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A Brief History of Infinity is a serious, in depth study of man's evolving concept of infinity. Paolo Zellini's thoughtfully examines and distills the ideas of philosophers, theologians, writers, and mathematicians.
Zellini's terse style requires close attention and I found it necessary to reread many sections. Zellini begins with Aristotle's negative notion of infinity (apeiron) as an incomplete and unrealized potentiality, and demonstrates that Aristotle view largely explains the inability (or refusal) of Greek mathematicians to introduce a concept of an actual, or real infinity.
The inexhaustibility of the unlimited and the impossibility of finding an absolute minimum or maximum became focal points of discussion in Oxford and Paris in the fourteenth century. Discourse on these topics remained important in the Renaissance, continued with Leibniz and Newton, and culminated in the nineteenth century with Cauchy's and Weierstrass' definitive formulation of infinitesimal calculus.
Having less familiarity with philosophy, I found it profitable to skip for a short period to later chapters that more directly addressed mathematical infinity, a topic of especial interest to me. These chapter included The Principle of Indiscernibles - Classes; The Actual Infinite - Indefinite and Transfinite; and The Antinomies, or Paradoxes of Set Theory.
Paolo Zellini's sources are wide ranging, almost intimidatingly so. We readers encounter the philosophical thoughts of the Platonists, Aristotle, the Pythagoreans, Anaximander, the Chaldeans, Duns Scotus, St. Thomas Aquinas, Giordano Bruno, Nicholas of Cusa, Raymond Lull, Descartes, Leibniz, Goethe, Kant, Hegel, Russell, Simone Weil, Quine, Popper, Wittgenstein, and many others. Similarly, on the literary front we meet Cervantes, Kafka, Borges, Musil, and others.
Mathematicians are prominent also. Zellini discusses the provocative ideas of Descartes, Newton, Leibniz, Dedekind, Poincare, Cauchy, Weierstrass, Bolzano, Frege, Du Bois-Raymond, Cantor, Russell, Whitehead, Godel, Von Neumann, Zermelo, Skolem, Brouwer, and many others.
In the end Paolo Zellini's analysis leans away from Cantor's actual mathematical infinity and toward a potential infinity, somewhat in accordance with Brouwer's finite constructive methods (intuitionism).
Key Idea: I was intrigued with Hermann Weyl's conciliatory observation: the infinite is intuitively accessible as an indefinitely open field of possibility, and in this respect would seem analogous to a series of numbers that can be extended unlimitedly. Yet completeness, the so-called actual infinite, lies beyond our reach. Nonetheless, the demands for a totality impel the mind to imagine the infinite, using some symbolic construction, as a closed entity. Hence, the primary philosophical interest of mathematics should consist in attaining a fundamental understanding of these symbolic constructions.