Woosuk Park.*Philosophy’s Loss of Logic to Mathematics: An Inadequately Understood Take-Over

Abstract

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.