Volume 19 Number 17
Produced: Wed Apr 5 22:54:41 1995
Subjects Discussed In This Issue:
Codes
[Mike Gerver]
Uses of Mathematics ?
[Sylvain Cappell]
Vav DeGichon: A Flawed Numerology?
[Mechy Frankel]
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From: <GERVER@...> (Mike Gerver)
Date: Wed, 29 Mar 1995 3:38:04 -0500 (EST)
Subject: Codes
Harold Gans' use of the Heisenberg uncertainty principle, chaos
theory, and Godel's incompleteness theorem to show that it is impossible
to predict the future (and that the results reported by Witztum et al
therefore show the divine origin of the Torah) reminds me of a story I
heard from Prof. Binyamin Frankel of Hebrew University. Prof. Frankel's
father was Prof. Avraham Frankel, a mathematician who made important
contributions to set theory in the early part of this century (e.g. the
Zermelo-Frankel axioms). When he was growing up, Binyamin Frankel would
hear his father and colleagues arguing about _why_ one plus one is two.
Binymain thought this was such a ridiculous question that he resolved
not to go into pure mathematics, or any field close to it. He ended up
going into experimental applied physics, and made a very successful
career of it.
Now I am not entirely sympathetic to Binymain Frankel's point of
view in this story. Although I am an applied physicist myself, I enjoy
reading popular articles on set theory, Godel's proof, chaos theory,
foundations of quantum mechanics, etc. I don't consider the questions
dealt with in these articles as ridiculous or insignificant. Indirectly,
applied physics often depends on statistical mechanics, whose
foundations depend on chaos theory, and applied physics always depends
on arithmetic, which depends on set theory and the Zermelo-Frankel
axioms. On the other hand, I think Binymain Frankel would justifiably
think it peculiar if he were giving a talk on x-ray diagnostics for
tokamaks and someone in the audience made a comment on it, using the
Zermelo-Frankel axioms to prove a point.
The results reported by Witztum et al in the Aug. 1994 issue of
Statistical Science raise real questions about whether and how these
results can be explained without invoking miracles and the divine origin
of the Torah. These questions have about as much to do with Heisenberg,
Godel, and chaos theory as set theory has to do with x-ray diagnostics.
By bringing up Heisenberg, Godel, and chaos theory, Gans is distracting
people from dealing seriously with the real issues.
Having said this, I really ought to explain what I think the real
issues are. I will do this briefly, but like Gans, I cannot go into any
detail in the space available. (This is not just an excuse for being too
lazy to write it up. I did try writing it up a couple of months ago, but
couldn't get it into less than 23K. Evidently this was too long for
mail-jewish, since it was never run.) I will be happy to provide more
details, off-line, for anyone interested.
The paper by Witztum et al actually has two separate surprising
results. 1) The distribution of c(w,w') is far from uniform in the
interval (0,1), being instead heavily skewed toward low values, when w
is a name from the list of famous rabbis and w' is the yahrzeit date of
that person. 2) The distribution is much less skewed when the lists of
names and dates are randomly permuted. In order to explain these
results without invoking miracles, we need a "natural" explanation for
both (1) and (2). Several possibilities come to mind, and I plan to
check them out and see if they work.
Result (1) might be explained if there were long range order in the
distribution of letters in the text of Breishit [Genesis]. For example,
suppose that certain letters of the alphabet are more likely to occur in
even positions in the text (letter #2, #4, #6, etc.) while other letters
are more likely to occur in odd positions (letter #1, #3, #5,...). This
will make certain words more likely to occur as unperturbed ELS's in the
text (i.e. (x,y,z) = (0,0,0)) than as perturbed ELS's (with other values
of x,y,z), while other words will be less likely to occur as unperturbed
ELS's. If both the name w and the date w' are from the set of words that
are more likely to occur as an unperturbed ELS, and if w and w' both
span a significant fraction of the text of Breishit, then c(w,w') will
tend to be closer to zero than to one. If w or w' is from the set of
words that is less likely to occur as an unperturbed ELS, then it may
not occur at all, and c(w,w') for his pair will be eliminated from the
distribution. The resulting distribution of c(w,w') for all the names
and dates would then be skewed toward low values, even if there is
nothing special about the names and dates. The effect would go away if
the words or the sentences were randomly scrambled, since this would
destroy the long range order. I do not know if the text of Breishit has
this kind of long range order, but I hope to find out. If it does, this
would be an interesting fact, but it would not be evidence for the the
divine origin of the text, since it would be quite possible for a human
being to create such a text without the use of a computer.
Result (2) seems at first to imply that the author of the text could
predict the future, that the text "knows" which rabbis would die on
which dates. A more reasonable way to look at it, though, is that the
rabbis in the list were not dying on random dates, but were
preferentially dying on certain dates, depending on their
names. Actually, it is clear that the yahrzeit dates on the list are not
randomly distributed. Memorable dates such as Rosh Chodesh, Yom Tov,
Chol Hamoed, Chanukah, etc. make up about one third of the dates on the
list, although they only make up about one sixth of the dates on the
calendar. Most likely this is because memorable yahrzeit dates are more
likely to be remembered, and because a yahrzeit date that is close to
Rosh Chodesh, for example, may get changed to Rosh Chodesh if it is
passed on as an oral tradition before being written down. This
particular effect does not seem to depend on the person's name, but
there may well be other effects that do. For example, I have found that
of the 11 people on the list who are known as "Ba'al ha..." followed by
the name of a sefer they wrote, 7 of them died in Shevat or Adar, while
you would only expect this to be true of 2 of them. A possible
non-supernatural explanation for this might be that people who devoted
all of their energy to writing one major sefer would be more likely to
neglect their health, or to neglect their parnassa [making a living] and
not have enough money for food and fuel, and be more likely to die in
the winter. I don't know whether patterns of this sort are enough to
account for the results reported by Witztum et al, but I hope to find
out.
Again, I apologize for the terse explanations in the last two
paragraphs, which are probably incomprehensible to anyone who has not
spent a lot of time working on this problem. I will be happy to send a
more lengthy explanation to anyone who wants it.
I must add that I find some of Sylvain Cappell's comments on the
codes (also in v18n95) unreasonable, although he may well be right about
the sociology of codes enthusiasts. Surely it is not relevant what the
reputation of "Statistical Science" is for publishing questionable
papers, or how many emininent professors agree or disagree with the
paper. The only things that are relevant are whether the results
reported in the paper are correct, and whether there is a
non-supernatural explanation for them. These questions can only be
answered by verifying the results, and by thinking of possible
explanations and testing them. The fact that the paper was published in
a refereed journal is relevant only in that it makes the paper plausible
enough that it is worthwhile to check it out, rather than to dismiss it
as a crank paper or a fraud. Of course, investigating these things is no
substitute for studying the content of the Torah. But the results
reported by Witztum et al are so surprising, if true, that they cry out
for an explanation, like any other surprising physical or mathematical
phenomenon, and it is in that spirit that I am drawn to investigate
them.
Mike Gerver, <gerver@...>
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From: <cappell@...> (Sylvain Cappell)
Date: Mon, 6 Mar 95 23:46:31 EST
Subject: Uses of Mathematics ?
Subsequent to my posting refering to the two new contending systems of
mathematically reading texts, there have been many postings and I have also
received curiously analogous emails from afficinados of both the "topological
codes" and the "statistical codes." These generally argued that their own
distinctive way of creating an innovative approach to religion and text is
based on a valuable use of mathematics, while the other is devoid of
scientific interest. While I had not disputed, analyzed or compared the
contrasting claims of either of these systems of mathematical approaches to
religious insight, I am afraid that I really am not yet able to see all the
sharp contrasts between the "topolgical codes" and "statistical codes" that
both groups seem to insist on. Indeed, enthusiasts for opposed escoteric
systems are often prepared to explain how their particular novel methodology
merits special consideration and is to be totally and absolutely
distinguished from what they view as the other's pseudoscience, whose
fallacies are all too apparent to them. Both systems, after all, while
making some references to historical Rabbinic Judaism, do in fact share
being based upon radically new departures in religious epistomolgy, in
both cases based on mathematically inspired methodologies or languages
which may, at least in principle, be applied or adapted to the texts of
many religions.
In both the "topolgical codes" and "statistical codes" groups I
indeed have some good and deeply valued mathematical friends who may
be involved, or more precisely, are perhaps just intruiged or attracted
by the astonishing vast claims being asserted for the power or novelty of
their respective speculative mathematically based approaches to religion and
texts. Indeed, in the case of the "topological codes" ( which I have looked
at further and which despite being a researcher in topology still can not
personally understand anything of ) there has been interest expressed
by a mathematical friend who is a recognized expert in topology. However,
supporters of the "statistical codes" feel that in assessing scientifically
the claims of the "topological codes" this is perhaps not really relevant,
as in the final analysis, apparently it was not in any case argued that these
notions of geometrically reading novel meanings into religious texts bear
any direct relation to conventional topology as commonly practiced by
mathematicians. The "statistical code" novelties, on the other hand, while
it has intruiged several fine pure mathematical friends, has not
apparently seriously interested any leader in statistical research, a subject
deservedly famous for having all too often trapped even distinguished
nonspecialist scientists, and the only very distinguished and disinterested
academic researcher on statistics that I know of who has examined this
gives it no credit.
Some writers referred to the publication of a paper in a peer-reviewed
journal as demonstrating the absolutely superior claim of "statistical codes"
to be a scientifically based approach to religion, unlike the "topological
codes". This doesn't really seem quite fair, as it is commonly known to all
serious scientists, that the mere publication of a paper in one of the several
hundred thousand peer-reviewed scientific journals currently in print does not
imply that the paper is correct or that it represents even the presumptive
currently accepted scientific consensus. ( As a particularily widely discussed
example of this commonplace, recall the controversy a few years back when the
editor of Nature, one of the great journals of science, published a paper on
th claims of the pseudoscience of homeopathy. ) Some colleagues doing research
in statistics have said that, alas, that is notoriously true of some
papers that appear in the journal containing the paper on which the adepts of
the "statistical codes" base their claim to have a more scientificly based
approach to religious texts. In any case, to be fair, the proponents of the
"topoogical codes" may analogously also come to seek the "confirmation" of
trying to get a scientific publication.
There are of course important epistomological differences immediately
apparent between the two schools of applying mathematics to religion. To the
perhaps mystically or artistically inclined devotees of the "topological
codes," it may seem to offer practitioners new cosmic meanings ( perhaps
related to those some have usually sought in some other Eastern religions
and religious traditions ), however apparently alien to conventional Judaism
or standard science. The "statistical codes" approach, on the other hand,
by its nature tends rather to, in effect, deprecate the traditional centrality
ascribed to meaning in Jewish texts. ( In this connection, some respondents
expressed concerns about what may ensue when "statistical codes"
come to be invoked by propagandists for some other religions, who perhaps,
as they may have greater funding, will use even bigger computers in
developing statistical "proofs" of religious points. Indeed, I have been
told by Russian colleagues that statistical "proofs" had earlier been used
by Russian Orthodox Christian mathematical mystics in the former Soviet Union,
though at the time they did not have access to poweful computers. Happily,
it should not prove necessary to invest in ever bigger computers to defend
Judaism from any"statistical attacks"; a powerful dose of Jewish humor should
suffice. )
Of course, conventional mathematics is hardly likely to be damaged by
all this, or even to ever take notice of any disputes between these two
mathematical approaches to religion and texts.
Both these novel approaches to applying mathematics to religion and
texts do, in fact, raise some perplexing and deeply disturbing sociological
questions. Some of these difficult sociological problems will, in the Jewish
context, no doubt be the focus of some serious academic researches: Why at
this particular time are some apparently favoring radical
innovations in place of the traditional centrality of meaning in the texts ?
Are some sadly reduced to viewing texts as formal or mathematical
cribs ? From whence do some get the novel feeling that the texts are
somehow in need of scientific endorsements or expansion in outlook or of
the "prestige" of invoking fancy sounding mathematical terminology ? Is
there no basis for concern that intelligent, well-educated and sensitive
Jews sadly unfamiliar with Jewish texts may be repelled from even looking
at them when turned off by what they may view as formalistic, boring, puerile
or profoundly nihilistic interpretations ?
To ask again, as none of the many replies and postings from either
school responded to this: Why shouldn't we
be confident that the reasons for becoming interested in Jewsh texts can be
drawn, as always, from their uniquely profound and exciting ideas, legal
codes, traditions, history, stories, poetry, wisdom and values ? Those are
all things that mathematics, wonderful and beautiful as it is, can make no
claims of providing.
Professor Sylvain Edward Cappell
Courant Institute of Mathematical Sciences, New York University
251 Mercer Street
New York, N.Y., 10012 <cappell@...>
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From: Mechy Frankel <frankel@...>
Date: Tue, 28 Mar 1995 11:57:05 -0500 (EST)
Subject: Vav DeGichon: A Flawed Numerology?
1. As I was leining last week's parsha, i was reminded of a problem
which i've not found a satisfying solution for and wondered whether
anybody else might have run across a decent explanation.
2. When reading parshas Shimini, we notice on that the vav in the word
"gichon" in Vayikra 10:42 is written extra large because, as many
chumashim note on the spot, it marks the halfway point, by letter count,
of the full torah text. Of course this simply follows a practice
prescribed by the post-talmudic composition, Meseches Sofirim (ch 9)
which directs a graphical marking of this numerical milestone. Meseches
Sofirim, in turn, is simply iterating the gemara Kedushin 30a which
uncontestedly asserts, amongst other things, that the vav of gichon is
the halfway point.
3. The problem with all this of course is that the vav of gichon is most
assuredly not the the torah's halfway point. In fact its not even close,
being almost 5000 places off the true letter count midpoint (to be found
in the entirely unremarked Vayikra 8:28) So what's going on here? To
scale this error in perspective we should consider that the torah
overall has a bit over 300,000 letters - so we're talking of a 1.6%
error, and for a simple counting problem, this (as we say in DC) should
not be considered close even for government work.
4. Some possible solutions include the following:
a) The gemara in Kedushin 30a records a tradition in the name of R. Yosi
that "inhu bikei bechaser veyeser veanan lo bekieanun" i.e. that already
in talmudic times people were apparently not completely certain of the
maleh and chasers (plene and defectives). Thus the text the gemara is
referring to could have had a much different distribution of vavs and
yuds than our modern chumash, and thus the vav of gichon could have been
the real halfpoint of their text, but not ours.
This, however, is quite problematic since i) Why should the author of
Meseches Sofirim written close to, or in, the period of prime activity
by the Tiberian Massoretes, and thus presumably had a version of the
torah pretty similar to ours (hard to believe there would be 5000
chaser-maleh differences by that point) repeat this incorrect assertion
which must have disagreed with the text now before him? ii) Meseches
Sofirim also repeats the gemara Kedushin 30a assertion that the words
"darosh dorash" in Vayikra 10:16 mark the halfway point in the torah by
word count (and is so noted in many chumashim today). This too is
manifestly incorrect, since the real halfway word is "mizbeach" in 8:15)
being off by about 900 words (actually about 61 pesukim and guestimating
a 15 word average). And this inconsistency could not be explained away
by appealing to uncertainty in chaser-maleh since that would have no
effect at all on the word count. iii) Even the general talmudic
uncertainty of chaser-maleh is by no means a given since R. Meier
testified in Sotah 20a that he himself was quite expert on these matters
("lo mibaieh bichaseros veyeseiros debaki ana"), and the very notion
that that the talmud would inform us that vav of gichon was the halfway
point, or expound the halacha that a sefer torah which had a maleh
writen as chase or the reverse was pasul (a halacha cited in Menachos
29b and surprisingly enough articulated by the very R. Yosi who declared
"anan lo bekianin' above, so clearly something is going on here) would
indicate that chazal in fact had a firm grasp of the text.
b) The text of the gemara in Kidushin 30a is simply corrupt from a very
early period, and Meseches Sofirim was mislead by the corrupt text. This
solution was offered by R. Y. Shor (Mishnas R. Yaacov, ch.4) but it's
difficult to accept that Maseches Sofirim would so slavishly repeat a
corrupt girsah when there was such a wide and obvious disagreement with
the text which must have been in front of him at that point.
c) R. Eliyahu Posek (in Piskei Eliyah, chelel 3, siman 1) suggests that
the halfway of letters in Kidushin 30a doesn't refer to all letters but
only the "different" or problematic ones. i.e. if we look only at
letters involved in maleh-chaser, kri-kesiv, special large or small
letters, etc. then vav of gichon will be halfway through that list,
similarly the word pair "darosh dorash" is halfway through a list
comprising only remarkable words, (such as tishagalna-tishcavneh). This
purely pilpulic response is not very convincing and in any event does
not explain why the vav in gichon should be any more the midpoint of
such a list than any other letter in the word gichon.
d) R. Reuven Margolis (in Hamikra VeHmesorah, and where I also initially
found the last two references) suggests that it is necessary to also
count blank spaces in the torah text, which are also prescribed by
tradition. e.g. spaces are prescribed between "open" and "closed"
parshas, within the text of the various shirim, etc. and the vav of
gichon would mark the halfway point in prescribed letter spaces, some of
which are unoccupied. This argument is made by its author somewhat more
persuasively than I've summarized it here but, aside from its pilpulishe
flavor, ultimately fails entirely to explain the "darosh dorash" word
count problem.
5. There are also problems with the talmudic (Kedushin 30a again) count
of the number of pesukim at 5885 which is off by about 40 from the
current arrangement. This, however, has long been remarked by chachamim
(see e.g. gilyon hashas to Berachos 7a, also the Yalkut Shimoni to Ekev
247 has a girsa referencing a 8455 pasuk count) who seem willing to
entertain a talmudic textual emendation here. In any event, this
generally would not seem to be as a big a problem as the letter or word
count discrepency since there are already indications from the talmud
itself that different communities split up some pasukim differently,
with different counts. e.g. kidushin 30 "ki asa R. Acha.. amar
bimaarava paskei leih lihai kera litlasa pisukei" ("when R. Acha came he
said in Israel this pasuk is split into three separate pesukim..").
5. So there you have it. I'm stuck. Any good references or ideas?
Mechy Frankel H: (301) 593-3949
<frankel@...> W: (703) 325-1277
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End of Volume 19 Issue 17
