Abstract
Roshdi Rashed’s work illustrates perfectly what can be a conscious and cautious practice of reflection, with the purpose of setting history of science (and mathematics) on renewed and deeper grounds (See the introduction, Problems of method: history of science between history and epistemology, in Classical Mathematics from Al-Khwarizmi to Descartes , 2014, (Rashed 2014).). This entails the methodical operations that he enumerates, such as enlargement towards undermined or ignored traditions (Chinese, Arab, Indian, etc.), careful and reasoned decompartmentalization of disciplines, correlative changes of periodization (without which the critique of scientific ideology and ideology of scientists would risk of falling back into some counter-ideological history, particular or general). (Europeocentrism for instance is twofold: promotion of the ambiguous and disputable notion of “western science” and ignorance or “minorization” of the contributions of non-western traditions. Cf. (Rashed 1984) and appendices in The Notion of Western Science: “Science as a Western Phenomenon” and “Periodization in Classical Mathematics” (Rashed 2014).) Among mathematicians, Gian-Carlo Rota is certainly both exceptional and, for this reason, exemplary. By choosing this perspective as a tribute, I hope that Roshdi Rashed will consider my comments not too unworthy. For any philosopher of science not insensitive to history of science, and for any historian not completely allergic to philosophical reflection, studying Rota’s contribution in the fields of logic and phenomenology reveals itself instructive and fruitful. Contrary to dominant trends amongst his colleagues, in his own way, Rota showed a strong and continuous interest in logic, history of science and philosophy.
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- 1.
Crapo and Senato (2001). D. A. Buchsbaum, Resolution of Weyl modules: the Rota touch, in H. Crapo, D. Senato (Eds) Algebraic Combinatorics and Computer Science A Tribute to Gian-Carlo Rota, Springer, 2001, p. 97-sq.
- 2.
Crapo and Senato (2001: 4).
- 3.
Rota (2001, p. 51 sq.) Twelve problems in probability no one likes to bring up, in Algebraic Combinatorics and Computer Science, H. Crapo et alii. (Eds.), Springer-Verlag, Milano, 2001.
- 4.
About his indefectible interest in logic, despite his teachers in mathematics, cf. his narrative about Church’s lectures in Indiscrete Thoughts, (Rota 1997b, p. 4–7); on his appraisal of the discipline as such, against what he calls, following Kemeny, “the mathematician’s bigotry”, see (Rota 1997b, pp. 7, 105, 123) (Rota 1990a).
- 5.
Despite and because of the strong eidetic background of his approach, Rota refutes expressly the classical problematic of mathematical existence in terms of Platonism (Frege, Gödel, etc.) in favour of an “eidetic perspectivism”, which has strong affinities with Husserl’s transcendental idealism. (See below, Sect. 2.) But we can anticipate on this particular point: “To the mathematician, an axiom system is a new window through which the object, be it a group, a topological space or the real line, can be viewed from a new and different angle that will reveal heretofore unsuspected possibilities. In saying this, we are acknowledging the actual practice of mathematics, and we are not arguing for Platonism: we are not in the least concerned with the problem of existence of mathematical objects, any more than the grammarian is concerned with the problem of existence of the verb or the adjective”. (Rota et al. 1988, p. 382). (See Rota 2000, p. 93; Rota 1991, p. 133–138.)
- 6.
Rota 1991, p. 247.
- 7.
Husserl (1950a, p. 170). See full quote below. Rota (1991, p. 136), insists on the fact that this applies to mathematics too: “Thus it appears that the activity of the mathematician is not that of proving theorems, but another one: that of proving that all theorems are intuitively evident, by an evidence that shall be as close as possible to Kant’s ideal of an analytic a priori statement.”
- 8.
“Psychologists have prescribed in turn sexual release, wonder drugs and primal screams as the cure for common depression, while preachers would counter with the less expensive offer to join the hosannahing chorus of the born-again. It goes to the credit of mathematicians to have been the slowest to join this movement. Mathematics, like theology and all free creations of the Mind, obeys the inexorable laws of the imaginary, and the Pollyannas of the day are of little help in establishing the truth of a conjecture. One may pay lip service to Descartes and Grothendieck when they wish that geometry be reduced to algebra, or to Russell and Gentzen when they command that mathematics become logic, but we know that some mathematicians are more endowed with the talent of drawing pictures, others with that of juggling symbols and yet others with the ability of picking the flaw in an argument. We often hear that mathematics consists mainly in ‘proving theorems.’ Is a writer’s job mainly that of ‘writing sentences’? A mathematician’s work is mostly a tangle of guesswork, analogy, wishful thinking and frustration, and proof, far from being the core of discovery, is more often than not a way of making sure that our minds are not playing tricks.» (Rota et al. 1992, p. 154).
- 9.
The narrative continues: “Several years ago I stated this opinion in a preprint I sent to Gödel , and he wrote me back a letter, the only letter I have ever received from him: ‘Dear Professor Rota, You are wrong. Husserl is not the greatest philosopher of all times. He is the greatest philosopher since Leibniz . Sincerely yours,’. Years later I understood what Gödel meant, after his writings in phenomenology were published, in the third volume of his collected papers. Gödel believed Husserl to be the philosopher who brought to a successful completion the program inaugurated by Immanuel Kant, which Kant and his successors had failed to carry through. ‘Husserl is the real Kant’, Gödel writes.” (ibid.)
- 10.
Quantum probability followed another path, by giving the proof of the experimental necessity of non-Kolmogorovian models (Accardi 1981a, b), by showing that Bayes’ definition of “conditional” was the hidden axiom of probability (an analogue to Euclide’s 5th Postulate) (Accardi and Fedullo 1981) responsible for the difference between the Kolmogorov and the quantum mathematical model; by giving the list of axioms unifying classical and quantum probability and producing non-trivial models (Accardi 1982a, b) and finally by dissipating the so-called paradoxes of quantum theory (Accardi 1993, 1999, 2003), and proposing a presentation fully compatible with standard logic.
- 11.
Same observation in Weyl (1968, p. 705): “Each field of knowledge, when it crystallizes into a formal theory, seems to carry with it its intrinsic logic which is part of the formalized symbolic system and this logic will, generally speaking, differ in different fields.” (Weyl, Gesammelte Abhandlungen. Band, III K. Chandrasekharan (ed.), Springer Verlag, Berlin, 1968, p. 705.)
- 12.
Sharing Hilbert’s conception of the relation between models and axiomatization, hence the so-called formalist approach to syntax and conception of formalization, Accardi (1981a, b) considers that Kolmogorov only presented a “mathematical model” for probability, comparable to Descartes geometry, and that one had to wait for De Finetti or Ramsey to get a first real but still incomplete axiomatization of classical probabilities, comparable to Hilbert’s axiomatization of Euclidian geometry. In this perspective, this axiomatization remained incomplete, because some principles were neither deducible from the axioms, nor exhibited as an axiom. This was the case for countable additivity. Another limit is the inadequacy of Kolmogorov’s model and its axiomatizations to handle probabilities in quantum physics, which required a new mathematical model and a new axiomatization. According to Accardi, Von Neumann only proposed a few steps toward a construction of a mathematical model (taking into account an analogue of space probability and random variable). The full project was only accomplished by quantum probability in his sense, by taking into account stochastic processes, quantum joint probabilities, independence, conditioning. This enlargement is compared to those of non-Euclidian geometries. — In order to avoid major misunderstandings and as it will appear clearly in the following, let us recall that, first, in the case of geometry as in the case of probability, Rota has a quite different understanding of the relation between models and axiomatics, semantics and syntax, and, secondly, that by quantum probability he does not aim at quantum probability in Accardi’s sense, but essentially von Neumann attempts to propose a syntax (not a model) for quantic probability; thirdly, that contrary to a limited view of combinatorics, consideration of continuity are not excluded from it and it was precisely the purpose of Rota and Klain’s “geometric probability” to extend enumeration to “the assigning of invariant measures” (Rota and Klain 1997).
- 13.
See Formal and Transcendental logic, English tr. p. 48 and p. 218.
- 14.
A \( \sigma \)-algebra of a set \( \Omega \) is the collection \( \varSigma \) of subsets of \( \Omega \) satisfying the following conditions whatever the the number (from 0 to enumerable infinite) and combinations of operations implemented. These operations, in the case of a \( \sigma \)-algebra, are: complement, union and intersection.
- 15.
The ring R generates Ω and the measure function μ, which maps from R onto the interval [0, + ∞]. Caratheodory’s extension theorem affirms the existence of a measure μ′ from the \( \sigma \)-algebras generated from R onto the same interval [0, + ∞], extending μ. And if m is s-finite, μ′ is unique and s-finite. Cf. Caratheodory (1948), and its presentation, in Halmos (1950, p. 54); and, more recently, Kallenberg (2002, p. 455).
- 16.
What about “pointless topology”? According to Rota (1997, p. 220) “the term ‘pointless topology’ goes back to von Neumann” and led to “serious misunderstandings”. Johnston (1983, p. 41) refers it back to Hausdorff’s choice of the notion of “open set” as primitive “in the study of continuity properties in abstract spaces”, but it is Stone [1934] which started exploiting “the connection between topology and lattice theory”. Affinity between Johnstone’s approach and Rota’s are obvious.
- 17.
Published in the Notices of the AMS, 1997, Volume 44, Number 11.
- 18.
A functor extends morphisms to categories themselves; a morphism being usually a function between two sets with the same structure (field, ring, group, Boolean algebra, lattices, etc.). In category theory, the function relates eventually two or more different mathematical objects (structures), without presupposing necessarily any isomorphism.
- 19.
See quote above, Discrete Thoughts, p. 154; and: “One wonders why category theory has aroused such bigoted opposition. One reason may be that understanding category theory requires an awareness of analogies between disparate mathematical disciplines, and mathematicians are not interested in leaving their narrow turf.” (Rota 1997b, p. 220)
- 20.
“The axiomatization of the notion of category, discovered by Eilenberg and Mac Lane in the forties, is an example of beauty in a definition, though a controversial one. It has given rise to a new field, category theory, which is rich in beautiful and insightful definitions and poor in elegant proofs. The basic notions of this field, such as adjoint and representable functor, derived category, and topos, have carried the day with their beauty, and their beauty has been influential in steering the course of mathematics in the latter part of this century; however, the same cannot be said of the theorems which remain clumsy.” (Rota 1997b, p. 220)
- 21.
Same statement a little more passionate and anecdotic in another indiscrete thought: “Algebraic geometry has been the bottom line of mathematics for almost one hundred years; but perhaps times are changing. The second story is more sombre. One day, in my first year as an assistant professor at MIT, while walking down one of the long corridors, I met Professor Z, a respected senior mathematician with a solid international reputation. He stared at me and shouted, ‘Admit it! All lattice theory is trivial!’ I did not have the presence to answer that von Neumann’s work in lattice theory is deeper than anything Professor Z has done in mathematics. Those who have reached a certain age remember the visceral and widespread hatred of lattice theory from around 1940 to 1979; this has not completely disappeared.” (Rota 1997b, p. 52)
- 22.
Same position in 1973: “The theory of distributive lattices is richer than the better known theory of Boolean algebras; nevertheless it has had an abnormal development, for a variety of reasons of which we shall recall two. First, Stone’s representation theorem of 1936 for distributive lattices closely imitated his representation theorem for Boolean algebras, and as a consequence turned out to be too contrived (I have yet to find a person who can state the entire theorem from memory.) Second, a strange prejudice circulated among mathematicians, to the effect that distributive lattices are just Boolean algebra’s weak sisters. More recently, the picture seems to have brightened. The definitive representation theorem for distributive lattices has been proved by H. A. Priestley; it extends at long last to all distributive lattices the duality ‘distributive lattice- partially ordered sets’, first noticed by Birkhoff for finite lattices. Strangely, Nachbin’s theory of ordered topological spaces had been available since 1950, but nobody before Priestley had had the idea of taking a totally disconnected ordered topological space as the structure space for distributive lattices.” (Valuation Ring of a distributive Lattice, Proceedings of Houston Lattice Theory Conference, Houston, 1973a) (Rota 1973a, p. 577).
- 23.
Théorie et applications des treillis, in Annales de l’Institut Henri Poincaré , Gauthier-Villars, Paris, (Tome 11, n°5 (1949), p. 227–240).
- 24.
We are following here Halmos’s definition and can justify this by its own comment: “Although it looks slightly different, this definition is, in fact, equivalent to the usual one: a Boolean algebra is, after all, a ring and a Boolean ideal in the present sense is the same as an ordinary algebraic ideal. Each of the two concepts (filter and ideal) is, in a certain sense, the Boolean dual of the other. (Some authors indicate this relation by using the term dual ideal instead of filter). Specifically, if P is a filter in a Boolean algebra A, and if M is a set of all those elements p of A for which \( p^{\prime } \in {\text{P}} \) [read: not p belongs to P], then M is an ideal in A, the reverse procedure (making a filter out of an ideal) works similarly. This comment indicates the logical role of the algebraically more common concept; just as filters arise in the theory of provability, ideals arise in the theory of refutability. (A proposition \( p \) is called refutable if its negation \( p^{\prime } \) is provable. (…) Logic is usually studied from the “1” approach i.e., the emphasis is on truth and provability, and consequently, on filters.” (P. Halmos, The Basic Concepts of Algebraic Logic, 1956, p. 371–372.) A Boolean logic will be a pair (A, M) where A is a Boolean algebra and M is a Boolean ideal in A, with A (set of all Boolean propositions) and M (refutable propositions) and eventually M* (filter or dual-ideal) refutable propositions.
- 25.
“Traditionally, the algebraic properties of Boolean algebras are reduced to those of Boolean rings by a well known construction. A Boolean ring, however, has the double disadvantage of having torsion, and of not being applicable to the richer domain of distributive lattices. In this paper we describe another construction, or functor, called the valuation ring, which associates to every distributive lattice L a torsionless ring V(L) generated by idempotents. The lattice L can be recovered by giving a suitable order structure to the valuation ring V(L), and thus the entire theory of distributive lattices is reduced to that of a simple class of rings. For example, the representation theory of distributive lattices is subsumed to that of valuation rings, where standard methods of commutative algebra apply.” (Rota 1973a, p. 575)
- 26.
While revising this paper, thanks to Mioara Mugur-Schächter, I got acquainted with Olivia Caramello’s work, which shares with Rota a comparable interest in logic. I see also some affinity between the way she uses in a freer spirit Grothendieck toposes and the way Rota uses lattices to build “bridges” between different fields of mathematics, not to speak of their similar scepticism vis-à-vis the pretension of set theory or category theory to be the ultimate language of mathematics (semantics or syntax). “Considered the importance of building bridges between distinct mathematical branches, it would be highly desirable that Logic would not just serve as a tool for analyzing analogies already discovered in Mathematics but could instead play an active role in identifying new connections between existing fields, as well as suggesting new directions of mathematical investigation. As it happens, we now have enough mathematical tools at our disposal for trying to achieve this goal. By providing a system in which all the usual mathematical concepts can be expressed rigorously, Set Theory has represented the first serious attempt of Logic to unify Mathematics at least at the level of language. Later, Category Theory offered an alternative abstract language in which most of Mathematics can be formulated and, as such, has represented a further advancement towards the goal of ‘unifying Mathematics’. Anyway, both these systems realize a unification, which is still limited in scope, in the sense that, even though each of them provides a way of expressing and organizing Mathematics in one single language, they do not offer by themselves effective methods for an actual transfer of knowledge between distinct fields. On the other hand, the principles that we will sketch in the present paper define a different and more substantial approach to the unification of Mathematics. Our methodologies are based on a new view of Grothendieck toposes as unifying spaces, which can serve as ‘bridges’ for transferring information, ideas and results between distinct mathematical theories. (…) In this paper, I give an outline of the fundamental principles characterizing my view of toposes as ‘bridges’ connecting different mathematical theories and describe the general methodologies which have arisen from such a view and which have motivated my investigations so far. (…). These principles are abstract and transversal to the various mathematical fields, and the application of them can lead to a huge amount of surprising and non-trivial results in any area of Mathematics, so we hope that the reader will get motivated to try out these methods in his or her fields of interest.” (Caramello 2010, p. 3–4)
- 27.
See above: note 13 and 15.
- 28.
We learn from Rota, via S. Ulam, that von Neumann himself, without giving up his intimate conviction about the inadequacy of naive set theory to contribute to the foundation of quantum mechanics, admitted his own failure to give an alternative foundation. “Most of von Neumann’s work in pure mathematics (rings of operators, continuous geometries, matrices of high finite order) is concerned with the problem of finding a suitable alternative to Boolean algebra, compatible with the uncertainty principle, upon which to found quantum theory. Nevertheless, the mystery remains, and von Neumann could not conceal in his later years a feeling of failure over this aspect of his scientific work (personal communication from S. M. Ulam).” (Rota et al. 1992, p. 168; Rota 1987a, p. 26). But Rota (1997: 1444) “Haiman’s proof theory for linear lattices brings to fruition the program that was set forth in the celebrated paper “The logic of quantum mechanics”, by Birkhoff and von Neumann. This paper argues that modular lattices provide a new logic suited to quantum mechanics. The authors did not know that the modular lattices of quantum mechanics are linear lattices. In the light of Haiman’s proof theory, we may now confidently assert that Birkhoff and von Neumann’s logic of quantum mechanics is indeed the long-awaited new ‘logic’ where meet and join are endowed with a logical meaning that is a direct descendant of ‘and’ and ‘or’ of propositional logic.” (Rota 1997a, p. 1444)
- 29.
While working on this paper, Luigi Accardi informed me that: “Rota è legato, sia pure indirettamente alla nascita della probabilità quantistica poiché, verso la fine degli anni 1970 (o primissimi 1980) egli tenne un corso di probabilità a Pisa e mi invitò a fare un seminario in quell’ambito. Nei pochi giorni che rimasi a Pisa decisi di leggere il lavoro di Bell sulla sua nota disuguaglianza, che molti mi avevano segnalato come un contributo fondamentale ai fondamenti della MQ. Era da molti anni che riflettevo sul problema sollevato da Feynman, relativamente all’esperimento delle due fenditure, e ricordo ancora oggi l’eccitazione che provai quando, nella Biblioteca della Scuola Normale Superiore di Pisa, intuii che sia questa analisi, sia la disuguaglianza di Bell, erano basate sulla stessa ipotesi matematica, usata implicitamente da entrambi gli autori: la possibilità di descrivere dati statistici, provenienti da esperimenti mutuamente incompatibili, all’interno di un unico modello Kolmogoroviano. ” (Personal communication, 6th of July, 2015).
- 30.
See the “natural” although somewhat idealized chameleon of L. Accardi (Urns and Chameleons). We have something similar with the embedment of quantum experiments into a non-differentiable and fractal space-time proposed by L. Nottale; many macro-physical and classical phenomena appeared describable in terms of periodical peaks of probabilities modelled (as a “Schrödinger’s flower”) (Nottale 2004), Daniel M. Dubois Editor, American Institute of Physics Conference Proceedings, 718, 68–95 (Nottale 2004, p. 68–95)). —Departing from quantum probabilities, see the new epistemological approach promoted by Mioara Mugur-Schächter (Mugur-Schächter 1994 and 2009), involving not only a new layer for QM, but also another deeper concept of probability (Mugur-Schächter 2014) and a new general epistemology (Mugur-Schächter 2002).—Another direction, stripped of the usual mathematical apparatus of QM, is represented by the search for anthropological or psycho-neurological quantum phenomena: M. Bitbol, La structure quantique de la connaissance individuelle et sociale, in Theorie quantique et sciences humaines, 2009, CNRS Editions, Paris). Last: Pierre Uzan, Conscience et physique quantique, Mathesis, Vrin 2013.
- 31.
In its presentation from 1973, the analogy between probability and predicate logic is mediated through the following steps: assignment of a probability measure m to the canonical idempotents; definition of a lattice space norm on the valuation ring by using the linear functional |Sa(e)e| = S|a(e)|m(e); complementation of the resulting normed linear space which can be seen as the space of all integrable functions; representation of the averaging operator as a conditional expectation operator, which once relaxed from the restrictive condition that “every element of the range be finite-valued”, can apply universally). (Rota 1973a, p. 622, 1960). We read page 607: “An averaging operator on a valuation ring V(L) is a linear operator A such that: (1) Au = u, Az = z. (2) A(fAg) = Af Ag. (3) If f is in the monotonic cone, so is Af. Sometimes these operators go by the name of Reynolds operators. In probability, they are called conditional expectations.”
- 32.
Despite Rota’s somewhat severe judgment about category theory, it seems that some of his insights are founding a (maybe) successful realization in the work of Olivia Caramello (Caramello 2010). Her purpose, as exposed in The Unification of Mathematics via Topos Theory, is precisely: “to present a set of principles and methodologies which may serve as foundations of a unifying theory of Mathematics”, “based on a new view of Grothendieck toposes as unifying spaces being able to act as ‘bridges’ for transferring information, ideas and results between distinct mathematical theories”. Neither this use of classifying toposes, nor Rota’s construction of cryptomorphism as “bridges” are trivial and limited to translating mathematical theories into one another, but include logical and syntactical properties and mathematical properties (in Rota’s case, for instance, between logical quantification and probability, and in Caramello’s, between completeness and joint dip). For similar reasons, by standing at the crossroads of mathematics and logics and challenging their respective and exclusive pretention to fix the formal frame of scientific rationality, Rota and Caramello contributions catalyse negative reactions from the part of “pure” mathematicians and “pure” logicians.
- 33.
Rota 1991, p. 134 & 135, respectively.
- 34.
“In 1973, Rota exploited the idea of averaging operators in the radically different context of valuation rings of a distributive lattice in [50] [i.e. here (Rota 1973a)]. This paper is fascinating because as Rota wrote, ‘the method of presentation is deliberately informal and discursive’. Much of the reasoning is by analogy, but not in the structuralist or Bourbakian sense, and many of the results are only heuristic. The aim is grandiose, nothing less than the ‘linearization of logic’.” (J. Dhombres, Reynolds and Averaging Operators, in Gian-Carlo Rota, On Analysis and Probability, Selected Papers, Birkhaüser 2003) (Dhombres 2003).
- 35.
Reynolds operator is an averaging operator, i.e. a heuristic way of simplifying Navier-Stoke equations that reveal fundamental for hydrodynamic (description of gas, fluids, etc. flows) and beyond. Their form is quite complex, combining three quantities of the material elements (mass, moment, energy: whose density is measured: r(r, t); velocity field: u(r, t) and energy density per mass-unity e(r, t). For more details see: Rota, On the Passages from the Navier–Stokes Equations to the Reynolds Equations (Rota 2003, p. 137–139). From the start, Rota’s interest is the representations (i.e. models) of the Reynolds operator, in ergodic theory, spectral theory, and probability theory. The simplified form writes: R (fg) = RfRg + R {(f – Rf) (g – Rg)}. The passage from N-S.’s operator to Reynold’s is smoothed, so to speak, by considering the functions of Navier-Stokes equations as random functions. (NB. The averaging form is: A[f(Ag)] = (Af)(Ag), with f, g two functions belonging to a commutative algebra.) Reynolds operator is thus seen as a conditional expectation operator of form: ES′ (uES′v) = ES′uES′v.
- 36.
- 37.
“In 1973, Rota exploited the idea of averaging operators in the radically different context of valuation rings of a distributive lattice in [50] [here, under (Rota 1973a)]. This paper is fascinating because as Rota wrote, ‘the method of presentation is deliberately informal and discursive’. Much of the reasoning is by analogy, but not in the structuralist or Bourbakian sense, and many of the results are only heuristic. The aim is grandiose, nothing less than the ‘linearization of logic’”, Reynolds and Averaging Operators, in Gian-Carlo Rota, On Analysis and Probability, Selected Papers, Birkhaüser, 2003, p. 163 (Dhombres 2003, p. 163; Rota 1990b).
- 38.
“I was too young and too shy to have an opinion of my own about Church and mathematical logic. I was in love with the subject, and his course was my first graduate course. I sensed disapproval all around me; only Roger Lyndon (the inventor of spectral sequences), who had been my freshman advisor, encouraged me. Shortly afterward he himself was encouraged to move to Michigan. Fortunately, I had met one of Church’s most flamboyant former students, John Kemeny, who, having just finished his term as a mathematics instructor, was being eased—by Lefschetz’s gentle hand—into the philosophy department.” (Rota 1997b, p. 7).
- 39.
Rota’s preface to P. J. Davis and R. Hersh’s book on The Mathematical Experience, Birkhaüser, 1981, p. xix. “They have opened a discussion of the mathematical experience that is inevitable for survival. Watching from the stern of their ship, we breathe a sigh of relief as the vortex of oversimplification recedes into the distance” (Rota 1981); see Husserl, in Formal und Transcendental Logic, § 60 and my comment on this parallel enlargement of the concepts of experience and evidence (Lobo 2006, p. 147–148; p. 160 sq.).
- 40.
Husserl writes: “eine Idee als zugehörige, die aber nie letzte ist, sondern Anhieb, in gewisser Weise Darstellung der im Unendlichen liegenden und unerreichbaren Idee, von der nur die Form als absolut Norm aller Konstruktion der Anhiebe gegeben ist.”
- 41.
Gödel’s first theorem states that there will always be, at least, arithmetic problems, which are syntactically expressible and intuitively solvable, whose truth is not demonstrable within the formal system.
- 42.
- 43.
Lobo (2012, p. 172–185).
- 44.
With good reasons, in the footsteps of Husserl’s critique of objectivism, Rota will later designate them as “items” instead of “objects” or “identities”, claiming the “end of objectivity”, i.e. “the end of the objectivistic conception of experience” (Rota 1991, p. 138).
- 45.
For the method of analogizing for the investigation of affective and volitional intentionality (and its correlates: values and goods) as well as the extraction of their corresponding formal structures see the Lessons on ethics and theory of values (Husserl 1988, pp. 37–38; 41–44, 45–50 etc., 2009, pp. 111–112, 115–119, 120–126).
- 46.
“Every field of mathematics has its zenith and its nadir. The zenith of logic is model theory (we do not dare state what we believe will be its nadir). The sure sign that we are dealing with a zenith is that as we, ignorant and dumb non-logicians, attempt to read the stuff, we feel that the material should be rewritten for the benefit of a general audience”.
- 47.
A new paradise was opened when Paul Cohen invented forcing, soon to be followed by the reform of the Tarskian notion of truth, which is the idea of Boolean-valued models. Of some subjects, such as this one, one feels that an unfathomable depth of applications is at hand, which will lead to an overhaul of mathematics.” (Rota 1997b, p. 218).
- 48.
Let us give the full quote of note 22: “We were turned off category theory by the excesses of the sixties when a loud crowd pretended to rewrite mathematics in the language of categories. Their claims have been toned down, and category theory has taken its modest place side by side with lattice theory, more pretentious than the latter, but with strong support from both Western and Eastern Masters.—One wonders why category theory has aroused such bigoted opposition. One reason may be that understanding category theory requires an awareness of analogies between disparate mathematical disciplines, and mathematicians are not interested in leaving their narrow turf” (Rota 1997b, p. 220).
- 49.
Ever since theoretical computer scientists began to upstage traditional logicians, we have watched the resurgence of nonstandard logics. These new logics are feeding problems back to universal algebra, with salutary effects. Whoever believes that the theory of commutative rings is the central chapter of algebra will have to change his tune. The combination of logic and universal algebra will take over.” (Rota 2008, p. 218–219) (see also Rota 1985, 1986b)
- 50.
“It has always been difficult to take quantum logic seriously. A malicious algebraist dubbed it contemptuously ‘poor man’s von Neumann algebras’. The lattice-theoretic background made people suspicious, given the bad press that lattice theory has always had.” (Rota 1997b, p. 218–219).
- 51.
“A more accomplished example of Desperationsphilosophie than the philosophy of quantum mechanics is hard to conceive. It was a child born of a marriage of misunderstandings: the myth that logic has to do with Boolean algebra and the pretence that a generalization of Boolean algebra is the notion of a modular lattice. Thousands of papers confirmed to mathematicians their worst suspicions about philosophers. Such a philosophy came to an end when someone conclusively proved that those observables, which are the quantum mechanical analogues of random variables, cannot be described by lattice-theoretic structure alone, unlike random variables.—This debacle had the salutary effect of opening up the field to some honest philosophy of quantum mechanics, at the same level of honesty as the philosophy of statistics (of which we would like to see more) or the philosophy of relativity (of which we would like to see less).” (Rota 1997b, p. 219).
- 52.
Rota/Baclawski give the following advice to their students: “The purpose of this course is to learn to think probabilistically. Unfortunately the only way to learn to think probabilistically is to learn the theorems of probability. Only later, as one has mastered the theorems, does the probabilistic point of view begin to emerge while the specific theorems fade in one’s memory: much as the grin of the Cheshire cat” (Rota/Baclawski 1979, p. vii).
- 53.
- 54.
Another paper on this particular subject (La reforme husserlienne de la logique selon Rota), previously given at a conference on Rota, at the Istituto Veneto, in 2014, is to appear simultaneously in the next volume of the Revue de Synthèse (eds. C. Alunni & E. Brian), Springer.
- 55.
Cf. Franz Brentanos Reform der Logik, Wilhelm Enoch, in: Philosophische Monatshefte, 29, 1893, pp. 433–458.
- 56.
Albino Lanciani, Analyse phénoménologique du concept de probabilité, Hermann, 2012.
- 57.
- 58.
“The examples that Husserl and other phenomenologists developed of this genetic reconstruction, admirable as they are, came before the standard of rigor later set by mathematical logic, and are therefore insufficient to meet the foundational needs of present-day science. In contemporary logic, to be is to be formal. It falls to us to develop the technical apparatus of genetic phenomenology (…) on the same or greater a standard of rigor than mathematical logic.” (Rota et al. 1992, p. 171)
- 59.
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Acknowledgments
I am grateful to them more than I can say to all the friends and colleagues, who take the time to read and comment this paper. This paper has benefited from the careful readings and/or the criticisms of Luigi Accardi, Françoise Balibar, Franck Jedrzejewski, Pierre Kerszberg, Didier Vaudène, Mark van Atten and Maria Villela-Petit da Penha. Last but not least, I am very thankful to Marian Hobson who helped greatly to convert my prose into proper English. Thank you to Pierre Giai-Levra who gave a final, generous and acute look at the last version of this paper. Remaining mistakes and shortcomings are attributable to me. I am also very grateful to Hassan Tahiri who organized this unforgettable conference in Lisbon at the Centro de Filosofia das Ciências, for his patience in waiting for the final version of this paper.
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Lobo, C. (2018). Some Reasons to Reopen the Question of the Foundations of Probability Theory Following Gian-Carlo Rota. In: Tahiri, H. (eds) The Philosophers and Mathematics. Logic, Epistemology, and the Unity of Science, vol 43. Springer, Cham. https://doi.org/10.1007/978-3-319-93733-5_8
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