This book grew out of a graduate student paper [261] in which I set down some criticisms of J. R. Lucas' attempt to refute mechanism by means of G6del's theorem. I had made several such abortive attempts myself and had become familiar with their pitfalls, and especially with the double edged nature of incompleteness arguments. My original idea was to model the refutation of mechanism on the almost universally accepted G6delian refutation of Hilbert's formalism, but I kept getting stuck on (...) questions of mathematical philosophy which I found myself having to beg. A thorough study of the foundational works of Hilbert and Bernays finally convinced me that I had all too naively and uncritically bought this refutation of formalism. I did indeed discover points of surprisingly close contact between formalism and mechanism, but also that it was possible to under mine certain strong arguments against these positions precisely by invok ing G6del's and related work. I also began to realize that the Church Turing thesis itself is the principal bastion protecting mechanism, and that G6del's work was perhaps the best thing that ever happened to both mechanism and formalism. I pushed these lines of argument in my dis sertation with the patient help of my readers, Raymond Nelson and Howard Stein. I would especially like to thank the latter for many valuable criticisms of my dissertation as well as some helpful suggestions for reor ganizing it in the direction of the present book. (shrink)

Husserl’s first work formulated what proved to be an algorithmically complete arithmetic, lending mathematical clarity to Kronecker’s reduction of analysis to finite calculations with integers. Husserl’s critique of his nominalism led him to seek a philosophical justification of successful applications of symbolic arithmetic to nature, providing insight into the “wonderful affinity” between our mathematical thoughts and things without invoking a pre-established harmony. For this, Husserl develops a purely descriptive phenomenology for which he found inspiration in Mach’s proposal of a “universal (...) physical phenomenology.” To account for applications to any domain, Husserl envisages a theory of all possible deductive systems, which he develops extensively in his Göttingen lectures wherein he engages with Hilbert’s work on deductive systems for geometry, real arithmetic, and physics. This leads Husserl to formulate claims of decidability and proofs of completeness for various arithmetics that result from his analysis of Kronecker’s general arithmetic. Careful attention to these proofs seem to show that Husserl was not oblivious to problems that underlie our incompleteness theorems, namely that of showing that some inversions of his algorithmic arithmetic are undefined. His growing preoccupation with the issue of a pre-established harmony between mathematical thought and reality motivate his pursuit of a “supramathematics” of all possible complete theory forms to demystify such harmony, by having such a form on hand for describing any empirical domain. He soon decides that a transcendental idealism of nature will reveal the wonderful affinity of thoughts and things comprising such harmony, to be a wonderful “parallelism of objective unities and constituted manifolds of consciousness.” But the paradoxes of logic and set theory cloud the clarity of mathematics, which Weyl would restore with Brouwer’s intuitionism and Hilbert with his metamathematics. Husserl informed Weyl that his student Becker had formulated a phenomenological foundation not only for Weyl’s generalization of relativity theory but also for the Brouwer-Weyl continuum. But Weyl eventually rejected much of Becker’s work, especially when it became clear that his phenomenological intuitionism could not account for the success of Hilbert’s transfinite mathematics in quantum physics. Becker responded to this “crisis of phenomenological method” with his mantic phenomenology celebrating the magic of mathematical mysticism, which Husserl finally rejects in favor of a pluralistic phenomenology of mathematics and nature. (shrink)

The metamathematical theorems of Gödel and Church are frequently applied to the philosophy of mind, typically as rational evidence against mechanism. Using methods of Post and Smullyan, these results are presented as purely mathematical theorems and various such applications are discussed critically. In particular, J. Lucas's use of Gödel's theorem to distinguish between conscious and unconscious beings is refuted, while more generally, attempts to extract philosophy from metamathematics are shown to involve only dramatizations of the constructivity problem in foundations. More (...) specifically, philosophical extrapolations from metamathematics are shown to involve premature extensions of Church's thesis. (shrink)

Husserl’s lifelong interest in Kant eventually becomes a preoccupation in his later years when he finds his phenomenology in competition with Neokantianism for the title of transcendental philosophy. Some issues that Husserl is concerned with in Kant are bound up with the works of Lambert. Kant believed that the role played by principles of sensibility in metaphysics should be determined by a “general phenomenology” on which Lambert had written. Kant initially believed that man is capable only of symbolic cognition, not (...) intellectual intuition. Lambert saw an increasing need in mathematics for symbolic cognition as exemplified in his proofs of the irrationality of π and e. Kant takes from Leibniz and Lambert an unrestricted notion of construction, allowing him to view mathematics as constructing its concepts in intuition, while Lambert’s proofs convince him that all mathematical problems are eventually solvable. Husserl criticizes Kant’s intuitionism for its inadequate accommodation of meaning to intuition, which he redresses with his theory of categorial intuition. This may improve on Kant’s intuition of the axiom of parallels but not so clearly on his spatial intuition. Husserl opposes Kant’s view of space as the form of outer intuition with his own view of it as the form of things. Husserl’s exploration of the geometry of visual space, which involves his earliest uses of epoché and reduction, converges however with Hilbert’s logical analysis of Kantian spatial intuition in leading to Euclidean spatial judgments. Hilbert’s analysis leads him to affirm the solvability of all well posed mathematical problems, a thesis complicated by the outbreak of logical paradoxes. Untroubled by such paradoxes, Husserl develops a supramathematics of all possible deductive systems whose completeness implicitly would also provide solutions of all such problems. Husserl’s full transcendental turn coincides with his realization that in effecting his Copernican turn, Kant was really the first to detect the “secret longing” of modern philosophy for a phenomenological clarification of being. Husserl now finds that Kant’s transcendental deductions presuppose a pure ego not adequately analyzed by Kant that survives the phenomenological reduction. Husserl’s idealism leads him to develop his own intuitionism, which adds to Kant’s, a “method of clarification” of mathematical concepts intended to clarify difficult impossibility proofs, but neither Husserl nor Kant base arithmetic on time. Husserl’s critique of the room left in Kant’s idealism, for things in themselves, leads to his own monadological solution of the problem of intersubjectivity. Husserl’s final judgment on Kant is that his form of transcendental idealism did not enable him to achieve absolute subjectivity through a genuine transcendental reduction. (shrink)

Analytic functions, typified by reflexes, use stable inherited structures which are organized for adjustments in behavioral space and time. Individually acquired integrative functions, typified by conditioning, symbolism, and reasoning, are wholly internal and may use any nervous tissue not differentiated for other specific activities. They must be defined relati‐vistically. The highest members of both classes are in the cerebral cortex, the first in the relatively stable projection centers, the second in the more labile areas of elaboration where there is no (...) inflexible localization of function. When consciousness emerges within the second class, the mechanism employed probably must be described in accordance with still unknown relativistic principles. (shrink)

“All experience is an arch wherethro’ gleams” an untravell'd world and through which come the joyous adventures of life and also grief and pain. Since all that we know and hope to know and think we know must come through this arch and since the primary task of science is the validation and enlargement of knowledge, science is vitally interested in this experience and its interpretation. This interest stems not from the philosopher's epistemology but it is strictly operational. We want (...) to know what kind of experience yields reliable knowledge when checked against other experience and applied in practical adjustment to things as they are and to prediction of the probable future course of events. In short, the field of natural science embraces everything that comes within the range of human experience. Its specific functions are to codify this mixed experience, to search out those uniformities which yield classification, verification and prediction, and from these observed uniformities to construct by sound reasoning verifiable laws, hypotheses and theories, that is, to find the meaning of the experiences had. First we want to know the facts, but what we are after in the upshot is the interpretation of the facts. (shrink)

In an interesting article on Kierkegaard and the “Existential” Philosophy, contributed to the number of Philosophy for July 1941, Miss Dorothy Emmet counselled her readers to make themselves acquainted with the Journals of the famous Danish thinker, now rendered accessible to Englishmen ignorant of his language by the translation of Mr. Dru. I have taken her advice and am grateful to her for it. I am not indeed convinced that this self-revelation of a remarkable personality can be ranked among the (...) great autobiographies of the world. It seems to me to fall, if only on account of its lack of unity as a work of literary art, below the standard which such an estimate of it would imply. But it is undoubtedly both impressive in itself and important in view of the influence which its author, little known in his lifetime beyond a circle of the parochial narrowness of which he was painfully conscious, has come, long after his death, to exert upon the spiritual and intellectual life of Europe to-day. The present article, however, will not be concerned with the exposition or criticism of this or other writings of Kierkegaard, but with thoughts, more or less suggested by my reading of him, respecting the notion of God and its significance for philosophy. (shrink)

In the first part of this article I propose to describe two strongly contrasted situations in the world of thought, one of fifty years ago and the other of to-day. In the second I shall submit to my readers some reflections suggested by the contrast between the two.

Just three years ago I contributed to this Journal a few remarks on the problem of the relation of “God” to “the World.” I propose in the present article to add some observations on the closely connected problem of the relation of “God” to “Man”; especially in view of the theory, by no means a new one, but at the moment much in evidence , that when we speak of God what we have really in mind is our own human (...) nature or some part or aspect of it, imaginatively objectified as a distinct or independent reality. (shrink)

I Suppose that most believers in God, if asked what is the relation of God to the world, would reply that he is its Creator and its Lord. But, like all the language in which we express our religious convictions, the language of this reply is plainly metaphorical. It calls up the picture of a human artificer or artist, the image of a human ruler or proprietor. And yet it needs but little reflection to perceive that there must be essential (...) differences between a human artificer or artist, who has a material to work in ready to his hand, and One who makes all things, as it is said, “ out of nothing “; between a human ruler or proprietor. (shrink)

Many of us have lately been reading, hearing or listening in to the present Master of Trinity's Presidential Address to the British Association's meeting at Oxford; and perhaps some reflexions suggested thereby may not be out of place in this number of Philosophy.

This year is being celebrated by a large number of our fellow-countrymen as the centenary of a movement, associated with the name of the University of Oxford, of which, although in its first stage it might easily be mistaken—and has often been mistaken—for a mere wave of theological and ecclesiastical reaction within the Established Church of England, the attentive historian of the nineteenth century must take account as in fact a very powerful influence in the religious and, no less really (...) though to a less degree, in the social and political life of the whole nation. Considerable, however, as is the importance which may justly be attributed in other respects to what is known as the Oxford Movement, the professed student of philosophy may be excused if he is chiefly struck by the apparent remoteness of its original leaders from the currents of speculative thought characteristic of the period in which it began its course. There were perhaps among them only two who can be named as contributors to philosophical literature in the technical sense now commonly borne by the term “philosophical”; and the contributions even of these two can scarcely be said to have taken their place among the works to which an ordinary teacher of philosophy would be likely to direct the attention of his pupils. To these two, however, John Henry Newman and William George Ward, I propose to devote here a few pages which may be found not without interest to readers of Philosophy. (shrink)

The division of cognitive labor is fundamental to all cultures. Adults have a strong sense of how knowledge is clustered in the world around them and use that sense to access additional information, defer to relevant experts, and ground their own incomplete understandings. One prominent way of clustering knowledge is by disciplines similar to those that comprise the natural and social sciences. Seven studies explored an emerging sense of these discipline‐based ways of clustering of knowledge. Even 5‐year‐olds could cluster knowledge (...) in a manner roughly corresponding to the departments of natural and social sciences in a university, doing so without any explicit awareness of those academic disciplines. But this awareness is fragile early on and competes with other ways of clustering knowledge. Over the next few years, children come to see discipline‐based clusters as having a privileged status, one that may be linked to increasingly sophisticated assumptions about essences for natural kinds. Possible mechanisms for this developmental shift are examined. (shrink)