Saturday, July 24, 2010

Gwiazda on Swinburne: The Principle P

This post is part of my series on Jeremy Gwiazda's criticisms of Richard Swinburne's argument for the existence of God. For the essential introduction and background to Swinburne's argument, see here.

To briefly recap, Swinburne claims that the probability of God (h) given certain empirical evidence (e) and background tautological evidence (k) is high. Or, at least, higher than any of the alternatives. This argument makes use of Bayes' theorem:

  • Pr(h|e&k) = Pr(e|h&k) Pr(h|k) / Pr(e|k)

We have seen in previous entries that one of the things Swinburne relies upon to support his claim that Pr(h|e&k) is high is the argument that intrinsic probability of theism, or Pr(h|k), is high due to God's simplicity.

To support the argument for God's simplicity, Swinburne appeals to something he calls the Principle P:
Hypotheses attributing infinite values to properties of objects are simpler than ones attributing large finite values.
God's three key properties are his knowledge, freedom and power, and according to orthodox theism these properties are infinite in their values. On principle P, this would make God the simplest possible being.

Principle P does a lot of heavy lifting for Swinburne. It allows him to rule out, as profoundly unlikely, any explanation that appeals to a finite or limited god. This undercuts some of Hume's classic objections to theistic explanations. Given the heavy lifting that P is doing, it is essential to investigate whether or not it is justified. That is what Gwiazda does and that is what we will do in this post.

Four Arguments for P
Gwiazda identifies four justifications of P, sprinkled throughout Swinburne's oeuvre. We will respond to each in sequence.

i. Mathematical Simplicity
The first justification comes from the mathematical understanding of simplicity. Swinburne argues that "a  law is mathematically simpler than another in so far as the latter uses terms defined by the terms used in the former but not vice versa". So, for example, addition is said to be simpler than multiplication because the latter operation depends on the concept of addition.

Applying this to infinity, Swinburne argues that we can understand what the idea of an infinite quantity (or quantity without limit) is without needing to understand the specific large numbers, such as a googolplex, that are part of that infinite quantity.

This, as Gwiazda points out, is a rather silly argument. For we can also understand what a googolplex is without understanding what infinity is. Neither is simpler than the other, even if we accept Swinburne's definition of mathematical simplicity.

ii. Scientific Practice
The second justification of the principle P comes from the history of scientific practice. Swinburne uses one example over-and-over again. It is the example of the speed of light.

Swinburne argues that scientists -- or natural philosophers as they were then known -- originally preferred to posit an infinite velocity for the speed of light. It was only later, when other considerations came into play, that the finite value was arrived at. This, he thinks, is suggestive of the simplicity of an infinite value.

Gwiazda points out that Swinburne is being selective in his treatment of scientific history. Although it is true that figures like Aristotle and Descartes preferred the infinite value for the speed of light, it is equally true that Empedocles, Francis Bacon and Galileo preferred a finite value. Furthermore, it is surely not an insignificant point that those preferring the infinite value turned out to be wrong.

There are a few more points to be made here. First, Swinburne would need a much broader survey of scientific history to justify his claim -- one example won't cut it. Second, Gwiazda thinks that there might be psychological reasons why we prefer infinite values over finite values. For instance, it might be due to an unwillingness to admit to imprecision in our measurements; or it might be due to a primacy-recency effect in which the limits or endpoints of a sequence are remembered with greater felicity than the intermediate points.

I wouldn't overemphasis those psychological observations myself.

iii. Finite Limitations Need Explanation
On occasion, Swinburne appeals to the notion that a finite limitation is begging for explanation, in a way that limitlessness does not. Gwiazda thinks that this would need to be developed into an actual argument if it is to succeed. There is no further detail given in Swinburne's work (according to Gwiazda anyway).

iv. Zero and Infinity are "Neat"
On other occasions, Swinburne appeals to the notion that zero, one and infinity are mathematically neater and simpler quantities than all other finite numbers. The idea here is that 0 and 1 must be understood jointly, because one can't have the concept of zero-ness without also having the concept of one-ness but one could have the concept of one-ness without having the concept of two-ness. Likewise, as discussed earlier, Swinburne thinks that infinity can be understood without needing to understand other finite quantities.

Again, there is little support for this intuition and it could even be developed into an objection to Swinburne's understanding of God. We will do this in a moment.

On the whole, Swinburne's support for P is pretty thin. Next, Gwiazda develops two arguments for the falsity of P.

2. Why P might be false
The first argument against P focuses on the fact that the postulation of infinite values is often a sign of complexity and error in scientific explanations. This is a direct refutation of Swinburne's second argument for P.

Gwiazda uses the example of singularities in cosmology to make his case. A singularity is a point at which some measurement takes on an infinite value. The best-known examples are the points of infinite density found in black holes and at the origin of the universe (Big Bang). Scientists are not particularly happy with these infinite values. Indeed, they usually take them as indications of errors in their present theories.

The second argument against P takes onboard Swinburne's claim about the simplicity or neatness of zero, one and infinity. If Swinburne is right when he makes this claim, it would seem to follow that a God with no freedom (or knowledge or power) would be just as simple as one with infinite freedom. It would also seem to follow that a God with one unit of freedom (or knowledge or power) would be just as simple as an infinite God.

This is surely not something Swinburne would wish to entertain. But if that's the case, he will have to do a lot more work ironing out his definition of simplicity and his arguments in favour of an infinite God.

That's it for this post. In the next entry we will consider Swinburne's understanding of the relationship between God's three key properties (freedom, knowledge and power) and how this understanding further damages his argument for the infinite and simple God. 


  1. While Gwiazda raises excellent points, it does appear to me that "infinity" is simpler under certain circumstances. For example, to the best of our knowledge the spatial structure of the universe is either a huge sphere (i.e the 3d surface of a 4d ball) or an infinite plane (i.e. extending infinitely in 3d and flat in 4d). Assuming no multiverse, I for one would be more comfortable with a flat infinity than with a bound sphere. The infinity seems to comprise less of a mystery than the finite value. "Why is the sphere THIS size" seems to be more puzzling than "Why is space infinite in extent".

    Granted, I can't formulate this thought very well, much less present a cogent argument for it. And I certainly don't think it applies to God, I just think that more clarity on these issues is needed.


  2. Yeah, I agree that infinity could be simpler.

    A guy called Russell Standish released a book a few years back called the Theory of Nothing in which he argued that a infinite universe would contain zero information and so would be the simplest possible entity.

    This is based on Max Tegmark's mathematical universe hypothesis which is essentially a version of modal realism (I think). I haven't gone into this material in much detail so I can't comment on the strength of the arguments.

  3. John,

    what program do you use to create your charts?



  4. I must say that from a purely practical perspective, this idea that infinity is "more simple" than a large finite number is complete and total insanity that should never be uttered with a straight face. As far as I am aware, we have found exactly two instances of infinity in the universe: the infinite density of black holes, which, as you mention, nobody likes due to their discontinuous nature, and the infinite space of the universe itself which is only infinite in a kind of cheating respect (i.e, the same way that the playing field in a game of Asteriods is "infinite").

    Large finite numbers? Well, we've got those all over the place. The mass of the sun is large and finite; the surface area of the Earth is large and finite; the distance to Alpha Centauri or any other point in the universe is large and finite; the average velocity of the Milky Way Galaxy is large and finite.

    Thus, anyone claiming with a straight face that the prior probability of something having an infinite attribute is greater than the prior probability of something having a large but finite attribute is trying to sell you something - and I'm entirely unsurprised to find that the man who came up with the Principle of Credulity is a salesman.