Torricelli elaborated the theory of ballistics as part of Galileo's theory of motion. In 1647 he had an interesting exchange of letters with G. B. Renieri, from Genoa, who complained that some experiments he had made with guns contradicted Galileo's theory. The correspondence discloses some fundamental issues of the Seventeenth century Scientific Revolution, the main one being to what extent mathematics can be applied to physics. Torricelli's view on this issue is ambivalent. He defends Galileo's kinematics as the (...) correct description of reality yet says that mathematics does not describe reality. It is suggested that this ambivalence is an outcome of the climate of uncertainty which followed Galileo's trial. (shrink)

An analysis of the counter-intuitive properties of infinity as understood differently in mathematics, classical physics and quantum physics allows the consideration of various paradoxes under a new light (e.g. Zeno’s dichotomy, Torricelli’s trumpet, and the weirdness of quantum physics). It provides strong support for the reality of abstractness and mathematical Platonism, and a plausible reason why there is something rather than nothing in the concrete universe. The conclusions are far reaching for science and philosophy.

The seventeenth century was an important period in the conceptual development of the notion of the infinite. In 1643, Evangelista Torricelli (1608-1647)—Galileo’s successor in the chair of mathematics in Florence—communicated his proof of a solid of infinite length but finite volume. Many of the leading metaphysicians of the time, notably Spinoza and Leibniz, came out in defense of actual infinity, rejecting the Aristotelian ban on it, which had been almost universally accepted for two millennia. Though it would be another (...) two centuries before the notion of the actually infinite was rehabilitated in mathematics by Dedekind and Cantor (Cauchy and Weierstrass still considered it mere paradox), their impenitent advocacy of the concept had significant reverberations in both philosophy and mathematics. In this essay, I will attempt to clarify one thread in the development of the notion of the infinite. In the first part, I study Spinoza’s discussion and endorsement, in the Letter on the Infinite (Ep. 12), of Hasdai Crescas’ (c. 1340-1410/11) crucial amendment to a traditional proof of the existence of God (“the cosmological proof” ), in which he insightfully points out that the proof does not require the Aristotelian ban on actual infinity. In the second and last part, I examine the claim, advanced by Crescas and Spinoza, that God has infinitely many attributes, and explore the reasoning that motivated both philosophers to make such a claim. Similarities between Spinoza and Crescas, which suggest the latter’s influence on the former, can be discerned in several other important issues, such as necessitarianism, the view that we are compelled to assert or reject a belief by its representational content, the enigmatic notion of amor Dei intellectualis, and the view of punishment as a natural consequent of sin. Here, I will restrict myself to the issue of the infinite, clearly a substantial topic in itself. (shrink)

This paper discusses the possibility of an absolute vacuum - a space without any substance. The motivation of this study is the contrast between most philosophers, up to Descartes, who stated that a vacuum was impossible, and the 17th century change of outlook, when the possibility and effective existence of the vacuum was accepted after the experiments of Torricelli and Pascal. This article attempts to show that, contrary to the received opinion, the acceptance of an ether is preferable to (...) the acceptance of a vacuum for several reasons. First: it is impossible to provide an empirical proof of the non-existence of the ether; second, an absolute vacuum is unthinkable; third, the ether concept is useful for the understanding of physical phaenomena; and fourth, the hypothesis of an ether in apparently void spaces is useful for the future development of science. The paper also endeavours to show that no recent advance of science changed those conclusions and that no future development can change them. (shrink)

In this article, some classical paradoxes of infinity such as Galileo’s paradox, Hilbert’s paradox of the Grand Hotel, Thomson’s lamp paradox, and the rectangle paradox of Torricelli are considered. In addition, three paradoxes regarding divergent series and a new paradox dealing with multiplication of elements of an infinite set are also described. It is shown that the surprising counting system of an Amazonian tribe, Pirah ̃a, working with only three numerals (one, two, many) can help us to change our (...) perception of these paradoxes. A recently introduced methodology allowing one to work with finite, infinite, and infinitesimal numbers in a unique computational framework not only theoretically but also numerically is briefly described. This methodology is actively used nowadays in numerous applications in pure and applied mathematics and computer science as well as in teaching. It is shown in the article that this methodology also allows one to consider the paradoxes listed above in a new constructive light. (shrink)