Mathematical Proof that God Does Not Exist?

I was recently scanning through my copy of The Cambridge Companion to Atheism and came across what purports to be a proof that an omniscient being cannot possibly exist. Since God is traditionally considered to be an omniscient being, this essentially amounts to a proof that God does not exist.

Following along with the text, I will proceed with a reductio proof that omniscient beings cannot logically exist. Assume that there is an omniscient being. Such a being would, by definition, know the set of all truths. Therefore, there must exist a set of all truths; call this T. Now, consider the power set of T (the set of all subsets of T); call this PT. By Cantor's theorem, PT will always contain "more" members than are in T (technically, the cardinality of PT will always be greater than T.) But for each member x of PT, there is at least one propositional truth; namely, that x is in PT. So, there must be some truths which are not in T. Therefore, T cannot be the set of all truths. This is a contradiction -- and we must conclude that the set of all truths does not exist. This implies that omniscient beings cannot logically exist.

Christians Can Change Their Minds on Homosexuality?

My mother recently sent me a Huff Post article on Facebook entitled Christians Can Change Their Minds on Homosexuality. Of course, this is a topic I've blogged on recently here and here. Nonetheless, I decided to write a response. What follows is an edited version of my response to that article:

Virginia Tech/Liberty U Debate on the Existence of God

Objective Morality without God?

What God Would Fancy

A number of Christians (or otherwise religious people) are astounded at how well designed our universe appears to be for human life; that God must rather like us for making a universe like ours. Well, I don't think this is right at all; as far as I can tell, if God exists, He rather fancies diffuse clouds of hydrogen gas.

Edit: Apparently, I'm not alone in this thought. Philosopher Brad Weslake of the University of Rochester sent me this quote from J.B.S. Heldane: "The Creator would appear as endowed with a passion for stars, on the one hand, and for beetles on the other, for the simple reason that there are nearly 300,000 species of beetle known, and perhaps more, as compared with somewhat less than 9,000 species of birds and a little over 10,000 species of mammals. Beetles are actually more numerous than the species of any other insect order. That kind of thing is characteristic of nature..."

"In God We Teach" Documentary

I recently came across a documentary entitled "In God We Teach". It's absolutely fantastic and well-balanced, though I would have liked to have seen a little more information about the legal issues involved. The documentary covers the events at a high school in Kearny, NJ, involving an Establishment Clause lawsuit. The documentary is free for non-commercial use and is available on YouTube. For those who have followed a number of these kinds of cases in the past, the events will probably seem all too familiar (it's strikingly similar to what we've seen in Ahlquist v Cranston, Damon Fowler's situation last year, and the on-going litigation concerning Giles County, Virginia.) Take a look:

                              

You can access the website for the documentary here.

A Polemic Against The Wager

We are told that we should worship God because if we do not, we risk burning in Hell. This is something called Pascal's Wager; that we should bet on God existing, because the potential consequences of not believing are so large.

This fails in a number of ways.

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.

A Basic Guide To Converting Me

Edit: Here's precisely how to be unsuccessful at converting me (note: the messages at the bottom of the video were not created by me):

 It has been an interesting intellectual exercise for me to try to posit what would convince me that God exists. That is to say, is there a hypothetical situation in which I would be compelled to believe that there is, in fact, a god or gods? Of course, I am not presently a religious person, so, as one would surmise, all attempts to provide reasons why one should believe have failed to convince me (if they had succeeded, I would be religious.)

What I would like to address here is two-fold: first, why do religious conversion attempts (specifically those made by Evangelical Christians) fail to convince me of anything and, second, what would convince me that a god (or gods) exist.

On Purpose and Meaning

A few people have asked me recently what I think the purpose or meaning of life is as an atheist. They would be correct to say that atheism itself cannot inform me about the meaning or purpose of life, since atheism is defined as merely being a lack of belief in any gods (it should be pointed out that mere theism -- that is, the mere belief in a god or gods -- cannot inform someone about the meaning or purpose of life either. Only particular theistic doctrines or philosophies can inform on that level.) Nonetheless, I do have an answer that I believe in with regards to this question, and I would happily share it.

What does God need with your money?

At the end of Star Trek 5, Captain Kirk and his crew meet an entity who claims to be God. The entity asks Kirk for his starship, to which Kirk replies, "What does God need with a starship?" I just finished watching the 1972 documentary Marjoe, and it has me wondering, "What does God need with your money?"

The answer to both questions? Absolutely nothing.

Click here to take a look.

Doubt & Scientific Certainty

 "Doubt is not a pleasant condition, but certainty is absurd." -- Voltaire (François-Marie Arouet)

My friend Jackie asked me today to explain why I said that I was 99.999% certain there was no God, but not 100% certain. I told her that, as a good scientist, I don't think that we can ever be 100% certain about anything, to which she replied that this doesn't make sense. I sympathise with the fact that she didn't quite understand what I was saying. I think this is a counter intuitive concept for many people and I often see students confused by this idea. Therefore, I thought that I would provide an explanation here as to why I don't think we can ever be 100% certain about almost anything.

On the Cosmological Argument

There is a certain deductive argument for the existence of God that is particularly more popular than others. Unlike other purely deductive arguments for God's existence, the Cosmological Argument is one which is readily used by the laity, albeit not under that name. Since the time of Saint Thomas Aquinas, it has been an established part of Catholic doctrine. In fact, there is a certain Catholic tradition which regards the existence of God as something which is deductively provable without reference to the Bible, church authorities, or faith. Instead, from mere recourse to logic alone, the doctrine states that we should be able to deduce God's existence. Several such  deductive arguments were proposed by Augustine, Aquinas, Anselm, and other scholars from the medieval period, often drawing from earlier work (most notably Aristotle and Plato.) Later, inductive and/or abductive arguments joined the fray, including William Paley's 19th century divine watchmaker argument (which would lead to its modern incarnation as Intelligent Design.)