Metamathematics and the Philosophical Tradition is the first work to explore in such historical depth the relationship between fundamental philosophical quandaries regarding self-reference and meta-mathematical notions of consistency and incompleteness. Using the insights of twentieth-century logicians from Gödel through Hilbert and their successors, this volume revisits the writings of Aristotle, the ancient skeptics, Anselm, and enlightenment and seventeenth and eighteenth century philosophers Leibniz, Berkeley, Hume, Pascal, Descartes, and Kant to identify ways in which these both encode and evade problems of (...) a priori definition and self-reference. The final chapters critique and extend more recent insights of late 20th-century logicians and quantum physicists, and offer new applications of the completeness theorem as a means of exploring "metatheoretical ascent" and the limitations of scientific certainty. Broadly syncretic in range, Metamathematics and the Philosophical Tradition addresses central and recurring problems within epistemology. The volume’s elegant, condensed writing style renders accessible its wealth of citations and allusions from varied traditions and in several languages. Its arguments will be of special interest to historians and philosophers of science and mathematics, particularly scholars of classical skepticism, the Enlightenment, Kant, ethics, and mathematical logic. (shrink)

The basic purpose of this essay, the first of an intended pair, is to interpret standard von Neumann quantum theory in a framework of iterated measure algebraic truth for mathematical (and thus mathematical-physical) assertions — a framework, that is, in which the truth-values for such assertions are elements of iterated boolean measure-algebras (cf. Sections 2.2.9, 5.2.1–5.2.6 and 5.3 below).The essay itself employs constructions of Takeuti's boolean-valued analysis (whose origins lay in work of Scott, Solovay, Krauss and others) to provide a (...) metamathematical interpretation of ideas sometimes considered disparate, heuristic, or simply ill-defined: the collapse of the wave function, for example; Everett's many worlds'-construal of quantum measurement; and a natural product space of contextual (nonlocal) hidden variables. (shrink)

Model-theoretic 1-types overa given first-order theory T may be construed as natural metalogical miniatures of G. W. Leibniz' ``complete individual notions'', ``substances'' or ``substantial forms''. This analogy prompts this essay's modal semantics for an essentiallyundecidable first-order theory T, in which one quantifies over such ``substances'' in a boolean universe V(C), where C is the completion of the Lindenbaum-algebra of T.More precisely, one can define recursively a set-theoretic translate of formulae νNϕ of formulae ν of a normal modal theory Tm based (...) on T, such that the counterpart `ξi' of a the modal variable `xi' of L(Tm) in this translation-scheme ranges over elements of V(C) that are 1-types of T with value 1 (sometimes called `definite' C-valued 1-types of T).The article's basic completeness-result (2.13) then establishes that ϕvarphi; is a theorem of Tm iff [[νN(ϕ) is aconsequenceof νN(Tm) for each extension N of T which is a subtheory of the canonical generic theory (ultrafilter) u]] = 1 – or equivalently, that Tm is consistent iff[[there is anextension N of T such that N is a subtheory of the canonical generic theory u, and νN(ϕ) for all ϕ in Tm]] > 0.The proof of thiscompleteness-result also shows that an N which provides a countermodel for a modally unprovable ϕ – or equivalently, a closed set in the Stone space St(T) in the sense of V(C) – is intertranslatable with an `accessibility'-relation of a closely related Kripke-semantics whose `worlds' are generic extensions of an initial universe V via C.This interrelation providesa fairly precise rationale for the semantics' recourse to C-valued structures, and exhibits a sense in which the boolean-valued context is sharp. (shrink)

A ground-motive for this study of some historical and metaphysical implications of the diagonal lemmas of Cantor and Gödel is Cantor's insightful remark to Dedekind in 1899 that the Inbegriff alles Denkbaren (aggregate of everything thinkable) might, like some class-theoretic entities, be inkonsistent. In the essay's opening sections, I trace some recent antecedents of Cantor's observation in logical writings of Bolzano and Dedekind (more remote counterparts of his language appear in the First Critique), then attempt to relativize the notion of (...) Inkonsistenz to self-sufficient theories T which interpret themselves. In effect, I argue that Gödel's diagonal lemma suggests a sense in which metatheoretic notions of proof, well-foundeness and satisfaction are object-theoretically inkonsistent. With respect to Cantor's Inbegriff, for example, the lemma yields that any object-theoretic reconstruction of thinkability generates an antidiagonal sentence , which one can paraphrase asSelf-referential application of the assertion that. (shrink)

SummaryThis essay explores analogies between classical notions of pyrrhonist skepticism and reflexive phenomena of twentieth‐century metamathematics. In a theoretical framework T, for example, one may interpret1T's appearances () as its axioms;2evident and inevident assertions () in the language L of T as sentences 0 which are decidable and undecidable in T; and3skeptical self ‐doubt about T in L as T's Godel‐sentence γ .These analogies complement another one, between pyrrhonist ‘modes’() of indefinite semantic regress (), and recurrent appeals to ‘new’ metatheories (...) T', T., to consider ‘new’, identify alternate interpretations, and adjudicate claims of consistency.These analogies are also reflexive in a wider sense. ‐ are stated for theories T, T', T., in some unspecified metatheory t̄–to which they also apply. This wider reflexivity suggests that ‘problems' of skeptical self‐doubt are not vicious or otiose, but significant aspects of the phenomena of indefinite semantic regress.RésuméCet essai explore quelques analogies entre les notions classiques du scepticisme pyrrhonien et les phénomenes réflexifs de la métamathématique du vingtième siècle. Dans un cadre théorique T du premier ordre, par exemple, on peut interpréter les apparences de T (), comme des axiomes; les assertions évidentes et non évidentes (), dans la langue L de T, comme les énoncés ø decidables et indecidables dans L; et le doute sceptique de ‘soi au sujet de T dans L, come ľénoncé ‐γ de Gödel de T .Ces analogies en complèment une autre, entre les ‘modes' pyrrhoniens() dela régression sémantique indéfinie(), et les appels récurrents à de ‘nouvelles' métathéories T', T., pour considérer de ‘nou‐veaux’, identifier des interpretations alternatives et juger des questions de consistance.Ces analogies sont aussi réflexives dans un sense plus large. ‐ se constatent pour les théories T, T T., dans une métathéorie indéterminée T –aG laquelle ilss'appliquent à nouveau. Cette réflexivité plus étendue suggère que les problèmes de doute sceptique de soi‘ ne sont pas vicieux ou vides, mais représentent au contraire des aspects signifiants des phénomènes de régression sémantique indéfinie.ZusammenfassungIn der vorliegenden Arbeit werden einige Analogien zwischen klassischen Begriffen des pyrrhonischen Skeptizismus und reflexiven Phänomenen der Metamathematik des zwanzigsten Jahrhunderts untersucht. In einem theoretischen Rahmenwerk T kann man die folgenden Interpretationen geben:1die Erscheinungen () von T als dessen Axiome;2evidente und nicht‐evidente Aussagen () in der Sprache L von T als Sätze ø, die entscheidbar, bzw. unentscheidbar sind; und3den skeptischen Selbst zweifel über T in L als den Güdel‐Satz γ für T .Diese Analogien sind complementär zu einer anderen, nämlich derjenigen zwischen den Pyrrhonischen Modi () des semantischen regresssus in indefinitum (), und dem wiederholten Rekurs auf ‘neue’ Metatheorien T', T, …, in denen man ‘neue’ berücksichtigt, alternative Deutungen erwägt und Fragen der Widerspruchsfreiheit beurteilt.Diese Analogien sind auch reflexiv in einem weiteren Sinne. ‐ werden für T, T, T, …, in einer Metatheorie T – bestimmt, für die sie auch gelten. Diese weitere Reflexivität legt nahe, dass ‘Probleme’ des skeptischen ‘Selbsf zweifels weder zu circuit vitiosi führen noch Scheinprobleme sind, sondern beachtenswerte Aspekte der Phänomene des indefiniten semantischen Regresses darstellen. (shrink)

In the the passage just quoted from theDialogues concerning Natural Religion, David Hume developed a thought-experiment that contravened his better-known views about chance expressed in hisTreatise and firstEnquiry.For among other consequences of the eternal-recurrence hypothesis Philo proposes in this passage, it may turn out that what the vulgar call cause is nothing but a secret and concealed chance.

What particular privilege has this little Agitation of the Brain which we call Thought, that we must make it the Model of the whole Universe? (Hume, Dialogues Concerning Natural Religion, 1976, p. 168)******...at once it struck me, what quality went to form a Man (sic) of Achievement especially in Literature and which Shakespeare possessed so enormously—I mean Negative Capability, that is when someone is capable of being in uncertainties, Mysteries, doubts without any irritable reaching after fact and reason. (Keats 1959, (...) p. 261)******Die menschliche Vernunft hat das besondere Schicksal in ihrer Gattung ihrer Erkenntnisse: dass sie durch Fragen belästigt wird, die sie nicht abweisen kann; denn sie sind ihr durch die Natur der Vernunft selbst aufgegeben, die sie aber auch nicht beantworten kann; denn sie übersteigen alles Vermögen der menschlichen Vernunft. (Opening lines of the Kritik der reinen Vernunft (1956), A VII; the italics in the text are mine)Human reason has the particular fa .. (shrink)

In the the passage just quoted from the Dialogues concerning Natural Religion, David Hume developed a thought-experiment that contravened his better-known views about "chance" expressed in his Treatise and first Enquiry. For among other consequences of the 'eternal-recurrence' hypothesis Philo proposes in this passage, it may turn out that what the vulgar call cause is nothing but a secret and concealed chance. (In this sentence, I have simply reversed "cause" and "chance" in a well-known passage from Hume's Treatise, p. 130). (...) In the first eight sections of this essay, I develop one topological and model-theoretic analogue of Hume's thought-experiment, in which 'most' ('A-generic') models M of a 'scientific' theory U are both 'eternally recurrent' and topologically random (in a sense which will be made precise), even though they are 'inductively' defined, via a step-by-step ('empirical'?) procedure that Hume might have been inclined to endorse. The last aspect of this model-theoretic thought-experiment also serves to distinguish it from simpler measure-theoretic prototypes that are known to follow from Kolmogorov's Zero-One Law (cf. the Introduction, 5.2, 6.1 and 6.7 below). In the last three sections, I will argue more informally (1) that the metamathematical thought-experiments just mentioned do have a genuine metaphysical relevance, and that this relevance is predominantly skeptical in its implications; (2) that such 'nonstandard' instances of semantic underdetermination and 'pathology' seem to be the metatheoretic rule rather than the exception; and therefore, (3) that metamathematical and metatheoretic 'malign-genius' arguments are quite coherent, contrary (e.g.) to assertions such as that of Putnam (1980), pp. 7-8. In the essay's conclusion, finally, I assimilate (2) and (3) to the familiar datum that 'simplicity', rather than 'pathology', has more often than not turned out to be an anomalous 'special case' in the historical development of scientific and mathematical ontology. (shrink)

Model-theoretic 1-types overa given first-order theory T may be construed as natural metalogical miniatures of G. W. Leibniz' ``complete individual notions'', ``substances'' or ``substantial forms''. This analogy prompts this essay's modal semantics for an essentiallyundecidable first-order theory T, in which one quantifies over such ``substances'' in a boolean universe V(C), where C is the completion of the Lindenbaum-algebra of T.More precisely, one can define recursively a set-theoretic translate of formulae N of formulae of a normal modal theory Tm based on (...) T, such that the counterpart `i' of a the modal variable `xi' of L(Tm) in this translation-scheme ranges over elements of V(C) that are 1-types of T with value 1 (sometimes called `definite' C-valued 1-types of T). (shrink)