"The trouble with the world is that the stupid are cocksure and the intelligent are full of doubt." -- Bertrand Russell

Friday, February 3, 2012

Playing with Aquinas

In Philosophy of Religion, we've been discussing Aquinas' arguments for the existence of God. After class today, I showed the professor a few objections that I had arrived at which he confessed to not having heard of or even thought about before. I thought that I'd share them here.

Aquinas thought there were only a finite number of events in the past and that, since there are only a finite number of events in the past, there must have been a first event. Of course, from the existence of a first event, he wants to claim that there must have been an uncaused cause -- something non-physical which causally determines the existence of the universe (non-physical because he commits himself to the view that all physical things require causes. Thus, by Aquinas' lights, anything that is non-caused must be non-physical.) He identifies this non-physical thing as God.

This has me asking the following.

Do all events have a finite amount of time between them? If they do, then the universe can have some finite age. If they do not, it's possible to have a set of past events between which there is, at best, an infinitesimal amount of time. If this latter situation occurs, then adding together all past time intervals over a finite amount of events gives us a set of measure zero. In other words, no time would have past at all. There is a pretty good reason to think that the latter is the case. Under a certain view about what causation is, we can imagine Aquinas to be talking about each space-time slice causing the next space-time slice. That is to say, the conditions at t_n are responsible for causing those at t_n+1. Furthermore, that events are just the temporal slices. Objections pertaining to relativity are just complications; we can imagine constructing something like this from the vantage point of any inertial frame that you wish. Thus, if Aquinas commits himself to the view that there are a finite number of events in the past and to a certain view about what an event is, then we can construct a pretty strong reductio against him.

But Aquinas is really trying to argue that there must have been a first cause, not that there must have been a finite number of events in the past. If we take some closed subset of the real line, there will be a first element on the line. Thus, even with an uncountably infinite number of past events, we do not have to commit ourselves to the non-existence of a first cause.

However, this is problematic for the following reason. We could have just as well taken an open interval on the real line and mapped it to events. Explicitly:

Construct some finite open sub-set of the real line. Call this L. Now, map the points along L to points along the time line T representing the continuum of all past events. The successor relation on L corresponds to the causal relation along T. Since each member of L has a successor and there is no first member (by construction), all events on T have causes and there is no first cause.

The Hume-Edwards principle states:

If the existence of every member of a set is explained, the existence of that set is thereby explained. (From Pruss' "The Hume-Edwards Principle and the Cosmological Argument")
Now, if Hume-Edwards is correct -- and I think it is -- then an adequate explanation for the universe can be given by citing the causes for each member of T in terms of some other member of T. Since there is no first cause on T, but each member of T has both a successor and a preceding element on T, an adequate explanation of the universe can be given without citing supernatural causes (provided that the universe envisioned in this thought experiment corresponds to our actual universe.)

Of course, we have no reason to think that there is no actual first event, but what this does adequately show is that a large space of possible defences of Aquinas are insufficient to establish the existence of a first cause.


  1. The third para (starting with "Do") is a bit messy and I'm not sure what the point is. How did we arrive at the final "Thus...". Clearly Aquinas is relying on the fact that his set is finite to find a least element, but what's wrong with this?

  2. Bruce -- You can't just automatically decide that a finite set of events have a finite amount of time that elapses between them. If there is no time between events -- which *is* the case under certain models of what causation is -- then you get a set of measure zero and no time would have elapsed whatsoever.

    But clearly time has elapsed in our universe.

    Therefore, unless Aquinas can prove to us that there must always be a finite amount of time that elapses between events, then Aquinas conclusion does not follow from his premises.

  3. I'm not sure if I understand how you're defining an event. Which of the following (if any) do you object to in a model of events?

    1) Events are a finite ordered sequence of points on the real (time)line.

    2) Distinct events cannot occur at the same time.

    3) Time between events is the difference of their values.

  4. None of those three. And, in fact, my counter argument depends on those three being true.

    If these are something like points on the real line,then we can easily find points for which the distance between them is zero (or, rather, has measure zero.) This is allowed because, in technical language, the real line is both dense and compact. Do you know what these words mean? And do you know what it means for a set to have measure zero?

  5. Yep, I know what those mean. Finite sets are trivially measure zero (in fact, countable sets), but what I think you mean here is that we want all the events to be contained in one connected set with measure zero. Otherwise time would stop and start on us!

    I don't think you can find two distinct points whose distance is 0. If you have a != b, and without loss of generality a < b then b-a > 0. Of course, you can make them arbitrarily close but there will always be a finite distance between them.

  6. I think you could have an infinitesimal distance between them and them be distinct. Especially if you allow me to wander off into non-standard arithmetic.

    But suppose you could not, and you just get a vanishingly small distance between them. My point still stands, and obviously so. You get a universe that has only existed for a small amount of time with an arbitrarily large number of events having transpired.

    By the way, I don't see why events can't be arbitrarily close together in the same sense as points on the real line are arbitrarily close together. If you really want there have only been a finite number of events in the past, you would need to disprove this case (since we can have an infinite number of distinct points on a finite sub-interval of the real line.)

  7. My understanding of your original argument was that
    1) Events might have no time between them.
    2) If so, everything would have happened in no time at all.
    3) Everything has taken some time to happen, contra 2.

    Is this correct? 1 seems crucial here.

    Another point of clarification, we're not saying that every point on our interval is an event right?

  8. Yeah, that's correct... But under your objection, we can just do this:

    1) Events might have vanishingly small times between them.
    2) If so, everything would have happened in a vanishingly small amount of time.
    3) Everything has taken a non-vanishingly small amount of time to happen, contra 2.

    Not every point along the real line need be an event, but each point *could* be an event. That means that the events could be dense in the same sense as the point are.

  9. There are still mathematical problems in your formulation. Unless you mean the words "dense" and "vanishingly" in a strictly colloquial sense, they are inconsistent with mathematical usage. Finite sets cannot be dense.

    Besides these mathematical quibbles, I'm not sure how your argument refutes Aquinas. He's saying there's a finite set of events. He could in principle spell them all out in a giant list. You're saying this leads to absurdity because if it is the case that (1) then (2) which is clearly false. But clearly (1) is false. The giant list that details the events already says when they happened.

  10. Ah, but *I* never agreed that the set of events was finite or countable. Aquinas assumes that, and it's part of what I take issue with.

  11. Besides, it's not clear to me how (1) is false. Events are points along a continuum; what I should have said is that the continuum is dense, and that the space between the points can be arbitrarily small. Furthermore, I see no reason for the events to even be countable, though we might have the intuition that they should be.

    Now, if Aquinas could prove that all past events are finitely many, then it's trivial that the set of past events has an earliest member. This seems to be your point, but it relies on an antecedent that I would take issue with. Aquinas simply assumes -- as you seem to want to do -- that the set of past events is finitely many. My argument is that the set of past events need not be finitely many, countable, or have finite durations between them. All three of these possibilities directly contradict Aquinas.

  12. Aquinas assumes there are finite events and you are attempting to construct a reductio argument against it. To do so you accept his assumption and show that it leads to bad things.

    You're trying to assert that events behave as (1) or constitute a continuum. But what is an event that you're talking about here? Since we're aiming for a reductio, we're operating under his assumptions, namely that events are defined as the elements of his finite list. By looking at the list we can plainly see that (1) is not the case.

    Of course, none of this proves he is right. It merely shows that your argument fails as a reductio to prove him wrong.