Seufz

[This is a proof-read copy of http://www.jfsowa.com/peirce/putnam.htm]

Peirce the Logician

Following is an excerpt of the last 5 pages of an article that was originally published in Historia Mathematica, vol. 9, 1982, pp. 290-301. It was reprinted in H. Putnam, Realism with a Human Face, Harvard University Press, 1990, pp. 252-260.

When I started to trace the later development of logic, the first thing I did was to look at Schröder's Vorlesungen über die Algebra der Logik. This book, which appeared in three volumes, has a third volume on the logic of relations (Algebra und Logik der Relative, 1895). The three volumes were the best-known logic text in the world among advanced students, and they can safely be taken to represent what any mathematician interested in the study of logic would have had to know, or at least become acquainted with in the 1890s.

As the title suggests, the approach was algebraic (Boole's logic, as we saw, grew out of abstract algebra), and the great problem was to develop a logic of Relative (that is, relations). (The influence of the German word Relativ is, perhaps, the reason Peirce always wrote "relatives" and not "relations.") Peirce, although himself a member of the algebraic school (he criticized himself for this in his correspondence), had reservations about Schröder's close assimilation of logical problems to algebraic ones. "While I am not at all disposed to deny that the so called 'solution problems', consisting in the ascertainment of the general forms of relatives which satisfy given conditions, are often of considerable importance, I cannot admit that the interest of logical study centers in them," Peirce wrote. And "Since Professor Schröder carries his algebraicity so very far, and talks of 'roots', 'values', 'solutions', etc., when, even in my opinion, with my bias towards algebra, such phrases are out of place..." But my purpose in consulting this reference work was narrower; I simply wished to see how Schröder presented the quantifier.

Well, Schröder does mention Frege's discovery, though just barely; but he does not explain Frege's notation at all. The notation he both explains and adopts (with credit to Peirce and his students, O. H. Mitchell and Christine Ladd-Franklin) is Peirce's. And this is no accident: Frege's notation (like one of Peirce's schemes, the system of existential graphs) repelled everyone (although Whitehead and Russell were to study it with consequential results). Peirce's notation, in contrast, was a typographical variant of the notation we use today. Like modern notation, it lends itself to writing formulas on a line (Frege's notation is two-dimensional) and to a simple analysis of normal-form formulas into a prefix (which Peirce calls the Quantifier) and a matrix (which Peirce calls the "Boolean part" of the formula).

Moreover, as Warren Goldfarb has emphasized in a fine paper on the history of the quantifier, the Boolean school, including Peirce, was willing to apply logical formulas to different "universes of discourse," and Peirce was willing (unlike Frege) to treat first-order logic by itself, and not just as part of an ideal language (with a fixed universe of discourse, namely, "all objects," for Frege). In fact — and this may be surprising to others as it was to me — the term "first-order logic" is due to Peirce! (It has nothing to do with either Russell's theory of types or Russell's theory of orders, although the way Peirce distinguished between first-order and second-order formulas — by whether the "relative" is quantified over or not — obviously has something to do with logical type.) In summary, Frege tried to "sell" a grand logical-metaphysical scheme with a dubious ontology, while Peirce (and, following him, Schröder) was busy "selling" a modest, flexible, and extremely useful notation.

The success they experienced was impressive. While, to my knowledge, no one except Frege ever published a single paper in Frege's notation, many famous logicians adopted Peirce-Schröder notation, and famous results and systems were published in it. Löwenheim stated and proved the Löwenheim theorem (later reproved and strengthened by Skolem, whose name became attached to it together with Löwenheim's) in Peircian notation. In fact, there is no reference in Löwenheim's paper to any logic other than Peirce's. To cite another example, Zermelo presented his axioms for set theory in Peirce-Schröder notation, and not, as one might have expected, in Russell-Whitehead notation.

One can sum up these simple facts (which anyone can quickly verify) as follows: Frege certainly discovered the quantifier first (four years before O. H. Mitchell, going by publication dates, which are all we have as far as I know). But Leif Erikson probably discovered America "first" (forgive me for not counting the native Americans, who of course really discovered it "first"). If the effective discoverer, from a European point of view, is Christopher Columbus, that is because he discovered it so that it stayed discovered (by Europeans, that is), so that the discovery became known (by Europeans). Frege did "discover" the quantifier in the sense of having the rightful claim to priority; but Peirce and his students discovered it in the effective sense. The fact is that until Russell appreciated what he had done, Frege was relatively obscure, and it was Peirce who seems to have been known to the entire world logical community. How many of the people who think that "Frege invented logic" are aware of these facts?

The example of Löwenheim shows something else: metamathematical work (of a certain kind) did not have to wait for Russell and Whitehead to make Frege's work known (and to extend it and repair it). First-order logic (and its metamathematical study) would have existed without Frege. (Zermelo even denied that his set-theoretic work depended on Whitehead and Russell; he claimed to have been aware of the "Russell paradox" on his own.)

The Peircian Influence on Whitehead and Russell

Still, I thought, Russell and Whitehead themselves certainly learned their logic from Frege. To check this I turned to Russell's autobiographical writings. The result was frustrating. In My Philosophical Development, Russell describes the impact that meeting Peano had upon his logical development. Strangely enough, he does not mention the quantifier, which seems so very central from our present point of view, at all. Peano taught Russell what was a commonplace in the Peirce-Schröder logical community, the difference in logical form between all men are mortal and Socrates is a man. And it is clear that one of the notations used in Principia for a universally quantified conditional — writing the variable of quantification under the sign of the conditional — came from Peano. But the quantifier as such is not something that Russell singled out for discussion (unless there is something in the unpublished Nachlass in the Russell Archives in Ontario). Even when Russell discusses his debt to Frege (in a peculiar way: Russell is unstinting in his praise of Frege's genius, but claims to have thought of the definition of number quite independently), he does not mention the quantifier. Principia is no more help on this score, although there is an indication in it that most of the specific notations were invented by Whitehead rather than Russell.

[Footnote: Subsequent to writing this essay I discovered that, in "Whitehead and Principia Mathematica," Mind (1948), p. 137, Russell says that Whitehead contributed the notion for the universal quantifier.]

Since I have mentioned Peano, I should remark that he was not only well acquainted with Peirce-Schröder logic, but he had actually corresponded with Peirce.

In desperation, I looked at Whitehead's Universal Algebra. This is a work squarely in the tradition to which Boole, Schröder, and Peirce belonged, the tradition that treated general algebra and logic as virtually one subject. And here, before Whitehead worked with Russell, there is no mention of Frege, but there is a citation of "suggestive papers" by Peirce's students O. H. Mitchell and Christine Ladd-Franklin. The topic, of course, is the quantifier.

In sum, Whitehead certainly came to his knowledge of quantification through "Peirce and his students." On the other hand, the axioms in Principia are almost certainly derived from Frege's Begriffsschrift; Peirce gave no system of axioms for first-order logic, although his existential graphs" are a complete proof procedure for first-order logic (an early form of natural deduction).

I have, if anything, minimized Frege's contribution and played up the Boolean contribution for reasons which I have explained. But to leave matters here would be as unjust to Frege and to a third tradition, the Hilbert tradition (proof theory), as Quine's unfortunate remark was to the Boolean tradition.

Frege's work is sometimes disparaged today (I mean Frege's logical achievement; Frege's stock as a philosopher has never been higher), though not, of course, by Quine. It is conceded that Frege was far more rigorous and, in particular, far more consistently free of use-mention confusions than other logicians; but such domestic virtue is no longer felt to be impressive. The central charge laid against his work (and that of Whitehead and Russell) is that what they called logic is not logic but "set theory," and that reducing arithmetic to set theory is a bad idea.

This raises philosophical issues far too broad for this essay. But let me just make two comments on this: (1) Where to draw the line between logic and set theory (or predicate theory) is not an easy question. The statement that a syllogism is valid, for example, is a statement of second-order logic. (Barbara is valid just in case

(F)(G)(H)((Fx ⊃ Gx) ∧ (Gx ⊃ Hx) ⊃ (Fx ⊃ Hx)), 

for example.) [Ed.The parentheses ( ) are supposed to remind one of the O in omnis. This was the notation Peano introduced for the universal quantifier. See J. F. Sowa on Peirce's MS 514] If second-order logic is "set theory," then most of traditional logic thus becomes set theory." (2) The full intuitive principle of mathematical induction is definitely second-order in anybody's view. Thus there is a higher-order element in arithmetic whether or not one chooses to "identify numbers with sets" (just as Frege realized).

But, philosophical questions aside, Frege certainly undertook one of the most ambitious logical investigations in all history. Its enormous sweep made it (after its repair by Whitehead and Russell, and its translation into a notation resembling Peirce's) a great stimulus to all future work in the field. The Hilbert school certainly put it in the center of their proof theoretic investigations: Gödel's most famous paper, after all, bears the title "On Principia Mathematica and Related Systems," That all its achievements could be imitated successfully by the Cantorians (Zermelo and von Neumann) does not take away either its priority or its influence. If Peirce and Schröder were the cutting edge of the logical world prior to Russell and Whitehead's Principia Mathematica (or a cutting edge — the Hilbert school was already under way), after the appearance of Principia their work lost its importance — or lost it except for one important thing: its influence on Hilbert, who followed Peirce in separating off first-order logic from the higher system for metamathematical study.

Principia in turn was to lose its cutting-edge position when interest shifted from the construction of systems (and the derivation of mathematics within them) to the metamathematical study of properties of systems. Nothing remains forever the cutting edge in a healthy science. But a fair-minded statement of the historical importance of the different schools of work, a statement that does justice to each without slighting the others, should not be impossible. Such a statement was given by Hilbert and Ackermann:

The first clear idea of a mathematical logic was formulated by Leibniz. The first results were obtained by A. de Morgan (1806-1876) and G. Boole (1815-1864). The entire later development goes back to Boole. Among his successors, W. S. Jevons (1835-1882) and especially C. S. Peirce (1839-1914) enriched the young science. Ernst Schröder systematically organized and supplemented the various results of his predecessors in his Vorlesungen über die Algebra der Logik (1890-1895), which represents a certain completion of the series of developments proceeding from Boole.

In part independently of the development of the Boole-Schröder algebra, symbolic logic received a new impetus from the need of mathematics for an exact foundation and strict axiomatic treatment. G. Frege published his Begriffsschrift in 1879 and his Grundgesetze der Arithmetik in 1893-1903. G. Peano and his co-workers began in 1894 the publication of the Formulaire des Mathematiques, in which all the mathematical disciplines were to be presented in terms of the logical calculus. A high point of this development is the appearance of the Principia Mathematica (1910-1913) by A. N. Whitehead and B. Russell. Most recently Hilbert, in a series of papers and university lectures, has used the logical calculus to find a new way of building up mathematics which makes it possible to recognize the consistency of the postulates adopted. The first comprehensive account of these researches has appeared in the Grundlagen der Mathematik (1934 1939), by D. Hilbert and P. Bernays.

If Quine had produced a statement like this in his book, I should not have had a topic for this essay!

La nature de la nature

[Auszug aus La Méthode, Tome 1, Éditions du Seuil, 1977, S. 11]

Nous savons depuis d'un demi-siècle que ni l'observation microphysique, ni l'observation cosmo-physique ne peuvent être détachées de leur observateur. Les plus grands progrès des sciences contemporaines se sont effectués en réintégrant l'observateur dans l'observation. Ce que est logiquement nécessaire: tout concept renvoie non seulement à l'objet conçu, mais au sujet concepteur. Nous retrouvons l'évidence qu'avait dégagée il y a deux siècles le philosophe-évêque: il n'existe pas de "corps non pensés"  ["L'esprit, ne prenant pas garde à lui-même, s'illusionne et pense qu'il peut concevoir et qu'il conçoit effectivement des corps existants non pensès ou hors de l'esprit, quoiqu'en même temps ils soient saisis et existent en lui", Berkeley, A treatise concerning the principles of human knowledge].  Or l'observateur qui observe, l'esprit qui pense et conçoit, sont eux-mêmes indissociables d'une culture, donc d'une société hic et nunc. Toute connaissance, même la plus physique, subit une détermination sociologique. Il y a dans toute science, même la plus physique, une dimension anthropo-sociale. Du coup, la réalité anthropo-sociale se projette et s'inscrit au coeur même de la science physique.

Tout cela est évident. Mais c'est une évidence qui demeure isolée, entourée d'un cordon sanitaire. Nulle science n'a voulu connaître la catégorie la plus objective de la connaissance: celle du sujet connaissant. Nulle science naturelle n'a voulu connaître son origine culturell. Nulle science physique n'a voulu connaître sa nature humaine. La grande coupure entre les sciences de la nature es les sciences de l'homme occulte à la fois la réalité physique des secondes, la réalité sociale des premières. Nous nous heurtons à la toute-puissance d'un principe de disjonction: il condamne les sciences humaines à l'inconsistance extra-physique, et il condamne les sciences naturelles à l'inconscience de leur réalité sociale. Comme le dit très justement von Foerster, "l'existence de sciences dites sociales indique le refus de permettre aux autres sciences d'être sociales" (j'ajoute: et de permettre aux sciences sociales d'être physiques)... (von Foerster, 1974)

Or toute réalité anthropo-sociale relève, d'une certaine façon (laquelle?), de la science physique, mais toute science physique relève, d'une certaine façon (laquelle?), de la réalité anthropo-sociale.

Dès lors, nous découvrons que l'implication mutuelle entre ces termes se boucle en une relation circulaire qu'il faut élucider:

physique -------> biologie  ----------> anthropo-sociologie
      |                                          |
      ---<-----------------------<---------------< 

Mais, du même coup, nous voyons que l'élucidation d'une telle relation se heurte à une triple impossibilité:

1. Le circuit

physique-biologie-anthropo-sociologie
       |                      |
        ---<---------<--------<

envahit tout le champ de la connaissance et exige un impossible savoir encyclopédique.

2. La constitution d'une relation, là où il y avait disjonction, pose un problème doublement insondable: celui de l'origine et de la nature du principe qui nous enjoint d'isoler et de séparer pour connaître, celui de la possibilité d'un autre principe capable de relier l'isolé et le séparé.

3. Le charactère de la relation

physique ---> anthropo-sociologie
     |                    |
    ---<-------<----------<

prend figure de cercle vicieux, c'est-à-dire d'absurdité logique, puisque la connaissance physique dépend de la connaissance anthropo-sociologique, laquelle dépend de la connaissance physique, et ainsi de suite, à l'infini. Nous avons là non pas une rampe de lancement, mais un cycle infernal.

Nous nous heurtons donc, après ce premier tour de piste, à un triple mur: le mur encyclopédique, le mur épistémologique, le mur logique. En ces termes, la mission que j'ai cru devoir m'assigner est impossible. Il faut y renoncer.

L'école du Deuil

C'est precisement ce renoncement que nous enseigne l'Université. L'école de la Recherche est une école du Deuil.

Tout néophyte entrant dans la Recherche se voit imposer le renoncement majeur à la connaissance. On le convainc que l'époque des Pic de la Mirandole est révolue depuis trois siècles, qu'il est désormais impossible de se constituer une vision et de l'homme et du monde.

On lui démontre que l'accroissement informationnel et l'hétérogénéisation du savoir dépassent toute possibilité d'engrammation et de traitement par le cerveau. On lui assure qu'il faut non le déplorer, mais s'en féliciter. Il devra donc consacrer toute son intelligence à accroître ce savoir-là. On l'intègre dans une équipe spécialisée, et dans cette expression c'est "spécialisé" et non "équipe" qui est le terme fort.

Désormais spécialiste, le chercheur se voit offrir la possession exclusive d'un fragment du puzzle dont la vision globale doit échapper à tous et à chacun. Le voila devenu un vrai chercher scientifique, qui oeuvre en fonction de cette idée motrice: le savoir est produit non pour être articulé et pensé, mais pour être capitalisé et utilisé de façon anonyme.

Les questions fondamentales sont renvoyées come question générales, c'est-à-dire vagues, abstraites, non opérationelles. La question originelle que la science arracha à la religion et à la philosophie pour l'endosser, la question qui justifia son ambition de science: "Qu'est-ce que l'homme, qu'est-ce que le monde, que est-ce que l'homme dans le monde?", la science la renvoie aujourd'hui à la philosophie, toujours incompétente à ses yeux pour éthylisme spéculatif, elle la renvoie à la religion, toujours illusoire à ses yeux pour mythomanie invétérée. Elle abandonne toute question fondamentale aux non-savants, a priori disqualifiés. Elle tolère seulement qu'à l'âge de la retraite, ses grands dignitaires prennent quelque hauteur méditative, ce dont se gausseront, sous les cornues, les jeunes blouses blanches. Il n'est pas possible de faire communiquer ses connaissances avec sa vie. Telle est la grande leçon, qui descend du Collège de France aux collèges de France.

Le Deuil est-il nécessaire? L'Institution l'affirme, le proclame. C'est grâce à la méthode qui isole, sépare, disjoint, réduit à l'unité, mesure, que la science a découvert la cellule, la molécule, l'atome, la particule, les galaxies, les quasars, les pulsars, la gravitation, l'électro-magnétisme, le quantum d'énergie, qu'elle a appris à interpréter les pierres, les sédiments, les fossiles, les os, les écritures inconnues, y compris l'écriture inscrite sur ADN. Pourtant, les structures de ces savoir sont dissociées les unes des autres. Physique et biologie ne communiquent aujourd'hui que par quelques isthmes. La physique n'arrive même plus à communiquer avec elle-même: la science-reine est disloquée entre micro-physique, cosmo-physique et notre entre-deux encore apparemment soumis à la physique classique. Le continent anthropologique a dérivé, devenant une Australie. En son sein la triade constitutive du concept d'homme individu-espèce-societé est elle-même totalement disjointe, comme nous l'avons vu (Morin, 1973) et le reverrons. L'homme s'émiette: il en rest ici une main-à-outil, là une langue-qui-parle, ailleurs un sexe éclaboussant un peu de cerveau. L'idée d'homme est d'autant plus éliminable qu'elle est minable: l'homme des sciences humaines est un spectre supra-physique et supra-biologique. Comme l'homme, le monde est disloqué entre les sciences, émieté entre les disciplines, pulvérisé en informations.

Aujourd'hui, nous ne pouvons échapper à la question: la nécessaire décomposition analytique doit-elle se payer par la décomposition des êtres et des choses dans une atomisation généralisée? Le nécessaire isolement de l'objet doit-il se payer par la disjonction et l'incommunicabilité entre ce qui est séparé? La spécialisation fonctionelle doit-elle se payer par une parcellarisation absurde. Est-il nécessaire que la connaissance se disloque en mille savoirs ignares?

Or, que signifie cette question, sinon que la science doit perdre son respect pour la science et que la science doit interroger la science? Encore un problème que, apparemment, ajoute à l'énormité des problèmes que nous contraint à renoncer. Mais c'est précisément ce problème que nous empêche de renoncer à notre problème.

Comment, en effet, céder à l'ukase d'une science où nous venons de découvrir une gigantesque tache aveugle? Ne faut-il pas penser plutôt que cette science souffre d'insuffisance et de mutilation?

Mais alors, qu'est-ce que la science? Ici, nous devons nous rendre compte que cette question n'a pas de réponse scientifique: la science ne se connaît pas scientifiquement et n'a aucun moyen de se connaître scientifiquement. Il y a une méthode scientifique pour considérer et contrôler les objets de la science. Mais il n'y a pas de méthode scientifique pout considérer la science comme objet de science et encore moins le scientifique comme sujet de cet objet. Il y a des tribunaux épistémologiques que, a posteriori et de l'extérieur, prétendent juger et jauger les théories scientifiques; il y a des tribunaux philosophiques où la science est condamné par défaut. Il n'y a pas de science de la science. On peut même dire que toute la méthodologie scientifique, entièrement vouée à l'expulsion du sujet et de la réflexivité, entretient cette occultation sur elle-même. "Science sans conscience n'est que ruine de l'âme", disait Rabelais. La conscience qui manque ici n'est pas la conscience morale, c'est la conscience tout court, c'est-à-dire l'aptitude à se concevoir soi-même. D'où ces incroyables carences: comment se fait-il que la science demeure incapable de se concevoir comme praxis sociale? Comment est-elle incapable, non seulement de contrôler, mais de concevoir son pouvoir de manipulation et sa manipulation par les pouvoirs? Comment de fait-il que les scientifiques soient incapables de concevoir le lien entre la recherche "désinteressée" et la recherches de l'intérêt? Pourquoi sont-ils aussi totalement incapables d'examiner en termes scientifiques la relation entre savoir et pouvoir?