1The Constructed Objectivity of Mathematics and the Cognitive Subject1

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1The Constructed Objectivity of Mathematics and the Cognitive Subject1 Giuseppe Longo CNRS et D?pt. de Math?matiques et Informatique ?cole Normale Sup?rieure 45, Rue d'Ulm, 75005 Paris e-mail: ÇThe problems of Mathematics are not isolated problems in a vacuum; there pulses in them the life of ideas which realize themselves in concreto through out human endeavours in our historical existence, yet forming an indissoluble whole transcend any particular scienceÈ [Hermann Weyl, 1949]. Introduction This essay concerns the nature and the foundation of mathematical knowledge, broadly construed. The main idea is that mathematics is a human construction, but a very peculiar one, as it is grounded on forms of invariance and conceptual stability that single out the mathematical conceptualization from any other form of knowledge, and give unity to it. Yet, this very conceptualization is deeply rooted in our acts of experience, as Weyl says, beginning with our presence in the world, first in space and time as living beings, up to the most complex attempts we make by language to give an account of it. I will try to sketch the origin of some key steps in organizing perception and knowledge by mathematical tools, as mathematics is one of the many practical and conceptual instruments by which we categorize, organise and give a structure to the world.

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The Constructed Objectivity of Mathematics and theCognitive Subject1Giuseppe LongoCNRS et Dpt. de Mathmatiques et Informatiquecole Normale Suprieure45, Rue d'Ulm, 75005 Parishttp://www.dmi.ens.fr/users/longo e-mail: longo@dmi.ens.frÇThe problems of Mathematics are not isolated problems in avacuum; there pulses in them the life of ideas which realizethemselves in concreto through out human endeavours in ourhistorical existence, yet forming an indissoluble whole transcendany particular scienceÈ [Hermann Weyl, 1949].IntroductionThis essay concerns the nature and the foundation of mathematical knowledge, broadlyconstrued. The main idea is that mathematics is a human construction, but a very peculiarone, as it is grounded on forms of "invariance" and "conceptual stability" that single out themathematical conceptualization from any other form of knowledge, and give unity to it. Yet,this very conceptualization is deeply rooted in our "acts of experience", as Weyl says,beginning with our presence in the world, first in space and time as living beings, up to themost complex attempts we make by language to give an account of it.I will try to sketch the origin of some key steps in organizing perception and knowledgeby "mathematical tools", as mathematics is one of the many practical and conceptualinstruments by which we categorize, organise and "give a structure" to the world. It isconceived on the "interface" between us and the world, or, to put it in husserlianterminology, it is "designed" on that very "phenomenal veil" by which, simultaneously, theworld presents itself to us and we give sense to it, while constituting our own "self".1 To appear in "Epistemolgy of Physics and of Mathematics", M. Mugur-Schachter editor,Kluwer, 2001.1

The mathematical structures are literally "drawn" on that veil and, as no other form ofknowledge, stabilize it conceptually: geometric images and spaces, or the linguistic/algebraicstructures of mathematics, set conceptual "contours" to relevant parts of the enormousamount of information that arrives upon us. Yet, this drawing is not arbitrary, as it isgrounded on key regularities of the world or that we "see" in the world. That is, on theseregularities that we forcibly single out by "reading" them according to our own search orprojection of similar patterns, as living beings: symmetries, physical and biologicalsymmetries, or the connectivity and continuity of space and time, for example.Intersubjectivity and history add up to the early cognitive processes; they modify ourforms of "intuition", including mathematical intuition, which is far from being stable inhistory. Indeed, mathematical intuition is constructed in a complex historical process, whichbegins with our biological evolution: the analysis of "intuition", not as a "magic" orinspeakable form of knowledge, but as a relevant part of human cognition, is one of the aimsof this ongoing project. A project which can be named a "cognitive foundation ofmathematics", as opposed to, or more exactly, complementing the metamathematical analysisof foundations largely developed in this century. Indeed, the foundational analysis ofmathematics cannot be only a mathematical challenge, as proposed by Frege and Hilbert'smathematical logic: Hilbert's metamathematics uses mathematical methods and, by this, itbecame part of mathematics, the very discipline whose methods or whose whole it wassupposed to found. Mathematical logic gave us an essential analysis of (logic/syntactic)proof-principles, and more it is giving: yet, we also need to go further and evidentiate "whatis behind" these linguistic principles, their meaning as rooted in our practices of life.Persisting only on the proof-theoretic, thus mathematical, analysis of mathematics, wouldleave us in a cognitive deadlock, actually in a philosophical or even conceptual vicious circle:one cannot found mathematical methods and tools by mathematical methods and tools. Forexample, no mathematical methods and tools can prove their own "consistency", which isthe metamathematical way to assure meaning to a mathematical theory. Sufficientlyexpressive, finitary theories, such as Arithmetic, have no finitary consistency proof; SetTheories which can represent a given infinity, need a larger one to be proven consistent.This is not a limitation to mathematics: in order to check the correctness of certain conceptualtools it is rather to be expected that one should "step back" from them and use different tools.Gdel's second incompleteness theorem proves exactly this: by Gdel's representationlemma, one can describe or encode the metatheory of arithmetic within arithmetic itself,which is thus part of the latter, and, then, prove that consistency is unprovable by thatarithmetized metatheory. That is, the finitistic and mathematical metatheory of arithmetic istoo weak to do the least job it was invented for, the proof of the consistency of Arithmetic,since it is given by the same finitary tools as Arithmetic itself.And there begins the infinite regression of "relative consistency" results: if one wants tobe sure that a given formal theory has a meaning (it is consistent), then one has to construct a2

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