Bibliography of A. A. Fraenkel (p. ix-x)--Axiomatic set theory. Zur Frage der Unendlichkeitsschemata in der axiomatischen Mengenlehre, von P. Bernays.--On some problems involving inaccessible cardinals, by P. Erdös and A. Tarski.--Comparing the axioms of local and universal choice, by A. Lévy.--Frankel's addition to the axioms of Zermelo, by R. Mantague.--More on the axiom of extensionality, by D. Scott.--The problem of predicativity, by J. R. Shoenfield.--Mathematical logic. Grundgedanken einer typenfreien Logik, von W. Ackermann.--On the use of Hilbert's [epsilon]-operator in scientific theories, (...) by R. Carnap.--Basic verifiability in the combinatory theory of restricted generality, by H. B. Curry.--Uniqueness ordinals in constructive number classes, by H. Putnam.--On the construction of models, by A. Robinson.--Interpretation of mathematical theories in the first order predicate calculus, by T. Skolem.--The elementary character of two notions from general algebra, by R. Vaught.--Foundations of arithmetic and analysis. Axiomatic method and intuitionism, by A. Heyting.--On rank-decreasing functions, by G. Kurepa.--On non-standard models for number theory, by E. Mendelson.--Concerning the problem of axiomatizability of the field of real numbers in the weak second order logic, by A. Mostowski.--Non-standard models and independence of the induction axiom, by M. O. Rabin.--Sur les ensembles raréfiés de nombres naturels, par W. Sierpinski.--Philosophy of logic and mathematics. Remarks on the paradoxes of logic and set theory, by E. W. Beth.--Logique formalisée et raisonnement juridique, par R. Feys.--Im Umkreis der sogenannten Raumprobleme, von H. Freudenthal.--Process and existence in mathematics, by H. Wang. (shrink)

Ernst Fraenkels Neopluralismus zählt zu den 'Klassikern'. Worin zeigt sich seine Anschlussfähigkeit? Antipluralistische Traditionsbestände sind wirkmächtig geblieben. 'Lobbyismus' hat immer noch einen schlechten 'Beigeschmack'. Das Problem von Grundkonsens und Streit stellt sich insbesondere bei politischen Gegnern, die mit 'Freund-Feind-Ideologien' agieren.

First published in 1999. Routledge is an imprint of Taylor & Francis, an informa company.

Periodic tables (PTs) are the ‘ultimate paper tools’ of general and inorganic chemistry. There are three fields of open questions concerning the relation between PTs and physics: (i) the relation between the chemical facts and the concept of a periodic system (PS) of chemical elements (CEs) as represented by PTs; (ii) the internal structure of the PS; (iii)␣The relation between the PS and atomistic quantum chemistry. The main open questions refer to (i). The fuzziness of the concepts of chemical properties (...) and of chemical similarities of the CE and their compounds guarantees the autonomy of chemistry. We distinguish between CEs, Elemental Stuffs and Elemental Atoms. We comment on the basic properties of the basic elements. Concerning (ii), two sharp physical numbers (nuclear charge and number of valence electrons) and two coarse fuzzy ranges (ranges of energies and of spatial extensions of the atomic orbitals, AOs) characterize the atoms of the CEs and determine the two-dimensional structure of the PS. Concerning (iii), some relevant ‘facts’ about and from quantum chemistry are reviewed and compared with common ‘textbook facts’. What counts in chemistry is the whole set of nondiffuse orbitals in low-energy average configurations of chemically bonded atoms. Decisive for the periodicity are the energy gaps between the core and valence shells. Diffuse Rydberg orbitals and minute spin–orbit splittings are important in spectroscopy and for philosophers, but less so in chemical science and for the PS. (shrink)

Strong Turing Determinacy, or ${\mathrm {sTD}}$, is the statement that for every set A of reals, if $\forall x\exists y\geq _T x (y\in A)$, then there is a pointed set $P\subseteq A$. We prove the following consequences of Turing Determinacy ( ${\mathrm {TD}}$ ) and ${\mathrm {sTD}}$ over ${\mathrm {ZF}}$ —the Zermelo–Fraenkel axiomatic set theory without the Axiom of Choice: (1) ${\mathrm {ZF}}+{\mathrm {TD}}$ implies $\mathrm {wDC}_{\mathbb {R}}$ —a weaker version of $\mathrm {DC}_{\mathbb {R}}$.(2) ${\mathrm {ZF}}+{\mathrm {sTD}}$ implies that every (...) set of reals is measurable and has Baire property.(3) ${\mathrm {ZF}}+{\mathrm {sTD}}$ implies that every uncountable set of reals has a perfect subset.(4) ${\mathrm {ZF}}+{\mathrm {sTD}}$ implies that for every set of reals A and every $\epsilon>0$ :(a)There is a closed set $F\subseteq A$ such that $\mathrm {Dim_H}(F)\geq \mathrm {Dim_H}(A)-\epsilon $, where $\mathrm {Dim_H}$ is the Hausdorff dimension.(b)There is a closed set $F\subseteq A$ such that $\mathrm {Dim_P}(F)\geq \mathrm {Dim_P}(A)-\epsilon $, where $\mathrm {Dim_P}$ is the packing dimension. (shrink)

Hume's explicit pronouncements about truth are few and unenlightening. In a well-known passage near the beginning of Book III of the Treatise he writes that ‘Reason is the discovery of truth or falsehood. Truth or falsehood consists in an agreement or disagreement either to the real relations of ideas, or to real existence and matter of fact.’ Hume's main concern in this passage, however, is not with the concept of truth, but with his thesis that moral distinctions are not derived (...) from reason: he introduces his reference to truth only with a view to showing that our ‘passions, volitions and actions … being original facts and realities, complete in themselves,’ are not susceptible of the agreement and disagreement spoken of, and therefore cannot be said to be true or false, in conformity with or contrary to reason. The account of truth given here is not elaborated, and perhaps not even thought to need elaboration. Similarly with another passage a few pages earlier in which Hume says that ‘Truth is of two kinds, consisting either in the discovery of the proportions of ideas, considered as such, or in the conformity of our ideas of objects to their real existence’. Here again his interest is not in truth itself but in ‘curiosity, or the love of truth,’ the passion which, Hume says, ‘was the first source of all our inquiries’. Hume seems to take it for granted that nothing more needs to be said about what it is to discover ‘the proportions of ideas, considered as such,’ or about the circumstances in which we can speak of there being ‘conformity’ or the lack of it between our ideas of objects and their real existence. So far as he is concerned the central point to grasp is the distinction between propositions which have to do with relations of ideas and those which express, or purport to express, matters of fact. Once this distinction is clear, the nature of truth is supposed to be plain. (shrink)

Few major philosophers show evidence of having studied the works of their predecessors with special care, even in cases where they were subject to particular influences which they were ready to acknowledge. Hume knew that he was working in the tradition of ‘some late philosophers in England, who have begun to put the science of man on a new footing’—‘Mr Locke, my Lord Shaftsbury, Dr Mandeville, Mr Hutchinson, Dr Butler, &c.’ But there is not much sign in the Treatise or (...) elsewhere in Hume's writings of any close acquaintance with the works of these authors; the presumption must be that he had read them at some time and extracted the main ideas, but was not in the habit of returning to their texts. He had something more important to do, namely to work at philosophical problems of his own. Similarly Kant, though he said that the Critique of Pure Reason was not meant to be ‘a critique of books and systems, but of the faculty of reason in general’, had clearly felt the impact of the thought of some important past philosophers, but equally had never spent much time in finding out just what these philosophers had to say. Plato, Aristotle, Descartes, Locke, Leibniz and Hume all get fairly frequent mention in his pages. But Kant takes his knowledge of Plato and Aristotle from J. J. Brucker's Historia critica philosophiae, a six-volume compilation which first appeared in 1742, or from doubtful sources such as Mendelssohn's doctored translation of the Phaedo, and though he doubtless knew the more recent authors at first hand clearly felt no need to study them in any depth. This was true even of writers to whom he attributed a particular importance, such as Leibniz and Hume. The references to Hume in the Critique and Prolegomena are all disappointingly general, and though the summary of Leibniz's philosophy in the section called ‘The Amphiboly of Concepts of Reflection’ has a certain force, it is not documented with references to Leibnizian texts. Kant knows that there is a difference between the views of the historical Leibniz and those which constituted the ‘Leibnizian-Wolffian system’ of his successors. But he is not very curious about the difference, or inclined to explore it. (shrink)

In writing on a subject in which the most significant words have been used in quite different senses by modern authors, I think it most prudent to begin by defining my terms. By rhythmical prose I mean all prose in which the writer consciously follows a definite scheme in order to obtain particular cadences at the close of the period or within it, and this whether the favoured cadences are marked by quantity or by accent. I subdivide rhythmical prose into (...) metrical and accentual. The latter term explains itself. In spite of Aristotle's familiar opposition of ṕÅθμóσ and μτρ¿ν, I use the word ‘metrical’ of quantitative prose, both because the term ‘quantitative prose’ seems to me cumbrous, and because to the speakers of any modern language metre is the natural opposite of stress accent, whether in prose or in verse; and Aristotle's terminology, which takes no account of stress accent, may reasonably be displaced by the modern distinction. (For the beginnings of this later usage in post-classical theoreticians cf. (shrink)

Effort is then well-nigh indescribable. Not wholly so, else it would be meaningless. Description is a matter of degree: who can fully describe red or wet? To be sure, description comes down in the end to the pointing to certain given qualities or relations or events which are just there. All connotation rests on denotation, though it may be something more. But the unique positive thing about effort is its originality; to which indeed we can point, since every one experiences (...) it in himself. Yet common to all men though it is, it is so intangible! more spirit than body, more force than substance. We should perhaps best call it not a datum but a factum, or better still, a fiat, a doing rather than a deed, yet not merely a becoming, for we see things coming to be, we hear a melody progressing and touch a moving object. Effort, dynamic as it is, is not just the process, rather it is--we must re-iterate--the start, the initiative of the process. Nor is the experience of it properly called mystical. Even if mystical experience were indescribable, as has been wrongly alleged, these would be two quite different indescribables, the one receptive, the other originative. All we can say perhaps is that effort is analogous to the push that starts the ball rolling, not the motion of the ball, not the acceleration, but the enforcement of these. Its witness does not lie in its value as explanation of the motion; the witness is direct, the immediate experience which every man has. It seems to be no object of scientific investigation in the technical sense. It cannot be measured by itself, as acceleration and motion are measured. That is why physics has abandoned the old notion of force, the physical analogue of effort, replacing it by mass-acceleration. That is why the rationalists in modern philosophy look askance at this direct experience of man, so intangible that we can draw from it no logical implications. For it has no structure. Its characteristic is not clearness nor distinctness. We might be tempted to say it is like a point in space, too small to be envisaged, yet not a zero; but it is more like a dot or spark, so small as to be just seen, so small as to have no visible spread, no content that we can analyze to draw out rational consequences connecting it with other things. Here too we see why the contemplative thinker seldom believes in free choice, which the natural man naturally accepts. Free choice is the kind of thing that might well belong to this experience; free choice is origination. Not necessarily that all effort is freely made. The baby exercises no free choice when he sucks hard at the nipple, nor the adult when he wrestles for his life with an enemy. But they want, need, desire; that is the point. Desire--that is the essence of effort. Desire has the originative quality; it comes as it were out of one's self and from no other source, it is what we want, our self insisting on its own. But the business is so deep within that it is easily overlooked, or deliberately excluded from the gaze, and without a qualm. We might say it is too near to be noticed, or too deep down to obtrude itself, as the water in the well is easy to see, but not the spring at the bottom of the well. Effort disappears from the scene when the thinker is calmly gazing; but when he acts it is felt, and strongly felt. It is in fact the very essence of the self, the well-spring of its worth and its growth. A man is what he does. His worth lies in what he makes. All that he is responsible for, all that he earns, is credited with, or guilty of, comes from this innermost source. So thinks the practical man, so we all believe when we are conducting our lives in intercourse with one another and with nature. But for the contemplative attitude there is here too little of clear articulation, of object-matter for analysis; no use can be made of it in a deductive system. So the system-builders discard it. (shrink)