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Topic: Re: Non-Euclidean Arithmetic
Replies: 5   Last Post: Sep 15, 2012 9:30 AM

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kirby urner

Posts: 3,690
Registered: 11/29/05
Re: Non-Euclidean Arithmetic
Posted: Sep 14, 2012 12:46 PM
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On Thu, Sep 13, 2012 at 7:36 PM, Paul Tanner <> wrote:

<< snip >>

> Well, since as I pointed, since a standard belief in God implies a
> belief in some sort of mathematical realism, denigrating an and all
> mathematical realism as "an infantile philosophy of little repute,
> believed in by lesser minds through the ages" implies denigrating a
> standard belief in God as "an infantile philosophy of little repute,
> believed in by lesser minds through the ages."

A lot of believers don't self style as uber-logical and would not
follow you in your claim that belief in God entails God believing in
real numbers or creating in advance every board game, like Monopoly.

The standard model is more that humans are deluded and sinful (fallen)
and most of what they think is beneath God. Exactly how and what God
thinks, as opposed to mere humans, is usually a topic shied away from
by the faithful.

Those who claim what must follow from a belief in God, who assert God
knows how many angels might dance on the head of a pin even if humans
have no clue, are the ones with more hubris. They set themselves up
as authorities and presume to lecture others on the mind of God.

> Just making sure that you know that you are insulting most people
> through what seems to be your militant atheism.

It doesn't make me an atheist to think poor deluded creatures such as
yourself and myself, barely able to think at all, let alone logically,
are unable to share coherently about the content of a mind we can't

Based on your postings here, you have a difficult time even fathoming
the mind of a peer human being. You immediately distort and twist
what you read, as many besides me have pointed out (not that you're
alone in doing this, it's an all-too-human trait).

If I were in need of a priest to explain the mind of God to me (as a
Quaker, I tend to not go through priests), I don't think I'd be
looking for a scholastic who goes on about how God knew about the
surreal numbers before Conway did, even back when Adam and Eve were
yakking with that snake.

By the way, what does Satan know (in a "standard belief in God" you
often find lesser deities -- a kind of polytheism)? Does Satan know
about the real numbers in advance, or just God in your model? Or does
Satan only find out about these things after the fact, by watching
over humans' shoulders? Just curious.

> Note: I am not arguing nor I am going to argue against atheism, even
> your evident militant kind. Just know that are you insulting most
> people.

I'm having a hard time understanding how this is relevant to anything.
Many believers believe in angels, and in some of the older
well-established theologies the angels were antagonistic towards
humans and thought God would be better served if he abandoned
humanity. "Humans are a failure, why not just love us?" is their
attitude. Sounds like ETs today. Like Martians or something.
Hostile, and more intelligent to boot.

In this model, perhaps the real numbers were introduced by the angels
as a deception. God knows they're bogus and not likely to last as an
institution beyond the year 3000 except in the writings of those
trying to stay backward compatible. Our current investment in real
number talk is considered "quaint" and "antiquarian" by the humans of
3000, He already knows that. He sees humans the way the angels do, as
deluded and stupid, but he's omniscient and knows by 3000 we'll
actually know a thing or two.

>> I think
>> it my civic duty to cast aspersions on these inferior memes trafficked
>> in by the purveyors of ignorance.

> More dumping on a standard belief in God, I see.

I do have my standards. Not every foolish belief system gets equal
time between my ears. Life is short.

>> I'm sure if they need to bolster their beliefs, they can do so on
>> their own without my assistance. The mangled "philosophy" you present
>> is so unbelievable on the face of it that I wouldn't no where to
>> begin.


I wouldn't "know" -- meant to go through here and catch the typos but
you beat me to it with a reply.

> "Mangled"?
> It's mangled only in the mind of perhaps some militant atheists.
> I'm only pointing out the obvious, which is that a standard belief in
> God implies some sort of mathematical realism.

Once humans have built aqueducts, freeways, skyscrapers, and invented
chess, then these things exist.

In your model, God already knows about freeways and aqueducts
beforehand. He apparently knows Adam and Eve will sin. He knows
humans will suffer terribly and be tortured. Their misery will be
legendary. This may be why the angels want Him to call it off. From
their point of view, He's probably insane in His old age and humans
are a symptom that's He's really gone of the rails. Satan tries to
reason with Him. "Just let them blow themselves up and put themselves
out of their own misery, let me help" he begs.

Remember the God of the Old Testament seemed to react to events as if
humans had freedom to disobey Him. Indeed, the Standard Model that
I'm familiar with gives humans free will, the ability to err, the
ability to sin. Ergo, they came up with a vain and self-aggrandizing
"mathematics" which they strut and puff about, pathetic little
creatures that they be, a steaming pile of sinfulness. God "knows"
about this BS but He's hardly a fan of it. But He wants humans to
learn from their own mistakes, like any good parent. He knows by 3000
we'll have abandoned these childish fantasies.

> If you deny this obvious fact, then perhaps you could explain how the
> former does not imply the latter, and so perhaps you could explain how
> this denigrating of mathematical realism is not denigrating a standard
> belief in God.

I've been giving some different pictures of belief systems, but you'll
be wanting to retain control over what "standard belief in God" means
i.e. all this stuff about Satan, leader of the angels (the most
misanthropic angels -- guardian angels have gone over to the humans,
so yeah, a kind of civil war) will be dismissed by you as

You will thereby insult all the people with non-standard beliefs, but
you apparently don't mind being insulting as long as what you're
defending is "standard". I'd say this marks you as having a strong
herd instinct, a desire to baa with the most sheep. You don't
encourage heresies. You're probably not a witch burner, but you might
be sympathetic to remedial catechism for those straying from

>> It's very possible to believe in God and yet to think human beings and
>> their pathetic "real numbers" are not godly in any way.

> Hello? Mathematical realism is simply the notion that mathematical
> objects exist outside being thought of by some human mind. Period.

Is a rock a mathematical object? How about a planet? A star? Sure,
why not. You have offered no definition of a mathematical object and
I'm seeing a lot of exquisite processing going on, energy
transformations, according to conservation laws. Math at work,
everywhere, objectively.

Humans didn't know they were on a planet for a long time, or at least
that wasn't a "standard belief". Geology was not deeply understood.
So in that sense, sure, mathematical objects abound, in physical
energetic form, outside of humans knowing about them.

Humans have barely ventured to the bottom of the oceans. What new
mathematical objects will they discover? God already knows what's
there (according to your "standard model", since He designed
everything, but His humans have lots of learning ahead of them.

> That means that if the standard belief in God holds, which includes
> the notion of God as Logos (literally, Logic), that any thought of any
> created being already is known by the Creator - and this holds in
> theism as well as pantheism, then some sort of mathematical realism
> holds. I just don't see why you don't see that if the God of
> scholastic theology of all the major religions exists, then some sort
> of mathematical realism holds. The operative word is "outside" or

Some sort of mathematical realism is acceptable to me. Rocks, stars,
ETs, maybe angels, all exist without humans having a clue about most
of them. Humans have very limited experience and there's much more to
invent and discover.

New generations of mathematician are rising up and won't want to stop
with the enterprise of inventing (e.g. quadray coordinates).

We will have more and more mathematical objects to play with. You say
God already knows about all of them, and He simply withholds this
knowledge out of respect for Free Will (standard model includes that).
Or maybe you don't believe in free will as you've not mentioned it.
It's a hole in your logic as admitting sin means admitting we might be
deluded in believing so strongly in our mathematical objects.

He sees us barking up wrong trees, including in math where
misconceptions may persist for centuries.

> "beyond" the human mind. If such a God of scholastic theology is in
> any way "outside" or "beyond" the human mind, then some sort of
> mathematical realism holds. I repeat: I just don't see why you don't
> see this.

You think God already knew how to play Chess even before humans
invented it. You think that's the standard model, like if I go to
some Jesuit or random Imam and say "did God know chess at the time of
the Big Bang?" that they will answer "Of course, this is the standard
belief, as Paul Tanner was trying to tell you on math-teach, you
should have listened to him."

This is a somewhat empirical claim that you're making, falsifiable and
therefore testable. I can start polling standard believers and asking
them if they think God knew about strip poker and darts even when
humans were still just hunting and gathering and had barely taken up
smoking. He knew about nicotine and the power of advertising. He
knew that someday people would use these mathematical objects to
addict themselves and get cancer. That's "mathematical realism" as
cancer cells are obviously "math objects" that obey the laws of math.

I do agree with you that there is much that is real that humans have
no clue about. Saying "God already knows" is an added expression, a
literary trope, though it's not part of the common core standards and
not part of the belief system of any recognized formal mathematics
today (no axioms deal with God in any formal system I know about -- we
can check Wikipedia).

God did not decree that atheists or even Satan cannot do math (right?
you're posing as an authority so maybe you know). Probably Satan is
way better at math than any human. When the snake was talking to Eve
in the Garden of Eden, he said "just do the math" and she did,
realizing that knowledge of good and evil would probably be helpful in
the long run, even if there was a price to pay.

Eve was smart to listen to an angel and defy God at this point, as His
plan for humanity was to pen them in this petting zoo of a Garden and
not let them invent much of anything. That was his "over protective
parent" phase. He's moved on since then, performing fewer and fewer
miracles according to the standard belief system (or maybe you know
better, given your priestly pose).

> And before you start to make even more mistakes as to what so many
> people actually belief about what they worship, why don't you ask
> people who actually believe in an infinite God whether they think that
> this infinite God they worship is so dumb that this infinite God has
> never thought of irrational numbers - or is capable only of thinking
> of natural numbers or whatever.
> And "pathetic "real numbers""?

The real numbers are but a subset of the complex numbers, which
contain solutions to more polynomials and contain operations that make
rotation much easier. The complex numbers are much more admirable and
interesting than the reals as a subset. C > R > Q > Z > N.

Now that we have the complex numbers, I think we can rename the reals
to "simple numbers" if we like i.e. simples + imaginaries = complex.

Of course this won't catch on; we're pretty much stuck with the word
"real". That's too bad. "Complex" sounds complicated and makes
people shy away, and yet there's so much good math there. We've
inherited some ethnicity's vocabulary for these ideas that no longer
serves our best interests I'd say.

I'm sorry if God has to know all this garbage plus even more garbage
that hasn't been invented yet. You'd think humans knowing it would be
sufficient i.e. why a redundant universe in which everything has to
exist twice, once in the mind of man and a second time in the mind of

In non-standard beliefs, God is an internalized aspect of humanity,
like a spark or light. There's intelligence there, an internal
teacher (this was St. Augustine's idea). The standard ego or cogito
surrounds this light and is the human psyche. It looks to the light
for guidance. However none of the thoughts or beliefs that come to
the cogito are "in the mind of God" necessarily as God is not
projected as having human-like thoughts and emotions (to imagine God
is idolotry and is considered sinful i.e. a waste of time).

God is not ignorant in this non-standard model, just doesn't need to
think thoughts because thoughts result from lag, falling behind, being
aberrational vis-a-vis this very instant (eternity). "I think
therefore I am" is what humans say, but not God. The human psyche is
separate from God and dependent on God for intelligence. That's
imagery I've come across quite a bit. Check Aldous Huxley or Meister

> Boy, it seems that since I've utterly destroyed your position on the
> merits, your "argument" has become nothing but throwing whatever
> insults you can think of towards what you so clearly hate.

I was thinking something similar. Now that I've shown how repeated
addition is quite competent to model two irrationals coupling and
begetting a child irrational, smoothly carrying the analogy from whole
numbers through fractions through R, you've resorted to some weird
quasi-religious set of heresies and/or standard models or whatever.
This is extremely ethnic and parochial, to invoke flash-in-the-pan
beliefs and fantasies and cast me as insulting towards them. This is
how you manage defeat apparently, by going off the deep end with

>> In humans, you're dealing with an inferior life form of less than
>> average intelligence (as ETs go) who have indulged in a kind of brain
>> rot we today call "mathematics".

> This is true evidently only in the mind of some militant, bitter atheists.

No, I think truly God-fearing individuals are capable of thinking of
humans as inferior to the angels. No human mathematician, living or
dead, has ever been as good as Satan at math.

>> It's nothing to be proud of, and if
>> we were more faithful, better people, we would have a much more
>> adequate mathematics than we do today.

> Why don't you show us what this "much more adequate mathematics" would be?

Well, I've been working with the ideas of the Transcendentalist -
Unitarian (Medal of Freedom winner, 11 PhDs etc.) to bring forward the
idea that "the cube" may not be always the best or most exalted unit
to use where volume is concerned.

I'm impressed that the regular tetrahedron divides evenly, with no
reminder, volume-wise, into the octahedron with the same edge length
(4 times) and that these two together fill space in a manner that
aligns with unit radius spheres packing at maximum density.

I'm fairly sure that these concepts will be more mainstream as they're
"with the grain" of the zeitgeist (Holy Ghost).

>> I'd say that's closer to my thinking than mindless obeisance to one
>> particular lineage of human ancestor and their one particular "cruft
>> wagon" (aka "dogma cart" aka "belief system").

> I think someone needs to speak for himself on this "mindless
> obeisance" stuff and his evident worship of Wittgenstein.

I *was* speaking for myself, saying "closer to my thinking". Sheesh.

I do point back to this list from the Wittgenstein list, in case
readers there want to see what I'm up to:

>> Of browbeating people into believing just the right stuff to make
>> advances in that stuff, is par for the course. It's what we do.
>> Mostly, that's what we call "education" (a brainwashing in the local
>> belief system).

> So brainwashing is what you think of what you do?

"Our browbeating...." typo, sorry.

>>> OK, but make that "repeated addition repeatedly redefined" while
>>> recalling that scaling has no problem and requires no redefinition
>>> from day one.

>> I never did any redefining.

> This is not true. You need to stick to the truth. And so we see that
> what I'm doing is necessary: Some of those who push the repeated
> addition agenda so while refusing to be up front about what they do.

What I mean is I came to this thread with a repeated addition meme
already intact. All the "redefining" as you call it happened years
ago. Now that I'm finished redefining for your benefit as well, we
understand it more as a standard God would understand it, as complete
and in no further need of jiggering.

However, I'm not super proud of this model because (a) the complex
numbers are where we should really be focused, and stopping with the
Reals is shortchanging everyone and failing to meet standards for K-12
(if you haven't worked with complex numbers by grade 12, your school
is the pits) and (b) so many multiplication games have nothing to do
with the Reals or its subsets and to dwell in Reals and its subsets
exclusively is a terrible and shockingly poor way to teach math and
should never be done in any school with any pretense of being a

I know that sounds a bit harsh and the floating schools in Bangladesh
(schools on boats, where the waters are rising) may not get to complex
numbers with everyone. I don't fault them for that. On the other
hand, these schools offer Internet connectivity and in principle
access to complex numbers. We may assume the people of Bangladesh
will be getting more of that content as time goes on. I'm in touch
with some of the teachers there, by one or two degrees of separation.

>> Approximation is not a new idea. It was there from the beginning,
>> with estimation, which Devlin is not against.

> Not so, since again, this is about math education and the progression
> through the number systems in question.

So, since again, this is about math education and the progression
through the number systems in question.

Estimation and approximation are there from the beginning and continue onward.

Also, remember we want to talk about the IEEE floating point standard
quite a bit. That's a more important number type in STEM than most.

In other words, in going from N to Z to Q to R to C, we haven't yet
really given a strong enough concept of "number types" or "types" in
general. We've laid some groundwork, but there are many more turns of
the spiral to go.

Thanks to computer languages and their "types", K-16 mathematics is
going to keep morphing. The string type is going to be used a lot
more. People who think mathematics (including multiplication) is just
"about numbers" are truly living in the dark ages.

>> I have to redefine things *for you*.
> No, again, since this is about math education, you have to redefine
> things for the students as they progress through the number systems in
> question.

Right, this pie on the table has a volume of 3 in 5th grade, 3.14 in
8th grade, and pi in 9th grade.

We divide the pie into 3 slices in 5th grade, 3.14 slices in 8th
grade, and have slices of 1/pi in 9th grade (maybe e of them, as
discussed above).

That corresponds to the N, Q, R progression.

But we don't stop there. The real numbers are not the end of the
story by any means and I'm not just talking about pinhead math
(infinitesimals). I'm talking about the complex numbers and the
pretty fractals you can make with them.

The students will discover that they've been using C all along, since
all of their numbers are subsets of C.

Quaternions are also math objects. Given how computer languages make
things more concrete, more graspable, we may get to those more by
2020, even in standard STEM. Not just for math majors or physics
majors. Less hyper-specialization. Less institutionalization of
disciplinary boundaries in support of a corrupt over-specialization
program with which the universities became complicit.

>>> [In reply to what some of the greatest mathematicians in history have
>>> done on the topics of the infinite and the continuum, Kirby writes:]

>>>> "They" -- you're sound so awed by authority.
>> Mathematicians are not universal in their assessment of one another's
>> work. The laity tends to be uniformly awed, but once you start
>> reading the history, you find that they squabble. Kronecker thought
>> Cantor was just silly, indulging in a kind of insanity and calling it
>> math. People jumped on that bandwagon though.

> And Kronecker as a denier was judged by history to be on this issue a crank.

It's easy with the benefit of hindsight to see who wins the popularity
contests. People were slow to jump on my bandwagon (with the
tetravolumes and stuff) precisely because they were hanging back to
see who would be the cranks and who would be the fastest races horses,
the ones to go with, to bet on.

The fact remains, and this was my point: you don't know what's going
to be dismissed as garbage or as a detour in the future. There is no
"they" that's all on the same page. You cannot appeal to "them" as a
unified priest-cast authority. Or rather, you can try, and you often
do. But it's not a good way to make your case.

> Yes, even a great mind can be a crank on some issue.

The fact that you have an algorithm for pairing fractions with natural
numbers, and therefore say they're "countable", but don't have a
similar algorithm for "real numbers"... there are many ways to go from
these basics.

Do you want to invent a notion of "cardinality" and sort sets by the
"countability"? I don't think anyone dictated we had to go that way.
I don't think evolution in mathematics is "forced" by any "one truth".
No, you have creative individuals who make up new language games, and
if it turns out they're playable by others, like Sprouts, or Game of
Life, or Tower of Hanoi, then you're well on your way to
institutionalizing some new math, that's all it takes.

Another board game for the closet. More clutter for the "mind of God".

> Throughout history, many great minds' accomplishments were denied at
> the time by sometimes even other otherwise great minds, and these
> deniers all were in the end judged by history to be at least on those
> issues cranks.

It goes back and forth. "History" is just more people with opinions
and what gets dismissed as crankery can come back later as respected.

In general, there's continued disagreement and maths are not
monolithic. The idea that maths is a singular domain of universal
agreement is a myth.

> It's too bad that you are throwing your lot in with the denying cranks
> of history.

In your language, "denying" rings with "science denying" and you're
this heroic soldier for God who defends the great minds and their
non-crank accomplishments against the ignorant "deniers".

I agree that healthy egos need to see themselves as heroic. Whatever
gets you through the day, whatever floats your boat.

> I see a double standard: You deny and/or denigrate the math you
> dislike, math based on that which is not computable (almost all real
> numbers) and infinity and continua, but you turn around and have a
> problem with what you perceive others to be doing, which is the same
> to math you like, math based on that which is computable and finitude
> and the discrete.

It's not really a double standard. I think people should exercise
their judgement and tastes and be clear about what they think is
beneficial, what's toxic and so on.

I think chess is a great game (Go too) but if it were imposed as a
mandatory thing, and if kids cried themselves to sleep because they
couldn't go on in school given their chess skills weren't up to snuff,
then I'd consider this an abuse of chess, similar to how calculus is
abused today, as a weeder / thinner.

I'm generically a "live and let live" liberal with high tolerance for
difference. I think Mormons should be free to have many wives and the
Taliban should have their own little Disneyland somewhere to practice
their faith, or pockets spread around, compounds, same as Jewish and
Christian sects with weird practices and beliefs.

I'm a secularist in thinking the best way to keep peace among ethnic
communities is to given them wide latitude to do their own thing. But
without oppressing others too much.

It's a fine line where we cross over to "oppressing others too much".
If someone escapes a Mormon compound and doesn't want to go back, I'm
all for offering asylum.

Likewise, I would like to offer more refuges to American kids, as I
think their culture easily crushes and destroys them (literally) in a
cruel and harsh manner. It sends them to die overseas by preying on
reflexes going back to the Indian Wars. There's not much soul
searching. The American people are slow to learn from history, as
Gore Vidal pointed out, because they revel in collective amnesia.

So I try to get support, including from the Feds, for a new brand of
school that's more international / cosmopolitan. I show the
curriculum I'd offer (lots of outdoor stuff, lots of physical work /
activity -- so more like the military in some ways).

I don't say I'm working alone in all this. I have peers. I write
about what I'm doing and with whom, quite a bit. I believe in a
certain level of transparency.

And I'm not really that anti the Real Numbers. I'm more against
dogmatic fundamentalism in all its forms and I find it in mathematics
sometimes, among its practitioners. As a secularist, I don't like it
when the religious crazies gain too much power, so I set aside some
time in my day to counter the brain rot they try to spread. Others do
that too and I'm grateful for their efforts. We've seen what happens
when organized religions gain too much control and it's far from
pretty, humans being what they are.

> (I say "perceive" because I don't think anyone is actually denying
> and/or denigrating the math in question you like. No, I'm not doing
> that. I'm just pointing out the limitations it has with respect to the
> math you don't like, math based on that which is not computable
> (almost all real numbers) and infinity and continua.)

I'm fine with an analog math track continuing its merry way using the
curriculum it already has or creates for itself. I've just wanted to
engage in some green field development along a parallel track, the
digital math track, where we'd finally get more of a footprint for our
free/open tools and the skills needed to use them.

In spending more time on SQL, Venn Diagrams, totatives/totients, group
theory, RSA, ASCII vs. Unicode, non-numeric and semi-numeric
algorithms, I'm definitely countering what I see as over emphasis on a
1900s skill set that is in less relative demand.

We need sysadmins who know sysops more than we need more people good
at integrating by parts. We need people who know what tcp/ip is and
how it works. The schools have been too slow and complacent with
their scientific calculators and the curriculum ideas of the 1950s and

Complex numbers should have gotten a huge shot in the arm from
fractals, along with more dynamical systems, but that's still left to
college teachers for the most part.

I think we'll be more effective recruiters for STEM if we stop
upholding the older standards as the most relevant. I craft
curriculum segments that are deliberately cross-disciplinary, e.g.
linking 1, 12, 42, 92, 162... to the morphology of the virus and the
geodesic dome and the face-centered aka cubic close packing matrix.

These are not ideas that all originate with me. I've been surveying
my culture and planet and made some decisions about how to invest my

I don't think the Real Numbers need more champions. There's a vast
army developing that area and passing that torch. I'm polemical
against the analog math track merely to heighten a sense of contrast
in order to bring a parallel digital track into sharp relief.

Digital or computational math is in the lineage of Discrete Math. I
wrote about how computer science teachers converged on Aug 7 for a
workshop a couple years ago (2009?) to plan this other track, and
Discrete Math standards were handed around, something already familiar
to the Oregon legislature.

People aren't used to digital / discrete math concepts as these were
not the focus in their own schooling. A certain amount of polemics,
or "techno-invective" as Ed Applewhite called it (a mentor / coach) is
required to get the new ball rolling.

I don't think you're easily able to follow the battle the way I see
it, having not done the same homework, and you confuse my distaste for
an overbearing under-informing precalc-calc with "science denying"
and/or "God denying", your pigeon-hole for just about anything you
don't well understand.

I'm standing back to show you more of the map, to provide overview,
but I doubt it will do much good.

I do think what I'm doing is "with the grain" of the zeitgeist which
could be translated into the thought that I think God is on my side.
However, as a secularist who thinks cults, religions, ethnicities
should share the planet and play nice with one another, I'm not eager
to invoke God in my arguments.

I'm not a segregationist like Haim who thinks every culture should
stick to some fixed location, as enforced by some global police.

A lot of atheists and humanists of integrity and intelligence would be
turned off by such "God approves of this message" blabber. I don't
want to disappoint them by sounding like a "standard believer". I owe
it to my peers not to debase myself in that way.

>> The proof God does not exist is that existence is precisely that
>> imperfection which makes one less than eternal. If God exists, then
>> He's not really God, as "existence" is a big step down from exalted.
>> Mere existence is too lowly for a true deity. If God exists, we
>> should pity Him as a poor slob.

> I know, I know, there is all this that the term "existence" is not the
> proper term to use, and all that.

This sounds like a complete cave-in and retraction of everything
you've been going on about re existence and realism, but I'm sure you
don't see it that way and there's no need to keep defending your
theology (beliefs about Theo) to me. I should let you get back to
your simple numbers and analog math track, as conventional and
standard as you want to make it. Go be a hero for science. No need
to mess with me 'n Satan etc.


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