Edward Halper

ABSTRACT In Greek Mathematical Thought and the Origin of Algebra, Jacob Klein contrasts ancient Greek philosophy's direct engagement with things through arithmetic with the ancient science of numeric calculation, logistic. By chronicling the later development of logistic, by means of increasing symbolization, ultimately into algebra, he argues that logistic has come to displace arithmetic and, thereby, to submerge the ontological issues at the center of Greek thought. This article argues, first, that Klein's target is Ernst Cassirer's notion of number as a “symbolic function”—what Cassirer would later call “symbolic form.” In Substance and Function Cassirer argues that number constitutes a conceptual structure progressively extended by thought so as to include the results of different sorts of calculation (e.g., negative numbers, irrational roots, and infinite numbers) within itself and, thereby, to be a closed system. Drawing upon Dedekind, Cassirer claims that no individual number can be understood apart from the entire conceptual structure. This article argues, second, that Cassirer recognizes, as it were, a different set of things, namely, relations, and that it is possible to engage these “things” directly by recognizing the ontological issues they raise.

IT IS well-known that Plato and Aristotle disagree on the possibility of knowledge of nature. Plato maintains that knowledge, in contrast with belief, is never mistaken, that the objects of knowledge are always the same and never becoming, and that what we sense is always becoming. He concludes that knowledge is possible only of objects that are unchanging and separate from sensibles, i.e., the forms. Aristotle rejects this conclusion and recognizes knowledge of sensibles. Surprisingly, though, he accepts Plato's assumptions. He too maintains that knowledge is not sometimes true and sometimes false, but always true ; he distinguishes the sensibles from the unchanging eternal beings ; and he asserts that the objects of knowledge "always are or are for the most part", and occasionally he even claims that they cannot be otherwise. The problem is, how can Aristotle accept Plato's assumptions about the nature and objects of knowledge and still maintain that knowledge of nature is possible?

Maimonides’ claim, in Guide of the Perplexed I.68, that our intellect, like God’s, becomes one with the object it knows would seem to be at odds with his injunction to his readers to set their “thought to work on the first intelligible” and to “rejoice in what [it] apprehends”. The former passage supposes that we grasp individual essences by themselves, whereas the latter supposes that such essences are known only through their first cause. Since we cannot grasp the first cause, God, we cannot, apparently, grasp anything else or rejoice in it. After briefly sketching the solutions of Aristotle and Avicenna, this paper argues that Maimonides resolves these issues by distinguishing the metaphysical grasp of a single essence from the grasp of its relations - due to God’s will - with other essences and, more specifically, the metaphysical grasp of our own human essence from the grasp of ourselves through our physical and moral relations with the world. Insofar as we know ourselves by knowing everythi...

Drawing principally on the Symposium, this paper argues that humor in Plato’s dialogues serves two serious purposes. First, Plato uses puns and other devices to disarm the reader’s defenses and thereby allow her to consider philosophical ideas that she would otherwise dismiss. Second, insofar as human beings can only be understood through unchanging forms that we fail to attain, our lives are discontinuous and only partly intelligible. Since, though, the discontinuity between expectation and actual occurrence is the basis for humor, Plato can use humor to express who we are as human beings.