NOW AVAILABLE IN THE ROAD NOT TAKEN, ON HUSSERL'S PHILOSOPHY OF LOGIC AND MATHEMATICS!

This is a preprint version of the paper that appeared in definitive form Experience and Analysis, Erfahrung und Analyse (Proceedings of the International Wittgenstein Conference on held in Kirchberg am Wechsel, August 2004), M. E. Reicher and J. C. Marek (eds), Vienna: ÖBV&HPT Verlag, 2005, pp. 30-38. The published version should be consulted for all citations.

Introduction

It is well known that Quine argued that modern empiricism was to a large extent conditioned by an ill-founded belief in a fundamental cleavage made by Kant between analytic truths, which are grounded in meaning independently of matters of fact, and synthetic truths, which are grounded in fact. In "Two Dogmas of Empiricism," Quine argued that it was a folly to look for such a boundary and that the idea that there was such distinction to be drawn at all was "an unempirical dogma of empiricists, a metaphysical article of faith." (Quine 1953)

It is much less well known that Husserl repudiated the same distinction and that phenomenology was to a large extent conditioned by his determination to overcome the destructive impact of Kant's theory. So, at this intersection of experience and analysis, we find ourselves at yet another crossroads between analytic philosophy and phenomenology. For the original cleavage between the two schools was to a large extent conditioned by reactions to Kant's cleavage.

Husserl's theory of manifolds was an important part of his answer to problems he detected. In what follows, I study its development in terms of the evolution of his ideas about empiricism and analyticity. To provide a context for integrating his theory into mainstream philosophy, I establish connections between it and work on axiom systems, truth in structures, and model theory.

Against Kantian Empiricism and Analyticity

Bolzano, Weierstrass, Brentano, Cantor, Frege, and Husserl all spotted dangers in Kant's distinction and steered their thought in another direction. "Bolzano not only gave the definitive criticism of Kant's concept of analyticity," P. Simons has pointed out, "he also proposed a definition of his own which is almost totally acceptable even by today's standard, and which anticipates Quine, among others. According to Bolzano, those propositions (in themselves) are analytic of which at least one constituent concept (idea in itself) can be varied salva veritate aut falsitate. Of analytic propositions, logically analytic ones are those of which all non-logical concepts can be varied without change of truth-value." (Simons 1992; Bolzano 1837)

Bolzano's pioneering work to rebuild intuitively accepted proofs of theorems in a rigorous way solely on the basis of arithmetical and logical concepts paved the way for Weierstrass, who taught that arithmetic could be built up in a purely logical fashion from the concept of whole number, which would free analysis from insidious appeals to intuitions of space imported into it since Kant had declared mathematical propositions synthetic a priori. (Sebestik 1992; Kline 1972; Jourdain 1991; Demopoulos 1994)

Brentano saw Kant as having been "misled into a false definition of analytical judgement, according to which an affirmative judgement is supposed to be analytic if its predicate is included in the concept of its subject," an error Brentano saw as being "connected with… the disastrous illusion that mere analytic judgments do not add to our knowledge." Kant himself, Brentano pointed out, inadvertently offered striking evidence against it when he maintained that logic was "supposed to be purely analytic and yet truly a science, and hence an enrichment of our knowledge." (Brentano 1911)

Inventor of set theory, Cantor was an avowed enemy of psychologism, empiricism, positivism, naturalism, sensualism, skepticism, and Kantianism, which he saw as wrongly locating the sources of knowledge and certainty in the senses or in the "supposedly pure forms of intuition of the world of presentation." Cantor wanted to free mathematicians to engage in strictly arithmetical forms of concept formation and open the way to a new, abstract realm of ideal mathematical objects, which could not be directly perceived or intuited. Much of the freedom he felt he enjoyed came from distinguishing between an empirical treatment of numbers and Plato's pure, ideal arithmoi eidetikoi. (Hill and Rosado Haddock 2000)

Frege held that arithmetic was analytic and that analytic statements could be informative. He denounced what was empirical as subjective, psychological, and fleeting (Frege 1884). He was, as Benacerraf has stressed, no empiricist and establishing the analyticity of arithmetical judgments was not his way of defending empiricism against Kantian attack. (Benacerraf 1981)

Repudiating Kant's Distinction

It was Weierstrass who awoke in Husserl a desire to seek radical foundations for knowledge. Husserl saw him as aiming to lay bare the original roots, elementary concepts, and axioms upon which the whole system of analysis might be deduced in a completely rigorous, perspicuous way. At the end of his career, Husserl said he had tried to do for philosophy what Weierstrass had done for mathematics. (Schuhmann 1977)

Uncertain as to whether to pursue a career in mathematics or in philosophy, the profound impact of Brentano's insightful, clear, rigorous, precise, objective analyses, and his ability to transform unclear beginnings into clear thoughts and insights made Husserl into a philosopher (Hill 1998). Phenomenology began to take shape during his fifteen years in Cantor's company. (Hill and Rosado Haddock 2000)

Husserl left no doubt about the role that his dissatisfaction with Kant's distinction played in the development of phenomenology. He considered Kant's logic utterly defective. Kant had not understood the nature and role of formal mathematics, and the way in which he had defined the concept of analyticity was totally inadequate and even utterly wrong. Not only had he never seen how little the laws of logic are all analytic propositions in the sense of his own definition, but he failed to see how little his dragging in of an evident principle for analytic propositions helped clear up the achievements of analytic thinking. (Hill 1991)

So, Husserl set out to develop the proper concept of the analytical and to discover the boundary separating genuine analytic ontology from material ontology essentially distinct from it. He said that it was his study of Leibniz' truths of reason and truths of fact and a new keen awareness of the contrast between Hume's matters of fact and relations of ideas and Kant's analytic and synthetic judgments that undermined his early confidence in empirical psychology, set the stage for his conversion from it, and played an important role in the formulation of the positions he later took. (Husserl 1975)

Abandoning Empirical Psychology

Initially inspired by Brentano, whose ideal was most nearly realized in the exact natural sciences, Husserl first tried to anchor arithmetical concepts in direct experience (Husserl 1891). However, empirical psychology never satisfied Husserl. It proved useful in investigating the origin of mathematical presentations, or elaborating practical methods that were psychologically determined, but when it came to going from the psychological connections of thinking to the unity of theory, it could establish no true unity and continuity. (Husserl 1900-01; Husserl 1975)

During the 1890s, Husserl began veering towards an objective logic, where truth was an analysis of essences or concepts. He concluded that the ultimate meaning and source of all objectivity making it possible for thinking to reach beyond contingent, subjective, human acts, and to lay hold of objective being in itself was found in ideality and in the ideal laws defining it. Everything that was purely logical, was an "in itself," something ideal having nothing to do with acts, subjects, or empirical persons belonging to actual reality. The entire overthrowing of psychologism through phenomenology, he said, showed that his analyses in On the Concept of Number and Philosophy of Arithmetic would have to be seen as pure, a priori analysis of essence (Husserl 1929; Husserl 1975). In 1905 he wrote to Brentano: "The empirical sciences --natural sciences, --are sciences of 'matters of fact'.... Pure Mathematics, the whole sphere of the genuine Apriori in general, is free of all matter of fact suppositions.... We stand not within the realm of nature, but within that of Ideas, not within the realm of empirical. . . generalities, but within that of the ideal, apodictic, general system of laws, not within the realm of causality, but within that of rationality.... Pure logical, mathematical laws are laws of essence…." (Husserl 1905)

Husserl came to see Bolzano's theories about presentations and propositions in themselves as an early attempt to provide a unified presentation of the domain of pure ideal doctrines and as providing a complete plan of a pure logic. He had initially viewed them as, curious conceptions, unintelligible, mythical entities suspended between being and non-being, but it suddenly became clear to him that Bolzano had not hypostasized them. Rather, they enjoyed the ideal existence or "validity" characteristic of universals. (Husserl 1975; Husserl 1994)

Husserl now saw Bolzano's presentations in themselves as "what were ordinarily called, the 'senses' of statements, what is said to be one and the same when, for example, different persons are said to have asserted the same thing, or what scientists simply call a theorem (for example the theorem about the sum of the angles in a triangle) which no one would think of as being someone's experience of judging and that this identical sense could be none other than the universal… present in all actual assertions having the same sense which makes possible the identification in question, even when the descriptive content of individual experiences of asserting varies considerably in other respects." The ideal entities so unpleasant for empiricistic logic and so consistently disregarded by it, Husserl came to insist, were not artificially devised either by himself or by Bolzano. They were given beforehand by the meaning of the universal talk of propositions and truths that is indispensable in all the sciences. And that indubitable fact had to be the starting point of all logic, for science was a web of theories, and so of proofs, propositions, inferences, concepts, meanings, and not of lived experiences. (Husserl 1908-09; Husserl 1994)

Analyticity without Empiricism

Husserl said that his fight against logical psychologism was meant to serve no other end than the supremely important one of making pure, analytic logic visible in its purity and ideal particularity (Husserl 1929). He came to hold that the only concrete, fruitful way of explaining analyticity was in stressing that in purely logical, formal, analytic propositions or laws, the variables are indefinite, the terms can vary completely freely and arbitrarily. Purely arithmetical theories, the purely analytical theories of mathematics, the traditional theory of syllogism, the pure theory of cardinal numbers, of ordinals, Cantorian sets, and so on were purely logical because their basic concepts expressed reasoning forms free of any cognitive content and could not be had through sensory abstraction. No epistemological reflection was required. (Husserl 1902/03; Husserl 1994)

He wrote of the highly important divorcing of the formal and the factual, or material, spheres of being. As examples of purely formal concepts based on the empty idea of something in general and connected with it through the axioms of formal ontology, Husserl listed something, one, object, property, relation, plurality, cardinal number, order, ordinal number, whole, part, which he contrasted with material concepts like house, tree, color, sound, spatial figure, sensation, feeling, smell, intensity, etc., which express something factual or sensory. (Husserl 1900-01)

Analytically necessary propositions are propositions which are true completely independently of any particular facts about their objects, of any actual matters of fact, of the validity of positing their existence. Analytic laws are universal propositions containing nothing but concepts as formal concepts and hence are free of any explicit or implicit positing of the existence of individuals. Analytic laws stand in contrast with particular instances of them that result when concepts regarding matters of fact and particular thoughts positing individual existence are introduced. He considered the difference between what was merely formal and without factual content easy to see in the difference between laws like that of cause and effect, which is about changes real things undergo, or laws about qualities, intensities, extensions, limits, forms of relations, which he contrasted with an analytically necessary proposition like 'There cannot be a father if there is no child,' or purely analytic generalizations like 'A whole cannot exist without parts.' It would be a formal analytical contradiction in terms, to call something a part if there was no whole to which it belonged. (Husserl 1900-01)

Analyticity and Manifolds

By drawing the boundary line existing a priori between mathematics and the natural sciences, Husserl believed he was delimiting and expanding the domain of the analytical in keeping with the most recent discoveries in mathematics, notably those concerning axiomatization and manifolds. (Husserl 1900-01; Husserl 1929)

When he abandoned the second volume of the Philosophy of Arithmetic on the logic of the deductive sciences, he began pushing his thinking beyond the mathematical realm towards a universal theory of formal deductive systems. Volume one had harsh words for Frege's project to found arithmetic on formal definitions out of which all its theorems could be deduced purely syllogistically Husserl only ever retracted those criticisms of Frege's logic. (Husserl 1891; Husserl 1900-01)

Husserl saw in the nascent mathematical theory of manifolds a partial realization of his ideal of a science of possible deductive systems (Husserl 1900-1901; Husserl 1929). In the early 1890s he compared Cantor's definition of a set or manifold as an aggregate of any elements combined into a whole by a law with Riemann's, or kindred ones, for which manifolds are aggregates of elements that are not just combined into a whole, but are ordered and continuously interdependent. Husserl defined "order" as a concatenation having the special property that each member possesses an unambiguous position in the narrow sense in relation to any arbitrary one and can thus be unequivocally characterized by the mere form of the direct or indirect connection with the last one. Manifolds, he stressed, are not mere aggregates of elements without relations. It is precisely the relations that are essential and distinguish them. (Husserl 1983)

Such reflections led Husserl to the detect a certain natural order in formal logic and broaden its domain to include two layers above the traditional formal logic of subject and predicate propositions and states of affairs, which deals with what might be stated about objects in general from a possible perspective. In the second layer, it was no longer a question of objects as such about which one might predicate something, but of investigating what was valid for higher order objects dealt with in an indeterminate, general way, not as empirical or material entities, and determined in purely formal terms, removed from acts, subjects, or empirical persons of actual reality. This is an expanded, completely developed analytics where one reasons deductively with concepts and propositions in a purely formal manner since each concept is analytic and each procedure purely logical. (Husserl 1906/07; Husserl 1917/18)

The third layer is that of the science of deductive systems in general, the theory of manifolds, theory forms, logical molds totally undetermined as to their content and not bound to any possible concrete interpretation. Here it is a matter of theorizing about possible fields of knowledge conceived of in a general, undetermined way, simply determined by the fact that the objects stand in certain relations that are themselves subject to certain fundamental laws of such and such determined form, are exclusively determined by the form of the interconnections assigned to them that are themselves just as little determined in terms of content as are the objects. (Husserl 1906/07; Husserl 1917/18)

Through axiom forms a manifold of anything whatsoever is defined in an indeterminate, general way. A certain something must by definition stand in a certain relationship to something else in the defining manifold. A set of axioms of such and such a form that are consistent, independent, and purely logical in that they obey the principle of non-contradiction yields the set of propositions belonging to the theory of such and such a form to be developed. The form exists insofar as it is correctly defined, insofar as the axiom forms are ordered in such a way as to contain no formal contradictions, no violation of analytic principles. Whether axioms as truths have existence in any objective, real, or ideal spheres corresponding to the prescribed form is left open. On the basis of the definition of the manifold, we can deduce conclusions, construct proofs, and it is then certain a priori that anything obtained in this way will correspond to something in our theory. (Husserl 1906/07; Husserl 1917/18)

For Husserl, this science of forms of possible theories was a field of free, creative investigation made possible once form was emancipated from content. Once one discovers that deductions, series of deductions, continue be meaningful and remain valid when another meaning is assigned the symbols, one is freed to reason completely on the level of pure forms. One can vary the systems in different ways. One finds ways of constructing an infinite variety of forms of possible disciplines. (Husserl 1906/07; Husserl 1917/18)

Hilbert, Bourbaki, Model Theory.

As more pieces of Husserl's theory of manifolds become available, it is apparent that it fits the definition of that systematic study of classes of structures defined by axioms now known as model theory. As Wilfred Hodges explained "Truth in a Structure": "algebraists and geometers have often found themselves studying certain objects which we now call structures, or more loosely models…. a structure is a collection of elements together with certain labelled relations…defined on those elements… When we define a class of structures by giving a set of laws which the structures must obey, these laws are called axioms for the class of structures." (Hodges 1985/86; Demopolous 1994b; Hintikka 1988)

Hilbert, an enthusiastic supporter of Husserl, studied structures in an abstract way, for their own sake, without specifying the nature of the objects subject to the operations whose rules were described. Famous for having said that it must be possible to replace in all geometric statements the words 'point,' 'line,' 'plane,' by 'table,' 'chair,' 'mug,' Hilbert freed the axiomatic method from the problem of relating to the existence of objects by demonstrating that justifying a system was not a matter of attaching the axioms to empirical facts, but of demonstrating its consistency. His axiom systems were not systems of statements about a subject matter, but systems of conditions for a relational structure, a form, an abstract object taken as the immediate object of the axiomatic theory (Bernays 1922; Bernays 1967; Weyl 1944; Revue 1993; Gray 2000; Cassou-Noguès 2001). Husserl considered the kinship between his manifolds and Hilbert's axiom systems to be evident. (Hill and Rosado Haddock 2000).

For the Bourbaki mathematicians, a mathematical structure is a set of elements whose nature is not specified. In defining a structure, one or several relations involving these elements are given and it is postulated that the given relation or relations satisfy certain conditions, which are spelled out, and which are the axioms of the structures envisaged. The axiomatic theory of a given structure is studied by deducing the logical consequences of its axioms, while excluding all other hypotheses about the elements considered, notably, any hypothesis concerning their special nature. (Bourbaki 1971)

Bourbaki wrote: "From the axiomatic point of view mathematics appears on the whole as a reservoir of abstract forms --the mathematical structures and it sometimes happens, without anyone really knowing why, that certain aspects of experimental reality model themselves after certain of these forms, as if by a sort of preadaptation. It cannot be denied… that the majority of these forms had a well-determined intuitive content at the beginning; but it is precisely by voluntarily emptying them of this content that it has been possible to employ them with all their potential efficacity and to render them capable of new interpretation and a complete fulfillment of their elaborative role." (Bourbaki 1971)

Bourbaki sees the essential goal of the axiomatic method as concerning the deep-lying intelligibility of mathematics. It teaches one to search for the deeper reason for its discoveries, the common ideas buried under the external apparatus, to single them out and to display them by working to dissociate the principle lines of reasoning figuring in the demonstrations of a theory, taking each of them in isolation and, considering it an abstract principle, unfolding its consequences. It then returns to the theory under study, recombines its components and studies their interactions. (Bourbaki 1971)

Bourbaki likened the inner vitality of mathematics to "a great city whose suburbs never cease to grow in a somewhat chaotic fashion on the surrounding lands, while its center is periodically reconstructed, each time following a clearer plan and a more majestic arrangement, demolishing the old sections with their labyrinthine alleys in order to launch new avenues toward the periphery, always more direct, wider and more convenient." He anticipated progress in the invention of new fundamental structures through the revealing of the fecundity of new axioms or new combinations of axioms. (Bourbaki 1971)

Conclusion

Husserl's theory of analyticity and manifolds was his project for limning the true and ultimate structure of reality through an austere scheme of axiomatization that knows no acts, subjects, or empirical persons or objects belonging to actual reality. Once the pieces of his theory are sewn together and then sewn on where they belong in philosophy, we can experiment with it as an alternative to Fregeo-Russello-Quineo methods of logics rooted as they are in an unworkable theory of reference to objects and identity. (Hill 1997; Hill 2004)

People spooked by intensions need not apply, however. For Husserl's method for finding clarity with respect to the central traits of reality means fraternizing with creatures of darkness, philosophizing in a metaphysical jungle of essentialism, fending off charges of trafficking in a curiously idealistic ontology that repudiates material objects, stomaching the odium of the a priori. (Quine 1947; Quine 1960; Quine 1976; Russell 1956)

As a disciple of Brentano, Husserl experienced emotions like those of Quine and Russell alluded to above, but finally found empirical psychology unable to provide the unity and continuity indispensable to science and knowledge in general. He surely would have also fled a fragmented world of rabbit parts, river stages and kinship, where the ontologies of physical and mathematical objects are but myths relative to an epistemological view (Quine 1960; Quine 1969). He overcame his feelings when he came to see intensions as the senses of statements, what is said to be the same when different persons are said to have asserted the same thing, what scientists call a theorem, the universal present in all actual assertions having the same sense, which makes identification possible, even when the content of individual experiences of asserting varies considerably (Husserl 1994). He developed a logical point to view in conformity with that. Now, what's so bad about that?

Literature

Benacerraf, Paul 1981 "Frege: The Last Logicist", in W. Demopoulos (ed.), Frege's Philosophy of Mathematics, Cambridge: Harvard University Press, 1995, 41-67.

Bernays, Paul, 1922 "Hilbert's Significance for the Philosophy of Mathematics", in Paolo Mancosu (ed.), From Brouwer to Hilbert, New York: Oxford University Press, 1998, 189-97.

Bernays, Paul 1967 "Hilbert, David", The Encyclopedia of Philosophy, P. Edwards (ed.), vol. 3, New York: MacMillan Publishing Co., 496-504.

Bolzano, Bernard 1837 Theory of Science, Berkeley CA: University of California Press, 1972, (partial translation by R. George).

Bourbaki, Nicolas 1971 "The Architecture of Mathematics", in F. Lelionnais (ed.) Great Currents of Mathematical Thought, vol. 1 Mathematics: Concepts and Development, New York: Dover, 23-43.

Brentano, Franz 1911 "On Attempts at the Mathematicization of Logic", Psychology from the Empirical Standpoint. New York: Humanities Press, 1973, 301-06 (section X of the 1911 appendix).

Cassou-Noguès, Pierre 2001 Hilbert, Paris: Les Belles Lettres.

Coffa, Alberto 1982 "Kant, Bolzano and the Emergence of Logicism", in W. Demopoulos (ed.), Frege's Philosophy of Mathematics, Cambridge: Harvard University Press, 1995, 29-40.

Demopoulos, William 1994a "Frege and the Rigorization of Analysis", in W. Demopoulos (ed.), Frege's Philosophy of Mathematics, Cambridge: Harvard University Press, 1995, 68-88.

Demopoulos, William 1994b "Frege, Hilbert, and the Conceptual Structure of Model Theory", History and Philosophy of Logic 15, 211-25.

Dieudonné, Jean 1971 "David Hilbert (1862-1943)", in F. Lelionnais (ed.) Great Currents of Mathematical Thought, vol. 1 Mathematics: Concepts and Development, New York: Dover, 304-11,

Frege, Gottlob 1884 Foundations of Arithmetic, Blackwell, Oxford, 1986.

Gray, Jeremy, J 2000 The Hilbert Challenge, Oxford: Oxford University Press.

Hill, Claire Ortiz 1991 Word and Object in Husserl, Frege, and Russell, The Roots of Twentieth Century Philosophy, Athens OH: Ohio University Press.

Hill, Claire Ortiz 1997 Rethinking Identity and Metaphysics, On the Foundations of Analytic Philosophy, New Haven: Yale University Press.

Hill, Claire Ortiz 1998 "From Empirical Psychology to Phenomenology: Husserl on the Brentano Puzzle", The Brentano Puzzle, R. Poli (ed.), Aldershot: Ashgate, 151-68.

Hill, Claire Ortiz 2004 "Reference and Paradox", Synthese138, 2004, 207-32.

Hill, Claire Ortiz and G. E. Rosado Haddock 2000 Husserl or Frege? Meaning, Objectivity, and Mathematics, Chicago: Open Court.

Hodges, Wilfred 1985/86 "Truth in a Structure", Proceedings of the Aristotelian Society, n.s. 86, 135-51.

Hintikka, Jaakko 1988 "On the Development of the Model-theoretic Viewpoint in Logical Theory", Synthese 77, 1-36.

Husserl, Edmund 1891 Philosophy of Arithmetic, Psychological and Logical Investigations with Supplementary Texts from 1887-1901 (Dallas Willard, trans.), Dordrecht: Kluwer, 2003.

Husserl, Edmund 1900-01 Logical Investigations, New York: Humanities Press, 1970.

Husserl, Edmund 1902/03 Allgemeine Erkenntnistheorie 1902/03, E. Schuhmann (ed.), Dordrecht: Kluwer, 2001.

Husserl, Edmund 1905 "Husserl an Brentano, 27. III. 1905", Briefwechsel, Die Brentanoschule I, Dordrecht: Kluwer, 1994.

Husserl, Edmund 1906/07 Einleitung in die Logik und Erkenntnistheorie, Hua vol. XXIV,

The Hague: M. Nijhoff, 1984.

Husserl, Edmund 1908/09 Alte und Neue Logik, Vorlesung 1908-09, E. Schuhmann (ed.), Dordrecht: Kluwer, 2003.

Husserl, Edmund 1917/18 Logik und allgemeine Wissenschaftstheorie, Hua vol. XXX. Dordrecht: Kluwer, 1996.

Husserl, Edmund 1929 Formal and Transcendental Logic, The Hague: M. Nijhoff.

Husserl, Edmund 1975 Introduction to the Logical Investigations. The Hague: M. Nijhoff.

Husserl, Edmund 1983 Studien zur Arithmetik und Geometrie, Texte aus dem Nachlass (1886-1901), Hua vol. XXI, The Hague: M. Nijhoff.

Husserl, Edmund 1994 Early Writings in the Philosophy of Logic and Mathematics, Dordrecht: Kluwer.

Jourdain, Paul 1991 "The Development of the Theory of Transfinite Numbers", Selected Essays on the History of Set Theory and Logic. Bologna: CLUEB.

Kline, Morris 1972 "The Instillation of Rigor in Analysis", Mathematical Thought from Ancient to Modern Times, Oxford: Oxford University Press, 947-78.

Quine, Willard 1947 "The Problem of Interpreting Modal Logic", Journal of Symbolic Logic 12,2, June, 43-48.

Quine, Willard 1953 "Two Dogmas of Empiricism", From a Logical Point of View (2nd rev.), New York: Harper & Row, 1961, 20-46.

Quine, Willard, 1960 Word and Object, Cambridge: M.I.T. Press.

Quine, Willard, 1969 Ontological Relativity and Other Essays, New York: Columbia University Press.

Quine, Willard 1976 Ways of Paradox, Cambridge: Harvard University Press

Revue Internationale de Philosophie, Hilbert, 1993 47, 186, 4.

Russell, Bertrand 1956 Logic and Knowledge, London: Allen & Unwin.

Schuhmann, Karl 1977 Husserl-Chronik, The Hague: M. Nijhoff.

Sebestik, Jan 1992 Logique et mathématique chez Bernard Bolzano, Paris: Vrin.

Simons, Peter 1992 "Wittgenstein, Schlick and the A Priori," Philosophy and Logic in Central Europe from Bolzano to Tarski, Selected Essays," Dordrecht: Kluwer, 361-76.

Weyl, Hermann 1944 "David Hilbert and his Mathematical Work," Bulletin of the American Mathematical Society 50, 612-54.