Prometheus and Mega Lists, April 1999 (Part Eleven)
Date: Sun, 11 Apr 1999 08:28:09 -0700
To: megalist@brokersys.com
From: Kevin Langdon <kevin.langdon@polymath-systems.com>
Subject: [MegaList] Reply to Chris Langan, 4/11/99 (Part One)
At 12:27 AM 4/10/99 -0400, Chris wrote:
> Okay, everybody, break out the Dom Perignon and raise
> your glasses to Kevin Langdon, who seems to be setting a
> new personal record for obtusity and obstinacy. His last
> messages underline as never before what he has proven
> countless times already - namely, that while he sometimes
> barely has a point when it comes to statistical psychometrics,
> he is almost always dead wrong when it comes to anything
> else, and will never, ever admit it. But this time it's even
> better. Now Kevin is challenging the entire scientific,
> mathematical and philosophical world over the meaning
> of "isomorphism" in the context of formalized theories! He
> can't win, of course...not today, not tomorrow, not when the
> burnt-out sun is a dense ball of ash spinning like an icy top
> in the blackness. But that's what makes it all so...remarkable!
isomorphism, n.: a one-to-one correspondence between
two mathematical sets; esp. a homomorphism that is
one-to-one
homomorphism, n. a mapping of a mathematical group,
ring, or vector space onto another in such a way that the
result obtained by applying an operation to elements of the
domain is mapped onto the result obtained by applying the
operation to their images in the range.
--*Webster's New Collegiate Dictionary*
[I have only included the mathematical definition
for each term., in accordance with Crhis' usage]
What I said, in my message was quoted by Chris in reponse to
his statement:
[Intervening material snipped.]
>>> An isomorphism is a bijective (one-to-one onto)
>>> correspondence or similarity mapping between, e.g., a theory
>>> T and its object universe T(U). In the absence of a clear
>>> difference relation between the range and domain of an
>>> isomorphism, it can be contracted so that range and domain
>>> are coincident.
>> An isomorphism is just a one-to-one correspondence. Chris is
>> seems to think that any one-way mapping implies a two-way
>> mapping, which isn't necessarily true.
> An isomorphism is surjective (onto) as well as injective (1-to-
> 1). It is virtually always reversible unless it is explicitly
> stipulated, for usually artificial reasons, that it is only one-way.
> When it comes to the topic at hand - the correspondence
> between a valid theory and its object universe - the isomorphism
> is always two-way. That's what the "valid" means. To reiterate,
> the theory "selects" its own object universe (which may not be
> as large as the one that the theorist originally had in mind).
I'm tired of saying that things are unclear on my own. Therefore,
I'm going to indicate certain points regarding which an empirical
determination of general understanding or nonunderstanding
seems desirable. This will be test point #1. I will also ask you at
other test points whether you follow what Chris is saying.
> Now I'm going to prove that if you don't introduce an artificial
> directional constraint, a one way isomorphism M:A-->B always
> implies a two-way isomorphism M':A<-->B. Take two n-element
> sets A and B, one the source and one the target of M. Draw the
> one-way correspondences between the elements of A and those
> of B as one-way arrows: M = {a1-->b1, a2-->b2, ..., an-->bn}.
> To get the two-way isomorphism M', just replace each one-way
> arrow with a two-way arrow: M' = {a1<-->b1, a2<-->b2, ... ,an
> <-->bn}. This proof generalizes easily to higher-order relations
> of elements.
And 1 + 1 = 2.
> It's for this reason that all logic and abstract algebra texts define
> an isomorphism as follows: "An isomorphism is a
> homomorphism that is bijective, or one-to-one onto (injective
> and surjective)."
That vocabulary is a little different from that which was used
when I was in college, but it should be clear that we're all
talking about the same thing--a one-to-one correspondence
between two mathematical objects. What Chris is saying is that
there is a metaphysical relationship between a cognitive model
and that which is modeled. I say he hasn't proven it.
Let's make this test point #2: do *you* believe this is proven?
> Give it up, Kevin. You can't win, and you're boring the hell out
> of me and everybody else.
There's nothing to give up. I'm just pointing out the lack of a proof
of Chris' fundamnetal proposition.
>>> In this sense, the abstract structure of a valid theory (as
>>> opposed to the pattern of neural potential encoding it within
>>> a brain) is virtually identical to its object universe.
>> What do you mean "virtually"?
> "Virtually": for all practical purposes.
How do you know what purposes will turn out to be practical?
> Act (or reason) as if it's true, and you can't be tripped up.
Famous last words...
>> What I've seen so far of the CTMU does not justify either the
>> existence of the relationship claimed between cognition and reality
>> or Chris' claims for the importance of his theory.
> Nonsense. The contractible isomorphism M:T<-->U(T) establishes
> the relationship; you're just in a state of denial. And by your own
> admission, you can't understand what I've written of the CTMU, so
> you can't say what it might or might not have established.
I can't make a silk purse out of a sow's ear, but that's not a deficiency
on my part.
>> > This is what always happens with Kevin; even when I quote
>> > known authorities (e.g., the given definition of "model" comes
>> > straight from Rogers' North-Holland textbook on logic and
>> > formalized theories), Kevin uses one ear as an entrance and the
>> > other as an exit, blinks and drools a little, and bellows out his
>> > unreasoning accusation of crankery
>> No matter where Chris gets his raw material, the way he puts
>> things gets them all tangled up. English, Chris. Plain English.
> There comes a point at which any philosophical discussion exceeds
> your evidently primitive notion of "plain English", Kevin. If I were
> using the relevant math symbology here, you'd *really* be hurting.
I don't think that you care much about that--so why *aren't* you
using "the relevant math symbology"?
<snip>
>> And how can I be a "philosopher"? I haven't been elected by the
>> ISPE.
> No. But you did manage to get yourself booted out of it on grounds
> of inveterate Nietzscheism.
The five founders of TNS were kicked out of the ISPE for founding
TNS to provide ISPE members with a democratic alternative. Only
someone intent on making points at the expense of honesty would
cite this as anything but a badge of honor.
>> > Some of you may now be thinking along the following lines.
>> > "Langan has a point. Brilliant as they can be, modern scientists
>> > do tend to have a muddled notion of the relationship of
>> > cognition and objective reality; that much is obvious from the
>> > fact that there are so many paradoxes in the foundations of
>> > mathematics and physics. And the logic he's using seems so
>> > simple and obvious as to be trivial. So if he can build a theory
>> > out of these simple connections, why shouldn't we give it an
>> > impartial look? After all, he is a member of the Mega Society.
>> > What's Kevin Langdon's problem, anyway?"
>> And some of you may *not* be thinking along those lines.
> Right. And at this point, I'm 100% safe in saying that if not, then
> one of the following is true of the person in question. (1) He hasn't
> been following this exchange. (2) He's a dummy. (3) He's a
> Langdonoid, blasting himself in the foot with a ray gun and looking
> forward to a family reunion on Planet X.
Speaking of shooting oneself in the foot, Chris keeps on insulting
his readers-and then he's surprised when nobody wants to march
with his "band."
>> > Only Kevin can answer that question, and he has never done
>> > so honestly. But no matter what his problem may be, it is
>> > clear that his style of thought, and his summary rejection of
>> > the work of qualified Mega Society members, is not to be
>> > emulated. Indeed, his chronic irrationality has the potential
>> > not only to keep us eternally in obscurity, but to make
>> > everyone in the group look like a fool...like a twittering fife,
>> > peeping piccolo or tooting tuba in the goose-stepping
>> > "marching band" that he perceives whenever he arrogantly
>> > surveys his imagined dominion.
>> Projection.
> Did you or did you not call the Mega Society a "marching band"?
> Tell the truth, now...some of us still have the issue in which you
> printed it.
But most of us won't go to the trouble of looking it up, so I've
reproduced it here (from *Noesis* #136, December 1997, p. 17):
Chris Langan complained that the unofficial Mega Society
Web sites don't present his version of how things are in the
society. But what is going on here is sort of like a marching
band with a tuba player who marches off in one direction
with his little dog (unfortunately not toilet-trained) while
everyone else goes in another (because the tuba player never
listens to any of the other instruments and marches with his
eyes closed). Naturally, an impartial observer asked
"Where's
the band?" will not point to the wayward tuba player.
Extending the analogy further, the tuba player is always playing
wrong notes and he keeps trying to trip the other members of the
band (without much success because he's very clumsy). It should
also be noted that the members of the band are volunteers, as are
the band leaders, and have not been leaned on by the Mafia.
<snip>
>> The "chronometric" tests that Jensen and other investigators have
>> developed use a much more radically different methodology
>> than the high-range power tests, but they correlate moderately
>> well with the standard tests.
> Seriously, have you ever wondered why that is? I mean, isn't speed
> the main factor in a chronometric test?
According to Jensen, the personal tempo factor does not correlate
with speed of neural processing. The former predominates in timed
IQ tests and the latter is the main ingredient in power tests. Why
should the latter relation hold? Because human memory is volatile.
You've got to be able to set a problem up fairly quickly for short-
term memory to be able to hold onto it long enough for you to see
the solution.
>> The "statistical cover charge" is publication in reputalble
scientific
>> journals of data on our tests and successfully defending it, and we
>> haven't done that yet, though a few academic psychometricians
>> have taken an interest in the new tests.
> Not quite. The statistical cover charge equals the emergence of
> several power-IEQ tests that correlate just as well with standard IQ
> tests as does the average IQ test itself.
You're assuming that the "average IQ test" is an optimal instrument
for measuring *g*; this is not the case. It would make more sense to
look for correlations with the *Raven Advanced Progressive Marrices*,
which is known to be an almost-pure test of *g*. (Unfortunately, there
seem to be problems with the norming of the Raven; sources don't
agree on the principal test parameters, but that's not a problem when
we're just looking at the correlation of raw scores with another test.)
> We'll be asked to produce statistics that are clear and extensive
> enough to unequivocally counter arguments that could presently be
> constructed on the weakness of IQ-IEQ correlations across the
> entire power-IEQ genre...i.e., the genre characterized by the
> methodological distinction on which they'll focus.
Absolutely.
> And time constraints aren't even the sole methodological issue.
> There's the related issue of item complexity. Surely Jensen realizes
> that you can't just monkey around with the time constraint and
> neglect the ramifications of using extremely complex items...items
> much more complex than those on any timed test. There are simple
> mechanistic arguments to the effect that different mental parameters
> may be involved (and mechanistic arguments are generally superior
> to statistical ones).
I've quoted Jensen with regard to time contstraints. He's also stated
that the crucial distinction is between items with no complexity at
all (he gave the example of the "Making X's" test, in which the
subject is simply asked to make rows of X's) and tests involving
more complex mental processes. From that point of view, there is
little difference between standard IQ tests and the high-range power
tests.
>>>>> Until we achieve a mechanical explanation of intelligence,
>>>>> *g* will remain a statistical construct defined on tests
>>>>> incorporating time constraints.
>>>> According to Jensen, tests with severe time constraints have
>>>> no significant *g* loading.
>>> What does Jensen mean by "severe"? I've talked to several
>>> smart people who have failed to finish standard g-loaded IQ
>>> tests, usually because they've gotten hung up on distractions
>>> or what they see as ambiguous items. That constitutes de facto
>>> speed loading in conjunction with *g* loading. But maybe I'm
>>> not understanding you properly. In any case, because power-
>>> IEQ tests are not in use by the psychometric community, Dr.
>>> Jensen has little or no data on them. So we have to be cautious
>>> about extending his remarks to cover them.
>> No extension is required. Jensen is comparing standard tests
>> given with and without time limits short enough that most testees
>> can't finish them in the time allowed. The closest thing to a timed
>> high-range power test is the Cattell Culture Fair, which is very
>> much a standard tests, designed by one of the most eminent
>> academic psychometricians.
> Unfortunately, when you deal with "most testees", you're dealing
> with the middle part of the curve, and this may not generalize to
> super-complex tests for superintelligent people.
As the super-complex tests are not timed, this is academic.
> And unless I'm mistaken, the CCF is still timed, which introduces
> the factor of mental endurance. When taking the CCF, you can't
> get up, take a walk, clear your mind, and "start fresh" after your
> brain has had time to do a certain amount of subconscious
> processing. Again, this falls under the rubric of "methodology"
> ...how test constraints can in principle affect performance.
Yes, indeed. The CCF allows only 12.5 minutes to complete 50
items. As I said, it's very much a standard test.
>> Although Jensen has found a correlation between psychometric
>> *g* and neural processing speed, this is not the whole story.
>>
>> The effect of the time limit on test scores should be known
>> for every timed test. However, this information is commonly
>> lacking in test manuals. Investigations have shown that, when
>> the items are evenly graded in difficulty and have plenty of
>> "top" (i.e., very difficult items), and the test is not too long for
>> the time available (i.e., the fast students can finish although
>> they reach their difficulty ceiling before the end of the test),
>> giving subjects additional time beyond the prescribed time
>> limit adds very little to the score and has little effect on the
>> rank order of subjects' scores.
> Excuse me, but after staring stupidly at this for a couple of minutes,
> I still find it a bit confusing. So I'm going to request your
> explanation as an experienced statistical psychometrician. It looks
> to me like Dr. Jensen is making a judgment on whether a test is
> "not too long for the time available" on the basis of the performance
> of "fast students". Then he's proposing to confirm this judgment by
> lifting the time constraint and seeing if the rankings change.
Take it simply. If the fast students can't finish the test, whatever
discrimination might have been provided by the harder problmes at
the end is wasted. But that's not as important as the empirical test of
whether the rankings remain the same. A test author could make no
a priori assumptions at all about time limits and simply administer
his test with varying limits, watching for the point where a short
time limit actually degrades the correlation of scores with scores on
the same test administered without any time limit.
> Given equal item weights and equal prorated error and correction
> rates, and given that "difficulty ceiling" (power) is independent of
> speed, the rankings *will* change if low-speed but high-power
> students now have a chance to get to problems they could have
> solved but didn't have time for the first time around, *unless* all
> the higher-scoring students also solve enough additional problems
> to maintain their places in the rankings.
Anything that changes the relative importance of speed and power
can be expected to change the rankings.
> That would seem to mean that EITHER (a) fast higher-scoring
> students didn't really hit their "difficulty ceilings" (otherwise, slow-
> but-brighter students would pass some of them once the time
> constraint is lifted),
"Difficulty ceiling" is relative. A problem that's very difficult with a
severe time constraint may be only moderately difficult without time
pressure.
> OR (b) low-speed students always turn out NOT to be able to
> solve enough additional problems to change their rankings. But
> (b) would constitute a de facto equation of speed and power,
> violating the initial assumption that speed and power are
> independent (the fact that this is seen as a basis to *infer* that
> speed and power are independent merely compounds my
> confusion).
Speed and power are uncorrelated factors. When time limits are
not severe enough to have much effect on rank ordering, the
power factor predominates, as it does in untimed administrations,
but when the rank ordering is affected, the power factor that we
are attempting to measure has been contaminated by the speed
factor.
> Since Dr. Jensen is a genius and an outstanding scientist, I
> must be missing something. Otherwise, I'd have to conclude,
> with all due respect, that standard tests really afford no way of
> evaluating the true"difficulty ceiling" of a given subject (in fact,
> this is exactly what power-IEQ tests alone are designed to
> determine).
Every test has a ceiling, and a sub-ceiling point above which it
can no longer discriminate effectively. If the testee is too smart
for the test, it won't assess that testee's "difficulty ceiling."
> After all, we've all encountered problems that we think at first
> sight are too hard to solve, and therefore skip, only to realize
> later on that we're easily capable of solving them (e.g., with a
> little *persistence*).
Yes. No one is denying this.
[Additional passage covering the same ground snipped.]
> So why are you norming power tests using speed tests? At the
> risk of repeating myself, if a test has a time limit, it is to some
> extent a speed test.
When a test is anchor normed, only the closest comparable
instruments can be used for norming purposes. If they're not
completely parallel, we must make do with this imperfection.
Previous scores are taken as estimates of *g*.
The speed factor is indeed problematic, because the bulk of the
norming data is from timed tests. Other contaminating factors
are less problematical because when many tests are examined
the *g* factor they have in common predominates. But a factor
shared by the bulk of the norming data cannot be filtered out
this way.
>>> But they can assume a dependency relation determined by
>>> neural organization and cognitive mechanics in the brain, as
>>> well as by particulars of neural and cognitive development.
>>> Since you know virtually nothing about cerebral organization
>>> and mechanics, you can't be making blanket statements whose
>>> validity actually depends on a particular set of structural,
>>> functional and developmental hypotheses.
>> Your assumption that you know more than I do about "cerebral
>> organization and mechanics" is not corroborated by any evidence
>> I'm aware of.
> Get real. Anybody with an elementary knowledge of neural nets
> knows more about cerebral mechanics than you do.
People I know who are working on neural net theory are hardly
experts on the human psyche. It's a theory. What's important, even
for *you*, Chris, is whatever understanding you have of experience
*from the inside*.
> Unless, of course, you'd like to tell us that despite all indications
> to the contrary, you're conversant with network architectures and
> algorithms.
Very familiar with the ones inside me. Only a nodding acquaintence
with the academic specialty and business applications.
>> But the statistical conclusion that speed and power are independent
>> factors is based on an analysis that treats the brain as a black box,
>> examining the inputs and outputs. This conclusion stands even if
>> what's going on in the brain is something like the little man who
>> turns on the light when you open your refrigerator door.
> Intelligence is modeled as a black box because there is no a priori
> reason to assume a specific structure, e.g., a structure (including
> architecture and programming) that relates speed and power. How
> you can interpret this to mean that speed and power are necessarily
> independent no matter what the box specifically contains is
> something known only to you. And I suspect it always will be.
You're unclear on the concept of factor analysis.
<snip>
>>> Although it violates your religion, read that essay I wrote in
>>> Noesis/ECE 141. In it, I talk about noise, neural time and space,
>>> and the possibility that sufficiently complex power-IEQ tests may
>>> force the brain to adapt, demanding the subtle modification of
>>> neural pathways over time. In contrast, IQ tests utilize existing
>>> pathways exclusively (that's why one can do them so much
>>> faster).
>> If not all, at least many roads lead to Rome. Solving the complex
>> problems on my tests requires the same kind of mental stretching
>> that is needed for making scientific discoveries about the world.
> To some extent, yes. But it's also true that if one really has what it
> takes to make discoveries about the world, one is apt to be doing so
> instead of taking tests. Again, motivation and persistence enter the
> picture. That's why we should be looking at applicants' records of
> solving hyperdifficult real-world problems. In some cases, we could
> even let the Nobel Committee help us out with qualifying new
> members. For what it's worth, this might also help us get where
> we're going in the world.
That's a slippery slope I'm not interested in staring down.
<snip>
>>> ... If you read my former message to the end, you should
>>> already know that cognition distributes over "objective
>>> reality" by isomorphism.
>> I already know that you *say* it does.
>>> Read on, and this should become clearer to you.
>> I doubt it.
> Well, there it is. Kevin Langdon steadfastly maintains that
> 1. His definition of "isomorphism" is superior to that found in
> the most advanced logic and algebra texts.
> 2. Isomorphisms are one-to-one (injective), but not necessarily
> onto surjective).
>3. Isomorphisms only go in one direction.
1. "My" definition of "isomorphism" is isomorphic to that found
in math books and in Webster's.
2. I didn't say this.
3. I specifically said that an isomorphism is a one-to-one
correspondence, and *not* a unidirectional mapping, as is
clear from my quoted remarks at the beginning of this message.
> At this juncture, I just have to ask: is there a single member of the
> Mega Society who agrees with Kevin on these points (1-3)? Be
> careful - this will go on your permanent record!
When Chris gives a multiple-choice test, the alternatives are always
slanted.
(Continued in Part Two)
Kevin Langdon
Date: Sun, 11 Apr 1999 18:06:16 -0400 (EDT)
From: Langan <clangan@suffolk.lib.ny.us>
To: megalist@brokersys.com
Subject: Re: [MegaList] Reply to Chris Langan, 4/11/99 (Part One)
Rather than type a new introduction, I'll just leave the old one in place
(it still applies).
> > Okay, everybody, break out the Dom Perignon and raise
> > your glasses to Kevin Langdon, who seems to be setting a
> > new personal record for obtusity and obstinacy. His last
> > messages underline as never before what he has proven
> > countless times already - namely, that while he sometimes
> > barely has a point when it comes to statistical psychometrics,
> > he is almost always dead wrong when it comes to anything
> > else, and will never, ever admit it. But this time it's even
> > better. Now Kevin is challenging the entire scientific,
> > mathematical and philosophical world over the meaning
> > of "isomorphism" in the context of formalized theories! He
> > can't win, of course...not today, not tomorrow, not when the
> > burnt-out sun is a dense ball of ash spinning like an icy top
> > in the blackness. But that's what makes it all so...remarkable!
>
> isomorphism, n.: a one-to-one correspondence between
> two mathematical sets; esp. a homomorphism that is
> one-to-one
> homomorphism, n. a mapping of a mathematical group,
> ring, or vector space onto another in such a way that the
> result obtained by applying an operation to elements of the
> domain is mapped onto the result obtained by applying the
> operation to their images in the range.
> --*Webster's New Collegiate Dictionary*
> [I have only included the mathematical definition
> for each term., in accordance with Crhis' usage]
Points to notice.
1. Above, Kevin presents a partial definition of "isomorphism" along with
the technical definition of "homomorphism". Additionally, he fails to
notice that the given partial definition of isomorphism tacitly implies
that it is onto (surjective). [I warned Kevin about relying on Webster's
in this kind of discussion.]
2. Kevin's assertion that isomorphisms can be one-way makes no difference
whatsoever to the thesis under discussion.
3. The proper algebraic definition of isomorphism goes something like
this:
If <A,*>,<A',*'> are monoids, m:A-->A' a map of A into A' such that m(a*b)
= ma *' mb for each a,b in A, then m is a homomorphism of the monoid A
into the monoid A'. If m:A-->A' is a homomorphism, then m is:
a monomorphism if it is injective (one-to-one)
an epimorphism if it is surjective (onto)
an isomorphism if it is bijective (one-to-one onto).
This is taken from the nearest abstract algebra text I could grab,
"Algebra" by Goldhaber and Erlich (Macmillan, 1970). Notice that since
reality has algebraic structure, consisting not just of sets but of
operations among their elements, we have to talk in terms of algebra. So
as we see, when Kevin says "isomorphism", what he really means is
"monomorphism" (even if he claims that in some strange way, this means
that he means "isomorphism" after all).
Kevin's confusion comes down to this. In the context of simple sets, we
can get away with saying that an isomorphism is "one-to-one" because, if
this is applied to both sets, surjection (onto) is logically implied. That
is, if one has two sets A and B and says that there is a 1-to-1 mapping
between them, one is saying that for every element of either set there is
another (unique) element in the other. It follows that the sets A and B
contain an equal number of elements, and that the mapping is surjective
(onto) no matter what its direction.
Thus, Kevin has not only failed to properly track the logical implications
of his set-theoretic definition of "isomorphism" in the context of set
theory itself, but he has failed to account for the additional structure
that algebra brings to set theory. Reality includes not just sets, but
algebraic operations within them. And that's the first installment of
Kevin's math and reality lesson for today.
> What I said, in my message was quoted by Chris in reponse to
> his statement:
>
> [Intervening material snipped.]
>
> >>> An isomorphism is a bijective (one-to-one onto)
> >>> correspondence or similarity mapping between, e.g., a theory
> >>> T and its object universe T(U). In the absence of a clear
> >>> difference relation between the range and domain of an
> >>> isomorphism, it can be contracted so that range and domain
> >>> are coincident.
>
> >> An isomorphism is just a one-to-one correspondence. Chris is
> >> seems to think that any one-way mapping implies a two-way
> >> mapping, which isn't necessarily true.
>
> > An isomorphism is surjective (onto) as well as injective (1-to-
> > 1). It is virtually always reversible unless it is explicitly
> > stipulated, for usually artificial reasons, that it is only one-way.
> > When it comes to the topic at hand - the correspondence
> > between a valid theory and its object universe - the isomorphism
> > is always two-way. That's what the "valid" means. To reiterate,
> > the theory "selects" its own object universe (which may not be
> > as large as the one that the theorist originally had in mind).
>
> I'm tired of saying that things are unclear on my own. Therefore,
> I'm going to indicate certain points regarding which an empirical
> determination of general understanding or nonunderstanding
> seems desirable. This will be test point #1. I will also ask you at
> other test points whether you follow what Chris is saying.
Once again, folks, Kevin is asking whether you agree with him regarding
the term "isomorphism" in the context of formalized theories. So crack
your knuckles and get going on those keyboards! Kevin wants to identify
you as a member of his cult, and I want to know what level of rationality
I'm dealing with on this list.
> > Now I'm going to prove that if you don't introduce an artificial
> > directional constraint, a one way isomorphism M:A-->B always
> > implies a two-way isomorphism M':A<-->B. Take two n-element
> > sets A and B, one the source and one the target of M. Draw the
> > one-way correspondences between the elements of A and those
> > of B as one-way arrows: M = {a1-->b1, a2-->b2, ..., an-->bn}.
> > To get the two-way isomorphism M', just replace each one-way
> > arrow with a two-way arrow: M' = {a1<-->b1, a2<-->b2, ... ,an
> > <-->bn}. This proof generalizes easily to higher-order relations
> > of elements.
>
> And 1 + 1 = 2.
Finally, Kevin manages to locate and correctly parrot a mathematical fact.
> > It's for this reason that all logic and abstract algebra texts define
> > an isomorphism as follows: "An isomorphism is a
> > homomorphism that is bijective, or one-to-one onto (injective
> > and surjective)."
>
> That vocabulary is a little different from that which was used
> when I was in college, but it should be clear that we're all
> talking about the same thing--a one-to-one correspondence
> between two mathematical objects. What Chris is saying is that
> there is a metaphysical relationship between a cognitive model
> and that which is modeled. I say he hasn't proven it.
I took my definition from a 1970 abstract algebra text in wide use in the
early 70's, so (unless Kevin is a fossil) the terminology was current
among logicians and mathematicians near the time Kevin was in college.
A model is an isomorphism M:T<-->U(T) (full definition) of a theory T and
its object universe U(T). The model M, because it has been constructed to
embody the theory T and possesses real, concrete existence in the object
universe U(T) of T, is an intersect of T and U(T). This is not what Kevin
calls a "metaphysical relationship"; it is an identity relationship of
scope limited to the model itself. It's analytic...it follows from the
proper definition of "model" (which again, I've taken from Mathematical
Logic and Formalized Theories, Rogers, 1971).
Give it up, Kevin. You can't win.
> >>> In this sense, the abstract structure of a valid theory (as
> >>> opposed to the pattern of neural potential encoding it within
> >>> a brain) is virtually identical to its object universe.
>
> >> What do you mean "virtually"?
>
> > "Virtually": for all practical purposes.
>
> How do you know what purposes will turn out to be practical?
By reliance on logical tautologies that cannot be violated because they
are necessary rational ingredients of experience.
>
> >> What I've seen so far of the CTMU does not justify either the
> >> existence of the relationship claimed between cognition and reality
> >> or Chris' claims for the importance of his theory.
>
> > Nonsense. The contractible isomorphism M:T<-->U(T) establishes
> > the relationship; you're just in a state of denial. And by your own
> > admission, you can't understand what I've written of the CTMU, so
> > you can't say what it might or might not have established.
>
> I can't make a silk purse out of a sow's ear, but that's not a deficiency
> on my part.
The problem is, *your brain* is a sow's ear.
> > There comes a point at which any philosophical discussion exceeds
> > your evidently primitive notion of "plain English", Kevin. If I were
> > using the relevant math symbology here, you'd *really* be hurting.
>
> I don't think that you care much about that--so why *aren't* you
> using "the relevant math symbology"?
At this point, I am.
>
> <snip>
>
> >> And how can I be a "philosopher"? I haven't been elected by the
> >> ISPE.
>
> > No. But you did manage to get yourself booted out of it on grounds
> > of inveterate Nietzscheism.
>
> The five founders of TNS were kicked out of the ISPE for founding
> TNS to provide ISPE members with a democratic alternative. Only
> someone intent on making points at the expense of honesty would
> cite this as anything but a badge of honor.
If that were true, then you wouldn't be the bugaboo of TNS today.
>
> >> > Only Kevin can answer that question, and he has never done
> >> > so honestly. But no matter what his problem may be, it is
> >> > clear that his style of thought, and his summary rejection of
> >> > the work of qualified Mega Society members, is not to be
> >> > emulated. Indeed, his chronic irrationality has the potential
> >> > not only to keep us eternally in obscurity, but to make
> >> > everyone in the group look like a fool...like a twittering fife,
> >> > peeping piccolo or tooting tuba in the goose-stepping
> >> > "marching band" that he perceives whenever he arrogantly
> >> > surveys his imagined dominion.
>
> >> Projection.
>
> > Did you or did you not call the Mega Society a "marching band"?
> > Tell the truth, now...some of us still have the issue in which you
> > printed it.
>
> But most of us won't go to the trouble of looking it up, so I've
> reproduced it here (from *Noesis* #136, December 1997, p. 17):
>
> Chris Langan complained that the unofficial Mega Society
> Web sites don't present his version of how things are in the
> society. But what is going on here is sort of like a marching
> band with a tuba player who marches off in one direction
> with his little dog (unfortunately not toilet-trained) while
> everyone else goes in another (because the tuba player never
> listens to any of the other instruments and marches with his
> eyes closed). Naturally, an impartial observer asked "Where's
> the band?" will not point to the wayward tuba player.
>
> Extending the analogy further, the tuba player is always playing
> wrong notes and he keeps trying to trip the other members of the
> band (without much success because he's very clumsy). It should
> also be noted that the members of the band are volunteers, as are
> the band leaders, and have not been leaned on by the Mafia.
My position regarding the above excerpt has always been that the little
martinet waving the conductor's baton in front of the band can only be
the one who* sees* a marching band when he looks at the group in question.
Don't be misled by Foghorn's casual analogies, folks. Kevin's real
thoughts and intentions always come percolating to the surface despite his
best efforts to conceal them. And as his analogy shows, his psychological
idiosyncrasies happen to include a John Phillip Souza complex.
Okay, that completes the section of Part 1 in which Kevin is 100% full
of turkey guano. Now we move on to psychometrics, where Kevin can at times
provide useful information. I'm actually impressed with a little of what
he says here.
> > <snip>
>
> >> The "chronometric" tests that Jensen and other investigators have
> >> developed use a much more radically different methodology
> >> than the high-range power tests, but they correlate moderately
> >> well with the standard tests.
>
> > Seriously, have you ever wondered why that is? I mean, isn't speed
> > the main factor in a chronometric test?
>
> According to Jensen, the personal tempo factor does not correlate
> with speed of neural processing. The former predominates in timed
> IQ tests and the latter is the main ingredient in power tests. Why
> should the latter relation hold? Because human memory is volatile.
> You've got to be able to set a problem up fairly quickly for short-
> term memory to be able to hold onto it long enough for you to see
> the solution.
Sounds reasonable.
>
> >> The "statistical cover charge" is publication in reputalble
scientific
> >> journals of data on our tests and successfully defending it, and we
> >> haven't done that yet, though a few academic psychometricians
> >> have taken an interest in the new tests.
>
> > Not quite. The statistical cover charge equals the emergence of
> > several power-IEQ tests that correlate just as well with standard IQ
> > tests as does the average IQ test itself.
>
> You're assuming that the "average IQ test" is an optimal instrument
> for measuring *g*; this is not the case. It would make more sense to
> look for correlations with the *Raven Advanced Progressive Marrices*,
> which is known to be an almost-pure test of *g*. (Unfortunately, there
> seem to be problems with the norming of the Raven; sources don't
> agree on the principal test parameters, but that's not a problem when
> we're just looking at the correlation of raw scores with another test.)
As I've pointed out, *g* is a statistical construct derived from data
generated exclusively by IQ tests. Now you seem to be treating it as an
objective mental attribute, saying that it exists in its own right (rather
than as a relationship of the mind to the specific devices from whose data
it was abstracted). Maybe you're right; maybe you aren't (I suspect that
you agree with me that g is probably nothing but basic neural efficiency).
But I suggest that you actually provide an exact functional definition of
it before proceeding.
> > We'll be asked to produce statistics that are clear and extensive
> > enough to unequivocally counter arguments that could presently be
> > constructed on the weakness of IQ-IEQ correlations across the
> > entire power-IEQ genre...i.e., the genre characterized by the
> > methodological distinction on which they'll focus.
>
> Absolutely.
>
> > And time constraints aren't even the sole methodological issue.
> > There's the related issue of item complexity. Surely Jensen realizes
> > that you can't just monkey around with the time constraint and
> > neglect the ramifications of using extremely complex items...items
> > much more complex than those on any timed test. There are simple
> > mechanistic arguments to the effect that different mental parameters
> > may be involved (and mechanistic arguments are generally superior
> > to statistical ones).
>
> I've quoted Jensen with regard to time contstraints. He's also stated
> that the crucial distinction is between items with no complexity at
> all (he gave the example of the "Making X's" test, in which the
> subject is simply asked to make rows of X's) and tests involving
> more complex mental processes. From that point of view, there is
> little difference between standard IQ tests and the high-range power
> tests.
You're ignoring the noise factor (how the brain deals with noise generated
by more associative complexity) and the notion that the brain may be
forced to adapt (modify synaptic weights) to solve extremely complex
problems. The modification of synaptic weights is governed by the first-
order learning function, which is not equivalent to g (basic neural
efficiency). A second-order learning function equivalent to a "second
order of g", governing the modification of the first-order learning
function, may also be involved (and so on through higher orders of g).
Jensen's analysis does not allow for all possible constraints affecting
these functions. So what you seem to be saying is that a certain amount of
first-order g, or basic neural efficiency, is necessary but (to our
knowledge) insufficient for the solution of hypercomplex problems. But the
transition from simple g to higher orders of g - and I ask that you humor
my possibly nonstandard terminology here - is not guaranteed to be linear.
So how can we assume a linear extrapolation from IQ?
Most of the preceding paragraph is presented as conjecture, of course,
not as fact. But we have to allow for all possibilities if we want to be
scientific.
> >> Although Jensen has found a correlation between psychometric
> >> *g* and neural processing speed, this is not the whole story.
> >>
> >> The effect of the time limit on test scores should be known
> >> for every timed test. However, this information is commonly
> >> lacking in test manuals. Investigations have shown that, when
> >> the items are evenly graded in difficulty and have plenty of
> >> "top" (i.e., very difficult items), and the test is not too long
for
> >> the time available (i.e., the fast students can finish although
> >> they reach their difficulty ceiling before the end of the test),
> >> giving subjects additional time beyond the prescribed time
> >> limit adds very little to the score and has little effect on the
> >> rank order of subjects' scores.
>
> > Excuse me, but after staring stupidly at this for a couple of minutes,
> > I still find it a bit confusing. So I'm going to request your
> > explanation as an experienced statistical psychometrician. It looks
> > to me like Dr. Jensen is making a judgment on whether a test is
> > "not too long for the time available" on the basis of the performance
> > of "fast students". Then he's proposing to confirm this judgment by
> > lifting the time constraint and seeing if the rankings change.
>
> Take it simply. If the fast students can't finish the test, whatever
> discrimination might have been provided by the harder problmes at
> the end is wasted. But that's not as important as the empirical test of
> whether the rankings remain the same. A test author could make no
> a priori assumptions at all about time limits and simply administer
> his test with varying limits, watching for the point where a short
> time limit actually degrades the correlation of scores with scores on
> the same test administered without any time limit.
>
> > Given equal item weights and equal prorated error and correction
> > rates, and given that "difficulty ceiling" (power) is independent of
> > speed, the rankings *will* change if low-speed but high-power
> > students now have a chance to get to problems they could have
> > solved but didn't have time for the first time around, *unless* all
> > the higher-scoring students also solve enough additional problems
> > to maintain their places in the rankings.
>
> Anything that changes the relative importance of speed and power
> can be expected to change the rankings.
>
> > That would seem to mean that EITHER (a) fast higher-scoring
> > students didn't really hit their "difficulty ceilings" (otherwise,
slow-
> > but-brighter students would pass some of them once the time
> > constraint is lifted),
>
> "Difficulty ceiling" is relative. A problem that's very difficult with a
> severe time constraint may be only moderately difficult without time
> pressure.
That's interesting.
>
> > OR (b) low-speed students always turn out NOT to be able to
> > solve enough additional problems to change their rankings. But
> > (b) would constitute a de facto equation of speed and power,
> > violating the initial assumption that speed and power are
> > independent (the fact that this is seen as a basis to *infer* that
> > speed and power are independent merely compounds my
> > confusion).
>
> Speed and power are uncorrelated factors. When time limits are
> not severe enough to have much effect on rank ordering, the
> power factor predominates, as it does in untimed administrations,
> but when the rank ordering is affected, the power factor that we
> are attempting to measure has been contaminated by the speed
> factor.
We can only say for sure that speed and power are uncorrelated in the
range Jensen's IQ tests are measuring (because of the possibility that
noise-suppression mechanisms and higher-order learning functions are
involved at extreme levels).
>
> > Since Dr. Jensen is a genius and an outstanding scientist, I
> > must be missing something. Otherwise, I'd have to conclude,
> > with all due respect, that standard tests really afford no way of
> > evaluating the true"difficulty ceiling" of a given subject (in fact,
> > this is exactly what power-IEQ tests alone are designed to
> > determine).
>
> Every test has a ceiling, and a sub-ceiling point above which it
> can no longer discriminate effectively. If the testee is too smart
> for the test, it won't assess that testee's "difficulty ceiling."
Are you saying that if a subject does not reach the difficulty ceiling of
a standard IQ test, he will be able to solve none of the more difficult
problems on one of your tests?
>
> > After all, we've all encountered problems that we think at first
> > sight are too hard to solve, and therefore skip, only to realize
> > later on that we're easily capable of solving them (e.g., with a
> > little *persistence*).
>
> Yes. No one is denying this.
>
> [Additional passage covering the same ground snipped.]
>
> > So why are you norming power tests using speed tests? At the
> > risk of repeating myself, if a test has a time limit, it is to some
> > extent a speed test.
>
> When a test is anchor normed, only the closest comparable
> instruments can be used for norming purposes. If they're not
> completely parallel, we must make do with this imperfection.
> Previous scores are taken as estimates of *g*.
>
> The speed factor is indeed problematic, because the bulk of the
> norming data is from timed tests. Other contaminating factors
> are less problematical because when many tests are examined
> the *g* factor they have in common predominates. But a factor
> shared by the bulk of the norming data cannot be filtered out
> this way.
You sound like there exist reliable factor analyses of power IEQ tests. Is
that true?
> >> Your assumption that you know more than I do about "cerebral
> >> organization and mechanics" is not corroborated by any evidence
> >> I'm aware of.
>
> > Get real. Anybody with an elementary knowledge of neural nets
> > knows more about cerebral mechanics than you do.
>
> People I know who are working on neural net theory are hardly
> experts on the human psyche. It's a theory. What's important, even
> for *you*, Chris, is whatever understanding you have of experience
> *from the inside*.
First, brains consist of neurons, and are therefore neural networks. So
certain aspects of the basic mechanics are present in the most general
neural models. And second, do you mean that you have to belong to the
"Neural Net Club" in order to examine neural nets, even when the Internet
offers many good simulation packages for nothing? You must be very unhappy
outside of academia, Kevin. Why don't you try to go back? The worst that
could happen is that they'll bleed you dry and make you take a lot of
courses about things you already know.
Meanwhile, I'm on the inside of any field I *want* to be inside of,
intellectually speaking. That's what a high IQ is good for. And what I'm
saying to *you*, Kevin Langdon, is that I know more about the subject than
you do. Probably a *lot* more!
>
> >> But the statistical conclusion that speed and power are independent
> >> factors is based on an analysis that treats the brain as a black box,
> >> examining the inputs and outputs. This conclusion stands even if
> >> what's going on in the brain is something like the little man who
> >> turns on the light when you open your refrigerator door.
>
> > Intelligence is modeled as a black box because there is no a priori
> > reason to assume a specific structure, e.g., a structure (including
> > architecture and programming) that relates speed and power. How
> > you can interpret this to mean that speed and power are necessarily
> > independent no matter what the box specifically contains is
> > something known only to you. And I suspect it always will be.
>
> You're unclear on the concept of factor analysis.
No, you're unclear on the meaning of "black box". And factor analyses of
IQ test data do not amount to an exhaustive categorization of neural
architecture and programming. You rely too much on statistics. They can
get you only so far. If Einstein had your fixation on statistical methods,
we'd have boxcars full of experimental scatter diagrams instead of General
Relativity. We need a *real* theory of intelligence.
>
> <snip>
>
> > To some extent, yes. But it's also true that if one really has what it
> > takes to make discoveries about the world, one is apt to be doing so
> > instead of taking tests. Again, motivation and persistence enter the
> > picture. That's why we should be looking at applicants' records of
> > solving hyperdifficult real-world problems. In some cases, we could
> > even let the Nobel Committee help us out with qualifying new
> > members. For what it's worth, this might also help us get where
> > we're going in the world.
>
> That's a slippery slope I'm not interested in staring down.
Better get used to it. What it means is that at some level of selection,
your tests may actually select for an *inability* to make the kind of
great contributions you're talking about. That is, they may select for
people who can't tell important problems from trivial ones (from a
social-historical viewpoint). But distinguishing the important from
the trivial is what enables real geniuses to get to the heart of
super-complex real-world problems that may never be approximated by your
tests.
>
> <snip>
MY FIRST LIST:
> > Well, there it is. Kevin Langdon steadfastly maintains that
> > 1. His definition of "isomorphism" is superior to that found in
> > the most advanced logic and algebra texts.
> > 2. Isomorphisms are one-to-one (injective), but not necessarily
> > onto surjective).
> >3. Isomorphisms only go in one direction.
KEVIN'S REVISED LIST:
> 1. "My" definition of "isomorphism" is isomorphic to that found
> in math books and in Webster's.
> 2. I didn't say this.
> 3. I specifically said that an isomorphism is a one-to-one
> correspondence, and *not* a unidirectional mapping, as is
> clear from my quoted remarks at the beginning of this message.
MY SECOND, DE-REVISED LIST:
1. No, it isn't. [Webster's, maybe, but that isn't a math book.]
2. Yes, you did. And you still are.
3. You specifically said that I erred in supposing that an isomorphism was
necessarily bijective. And you said that an isomorphism is one-to-one. So
what you meant was that an isomorphism is not necessarily onto.
>
> > At this juncture, I just have to ask: is there a single member of the
> > Mega Society who agrees with Kevin on these points (1-3)? Be
> > careful - this will go on your permanent record!
>
> When Chris gives a multiple-choice test, the alternatives are always
> slanted.
From Kevin's slanted viewpoint, that's quite true.
>
> (Continued in Part Two)
>
Chris Langan