Algebraic Concepts Characterized / Distinguishing the General from the Generic / Pre-specificity / The Alphabetic Absolute / Thinking as an Algebraic Mechanist

Equation

by Vera Bühlmann

author’s manuscript.

The mathematical notion of the equation is first documented in the 16th century, when it seems to have been introduced as what we would today call a terminus technicus for organizing the practice of equalizing mathematical expressions. It seems to have been introduced to European Renaissance science and philosophy together with algebra: an equation is the mathematical form for rationalizing and reasoning identity. The term “algebra” comes from Arabic al-dschabr for the „the fitting together of broken parts“, and its first appearance is usually referenced to the title of the Persian scholar al-Chwarizmi’s book Al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-ʾl-muqābala (The Compendious Book on Calculation by Completion and Balancing). Two things are important to point out right ahead: the mathematical term of an equation references a mathematical form for stating identity, and it does so precisely by not assuming identity to be given as a whole. In this regard, it crucially differs from the identity notion in philosophy – it helps to reason and rationalize identity, but in the original sense of Greek mathema, literally “that which can be learnt”, and mathematics for “all that pertains to what can be learnt”. Through this emphasis on learning (rather than knowing), and hence on mathematics as an art, the equational notion of identity is always already in pact with the mathematical irrational (the infinitary). It remains indetermined with regard to the whether identity as a postulated principle is to be regarded as a logical device, or whether it there is to be assumed a substantial reality of this principle in nature that can empirically be studied in physics[1] (–> invariance). This indeterminedness is indeed the key aspect which Michel Serres attributes to algebra for the advent of modern science and its paradigm of experimentation at large: Experimentation, invention, consist in making the cypher under which nature hides appear, he maintains. “At the beginning of the seventeenth century, when what we came to call the applied sciences first made their appearance, a theory spreads that one can find in several authors, although none of them is its sole source, which seeks to account for a harmony that is not self-evident.” Rather than looking at conceptual identity as the provider for self-evident harmony to look for in nature, what begins to spread, since Galileo and certainly with Descartes, Leibniz, Pascale, Fontenelle, is “the idea that nature is written in mathematical language.” But Serres immediately points to the insufficiency of the term language here; he points out the constitutive role of algebra for the role of mathematics in experimentation, and specifies that “[l]n fact mathematics is not a language: rather, nature is coded. The inventions of the time do not boast of having wrested nature’s linguistic secret from it, but of having found the key to the cypher. Nature is hidden behind a cypher. Mathematics is a code, and since it is not arbitrary, it is rather a cypher.” Serres’ speaking of cypher here is to be taken in mathematical sense: cypher is a term for how, in mathematical notation, naught can be expressed. It literally meant zero, from the Arabic term ṣifr, for zero. A cipher constitutes a code that affords encryption and decryption such that once the operations have been performed, the “text” or “message” – nature, in Serres’ cited passage – that it envelops, has not been affected by these operations. Algebra, as the art of speculative completion and balancing, experimentally searches for the code without having it to start with: the equational notion of identity hence is capable of organizing the practice of equalizing mathematical expressions in experimental manner. “Now, since this idea [that the harmony to be seeked is not self-evident but depends upon experiment, VB] in fact constitutes the invention or the discovery,” Serres continues, “nature is hidden twice. First under the cypher. Then under a dexterity, a modesty, a subtlery, which prevents our reading the cypher even from an open book. Nature hides under a cypher. Experimentation, invention, consist in making it appear.”[2

This emphasis on an equational identity notion, whose determination correlates with its articulation in the characters of a cipher and by the rules of a code, bears two great promises: (1) it affords a thinking that is capable to leaving its object – that which it envelops in code and makes appear – unaffected, and thus gives new support to a scientific notion of objectitivity; (2) this thinking proceeds algorithmically, formally, and hence can be externalized into mechanism that can perform it decoupled from a human cogito, but at the same time this does not liberate thought from mastership and literacy, for “reading” this cypher behind which “nature hides” crucially depends upon dexterity, modesty, and subtlery. In other words: a reasoning that can be externalized into a mechanism, and hence render obvious a not self-evident harmony (an interplay of parts that function well, work together fittingly, etc), must be considered strictly decoupled from any notion of truth. Equational identity is genuinely abstract. The Introduction to Mathematics written by Alfred North Whitehead is an exemplary text elaborating on the notion and role of mathematical abstractness.[3] How this role can be played today by a mathematical notion of information remains a largely open issue to this day. [4]

This article indexes Alfred North Whitehead’s Treaties on Universal Algebra from 1898 as the moment in which algebraic abstractness begins to find a novel embodiment in “information”.[5] It will trace some of the “genetical” heritage of mathematical abstraction whose lineages conflue here.[6] Until the late 19th century, algebra was used almost synonymously with a theory of equations, and its symbolical notion was thought to encode quantity in its classical double-articulation as magnitude (metrical, answering to how much?, presupposing a notion of unit) and multitude (countable, answering to how many? presupposing a notion of number). When Alfred North Whitehead wrote his Treaties, this had changed: with Cantor’s countable infinities (among many other factors that had been contributed, to name only a few of the most important names, by William Rowan Hamilton, Richard Dedekind, George Boole, Hermann Günther Grassmann,), the classical distinction between multitude and magnitude gave way to a more abstract distinction between ordinality (answering to the howmaniest?) and cardinality (how many?).[7] The generalized quantity notion was now that of ‘sets’; the status of mathematics with regard to philosophy and the natural sciences, but also with regard to linguistic form (structuralism) and literary form (e.g. in Wittgenstein’s notion of ‘natural language’ as ‘mathematical prose’) was profoundly unsettled thereby. This is what it means to say that mathematics is no longer concerned with quantity, but with symbolical systematiciy. When Whitehead wrote his Treatise on Universal Algebra, algebra needed to be addressed by means of what he suggested to think of as „a comparative study,” because it had given rise to “various Systems of Symbolic Reasoning“.[8] And those Systems of Symbolic Reasoning, as Whitehead calls them, had been looked upon „with some suspicion“ by mathematicians and logicians alike – as Whitehead puts it: „Symbolic Logic has been disowned by many logicians on the plea that its interest is mathematical, and by many mathematicians on the plea that its interest is logical“.[9]

The practice of equalizing mathematical expressions, which gave rise to the notion of equation, meant that arithmetics could not only be done with numbers (arithmos) understood in an Aristotelian sense (as ontological science), but with what we could call ‘lettered/characterized numbers’ that entailed an intermediary symbolical-notational formality (codes or poly-tomistic alphabets, elements that are not non-divisible, atomic, but partitionable in many ways). The decisive aspect is not that letters of the alphabet were newly used in mathematics[10]; it is that alphabetic letters began to be used for the notation of numbers in a manner that changed the concept of number: numbers, now, could be articulated as an interplay between variable parts and constant parts. This was not the case in antiquity. Here, numbers were always determinate numbers of things, while the algebraic concept of number works upon what is a ‘given’ only in the form of a metrical measurement point. Eventually, this novel manner of thinking about numbers gave rise to sophisticated procedures of estimation like stochastic interpolation and extrapolation. Mathematics thereby came to be seen as an activity, an intellectual and practical art, and the resulting geometry was referred to not as ‘elementary’ (stochastiké, in the tradition of Euclid’s Elements), but as ‘analytical’, ‘specious’, and eventually as ‘population based’ (modern stochastics, probabilistics).[11] The notion of a mathematical object was called by the early algebraists la cosa, the unknown – or not exhaustively known – ‘thing’.[12]

This novel notion of the object triggered in philosophy (and in politics) the inception of concepts of sufficient reason on the one hand, and of absolutism, literally meaning “unrestricted; complete, perfect”; also “not relative to something else,”[13] on the other hand. Not relative to something else meant for mathematics that the role of proportion (A is to B as C is to D) as the classical paradigm for analysis (literally the dissolving, from Greek –lysis, for “a loosening, setting free, releasing, dissolution,” from lyein “to unfasten, loose, loosen, untie” what is analog, from Greek analogon, from ana “up to” and logos “account, ratio”) was generalized, and thereby also relativized; proportion was now addressed as ‘proportionality’, and reason was now relative to conditions of possibility and the inclinations of dispositions. The practices of equalizing mathematical expressions unfolds in this generalized role of proportion as proportionality, and the notion of ‘equation’, with the symbolic forms of organizing these practices can be understood as the technical term to express this relativization of the analogical structure of proportion. It introduced a novel art, the ars combinatoria, and the practice of algebraically equalizing mathematical expressions culminated with Newton’s and Leibniz’ infinitesimal calculus as a novel mathesis universalis (a universal method) which triggered a fierce dispute in the 18th century between philosophical Rationalism (baroqu’ish and ‘orthodox’ in spirit) and Empiricism (reformationist and ‘modernist’ in spirit). Immanuel Kant’s notion of the transcendental, together with his program of critique for philosophy, eventually relaxed the disputes (temporarily)[14]. Algebra, as the theory of equations, was now to provide insights not about the nature of elements immediately, but in rules that can be deduced from Natural Laws that reign in physics. Mechanics came to be seen as a particular case of a more general physics, including dynamics and soon thereafter also thermodynamics. It was now the formulation of these laws (no longer that of mechanical principles) that was to be stated in the form of equations, accessible critically through empirical experiment coupled with exact conceptual reasoning, and hence decoupled from an affirmation of any metaphysical (and theological) assumptions in particular.[15] [16]

[–> negentropy; –> invariance].

**************************************

[1] Cf. Jacques Monod, Chance and Necessity, An Essay on the Natural Philosophy of Modern Biology,

Vintage Books, New York 1972 [1970].

[2] Serres, Birth of Physics, ibid., p. 140.

[3] For an extensive discussion of mathematic’s abstractness cf. Alfred North Whitehead: An Introduction to Mathematics, Williams and Norgate, London 1917, especially the first chapter „The abstract Nature of Mathematics“.

[4] Michel Serres, “Les nouvelles technologies : révolution culturelle et cognitive”, Conférence de Michel Serres lors du 40è anniversaire de l’INRIA en 2007, available online: https://www.youtube.com/watch?v=ZCBB0QEmT5g; cf. also the manuscript “Information and Thinking” of his lecture at the Philosophy after Nature Conference 2014 in Utrecht, forthcoming in the conference’s proceedings edited by Rosi Braidotti and Rick Dolphjin.

[5] Alfred North Whitehead, Treatise on Universal Algebra with applications, Cambridge University Press, Cambridge 1910.

[6] A remarkable study on the Cogito in terms of a materialist genetic heritage is: Anne Crahay, Michel Serres. La Mutation du Cogito. Genèse du Transcendental Objectif, De Boeck, Brussels 1988.

[7] cf. Sören Stenlund, The Origin of Symbolic Mathematics and the End of the Science of Quantity, Upsala University Press 2014 (available online: http://www.divaportal.org/smash/get/diva2:709492/FULLTEXT01.pdf)

[8] ibid. vi.

[9] ibid.. Cf. also the discussion of how Universal Algebra proceeded and evolved until the 1960ies by George Grätzer: Universal Algebra, The University Series in Higher Mathematics, D. van Nostrand Company Inc., Princeton 1968; as well as for a discussion of the subject’s developments since the 1950ies until 2012: Fernando Zalamea, Synthetic Philosophy of Contemporary Mathematics, transl. by Zachary Luke Fraser, Urbanomics 2012; also cf. the appreciation and critique by Giuseppe Longo, “Synthetic Philosophy of Mathematics and Natural Sciences. Conceptual analyses from a Grothendieckian Perspective. Reflections on “Synthetic Philosophy of Contemporary Mathematics” by Fernando Zalameo, available online at Giuseppe Longo’s institutional page CNRS, Collège de France et Ecole Normale Supérieure, Paris: http://www.di.ens.fr/users/longo/files/PhilosophyAndCognition/Review-Zalamea-Grothendieck.pdf

[10] Indeed, the separation of a particular notational system for mathematics is a rather recent development compared to the history of mathematics; it is a bifurcation after many centuries of using one and the same script for linguistic as well as mathematical articulations. Cf. Gerd Schubring, “From Pebbles to Digital Signs: The Joint Origin of Signs for Numbers and for Scripture, Their Intercultural Standardization and Their Renewed Conjunction in the Digital Era” in Vera Bühlmann, Ludger Hovestadt, Symbolizing Existence, Metalithicum III, Birkhäuser, Vienna, 2016 (in press).

[11] Cf. Rosa Massa Esteve, “Symbolic language in early modern mathematics: The Algebra of Pierre Hérigone (1580–1643)“, Historia Mathematica, Elsevier, Volume 35, Issue 4, November 2008, 285–301.

[12] Jules Vuillemin gives a fabulous account of these complex interrelations in his seminal book La Philosophie de l’Algèbre, Tome I : Recherches sur quelques concepts et méthodes de l’Algèbre moderne (Paris: 1962).

[13] cf. www.etymonline.com; also the article by Wilhelm G. Jacobs, „Das Absolute“, in: Hans Jörg Sandkühler (Ed.), Enzyklopädie Philosophie, Felix Meiner Verlag, Hamburg 2010, p. 13-17.

[14] It seems not entirely implausible, at least, to think about the early 20th century foundational crisis as a continuation of just these disputes on higher levels of abstraction. For a largely unbiased account and a serious and open-minded suggestion of how to approach the dilemma, cf. Hermann Weyl, The Continuum: A Critical Examination of the Foundation of Analysis, (1918).

[15] Due to its brevity, this summary follows the lineage in the critical tradition what has turned out as the predominant one, linking 20st century analytical as well as continental philosophy; it thereby understates the ideas of Leibniz, Lambert, and others, who maintained that algebra, by its calculatory and symbolic methods, could actually be seen as opening up the closedness of classical logics in the Aristotelian tradition, thereby introducing an ars inveniendi, an art of invention into logics – an approach that was still pursued by figures as eminent for the 19th century development of algebra as Charles Sanders Peirce (abduction) or Richard Dedekind (abstraction as a creative act). Against these ideas, Kant famously stated: „Die Logik ist […] keine allgemeine Erfindungskunst und kein Organon der Wahrheit; – keine Algebra, mit deren Hülfe sich verborgene Wahrheiten entdecken ließen“ (Immanuel Kant, Logik. Ein Handbuch zu Vorlesungen (1800), Friedrich Nicolovius: Königsberg; Akademie-Ausgabe, Bd. 9, 1–150, A 17, here cited in: Volker Peckhaus, „Die Aktualität der Logik als Organon“, in: Günter Abel (Ed.), Kreativität : XX. Deutscher Kongress für Philosophie, 26.-30. September 2005 an der Technischen Universität Berlin : Kolloquienbeiträge, Hamburg: Felix Meiner Verlag, 2006. For an introduction of the conflicts the unsettled status of algebra triggered in the empirical sciences themselves, cf Elisabeth Stengers’ Cosmopolitics I, University of Minnesota Press 2010 [2003], especially book II The Invention of Mechanics, Power and Reason, p. 68ff, therein „The Lagrangian Event“ (112ff.) and „Abstract Measurement: Putting Things to Work“ (129ff.).

[16] Cf. the seminal study on the implications of this for axiomatics by Robert Blanché, L’Axiomatique, Presses Universitaires de France, Paris 1980 [1955]; there is an English edition of this book by Geoffrey Keene, Monographs in modern logic, The Free Press of Glencoe, New York 1962, but it excludes the crucial two chapters with which Blanché ends his study, discussing the implications for science and for philosophy. Cf. on the genealogy of philosophical notions of necessity and contingency, and the relatively recent upheavels with regard to this genealogy, als Jules Vuillemin’s studies Necessity or Contingency. The Master Argument, distributed for Center for the Study of Language and Information by The University of Chicago Press, Chicago 1996; and What Are Philosophical Systems, Cambridge University Press, Cambridge 2009.

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