Plato and Aristotle both attribute great significance to the reality, goodness, and importance for reasoning of limits. Aristotle staunchly opposes assertions of really existing infinity, or of an infinite regress of reasons.
(Despite the opinions of some mathematicians, all that modern mathematics truly requires is “potential” infinity — the ability to construct something arbitrarily large or small through arbitrarily many definite acts of extending, dividing, or whatever of some definite thing. This is consistent, for instance, with the modern grounding of the infinities of calculus in analysis based on the concept of limits. On the frontiers of mathematical research, homotopy type theory and univalent foundations allow the most extravagant “classical” mathematics of higher infinities to be expressed in terms of definite constructions.)
Aristotle’s position on this is closely tied to his central concepts of ends and the good (see also Aristotle on Explanation).
“And since that for the sake of which something is is an end, and this sort of thing is what is not for the sake of anything else, but they are for the sake of it, then if there is any such last thing, there will not be an infinity, but if there is no such thing, there will be nothing for the sake of which it is. But those who make there be an infinite are unaware that they abolish the nature of the good. (Yet no one would make an effort to do anything if he were not going to come to a limit.) And there would not be intelligence among beings; for what has intelligence always acts for the sake of something, and this is a limit” (Metaphysics book II chapter 2, Sachs tr., p. 31).
This is also central to his argument for the existence of a first cause.
The emphasis on a kind of finitude here should not be taken to imply any dogmatic attachment to particular formulations or representations, such as Hegel for instance objected to. In the following chapter, in part echoing the beginning of the Nicomachean Ethics, Aristotle notes, “Some people expect everything to be said with precision, while others are annoyed by precision, either because they can’t keep the connections straight or because of its hairsplitting pettiness. For precision does have something of this sort about it, so that, just as in business agreements, so also in reasoning it seems to people to be ungenerous. For this reason one must have been trained in how one ought to receive each kind of argument, since it is absurd to be searching at the same time for knowledge and for the direction to knowledge; and it is not possible to get either of the two easily” (pp. 32-33).
“Courses of lectures go along with one’s habits; for in the way that we are accustomed, in that way we think it fitting for something to be said, and what departs from this does not seem the same, but through lack of acquaintance seems too obscure and alien. For we are used to what is familiar. And what great strength the customary has, the laws show, in which mythical and childish things are of greater strength than knowing about them, because of custom” (p. 32).
This last remark clearly shows that Aristotle’s emphasis on the shareability and actual sharedness of values does not at all mean he assumes that what is held to be authoritative by one’s own community or society is always right. Unlike Socrates, who after his eloquent defense meekly accepted his city’s ignorant condemnation, Aristotle at one point fled Athens “lest the Athenians sin against philosophy twice”. This lesson is crucial to the understanding of Hegel as well.
The Postmodern Peripatetic 