Neither ancient nor modern: Wallis and Barrow on the composition of continua. Part two: The seventeenth-century context: The struggle between ancient and modern

Abstract

In the the first part of this article* it was shown that neither John Wallis nor Isaac Barrow fits precisely into the modern or traditional categories commonly used by historians. Yet other kinds of tension between tradition and innovation did exist in the wider intellectual community of early modern England that might have some bearing on the state of mathematics. We therefore need to explore how the opposition between ancient and modern was expressed in the seventeenth century. First, there was the religiously based belief in the decay of nature; supporters of this belief regarded traditional methods to be superior. The defenders of the moderns, on the other hand, did not consider the Bible to rule out contemporary improvements on ancient techniques. But the very idea that knowledge could be advanced, particularly beyond the classical works, was still new and strange in this period.1 Second, the dispute surrounding educational reform included the rejection of traditional educational methods. Social and economic pressures led reformers to propose an increased concern with utility and practical methods, which might increase employment. Once the context of the tension between the supporters of the ancients and the supporters of the moderns has been explored, we can ascertain whether Wallis and Barrow stood on different sides in the conflict, and how this may have have influenced their mathematics.

“In Nature as in Geometry”: Du Châtelet and the Post-Newtonian Debate on the Physical Significance of Mathematical Objects

“…22 See e.g. Leibniz (1863, IV:451-2; V:178-9) and the discussions in Hill (1996) andDe Risi (2021, 15-17). 23 The Port-Royaliens define abstraction in terms of analysis into parts or aspects (Arnauld and Nicole 1996, 37).…”

Ιστορική Εξέλιξη Ερμηνείες Και Διδακτικές Προσεγγίσεις Της Έννοιας Του Απειροστού

“…ΙΙ, σελ.28., εκδόσεις του Διεθνούς Δημοκριτείου Ιδρύματος, Ξάνθη 1982 29. Heath, T.L., ο.π.σελ., 38.BΗ απόδειξη του δευτέρου μέρους της πρότασης έχει ως εξής30 : Εάν υπήρχε μια ευθύγραμμη γωνία μικρότερη από τη γωνία που σχηματίζεται από την περιφέρεια CHA και την εφαπτομένη AE, τότε η περιοχή μεταξύ της περιφέρειας και της ευθείας θα παρεμβάλλονταν μια ευθεία γραμμή ΑF και θα σχηματιζόταν μια γωνία αποτελούμενη από ευθείες πλευρές, η οποία θα ήταν μεγαλύτερη από τη γωνία που σχηματίζεται από την ευθεία γραμμή ΒΑ και την περιφέρεια και ταυτόχρονα μικρότερη από τη γωνία που σχηματίζεται από την περιφέρεια και την ευθεία γραμμή ΑΕ. Αλλά έχει αποδείξει ήδη ότι τέτοια ευθεία γραμμή δεν μπορεί να παρεμβληθεί 31 .…”