WHITROW AND POPPER ON THE IMPOSSIBILITY OF AN INFINITE PAST
William Lane Craig, British Journal for the Philosophy of Science, 1979, Vol 30 No 2, pp 165-170
[references not included here; purchase the full text at http://bjps.oxfordjournals.org/cgi/reprint/30/2/165]
It is gratifying to find Drs Whitrow and Popper re-opening the debate on whether it is philosophically tenable to hold that our universe is temporally infinite in the past. Even a little research in this field reveals that the question is anything but settled, and the issues involved are so gripping that they merit further discussion.
I HISTORY OF THE PROBLEM
Whitrow's necessarily abbreviated history of the argument against infinite temporal regression correctly traces its roots to John Philoponus (d. 580?); but it omits what is undoubtedly the most significant phase of the argument's history, namely its formulation and development by the medieval Arabic and Jewish philosophers. Anxious as they were to vindicate contra Greek philosophical thought the Biblical and Qur'ānic doctrine of creation, these thinkers employed Philoponus's arguments in numerous forms. The most significant Arabic proponents of the argument were perhaps the philosopher al-Kindī (801-873) and theologian al-Ghāzālī (1058-1111); of the Jewish thinkers one would probably name Saadia ben Joseph (882-942) as the most important. These thinkers employed basically two sorts of arguments to demonstrate temporal finiture of the universe. First, the argument from the impossiblity of the existence of an actual infinite:
An actual infinite cannot exist.
An infinite temporal regress of events is an actual infinite.
∴ An infinite temporal regress of events cannot exist.
Second, the argument from the impossibility of completing an actual infinite by successive addition:
An actual infinite cannot be completed by successive addition.
The temporal series of past events has been completed by successive addition.
∴ The temporal series of past events cannot be an actual infinite.
These arguments became the centre of hotly disputed controversy in Arabic, Jewish, and finally Christian medieval thought. The second form of the argument ultimately became immortalised, as Whitrow notes, in the thesis of Kant's first antinomy concerning time.
2 KANT'S THESIS
Whitrow has contended that the argument presented in Kant's thesis is basically sound and has offered his own version of the proof. But since Whitrow's own version appears to me to be somewhat less satisfactory than Kant's and since it is only the latter that Popper directly criticises, I shall confine my comments to the argument contained in the antinomy itself.
It is noteworthy that Popper summarises Kant's thesis as arguing that 'an actual infinity of instants' cannot elapse. One might be tempted to regard this as a minor slip, except that Popper goes on to clarify that Kant's argument can only therefore establish that 'time must have a beginning, rather than the world must have a beginning in time'. Now this is simply faulty historical exposition. A close examination of the antinomy makes it evident that Kant is not arguing for a beginning of time itself, but for a beginning of the universe in time. Kant here conceives of time in an absolute, Newtonian sense as a continuum in which events exist, but which itself exists independently of events. This is even more evident in the antithesis, where he speaks of time's still existing prior to the beginning of the world. Kant's argument concerns, not instants of time, but states of things, and Kant's view of time suggests that in the thesis he is speaking of temporal states as determined by the series of events. The states of time are differentiated by the events occurring in time, and thus prior to the first event would exist an undifferentiated temporal state. Kant's thesis attempts to prove that the present event/state could never arise if it were preceded by an infinite number of events/states.
Popper's second clarification of the thesis is equally mistaken, for he asserts that 'Obviously, Kant's argument rests on the assumption that there cannot exist . . . a "completed" or "actual" infinity; an assumption he shares with Aristotle.' This assertion accurately describes the first sort of argument employed by the Arabic and Jewish thinkers, but it has nothing to do with the second, which is the version adopted by Kant. The second argument does not deny the possibility of the existence of an actual infinite; rather it denies that an actual infinite can be formed or completed by successive addition. The principle common to each thesis in Kant's antinomies is that the conditioned presupposes all its conditions, and the common pattern of the argument is:
The conditioned can only arise when its conditions are complete.
If they are infinite they can never be complete.
∴ They cannot be infinite.
The problem in Kant's thesis is how an actual infinite can be formed, not just whether it can exist. That is why there is no antinomy about future events, since this series can never be completed nor is it the condition of the present event.
3 POPPER'S CRITIQUE
Popper responds that if the world has existed forever, then the set of all past events is actually infinite, but the series of past events is potentially infinite. This may be seen by numbering the events from any moment backwards in time, thus forming a potential infinite:
Past <— - - - - - - - - - - - - - - - Any given moment
..., 5, 4, 3, 2, 1, 0
This, the problem of an actual infinite being formed by successive addition does not arise. This problem arises only if we illicitly assume a point infinitely distant in the past and then ask how we arrived at the present moment.
This response, which has been tirelessly repeated by critics of Kant's thesis, appears to me to be fundamentally in error. Regarding the series of past events as a potential infinite is possible only by beginning at any given moment and mentally regressing into the past. But the series of events itself is really progressing in time, in the sense that it increases with each new event that happens. Thus, the real series of events, if infinite, would be an actual infinite completed by successive addition:
Past - - - - - - - - - - - - - - —> Any given moment
..., -5, -4, -3, -2, -1, 0
Popper confuses the mental regress of counting with the real progress of the series itself. Numbering the series from the present backwards only shows that if there are an infinite number of past events, then we can denumerate an infinite number of past events. But the problem is, how can this infinite collection of events come to be formed by successive addition? Kant does not assume an infinitely distant beginning. On the contrary, the very fact that the series has an end but no beginning is what makes it so inconceivable. If the series really had a beginning, but no end, then we could rightly conceive of it as a potential infinite, as we do the future:
Any given moment - - - - - - - - - - - —> Future
0, 1, 2, 3, 4, 5, ...
Such a series is always finite, but always increasing; it is an indefinite series. But the past may not be so regarded, for it is ex hypothesi infinite and not increasing in a backwards direction. To be a potential infinite, the series of past events would have to be finite, but always growing backwards. But by mentally beginning at the present and regressing in time, is the real series itself somehow transformed from a series with no beginning but an end into a backward-growing series with a beginning but no end? How we mentally conceive the series does not in any way affect the ontological character of the series itself as one having no beginning but an end, or in other words, as an actual infinite formed by successive addition.
Kant's argument, therefore, emerges unscathed from Popper's criticism. If there were an infinite number of events prior to the present event, then the present event could never arrive, since an actual infinite canoot be formed by successive addition. Therefore, the series of past events must be finite; in other words, a finite time ago, the universe began to exist.
4 CONCLUDING REMARKS
Kant's argument that the series of past events/states must be finite appears to be sound. This purely philosophical argument receives remarkable confirmation from current astrophysical theories concerning the expansion and thermodynamic properties of the universe. Contrary to Popper's allegation, these not only ensure the existence of a universal time reckoned from the origin of the universe several billion years ago until now, but also of physical and causal continuity of the series of past events back to the point of singularity and absolute origin.
WILLIAM LANE CRAIG
Universität München