Evans attempted to develop a classical unified field theory of gravitation and electromagnetism on the background of a spacetime obeying a Riemann-Cartan geometry. In an accompanying paper I, we analyzed this theory and summarized it in nine equations. We now propose a variational principle for a theory that implements some of the ideas that have been (imprecisely) indicated by Evans and show that it yields two field equations. The second field equation is algebraic in the torsion and we (...) can resolve it with respect to the torsion. It turns out that for all physical cases the torsion vanishes and the first field equation, together with Evans’ unified field theory, collapses to an ordinary Einstein equation. (shrink)

We show that there is a limit lemma for enumeration reducibility to 0 e ', analogous to the Shoenfield Limit Lemma in the Turing degrees, which relativises for total enumeration degrees. Using this and `good approximations' we prove a jump inversion result: for any set W with a good approximation and any set X< e W such that W≤ e X' there is a set A such that X≤ e A< e W and A'=W'. (All jumps are enumeration degree jumps.) (...) The degrees of sets with good approximations include the Σ0 2 degrees and the n-CEA degrees. (shrink)

In an almost forgotten passage of the Postenor Analytics (Bk I, ch. III) Aristotle argues against 'another school', according to which it is possible to proof things 'by each other and in a circle'. His logical refutation of this opinion became so dominant in the Western philosophical tradition, that the 'vicious circle' has always deemed a crime since. A scientific demonstration has to be built on firm premisses in order to deduce conclusions from them in a straight, ongoing proces, in (...) which one does not have to return to the premisses for a reconsideration of them on the basis of their implications. Although Aristotle does not mention names, he in fact reacts against Plato and the mainstream of Greek thought, which stresses the essential circularity of human argumentation. In the first, the historical part of the treatise the roots of this circular logic are unearthed in the work of Heraclitus and Parmenides, who both defended a ring structure of their metaphysical explication of nature. Truth is described as well-rounded, because you can follow it in a circle, passing through each of its links in turn, back to your startingpoint. The association of circular motion with thought is fully present in the work of Plato, who speeks many times about the 'periforas tès dianoias' and the essential interdependance of our ideas. Not the theory of an anonymous alternative school but Aristotle's own rectilinear formal logic was the exceptional, untraditional standpoint, which, moreover, was in conflict with his explication of divine thinking as 'noèsis noèseôs'. In modern times Hegel, Nietzsche and Heidegger were the main representatives of an anti-Aristotelian attitude. All three reject the logical prohibition of the circular argument, because it is inadequate to the life of concrete scientific thinking. The path of science always turns around towards its origins and results in a correction of its fundamental hypotheses on a higher level. Circularity is considered as a hallmark of sound logical procedure, as a characteristic of excellence in human understanding. The famous 'hermeneutic circle' is interpreted as the modern equivalent of the Hegelian 'Kreislauf and the Greek ' to par'allèlôn'. In the second, the more systematic part of the text the circular nature of our cognitive and scientific behaviour is demonstrated along three different lines. Reflection on the natural language with its finite stock of words, whose meanings are structurally connected, leads us to the so called semiotical circularity. Natural language is a complex network in which circular argumentation is not only unavoidable, but even the only means of explication. We have to work with the material contained in our dictionary, in which the items are defined by each other. The second reason is constituted by the system-orientation of science. In science we try to systematize our knowledge by means of formalisations and abstractions. The development of science can be characterised as a succession of systems, in which every element gets its meaning from the whole organisation. If we accept systematicity as the arbiter of truth, our logic is anti-fundamentalistic and the path of our thought is in fact circular (or better : spiralistic). The third adstruction of our theory is afforded by the coupling, that is realised in a rational proof. We draw conclusions from the coherence of our perceptions, from the reciprocal implications of our ideas. Never any of our ideas is without the repercussions of the movements elsewhere in the field. When we proof something, we also disproof and correct our startingpoints at the same time. The tradition of the circular nature of human thinking is very old, but not antiquated. The accusation of committing a fallacy in the case of circularity in normal and in scientific thinking is unfouded. It is now the formal logicians turn to defend his claim. (shrink)

Outside France the epistemology of G. Bachelard is unknown ; in France his influence is considerable, especially on philosophers like L. Althusser, M. Foucault, G. Canguilhem, J. Hyppolite, M. Serres, G. G. Granger, D. Lecourt and many others. Bachelard occupies a strategic point on the crossroads of all theoretical debates concerning science. The fact that he seems to give satisfactory answers on the problems which have risen after the breakdown of the logical-positivistic philosophy of science, justifies an exposition and evaluation (...) of his original contribution to philosophy. The author distinguishes the following items. 1. The determination of Bachelards philosophy as a scientific philosophy which is wedded to a history of the sciences, especially the natural sciences. 2. The 'systematicity' as the criterium of science against other, traditional, criteria like empiricalness, logical deducibility, correspondence with reality and so on. 3. The rectification-principle : the formation of a scientific system cannot be conceived otherwise than as the restructuring or reorganisation of the ruling system or system-sets. 4. The transformation of scientific knowledge shows many discontinuities in all its phases and branches. Bachelard calls them ruptures. 5. The translation of a theory into another, more coherent and comprehensive one, is baptised as a 'dialectisation' of the concept. By a dialectisation a system is both generalised and specified. Formal logic, which is based on identity and the principle of the excluded middle, is not able to interprete this dynamic aspect of scientific thinking. 6. Central in Bachelards philosophy is also the concept of recurrence. Each new organisation of the scientific system (global or partial) sheds new light on their history and their logical foundations. History has to be rewritten after each progress. Recurrence, however, also has systematic implications. Science always desimplifies its own evidences. 7. The scientist has to demolish the obstacles which he made himself by surcharging the content of the concepts not in use and by not assimilating them in the system. 8. Bachelards philosophy of science is both idealistic and realistic ; the phenomenology becomes phenomenotechnique under the hands and in the brains of the scientist. 9. He always denies (negativity) the earlier theories and objects by incorporating them in new relationships. 10. This constructive aspect of theory formation is the same in natural science and mathematics. Bachelard opposes the logical-positivistic idea that these are methodically dissimilar. In a critical commentary the author discusses the question of 'dialectical logic' in the sciences, in relation to some recent research in this field by I. Lakatos en Errol E. Harris. In his opinion the epistemology of Bachelard affords a creative renewal of the understanding of science, although further research is needed in many aspects. (shrink)

An accurate analysis of the text shows that the small treatises have a logical structure and a style which is in all aspects unspinozistic. The main points of difference are : a formalistic interpretation of mathematics‚ the opposition between mathematics and physics‚ slavish cartesianism‚ the presence of numerous pleonasms‚ carelessness of expression‚ parade of learning‚ prolixity‚ attention for irrelevant qualities of authors quoted‚ educational purpose. Together with De Vet’s demonstration that the author of SRR and RK is still alive in (...) 1693 (see Tijdschr. v. Filosofie 45‚ 1983) these arguments force us‚ to consign Spinoza’s authorship‚ recently still defended by Petry a.o.‚ to the realm of fancy. (shrink)

Vertaling (verkorte versie) van en commentaar op de wijsgerige verhandeling van de Nederlandse filosoof (1632-1677) over de betekenis van het christendom.