Academic journal article The Midwest Quarterly

Sending an Irishman "Realing": Constructive Realism and George Berkeley's Philosophy of Arithmetic

Academic journal article The Midwest Quarterly

Sending an Irishman "Realing": Constructive Realism and George Berkeley's Philosophy of Arithmetic

Article excerpt

I RECENTLY HAD the opportunity to reread George Berkeley's A Treatise Concerning the Principles of Human Knowledge (1710), published in the same year he was ordained an Anglican priest. You may remember Berkeley: he's the 18th-century British empiricist you laughed off stage in your first philosophy class because he argued for immaterialism, the view that material objects do not exist, except as ideas in a mind. Jonathan Swift shared your skepticism; he left Berkeley standing on a doorstep, claiming that the Irishman ought to have as little trouble walking through a closed door as an open door. Now, the striking thing I realized about Berkeley--the thing I want to share with you--is that as far as the philosophy of arithmetic is concerned, Berkeley begins in exactly the right place. Unfortunately, he makes several missteps along the way, and ends up holding an untenable view. But where Berkeley goes astray, my philosophy of mathematics called constructive realism stays on course, resulting in--if you'll pardon the bias--a very plausible philosophy of arithmetic.

We begin by developing Berkeleyan arithmetic, a nominalist, formalist, and finitist view. It is nominalist, because Berkeley denies that numbers are general ideas, or universals of any sort, corresponding to numerals. Instead, the numerals function like names that can stand for features of individual, particular ideas. Berkeleyan arithmetic is formalist, because he considers the numerals themselves--and operations on them--to be the objects of arithmetic. Finally, Berkeleyan arithmetic is finitist, because there will not be an infinity of numerals to serve as the objects of arithmetic. The development to follow will be neither as technical, nor as admirable, as the analysis given by Douglas Jesseph in his Berkeley's Philosophy of Mathematics. Moreover, it is intended to be largely descriptive, and not sharply critical. Berkeley already has a legion of critics, and the first task is to understand--and appreciate--Berkeleyan arithmetic.

Our starting point is Berkeley's somewhat problematical notion of an idea. An idea, whether of sensation or reflection, is the content of an experience. An idea of sensation is the content of a visual, auditory, or tactile experience, and an idea of reflection is a thought or mental image. Thus, as you gaze at this page, you are having a visual experience, and the content of that experience is a Berkeleyan idea of sensation. Then, as you close your eyes and summon up a mental image of this page before your "mind's eye," or when you think about the Kentucky Derby winner, Fusaichi Pegasus, the content of those experiences are Berkeleyan ideas of reflection.

Now, Berkeley's objection to a proposal made by John Locke about ideas is key to understanding how his philosophy of arithmetic is nominalist. Locke and Berkeley agreed that we have particular ideas of sensation and reflection, like your present visual experience or corresponding thoughts about it. But Locke proposed that we use a mental process called abstraction to form general ideas. As Locke understood them, general ideas are ideas that general terms stand for, much like the meanings of the words `white' or `horse'. Now, white is a feature of the particular visual experience you are having at present, and your thought of Fusaichi Pegasus is of a particular horse. However, the general terms `white' and `horse' mean what is common to all particular white things and all particular horses, and these ideas are formed by abstraction, by attending to and combining only the white quality--or the "horse-ish" quality--of many particular ideas of sensation and reflection (409-20).

The problem began with Berkeley's somewhat uncharitable interpretation of Locke's general ideas, and the mental process of abstraction that produces them. Sarcastically referring to them as "abstract ideas" because of their dubious origin, Berkeley turned Locke's general ideas into ideas we cannot have--not without great difficulty anyway. …

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