Infinity

  • Leonard Peikoff
    There is a use of [the concept] “infinity” which is valid, as Aristotle observed, and that is the mathematical use. It is valid only when used to indicate a potentiality, never an actuality. Take the number series as an example. You can say it is infinite in the sense that, no matter how many numbers you count, there is always another number. You can always keep on counting; there’s no end. In that sense it is infinite — as a potential. But notice that, actually, however many numbers you count, wherever you stop, you only reached that point, you only got so far. . . . That’s Aristotle’s point that the actual is always finite. Infinity exists only in the form of the ability of certain series to be extended indefinitely; but however much they are extended, in actual fact, wherever you stop it is finite.
  • Introduction to Objectivist Epistemology
    An arithmetical sequence extends into infinity, without implying that infinity actually exists; such extension means only that whatever number of units does exist, it is to be included in the same sequence.
  • The Objectivist Forum
    Every unit of length, no matter how small, has some specific extension; every unit of time, no matter how small, has some specific duration. The idea of an infinitely small amount of length or temporal duration has validity only as a mathematical device useful for making certain calculations, not as a description of components of reality. Reality does not contain either points or instants (in the mathematical sense). By analogy: the average family has 2.2 children, but no actual family has 2.2 children; the “average family” exists only as a mathematical device.