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Toward a History of Mathematics Focused on Procedures

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Abstract

Abraham Robinson’s framework for modern infinitesimals was developed half a century ago. It enables a re-evaluation of the procedures of the pioneers of mathematical analysis. Their procedures have been often viewed through the lens of the success of the Weierstrassian foundations. We propose a view without passing through the lens, by means of proxies for such procedures in the modern theory of infinitesimals. The real accomplishments of calculus and analysis had been based primarily on the elaboration of novel techniques for solving problems rather than a quest for ultimate foundations. It may be hopeless to interpret historical foundations in terms of a punctiform continuum, but arguably it is possible to interpret historical techniques and procedures in terms of modern ones. Our proposed formalisations do not mean that Fermat, Gregory, Leibniz, Euler, and Cauchy were pre-Robinsonians, but rather indicate that Robinson’s framework is more helpful in understanding their procedures than a Weierstrassian framework.

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Notes

  1. Some historians are fond of recycling the claim that Robinson used model theory to develop his system with infinitesimals. What they tend to overlook is not merely the fact that an alternative construction of the hyperreals via an ultrapower requires nothing more than a serious undergraduate course in algebra (covering the existence of a maximal ideal), but more significantly the distinction between procedures and foundations, as discussed in this Sect. 2.1, which highlights the point that whether one uses Weierstrass’s foundations or Robinson’s is of little import, procedurally speaking.

  2. Sherry (1987) argued that Berkeley’s criticism of the calculus actually consisted of two separate components that should not be conflated, namely a logical and a metaphysical one:

    1. (a)

      logical criticism: how can dx be simultaneously zero and nonzero?

    2. (b)

      metaphysical criticism: what are these infinitesimal things anyway that we can’t possibly have any perceptual access to or empirical verification of?

  3. Note that the modern Zermelo–Fraenkel (ZFC) framework definitely works as a foundational system, but no-one can adequately say why, for instance, ZFC is consistent to begin with (moreover, in a precise sense discovered by Goedel, this cannot even be answered in the positive).

  4. In point of fact, Euler is not seeking to ‘rigorise’ the calculus here, contrary to what Gray implies. Moreover, there is little indication that Euler found it problematic. He merely goes on to develop the calculus, e.g., by expanding trigonometric functions into series. It was the task of later generations to reshape his theses in a different setting.

  5. At http://mathoverflow.net/questions/242379 the reader will find many other examples.

  6. The proof exploits the assumption that there exists a set S of all things, and that a mathematical thing is an object of our thought. Then if s is such a thing, then the thought, denoted \(s'\), that “s can be an object of my thought” is a mathematical object is a thing distinct from s. Denoting the passage from s to \(s'\) by \(\phi \), Dedekind gets a self-map \(\phi \) of S which is some kind of blend of the successor function and the brace-forming operation. From this Dedekind derives that S is infinite, QED.

  7. In modern notation this can be expressed as \((\forall x,y>0)(\exists n\in {\mathbb N})[nx > y]\).

  8. In modern editions of The Elements this appears as Definition V.4.

  9. This can be translated as follows: “I use the term incomparable magnitudes to refer to [magnitudes] of which one multiplied by any finite number whatsoever, will be unable to exceed the other, in the same way [adopted by] Euclid in the fifth definition of the fifth book [of the Elements].”

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Acknowledgments

We are grateful to Paul Garrett for subtle remarks that helped improve an earlier version of the text. M. Katz was partially funded by the Israel Science Foundation grant number 1517/12.

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Authors and Affiliations

  1. Institute of Mathematics, Pedagogical University of Cracow, Kraków, Poland

    Piotr Błaszczyk

  2. IPPI, Moscow, Russia

    Vladimir Kanovei

  3. MIIT, Moscow, Russia

    Vladimir Kanovei

  4. Department of Mathematics, Bar Ilan University, Ramat Gan, 52900, Israel

    Karin U. Katz & Mikhail G. Katz

  5. Sobolev Institute of Mathematics, Novosibirsk State University, Novosibirsk, Russia

    Semen S. Kutateladze

  6. Department of Philosophy, Northern Arizona University, Flagstaff, AZ, 86011, US

    David Sherry

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  1. Piotr Błaszczyk
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Błaszczyk, P., Kanovei, V., Katz, K.U. et al. Toward a History of Mathematics Focused on Procedures. Found Sci 22, 763–783 (2017). https://doi.org/10.1007/s10699-016-9498-3

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