Metakides, North-Holland, , pp. Simpson and R. Freyd and A. Robertson and P. Hirst , Weak comparability of well orderings and reverse mathematics, Annals of Pure and Applied Logic 47 , pp. Meyer , Wither Relevance Arithmetic? Simpson and X. Sheard , Elementary descent recursion and proof theory, Annals of Pure and Applied Logic 71 , pp. Simpson , Issues and problems in reverse mathematics, in: Computability Theory and Its Applications, Contemporary Mathematics, volume , , Internal finite tree embeddings, in: Reflections on the Foundations of Mathematics: Essays in honor of Solomon Feferman, ed.
Metamathematics of comparability, in: Reverse Mathematics, ed. Simpson, Lecture Notes in Logic, vol. My Forty Years on His Shoulders. Logic, Vol. Selection for Borel relations, in: Logic Colloquium 01, ed. Godeharad Link, de Gruyter, , Enayat and Kossak, , Science, 20, , pp. Applications of Mathematics to Computer Science, in: Emerging syntheses in science, ed. Avigad , Combining decision procedures for the reals, Logical Methods in Computer Science 2 , Kieffer, J.
Avigad , A language for mathematical knowledge management, in: Computer Reconstruction of the Body of Mathematics, ed. An infinitesimal definition certainly leads to computationally meaningful formulas for the derivatives of the standard functions, even though classical. To emphasize the difference between the two types of numerical meaning, we introduce the following distinction. We will refer to the content of mathematics that is fully constructive from the bottom i.
Pre-LEM numerical meaning is, of course, exemplified by Foundations of constructive analysis Bishop, Example V. The volume calculations in Archimedes appear to rely on proofs by contradiction. Thus, a contradiction is derived from the assumption that the volume x of the unit sphere is smaller than two-thirds of the volume of the circumscribed cylinder, and similarly for greater than. The contradiction proves that the volume is exactly two-thirds. The proof appears to depend on the trichotomy for real numbers:. We start with the trichotomy, eliminate the possibilities on the right and on the left, to conclude that the possibility in the middle equality is the correct one.
Beeson Indeed, the theoretical entity called the infinitesimal. The equations are thereby placed in the context of a variational problem. Yet the inequality itself possesses indisputable numerical meaning. The inequality relates a pair of metric invariants of a Riemannian 2-torus T. The first is the least length, denoted S y s , of a loop on T that cannot be contracted to a point on T. The second invariant is the total area, denoted A r e a , of T. The inequality, S y s 2. Burago, who acquainted me with the results of Loewner, Pu, and Besicovitch. These attracted me by the topological purity of their underlying assumptions, and I was naturally tempted to prove similar inequalities in a more general topological framework Gromov, , p.
Our theory of infinitesimal variables [. Connes claims that his infinitesimals provide a computable answer to a probability problem outlined earlier in his text. Avigad of ergodic theory can be viewed as an analysis of post-LEM numerical meaning, in the context of ergodic theory. More generally, the technique of proof mining seeks to extract numerical content from non-constructive proofs, using logical tools, see U.
Kohlenbach and P. Oliva That is, one may use all the formal machinery, in particular, nonconstructive but formally valid existence statements such as the Bolzano-Weierstrass theorem , to prove, formally, real propositions, i. Stolzenberg concludes on the following optimistic note:. Now the constructivisation of classical results may succeed at any rate modulo strengthening the hypotheses or weakening the conclusions, see Subsection IV. But Bishop apparently believed that a Robinson infinitesimal resists constructivisation.
If both forms of constructivism discussed in Subsection V. To answer the question, we will start by pointing out that the burden of explanation is upon the constructivists on at least the following two counts: 1. How is it that one of their own, A.
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Bishop sometimes seems to identify his notion of numerical meaning, with. One can adopt the methodological goals of a search for greater numerical meaning,. Int merely being polite in conceding a point to Class? The fervor of Bishopian constructivism. Pourciau and published in the Studies in History and Philosophy of Science:. The faith that sustains the classical world view emanates from one belief more than any other: that mathematical assertions are true or false, independently of our knowing which.
Every conflict. This is not a belief so much as a hidden cause: it creates the world view— making, for example, proofs by contradiction appear self-evidently correct— but remains transparently in the background, unseen and unquestioned Pourciau, , p. Such an unquestioning stance naturally leads to calcification, as Pourciau continues:. Once created, the classical world view is sustained and calcified. Talk about sets, functions, real numbers, theorems and so on is taken by the classical mathematician as being literally about mathematical objects that exist independently of us, even though such talk, classically interpreted, has the appearance, and nothing more, of being meaningful Pourciau, , p.
The reader recognizes the pet constructivist term, meaning, already analyzed in Subsection III. In a literal sense, it is shattering. Avigad, while agreeing that [ w] e do not need fairy tales about numbers and triangles prancing about in the realm of the abstracta[,] notes that [ p] roof-theoretic analysis [. Giorello as follows. To elaborate, we argue as follows.
The contradiction proves that Pourciau does in fact think of LEM as mathematical phlogiston. However, we include it here for the benefit of a classically minded reader. Kuhnian revolution. Based on the loss of such assertions in an intuitionistic framework, Pourciau concludes that Intuitionism possessed a potential of developing into a unique mathematical Kuhnian revolution. Pourciau writes that. But from outside the classical paradigm, [. He argues that both Crowe and Dauben.
Pourciau, , p. Pourciau views such cumulativity as a necessary consequence of the classical i. LEM-circumscribed paradigm, suggesting that a Kuhnian revolution in mathematics would in fact be impossible, without first extirpating LEM. Consider, for example, the largely successful assault on infinitesimals in the aftermath of the rise of Weierstrassian epsilontics. A similar observation applies to infinitesimals themselves see Example V. Thus, Weierstrassian epsilontics constitute a,. Indeed, Robinson , p. Cleave , Cutland et al. This ongoing project appears to be a striking realisation of a reconstruction project enunciated by I.
Grattan-Guinness, in the name of Freudenthal : it is mere feedback-style ahistory to read Cauchy and contemporaries such as Bernard Bolzano as if they had read Weierstrass already. On the contrary, their own pre-Weierstrassian muddles need historical reconstruction Grattan-Guinness, , p.
The Brouwer— Hilbert debate captured the popular mathematical imagination in the s. Burgess discusses the debate briefly in his treatment of nominalism in Burgess, , p. Part of the attraction stems from a detailed lexicon developed by Bishop so as to challenge received classical views on the nature of mathematics. A constructive lexicon was a sine qua non of his success.
It may be helpful to provide a summary of such terms for easy reference, arranged alphabetically, as follows. Debasement of meaning is the cardinal sin of the classical opposition, from Cantor to Keisler, 95 committed with LEM see below. Fundamentalist excluded thirdist is a term that refers to a classicallytrained mathematician who has not yet become sensitized to implicit use of the law of excluded middle i. Idealistic mathematics is the output of Platonist mathematical sensibilities, abetted by a metaphysical faith in LEM see below , and characterized by the presence of merely a peculiar pragmatic content see below.
Integer is the revealed source of all meaning see below , posited as an alternative foundation displacing both formal logic, axiomatic set theory, and recursive function theory. The integers wondrously escape97 the vigilant scrutiny of a constructivist intelligence determined to uproot and nip in the bud each and every Platonist fancy of a concept external to the mathematical mind.
Pourciau in his Education Pourciau, appears to interpret it as an indictment of the ethics of the classical opposition. Yet in his. Bishop, , p. Bishop describes integrity as the opposite of a syndrome he colorfully refers to as schizophrenia, characterized by a number of ills, including a a rejection of common sense in favor of formalism, b the debasement of meaning see above , c as well as by a list of other ills— but excluding dishonesty. Now the root of integr-ity is identical with that of integer see above , the Bishopian ultimate foundation of analysis.
Brouwer sought to incorporate a theory of the continuum as part of intuitionistic mathematics, by means of his free choice sequences. Law of excluded middle LEM is the main source of the non-constructivities of classical mathematics. Numerical meaning is the content of a theorem admitting a proof based on intuitionistic logic, and expressing computationally meaningful facts about the integers.
It connotes an alleged lack of empirical validity of classical mathematics, when classical results are merely inference tickets Billinge, , p. Realistic mathematics. There are two main narratives of the Intuitionist insurrection, one anti-realist and one realist. The issue is discussed in Subsections VI. Since the sensory perceptions of the human body are physics-and chemistry-bound, a claim of such trans-universe invariance amounts to the positing of a disembodied nature of the natural number system transcending the physics and the chemistry. The anti-realist narrative, mainly following Michael Dummett , traces the original sin of classical mathematics with LEM, all the way back to Aristotle.
Generally speaking, it is this narrative that seems to be favored by a number of philosophers of mathematics. The latter requirement, in the context of mathematics, is a restatement of the intuitionistic principle that truth is tantamount to verifiability. What Dummett proceeds to say at this point, reveals the nature of his interest: but intuitionism represents the only sustained attempt by the opponents of a realist view to work out a coherent embodiment of their philosophical beliefs [ emphasis added— authors]. What interests Dummett here is the fight against the realist view.
What endears intuitionists to him, is the fact that they have succeeded where the phenomenalists have not: Phenomenalists might have attained a greater success if they had made a remotely comparable effort to show in detail what conse-. We hereby explicitly sidestep the debate opposing the realist as opposed to the super-realist, see W. Tait position and the anti-realist position. While Dummett chooses to pin the opposition to intuitionism, to a belief in an interpretation of mathematical statements as referring to an independently existing and objective reality Dummett, , p.
Avigad memorably retorts as follows: We do not need fairy tales about numbers and triangles prancing about in the realm of the abstracta Avigad, Turning now to the realist narrative of the intuitionist insurrection, we note that such a narrative appears to be more consistent with what Bishop himself actually wrote. In his foundational essay, Bishop expresses his position as follows: As pure mathematicians, we must decide whether we are playing a game, or whether our theorems describe an external reality.
The right answer, to Bishop, is that they do describe an external reality. In Bishop, , p. Kopell and G. Stolzenberg, close associates of Bishop, published a threepage. Similar views have been expressed by D. Bridges, see e. Bridges, , and, as we argue in Subsection VII.
The Interview with a constructive mathematician was published by leading constructivist F. Richman in In the Interview, Richman seems to reject any alternative to an anti-LEM constructivism, in his very first comment: the constructive mathematician dismisses classical mathematics as an exercise in formal logic, much like investigating the consequences of large cardinal axioms Richman, , p. It should be noted that fellow constructivist D.
This is typical of generalizations: the notion of a normal subgroup is equivalent to that of a subgroup in the context of the commutative law Richman, , p. Yet he quickly retreats to the safety of anti-LEM, observing that the law of excluded middle obliterates [ emphasis added— authors] the notion of positive content [. While little evidence is offered for such a counter-intuitive no pun intended assertion, it sets out the following hope.
If a numerical constructivist can, after many years, be led to abandon numerical constructivism and switch to anti-LEM; then also, at some future time, anti-LEM constructivists can perhaps be persuaded, by force of overwhelming evidence, to abandon anti-LEM and switch to numerical constructivism of the companion variety. The nature of such evidence will be discussed below. Richman writes that, even in the countable case, the centerpiece of the subject, the classification theorem for countable torsion abelian groups, cannot even be stated without ordinal numbers Richman, To Richman, the non-constructivity of the arguments appears to be compounded by the non-constructive formulation of the very statements of the results of torsion abelian group theory.
Richman apparently perceived a lack of numerical meaning of even the post-LEM kind, as discussed in Subsection V. What is the nature of the evidence in favor of a numerical constructivism of a companion variety? A mathematician working in the tradition of Archimedes, Leibniz, and Cauchy owes at least a residual allegiance to the idea that the most important mathematical problems are those coming from physics, engineering, and science more generally.
Is constructive mathematics part of classical mathematics? Tait argues that, unlike intuitionism, constructive mathematics is part of classical mathematics. In this sense, intuitionism is not a rival, but an offspring, of classical mathematics. To quote J. Avigad, [ t] he syntactic, axiomatic standpoint has enabled us to fashion formal representations of various foundational stances, and we now have informative descriptions of the types of reasoning that are justified on finitist, predicative, constructive, intuitionistic, structuralist, and classical grounds Avigad, This idea, as applied to intuitionism, was expressed by J.
Gray in the following terms: Intuitionistic logics were developed incorporating the logical strictures of Brouwer; constructivist mathematics still enjoys a certain vogue. But these are somehow contained within the larger framework [ emphasis added— authors] of modern mathematics. Gray, , p. In a similar vein, no less an authority than M. Heidegger wrote almost simultaneously with Kolmogorov quoted in Subsection II. Lest one should doubt whether Heidegger meant for his comments to apply to mathematics, he continues:. Lest one should doubt whether he had Brouwer in mind, Heidegger continues: In the controversy between the formalists and the intuitionists,.
What Heidegger appears to be saying is that if we take the supposed objects to be sets or integers, the issue becomes whether a primary way of. Yet, we will show that Heyting unequivocally sides with the view of intuitionism as a companion, rather than. The first part of the book is written in the form of a dialog among representatives of some of the main schools of thought in the foundations of mathematics.
The protagonist named. Class makes the following point: Intuitionism should be studied as part of mathematics. In mathematics, we study consequences of given hypotheses. The hypotheses assumed by intuitionists may in fact be interesting, but they have no right to a monopoly Heyting, , p. In other words, emptying our logical toolkit of the law of excluded middle is one possible foundational framework among others.
In this connection, S. One can legitimately pose the question whether Int is merely being polite in his response to Class. He does this, obviously, not as a way of disparaging intuitionism, but as a way of underscoring its intrinsic value as an intellectual pursuit, as distinct from a scientific pursuit.
Maddy claims that Heyting [. Int replies as follows: As to the mutilation of mathematics of which you accuse me, it must be taken as an inevitable consequence of our standpoint. It can also be seen as an excision of noxious ornaments, [. His comments show a clear recognition of the potency of the mutilation challenge, as well as a necessity to compensate for the damages. Heyting was able to, so to speak, rise above differences of Class and Int.
What would Heyting have thought of a rigorous justification of infinitesimals in the framework of classical logic? Already in , Heyting allowed his protagonist Form to cite a reference by A. Robinson in a comment dealing with the use of metamathematics for the deduction of mathematical results Heyting, , p.
Shapiro , in a retort to N. Here Heyting semantics refers to the constructive interpretation of the quantifiers discussed in Subsection II. Shapiro writes as follows: Let x be any predicate that applies to natural numbers. It is a routine theorem of classical arithmetic that. Shapiro, , p. Shapiro continues: Under Heyting semantics, this proof amounts to a thesis that there is a computable function that decides whether holds. So under Heyting semantics [. Namely, the constructive existence of y necessarily entails being able to find a computational procedure capable of identifying such a y as a function of x , and hence a decision procedure for the predicate.
Does this prove that classical mathematics and intuitionism cannot get along, as Shapiro puts it? In fact, the philosophical exchange between Tennant and Shapiro has an uncanny parallel— a quarter century earlier, in the exchange between Bishop and Timothy G. McCarthy, writing for Math Reviews It is not plausible to suppose that A is classically true false if and only if [ the proposition] LPO! A [ respectively,] LPO! Such seems to be the position adopted by the mature Heyting, as well, as we analyze in the next section.
Robinson Heijting, are fascinating and worth reproducing in some detail: 1. The two subjects become more and more intertwined. For a discussion of the transfer principle of non-standard analysis, see Appendix A. Heyting does not stop there, and makes the following additional points: 1. This is the case in differential geometry and in many parts of applied mathematics such as hydrodynamics and electricity theory, where physicists use infinitesimals without a twinge of conscience.
You showed how these theories can be made precise by your method. This mysterious object became lucid in the light of non-archimedean analysis [ emphasis added— authors]. On the subject of whether there is meaning after model theory, Heyting has this to say: 1. Once you had shown by the paradigm of the calculus how it can be used, many other applications were found by yourself as well as by many other mathematicians. In a sense your work can be considered as a return from this abstraction to concrete applications. The general non-constructive theory of non-standard models links its applications together into a harmoni[ ous] whole.
He recognizes that, while the model may be non-constructive, its applications may indeed be concrete. Such a thesis is consonant with what we called numerical constructivism in Subsection V. Such an awareness is therefore not at odds with a recognition of the coherence of post-LEM numerical meaning. It is at odds, however, with an anti-LEM radical constructivist stance, as enunciated by E.
Constructivism, physics, and the real world. In this section we deal with challenges to anti-LEM constructivism stemming from natural science. Bishop Bishop, , p. Here is a question that has bothered me ever since the first time I read Bishop. In what sense can your [ constructive] real line be used as a model for either space or time in physics? Richman, , p. It needs to be understood what Richman meant by this cryptic remark.
A classical mathematician believes that the circle can be decomposed as the disjoint union of the lower halfcircle with one endpoint included, and the other excluded and its antipodal image upper halfcircle. Meanwhile, the constructivist believes that it is impossible to decompose the circle as the disjoint union of a pair of antipodal sets.
Colyvan , or G. In fact, Bishop himself was challenged on the relationship to physics by G. Mackey, in the following terms: Consider the foundational question in physics: what is the real mathematics that the physicists are doing? Rather, it involves the relation of the results to the real world [ emphasis added— authors]. However, what Bishop is emphasizing in his essay is a notion of meaning. The kind of unabashedly LEM-dominated mathematics that a physicist practices is, as Bishop seems to admit matter-of-factly, successful in relating to the real world, and therefore apparently meaningful.
In his Constructivist manifesto, Bishop wrote that Weyl, a great mathematician who in practice suppressed his constructivist convictions, expressed the opinion that idealistic [ i. Bishop does not elaborate as to why he feels H. The challenge to constructivism stemming from the Quine-Putnam indispesability thesis has been extensively pursued by G. Hellman expresses an appreciation of the foundational significance of constructivism in the following terms: A turning point [.
Hellman a, p. However, one can recognize the coherence of the intuitionist critique of the foundations of mathematics, while rejecting the intellectual underpinnings of its insurrectional narrative, in its mutually contradictory realist and anti-realist versions, as discussed in Subsection VI. In the context of general relativity theory, Hellman argues that, very likely, the Hawking— Penrose singularity theorems for general relativistic spacetimes are essentially non-constructive Hellman, , see also Billinge, , see Subsection VIII. The philosopher of mathematics M.
The battle imagery is typical of the anti-realist type of insurrectional narrative, already discussed in Subsection VI. Meanwhile, the philosopher of mathematics G. Figure VIII. More generally, variational problems tend to be resistant to efforts at constructivizing, a point apparently acknowledged by Beeson when he writes that Calculus of variations is a vast and important field which lies right on the frontier between constructive and non-constructive mathematics Beeson, , p.
The problem is that the extreme value theorem is not available in the absence of the law of excluded middle. The extreme value theorem is at the foundation of the calculus of variations. As concrete examples, one could mention general existence results for geodesics, minimal surfaces, constant curvature mean surfaces, and therefore soap films and soap bubbles.
Question VIII. Classically, one has a description of the resulting object in terms of a minimizing possibly non-unique closed geodesic. Providing a constructive description is not immediate, as it depends on general results of the calculus of variations, as already described above. What Novikov makes excruciatingly clear is the disastrous effect of such a divorce on mathematics itself. A broad range of subjects illustrating the symbiotic relationship between the two fields can be found in Grattan-Guinness Grattan-Guinness emphasizes the role of analogies drawn from other theories,.
What is worse, the gap is growing wider, according to Novikov. Traditional mathematical foundations, whether classical or intuitionist, are proving to be inadequate for the job of accounting for the progress in physics. Accordingly, Novikov is critical of set-theoretic foundations, going as far as criticizing Kolmogorov himself, for systematic efforts to introduce a set-theoretic approach in secondary education. The Hawking— Penrose theorem in relativity theory has been the subject of something of a controversy in its own right, see G. Hellman , H. Billinge , and E. Hellman argued that the Hawking— Penrose theorem is an important result which very likely does not have a constructive analog.
Davies , p. The Hawking— Penrose singularity theorem can be thought of as a semi-Riemannian analog of the Myers theorem Myers, The latter is a result in Riemannian geometry. The result in question is a bound, modulo a suitable geometric hypothesis on the Ricci curvature, on the distance from a point to its conjugate locus. The traditional formulation of the Myers theorem relies upon the concept of a geodesic and a Jacobi field, both of which are solutions of variational problems.
Myers found a bound on the distance along a geodesic , to the nearest vanishing point, called a conjugate point, of a Jacobi field along. Returning to the Lorenzian case, note that Penrose does not even use the term singularity in formulating the result that came to be known as the Hawking— Penrose theorem.
The proof is indirect, i. Penrose describes one of the main ingredients in the proof as the Raychaudhuri effect. The latter is discussed in Penrose, , item 7. Penrose points out that [ f] or manifolds with a positive definite metric, essentially the same effect had been studied earlier by Myers Penrose, , p. Penrose places singularities. Note the significant i. Whether or not geodesic incompleteness of spacetime can be understood in terms of a suitable singular limiting object, boundary, or hole, is a separate question extensively discussed in the physics literature, see references cited in Wald, , p.
Furthermore, while a suitable bound on the Ricci curvature does allow one to prove the existence of limiting objects in a Riemannian context see Gromov , Theorem 5.
The following phrase appears in Billinge: a spacetime is singular if it is timelike or null geodesically incomplete,. Rather, the geodesics are, as Penrose puts it in Penrose, , p. The textbook contains the following remarks, which express the distinction between. We provide motivation for the notion of timelike and null geodesic incompleteness as a criterion for the presence of a singularity [.
Such an error could have been written off as a mere slip, were it not for the further evidence of insufficient background presented above. Tennant, writing on Logic, mathematics, and the natural sciences,. Tennant claims that such a constructivist version is adequate for applications. Tennant continues by explaining that to be able to produce all possible refutations [ emphasis added— authors] of empirical theories, the underlying logic can [. Furthermore he concedes: Nor do we intend to say anything about confirmation or probabilification of hypotheses by evidence Tennant, , p.
However, this does not alter the fact that expressing the Hawking— Penrose theorem let alone proving it remains an elusive goal, intuitionistically speaking, with the attendant loss of scientific insight provided by the theorem. The Hawking— Penrose theorem, described by Davies as extremely influential.
Figure A. In , Robinson delivered the Brouwer memorial lecture Robinson, on the subject of Standard and nonstandard number systems. Robinson concludes that the dynamic evolution of mathematics is an ongoing process not only at the summit [. We would like to suggest that the possibility of bridging the gap between constructivism and non-standard analysis as well as, indeed, the rest of classical mathematics can be analyzed in the context of the dichotomy of a numerical constructivism versus an anti-LEM constructivism see Section V.
Rival continua. A Leibnizian definition of the derivative as the infinitesimal quotient. We will denote such a B-continuum by the new symbol IIR. We illustrate the construction by means of an infinite-resolution microscope in Figure A. Hewitt in , see Hewitt, , p. To elaborate on the ultrapower construction of the hyperreals, let Q N. Let Q N C denote the subspace consisting of Cauchy sequences. The reals are by definition the quotient field. Meanwhile, an infinitesimalenriched field extension of Q may be obtained by forming the quotient.
See Figure A. Here a sequence h u n i is in F u if and only if the set. A more detailed discussion of the ultrapower construction can be found in M. Davis , Gordon et al. More advanced properties of the hyperreals such as saturation were proved later, see Keisler for a historical outline. Lightstone A discussion of infinitesimal optics is in K.
Stroyan , H. Keisler , D. Tall , L. Magnani and R. Dossena , and R. Magnani Ehrlich recently constructed an isomorphism of maximal surreals and hyperreals Ehrlich, Arkeryd , ; modeling of timed systems in computer science see H. Rust, ; mathematical economics see Anderson, ; mathematical physics see Albeverio et al. In , Hewitt reminisced about his. Nauk SSSR 22 [ ], Along the way I found a novel class of real-closed fields that superficially resemble the real number field and have since become the building blocks for nonstandard analysis.
I had no luck in talking to Artin about these hyperreal fields, though he had done interesting work on real-closed fields in the s. Only Irving Kaplansky seemed to think my ideas had merit. My first paper on the subject was published only in Hewitt, Hewitt goes on to detail the eventual success and influence of his text. We are grateful to Geoffrey Hellman for insightful comments that helped improve an earlier version of the manuscript, to Douglas Bridges for a number of helpful remarks, and to Claude LeBrun for expert comments on the Hawking— Penrose theorem. Pure and Applied Mathematics,.
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Luxemburg and S. Reprint of the second edition. With a foreword by Wilhelmus A. Princeton Landmarks in Mathematics. Educational Studies in Mathematics 69 , — Errett Bishop: reflections on him and his research. Contemporary Mathematics, MLQ Math. Number concepts underlying the development of analysis in 17— 19th Century France and Germany. Proceedings of the symposium held in Venice, May 16— 22,