numbers that are induced after the first infinity of natural numbers and are represented by + n (n representing the nth trans-finite ordinals. After an infinite number of these transfinite numbers a new series is induced, and these are followed by another series of numbers, Gentzen's consistency proof involves a triple induction of such transfinite ordinals. Obviously such a proof is not dealing with ordinary arithmatic, nor is it making use of a simple set of monomorphic axioms. This attempt goes back to a suggestion made by Hilbert about the possibility of a semi-formal system which has a rule of infinite induction based-on an infinity of premises. An informal system was not the original goal of the formalist approach, nor is it what most .people would assume they are working with in mathematics.
One attempt to get around Gödel’s proof is to dismiss it by saying axioms are not the result of deductive proof to begin with. That axioms are not provable deductively is why they are axioms. The question is not whether they are deductively provable, but whether they can be so selected as to make everything else in mathematics deducible from them. It could also be suggested that axioms are tested by the functioning of the system for which they form the basis. The same author claims that metamathematics is too rigid, it only needs to establish a high degree of dependability but does not need to be deductively valid in every situation. This is the same as saying that although arithmetic contains impredicative definitions it is still usable because its axioms obtain results and help organize mathematics. No one would deny that arithmetic has been productive, but is a pragmatic basis the foundation of arithmetic, and via arithmetic, of science? Wittgenstein has suggested that the fear of contradictions is a superstition and that a system that produces contradictions is still usable. This statement is clearly based on his concept of "language games" which fragments reality into a number of non-intersecting and even contradictory systems each described by their own "language," If this is what the above authors are suggesting, they may not worry about contradictions in mathematics, but they are allowing all of knowledge to fall apart.
Another insufficient alternative is to say that Gödel has simply shown that the ideal of a formalized system can not be reached, but the attempt is still valuable. Simply put, this means that mathematics is on the same level as a child trying to learn to swim — that he tries — is what counts. But is mathematics a discipline in which trying is what counts, or is the standard truth and falsehood or verifiability and falsifiability? Mathematics has been the model for science. However, science does not keep theorems because they were "good tries" but because they offer explanatory power. Contradictions within a system do not yield such.
The conclusions Gödel came to about his theorem should be noted. In the original paper, Gödel stated that Hilbert's ideal had not been refuted because Hilbert had desired a finite consistency proof and there might be such a proof outside of the system of Principia Mathematica. This begs the question as to whether it would be a finite proof of that system. In addition, Gödel himself suggests that similar theorems can be suggested for any system and since these theorems depend on the consistency of the system in which they are expressed, the consistency of any formalized system will be implied by a theorem provable by informal logic but not by any finite formal system. A second conclusion of Gödel is that the consistency of mathematics rests on Platonic realism. The axioms of mathematics describe some "well-determined reality." Based on this model of mathematics he suggests that there are axioms of mathematics yet to be discovered. If mathematics has an objective reality and is not just an artificial human construction then there is the possibility that its nature has not been fully recognized. Along the same line is the explanation that a hierarchy of systems are needed to explain mathematics. Gödel has taken a different route than those previously mentioned. Mathematics is not to be considered defective. It is to be seen as part of a larger reality which includes mathematical and formal logical deductions, but also includes higher levels of reasoning. The choice seems to be throw out all formal systems, including mathematics, because they are incomplete, or to accept some transcendence claims which could give human reason power to solve what may not be solved in the formal system.
The Nature of Nature and Human Nature
If transcendence is required to maintain mathematics it will have a definite effect on such statements as those referred to in the introduction of this paper. Knowledge does not exist in two completely unrelated Leibnizean categories of "truths of reason" and "truths of experience." The two can be interrelated by a transcendent mind. Mathematics formulated as a deductive logical system would be a "truth of reason." Yet mathematics is needed to handle that which is quantified in empirical science of "truths of experience." The two can not be exclusive categories. If transcendence is allowed, they may both be considered statements about a reality designed by a common Creator whether in the thought structures of the mind or in a natural world that can be analyzed by those thought structures.
The Positivistic reduction of all truth to that which is quantifiable is in trouble if elementary number theory is the means of quantification, Since Gödel’s theorem implies the consistency of number theory and Gödel’s theorem is provable by informal logic, or transcendent human reason, the consistency of elementary number theory requires transcendence to be proved, Transcendence is a metaphysical concept, and Positivism refuses to accept metaphysical statements because they are not quantifiable. As Ayer put it, they say nothing about the world, in the technical sense they are "nonsense."
It turns out that the system of quantification which was used to outlaw metaphysics requires metaphysics to prove its own consistency. The nature of mathematics formulated by the human mind to comprehend nature is such that it requires transcendence.
This has a direct bearing on the question of whether one can make a machine model for the human brain, Since Gödel’s theorem asserts a formula which is decidable by human reason but not within the system of arithmetic it asserts a difference in kind between the human mind and any formal system such as the mathematics with which a machine functions. Those who favor a mechanistic model for man have raised various objections to this assertion.
A first suggestion is that the theorem makes no difference as to a machine's abilities, one must just expect occasional wrong answers from the machine. One might suggest that the computer is in a "fallen" state and its reason has been damaged. More seriously one might ask now it will be decided that the computer has given a wrong answer. If it will require human reason to check out the answers given by the machine, then the difference between man and machine is affirmed.
Another possibility is to make the machine independent of one consistent logic. This amounts to a claim that the human mind is more complex than a machine. One way of making such a machine would be to allow it to entertain certain non-deductive inferences which would be dropped if they led to a contradiction. How the computer would decide which unproved assertions it would accept would create a problem as to which of two opposite and improvable propositions it would accept. What if it were to accept the false member of a pair of unprovable within the system propositions? Turing seems to suggest that humans transcend the ability of a single machine at a time. The idea seems to be that there could be another machine which includes as one of its axioms the Gödel theorem which is unprovable on another machine and since a theorem follows from itself the theorem would be provable in that system. Such a machine would still operate according to the specifications of a formal system for which a new Gödel theorem would be postulated unprovable by that machine. Even if an infinite set of axioms were allowed to cover the process of formulating new Gödel proofs, since the machine has to operate according to specifics, a finite rule would be needed that determined how such axioms would be formulated. A new Gödel proof could then be made for the new system including its rule for generating infinite axioms which would not be unprovable on the system. If the machine were allowed infinite specifications and changes it might solve a Gödel equation. But what would be meant by an infinite machine? Secondly, if mind is reducible to a finite brain, why would this be necessary? A second argument from Turing is that the machine may have limitations, but so does the human mind. The problem is not whether both have limitations, but whether the human mind transcends the machines limitations. Turing admits that the difference is not one of speed of operation, as a quantitative difference between man and machine might suggest. If the difference is not quantitative, then there is a qualitative difference in kind between mind and machine. It has been postulated that a machine that could be equipped with means to observe and make statements about its own structure would be capable of solving Gödel’s theorem. This ability to scan its own structure will also need to be formally described and therefore be subject to previously mentioned limitations. Another suggestion has been that a machine could be programmed to reject the last axiom in any proof that leads to an inconsistency. This is certainly not how the mind works, however. If one discovers that a certain means of proving is inconsistent, one tries not to use it for any proof, not just in the proof in which it leads to inconsistency. A man is not admired for his ability to change his premises every time he thinks they would lead to inconsistencies and then restore them later.
The construction of a Turing machine has been suggested as a way around Gödel’s theorem's limitations on a machine. A Turing machine is an idealized machine which has no specified structure, but which has a number of states corresponding to logical states, a tape that can be scanned, and a precise list of operations. If one defines intuition as tearing empirically, the Turing machine could be programmed to empirically observe itself and therefore be able to decide Gödel theorems. A formal system will have to state how the machine can learn induct ively, and a Gödel’s theorem can always be made for a formal system. Even if this limitation were not present, one might ask if it is by empirical observation that the human mind is able to solve Gödel’s theorem. Another route using a Turing machine is to program a Turing machine to correspond to all the operations a man is capable of carrying out. The machine would then be capable of carrying out any proof the human mind could also prove. This simply avoids the issue. Is it possible to make a machine that can be programmed to carry out all of human reason? Turing's model of the human mind and his model for how to test a computer against a human mind both assume a priori that the mind is a machine and create a system where no transcendence is allowed before the question is asked as to whether machine and mind are different. Turing makes a final attempt in trying to equate mind and machine by claiming that more complex machines may reach a "critical mass" at which point they will cease to be mechanically predictable and become creative. Turing does not explain why the machine would become supercritical simply because its size was increased. His comparison is with a fission pile which due to its nature jean become critical at a predictable point. This is not true of a computer. There is no reason for a computer with greater capacity to quit performing as it is programmed. The conditions under which any given part of the computer will operate will not be altered as they are for any section of an atomic pile. Even if this were to happen, one could claim that what was produced was no longer a machine. The question was whether minds and machines are the same, not how they are produced.
It would seem that there is a fundamental difference between the human mind and any machine. There is no way around the fact that a machine is a formal system and a Gödel’s theorem can be formulated for any formal system.
The human mind can solve the equation and the machine can not by its very nature. The mind then does have the ability to transcend any cause-and-effect system that operates on a mechanistic model of the universe (whether Walden II or Brave New World as contemporary models or Jonathan Edwards' predestination as an earlier one). The mind is not reducible to the equations of biochemistry and biophysics. Freedom and creativity are possibilities of the transcendent mind.
If one is interested in working through Gödel’s full proof, one should start by going directly to Gödel. Those who have tried to explain him make major changes in his methodology and symbolism. I have made one change and that is to change his sign to which is the standard way of signifying implication. Braithwaite's "Introduction" is helpful because he explains other symbols that Gödel uses in a non-standardized way. Extremely important is to make a separate copy of the meanings of his abbreviated German functions. Most of Gödel’s commentators recast his substitution formulae to make the function substituted into and the number sign which is substituted appear as a set of co-ordinates. This is to make comparison with the Richard Paradox easier.
Bibliography on Kurt Gödel
Ambrose, Alice and Lazerowitz, Morris. Fundamentals of Symbolic Logic. New York: Holt, Rhinehart, and Winston, Inc., 1962. Concludes that elementary number theory is incomplete. The consistency of arithmetic can not be proved within the system of arithmetic.
Angell, Richard. Reasoning and Logic. New York: Appleton-Century-Crofts, 1964. The complete axiomization of elementary number theory is not both consistent and complete.
Ayer, Alfred Jules. “Critique of Theology”. In Meaning and Knowledge, pp. 46-49. Edited by Ernest Nagel and Richard Brandt. New York: Harcourt, Brace, and World, Inc., 1965. States that all statements other than tautologies based on a prioris are only probable. Metaphysical assertions say nothing about the world.
Bachelard, Suzanne. A Study of Husserl’s Formal and Transcendental Logic. Evanston: Northwestern University Press, 1968. Implies that Gödel can be ignored as long as the system can still function in other area.
Bennett, Albert and Baylis, Charles. ...
Bochenski, J.M. A History of Formal Logic. Translated by Ivo Thomas. New York: Chelas Publishing Co., 1970. A less formal condensation by Gödel’s of his Original Theorem.
_______. A Precis of Mathematical Logic. Translated by Otto Bird. New York: Gordon and Beard, Science Publishers, Inc., 1959. Explains various symbols and functions as Gödel uses them.
Carnap, Rudolf. Introduction to Symbolic Logic and Its Applications. New York: Dover Publishers, Inc., 1958. As a result, a polymorphic model for math must be intrduced.
_______. Logical Foundation of Probability. London: Rutledge and Kegan Paul, LTD., 1951. Explains various functions used in Gödel’s proof.
Church, Alonzo. “A Note on the Entsheidungproblem.” The Journal of Symbolic Logic 1(1): 40-41. Establishes a general class of unsolvable equations similar to Gödel.
Church, Alonzo. “Correction to a Note on the Entsheidungproblem.” The Journal of Symbolic Logic 1(3): 101-102. A corrective to the above. Distinguishes between constructive and non-constructive proofs of undecidiabilty.
Chwistek, L. “A Formal Proof of Gödel’s Theorem”. The Journal of Symbolic Logic 4(2): 61-68. Symbolism is more complex than Gödel’s. Contains a helpful chart of variables in the back of the article.
Dreyfus, Hubert L. What Computers Can't Do: The Limits of Artificial Intelligence. 2nd ed. New York: Harper 6 Row Publishers; Harper Colophon Books, 1979.
Encyclopedia of Philosophy, 1972 ed. S.v. "Gödel’s Theorem," by J. Van Heijenoort. Gives a good historical background to the problem. Gödel’s symbols are greatly changed. Discounts any epistemological significance for the proof. Contains a good bibliography.
Feterma, Solomon. "Systems of Predictive Analysis." In The Philosophy of Mathematics. Edited Jaakko Hintikka. Oxford: Oxford University Press, 1969. The effects of Gödel’s theorem on set theory are analyzed.
George, F. H. Automation, Cybernetics and Society. London: Leonerd Hill, LTD., 1960. Sees no reason to posit Gödel’s proof as a problem to the construction of machine model for the brain. All you have to do is expect a wrong answer once in awhile.
Gödel, Kurt. On Formally Undecidable Propositions of Principia Mathematica and Related Systems. Translated by B. Meltzer and Introduction by K. Baraithwaite. New York: Basic Books, Inc., Publishers, 1962. A translation of Gödel’s original paper. The introduction paraphrases the proof and explains his more problematic symbolism. The proof is complicated by the use of abbreviations for the German names of thirty-four different expressions.
Henkim, Leon. "Completeness in the Theory of Types." In The Philosophy of Mathematics. Edited by Jaakko Hintikka. Oxford; Oxford University Press, 1969. Valid formulae can be made, but not formal ones.
Hilton, Alice Mary. Logic, Computing Machines and Automation. Washington: Spartan Books, 1963. Avoids the problem through fragmentation.
Jaki, Stanley. Brain, Mind and Computers. South Bend, Indiana; Gateway Editions, 1969. A comprehensive defense against those who identify the brain with a machine. Refutes various attempts to circumvent Gödel.
Kneale, William and Martha. The Development of Logic. Oxford; The Clarendon Press, 1968. Explains changes Gödel has make in standard symbolism. Connects Gödel’s work with Peano's axioms. Gives an extended explanation of the proof.
Kreisel, George. "Mathematical Logic: What Has it Done for the Philosophy of Mathematics." In The Philosophy of Mathematics. Edited by Jaakko Hintikka. Oxford: Oxford University Press, 1969. Gives Gödel’s interpretation of the effect his proof has on our conception of mathematics.
Lucas, J. R. "Minds, Machines, and Gödel." Philosophy 36(137):112-27. Defense of the mind against the possibility of any conceivable kind of machine model.
Mehlberg, Henryk. Logic and Language. Dordrecht: D. Reidel Publishing Co., 1962. Gives a good history of Gödel’s proof. Tries to formulate the limitations as to where it applies.
Nagel, Ernest and Newroan, James. "Gödel’s Proof," Scientific American 194 (6):71-86. A very extended history of the idea of a complete axiomization of mathematics. Gives a good non-technical explanation of Gödel’s proof. Is hesitant to think of this proof as a barrier to a machine model of the brain.
_______. Gödel’s Proof. New York; New York University Press, 1958. A further development of the above article. Gives helpful explanatory notes. Also treats the connection of Gödel’s proof to the Richard Paradox.
Putnam, Hilary. "Minds and Machines." In Minds and Machines, pp, 72-97, Edited by Alan Ross Anderson. Englewood Cliffs; Prentice-Hall, Inc., 1964. Claims that the mind can not transcend a Turing machine. Does not really deal with the issue.
Quine, Willard Van Orman. From a Logical Point of View. Cambridge: Harvard University Press, 1964. Mathematics is reduced to "myth-making" to continue to function. Compares Gödel’s proof to other paradoxes.
_______. Mathematical Logic. Cambridge; Harvard University Press, 1947. Due to Gödel, logical truth will always be informal.
_______. Methods of Logic. New York: Holt, Rinehart, and Winston, 1963, Contains a good bibliography. Applies Gödel’s proof to set theory.
Rogers, Hartley. "The Present Theory of Turing Machine Computability." In The Philosophy of Mathematics. Edited by Jaakko Hintikka. Oxford: Oxford University Press, 1969. A practical suggestion as to how a computer might function to avoid the problem Gödel presents.
Rosser, Barkley. "An Informal Exposition of Gödel’s Theorems and Church's Theorem." The Journal of Symbolic Logic 4(2): 53-60. Major emphasis is the system of propositions within which the proofs are constructed.
Sayre, Kenneth and Crosson, Fredrick. The Modeling of the Mind. Notre Dame: University of Notre Dame Press, 1963. Numerous articles pro and con on the relationship of mind to machine. Reprints Lucas's article.
Smart, J. J. C. Philosophy and Scientific Realism. London: Routledge and Kegan Paul, 1963. Sees no essential difference between mind and machine. Suggests that a machine could be designed to restructure itself to solve undecidable formaulae.
Smullya, Raymond. "Languages in Which Self-Reference is Possible." In The Philosophy of Mathematics. Edited by Jaakko Hintikka. Oxford: Oxford University Press, 1969. A good non-technical description of Gödel’s proof.
Strauss, James. "A Puritan in a Post-Puritan World." In Grace Unlimited. pp. 243-64. Edited by Clark Pinnock. Minneapolis: Bethany Fellowship, Inc., 1975. Uses Gödel’s proof to defend freedom and transcendence.
_______. “Modern and Contemporary Philosophy, Mathematical Origins of 20th Century Philosophy.” Syllabus, Lincoln Christian Seminary, 1980.
Tarski, Alfred, Introduction to Logic. Translated by Olaf Helmer. New York: Galaxy Books, 1965. Although arithmetic is incomplete it can function as though it were not.
Tarski, Alfred; Robinson, Andrzej; and Mostowski. Undecidable Theories. Amsterdam: North-Holland Publishing Company, 1968. Gives a simplified form of Gödel' s theorem and some of its corollaries.
Thomason, Richmond. Symbolic Logic: An Introduction. Toronto: The Macmillan Company, 1970. His diagrams tend to make explanation more difficult. Contains a good index of symbols.
Turing, A. M. "Computing Machinery and Intelligence." In Minds and Machines, pp. 4-30. Edited by Alan Ross Anderson. Englewood Cliffs; Prentice-Hall, Inc., 1964. Feels there is no real distinction between mind and machine in that both are limited.
Whitehead, Alfred North and Russell, Bertrand. Principia Mathematica. Cambridge: The University Press, 1970. The basis Gödel started from for the disproof of the idea that mathematics can be completely systematized.
Wilder, Raymond. Introduction to the Foundation of Mathematics. New York: John Wiley and Sons, Inc., 1967. Recasts and reviews Gödel’s proof.
Wittgenstein, Ludwig. Remarks on the Foundations of Mathematics. Translated by G. M. Anscomb. Cambridge: The Massachusetts Institute of Technology Press, 1967. Contradictions in mathematics are no problem to him as long as it is usable.
 Alfred Jules Ayer, "Critique of Theology," in Meaning and Knowledge, ed. Ernest Nagel and Richard Brandt (.New York: Harcourt, Brace and World, Inc., 1965), pp. 46-49.
 Ernest Nagel and James Newman, Gödel’s Proof (New York: New York University Press, 1958), p. 5. Gary North, ed. Foundations of Christian Scholarship (Essays in the Van Til Perspective) (Vallecito, CA; Ross House Books), 1979, pp. 159-188.
 Ibid, p. 42.
 Encyclopedia of Philosophy. 1972 ed., s.v. "Gödel’s Theorem," by J. Van Heijenoort.
 Nagel and Newman, Proof, p. 51.
 Ibid, p. 15.
 R. B. Braithwaite, Introduction to On Formally Undecidable Propositions of Principia Mathematica and Related Systems, by Kurt Gödel (New York: Basic Books, Inc., Publishers, 1962), p. 2.
 William and Martha Kneale, The Development of Logic (Oxford: The Clarendon Press, 1968), p. 724.
 Nagel and Newman, Proof, p. 43.
 Kneale, Logic, p. 714.
 Braithwaite, Introduction, pp. 2-3.
 Nagel and Newman, Proof, p. 13.
 Ibid, p. 27.
 Ibid, p. 32.
 Ibid, pp. 26-34.
 Ibid, p. 33.
 Encyclopedia, "Gödel," p. 349.
 Nagel and Newman, Proof, p. 25.
 Encyclopedia. "Gödel," p. 349.
 Braithwaite, Introduction, p. 4.
 Kneale, Logic, p. 390.
 Ibid, p. 469.
 Ibid, p. 470.
 Encyclopedia of Philosophy, 1972 ed., s.v. “Logical Paradoxes”, by John Van Heijenoort.
 Nagel and Newman, Proof, pp. 60-63.
 Kurt Gödel, On Formally Undecidable Propositions of Principia Mathematica and Related Systems (New York; Basic Books, Inc,, Publishers, 1962), pp. 38-41.
 Ibid, p. 38.
 Nagel and Newman, Proof, p. 71.
 Gödel, Proof, p. 46.
 Ibid, p. 49.
 Braithwaite, Introduction, pp. 4-5.
 Ibid, p. 7.
 Ibid, p. 12.
 Ibid, p. 37.
 Gödel, Proof, pp. 55-56.
 Ibid, pp. 33-34. The capitalized letters are abbreviations for German words for metamathematical functions or their Gödel numbers which are translated on these pages. To follow Gödel’s explanation requires a copy of this section or the functions and equations are meaningless.
 Ibid, p. 57.
 Braithwaite, Introduction, p. 27.
 Gödel, Proof, p. 57.
 Ibid, p. 39.
 Ibid, pp. 33-34.
 Ibid, p. 58.
 Ibid, pp. 33-34.
 Ibid, p. 58.
 Ibid, pp. 58-59.
 Ibid, p. 59.
 Barkley Rosser, "An Informal Exposition of Gödel’s Theorems and Church's Theorem," The Journal of Symbolic Logic 4(2):58.
 Kneale, Logic, p. 718.
 Ibid, p. 719.
 Nagel and Newman, Proof, p. 90.
 Gödel, Proof, p. 69.
 Nagel and Newman, Proof, p. 93.
 Ibid, p. 58.
 Gödel, Proof, p. 61.
 Braithwaite, Introduction, pp. 14-15.
 Hartley Rogers, "The Present Theory of Turing Machine Computability," The Philosophy of Mathematics, ed. Jaakko Hintikka (.Oxford: Oxford University Press, 1969), p. 141.
 Henryk MehLberg, Logic and Language (Dordrecht D. Reidel Publishing Co., 1962), p. 80.
 Rudolf Carnap, Introduction to Symbolic Logic and Its Applications (London: Rutledge and Kegan Paul, LTD., 1951), p. 174.
 Willard Quine, Methods of Logic (New York: Holt, Rhinehart, and Winston, 1963), p. 247.
 Richard Angell, Reasoning and Logic (New York; Appleton-Century -Crofts, 1964), p. 603.
 Rosser, "Exposition," p. 55.
 Gödel, Proof, p, 70.
 Quine, Methods, p. 244.
 Andrzej Mostowski, Sentences Undecidable in Formalized Arithmetic (Amsterdam: North-Holland Publishing Co., 1964), p. 3.
 Kneale, Logic, p. 724.
 Stephen Kleene, Introduction to Metamathematics (New York: D. Van Nostrand Co., Inc., 1952), p. 476.
 Encyclopedia, "Gödel," p. 355.
 Mehlberg, Logic, p. 83.
 Ibid. p. 84.
 Solomon Feterma, "Systems of Predictive Analysis," The Philosophy of Mathematics, ed, Jaakko Hintikka (Oxfords Oxford University Press, 1969), p. 97.
 Ludwig Wittgenstein, Remarks on the Foundations of Mathematics, trans, G. M. Anscombe (Cambridge: The M.I.T. Press, 1967), pp. 51-53.
 Suzanne Bachelard, A Study of Husserl's Formal and Transcendental Logic (Evanston: Northwestern University Press, 1968), p. 53.
 Gödel, Proof, p. 71.
 Abraham Fraenkel, "Epistemology and Logic," Logic and Language, ed, Henryk Mehlberg (Dordrecht: D, Riedel Publishing Co,, 1962), p. 8.
 Ibid., p. 9.
 Encyclopedia. "Gödel," p. 356.
 Ayer, "Critique," p. 47.
 F. H. George, Automation, Cybernetics, and Society (London; Leonerd Hill, LTD., 1960), pp. 87-88.
 Mary Alice Hilton, Logic, Computing Machines and Automation (Washington: Spartan Books, 1963), p. 384.
 J. R. Lucas, "Minds, Machines, and Gödel," Philosophy 36 (137): 119.
 A. M. Turing, "Computing Machinery and Intelligence," Minds and Machines, ed. Alan Ross Anderson (Englewood Cliffs: Prentice-Hall, Inc., 1964), p. 16.
 Lucas. "Minds," p. 117.
 Turing. "Machinery," p. 16.
 Hilary Putnam, "Minds and Machines," Minds and Machines, ed, Alan Ross Anderson (Englewood Cliffs: Prentice-Hall, Inc., 1964), pp. 72-73.
 Lucas. "Minds," p. 121.
 J. J. C. Smart, Philosophy and Scientific Realism (London: Routledge and Kegan Paul, 1963), pp. 128-29.
 Ibid, pp. 119-20.
 Putnam,:"Minds," p. 77.
 Turing, "Machinery," pp. 4-10 and 26-27.
 Ibid, p. 25.
 Lucas, "Minds," p. 126.
 See S. Jaki, Brain, Mind, and Computers, (South Bend, Indiana: Gateway Editions, 1969); and Dr. James Strauss, Modern and Contemporary Philosophy, "Mathematical Origins of 20th Century Philosophy," (Syllabus, Lincoln Christian Seminary, 1980), pp. 54ff.; and Hubert L. Dreyfus, What Computers Can't Do: The Limits of Artificial Intelligence, 2nd ed. (New York: Harper and Row Publisher's; Harper Colophon Books, 1979.