Reply to Chris Langan on Isomorphisms,
Models and Objectivity

Kevin Langdon

I began this message to the MegaList but then the discussion switched
over to the fire list and I never finished it--until now.

To: fire@prometheussociety.org, mega@yahoogroups.com
From: Kevin Langdon <kevin.langdon@polymath-systems.com>
Date: Thu, 05 Jul 2001 00:15:32 -0700
Subject: [mega] Reply to Chris Langan on Isomorphisms, Models, & Objectivity

On Sun, 11 Apr 1999 18:06:16 -0400 (EDT), Chris wrote:

> Subject: Re: [MegaList] Reply to Chris Langan, 4/11/99 (Part
> One)

>>> 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 Chris' 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.]

I certainly agree that an isomorphism, as defined in mathematics,
is necessarily onto; no set elements are not included in its range.
The key point is that an isomorphism is a mapping that preserves
all operations.

But over and above that, I understand you to be claiming a
metaphysical relationship between a model and the real-world
object which it represents, and that is what I do not admit to have
been established. I also question the way you use terminology,
which I sometimes find ambiguous.

> 2. Kevin's assertion that isomorphisms can be one-way makes
> no difference whatsoever to the thesis under discussion.

I never said that, but I agree that the "isomorphism" discussion
is not central to the real question at hand, which is whether
Chris' logical arguments imply anything about the real world.

> 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).

This "strange way," as was the case with equating "theory" and
"model," is common usage among scientists and mathematicians.
It's not my fault if they're sometimes less than rigorous.

> 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.

It's true that I had not explicitly stated the full mathematical
definition of an isomorphism. I took it as implicit context that
we're talking about the whole of each set.

> 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.

No problem to here, now that these terms have been defined. I often
don't know whether to look up a word or phrase Chris has used
because I don't know whether it's standard or Langanian language.

>>> 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.

If I were interested in founding a cult I sure as hell wouldn't select
my audience for intelligence.

I would appreciate readings from as many of you as possible on the
seven test points in my two messages of 4/11. It's not a question of
whether you're a Langdonoid, a Langanoid, or a hemorrhoid, but of
where the members of Mega stand on the particular propositions in
dispute.

<snip>

>>> 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.

I'm a fossil. I was in college in the early 1960's.

> 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.

Win what? I say that this is unclear. Chris has been saying that his
theory *is metaphysical* and that it *goes to the heart of everything*.
Naturally I'm suspicious of drawing conclusions about "model" and
"reality" that amount to anything more than "the model is constructed
with the intent of duplicating the structure of the modeled
phenomena." If that's what Chris means, I agree with him. If he
means something more, I say it hasn't been established.

>>>>> 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.

Here again, Chris is claiming that logic implies something about the
real world. Although things that are logically impossible seem to
be ruled out, for the most part, in the world we live in, it does not
follow that logic constrains the world to be exactly one way.

<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.

KEVIN'S SECOND, REREVISED LIST:

1. I've responded to this above.
2. Where?
3. That isn't what I said, either. You have a way of stating something and
grafting something else onto it. If I deny the "something else" that doesn't
imply that I'm denying the simple truth embedded in the formulation.
What I denied was that you've proven anything about the real world.


On Sun, 11 Apr 1999 18:45:14 -0400 (EDT), Chris wrote:

> Subject: Re: [MegaList] Reply to Chris Langan, 4/11/99 (Part Two)

>>> Gird thy loins, my fellow Megarians, as duty calls us to beat
>>> down the remainder of Kevin Langdon's brutal assault on the
>>> lexicon of logic, philosophy and higher mathematics.
>>>
>>> On Fri, 9 Apr 1999, Kevin Langdon wrote:

>>>>>> I agree that scientific theories do not generally include models
>>>>>> of the theoreticians themselves.

>>>>> And that's what makes them objective. The mental processes of
>>>>> the theorist are deliberately excluded from scientific theories.
>>>>> When you denied that such theories were objective for that
>>>>> reason, you were wrong.

>>>> That certainly doesn't make them objective. When a pro wrestler is
>>>> trash talking about how he's gonna put his opponent in the hospital,
>>>> his model of the situation doesn't include the mental processes of
>>>> the wrestler himself, but it would not be in accordance with standard
>>>> usage to describe his statements as "objective."

>>> Phenomenal focus is a necessary but insufficient condition for
>>> objectivity (do you understand this distinction?)

>> Approximately, but I'm not sure I have a precise handle on what you
>> mean by "phenomenal focus" or what other element you believe is
>> necessary for "objectivity."

> Phenomenal focus means empirical versus rational focus. Phenomena
> are "out there". Rational processes are usually thought to be "in here"
> ...unless you'd just like to agree with my original point, which is that
> they're both "out there" (and for that matter, "in here").

No, that's the point that hasn't been established.

>>> Trash-talking pro wrestlers fail the objectivity test for another reason,
>>> namely emotional and dramatic contamination. The "theories"
>>> promulgated by these wrestlers are deliberately infused with dramatic
>>> and emotional elements. This is a separate issue, and it has no bearing
>>> on what most of us refer to as "scientific theories".

>> That sinks your boat. Your messages are full of histrionics, hyperbole,
>> accusations, caricatures, gratuitous attribution of motives to others, and
>> putting words into other people's mouths. (Perhaps Chris will tell us
>> about the fish he sees from his "submarine.")

> Anybody but you could easily have separated the personal elements in
> my messages from the theoretical content. Try to stay in focus.

The personal elements are very revealing. Try to expand your focus.

>>>>> What you meant was that scientific theories lack *absolute
>>>>> certainty* due to the impossibility of perfect empirical confirmation
>>>>> in an inductive context relative to a given set of axioms, theorems
>>>>> and postulates. In other words, you don't properly understand the
>>>>> meaning of "objectivity".

>>>> What you meant is that I don't use the term the same way you do.

>>> What I meant was that you don't use it correctly (see above comment).

>> My definition of objectivity accords a lot more closely with standard
>> usage than Chris'. It's just seeing things as they are, the opposite of
>> subjectivity, which takes them in relation to an aspect of oneself as if
>> they had no independent existence.

> Objectivity entails (a) the exclusion of emotions, feelings, and so on
> from one's perspective; (b) a focus on pure perceptions (as opposed to
> emotions, feelings, prejudice, etc.).

This is not true. Objectivity entails the inclusion of everything, while being
attached to nothing.

> For the record, Kevin, I'm no longer interested in your opinions
> regarding "standard usage". We've already seen where it got you in
> the context of isomorphisms.

> This is a philosophical discussion, so we have to use sufficiently
> advanced terminology, not what you can regurgitate out of Webster's.

"Advanced" terminology is unnecessary and confusing if "elementary"
terms have not been defined and understood first--and "regurgitating"
something out of a math book is on exactly the same level.

>>>>>>> You're confusing non-objectivity with any subunary degree
>>>>>>> of confirmation. The concept of objectivity, which is
>>>>>>> defined by juxtaposition to subjectivity, is independent of
>>>>>>> degree of confirmation relative to a given axiomatization of
>>>>>>> a theory. [This is your first really bad mistake.]

>>>>>> Presumably, you are speaking of, e.g., certain mathematical
>>>>>> propositions which are analytically true. You seem to be
>>>>>> further asserting that there are such propositions regarding
>>>>>> the physical world, but you have not established this to my
>>>>>> satisfaction.

>>>>> This is the exact opposite of what was actually stated. Analytic
>>>>> truth, which boils down to syntactic consistency, is not subject to
>>>>> empirical confirmation. Empirical truth, on the other hand, is
>>>>> always subject to confirmation.

...which implies, as I said, that logic does not imply anything about the
real world.

<snip>

>>>>>>>> A model is a model. The real world is the real world. The
>>>>>>>> map is not the territory.

>>>>> Are you spacing out on me here? I'm trying to explain what a
>>>>> "model" is in the context of advanced logic and formalized
>>>>> theories. By definition, according to every scientist on this
>>>>> planet, a model is based on an isomorphism, and in fact an
>>>>> intersection, between theory and universe. That constitutes
>>>>> exactly what you say is missing: an identity relation between
>>>>> "map" (which in your lexicon corresponds to theory) and
>>>>> "territory" (the real world).

>>>> According to *you*, not to "every scientist on the planet." A
>>>> scientist *tries* to make his models correspond to the real
>>>> world, but most scientists would not agree with what you
>>>> wrote about an "identity relation."

>>> Every scientist on the planet, especially if he knows anything
>>> about logic, defines objectivity, model and isomorphism just like
>>> I do. The other ones, and I say this with all due respect, don't
>>> have meaningful opinions on the matter.

>> Chris always assumes that those who disagree with him don't have
>> meaningful opinions.
>>
>> Here's test point #4 (do you follow Chris' definitions?) and #5
>> (do you agree with them?).

> Let me get this straight. Are you attempting to turn technical
> semantics into a popularity contest to be decided by "Langdonoids
> R Us"? The only thing you can "test" by doing that is the cohesion
> of your cult, a topic in which I personally have no interest.

It's not a question of "popularity"; it's a question of confirming or
disconfirming *Chris' remarks* about what has been proven in the
eyes of others.

>>> But after all, they're not logicians; they're scientists. So they only
>>> have to *do* science, not explain what it is.

>> I agree with Chris that many scientists have never thoroughly
>> examined the roots of their discipline.

>>> First, you use "theory" and "model" as synonyms. That's what
>>> you might call a "nonstandard" equation.

>> I've seen a lot of world-class scientists use "theory" and "model"
>> interchangeably.

> Well, then, you've seen a lot of them use it sloppily. So have I...but
> again, what difference does that make? You've already admitted that
> "many scientists have never thoroughly examined the roots of their
> discipline". Some of the roots they neglect are logical.

But not *termino*logical. The purpose of language is to communicate;
"standard" usages are a means to an end. Sometimes less rigorous
language does the job at hand better.

>>> But to answer your question, we're talking about functional
>>> scientific theories here. And while a specific theory only
>>> corresponds to reality up to some limited degree of confirmation,
>>> its generalized cognitive substrate - e.g. logic and mathematics -
>>> corresponds perfectly to reality in a selective way. That is, they
>>> define the architecture and programming of the parallel distributed
>>> processor providing causality with an objective mechanism.

>> Test point #6. Who understands the paragraph above?

> Most qualified members, as opposed to unqualified members, probably
> understand that paragraph. But if anybody doesn't, so what? It was
> merely added as an abbreviated jolt to your memory, to refresh it
> concerning what I'd already expanded on in previous messages. Why
> are you so desperate to divert everyone's attention from the meat of
> this discussion?

Here is another example of Chris' assertions that he knows what others
are thinking. But, as usual, he's wrong. I'm just trying to find the beef.

>>>>> Of course, you might say something like this: "Isomorphism,
>>>>> shmorphism! Nothing whatsoever is reposed in reality; we just
>>>>> assume it has causal structure of its own and try to approximate
>>>>> that structure using our always-makeshift theories." But then I
>>>>> need merely point out that the objective causal structure in
>>>>> question has a certain minimal description, and this is just the
>>>>> structural intersect of all of these temporary theories: namely,
>>>>> information transduction, which we have already equated to
>>>>> generalized cognition. More specifically, it is distributed parallel
>>>>> computation, which is just what one gets by distributing any
>>>>> dynamic theory over any homogeneous local medium. Without
>>>>> this, objective causality would lack the means to function, and
>>>>> reality would be an unpredictable Langdonoid chaos for ever after.

>>>> I am not denying that there are *laws* governing physical and
>>>> psychic phenomena. But these laws must be discovered and verified
>>>> and this is an *empirical* matter.

>>> Did you, or did you not, just affirm the existence of physical and
>>> psychic laws? Because if you did, then you must have been making
>>> a rational statement, because you don't have the empirical data to
>>> identify the laws in question. This means that you are projecting your
>>> rational need for laws onto "objective reality". You're doing it, Kevin.
>>> You're doing it right now, and out of the other side of your mouth,
>>> you're denying that it can be done!

>> This is bizarre. I've observed that certain physical and psychic laws
>> have been demonstrated and established very firmly by many
>> experiments, including my own. There is no "projection" in this.

> You're bizarre, all right. First you said "I am not denying that there
> are *laws* governing physical and psychic phenomena." Presumably,
> you meant that because the existence of laws is obviously a general
> rational prerequisite for the existence of "phenomena". Then you said
> "But these laws must be discovered and verified and this is an
> *empirical* matter." Presumably, you meant that the general rational
> necessity of laws can only be refined into specific laws through
> observation, a point on which we quite agree.

What "rational necessity"? I don't presume that. The world *could*
(logically) be random and without any systematic regularities, in which
case there wouldn't be any scientists and they wouldn't be able to
observe or confirm anything. I *observe* that there are regularities;
it's an empirical conclusion.

> I was speaking, of course, about your apparent concession that laws
> are a rational condition for the existence of phenomena. I assumed you
> were capable of understanding this but had not followed the immediate
> implications. Perhaps I was mistaken. Would you now like to disown
> or modify this concession?

I agree that if there are phenomena and observers who can see them,
they must be governed by laws of some kind.

Although we don't know what forms other universes might assume, one
class of theoretical physical models predicts that there will turn out to be
many universes, each with its own set of laws.


Kevin Langdon