Abstract
The view of infinity as a metaphor, a basic premise of modern cognitive theory of embodied knowledge, suggests in particular that there may be alternative ways in which one could formalize mathematical ideas about infinity. We discuss the key ideas about infinitesimals via a proceptual analysis of the meaning of the ellipsis “...” in the real formula \(\hbox{.999\ldots = 1}\). Infinitesimal-enriched number systems accommodate quantities in the half-open interval [0,1) whose extended decimal expansion starts with an unlimited number of repeated digits 9. Do such quantities pose a challenge to the unital evaluation of the symbol “.999...”? We present some non-standard thoughts on the ambiguity of the ellipsis in the context of the cognitive concept of generic limit of B. Cornu and D. Tall. We analyze the vigorous debates among mathematicians concerning the idea of infinitesimals.
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Notes
See footnote 6 for a discussion of such an approach in the context of an infinitesimal-enriched continuum.
Such as the conception of an infinitesimal as represented by “.000...1” (with infinitely many zeros preceding the digit 1) being the difference between 1 and “.999...9” (with infinitely many digits 9); cf. Lightstone’s notation in footnote 8.
See also Sad, Teixeira and Baldino (2001, p. 286).
The term procept was coined by Gray and Tall (1991).
Each real number is accompanied by a cluster (alternative terms are prevalent in the literature, such as halo, but the term cluster has the advantage of being self-explanatory) of hyperreals infinitely close to it. The standard part function collapses the entire cluster back to the standard real contained in it. The cluster of the real number 0 consists precisely of all the infinitesimals. Every infinite hyperreal decomposes as a triple sum H + r + ε, where H is a hyperinteger (see below), r is a real number in [0,1), and ε is infinitesimal. Varying ε over all infinitesimals, one obtains the cluster of H + r. A hyperreal number H equal to its own integer part: H = [H] is called a hyperinteger. Here, the integer part function is the natural extension of the real one. Limited (finite) hyperintegers are precisely the standard ones, whereas the unlimited (infinite) hyperintegers are sometimes called non-standard integers. The limit L of a convergent sequence \(\langle u_n \rangle\) is the standard part st of the value of the sequence at an infinite hypernatural: L = st(u H ), for instance at \(H=[{{\mathbb N}}]\). In connection with .999..., I. Stewart notes that
[t]he standard analysis answer is to take ‘...’ as indicating passage to a limit. But in non-standard analysis there are many different interpretations. (Stewart, 2009, p. 176)
Some additional details are to be found in Katz and Katz (2010). Historically, there have been two main approaches to infinitesimals. The approach of Leibniz postulates the existence of infinitesimals of arbitrary order, while B. Nieuwentijdt favored nilpotent (nilsquare) infinitesimals (see Bell, 2009). Bell notes that a Leibniz infinitesimal is implemented in the hyperreal continuum of Robinson, whereas a Nieuwentijdt infinitesimal is implemented in the smooth infinitesimal analysis of F.W. Lawvere, based on intuitionistic logic.
The symbol is used in a different sense in projective geometry, where adding a point at infinity { ∞ } to \({{\mathbb R}}\) results a circle: \({{\mathbb R}}\cup \{\infty\} \approx S^1\).
Note that Δx = .9.. − 1 = − .0..;..01 in Lightstone’s notation (see Lightstone, 1972), where the digit “1” appears at infinite decimal rank H.
See Goldwurm (2001, p. 104) for biographical information on HaYisraeli.
The falsification problem is analyzed in Ehrlich (2006).
Historians of mathematics have noted the vitriolic tenor of Bishop’s criticism, see e.g., Dauben (1996, p. 139).
Namely, review of an earlier edition of Keisler (1986).
The description of Hilbert’s program as “formal finesse” has been objected to by many authors. Avigad and Reck (2001) provide a detailed discussion of the significance and meaning of Hilbert’s program.
Note that Schubring (2005) attributes the first systematic use of infinitesimals as a foundational concept to Johann Bernoulli.
See footnote 14 for a historical clarification.
An alternative infinitesimal-enriched intuitionistic continuum has been developed by Lawvere, see footnote 6.
The issue of the falsification of the history of the calculus was discussed by Lakatos, see main text at footnote 10.
In a letter to M. Vygodskiĭ, the mathematician Luzin questioned whether the Weierstrassian approach to the foundations of analysis “corresponds to what is in the depths of our consciousness . . . I cannot but see a stark contradiction between the intuitively clear fundamental formulas of the Integral calculus and the incomparably artificial and complex work of the ‘justification’ and their ‘proofs’ ” (Luzin, 1931). The publication of the text Fundamentals of Infinitesimal Calculus, by Vygodskiĭ, in 1931, provoked sharp criticisms. Luzin wrote his (two) letters to counterbalance such criticisms and took the opportunity to elaborate his own views concerning infinitesimals.
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Acknowledgements
We are grateful to David Ebin and David Tall for a careful reading of an earlier version of the manuscript and for making numerous helpful suggestions. We thank the editor of the article and the anonymous referees for an exceptionally thorough analysis of the shortcomings of the version originally submitted, resulting, through numerous intermediate versions, in a more focused text.
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Katz, K.U., Katz, M.G. Zooming in on infinitesimal 1–.9.. in a post-triumvirate era. Educ Stud Math 74, 259–273 (2010). https://doi.org/10.1007/s10649-010-9239-4
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DOI: https://doi.org/10.1007/s10649-010-9239-4