Objections To Set Theory As A Foundation For Mathematics
From set theory's inception, some mathematicians have objected to it as a foundation for mathematics. The most common objection to set theory, one Kronecker voiced in set theory's earliest years, starts from the constructivist view that mathematics is loosely related to computation. If this view is granted, then the treatment of infinite sets, both in naive and in axiomatic set theory, introduces into mathematics methods and objects that are not computable even in principle. Ludwig Wittgenstein questioned the way Zermelo–Fraenkel set theory handled infinities. Wittgenstein's views about the foundations of mathematics were later criticised by Georg Kreisel and Paul Bernays, and investigated by Crispin Wright, among others.
Category theorists have proposed topos theory as an alternative to traditional axiomatic set theory. Topos theory can interpret various alternatives to that theory, such as constructivism, finite set theory, and computable set theory.
Read more about this topic: Set Theory
Famous quotes containing the words objections, set, theory, foundation and/or mathematics:
“Miss Western: Tell me, child, what objections can you have to the young gentleman?
Sophie: A very solid objection, in my opinion. I hate him.
Miss Western: Well, I have known many couples who have entirely disliked each other, lead very comfortable, genteel lives.”
—John Osborne (1929–1994)
“We set up a certain aim, and put ourselves of our own will into the power of a certain current. Once having done that, we find ourselves committed to usages and customs which we had not before fully known, but from which we cannot depart without giving up the end which we have chosen. But we have no right, therefore, to claim that we are under the yoke of necessity. We might as well say that the man whom we see struggling vainly in the current of Niagara could not have helped jumping in.”
—Anna C. Brackett (1836–1911)
“Everything to which we concede existence is a posit from the standpoint of a description of the theory-building process, and simultaneously real from the standpoint of the theory that is being built. Nor let us look down on the standpoint of the theory as make-believe; for we can never do better than occupy the standpoint of some theory or other, the best we can muster at the time.”
—Willard Van Orman Quine (b. 1908)
“What is the foundation of that interest all men feel in Greek history, letters, art and poetry, in all its periods from the Heroic and Homeric age down to the domestic life of the Athenians and Spartans, four or five centuries later? What but this, that every man passes personally through a Grecian period.”
—Ralph Waldo Emerson (1803–1882)
“The three main medieval points of view regarding universals are designated by historians as realism, conceptualism, and nominalism. Essentially these same three doctrines reappear in twentieth-century surveys of the philosophy of mathematics under the new names logicism, intuitionism, and formalism.”
—Willard Van Orman Quine (b. 1908)