For Nadir and Mohd: the Subjectivist Approach to Probability Theory and the Scientific-Hermeneutic Approach to the Qur'an
Denis Giron
This post will focus on a specific topic that has come up in another
thread in SRI, which has been archived by Google here:
http://groups.google.com/groups?threadm=dd10d076.03091...@posting.google.com
The thread, consisting mostly of posts from Nadir Ahmed, Mohd Anisul
Karim, and myself, is on the subject of a debate I had with Nadir last
September with respect to the scientific-hermeneutic approach to the
Qur'an. For more information on the debate, audio of the debate, and a
partial transcript, visit the following website:
http://www.joes.com/home/ahmedgiron/
Nadir and Mohd (as well as various other proponents of the
scientific-hermeneutic approach to the Qur'an) have argued that the
odds of them being able to correlate Qur'anic statements with modern
science, conditional on the assumption that the Qur'an was authored by
a human being, are so small that we have no choice but to conclude
that a divine power was the author of the Qur'an. Of course, we know
this is false by virtue of the fact that we can correlate other texts
with modern science.
I, as well as many others before me, have argued previously on serval
occasions that the correlations peformed by proponents of the
scientific-hermeneutic approach to the Qur'an have a circularlity to
them. First it is assumed, rather than demonstrated, that the author
of the Qur'an intended for the passage in question to be a reference
to the scientific "fact" with which the passage is being correlated.
Then the proponent of the approach wonders aloud how the author of the
Qur'an could have known about such things, and such an appeal to
personal incredulity makes way for the conclusion that Allaah was, in
fact, the author of the Qur'an.
In response to the charge of this being nothing more than "the miracle
of reinterpretation," Nadir made an appeal to mathematics. He
presented eight statements in the Qur'an that he was able to correlate
with science, and then claimed that the chances of a mere mortal
writing these statements were 1 in 10^28, thus the only possible
conclusion is that a power greater than man was the author. It was
pointed out that Nadir has never given any justificatication for this
seemingly arbitrary figure. Furthermore, it was noted that even if we
accepted Nadir's unsupported numbers, we can cite perfectly natural
events that happen at even lower odds, thus his conclusion in favor of
a divine origin for the Qur'an is nothing more than a leap in logic.
It is here that Mohd entered, arguing that one cannot justify the
numbers put forth by Nadir in the sense that a proponent of the
frequentist approach to probability theory would do so (exempli
gratia: like giving a hard and clear figure like 1 in 52 for the
chances of the ace of spades being picked at random from a well
shuffled deck of cards). Mohd further argued that if one takes the
subjectivist approach to the theory of probability with regard to
Nadir's figures, they become relatively justified. Since then Nadir
has been going to various spots on the net asserting that his numbers
are justified because of "subjective probability," without any further
explanation. As for Mohd, he and I have continued to argue about
justification for whatever figures one presents. The subjectivist
approach does not hold that you can just assert any figures you want;
if there was no need to justify one's figures, the entire sports
betting industry would collapse in on itself.
Nonetheless, Mohd has asked that we employ Bayes' theorem. There is a
way in which we can consider the chances of the Qur'an being authored
by Allaah vis a vis the chances of the Qur'an being authored by a mere
mortal, conditional on the evidence that we can correlate statements
in the Qur'an with modern science, without considering any precise
figures at all. Here I will attempt to do just that, and analyze the
reasoning behind Mohd's appeal to probability. My arguments are
adapted from various arguments made by others (such as stuff I found
on texts relevant to the subject, or stuff I found in discussions
elsewhere on the net). For a basic introduction to conditional
probabilities, the subjectivist approach, and Bayes' theorem, consider
following entry in the Stanford Encyclopaedia of Philosophy:
http://plato.stanford.edu/entries/bayes-theorem/
Before I begin I would like to give an explanation of the variables
and equations I will employ in this post. The Stanford Encyclopaedia
of Philosophy uses 'PE(H)' (where the 'E' is rendered in subscript) to
denote "the probability of hypothesis H conditional on evidence E."
However, since I cannot employ HTML scripting over usenet, and thus
cannot render the 'E' of 'PE(H)' in subscript, I will scrap this
notation in favor of the more popular (and aesthetically pleasing, in
my opinion) 'P(H|E)', which denotes the same thing [I say more popular
because this is the mode of notation I have found in most textbooks
relevant to the subject - for one example, see Robert L. Winkler,
"Introduction to Bayesian Inference and Decision," (Holt, Rineheart
and Winston, 1972)]. Also, the "greater than or equal to" sign cannot
be expressed over usenet, so I will denote it with '_>', such that a
statement of the structure...
x _> y
...denotes "x is greater than or equal to y." The variables I will use
are 'S' for, roughly, "one can correlate statements in the Qur'an with
modern science," and 'A' for "the Qur'an was authored by Allaah"
(Nadir only described this as "a power greater than man" - the posters
in the FFI forum tried to jump on this, noting that it leaves open the
possibility that this could include Shaytaan, some other jinn, aliens,
unicorns, a professor from Hogwort's school of wizardry, or a Jedi
knight; I'm going to simply assume those other beings are not included
as possible authors of the Qur'an, and further assume that Allaah is
the only possible non-human author for the Qur'an). For any variable I
use, such as 'A', the negation of that variable will be expressed by a
tilde/squiggle, such as '~A'.
Now, without positively asserting any specific figures, the appeal to
probability put forth by Nadir and Mohd wishes to argue, at least, the
following:
P(A|S) > P(~A|S)
That is, given the evidence (i.e. 'S' - Nadir's ability to correlate
Qur'anic statements with science), the probability that the author of
the Qur'an is Allaah is greater than the probability that the author
of the Qur'an is not Allaah (or given the evidence, the divine origin
explanation is more probable than any natural explanation). The
question now becomes how we reach such a conclusion - we need a
starting point. Up front let me state that I'm willing to accept, a
priori, that...
P(S|A) > P(S|~A)
That is, it is more probable that a person would be able to correlate
statements in the Qur'an with science if the author was Allaah than if
the author was not. I have not proven this to actually be true, but
Nadir and Mohd assume it to be true (moreover they attempt to adduce
from this assumption that the author of the Qur'an is, therefore, a
divine power), and I'm willing to accept it as true. The reason I'm
willing to accept this as true, a priori (i.e. without even
considering any evidence), is because it just seems to make sense to
me that if the author of the Qur'an was Allaah, He would have had
greater ability to insert references to modern science than a human
author, hence the chances of someone being able to correlate a passage
with a corresponding scientific fact increases. Now that I have agreed
to accept this starting premise of the scientific-hermeneutic approach
to the Qur'an, we have to ask via what reasoning one moves from...
P(S|A) > P(S|~A)
...to...
P(A|S) > P(~A|S)
I will henceforth refer to the above as "the starting premise" and
"the conclusion," respectively. The move between propositions of these
structures is not intuitive. For example it is the case that the
probability of me pulling an ace of spades from a deck of cards is
significantly higher if I search and sift specifically for that card
(rather than picking at random) than is the probability of me getting
an ace of spades if I don't search and sift (i.e. if I just pick at
random); however it is not the case that the probability of me sifting
and searching is higher than not doing so (i.e. picking randomly)
conditional on the evidence that I pick an ace of spades. Conversely,
however, the odds of a book being authored by a human are higher if
the book is written in a human language, and also the odds of a book
being written in a human language are higher than not being such
conditional on the evidence that the author was human (though of
course this does not mean that, necessarily, all books written in
human languages were written by human authors).
So, while the move between propositions of the same structure as the
starting premise and the conclusion are never fool-proof, they are
possible; we just need a justification for such a move. With respect
to the scientific-hermeneutic approach to the Qur'an and the appeal to
probability, I will again note that, as an attempt to give us a way by
which we can move from the starting premise to the conclusion, Mohd
has asked that we take a Bayesian approach. As per the Standford
Encyclopaedia of Philosophy, the most basic form of Bayes' theorem is
as follows:
PE(H) = [P(H)/P(E)]PH(E)
Rewriting this using the relevant variables and the prefered form of
notation we get formulas for determining the two sides of the
inequality expressed in the starting premise:
P(S|A) = [P(S)/P(A)]P(A|S)
...and...
P(S|~A) = [P(S)/P(~A)]P(~A|S)
Thus we can rewrite the starting premise as follows:
[P(S)/P(A)]P(A|S) > [P(S)/P(~A)]P(~A|S)
I would ask that people remember from their precalculus studies that
if you have an inequality of the structure...
(x/y)z > (x/w)v
...you can divide both sides of the inequality by the variable common
to both ('x') to get:
z/y > v/w
So by appealing to this mathematical/logical rule, I will divide both
sides of the inequality expressed in the starting premise by the
variable common to both ['P(S)'] to get:
P(A|S)/P(A) > P(~A|S)/P(~A)
Now, with the intention of making another appeal to a
mathematical/logical truth, I would note that if an inequality of the
structure...
z/y > v/w
...is true, then the rearranged equation...
z/v > y/w
...is also true (feel free to plug in your own values to test this).
Thus I can rewrite the starting premise as:
P(A|S)/P(~A|S) > P(A)/P(~A)
Now recall that the conclusion is:
P(A|S) > P(~A|S)
Note that if you have an inequality of the structure...
x > y
...and this proposition is true, then the following is also true:
x/y > 1
So the conclusion can be rewritten as:
P(A|S)/P(~A|S) > 1
Notice that now the left-hand side of the inequality expressed in the
conclusion is identical to the left-hand side of the inequality
expressed in the starting premise. So we again ask how one moves from
the starting premise...
P(A|S)/P(~A|S) > P(A)/P(~A)
...to the conclusion...
P(A|S)/P(~A|S) > 1
Obviously you can only make that step if you presuppose that the
right-hand side of the inequality expressed in the starting premise...
P(A)/P(~A)
...is greater than or equal to one. In other words, one must assume
that the following is the case:
P(A)/P(~A) _> 1
This assumption can be rewritten as:
P(A) _> P(~A)
Now, here we have exposed a hidden premise in the reasoning. These are
prior probabilities, thus this assumed proposition holds, a priori,
that the likelihood of the Qur'an being authored by Allaah is equal to
or greater than the probability that the Qur'an was not authored by
Allaah. Why should we assume, a priori, that this is true? Think of
all the possible authors for the Qur'an; on an a priori basis, we
would have to revert to a frequentist approach and conclude that the
probability of the Qur'an being authored by a being other than Allaah
is greater. The hidden assumption just noted simply isn't a serious
way to approach a text. It is a duplicitous method by which we, a
priori, treat the Qur'an differently from any other book. A serious,
objective individual, who does not have an emotional investment in a
given book, will pull that book off a shelf and assume, a priori, that
the chances of this book being written by a human author are higher
than the chances of it being written by a divine power. What we see
then, is that proponents of this approach, on a very basic level, are
already leaning towards the Qur'an being the word of Allaah in their
hidden assumptions.
Interestingly, Nadir has recently started discussion regarding the
eight verses he called to witness on the FFI forum. A relevant section
of Nadir's page devoted to this subject is the following:
http://examinethetruth.com/ahmed_sina.htm
It is on the FFI forum that other posters have come up with another
approach. One variant of the approach that has come up would be
roughly as follows:
P(A|V) = P(A)/[P(A) + P(~A)P(M)]
First I would like to explain this formula, and why I think it
accurately captures what Nadir and Mohd are trying to say (and I will
simply proceed under the assumption that this formula is acceptable on
the grounds that it properly represents or successfully captures
Nadir's argument). P(A|V) represents the probability that Allaah is
the author of the Qur'an conditional on the evidence that Nadir has
brought to the table ('V' - the verses that Nadir has called to
witness). As for the equation, first let me note that the following is
true:
P(A) = P(A)/[P(A) + P(~A)]
The reason this is true is if you take an equation of the structure...
x = x/[x + ~x]
...where x is bound within [0,1] and ~x (the negation of x) is equal
to...
1 - x
...then the equation is true (try plugging in your own numbers). I
have it bound within [0,1] (i.e. it is between zero and one or equal
to zero or one) because with probabilities we say there is a 2%
chance, a 10% chance, a 50% chance, et cetera. So the chances of any
event (such as Allaah being the author of the Qur'an) have to be
somewhere within the range from 0% to 100% (it is simply not sensible
to say there is a less than 0% chance or more than 100% chance).
Now what has been done is, the true equation...
P(A) = P(A)/[P(A) + P(~A)]
...has been altered accordingly to:
P(A|V) = P(A)/[P(A) + P(~A)P(M)]
Above P(M) stands for the probability that man could write a something
that later generations could correlate with science. This definition
is very important. Suppose that P(M1) stands for the probability that
a man (in the seventh century) could have knowledge of the scientific
facts and theories which Nadir correlated with the Qur'an, and suppose
that P(M2) stand for the probability that a man could author a text
that later generations could correlate with such science (irrespective
of whether the author himself knew about such science, much less
intended the text to be a reference to such). In other words, these
are two very different propositions, though Nadir and Mohd, I feel,
have tried to avoid this key difference (Mohd has even tried to argue
that it is not possible to correlate a text with science if that was
not the author's intention), but it is quite significant, and I would
think that it is fairly obvious that
P(M2) > P(M1)
That is, the probability that a person could correlate a text with
science is greater (I would add much greater) than the probability
that the author of the text actually knew such science. So P(M) is the
same as P(M2) above, and is simply the probability that a man could
write a text (like the Qur'an, or some other text) that later
generations could correlate with science, regardless of whether that
was the author's intention or not. However, though P(M) seems
intuitively high to me, it reads as a prior probability, so I would
make it conditional on some evidence E (such as my ability to
correlate passages from the Bible, the Talmud, Daoist texts, the poems
of Walt Whitman and the writings of Saint Hilary of Poiters with
modern science, or the ability of others to do such with Hindu
scriptures, Buddhist scriptures, ancient creation myths, the poems of
Virgil, or the book of Mormon). So the equation would be rewritten as:
P(A|V) = P(A)/[P(A) + P(~A)P(M|E)]
So, to translate this, the probability that the Qur'an was authored by
Allaah, conditional on the evidence of the verses that Nadir
correlated with science, is equal to the prior probability that Allaah
is the author of the Qur'an, divided by the sum of the prior
probability that Allaah is the author of the Qur'an on the one hand
and the prior probability that Allaah is not the author of the Qur'an
multiplied by the probability that man could write a text that others
could correlate with science on the other. I know that is a tongue
twister, so I'll explain why this equation accurately represents
Nadir's argument.
Suppose first that I am an obstinate atheist who refuses, under any
and all conditions, to accept that Allaah even exists, and furthermore
I positively assert (and assume a priori) that Allaah does not exist.
In this case, the prior probability that Allaah is the author of the
Qur'an, P(A), would be zero, and the prior probability that Allaah is
not the author of the Qur'an would be one. Assuming P(M|E) is more
than zero (and we have good reason to believe it is more than zero),
then the probability that Allaah is the author of the Qur'an
conditional on the evidence is also zero.
However, suppose I was an obstinate Muslim who refused to consider any
possibility that Allaah was not the author of the Qur'an. In that case
P(A) would equal 1, and P(~A) would equal zero, thus the chances that
Allaah was the author of the Qur'an would be 1 (or 100%) - a sure
thing.
Thus far you see that this rests a great deal on what we presuppose
about Allaah being the author of the Qur'an, and you may wonder what,
then, is the point of P(M|E). Well, suppose now that P(A) is bound
within (0,1) (that is, it is between zero and one but not equal to
either of those figures). In this case, both P(A) and P(~A) have
values, and thus the result of the equation will rest on P(M|E).
For example, if we agree that P(M|E) = 0, that given the evidence,
there is no chance that a man could write a text that later
generations could correlate with science, then the only possible
explanation given the evidence is that Allaah is the author of the
Qur'an (according to this equation).
However, if P(M|E) = 1, that is it is a sure thing that man can write
verses that can be correlated with science, then the probability that
Allaah is the author of the Qur'an cannot be increased in light of the
verses called to evidence by Nadir, and the posterior probability that
Allaah is the author of the Qur'an is exactly the same as whatever we
assumed the probability of Allaah being the author of the Qur'an was a
priori.
So what we see is that this equation establishes a relationship
between what we assume a priori regarding Allaah being the author of
the Qur'an and the ability of mere mortals to write texts that can be
correlated with science. So, assuming P(A) and P(~A) both have values
above zero, the chances that Allaah is the author of the Qur'an
(conditional on the verses that Nadir has correlated with science) is
in direct proportion to the probability that man can write a text that
can be correlated with science. The lower P(M|E) is, the higher P(A|V)
is. This is precisely in line with Nadir's original argument, which
claims (without justification) that the probability of a man writing a
text that can be correlated with science is extremely low (a single
example could be, under Nadir's original figures, 0.0001, i.e. 1 in
10,000!).
However, also note that if P(M|E) is a sure thing, this does not lower
the a priori probability that Allaah is the author of the Qur'an
(which makes sense, because just because a text can be explained
naturally does not mean it necessarily does not have a divine origin,
hence the reason I always say that Nadir and Mohd have, thus far,
failed to prove the Qur'an is the word of God, but this does not mean
the Qur'an is therefore not the word of God).
Now, in light of the various pieces of evidence I and others have
brought to the table over the years in discussions on the Qur'an and
science in SRI, I think it is plainly obvious that the probability
that a man could write a text that others could correlate with science
(regardless of whether this was the author's intention or not) is
significantly high. It may even be a sure thing when we call to
evidence the seemingly limitless creativity of the human mind. Thus
objectively we would have to argue that probability that Allaah is the
author of the Qur'an conditional on Nadir's ability (or the ability of
other proponents of the scientific-hermeneutic approach) to correlate
certain verses with science is not significantly higher (if it is
higher at all) than the a priori assumption one has regarding the
probability that Allaah is the author of the Qur'an. Furthermore, the
higher the probability that a man could write a text that others could
correlate with science is, the farther the probability that Allaah is
the author of the Qur'an moves away from being a sure thing (assuming
one does not presuppose that Allaah being the author is a sure thing).
So the room for doubt becomes larger and larger. Nadir has said
repeatedly that the mathematics he made an appeal to "prove" (beyond
all doubt) that the Qur'an is the word of Allaah - he has said that no
one can dispute the mathematics (though he has yet to bring these
alleged mathematical proofs of his to sci.math and see what actual
mathematicians think). However, in light of the evidence, it is clear
that it is not a sure thing, thus Nadir has not proven that the Qur'an
is the word of God (especially not in the sense that he has claimed).
I understand this is a very long post, and for that I apologize. I
also admit that these are only my preliminary thoughts on the issue,
and further admit that I am far from being an expert on the subject of
probability theory, thus I am very open to objections. I look forward
to the comments of others (particularly Nadir and Mohd) with regard to
what I have written above.
-Denis Giron
http://freethoughtmecca.org/home.htm