TY - JOUR
AU - Franklin,, James
AB - Woosuk Park’s book discusses a number of disparate issues which are not on the agenda in the philosophy of mathematics but deserve to be. One is the Aristotelian realist option in the philosophy of mathematics. Since Frege, the philosophy of mathematics has been divided into two opposing schools on the nature of mathematical entities, Platonism and nominalism. ‘Full-blooded’ Platonism holds that entities like sets and numbers are classical acausal ‘abstract objects’, existing in a non-physical Platonic realm. Nominalism holds that mathematical entities do not exist at all and mathematics is merely a language, or manipulation of formal symbols or similar. That neglects the Aristotelian option, that mathematical entities exist in a way, but not as abstract entities, and that mathematics studies certain aspects of the real non-abstract (physical and other) world, for example its structural, or quantitative, or relational aspects [Franklin, 2014; Jacquette, 2014; Thomas, 2008]. As Park points out, the ‘science of quantity’ version of Aristotelian theory was once dominant. He praises in particular Biancani’s De Mathematicarum Natura Dissertatio [1615]. Of most interest is Biancani’s treatment of the question whether scientiae mediae (what we would call applied mathematics, like astronomy and optics) have perfect demonstrations. That is, can mathematics prove truths valid in the physical as well as the abstract world? Biancani’s defence of the affirmative is less than convincing; contemporary readers might be more impressed by Archimedes’ demonstration (or purported demonstration) of the law of the lever and Galileo’s proof that falling bodies cannot have speed proportional to the distance travelled. Park writes (p. 6) ‘By introducing Biancani and Aristotelian philosophy of mathematics to [the] forefront, I want to hint at the urgent need to reconsider the Aristotelian position in logic and mathematics, which disappeared almost completely from the scene without good reasons in the early twentieth century.’ While he does not develop a particular Aristotelian option himself, nor defend any existing one, he does give a careful account of one of the most embarrassing incidents for the contemporary Aristotelian school, Penelope Maddy’s defection. In her book Realism in Mathematics [1990] Maddy argued that sets are perceivable. For example, if I open the refrigerator and see that there are three eggs in it, I perceive a set of three eggs; ‘we can and we do perceive sets, and … our ability to do so develops in much the same way as our ability to see physical objects’ [1990, p. 58]. That is, as she notes, an Aristotelian realist position as it takes sets to be not abstract and acausal but physical and perceivable (and real). But in her Naturalism in Mathematics [1997] Maddy rejected that position and took a more Wittgensteinian (in effect nominalist) view of mathematics. Park follows through in some detail a debate between Maddy and Colyvan [2001, Ch. 5] over the implications of the indispensability of mathematics in physical science, and concludes that the main reason for Maddy’s change of mind was her increasing sense of a disanalogy between science and mathematics — where science postulates as few entities as possible, mathematics postulates as many as possible, such as huge infinite sets. That has little apparent relevance to obviously instantiated small mathematical structures such as sets of three eggs. Park argues, following [Baker, 2002] and [Dieterle, 1999] that Maddy has not made out a disanalogy between science and mathematics sufficient to impugn an Aristotelian realist interpretation of mathematics. The book’s title, Philosophy’s Loss of Logic to Mathematics refers to another issue Park addresses: the nature of the late nineteenth-century revolution in logic led by Frege and Russell with contributions from mathematicians. It is, as his subtitle says, ‘an inadequately understood take-over’. Although logic, one of the earliest mature sciences, made certain advances in the two thousand years after Aristotle, it did not produce a logical technique flexible enough for use in mathematics. What exactly was the nature of the impediments in traditional logic and of the new discoveries of Frege and others? Park has a suggestion to add to the usual mix. Undoubtedly one crucial advance was multiple quantifiers, applied to relations. The Cauchy–Weierstrass definition of continuity of a function begins ‘For all |$\varepsilon > 0$|⁠, there exists |$\delta > 0$| such that for all |$x$|⁠, if |$|x - c| <\delta $| then …’ What is logically difficult about that is the triple quantification, especially with the ‘there exists’ nested between two ‘for all’s. That cannot be done in single-quantifier traditional logic: the syllogism deals only in arguments with a single ‘all’ or ‘some’. The ‘for all |$x$|⁠, if |$|x - c| <\delta $| then …’ is a reminder that traditional logic does not represent ‘All |$A$|s are |$B$|s’ as ‘For all |$x$|⁠, if |$x$| is an |$A$| then |$x$| is a |$B$|’: as Russell emphasized with his talk of ‘propositional functions’, doing so is a great gain in the ability to express natural language in symbolic logical terms. It has led to the practice, very unnatural from a traditional perspective, of teaching propositional logic first and then moving to predicate calculus via propositional functions. An almost equally important development, mainly due to Dedekind and Frege, was that of being explicit about sets, so that ‘For all |$\varepsilon > 0$|’ is thought of as quantification over a pre-existing set (of positive real numbers). Without denying any of that, Park points out a more basic flaw in traditional logic, one which gave Frege considerable trouble but which had to be exposed and overcome before the developments just mentioned could gain traction. He expresses it — obscurely but keeping to Frege’s language — by saying that Frege distinguished between ‘falling under’ and ‘subordination’ but traditional logic failed to (and, Frege says, mathematicians failed to as well). The number 2 ‘falls under’ the concept ‘prime’ but ‘even prime’ does not fall under the concept ‘prime’ but is rather ‘subordinate’ to it. There is an obvious close parallel with the distinction between set membership and the subset relation (2 is a member of the set of primes but the set of even primes is a subset of the set of primes), but Frege regards concepts as prior to sets and so wishes to situate the distinction in concepts. It is quite true that traditional logic obscured the distinction by trying to assimilate the particular proposition (‘Socrates is a mortal’) to the universal proposition (‘All men are mortal’). It is also true that there is considerable temptation in mathematics to elide these distinctions, for example to teach formally that the |$XY$|-plane is a subset of three-dimensional space but in practice to expect students to visualize it as a part of three-dimensional space. Park argues that the total success of the late nineteenth-century revolution in logic has meant that we, its ‘fortunate heirs’, have forgotten what the questions were that exercised the pioneers. Since the philosophy of logic — in the basic sense of what logic is about — remains in a primordial state, a resumption of inquiry into these questions is indicated. Other issues that Park discusses include the parallels between Duns Scotus’s and Frege’s ontologies, the motivation of Gödel’s ontological proof of the existence of God and Zermelo and Hilbert on the axiomatic method and implicit definition. Park’s view of what are worthwhile issues is ‘outside the square’. Philosophers of mathematics will expand their minds by following his lead. Footnotes Orcid.org/0000-0002-9546-7273. References Baker, A. [ 2002 ]: ‘ Maximizing principles in mathematical methodology ’, Logique et Analyse 45 , 269 – 281 . WorldCat Biancani (Blancanus), G. [ 1615 ]: De Mathematicarum Natura Dissertatio . Bologna : Coechi . Klima G. , trans., in Mancosu P. , ed., Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century , pp. 178 – 212 . Oxford University Press , 1996 . Google Preview WorldCat COPAC Colyvan, M. [ 2001 ]: The Indispensability of Mathematics . Oxford University Press . Google Preview WorldCat COPAC Dieterle, J.M. [ 1999 ]: ‘ Mathematical, astrological, and theological naturalism ’ Philosophia Mathematica ( 3 ) 7 , 129 – 135 . Google Scholar Crossref Search ADS WorldCat Franklin, J. [ 2014 ]: An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure . Basingstoke : Palgrave Macmillan . Google Preview WorldCat COPAC Jacquette, D. [ 2014 ]: ‘ Toward a neoaristotelian inherence philosophy of mathematical entities ’ Studia Neoaristotelica 11 , 159 – 204 . Google Scholar Crossref Search ADS WorldCat Maddy, P. [ 1990 ]: Realism in Mathematics . Oxford University Press . Google Preview WorldCat COPAC Maddy, P. [ 1997 ]: Naturalism in Mathematics . Oxford University Press . Google Preview WorldCat COPAC Thomas, R.S.D. [ 2008 ]: ‘Extreme science: Mathematics as the science of relations as such’ in Gold B. and Simons R. , eds, Proof and Other Dilemmas: Mathematics and Philosophy , pp. 245 – 263 ). Washington, D.C. : Mathematical Association of America . Google Preview WorldCat COPAC © The Authors [2019]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)
TI - Woosuk Park.Philosophy’s Loss of Logic to Mathematics: An Inadequately Understood Take-Over
JF - Philosophia Mathematica
DO - 10.1093/philmat/nkz018
DA - 2019-10-01
UR - https://www.deepdyve.com/lp/oxford-university-press/woosuk-park-philosophy-s-loss-of-logic-to-mathematics-an-inadequately-1b3EidL4My
SP - 440
VL - 27
IS - 3
DP - DeepDyve
ER -