Date: Sun, 27 Feb 2000 14:11:36

Subject: An Inconsistent Triad

There are three propositions that many Objectivists believe -- and that Rand herself almost certainly believed -- that form an inconsistent triad. That is, they form a set that has the property that any two are consistent, but the addition of the third makes the set contradictory.

The first rarely gets stated, because it is taken for granted by almost everybody. It may well be that it is so thoroughly taken for granted, that it is not even recognized as a belief to which there might be some alternative. Objectivists take it for granted, but are hardly ever pushed to defend it, because the people they engage in discussion also take it for granted.

It is the assumption that time is non-cyclical. There is a
definite future and past, so it is not the case that, say, a future
event is the very same event as some past event. In fact, if time
were cyclical, every event would be in both the future and past of
every other (and even in its *own* future and past). There could
be, say, three events, A, B and C, with A preceding B, B preceding C
and C preceding A. (This is not to say that the cycle would *repeat*.
Rather, the very same event, occurring only once, would both precede
and follow itself.)

The second proposition is that every event has causal conditions.
What that means is that for every event, there are some prior
conditions such that, had they not occurred or been present, then the
event in question either would not have occurred or else would have
been different.^{1} Now, that's a weaker claim than that
every event has a cause, in the sense of *sufficient* conditions
for its occurrence, but nonetheless, if the latter is true (as I
think some Objectivists have unwisely maintained), it entails the
former, that there are causal conditions for every event.

The third proposition is that there are no actual infinities. This means that there are no sets or groups or collections with infinitely many actual or existing members and also that there are no properties of anything that actually exists that are present to an infinite degree. Infinity is, according to Objectivists, a concept of method, very useful for certain purposes, but nowhere actually instantiated. There may be potential infinities, such as finite sets that admit the addition of indefinitely many further members or indefinitely divisible magnitudes, but what is concretely real is always finite. (At a given time, a finite number of members has been added or a finite number of divisions has been performed.)

These three propositions, though they are all believed by many Objectivists, form an inconsistent triad. There is no way that all three can be true. Demonstrating this is simple. Take any two of them as premises and you can prove that the third is false.

If time is non-cyclical and every event has some causal condition,
then there must be an infinite series of events extending into the
past. That is, for every event, there will be at least one before it
to be or to be part of its causal condition. And, since time is being
assumed to be non-cyclical, the earlier event will itself require
another as a causal condition which has not already been accounted
for. To put it the other way around, if you suppose time to be
non-cyclical and every event to require a causal condition, you can
prove that any particular finite number suggested cannot be equal to
the total number of events; there will have to be at least one more
than any finite number to provide a causal condition for the earliest
member or members of the set of events. Since that argument works for
each finite number, the total number of events must not be finite.^{2}

If the non-cyclicity of time and the non-existence of actual infinities are both true, then there must be finitely many events in the past. But if there are finitely many events in the past, then at least one, the first (or more than one tied for first), must have occurred without causal conditions. (The existence of a first -- or a tie for first -- is guaranteed by the non-cyclicity of time.)

If the causal conditioning of every event and the non-existence of actual infinities are both true, then time must be cyclical, so that every event can have a causal condition without there being an infinite set of causal conditions.

For my money, the one I'd be most reluctant to give up is the non-cyclicity of time. I don't have any rock-ribbed proof that time is not cyclical. Still, accepting its cyclicity would upset a great deal else that I take to be well-founded. For example, we typically take events in the past to be settled and (at least some) events in the future not to be settled yet. It is settled (say) that I had coffee this morning, but not yet settled whether I will have coffee tomorrow morning. But if time is cyclical, then either the future is just as settled as the past or else the past is just as unsettled as the future.

If we're going to hang onto non-cyclicity, we have to give up at least one of the others. As far as I can see, no very strong argument has been offered either for the denial of actual infinities or for the causal conditioning of every event. It seems to me that neither of those claims is a necessary truth, so, if either of them is true, it is not because it couldn't have been false.

If asked to guess, I think it's somewhat more likely that not every event has a causal condition than that there are actual infinities, but I don't see any reason they couldn't both be true, i.e., that there are some events (at least one) that do not have causal conditions and that there are also actual infinities.

Rob Bass

rhbass@gmail.com

http://oocities.com/amosapient

===

In private e-mail, somebody asked me about Aristotle's objections to actual infinities and about the way that they were answered by mathematicians.

I'm not sure I can nail down which were Aristotle's objections and which came later without doing some research and text-checking. Anyhow, without attributing them to Aristotle, here are several that have been used at one time or other:

All alleged cases of actual infinities turn out, upon anlysis, to really be only potential infinities -- a series that may be added to or a magnitude that can be sub-divided indefinitely -- but that, at any given time, has only finitely many members or finitely many subdivisions. Aristotle could have used this as part of an argument that there can't be actual infinities because of a modal principle he accepted, that whatever is possible is (somewhere, somewhen) actual and, conversely, that what is not ever actual is also not possible. That kind of argument isn't available now, even if we had a convincing analysis of all putative cases of actual infinities, because nobody anymore seriously thinks Aristotle was right about the modal principle. (There may never be a solid gold mountain, but that doesn't make solid gold mountains impossible.)

Taking away one from an infinite set has to leave an infinite set -- otherwise, it wouldn't be infinite to begin with. But taking away one from an odd-numbered set leaves an even number, while taking one from an even-numbered set leaves an odd number. So infinity must be both even and odd (since you've got infinity before and after the subtraction of a single member).

There are 24 times as many hours as there are days, but in an infinite set of days, there are exactly as many days as hours.

All the objections I've seen to the possibility of actual infinities fall apart if you, first, bear in mind the obvious point that infinity is not just a (very large) finite number -- i.e., don't assume that infinity has properties, like being even or odd, that are well-defined only for finite numbers, second, don't confuse the counter-intuitive with the impossible, and third, remember Cantor's definition of an infinite set (roughly, a set such that it is possible to place a proper subset of itself in one-to-one correspondence with its superset).

None of that, of course, shows that there really are actual
infinities. But the arguments of Cantor and others showed, to my
satisfaction, anyhow, that none of the traditional objections to
actual infinities holds water. The concept, at least, is coherent.
Whether there are any examples of actual infinities is a different
question. (In my post, I pointed out that *if* two other
premises that many Objectivists accept are true, *then* there
must be an actual infinity of events extending into the past.)

===

Also, Richard Lawrence has available on-line an
(unimpressive) attempt to rebut my argument. Since I found it, I
composed a *Reply to Lawrence*.

1. Further, the causal conditions for an event must themselves
include at least one event. For suppose that the causal conditions
need not include any event. That is, suppose that there is some
event, *E*, and that, before the occurrence of *E*, there
is a complete set of its causal conditions, *C*, which includes
no event. How then does *E* arise from *C*? Isn't *that*
an event -- *C* giving rise to *E* -- that is distinct
from both *C* and *E*, but, contrary to the supposition, a
causal condition of *E*?

2. This needs a minor qualification. There might be a time that
is, as a whole, non-cyclical but that includes a cycle at its
beginning -- in which A causes B, B causes C, C causes A *and*
D, D causes E, and so on. Like so:

To avoid this complication, what would have to be added is that events may be compound -- one can be composed of two or more others, as two different drops of water falling may be part of the same rain-storm -- and that any finite set of events counts as a compound event, and therefore stands in need of some causal condition that is not a part of that finite set.