Leibniz on the Continuity of Space

Abstract

The present essay describes Leibniz's foundational studies on continuity in geometry. In particular, the paper addresses the long-debated problem of grounding a theory of intersections in elementary geometry. In the early modern age, in fact, several mathematicians had claimed that Euclid's Elements needed to be complemented with additional axioms in order to ground the existence of the intersection points between straight lines and circles. Leibniz was sensible to similar foundational issues in the Euclidean tradition, and dedicated several studies to investigate a good definition of the continuity of space, in order to ground a general theory of intersections of curves and surfaces. While Leibniz's researches on continuity in relation with the Calculus have been extensively studied in the past, the present essay deals with the less-known Leibnizian notion of a continuous space, as it is to be found in several unpublished writings preserved in Hannover. The subject widens as to encompass the relation between mereology and analysis situs, Leibniz's studies on a geometrical characteristics, and Leibniz's theory of space at large. Acknowledgements. I thank the Max Planck Institute for Mathematics in the Sciences and his director Jürgen Jost for providing me with the possibility of writing a good part of this paper while I was based in Leipzig, and more in general for having fostered important occasions of discussion with other Leibniz scholars. I also thank Richard Arthur, Eberhard Knobloch, Julien Narboux and Erich Reck, who have discussed with me this paper in several occasions; and Andrea Costa, Siegmund Probst and Javier Echeverría for sharing with me some unpublished papers by Leibniz.

Leibniz�s well-founded fictions and their interpetations

“…no point), even though there are more divisions of bodily parts than can possibly be expressed by any finite number. (De Risi [34], 2019; emphasis in the original). This use of the term "syncategorematic infinity" refers to matter or space, closely related to indefinite divisibility.…”

“…If any opponent tries to contradict this proposition, it follows from our calculus that the error will be less than any possible assignable error since it is in our power to make this incomparably small magnitude small enough for this purpose inasmuch as we can always take a magnitude as small as we wish. ( Meanwhile I have shown that these expressions are of great use for the abbreviation of thought and thus for discovery as they cannot lead to error, since it is sufficient to substitute for the infinitely small, as small a thing as one may wish, so that the error may be less 34 Note that Ishiguro's sentence "If magnitudes are incomparable, they can be neither bigger nor smaller" [50, p. 88] involves an equivocation on the term incomparable: if incomparable is taken to mean the definition from the theory of partially ordered sets, then this is a tautology (roughly "if magnitudes cannot be compared, then they cannot be compared"); if incomparable is taken to mean "any positiveinteger multiple is still less than any positive real" then Ishiguro's statement is mathematically incorrect, for the hyperreals are a totally ordered field, hence every element can be compared with any other. than any given amount, hence it follows that there can be no error.…”

Euclid After Computer Proof-checking

“…Other sources make clear that the Greeks regarded lines (curved or straight) as the traces of a motion. 9 In particular, most modern mathematicians, steeped in set theory from their youth, think that a line is equal to the set of its points. This was definitely not the Greek conception.…”

“…However, the default in Euclid, though not necessarily in all Greek geometry, is that lines are straight. We follow Euclid.8 A referee pointed out[29], which gives an axiomatic framework for a part of geometry in which the collapsible compass is preserved 9. This view of lines goes back at least to Aristotle; see[28, p. 79], where Proclus says Aristotle regarded a line as "the flowing of a point.…”

“…One may wonder why it took two millenia for this axiom to be formulated. See[9] for a penetrating historical and philosophical discussion of that question. In Pasch's statement, the first "between" here used "innerhalb" and so was strict; the second did not use "innerhalb" so was not strict betweenness.…”