Wednesday, July 28, 2010

Productive dialog without honest dialog

The Maverick Philosopher is at it again, this time pondering whether "whether any productive dialog with atheists is possible" in his quest for all that is true (meaning Conservative and Christian, naturally). Oddly, my first thought was that a productive dialog would be much easier if he didn't delete every comment he didn't like on from his blog, despite their being direct and polite, but we can't expect too much from Dr. Vallicella, I suppose.

The Maverick Philosopher reintroduces many of the same misunderstandings as in his Teapot post, to which I posted a reply. As he is a reasonably intelligent man, it is hard to attribute his misunderstandings to something other than a self-induced blindness. Another great example is his post on the liberals playing the race card, where both he and the article of Thomas Sowell's to which he links manage to over look that the two biggest race cards played recently were by Mark Williams and Andrew Brietbart. I guess the race card only offends Dr. Vallicella when liberals play it. So, I'll address the usual misunderstandings of Dawkins point below the fold.

"Some of Us Just Go One God Further"

I've seen this quotation attributed to Richard Dawkins. From what I have read of him, it seems like something he would say. The idea, I take it, is that all gods are on a par, and so, given that everyone is an atheist with respect to some gods, one may as well make a clean sweep and be an atheist with respect to all gods. You don't believe in Zeus or in a celestial teapot. Then why do you believe in the God of Isaac, Abraham, and Jacob?

There is one way in which all gods are on a par: there is no reliable evidence for any of them. However, the quote here is not a statement of some necessary ontological status regarding gods. It is a challenge to apply consistent standards of evidence. Given that you reject personal testimonies for Zeus and Anansi, that you find their histories insufficient to believe in them, why do you accept evidence for Yahoweh that is not better in any non-subjective fashion?

What Dawkins and the gang seem to be assuming is that the following questions are either senseless or not to be taken seriously: 'Is the Judeo-Christian god the true God?' 'Is any particular god the true God' 'Is any particular conception of deity adequate to the divine reality?'

I can see no reason to say these questions are senseless or trivial to Dawkins. If they were, he would not have devoted a considerable amount of time and energy to writing a book on them. Dawkins was already a popular author of books on evolution, and his aggressive stance in favor of atheism has cost him some of that popularity. From all indications, Dawkins sees this cost as well-invested, because the issue is both capable of being intelligently discussed and important to address.

The idea, then, is that all candidates for deity are in the same logical boat. Nothing could be divine. Since all theistic religions are false, there is no live question as to which such religion is true. It is not as if there is a divine reality and that some religions are more adequate to it than others. One could not say, for example, that Judaism is somewhat adequate to the divine reality, Christianity more adequate, and Buddhism not at all adequate. There just is no divine reality. There is nothing of a spiritual nature beyond the human horizon. There is no Mind beyond finite mind. Man is the measure.

Well, it does seem trivial that if there is no divine reality, you can't be closer to it with one model of it than another model. However, I have not read anything by Dawkins that says "Nothing could be divine", rather, 'Nothing is divine' seems to be a much better summary of his position. There is no ruling out of any possible type of supernatural, only noting that there is no reliable evidence in favor of any sort of supernatural entity. I have never read a claim he can disprove the existence of an infinite mind, merely that there is no reliable evidence for such a mind.

Of course, this does not rule out the ability to disprove the existence of a particular model of the supernatural, many of which fall apart simply because they are bronze-age creations of people who did not possess what we would recognize as a consistent philosophy, which are then shoe-horned into our modern thoughts. For example, the notion of an omnibenevolent being who engages in eternal conscious torture for temporally limited offenses is inherently self-contradictory. You don't need to rule out the existence of all possible gods to rule out the existence of that particular god.

That is the atheist's deepest conviction. It seems so obvious to him that he cannot begin to genuinely doubt it, nor can he understand how anyone could genuinely believe the opposite. But why assume that there is nothing beyond the human horizon?

In my case, I assume there is plenty beyond the human horizon. There are galaxies we have not even seen yet, ideas about the beginning and possible end of time itself, infinities of space, and whole manners of natural phenomena that exceed our horizon. I accept them because we have evidence that they do or at least can exist.

My return question: why assume there is something out there that intends to be found but fails to leave any reliable evidence pointing towards it?

The issue dividing theists and atheists can perhaps be put in terms of Jamesian 'live options':

EITHER: Some form of theism (hitherto undeveloped perhaps or only partially developed) is not only logically and epistemically possible, but also an 'existential' possibility, a live option;

OR: No form of theism is an existential possibility, a live option.

That's easy: theism is absolutely a live option for the majority of atheists. Many of us find it the preferred option. However, the universe does not run itself based upon our preferences. Theism is a live option, but it is not an evidenced option. You can't just wish gods into existence.

Theist-atheist dialog is made difficult by a certain asymmetry: whereas a sophisticated living faith involves a certain amount of purifying doubt, together with a groping beyond images and pat conceptualizations toward a transcendent reality, one misses any corresponding doubt or tentativeness on the part of sophisticated atheists. Dawkins and Co. seem so cocksure of their position. For them, theism is not a live option or existential possibility. This is obvious from their mocking comparisons of God to a celestial teapot, flying spaghetti monster, and the like.

So, are we to equate the careful considerations of The Maverick Philosopher with the bold declarations of Dawkins as playing the same role in the social movements dedicated to their respective views of the supernatural? No, I don't think so. Dawkins is not The Maverick Philosopher for atheists. To the degree that we would have leaders at all, he is the James Dobson, the Malcolm X, or the Deepak Chopra: a public persona pushing an agenda. I'm sure Dr. Vallicella knows better than I who the serious atheistic philosophers are.

For sophisticated theists, however, atheism is a live option. The existence of this asymmetry makes one wonder whether any productive dialog with atheists is possible.

Well, I don't recall there being a lot of productive dialog with the James Dobson's of the world, either. Perhaps if you seek dialog, it should be with someone seeking to dialog. Dawkins is an advocate.

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Tuesday, July 27, 2010

Putting the Grim Philosopher argument to sleep

Now that we understand just what sort of thing a maximal collection of consistent truths must be, we can get around to discussing the attempt to form a contradiction based upon its possible existence, which argument The Maverick Philosopher (Dr. Vallicella) seems determined to support.

In fact, The Maverick Philosopher is so intent on preserving this argument that my comments pointing out a summarized version of my last post are no longer present. That is, I posted the first comment on 7/14, it was gone by 7/15, I posted another comment on 7/18 in the morning, I checked back on 7/18 in the afternoon to see it there, it was gone on 7/19, I sent an email on 7/19 asking why my comment was removed, and so far I have received no response to that email. I should not be surprised The Maverick Philosopher prefers to deal the disproof of his argument by covering up the evidence. After all, this is a man who, despite the presence of a dictionary definition that specifies "A strong fear, dislike, or aversion" as the definition of 'phobia', insists that "A phobia is an irrational fear. If you use the word in any other way you are misusing it. " Apparently avoiding the inconvenient is one of his preferred methods of dealing with reality. So, I'll have to produce the reality regarding his proof on my blog below the fold, where he can avoid it by not coming here at all. For someone who shakes in his boots at the idea of Muslims and thinks it rational, discretion is undoubtedly the better part of valor.

Addressing the 4 points in order:
1. Cantor's Theorem states that for any set S, the cardinality of the power set P(S) of S > the cardinality of S. ... But the proof needn't concern us. It is available in any standard book on set theory.

Quite accurate, assuming the power set can be formed. That's part of the issue with this proof. When you get to a certain size, a collection goes from being a set to a proper class. Depending on the particular set theory, this means either that, for some proper class V, either P(V) cannot be constructed (NBG and ZFC) or it cannot be larger than V (MK). Either way, Cantor's Theorem does not apply.

2. Suppose there is a set T of all truths, ... But according to Cantor's Theorem, the power set of T is strictly larger than T. So there will be more of those truths than there are truths in T. It follows that T cannot be the set of all truths.

As I demonstrated in the last post, the size of T is the size of the ordinal numbers, which is a proper class. So, there cannot be a greater cardinality in the class of all subclasses of T, even if such a subclass were to exist.

3. Given that there cannot be a set of all truths, the actual world cannot be the set of all truths. This implies that possible worlds cannot be maximally consistent sets of propositions. I learned the Cantorian argument that there is no set of all truths from Patrick Grim. I don't know whether he applies it to the question whether worlds are sets.

Since we have no disproof of the notion that there can be a class of all truths (such a class would exist under any typical rendition of set theory), it turns out that worlds can indeed be maximal collections of consistent truths.

4. As far as I can see, the fact that possible worlds cannot be maximally consistent sets does not prevent them from being maximally consistent conjunctive propositions.

This position requires at least some reason to differentiate between the category of consistent sets and the categroy of consistent conjunctive proposition, in that there are member of one which are not naturally a part of the other. However, there is an obvious and natural one-to-one relationship between the category of consistent sets and the category of consistent conjunctions. Namely, {pi}i ε I <---> &i ε I(pi). In fact, if you define a partial order on the category of consistent conjuction by sayin that &i ε I(pi) <= &j ε J(pj) whenever, for each i ε I, pi ε {pj}j ε J, this relationship even preserves the partial order of the set category. This means any maximal consisent conjunction would map to a maximal collection of consistent propositions. So, if you say the maximal collections of sets does not exist, perforce the maximal consistent conjuction does not exist, either.

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Thursday, July 22, 2010

A primer on transfinitie recursion

I am going to discuss a transfinite recursion in this post, as a prelude to filling in the argument from last July pointing out that the attempted argument by Dr. Vallicella against the possible existence of a maximal collection of truths, to which I wrote a response last July. Today, that response seems inadequate, so I intend to do the proof more comprehensively. That way, in the future I can just link to the new post whenever Dr. Vallicella decides to raise this argument again, as he did last week.

A good background in ordinal numbers generally will be very helpful for understanding this post. I will not attempt to improve on the three-part series penned by Mark Chu-Carroll over at Good Math, Bad Math. For better or worse, I'll assume the reader understands that material and go from there below the fold.

Transfinite recursion is the process of defining a function over all of the ordinals. Recursive definitions are fairly common at the level of the finite. For example, the factorial function (identified by a bang, for example, 3!) comes up in pretty much any college level math class that mentions probability. Factorials can be defined recursively. You start off saying 0! = 1. Then, for any integer n > 0, n! = n * (n-1)!. Thus,
1! = 1 * 0! = 1 * 1 = 1.
2! = 2 * 1! = 2 * 1 = 2.
3! = 3 * 2! = 3 * 2 = 6.
4! = 4 * 3! = 4 * 6 = 24.
5! = 5 * 4! = 5 * 24 = 120.
This is not the only way to define n!, but it is a useful way. This definition consisted of two steps: the definition for 0, and the definition for any number given the value for the immediately previous number.

With transfinite recursion, the same basic idea applies: for any particular ordinal ω, you use the definitions of some or all the ordinals β where β < ω to define the function on ω. If you can do that as a single definition, so much the better, but often these definitions consist of three steps: define the function for the first ordinal (0), define the function for each ordinal that has a direct predecessor (the are called successor ordinals), and define the function for ordinals that have predecessors, but no direct predecessor (these are called limit ordinals). As you might have noticed, the first two steps are the same as we saw for the definition of factorial. I will be using transfinite recursion to start with a single truth t, and generate from that a maximal collection of consistent truths T that derive from t. "Maximal" means: given any collection Tx of truths derived from t, Tx is a subset of T.

The first step of the transfinite recursion is simple: T0 = {t}. T0 is very obviously derived from t.

Edit: In my discussion in the next paragraph, I glected to account for the fact that Tβ+1 will contain every truth in Tβ even without being unioned. This is corrected.

The next step will be to define Tβ+1 given some Tβ derived from t. Of course, I will use the definition offered by Dr. Vallicella in his proof, otherwise there would be no point to this. Tβ will consist of truths, {tβ,1, . . . , tβ,i, tβ,i+1, . . ., tβ,ω} that derive from t. Consider the power set P(Tβ) of Tβ. The truth tβ,1 in Tβ will be a member of some of Tβ's subsets but not of others. Thus, tβ,1 ε {tβ,1, tβ,2}, and tβ,1 ~ε { } are both truths. In general, for each subset s in the power set P(Tβ) there will be a truth of the form tβ,1 ε s or tβ,1 ~ε s. You can denote whichever of these statements is true by q(tβ,1,s). We then define Tβ+1 = {q(tβ,i,s) | tβ,i ε Tβ and s ε P(Tβ)} union {t}, and say that Tβ+1 is also derived from t. This is the process Dr. Vallicella uses to generate a larger set. Note that since every statement of Tβ is repeated in Tβ+1, Tβ is a subset of Tβ+1.

Before we move on, lets look at T1 and T2. T0 has two subsets, {} and {t}, and one element, {t}. t ε {t} (we can call this truth t1,1) and t ~ε {} (we can call this truth t1,2). Then, T1 = {t, t1,1, t1,2}. T1 has eight subsets: {}, {t}, {t1,1}, {t1,2}, {t, t1,1}, {t, t1,2}, {t1,1, t1,2}, and {t, t1,1, t1,2}. With three elements of T1 and eight subsets of T1, we can generate 24 true statements of the form t2,j, 1<= j <= 24, two of which were in T1 (left as an exercise for the interested reader). T2 will consist of these 24 statements, and t, for twenty-five truths. In case you are curious, there are 25*2^25+25 (i.e. 838,860,825) statements in T3.

The third step is simple. For any limit ordinal ω, Tω is the union of all the Tβ where β < ω. Since all the statements of each Tβ are derived from t by the process in the previous paragraph, every element of Tω is derived from t.

The (proper) class of all of all ordinals is commonly called Ω. So, for each ω ε Ω, we have defined Tω. T is the union of all the Tω over all the ω ε Ω. T is now the maximal collection of truths derived from t. Since for any ordinal ω, #(Tω) >= #(ω) (that is, Tω at least as many elements as ω), T has more elements than any ordinal. Therefore, T must be a proper class. I'll discuss what this means in my next post.

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Wednesday, July 21, 2010

At the old ball game

Recently, Son#1 went to see the Cardinals, by himself. His school's (marching?) band was scheduled to play there, and while he is not in that particular band, he wanted to be there to support them. Unfortunately we couldn't buy tickets through the band teacher due to pressing financial issues at that time. So, Son#1 went downtown to get his own tickets ahead of time. Needless to say, this opened up a flood of conflicting emotions in CharityBrow and me.

First was pride. Son#1 had never been further than a couple of miles from home on his own (he takes a bus to his job at a local department store). This time he took a bus to the MetroLink and then took the MetroLink to the stadium, some 15 or miles away. He never hesitated, showed no fear, and took everything in stride. He called regularly so CharityBrow and I would not worry. In fact, when it turned out the box office was closed on his first visit (a couple of days before the game, the Cardinals were out of town), he didn't get upset. He just turned around and came home. When he got home, he asked me what time he should leave on game day, and we went over the schedule together. On game day, he left on time, bought his ticket, found his seat, watched the game, and came home by a different route than the one he used to get to the game (due to the late-night buses running slightly differently). Ten years ago I had no idea he would ever be able to do these things. Heck, eleven years ago he was still in diapers at age seven.

Of course, there was a lot of fear on my part, as well. CharityBrow and I have always seen Son#1 as an easy target for predators. He trusts pretty much everyone he meets and thinks of them as his friend, at least until they do something mean. He'll hug girls he's never met, always with a big smile. He once left a bicycle unlocked across a park at night, because he never thought it might get stolen. So, we worried about someone offering him a ride and us never seeing him again.

For Son#1, I'm not sure it was even a big deal. Going to the game was a big deal, but the fact he did it on his own, not so much. In that way he's a lot like me. I started taking BiState (now MetroBus) to school in the fifth grade, and never though much of it. Wandering was always part of the fun. I would worry my parents greatly from time to time because of that. In that aspect, I suppose the parent's curse holds true: he is doing to me what I did to them. Should I feel good about that?

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Tuesday, July 20, 2010

Exploring the urban prairie

I work in East St. Louis, and on occasion move around the city from time to time either as a part of my job, or just to get a bite to eat. So, I get to see a fair share of urban prairie. As an urbanite, I find it rather depressing. I'll say a little more about it below the fold.

There is an interesting variety to the prairie. Sometimes the grass is so thick and high you'd never know anything had been built there at all. For example, along 13th between Exchange and Lynch there are multi-lot sized sections of meadow, which currently feature some wild, pretty, blue flowers (as of yesterday, anyhow). Other times, you can still see large patches of asphalt or concrete that have not yet successfully been buried, such as along State street between 81st and 85th. Presumably, much of this land belongs to the city. So, not only does the city need to pay for whatever upkeep there is on the property (the grass does need to be cut every month or so, at least along State Street), but it brings in no revenue. As more and more land becomes deserted, it becomes harder to support basic services for the remaining residents.

I wonder how long this can continue. Unlike East Coast cities, we still don't have land shortages here in the Midwest. So, O'Fallon, Shiloh, Troy, etc. are all growing, talking about getting new interstate exits, etc., while land much closer to St. Louis languishes. Eventually, the USA will have enough population pressure that this land will be used again. I wonder how long that will take.

Some of the land is apparently going to community gardens. In the next 50 years, we'll see full-scale farms? Can the latest 10 million help reverse the trend? I wonder if I will se a turnaround in my life time (based on family history, another 40 years or so).

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