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A Nilpotent Infinitesimal Extension of 91 C.A. Knudsen

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A Nilpotent Infinitesimal Extension of 91 C.A. Knudsen Logique & Analyse 157 (1997), 97-100 A NILPOTENT INFINITESIMAL EXTENSION OF 91 C.A. KNUDSEN Altogether with the eclosion of the epistemological crisis of the incommen- surability, the concept of infinite and infinitesimal made their appearance in the mathematical thought only to be subtly criticized by Zeno and to be eliminated, at least from geometry, by Eudoxus. They reappear in the 16th, 17th and 18th centuries and were used, among others, by Kepler, Cavalieri, Fermat, Newton, Leibniz and Euler. They got special attention in the first textbook on calculus by l'Hospital and Euler's "Introduction in Analisis Infinitorum". Made object of pastiche by Voltaire and derided by Berkeley they were filially substituted by the concept of limit in the Cauchy-Weier- strass efforts of "rigorization", remaining however as heuristic devices in the applications of the calculus to physical problems. In the 1960's Abraham Robinson, inspired in the so-called non-standard models of Arithmetic of T. Skolem, showed that Leibniz's ideas that arith- metic infinitesimals —infinitely small but non-zero (real) numbers— and infinite quantities were "useful fictions" could be fully vindicated, and that they lead to a novel and fruitful approach to Classical Analysis and others branches of mathematics as Probability Theory. The key to the method is, as Robinson calls attention, provided by the detailed analysis of the relation between mathematical languages and mathematical structures which is at the heart of contemporary model theory and was, by the way, Skolem ini- tial motivation —How the language we use to talk about a mathematical object characterizes the very object we are talking about? In the 1970's through the category-theoretic ideas of Lawvere and others it has become possible to realize the concept of nilpotent or geometric in- finitesimals which also can be traced back to Archimedes' Method. These infinitesimals arise as two distinct but related concepts. One from the device of regarding a circle as a polygon with an indefinitely large number of straight sides —linear infinitesimals. The other from the idea that a sur- face or volume may be regarded as the sum of indefinitely many numbers of lines or planes —indivisibles of the surface or volume. These ideas may be put down as two principles: (I) F o r any smooth curve and any point on it there is a non degenerate segment of the curve around the point which is straight. 98 C.A. KNUDSEN (II) T h e re is a "number" s 0 such that 6 It2 is= ea syO to. see that (I) actually implies (II). Both principles have been freely employed by mathematicians. Bell calls principle (I) the Principle of Local Straightness of Curves which is, of course, closely related to Leib- niz's well-known principle —Natura non facit saltus— and to what Bell calls Principle of Local Uniformity of Natural Processes which states that any of such processes may be considered as occurring at a constant rate over any sufficiently short period of time viz., over an infinitesimal instant there is no acceleration —instantaneous velocity sounds more familiar. Re- cently N.C.A. da Costa, A. Balan and J. Kouneiher have been trying to put invertible arithmetic infinitesimals and nilpotent infinitesimals in the same room through a beautiful construction of an ante-real ring. In this paper we try to model some of the ideas touched upon above in an elementary fashion. To begin with let us consider = : N –› l t : x(n) = x n with its usual structure of c1ommutative ring and the quotient 91 = where – is the "tail" equivalence relation: y E 91N ; – y 44 , 3 n '91 can be partia0lly ord:ered by n n o [x1 < [z]x 44• 3 n ( 0 n ) It is easy to see =th:a t thisn ordne r is well-defined and compatible with the quo tient algebraic structure of 91, which is an interesting algebraic creature. It y o x is a commutative partially ordered algebra extending 91. In it one can see ( ( nn )) < invertible and no. n invertible arithmetic infinitesimals. For instance [1] is z an invertible infinitesimal whose inverse is the infinite [ i ( n ) ] . ;x (2n[ x+ 1)l – s2nu c h t h a t m1 al; the N o+1 a n d -xc h( a2i nn ) Lrin —i .- i< L[nin l] - < [b , i o= f i i0shnowfs ithat,we can have "very small" arithmetic infinitesimals and "huge" n ' ] - niinfiinittesse. 91 is intimately related to the *FR-Robinson extensions. Since the ultrafilters contain the cofinites,< . .it is easy to see that there is a biunivocal sa i m a ln s e x a m p l e o f a n o n - i n v e r t i b l e i n f i n i t e s i A NILPOTENT INFINITESIMAL EXTENSION OF 9i 99 correspondence between the *Di-extensions and the maximal ideals of Consider now M 2(k) { ( z w :x,y,z,w and let L C M2(§k) xbe thYe set of matrices of the type ) Ox) where x E Di and a is an invertible arithmetic infinitesimal. Then L and L is closed with respect to thae usual operations of addition and multi- plication of M2(9l). The elements e = ( 0) a are, of course, nilpotent. For each x E Di we denote by 0 0 \ O the fiber over x, where i s the set of invertiblex infinitesimals of , and by an the set ) : a 9 Y=1 ( E ce If now we consider the functions f : R —} ) Di that behave locally like poly- nomials, (having Taylor elxJp ansions at least of order 2) we can extend them naturally to 8 ) : c r E f L L q If f ( x ) = I a k V=13. x a ka X ) k ((L X(1)= k=0 (a(1 0 )( X 0 We have then the following obvious k d e f i n e Theorem I (Principle of Infinitesimal Linearity) For any function g :9)? —› L, there exists a unique fiber Lb such that g((g8 ) ) = g ( ( 8 8 ) ) + ( 1 ) 4 1 ? ) ) ( a ° 8 ) ; ( b f i ? 7 ) E L b 100 C.A. KNUDSEN Let us say now that two points of L are infinitesimally close if they differ by an element of A function f : L —) L., is said to be continuous if it sends infinitesimally close points in infinitesimally close points. It follows from the above theorem that all functionsf: L are continuous. We can now define the derivative of a function f : L L . Given Ix let g : L be defined by g(( By the theorem there0 is a u nique b E 91 such that 8 ) This unique b is ca8lle)d the derivative o f f a t ( t 3 x ((h / 3 = i l 4 ) ( ) a n d a n )( i f((sx)3c d x w r i t t e n Itw is not difficult to show0 that up to elements of 9)1 we have the identities f ) tehat a p'pexar in) the basic1 rules of the differential calculus. h 3 x Departamento de Matemdtica a + = Universidade Federal do Para, Brasil v ) e ( g ) REFERENCES f 1 [1] Begll, J.L)., Infinitesimals,1 Synthese 75, 285-315,1988. [2] Da Costa, N.C.A.5, Balan, A., Kouneiher, J., Analyse Ante-Reelle, Boletim Soc. Par.N 0Ma tematica, 1996. [3] Go)ldblatt, R.I., To\ poi the Categorical Analysis of Logic, North-Hol- land, 1979. [4] Kn-udsen, C.A., ,Impasse Aritmo-Geométric° e a Evoluctio do Conceit° de Niimero, PublicaOes do IME—USP, 1985. [5] Kock, A., Synthetic Differential Geometry, London Mathematical So- ciefty Lecture Noctes, 1981. [6] Rescher, N., The Philosophy of Leibnitz, Prentice-Hall, 1967. [7] Ro(bins(on, A., N,on- Standard Analysis, North-Holland, 1966. h ? ( ) ( x ) ) ) ) + ( b 3 2 n 8 ).
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