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Plato's Ghost: the Modernist Transformation of Mathematics (Book Review)

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Plato's Ghost: the Modernist Transformation of Mathematics (Book Review) Digital Collections @ Dordt Faculty Work Comprehensive List 12-2009 Plato's Ghost: The Modernist Transformation of Mathematics (Book Review) Calvin Jongsma Dordt College, [email protected] Follow this and additional works at: https://digitalcollections.dordt.edu/faculty_work Part of the Christianity Commons, and the Mathematics Commons Recommended Citation Jongsma, C. (2009). Plato's Ghost: The Modernist Transformation of Mathematics (Book Review). Perspectives on Science and Christian Faith, 61 (4), 265. Retrieved from https://digitalcollections.dordt.edu/faculty_work/258 This Book Review is brought to you for free and open access by Digital Collections @ Dordt. It has been accepted for inclusion in Faculty Work Comprehensive List by an authorized administrator of Digital Collections @ Dordt. For more information, please contact [email protected]. Plato's Ghost: The Modernist Transformation of Mathematics (Book Review) Abstract Reviewed Title: Plato's Ghost: The Modernist Transformation of Mathematics by Jeremy Gray. Princeton, NJ: Princeton University Press, 2008. 515 pages. ISBN 9780691136103. Keywords book review, Plato's Ghost, modernism, Jeremy Gray Disciplines Christianity | Mathematics This book review is available at Digital Collections @ Dordt: https://digitalcollections.dordt.edu/faculty_work/258 Book Reviews O’Brien’s Spirituality in Nursing: Standing on Holy Ground has a high sense of the seriousness and value of (2003), this manuscript extends the conversation begun what it is trying to achieve. there in useful ways. This captures much of what Gray intends with the word, The chapter on ethical decision making seems incom- but he fleshes it out a bit further for mathematics. For plete. While an initial linkage between spirituality and him, modernism is a shift in professional mathematicians’ ethical decision making is asserted by the statement philosophical perspective that embraces an abstract ontol- “… spiritual issues are inextricably interwoven with the ogy and an epistemology that nearly dissolves into logic. kinds of ethical decisions that confront health care profes- The main conditions imposed on theorizing by modern- sionals and those for whom they care” (p. 331), this link- ist mathematicians are those of the formal axiomatic age could be more fully explicated. Might it be one’s method—concepts must be logically consistent and results conceptualization of person, beliefs about the purpose of rigorously derived, but otherwise mathematical creation health, or definition of nursing practice that introduce is completely free. spiritual issues into specific ethical dilemmas? Further, This outlook certainly typifies many foundational the ethical theories and principles are applied to prece- developments in mathematics around 1900, but Gray dent-setting cases rather than to the daily ethical dilem- argues that it is characteristic of mathematical practice mas that nurses encounter in their practice. Such an more broadly and that viewing this time period through approach distances this important topic from the every- the lens of modernism unifies a number of aspects of day experience of the nurse, and does not address the mathematics. question of how necessary support can be provided to these point-of-care practitioners. After an introductory chapter delineating his thesis in general terms, Gray divides his story into three main The second edition of Spiritual Dimensions of Nursing parts: (1) a pre-modern period (the nineteenth century Practice is a timely update that fulfills its specified goals. prior to about 1890, though he identifies Riemann and This book constitutes an excellent addition to the nursing Dedekind as mid-century precursors), (2) a period in and allied health literature. which modernism emerges, and (3) a time in which its Reviewed by Mary Molewyk Doornbos, Chairperson and Professor, outlook has become the accepted orthodoxy of profes- Calvin College Department of Nursing, 1734 Knollcrest Circle SE, sional mathematicians. The final three chapters are Grand Rapids, MI 49546. devoted to issues more on the periphery of mathematics (its relation to physics, attempts at popularization and writing its history, and its relation to language and psy- chology) and to some further mainstream developments between the two world wars. Within each main time MATHEMATICS period Gray follows a topical organization, looking at developments in four main fields: geometry, analysis, PLATO’S GHOST: The Modernist Transformation of algebra, and logic/set theory/foundations. Mathematics by Jeremy Gray. Princeton, NJ: Princeton From his past work, Gray is very conversant with University Press, 2008. 515 pages, glossary, bibliography, developments in geometry and analysis, and his treat- index. Hardcover; $45.00. ISBN: 9780691136103. ment of these topics is authoritative and informative. “Modernism,” like its younger cousin “postmodernism,” Modernism in geometry is associated with changing is one of those squirmy weasel-words that is difficult views on the nature of and developments within geome- topindownlongenoughtogainaclearandcogentview try (non-Euclidean geometry, projective geometry, of its meaning and referents. Perhaps these terms are Hilbert’s axiomatization of elementary geometry, Italian best used to describe whole families of attitudes and axiomatic geometry) as well as on geometry’s relation to beliefs. Nevertheless, historians and critics have felt com- science and everyday experience. In the field of analysis, fortable applying them to specific trends in the visual Gray distinguishes between early foundational efforts arts, architecture, literature, poetry, drama, film, music, (Cauchy’s arithmetization, Weierstrass’s rigorization) and theology, and philosophy. Modernism, in this sense, is later more abstract developments in analyzing the nature often pegged to certain developments around the turn and meaning of numbers (Dedekind on real numbers and of the twentieth century, especially in the arts. But what natural numbers, Cantor on transfinite ordinal and cardi- about science and mathematics? Are there substantial nal numbers). modernist trends in these fields? In Plato’s Ghost,the Gray also points out modernist developments in alge- distinguished historian of mathematics Jeremy Gray bra and the foundational fields of logic and set theory. investigates this possibility for mathematics. Algebra moved from more concrete concerns in solving A “core definition of modernism” is offered at the equations and finding regularities within number theory outset of the book: it is that “cultural shift” occurring to maneuvers of inventing new types of numbers for vari- between 1890 and 1930 which makes mathematics ous tasks (ideal numbers, quaternions, p-adic numbers). an autonomous body of ideas, having little or no In the twentieth century, modernism becomes entrenched outward reference, placing considerable emphasis in algebra with the structuralist approaches of Emmy on formal aspects of the work and maintaining a Noether and Bourbaki. complicated—indeed anxious—rather than a naive In the case of logic, two decades after an 1820s revival relationship with the day-to-day world, which is in Great Britain of traditional modes of deduction, the the de facto view of a coherent group of people, field was transformed by Boole and others into a branch such as a professional or discipline-based group that of algebra. It was later extended to include relations, Volume 61, Number 4, December 2009 265 Book Reviews quantifiers, and mathematical symbolism, and its relation has even deeper roots in early modern thought where to mathematics was inverted and refined by Frege and assertion of human autonomy arises as a major theme. Russell. In the early twentieth century under Hilbert’s Are twentieth-century developments a radical departure influence, logic became the tool of metamathematics, from these earlier developments, a genuine paradigm whose concern was the analysis of axiomatic theories for shift, as Gray asserts, or are they an intensification of consistency, completeness, and independence, becoming aspects of the same humanist spirit? Modernism’s histori- aligned in the end with set theory and abstract model cal lineage ought to be traced further back than Gray does theory. Promoting set theory as the ultimate foundation in Plato’s Ghost, to give us a more long-term perspective for mathematics provided the discipline with a self-con- on what is brand new and what might develop core tained modernist ontology. In discussing these develop- tendencies that had already become prominent when ments, Gray tends to rely more on other authorities than “modernism” was first self-consciously proclaimed by on his own work, but foundational aspects are probably mid-eighteenth-century thinkers. This may be asking for the best-known part of the story he is telling. more than can be comfortably proved in scientific or historical terms, but readers of these pages will likely Even applied areas of mathematics felt the drift acknowledge a responsibility to test the spirits, in intel- toward modernism. This helps us understand why lectual affairs as well as in spiritual and moral matters. Eugene Wigner, a leading physicist, would write in 1960 about “The Unreasonable Effectiveness of Mathematics What we have here, then, is an excellent and detailed in the Natural Sciences” as being a mysterious business. survey of how modernism took root in mathematics. As Gray notes, mathematical physics had given way first Plato’s Ghost
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