Infinite Past Time

An Analytical Contradiction in "Infinite Past Time"

The problematic character of infinite past time is revealed by a seemingly inescapable analytical contradiction in the very expression “infinite past time.”

If one splits the expression into its two component parts: (1) “past time” and (2) “infinite,” and attempts to find a common conceptual base which can apply to both terms (much like a lowest common denominator can apply to two different denominators in two fractions), one can immediately detect contradictory features. One such common conceptual base is the idea of “occurrence,” another, the idea of “achievement,” and still another, the idea of “actualizability.” Let us begin with the expression “past time.”

Past time can only be viewed as having occurred, or having been achieved, or having been actualized; otherwise, it would be analytically indistinguishable from present time and future time. In order to maintain the analytical distinction among these three interrelated ideas, present time must be viewed as “occurring,” or “being achieved,” or “being actualized”; and future time must be viewed as “not having occurred,” “not achieved,” and “not actualized.” The notion of “past” loses its intelligibility with respect to present and future if its meaning were to include “occurring,” “being achieved,” or “being actualized” (pertaining to the present); or “not having occurred,” “not achieved,” or “not actualized” (pertaining to the future). If past time is to retain its distinct intelligibility, it can only be viewed as “having occurred,” “achieved,” and “actualized.”

Now let us turn to the other side of the expression, namely, “infinite.” Throughout this Unit, I will view “infinity” within the context of a continuous succession because I will show that real time in changeable universes must be a “continuous succession of non-contemporaneous distension.” Now, infinities within a continuous succession imply “unoccurrable,” “unachievable,” and “unactualizable,” for a continuous succession occurs one step at a time (that is, one step after another), and can therefore only be increased a finite amount. No matter how fast and how long the succession occurs, the “one step at a time” or “one step after another” character of the succession necessitates that only a finite amount is occurrable, achievable, or actualizable. Now, if “infinity” is applied to a continuous succession, and it is to be kept analytically distinct from (indeed, contrary to) “finitude,” then “infinity” must always be more than can ever occur, be achieved, or be actualized through a continuous succession (“one step at a time” succession). Therefore, infinity would have to be unoccurrable, unachievable, and unactualizable when applied to a continuous succession. Any other definition would make “infinity” analytically indistinguishable from “finitude” in its application to a continuous succession. Therefore, in order to maintain the analytical distinction between “finitude” and “infinity” in a continuous succession, “infinity” must be considered unoccurrable (as distinct from finitude which is occurrable), unachievable (as distinct from finitude which is achievable), and unactualizable (as distinct from finitude which is actualizable). We are now ready to combine the two parts of our expression through our three common conceptual bases:

“Infinite..............................Past Time”

“(The) unoccurable................(has) occurred.”

“(The) unachievable...............(has been) achieved.”

“(The) unactualizable.............(has been) actualized.”

Failures of human imagination may deceive one into thinking that the above analytical contradictions can be overcome, but further scrutiny reveals their inescapability. For example, it might be easier to detect the unachievability of an infinite series when one views an infinite succession as having a beginning point without an ending point, for if a series has no end, then, a priori, it can never be achieved. However, when one looks at the infinite series as having an ending point but no beginning point (as with infinite past time reaching the present), one is tempted to think that the presence of the ending point must signify achievement, and, therefore, the infinite series was achieved. This conjecture does not avoid the contradiction of “infinite past time” being “an achieved unachievable.” It simply manifests a failure of our imagination. Since we conjecture that the ending point has been reached, we think that an infinite number of steps has really been traversed, but this does not help, because we are still contending that unachievability has been achieved, and are therefore still asserting an analytical contradiction.

Another failure of our imagination arises out of thinking about relative progress in an historical succession. Our common sense might say that infinite past history is impossible because an infinity is innumerable, immeasurable, and unquantifiable, making the expression “an infinite number” an oxymoron. But then we get to thinking that infinite history seems plausible because each step relative to the other steps is quantifiable in its progression; each step is subject to relative numeration. Therefore, it seems like history can really achieve an infinite number of steps.

However, as the above analysis reveals, this cannot be so because an infinity in a continuous succession must be unachievable or unactualizable as a whole (otherwise, it would be analytically indistinguishable from “finitude” in a continuous succession). Since, as has been said, past time must be achieved or actualized (otherwise it would be analytically indistinguishable from “present” and “future”), “infinite past time” must be an “achieved unachievable” or an “actualized unactualizable” (an intrinsic contradiction). Moreover, the expression “an infinite number” is also an intrinsic contradiction because “number” implies a definite quantity, whereas “infinity” implies innumerability (more than can ever be numbered). Therefore, infinite history and its characterization as “a completion of infinite time,” remains inescapably analytically contradictory.

This intrinsic analytical contradiction reveals the problematic character of the very idea of “infinite past time.” It now remains for us to show the inapplicability of this problematic idea to our universe, and indeed, to any really possible changeable universe. This step will give ontological (“synthetic”) significance to the analytical contradiction by showing that the condition of the real world (i.e., our real universe, or any really possible changeable universe) will contradict (and therefore resist) the application of this problematic idea to it. The result will be that no real universe could have had infinite past time.

Before we can proceed to this proof, we must first give an ontological explanation of real time^[1], and then show that this real time must be intrinsic to any changeable universe, and then explain Hilbert’s distinction between actual and potential infinities so that it will be clear that “infinite past time” (as defined) must be an actual infinity which Hilbert shows to be inapplicable to any reality to which the axioms of finite mathematics can be applied. The ontological proof against an infinity of past time will follow from this.

Please note that the a-priori synthetic proof is based upon the work of the famous mathematician David Hilbert (see “On the Infinite” in Hilary Putnam, ed Philosophy of Mathematics (Inglewood Cliffs, NJ: Prentiss Hall, 1977). What follows is a brief summary of Hilbert’s discovery. For a full explanation see NPEG, Chapter V, and PID Units 22 – 27.