EXPANDED EDITION (eBook): -/- Infinity Is Not What It Seems...Infinity is commonly assumed to be a logical concept, reliable for conducting mathematics, describing the Universe, and understanding the divine. Most of us are educated to take for granted that there exist infinite sets of numbers, that lines contain an infinite number of points, that space is infinite in expanse, that time has an infinite succession of events, that possibilities are infinite in quantity, and over half of the world’s population believes (...) in a divine Creator infinite in knowledge, power, and benevolence. -/- According to this treatise, such assumptions are mistaken. In reality, to be is to be finite. The implications of this assessment are profound: the Universe and even God must necessarily be finite. -/- The author makes a compelling case against infinity, refuting its most prominent advocates. Any defense of the infinite will find it challenging to answer the arguments laid out in this book. But regardless of the reader’s position, FOREVER FINITE offers plenty of thought-provoking material for anyone interested in the subject of infinity from the perspectives of philosophy, mathematics, science, and theology. -/- This electronic edition of FOREVER FINITE is offered free online for all and is expanded with content that does not appear in the published edition due to print production page limitations. From the Introduction to the Conclusion, Chapters 1–27 are identical to the published print edition, including the page numbering for the chapters. This electronic edition adds an appendix and associated content—specifically, updates to the book’s back matter (68 new endnotes, 17 new bibliographic references, 3 new figure credits, 14 new glossary terms) and front matter (an updated table of contents and this preface note). (shrink)

I discuss two questions: (1) would Duhem have accepted the thesis of the continuity of scientific methodology? and (2) to what extent is the Oxford tradition of classification/subalternation of sciences continuous with early modern science? I argue that Duhem would have been surprised by the claim that scientific methodology is continuous; he expected at best only a continuity of physical theories, which he was trying to isolate from the perpetual fluctuations of methods and metaphysics. I also argue that the evidence (...) does not support the conclusion that early modern doctrines about mathematics and physics are continuous with the subalternation of sciences from Grosseteste, Bacon, and the theologians of fourteenth-century Oxford. The official and dominant context for early modern scientific methodology seems to have been progressive Thomism, and early modern thinkers seem to have pitted themselves against it. (shrink)

This paper explores the nature, development and influence of the first English account of absolute time, put forward in the mid-seventeenth century by the ‘Cambridge Platonist’ Henry More. Against claims in the literature that More does not have an account of time, this paper sets out More's evolving account and shows that it reveals the lasting influence of Plotinus. Further, this paper argues that More developed his views on time in response to his adoption of Descartes' vortex cosmology and cosmogony, (...) providing new evidence of More's wider project to absorb Cartesian natural philosophy into his Platonic metaphysics. Finally, this paper argues that More should be added to the list of sources that later English thinkers – including Newton and Samuel Clarke – drew on in constructing their absolute accounts of time. (shrink)

One of the most pressing philosophical problems in early modern Europe concerned how the soul and body could form a unity, or, as many understood it, how these two substances could work together. It was widely believed that there were three (and only three) hypotheses regarding the union of soul and body: (1) physical influence, (2) occasionalism, and (3) pre-established harmony. However, in 1763, a fourth hypothesis was put forward by the French thinker André-Pierre Le Guay de Prémontval (1716–1764). Prémontval’s (...) hypothesis, given the grand name of “psychocracy” (i.e., the dominion or the rule of the soul), held that there was a real influence between soul and body, but that this was an immaterial kind of influence as opposed to the physical kind that had been entertained heretofore. Prémontval’s hypothesis is the focus of this paper. I shall begin by sketching out the details of Prémontval’s hypothesis (Section 1), then proceed to consider its claims to constitute a true fourth hypothesis distinct from the other three (Section 2), before closing by briefly considering two objections and the responses either that Prémontval himself made or that may be made on his behalf (Section 3). (shrink)

Many historical and philosophical studies treat infinity as an exclusively quantitative notion, whose proper domain of application is mathematics and physics. The main aim of this paper is to disentangle, by critically examining, three notions of infinity in the early modern period, and to argue that one—but only one—of them is quantitative. One of these non-quantitative notions concerns being or reality, while the other concerns a particular iterative property of an aggregate. These three notions will emerge through examination of three (...) central figures in the period: Locke, Descartes, and Leibniz. (shrink)

On Zeno Beach there are infinitely many grains of sand, each half the size of the last. Supposing Aristotle denied the possibility of Zeno Beach, did he have a good argument for the denial? Three arguments, each of ancient origin, are examined: the beach would be infinitely large; the beach would be impossible to walk across; the beach would contain a part equal to the whole, whereas parts must be lesser. It is attempted to show that none of these arguments (...) was Aristotle’s. Indeed, perhaps Aristotle’s finitism applied to magnitude only, not plurality. (shrink)

In two rarely discussed passages – from unpublished notes on the Principles of Philosophy and a 1647 letter to Chanut – Descartes argues that the question of the infinite extension of space is importantly different from the infinity of time. In both passages, he is anxious to block the application of his well-known argument for the indefinite extension of space to time, in order to avoid the theologically problematic implication that the world has no beginning. Descartes concedes that we always (...) imagine an earlier time in which God might have created the world if he had wanted, but insists that this imaginary earlier existence of the world is not connected to its actual duration in the way that the indefinite extension of space is connected to the actual extension of the world. This paper considers whether Descartes’s metaphysics can sustain this asymmetrical attitude towards infinite space vs. time. I first consider Descartes’s relation to the ‘imaginary’ space/time tradition that extended from the late scholastics through Gassendi and More. I next examine carefully Descartes’s main argument for the indefinite extension of space and explain why it does not apply to time. Most crucially, since duration is merely conceptually distinct from enduring substance, the end or beginning of the world entails the end or beginning of real time. In contrast, extension does not depend on any enduring substance besides itself. (shrink)

in 1612, lodovico cigoli completed a fresco in the pauline chapel of the basilica of santa maria maggiore in rome depicting apocalypse 12: “a woman clothed with the sun, and the moon under her feet.” he showed the crescent moon with spots, as his friend galileo had observed with the newly invented telescope. considerations of the orthodox view of the perfect moon as held by philosophers have led historians to ask why this clearly imperfect moon in a religious painting raised (...) no eyebrows. we argue that when considered in the context of biblical interpretation and the rhetoric of the counter-reformation, the imperfect moon under the woman's feet was entirely consistent with traditional interpretations of apocalypse 12. (shrink)

The ArgumentIn 1612, Lodovico Cigoli completed a fresco in the Pauline chapel of the Basilica of Santa Maria Maggiore in Rome depicting Apocalypse 12: “A woman clothed with the sun, and the moon under her feet.” He showed the crescent Moon with spots, as his friend Galileo had observed with the newly invented telescope. Considerations of the orthodox view of the perfect Moon as held by philosophers have led historians to ask why this clearly imperfect Moon in a religious painting (...) raised no eyebrows. We argue that when considered in the context of biblical interpretation and the rhetoric of the Counter-Reformation, the imperfect Moon under the woman's feet was entirely consistent with traditional interpretations of Apocalypse 12. (shrink)

It has been argued that there is a genuine conflict between the views of geometry defended by Hume in the Treatise and in the Enquiry: while the former work attributes to geometry a different status from that of arithmetic and algebra, the latter attempts to restore its status as an exact and certain science. A closer reading of Hume shows that, in fact, there is no conflict between the two works with respect to geometry. The key to understanding Hume's view (...) of geometry is the distinction he draws between two standards of equality in extension. (shrink)

Mathematics is obviously important in the sciences. And so it is likely to be equally important in any effort that aims to understand God in a scientifically significant way or that aims to clarify the relations between science and theology. The degree to which God has any perfection is absolutely infinite. We use contemporary mathematics to precisely define that absolute infinity. For any perfection, we use transfinite recursion to define an endlessly ascending series of degrees of that perfection. That series (...) rises to an absolutely infinite degree of that perfection. God has that absolutely infinite degree. We focus on the perfections of knowledge, power, and benevolence. Our model of divine infinity thus builds a bridge between mathematics and theology. (shrink)