Chapter 1.
DC: So you’re finally going to explain to me what a transcendent machine is.
Muse: That’s the plan.
DC: What is it then?
Muse: Even if I explained the simple version, without context it wouldn’t mean much to
you. I’m afraid this will take
longer than a few minutes.
DC:
Okay,
I’m prepared for that.
Muse: Where
do you want
me
to
begin?
DC:
You don’t know?
Muse: Too many variables.
DC:
At the beginning?
Muse
: The beginning by way of chronology, concepts, level of difficulty?
DC: Maybe a litt
le of each?
Muse: In that case, I suggest mathematics.
DC:
Because it’s
basic
, like language, conceptually abstract, and hard?
Muse: No. While it is
basic
,
it’s
conceptually easy.
Most important, it’s the crux around
which everything else we’ll talk about
revolves.
Without some
fundamental
understanding of mathematics, nothing else
we talk about
will make any sense.
DC:
Will
I need a refresher course in calculus before we begin?
Muse: Not at all. Most of the perceived difficulty with mathematics is human
imposed. All
the complicated symbols, names of processes, and rules relate to one type of math.
Math as a particular tool we’ve invented for lots of practical measurements we need
to get to the moon, and so forth.
Not the kind of math we’ll be talking much
about
here
.
DC: Then what type of math will we be speaking of?
Muse:
The type that, unlike the one we created to solve our
problems, eternally exists as
the
backbone of our universe. Of all universes.
DC: Whoa. You’ve already lost me.
Muse: Okay then, le
t’s begin at the beginning as you suggest.
To make it easier to
understand, I’ll break the topic into four areas and then discuss them in order. So
2
you can keep track, let me give the names of these areas to you at the beginning.
First, there’s the issue o
f humans not creating mathematics, but discovering it. Math
has always been here and always will be. Second, mathematics is not a tool to
understand reality, math is the reality which is constantly being explained to us
through Nature
, which is an approxim
ation
,
at best
,
of what we will discover is the
truth
: mathematics.
Third,
the mathematics that humans use is itself a mere
approximation of this truth.
The real math is a kind of logic or set of principles, not
the
numbers and formulae we currently employ to confuse us as much as enlighten
us.
Fourth, the mathematics to which I am referring here is timeless and does not
change. We may change our views of it, alter our perspectives on the manner in
which it reveals it
self to us, but mathematics is what some might call God.
Everything we are and experience around us finds its source in this God.
DC: So
,
religion plays a part?
Muse: Not really. Philosophy probably, but not religion.
DC:
Why bring up God then?
Muse: Jus
t to let you know the importance I place on mathematics. As I say, it’s critical to
everything we’re going to speak about here.
DC: Gotcha.
Muse:
The first point, then, is that humans discovered mathematics rather than creating it.
DC: Not really a new
idea.
Muse:
None of the ideas we’ll be discussing are really new. I should have said that up front.
What’s new here is the collection of them and the order in which I’ve built a concept.
That of the transcendent machine.
New, at least these days, is really
a matter of
combinations of things rather than a single idea that no one has thought of before.
DC: I get it. So this combination of ideas involves discovery rather than creation of
mathematics.
Muse: Yes.
DC: Didn’t Plato argue something like this?
Mu
se:
I
ndeed
so
.
For example, this quote from Book VII of the Republic:
3
The science (geometry) is pursued for the sake of the
knowledge of what eternally exists, and not of what comes for
a moment into existence, and then perishes. [
The Republic
of
Plato
Bo
ok VII, trans. by John Llewelyn Favies a
nd David James
Vaughan (1908
)]
DC: That’s good, but it doesn’t really prove he believed that the universe is a reflection of
mathematics rather than the other way around.
Muse: How about this then?
Wherefore also t
hese Kinds (elements) occupied different
places even before the universe was organised and generated
out of them. Before that time, in truth, all these were in a state
devoid of reason of measure, but when the work of setting in
order this Universe was bei
ng undertaken, fire and water and
earth and air, although possessing some traces of their own
nature, were yet disposed as everything is likely to be in the
absence of God; and inasmuch as this was then their natural
condition. God began by first marking t
hem out into shapes by
means of forms and numbers.
[
Timaeus
53ab, trans. R. G. Bury,
in
Plato: Timaeus, Critias, Cleitophon, Menexenus, Epistles
(1929), 125

7]
DC: Much better. The part “God began by first marking them out into shapes by means of
forms an
d numbers” brings religion back into it though.
Muse: It does. However, whether God is math, or whether God is not math, the fact is the
Platonists believed t
hat math was the source of the U
niverse as we know it.
DC: True. At least from these sources.
Muse: But there are many others. For example,
Euclid said that “The laws of Nature are but
the mathematical thoughts of God.” The astronomer Galileo Galilei observed in 1623
that the entire universe "is written in the language of mathematics
."
(Galileo Gal
ilei,
4
Il Saggiatore
, 1623)
Bertrand Russell noted
that “Mathematics takes us still further
from what is human, into the region of absolute necessity, to which not only the
actual world, but every possible world, must conform.”
And
his kind of thought
persi
sts. Roger Penrose, famed professor at the University of Oxford and still alive
states in his book
The Large, the Small, and the Human Mind
, likes to think of the
world more appropriately as “emerging out of the (“timeless”) world of
mathematics.”
(p. 2)
H
e further states that “One of the remarkable t
hings about the
behaviour of th
e world is how it seems to be grounded in mathemati
cs to a quite
extraordinary deg
ree of accuracy. The more we understand
about the physical
world, and the deeper we probe into th
e laws of nature, the more it seems a though
the physical world almost evaporates and we are left only with mathematics. The
deeper we understand the laws of physics, the more we are driven into this world of
mathematics and of mathematical concepts.” (p.
3)
D.C. But these are simply beliefs. There’s no proof.
Muse: It also explains many things though. When Penrose says “timeless,” we no longer
have to worry about what came before the Big Bang or after eternity.
In fact, most
problems simply vanish when we
consider that we’ve discovered and not invented
mathematics, and that it, rather than
Nature, is the true reality.
D.C.: But what about G
ö
del’
s Incompleteness Theore
m?
Muse: Ah, the one that says that no consistent system is capable of proving all truths
about
the relation of the natural numbers?
I could, of course, point out his use of the word
“natural” in his conclusion. Nature is an imperfect realization of true mathematics.
And, as we shall see later on in our discussion, numbers are an example of
ma
nkind’s contribution to mathematics, not necessarily the mathematics I am
speaking of here.
There are many other confusing things about the math we use.
Irrational, transcendental, and
imaginary numbers represent some examples. But,
again, we will encounte
r these examples later on in our discussion. Suffice it to say
here, that while I feel my arguments about mathematics persuasive, I grant you that,
so far at least, I have not proven my case, only reinforced it.
D.C.: I’m still waiting for proof.
5
Muse: I probably cannot give that to you. What is proof to me may not be proof to you. But I
will try anyway.
Consider for a minute that you are holding up an apple in your right
hand.
D.C.: Okay.
Muse: Now imagine that you’ve asked me how many apples you
have. What do you expect I
will say?
D.C.: One apple.
Muse: Correct. You have one something in your right hand. We prefer to call it an apple, so
we’ll say you have one apple.
D.C.: Yes.
Muse: Now, hold up another thing in your left hand. Say it’s an appl
e too. Also say you ask
me how many apples you are holding up.
D.C.: How many apples are you holding up?
Muse: One in each hand.
D.C.: Yes.
Muse: But wouldn’t you expect me to say two?
D.C.: Same difference.
Muse: No, it’s not. If I respond that you have
one apple in each hand, it’s the same as saying
you have one apple in your right hand.
D.C.: And one apple in my left hand.
Muse: But the fact that you’re holding up two apples in your mind is quite a different thing.
D.C.: How so?
Muse: Two apples requi
res that we set aside the differences between the two apples. That
we abstract them into a concept. After all, the apples aren’t identical.
D.C.: This is just nitpicking.
Muse: No, it’s not.
Abstraction is a critical matter. The apples aren’t identical. A
ctually, even
if they were absolutely identical, they would necessarily take up different spaces
and thus
not be identical
.
D.C.: Polemics. What has this got to do with mathematics as reality?
Muse: I’m getting there. Bear with me.
Our ability to do this
abstraction not only helps us
with language, it’s imperative for us to do numerical mathematics. And numerical
6
mathematics makes intelligence possible. Without it, we would not be able to
understand much of anything.
We relay on measurements for most ever
ything we
consider. From economics to
biology.
Without our ability to abstract things, even
language would be impossible.
We wouldn’t even have a word such as ‘apple.’
Abstraction is the benchmark of understanding Nature.
D.C.: It’s certainly important. B
ut why didn’t we create it for just such a purpose, rather
than, as you say, discover it?
Muse:
Because there were two apples long before humans. You’re not going to tell me you
think that they only became two apples when we invented the words and then
pro
vided numbers for more than one of them?
D.C.: So you’re saying that abstraction is what we need to be able to do in order to deal
intelligently with the universe and that the ‘two apples’ were here, shall we say, for
the picking all along.
Muse: I am say
ing that, and good humor for you.
D.C.: Then let me say that I am at least partially convinced of your claim that we discovered
mathematics rather than inventing it.
Muse: Let me clarify, though, that to a degree humans have created their own particular
b
rand of mathematics. An interpretation of the reality of mathematics that fits their
needs.
D.C.: You’re contradicting yourself.
Muse: No, I’m not. I’m qualifying myself. Remember that initially I set up four areas, the
first two we've generally discussed. Those are that humans discovered math, and
that math is reality. The third is that we’re not necessarily speaking about the math
that you were taught in school or that scientists use as tools.
This third area is a
condition to remember. That’s all.
You can imagine that our discovery of this
invisible, odorless, untouchable, reality would not be something we could
understand perfect
ly. We are imperfect. And thus we are
somewhat inept at
interpreting the precepts. Not that we’ve done a bad job, mind you. Just that there
are some important differences between our interpretation and the timeless reality
called mathematics.
D.C.: Such a
s?
7
Muse:
For example, this whole number thing.
D.C.: Number thing?
Muse: Yes. We treat mathematics as if it solely consisted of number manipulation. Numbers
of apples, and so on.
D.C.: But numbers are important.
Muse: They are.
Except
they do not in any w
ay represent the big picture. They are, as a
matter of fact, a very small portion of what mathematics, at least the mathematics I
now have you provisionally believing we discovered and that represents reality,
really is.
D.C.: If not numbers, what?
Muse:
What we are about to discuss, I will present in much more detail later on. So if I don’t
cover it thoroughly enough for you at this time, don’t worry, we’ll be coming back to
it.
D.C.: Thanks.
Muse:
Okay, then, numbers are increments. Right?
D.C.: Yes. One
, two, three, and so on.
Muse:
Yes
.
We call this discrete. If you say the number ‘two,’ for example, it’s discrete. It’s
an amount and we both understand that.
It’s also the same number two I might have
said yesterday, and it will be the same number two th
at I’ll say tomorrow if I want
to.
D.C.: Good humor.
Muse: Thanks.
Now here’s the problem. Leaving abstraction aside for the moment, Nature
is not discrete
. It’s constantly changing. We call it continuous.
If we abstract a rock,
call it that, and understand, both of us, what it is, it was a rock yesterday and will be
one again tomorrow. But in fact, Nature is always changing. That rock was not the
same yesterday as it is today or will be tomorrow. In fact, i
t’s not the same from
when I begin to say the word ‘rock’ until I finish say the word.
D.C.: Nitpicking again.
Muse: Not so. Let’s move away from apples and rocks.
How many real numbers exist
between the integer one and the integer two. Real numbers meani
ng decimal ones.
D.C.: By definition, I suppose there would be an infinite number.
8
Muse: Absolutely. No matter how small the number you give me after the integer one I can
give you a smaller one. Ad infinitum.
If you give me a one with twenty

five thousa
nd
zeros and then a one after it, I’ll give you a one with twenty

six thousand zeros and
then a one after it.
And furthermore, there’s an infinity of numbers between those
latter two numbers as well.
D.C.: Therefore, using real numbers, you could never ge
t from one to two.
Muse: Exactly. That’s called the ‘asymtotic paradox.’
Where using a half

the

distance model
for getting to
the finish line, you can never arrive.
D.C.: I’ve always wondered about that.
Muse:
Using discrete numbers to measure things ca
uses all manner of problems when
applied to Nature’s continuity.
The answer to the paradox is that things don’t work
in numerical ways. They work in continuous ways. Thus, numbers can deceive. So
many of our problems with understanding our universe resides
in numbers.
D.C.: So mathematics, the reality to which you’ve been referring, does not include numbers?
Muse: I didn’t say that. But numbers play a much diminished role.
They are good for
describing starting conditions or specific points in time.
But real
mathematics tends
more toward the math that Plato and the Greeks referred to. Geometry.
D.C.: You said that you’ll discuss this in more detail later on.
Muse: Yes. Absolutely. We’ve barely touched the surface.
D.C.: Good.
Though I do have a question.
Muse: What?
D.C.:
It would seem that the world of mathematics, at least part of that world is discrete or
numerical and Nature is continuous. Doesn’t that suggest that Nature is reality
rather than mathematics?
Muse: Good thinking. It’s not true, but you’v
e given this some thought.
D.C.: Thanks.
Muse:
Discreteness
causes
problems. Infinity is one of the most immediately noticeable of
them.
As practiced today, mathematics includes provisions for a variety of sizes of
infinities.
D.C.: How can the possibly be
?
9
Muse: Well, it has to be so. After all, one would have to admit that the even

numbered real
numbers between one and two, while infinite, must be smaller than all of the real
numbers between one and two.
D.C. I suppose so.
Muse: The great mathematician
Georg Cantor worked out many different sizes based on
logic.
D.C.: So, even though infinity means limitless, there have to be different sizes of
limitlessness?
Muse: Something like that.
Of course, all these problems go away when we put numbers in
their p
lace.
D.C.: Not humorous.
Muse: Sorry.
But, in short,
continuousness provides a perfect way for the truth to exist.
Thus, in this way, Nature actually helps us to better understand reality. Mathematics
depending on numbers alone, is not the mathematics of
truth. But I’ll expand on this
later.
D.C.: Good, I’m already getting tired.
Muse: We do have one last area to cover.
D.C.: Being?
Muse: That mathematics is timeless.
D.C.: Didn’t we already discuss that?
Muse: To a degree. But certainly not thoroughly
enough.
D.C.: What remains?
Muse:
To convince you of its truth.
D.C.: I’m convinced.
Muse: I fear you’re just saying that in order to take a nap.
D.C.:
All right
then. Convince me.
Muse:
I’m going to return to the math that Plato and the Greeks referred
to. Geometry.
Close your eyes and imagine a cube. A perfect cube.
D.C.: Okay.
Muse: Now, would you have to see a perfect cube in order to see that cube in your mind?
D.C.: I don’t know.
10
Muse: Have you ever seen a perfect cube?
D.C.: I think so, but
somehow I imagine you’ll tell me I haven’t.
Muse: You haven’t. Perfect cubes don’t exist in nature. In fact, nothing perfect exists in
Nature except
themselves. That is to say, a particular cube of sugar is perfectly itself.
But there’s no way for there to
be another perfect cube of sugar equivalent to the
first one.
Yet you can envision a perfect cube in your mind that will be the same
perfect cube yesterday, today, and tomorrow.
Mathematics is perfect. Nature is
approximation. Of what? Mathematics. It cou
ldn’t be the other way around.
D.C.: I suppose so.
Muse: Then we’ve covered quite a bit in a short period of time. Four ideas concerning
mathematics: discovered not invented, is true reality, is not completely understood
by humans, and is timeless. Etern
al.
D.C.: That wasn’t so difficult.
Muse: I’m glad you thought so. Einstein once wrote that “if you can’t explain it to a six

year
old, you don’t understand it yourself.”
D.C.: Thanks for that.
Muse: I didn’t really mean it the way it sounded. But it’s wh
at he said.
D.C.: And you’ll continue to expand on these thoughts as we proceed?
Muse: I will. You can be sure of that.
11
Chapter 2.
Muse: In this session I would really like to concentrate on this notion of what mathematics
is if it’s not a set of rela
tively simple numerical and discrete formulas.
D.C.: Speak for yourself. They’re not simple from my standpoint.
Muse: They would be if you considered that what mathematicians have constructed are
sets of tools, each requiring a language of principles and
symbols some of which
have different meanings under certain circumstances.
One can make almost
anything complicated by creating problems whose solutions require extensive
translations and cross

translations to solve.
Calculus, for example, is simply a
met
hod for finding speeds and areas by collecting ever

smaller segments of data
until any further collection produces the same general answer.
Of course, working
out the details according to so and so’s particular method can then become seriously
arduous,
eve
n
when the basic principle is simple.
D.C.: Isn’t calculus suppose
d to be continuous mathematics?
Muse: Now we’re getting into definitions that pertain to who’s using the terms. I have been
using the term ‘continuous’ to mean ‘non

numerical.’ In fact, the
term ‘continuous’ in
mathematics has a different meaning. There the term means a continuous function
that produces an output perfectly relative to its input. Continuous means ascending
or descending numbers of similar size. Thus, 1, 2, 3, 4, is continuous
, where
chaotic
behavior in numbers
is not. That is not my intended meaning of the word and I’m
glad you brought it up. My
use of ‘
continuous
’
mean
s
‘
analog.
’
That is, seamless. Non

numerical.
This will become a big deal as we proceed, and thus I will use the word
‘analog’ instead of ‘continuous’ from now on. And, to be
consistent
, I’ll use the word
‘digital’ for numerical.
The asymtotic paradox that I referred to in our previous
meeting is ther
efore digital.
Does this make sense?
D.C.: It does.
And I’m looking forward to see
ing
what a truly analog mathematics looks like.
Muse: Remember, though, I’m not dispensing with numbers here. They will play a role. Just
not
the
role
you may be
used to.
D.C.: Okay.
Muse: First, though, let me expand that quote from Einstein I gave last time we spoke.
12
D.C.: The one about how it’s simple even for a six

year old?
Muse: More exactly, that if you can’t explain it to a six

year old you don’t understand it
yourself.
In principle, mathematics is entirely based on addition.
D.C.: Just addition?
Muse: And here’s proof. Today’s digital computers fundamentally
—
and by that I
mean their
hardware
—
addition. Binary addition. Only zeros and ones. To get subtraction, this
hardware adds a positive and negative number. Four plus a minus two equals two.
Same as subtraction, but it’s addition. For multiplication, just add one of the
nu
mbers to itself the number of times the other number requires. And on and on.
Until we get to the most difficult and complex mathematics we can imagine, it’s all
just addition. Of course, you wouldn’t want to attempt to do what the computer
hardware does a
nd have to compute all of those additions. Those complex formulas
are meant to simplify the
complexity of all those additions.
D.C.: I get that. Of course, unless you take a few years to understand all of this simplicity,
you can’t understand any of it.
Mu
se: Yes. But it’s meant to simplify things. That’s the important point.
And since I’m going
to use some of these simplifications, called formulas, as we go along, it’s important
for you to constantly remember this.
D.C.: For what good it will do me, I wil
l.
Muse: All right, then, let’s proceed to
our discussion of numbers and mathematics.
D.C.: Where numbers have no role.
Muse: Not ‘no role,’ just a diminished role.
Most definitions of mathematics includes the
word number. Thus, it would be impossible for
me, even irrational of me, to speak of
math without including numbers. However, as we go along, you’ll find that discrete
numbers have less and less importance.
D.C.: Understood.
Muse: Then let’s begin with algebra, one of the most important branches of ma
th and upon
which, like addition, most of the rest of mathematics is built.
You already know from
your high school courses what algebra is, but let me present it in a slightly different
way.
Most significantly, algebra is an abstraction of an abstraction.
D.C.: You’ve already lost me.
13
Muse: No I haven’t. We’ve already discussed the fact that uses numbers for apples is an
abstraction that allows us to say that one plus one is two. The ‘ones’ and ‘twos’ here
are abstractions for the apples. Algebra then uses
letters in place of some of the
numbers. The second abstraction.
For example, ‘X plus Y = Z’ is a simple algebraic
formula
that abstracts discrete numbers into variables.
If we say that the X equals
one and the Z equals two, then by various manipulations,
which you probably won’t
need to actually do here, Y equals one. Right?
D.C. Yes. But now that you mention it, it does seem strange that our first abstraction is into
numbers and our second is into letters. Wouldn’t it save time by simply converting
the a
pples into letters.
Muse: True. And a good example of why our discoveries of mathematics has been a long
journey and has taken many historical side trips. It’s why I said that many human
interpretations of the true mathematics are not necessarily true math
ematics.
But
this leap from numbers to letters representing numbers called algebra is incredibly
significant, both for understanding math as we know it today, and the lessening of
the importance of numbers we’ve been discussing.
D.C.: Gotcha.
Muse: Good.
Then let me remind you also that a great deal of the mathematics we humans
use today is for analysis. To solve problems we couldn’t solve in other ways.
How far
away is that mountain?
How many little widgets does it take to make a certain big
widget?
How f
ast is something going at
a certain
instant?
Mathematics is perfect for
solving these and millions of other important questions that arise all the time.
Algebra and its descendants is the perfect aspect of math for working these
solutions out.
And, again,
algebra is an abstraction of an abstraction.
Fewer
numbers.
Therefore we can apply algebraic formulas to many different problems.
Not just apples. Anything.
Algebra is so entrenched in our mathematical minds that
we sometimes forget how incredibly importan
t it is in all of the math that builds
upon it.
And what it does, is diminish the significance of discrete numbers in favor of
‘process.’ The process of principle is what matters more than the individual
numbers
that
our letters represent.
14
D.C.: I’m guess
ing there’s more to this idea of
lesser importance placed on numbers than
algebra.
Muse: There is. But I wanted you to know how critically important this one is before
moving on.
D.C.: You have.
Muse: Good.
So we have two ideas so far. One I’ve just stat
ed and won’t repeat and the
other I stated earlier on and will repeat. The true mathematics I referred to in our
previous meeting refers to creating things and not just analysis. In fact, the
fundamental mathematics that we’ve discovered and interpreted, n
ot created,
primarily creates rather than analyzes.
This latter point clearly diminishes the role
of, for example, many aspects of geometry and calculus, for example, where the
approach is designed to discover speeds, areas, lengths of lines, and so on. Ev
en a lot
of algebra, such as the formula I just used as an example, while important to us as
humans, has little use in the mathematics of eternity we discussed last time. That
mathematics creates universes. And ends them as well. Creation, as we shall see,
is
as important, if not more important, than analysis. I cannot stress this more at this
point, so please remember it.
D.C.: I will.
Muse:
Then let’s return to the notions of digital and analog.
Digital refers to discrete
numbers that cannot, as we’ve discussed previously, precisely describe an analog
world. While digital mathematics can adequately describe measurements at a
certain point in time, they cannot describe them as time proceeds except
in an
approximate way.
While these approximations may be incredibly close to precise,
they are indeed not exactly precise. And the small differences, as will later see, can
make a big difference. We tend, with digital computers, to let the very small
diffe
rences between the two go because they work for all intents and purposes.
Watching a movie, for example. We’ve been able to fool our eyes into believing
they’re seeing analog motion when in fact we are seeing only twenty four frames
—
still pictures taken ov
er time
—
a second.
Our digital records require nearly forty

five
thousand samples per second to fool our ears, but these samples are still nothing
15
but numbers and, as such, do not truly represent analog sound. It’s just that we can’t
tell the difference. No
t that a difference doesn’t exist. If that makes sense.
D.C.: It does.
Muse: Remember now, as we begin to speak of more analog types of mathematics we’re not
going to be speaking of incremental development. That is, don’t expect numbers to
just move smoot
hly from one to another
upward or downward
. We already know
that doesn’t work with the half

the

distance to the goal line
model
.
If I have one
number and expect it to move to the next number incrementally, what’s the next
number going to be? No matter what
number you choose and no matter how small it
is, I will just find one smaller, and so on. Ad infinitum. That kind of process will just
not work.
So what we will see is numbers that won’t seem smooth in the sense that
they represent some kind of time progr
ession. Time doesn’t move incrementally
except possibly in the way we measure it with clocks and the like. Sundials do work
that way, but they are analog.
D.C.: I get it. Sort of.
You have just reinforced the difference between digital and analog. You
haven’t shown me the manner in which the digital can create analog.
Muse: I’m getting there.
Let me give you an example of a simple feedback system first, one
I’m sure you’ve encount
ered before but thought nothing about. This example
involves someone on a stage with a microphone standing in front rather than behind
the loudspeakers that amplify what the microphone picks up.
This creates a loop. In
essence, the person on stage says or
plays something into the microphone
that
turns
the sound into an electric current, amplifies it and sends the signal out of the
loudspeakers as sound again. This sound then, because the person on stage is
standing in front of the speakers, re

enters the mi
crophone, and so on. The result is
what some feel is an annoying whiny whistle
,
that many find obnoxious, some even
painful.
It’s called positive feedback and most performers avoid it like the plague.
In
contrast, many people, mostly composers and performe
rs, find it quite interesting
and have investigated many of the diverse sounds you can obtain using the process.
Even use it in their recorded music.
D.C.: I’ve experienced it.
Muse:
In mathematics, one way to produce the same effect is by using IFS.
16
D.C.
: Ifs?
Muse: Iterated functions systems. A fancy way of saying feedback. It’s an equation that
feeds its output back into its input
. This
process produces an infinite amount of
values for its variable and thus is the first time we’ve dealt with that pesky
notion of
‘time.’ Time, you see, is what’s at the root of analog versus digital. Digital, after all, is
perfectly happy, remaining the same forever. Two plus two is the same now as it
was before the universe existed and will be after it’s gone. Feedback co
nstantly
changes and thus represents change over time. This is what we’ll need to get on with
creation as opposed to analysis.
D.C.: It’s interesting that you make the distinction between analysis and creation. It’s the
difference between science and art,
no?
Muse: In some ways, yes. But scientists do create and artists do analyze as well. We will be
going into this in great depth in a later discussion.
D.C.: Good.
Muse: For now, let’s concentrate on this notion of time. And for that, let’s return to calc
ulus,
that supposedly difficult subject we've discussed before.
D.C.: I thought you said calculus would not be of much help since it is analytical.
Muse: I did say that. Good memory. However, it’s good for us to imagine a way in which we
can flip certain
types of calculus from analysis to creation. Particularly differential
calculus since it deals with time in many important ways.
Its very purpose is to
discover the instantaneous rate of change over time.
D.C.: And integral calculus joins small pieces of t
hings together to determine areas and
volumes.
Muse: Good, you’ve been studying.
Since you have, then you know that differential calculus
achieves its analytical goal by taking ever smaller points in time until a limit
becomes obvious and therefore can acc
omplish something very powerful, a number
value for the rate of change at a certain point in time. I could give you a formula as
demonstration, but we’d be lost in the special symbols when knowing what it does is
more important. Most significantly, differe
ntial calculus deals with time, just as the
IFS do.
D.C.: I should ask this now and get it out of the way, ‘What is time?’
17
Muse: Good question, though I don’t want to spend too much time on it or we’ll never
return to the transcendent machine. The truth i
s, that no one knows for sure what
time is.
We know that it’s a
n unfolding of events that moves like an arrow always
forward. We know from Einstein that it’s relative, not only in our different
perceptions of it, but in its relationship to the speed of lig
ht.
Beyond that, all I can
say is that as we proceed with IFS and differential calculus, we’ll be coming ever
closer to its meani
ng even tho
ugh we’ll probably never get there.
D.C.: Is there an equation for it in the true mathematics we’ve been discovering
over the
centuries.
Muse: No doubt so. But I don’t know the math other than including a variable ‘t’ for time in
formulas that do not give definitions for it.
D.C.: A fly in the ointment.
Muse: But not a big fly. We take time for what it is and move on.
Now let’s get back to
differential equations. They determine the instantaneous rate of change at a precise
time. Remember now, this is a ‘rate of change,’ not the current speed. This is an
important consideration.
D.C.: Got it.
Muse: Therefore, is we disc
over the instantaneous rate of change, what I’ll call IRC for the
time being, it’s telling us something about time. In essence it predicts things like the
future location of the object with the IRC. Maybe that prediction is microscopic,
since the IRC can i
tself change, but nonetheless, we have discovered something
important. With prediction comes analog motion. Time.
If we discover two IRCs for
the same object within microseconds of one another, even if they differ we can
determine a smooth change between t
hem. Thus, we are in a continuous mode
rather than a discrete mode. We’ve overcome the hurdle between numbers
—
digital
—
and continuous motion
—
analog.
D.C.: But this is analysis.
Muse: Yes, but it’s rather simple to turn the process around. Simply give the di
fferential
equation the values for its variables and it produces IRCs. Do it iteratively and we
produce smooth continuous motion.
D.C.: I get that.
Not so difficult after all.
18
Muse: So you see that mathematics built on numbers
—
digitally
—
can indeed create
time.
Or at least create motion in time. It can in fact cross the finish line rather than
constantly having to halve the distance between an object and its goal.
D.C.: Perfect.
Muse: Especially so since we’ll be discussing this in even more detail when we get to
speaking about the transcendent machine.
D.C.: When will that be?
Muse: Not too far away. We have to cover some more fundamental territories before we get
there. One of
those in non

linearity.
D.C.: A mouthful that.
Muse: In some ways an easy concept, and in other ways a very complex one as we shall see.
However, I think that subject should wait for our next discussion.
D.C.: So be it
19
Chapter 3.
D.C.: Nonlinear.
Mus
e: You remembered.
D.C.: I take our discussions seriously.
Muse: So do I. Strictly speaking, nonlinear means ‘not in a line’ and refers to those x,y
graphs that we all drew in high school. If x changes in direct proportion to y then
you get a straight line
when you connect the dots. You remember those.
D.C.: I do.
Muse: Then nonlinear simply means that x and y do not change in direct proportion. For
example, you might remember graphing an exponential curve, where y increases at
faster and faster rates
as
co
mpared with x. This would be one example of a
nonlinear process, though a relatively boring one.
D.C.: Boring?
Muse: Yes. It’s predictable.
D.C.: Predictable is boring?
Muse: Yes. At least when you compare it to the universe. If mathematics is reality and
Nature is an imitation of that reality, then mathematics must be able to accomplish
something non

boring. Unpredictable. After all, Nature is unpredictable.
D.C.: So now your telling me that we want math to create complex, non

boring, and
unpredictable th
ings.
Muse: Yes. And we actually have already discussed that when we spoke of Ifs. You
remember. Iterative functions. Sometimes called difference functions as opposed to
differential functions. When you give an initial value to an IF, it feedbacks on itsel
f
with the consecutive values produced by the feedback being chaotic, an important
word for us as we continue.
D.C.: Chaos is important?
Muse: Very, But we’ll get to that soon enough. For now, it’s just important for us to
remember that for mathematics to
create Nature, it has to produce chaos and Ifs are
one way to accomplish that. Not the only way, but one of the easier ways.
20
Differential equations can produce chaos as well, but that’s not as easy to explain, so
we’ll being with Ifs,
if
you don’t mind.
D
.C.: Cute.
Muse:
I’m going to use what is called the logistic equation as an example.
D.C.: I’m glad logic is a part of the process.
Muse: Actually, the use of the term here refers to a growth parabola that specifies how
population varies with time. That needn’t concern us here, since it has nothing to do
what I’m going to demonstrate. The formula is quite simple and I can say it in
Eng
lish rather than writing it out.
If you take a variable, say ‘x,’ and multiply it by
a
constant,
itself
,
and one minus itself
, and take that as the new form of x and iterate it
you get a sequence of numbers that proves quite interesting depending on the
va
lues
for x and the constant you use.
D.C.: You’re losing me.
Muse: No I’m not. I’ll say it again, and give you an example. I’m going to call the constant ‘c’
and the variable ‘x’ as I said. Let’
s say we give c a value of one and x a value of one
and compu
te the first iteration. One times one times one times one minus one or
zero
gives us zero
.
Using that zero as the new x we find that zero times zero times
one minus zero gives us zero. Using that as a new x and compute one times one
times one minus one or
zero equals zero.
I think you can see where this is going.
D.C.: An infinite number of zeros.
Muse: Right. No big deal. But let’s try a different number for x
—
say .5
—
and see what
happens. One times one times .5 times one minus .5 gives us
.5 times .5 equals
.25.
If
we keep iterating this we find that the results keep getting smaller and smaller
approaching zero, but never actually reaching it.
D.C.: Okay. I basically understand. But two things. One, so what? And two, could we
please
rever
t to formulas so I can see what you’re talking about?
Muse: To answer your first question, I’m not through yet. To answer your second question,
remember that you wanted me to revert. I didn’t force it on you. Mathematical
notation is meant to make things e
asier, not harder. Here’s the equation in terms
that you can, I think, understand:
21
C * X
n
* (1

X
n
) = X
(n + 1)
D.C.: I’m sorry I asked.
Muse: Don’t be. It’s simple once you understand the symbols. The C is the constant, the X
n
is
the current state of X,
and the X
(n + 1)
is the new X that replaces the current X as the
equation iterates.
All you need do is plug in the two pieces of data and off we go.
Here’s what I said before. Follow along as I say it. Let’s say we give c a value of one
and x a value of o
ne and compute the first iteration. One times one times one times
one minus one or zero gives us zero. Using that zero as the new x we find that zero
times zero times one minus zero gives us zero. Using that as a new x and compute
one times one times one m
inus one or zero equals zero.
The only thing that’s new
here is the
‘n’ and ‘n + 1’ subscripts, and all they reference is time. The ‘n + 1’ simply
means it’s a later version.
D.C.: I’ve got it now.
Muse: Good. Now for the answer to your first question. If
we keep the value for x at one or
smaller and fool with the constant c, we will discover many interesting things.
D.C.: I thought ‘c’ meant constant. Constant means it doesn’t change.
Muse: You’re right. We really call c in this case a parameter. This mea
ns that we tend to
change it less often than we do x in our formula.
We might, for example, set c to 2
and then move x between zero and one to find out what happens and then change c
to another number and move x around
, and so on.
D.C.: Can we change the
c to p in the formula then? I hate to be picky, but since I’m new at
this business, I’d like to make sure I understand this when we come back to it later
on.
Muse: Which we will, so I’ll do just that. Here’s the new version:
P * X
n
* (1

X
n
) =
ƒ
X
(n + 1)
Regardless of what you call the variable, however, the results remain the same.
D.C.: Got it.
But you’ve added something like an ‘f’ on the right side of the equation.
22
Muse: Sorry. My fault. Just used to including it. The ‘f’ simply means that the equati
on is a
function of X. For our purposes, this simply means that the equation should be
iterated as we’ve been discussing so far. Don’t let it disturb you. Just another of
those human interpretations of the eternal reality of mathematics.
D.C.: All right. I want to make sure I understand everything as we go along.
Muse: Of course. N
ow, here’s my response to your first question. As we change the
parameter value and experiment with the value of x we get some pretty interesting
results.
Most
of the time we get a set of values that settle down to a particular target
but never quite reaching it as we did with .5 earlier on. But as we slowly up the
value for P we discover that some strange things begin to occur. At the value 3.4, for
example, the
output sequence settles into a flip

flopping set of values that
nearly
repeats.
D.C.: Nearly?
Muse: Yes, the lower values are close as are the upper ones, but they don’t repeat exactly.
And I’m glad you asked that question. For when we move the parameter
value up to
3.75,
the values jump all over the map. Complete madness. There’s no predicting
what number will come next.
In fact, seeing only the output from the function one
cannot reverse engineer the results to find the function. The numbers are perfectl
y
deterministic, but the output is indeterministic. This has always fascinated me.
D.C.: Chaos?
Muse: I’m glad you remembered the term. That’s it, exactly.
It’s a fertile field of numbers,
not one of which is ever the same.
At least as far as we can tell.
One can only test
these things for a certain period of time. Given infinite time and infinite values,
repetition may occur. Who knows? But for our practical purposes, its one big soup of
numbers that will go on forever if we let it.
D.C.: Where does this
parabola you mentioned earlier come it. This population thing.
Muse: Ah, another good question. And good memory. Are you ready for some s
lightly new
ideas at this point?
D.C.: I am.
Muse: Then let’s get on with some diagrams, or what
mathematicians
call m
aps. Typically,
these maps are graphs with some value moving up or down the y or vertical axis as
23
time progresses from left to right.
However, we can also give the horizontal axis x a
different measurement besides time. For example, we could use the x axis
for x in
our formula and the y axis our parameter value and plot the values accordingly.
If
we do this for different values of P with each with several values for x, we find the
following maps helpful.
D.C.: Wow. The parabola that you promised me slowl
y turns to mush. That’s remarkable.
Muse: It is. Especially for such a simple equation.
There’s a lot of Nature in there, and that’s
why it’s important as I go about describing the transcendent machine.
And it’s all
nonlinear. But the type of nonlinear tha
t the term mostly covers is shown in the
lower right map.
And maybe that’s a good place to end our discussion for today.
24
Chapter 4.
Muse: Welcome. I’m glad you returned.
D.C.: Why wouldn’t I?
Muse: I thought you may
have lost heart given that we haven’t yet broached the question of
the transcendent machine.
D.C.: There’s that. But you’ve made all of this preamble very interesting so far. I’d like to
stick with it.
Muse: Good. We’ve made some real progress, I’d hate
to see it go for naught.
D.C.: Me too.
Muse: Okay, then, let’s proceed with attractors.
D.C.: Attractors?
Muse: Yes.
You may remember that when we were discussing the logistic formula last time,
many of the outputs moved for a
while only to close in on a
set value.
D.C.: Right.
Muse: These are called attractors. Because they seem to attract the data to a point, or in
some cases to
a set of points, and then remain there.
This didn’t happen to the
chaotic output. At least yet.
D.C.: Yet?
Muse: Well, there a
re many equations that with the right initial conditions begin chaotically
and then either settle into an attractor temporarily or even permanently. Sometimes
these attractors are quite complex and visually elegant when graphed in phase
space.
It seems lik
e a quirk, but there are infinitely many equations for producing
chaos and therefore infinitely many attractors out there. Many of them ring bells.
D.C.: What do you mean?
Muse: They remind of us of things we find in Nature.
Like leaves, for example.
So mu
ch so,
in fact, that many feel that the relationship between the two are not coincidental.
D.C.: But they aren’t the things themselves. They don’t grow
like leaves
. They don’t produce
offspring like life does.
Muse: No. Other than the fact that they give
the appearance of being a part of Nature, they
are simply models. But let’s leave this area of mimicry for the moment and get back
25
to the logistic equation and its attractors. Remember that the equation we were
using is:
P * X
n
* (1

X
n
) =
ƒ
X
(n + 1)
D.C.
: I remember. I even looked it up on the Internet and found several variants. Different
letters used, and particularly the right hand side of the equation on the left side.
Muse: All kinds of variations. Many reasons for this that I’ll not go into at the m
oment.
What’s important is that you remember the iterative part we discussed and the
impotance of the initial conditions.
D.C.: I do.
Muse: Well, the variations of inputs produces a lot more interesting things than I describes
to you last time. We then sa
w what I’m now calling the approach to a simple
attractor. Initially zero. And then other numbers.
Using other numbers of x and p
produce even more interesting things.
D.C.: Not
leaves.
Muse: No, not leaves. But in many ways just as interesting.
Take the initial values for P as
3.6 and x as .1 and you get a map of X
n
related to
X
(n + 1)
as shown here:
which is very interesting indeed.
You can see the four different points converging on a
single point
as successive rectangles.
D.C.: Interestin
g indeed, but why is it important?
Muse: Because every segment of the attractor imitates the first rectangle at continuously
smaller and smaller scales. It is what we call self

similar. You may have heard the
word fractal used.
26
D.C.: I have. Kind of a buzz
word.
Muse: It’s become so, yes.
It comes from the word fraction, referring to the fractional
equivalency of each of the diminishing in size rectangles here.
D.C.: So this is a fractal attractor?
Muse: Yes. Another term for this is ‘strange’ attractor. Si
nce not all attractors have this
quality, we call these ‘strange.’ Some prefer the word ‘chaotic’ attractor, but mostly
we use the word strange and so that’s what we’ll use here.
D.C.: So we have certain equations iterative equations that produces just pl
ain chaos, chaos
that zeros in on certain numbers called attractors, more intricate attractors
that
seem to cycle around points
, and then attractors that
are
strangely
self

similar. Have
I got it?
Muse: You do. Plain and simple.
But there’s even more. For
example, we can create what is
called a bifurcation diagram that plots the value of p against the points where x has
concentrated after several initial iterations
—
an attractor
—
a
nd get some interesting
results
and this is what you get.
This plotting turns out to be extremely revealing in terms of other
mathematical functions, but let’s not bother with those for the moment. Suffice it to
say, you’ve got the basic points we need to know at this point.
Any questions.
D.C.: Not so far. Excep
t, possibly that we’ve been working exclusively with numbers
and not
continuity.
So, even though we’re producing output rather than just analyzing
things, we’ve left the differential processes you discussed previously. We’re in the
digital world.
27
Muse: Gre
at observation. We are. Good for demonstration, but as we discussed earlier, not
the real thing.
Which brings me to partial differential equations.
D.C.:
Are you sure I’m ready for this?
I still haven’t got a good idea of differential equations
and we’re
on to these ‘partial’ things.
Muse: It’s not a particularly big leap. We can, in fact, discuss both
regular and partial
differential equations
at the same time since partial differential equations
—
we call
them PDEs
—
are just more elaborate multivariable di
fferential equations where we
don’t know enough information about the details to adequately solve them, except
in
straightforward
circumstances. What’s great about them, though, is that we can
use them to create output that exists in time. That is, this ou
tput c
hanges over time
just as our IFS
did, but more elegantly and continuously so.
D.C.: Okay.
Muse:
To
begin,
partial differential equations, like ordinary differential or integral
equations, are functional equations. That means tha
t the unknown, or unk
nowns,
we’
re trying to determine are functions.
This makes them hard to do, but relatively
easy to understand. And, for our uses, we’re not trying to discover the functions, we
will be plugging them in and using them to generate output. So we can just skip
the
hard part. Make sense?
D.C.: Sort of, at least I like the easy part.
Muse: Good.
Now to intimidate you and send readers scurrying for the bathroom. This
represents the dynamics of a pendulum under the influence of gravity:
Here is the
Benjamin

Ono equation representing internal waves in deep water:
As you can see, both of these equations contain symbols that we haven’t discussed. Some,
like ‘sin,’ for example, refer to common trigonometric operations. The ‘d’ references
28
something we
call ‘derivative’ or slope.
But both represent combinations of known
and unknown symbols that would take us several discussions to go through and
understand. But we’re not going to do so. All we want is to know what kinds of
functions or numbers these rep
resent so we can
rearrange them such that the
functions they contain
can
produce output.
D.C.: Wouldn’t it be better to understand them?
Muse: Sure. But for our purposes all we need are the kinds of data we’re going to work with
and then we can play with
these data as if they were knobs on a machine. In fact, as
we will presently
see
, they
wi
ll
become knobs on a machine. The transcendent
machine.
D.C.: Ah, so we’re coming closer to that.
Muse: We are.
But the important thing at this point is to understand
continuous functions
so that we have a basis for producing nonlinear output from a mathematical
initialization
.
While you might not understand all of the human created symbols and
processes, you do understand the concepts, and that’s what’s important.
From
this
point on, we can assume the math will do it’s job and we can tend to the
tuning of
systems of output that
results from our tinkering with the variables, parameters,
and constants that will alter that output in ways that we will find incredibly
intere
sting and important.
D.C.: Sounds great!
Muse: Believe me, it is.
Before we continue, however, let me review a couple of things for
clarification.
We first discussed functions in terms of numbers. You noticed that we
had not considered continuous function
s and so we followed that with differential
calculus and instantaneous
rate of change. We talked about nonlinear mathematics
and I demonstrated them in terms of discrete functions first and then continuous
functions later on. Back and forth. We could have
done it differently by doing the
entire numerical process first and then the continuous one. We didn’t, both because
you sort of dictated our direction and because it makes a certain sort of sense in the
back and forth manner. We must remember that we’ve c
hosen this latter way of
going about it so as not to mix the processes together.
D.C.: I understand.
29
Muse: Good. I figured you did, but I had to make sure. I don’t want the least bit of confusion
at this point.
D.C.: I can’t guarantee that. These partia
l differential equations are not completely clear to
me.
Muse: I understand that. Just think of them as
mathematical equation
s
th
at involve
two or
more independent variables, an unknown function (dependent on those variables),
and partial derivatives of t
he unknown function with respect to the independent
variables.
Whew. I hate to take this much further, but you asked for it. Here’
s the
simpl
est example I can think of:
Nothing much to go on here. Except we consider
to
be the function you need to discover to solve the equation. X and t are two variables,
here being time (t) and some kind of x (usually a spatial coordinate of some kind.
Our use of the word ‘partial’ here results from the idea that we can figure one
deriv
ative while keeping the other constant. Thus, as we attempt to solve the
equation, we’re using only partial derivatives.
D.C.: I’m sort of getting it. However, I’m still being thrown by the symbols.
Muse: And you will continue to be as they, the results
of human interpretations of the
processes involved, are generally inconsistent from mathematician to
mathematician. That’s why I’d rather help you understand the principles and not
take you through the draconian processes in detail.
D.C.: Got it.
Muse: Th
en let’s let this go for the moment, as we have the fundamentals that will serve us
well as we proceed. You might study some of this material
—
attractors, strange
attractors, and partial differential equations on the Web until we meet again, but
don’t let a
ll of the details confuse you. What we have here is a method for creating
incredible amounts of diverse
unordered
materials in which interesting
orders
appear, sometimes to stay and sometimes to appear and reappear momentarily.
And
30
creating these materials
in continuous ways means that they are not simply models,
but representations of the real thing.
D.C.: Representations?
Muse: Yes. We’re not quite to reality yet. We’ll get there next time I see you.
31
Chapter 5.
D.C.: What’s this?
Muse: I assume you’re
referring to the device sitting on the table next to you.
D.C.: I am. It looks curious. And old.
Muse: It’s both. But let’s begin and a bit later I’ll tell you what it is.
D.C.: Is it the transcendent machine?
Muse: Later. We have to get through some more
mundane issue first, so yoku’ll understand
better why it’s here.
D.C.: Okay. More of the usual?
Muse: No math this time. Now we’re going to speak of computers. The kind that you and I
both have and use all the time. We’ve got laptops, desktops, pads, pods
, phones, and
I’m no longer sure how many other things that we’ve developed that use
computational technology. We need to speak about these in some depth.
D.C.: I look forward to it.
Muse: Good. Let’s begin with
your standard laptop computer, though all
computers work
according to the same principles.
D.C.: I brought mine along. Do you want me to get it out for this part?
Muse: Okay. We’re not going to use it, but having it sitting there on your lap makes things
more real like. Notice I used the word
‘like’ just then. To reinforce the notion that
mathematics is reality, not Nature.
D.C.: You’re saying that my laptop is part of Nature?
Muse: In a way. We’re part of Nature and we made it, so it certainly belongs in the realm of
Nature. Right?
D.C.: I’ll
take your word for it. To me it’s not natural. But this is probably just a terminology
thing and not worth of our continuing.
Muse: I agree.
So take a look at it. A very impressive bit of technology wouldn’t you say?
After all, the first computers took up
whole rooms and couldn’t accomplish much of
anything that that laptop can.
And, considering the laptop you have there runs on
nothing but addition. That’s it.
D.C.: I remember we spoke of such things earlier on.
32
Muse: We did. As you probably already know
, the hardware of your laptop consists of on

off switches that use binary mathematics, only zeros and ones, to add numbers
together. Using transistors that take up very little space, these transistors can run
billions of additions per second. Maybe more th
an that by now.
It’s truly incredible.
D.C.: But, as you pointed out in an earlier conversation, it’s numerical. Digital. Discrete.
Muse: Yes.
Therefore, it’s good for modeling t
hings, but no matter how fast it
runs, it will
never catch up to continuous
movement. It will always be slitting things in half,
never getting to the finish line.
D.C.: I remember.
There must be a way to make computers run on as a continuum.
Muse:
There is. And it’s sitting in to the right of you. You noticed it when you walked
in.
D.C.: This is a computer?
Muse: Yes. An analog computer.
D.C.: Must be something new, I’ve never heard of one of these before.
Muse: Not new. Old. Computers began this way but quickly shifted to digital when we
discovered they ran faster, could be prog
rammed more accurately, and could
simulate continuous behavior so closely that one could not distinguish between the
two.
D.C.: So how does this guy work?
Muse: Well, to begin with, every formula you wish to solve has to be wired uniquely. By
wired here,
I mean that these things called patch cords have to be inserted into the
plugs on the front of the computer and configured in such a way that the math
proceeds as you want it too.
D.C.: Seems laborious.
Muse: Quite so.
In fact it takes quite a while to ju
st make certain you have all the cables
inserted correctly.
D.C.: How does it work?
Muse: Well, the little squiggles you see above each plug on the front of the machine
tells the
operator what it does. The inner wiring connects the plugs to resistors, ca
pacitors,
and so on, such that a proper mathematical function is carried out. There are
‘adders,’ ‘subtractors,’ ‘multipliers,’ and so on. All you need to do is to set them up
33
and then use the dials there, we call them ‘pots’ for potentiometers, to set the
values
for the variables and off you go.
D.C.:
But those ‘pots’ as you call them don’t even have any markings around them to ensure
accuracy.
Muse: Remember, we’re not after accuracy since this is not analysis. We’re after creating
something. Nature. The
electricity pushing though this machine to the oscilloscope
there to the right of the computer will tell us if we’re on the right track or not.
D.C.: Bu
t would
n’t a digital computer be far more accurate.
Muse: Accurate to my preferences. However, I have
no preferences. I’m trying to find a
perfect set of variable settings without any idea what the output would be. There’s
no way with nonlinear equations to determine the output, remember?
D.C.: I do. But it just seems a bit strange. I use my laptop for a
ccuracy and speed.
Muse: And you get just that. With this analog computer you’re not after those things. Here
we’re trying to try different continuous equations with some variable values in
hopes of discovering chaos and, maybe in the process, some interesting attractors
and
maybe a strange attractor or two.
D.C.: But hasn’t this been done before?
Muse: In fact it has. In fact, on the campus where we now sit. Back in 1977. But remember
there’s an infinite number of possible outcomes. We could sit here for eternity and
not ex
haust the possibilities, one of which might just be the one that surprises us the
most.
D.C.: In what way.
Muse: Saving that for another day. Again, lots of foundation work to go before then.
D.C.: Okay.
Muse: What’s important now is for you to realize t
hat that little machine that sits to your
left produces Nature in the form of electricity that we can see on the screen of the
oscilloscope to its right.
This electricity is as real as you or I, not the simulations that
digital computers create. That lapto
p is indispensible. It’s an incredible tool that
neither you nor I could possibly do without these days. But, even though its
machinery belongs to Nature, its output is not real. The output of the one to your left
34
is reality. What comes from it is the dire
ct result of mathematics, just as you and I
are.
D.C.: I get it, though it’s hard for me to think of myself as a product of a mathematical
equation.
Muse: Then don’t.
But understand, while we make very few analog computers these days,
those that we do mak
e are critical to certain kinds of studies. Turbulence, for
example, and other types of high chaotic Nature like air turbulence around jet
engines on the aircraft that we fly around in these days needs these guys because
they can create perfect examples of
the situations our airplanes face every day.
And,
of course, you and I need them for our quest for the transcendent machine.
D.C.: Yes.
And we use the analog computer for controlling the electrical flow to produce
chaotic output?
Muse:
Yes.
D.C.: I thin
k if you’d told me that before we’d begun I would have had serious doubts about
continuing.
Muse: I’m sure you would have. But, trust me, when we get through with all this you will
thank me. If not for showing you something incredibly interesting, at least
giving
you notions of your own for how to take these ideas and make them your own. If,
that is, you don’t by then consider me a total crackpot and forget the whole thing.
D.C.: I doubt that will happen. But who knows.
Muse:
Would you like to try using th
is computer rather than you laptop for a while?
D.C.: Sure.
Muse: Then let’s try programming something simple, like the logistic equation. Once we
have the patch cords in the right places, we can play with the pots and see what
kinds of chaos and attractor
s we can discover.
D.C.: Isn’t the logistic equation a digital rather than an analog one.
Muse: Yes. But interestingly enough, when we set it up on the analog computer including
the feedback, we can turn it into an analog equation of sorts.
D.C.: Wow. This
is terrific.
Muse: I agree.
35
36
Chapter 6.
D.C.: I see it’s still here.
Muse: Yes. And remember that what it’s producing is continuous nature and not numbers.
D.C.: Nature versus numbers.
Muse: And so now that you’ve got a feel for how to use this analog computer, and can at
least see what kinds of things it produces, we can begin to speak about
what we
might expect it to do given the perfect circumstances. After all, it’s already
producing
its own universe.
D.C.: What do you mean, its own universe?
Muse:
Well, one definition of universe that you’ll readily find in dictionaries is the realm in
which something exists.
D.C.: I’m used to it meaning everything that exists.
Muse: Everything nat
ural, yes. Okay, we won’t call it a universe then, we’ll call it a
microcosm then. It certainly fits that definition.
D.C.: Fine with me.
Muse:
Then I’ll ask you a question for a change. What do you think that the mathematics we
can produce with our patch
cords on this computer?
D.C.: Lot’s of interesting oscilloscopic pictures.
Muse: That’s it? I mean, given that mathematics created the entire universe, don’t you think
it might be able to do more than
create pictures?
After all,
an
oscilloscope is merely
and representation of what it’s capable of, not the actual thing. Also remember this
is electricity. In the raw, as actual nature, it can produce a shock you probably won’t
appreciate. Could
it
also produce other things?
D.C.: Not sure what you’re getting
at.
Muse: Well, let’s for a second imagine that it might produce natural things with which we
might not be familiar that could, in some ways, resemble more familiar things in our
own microcosm. For example, a rock. Now it couldn’t produce an actual rock
.
But
couldn’t it create something akin to a rock within the boundaries of its own world.
D.C.: World?
Muse: For some reason I like using that word more than microcosm or even universe.
37
D.C.: Okay. But a rock?
Muse: Not a rock as you and I know one, but so
mething similar. Could a non

changing
attractor be a rock

like thing in the electrical world we’re talking about?
D.C.: The attractors we’ve seen are approached but we never seem to get to them.
Therefore, I would doubt a rock would be a good analogy.
Muse
: Excellent word. And I agree attractors would not make perfect rocks in the electrical
world. What would?
D.C.:
Plox.
Muse: Plox? What are plox?
D.C.: Certain kind of attractors.
Muse: But the word has no meaning.
D.C.: It does now. That attractor on the
screen has attributes, characteristics, whatever. And
plox’s definition is something that has those things.
Muse: Excellent perception. But I’d hate to have to name every single attractor by a
separate name.
D.C.: But that’s what we do with Nature.
Muse:
Thank you.
D.C.: For what?
Muse: For teaching me something. This is truly great. You’ve really gotten into this electron
world. We’ll call it EW for electron world, if that’s okay with you.
D.C.:
Or maybe eWorld?
Muse: I’m sure that someone else has come up with that one by now for something else.
Too catchy. So let‘s stick with EW.
But let’s not attempt to name everything in the
EW or we’ll have to create dictionaries, languages, and God knows what else.
D.C.: I
agree.
Muse: We just have to remember that this world has its own Nature. And that it was
created with much the same mathematics as created the rest of the Universe that we
live in. Now we’re into the spirit of this process, let’s see where it takes us.
D
.C.: Sounds exciting.
Muse:
I’
m glad yo
u think so too, because the next step we’re going to take is not so much a
step as a leap.
It’s called abiogenesis.
38
D.C.: Abiogenesis?
Muse: Yes. The first letter ‘a’ means ‘no.’ As in ‘atypical’ or
‘
asymmetrical
,
’ m
eaning not
typical and not symmetrical. The second three letters ‘bio’ refer to biological, and
‘genesis
,’ well, refers to genesis. The
origin of something. So the word ‘abiogenesis’
means ‘life not originating from a biological source. Or, more specifical
ly, life not
beginning in a special instance. Lightning striking the primordial soup and all that.
As we use it today it then means life originating gradually and naturally over time,
and not suddenly. There’s no instant where before there was no life and
after there
is life. There’s plenty of goo that’s both life like and non lifelike at the same time.
D.C.: Lets to digest.
Muse: Yes, but there are clear examples of how this could be, even though for many people
the thought of it brings up all manner of d
ifficult to swallow notions because of their
prior beliefs in religion, and so forth.
D.C.: Could you give me an example of an example?
Muse: Sure. The first that comes to mind is viruses.
The little gizmos that make us so sick
and that we haven’t found a
way to counteract yet. Many biologists consider these
things non

life. Yet, at the same time, they contain RNA, a key ingredient of life. A
precursor of DNA which we all know is extremely important in life. And we know
that once viruses enter a living cel
l, they become clearly alive, reproduce, and all
that. So viruses seem to fall in a vague intermediate stage having characteristics of
both life and non

life.
D.C.: I guess I knew that.
Muse: Some scientists think that crystals are a close precursor of li
fe in that they
reproduce, grow, develop, and so on, while apparently not alive. And there are many
other less obvious examples of this slow development of life and who knows how
many have disappeared over time that would show abiogenesis more clearly.
D.
C.: I’ve got the idea, but I cannot see why it’s important. What’s this got to do with analog
computers?
Muse: Good question.
Remember, analog computers produce real things in our EW, our
electron world. We already have ploxes and who knows how many other
‘things’ in
39
this EW.
If life proceeds naturally from rocks and other things in our Natural world,
shouldn’t it be so that it could proceed naturally from the EW as well?
D.C.: Maybe, but wouldn’t that take billions of years, rather than just a few minutes?
Muse: I have no idea. Actually no one really does. After all, I’m presuming that we’ve
discovered ploxes occurring rather quickly in the EW, it might go faster than we
think. There’s just no way to know without simply trying it.
D.C.: This seems nuts, if
I can be frank.
Muse: You can be frank, though I’d a lot rather you be yourself.
D.C.: Cute. Don’t you think we’d better know what life is? Otherwise, how would we know if
we’ve found it?
Muse: Agreed. And that’s the million dollar question. Since no one s
eems to agree on a
definition, this might stop us right in our tracks.
D.C.: Plus any definition we might agree upon could prove incorrect after we find or don’t
find an example of it in EW.
Muse: Yes. But, I’m afraid, if we’re going to get anywhere near
to the transcendent machine,
we’re going to have to define life first.
D.C.: Then let’s get to it. Even though I must say that I didn’t expect
doing
this
when I
showed up today.
Muse: I bet not. But let’s begin anyway. And let’s limit ourselves to three
attributes of life.
Three things that life must have in order to exist. There will obviously be a zillion
smaller but related attributes that we will have to consider are subsumed within
these three, but so be it. You begin.
D.C.: Yikes. I have no idea
where to begin.
Muse: Okay, then. I’ll start. How about ‘metabolism?’
The physical and chemical processes
that
originate
, maintain, and
destroy something
.
D.C.: Interesting.
Would the planet Earth qualify as having a metabolism?
It certainly has
those cha
racteristics.
Muse: I suppose it would. So would a star. And zillions of other things. At the same time,
however,
this is but one of three characteristics of life. The other two may disqualify
these other non

living things.
D.C.: Okay. I’ll buy metabolism,
but let’s come back to it when we’ve finished. What’s next?
40
Muse:
How about ‘reproduction?’ That’s something all life has in common. And it would
remove Earth and most other things from the list of possibilities.
D.C.: It would. I like it. So, metabolism
and reproduction. And the third entry would be?
Muse:
Memory.
I don’t necessarily mean memory in terms of remembering things with our
minds. After all, amoebas don’t have mind but are alive. By memory here I mean
DNA types of memory as well. Even the simpl
est forms of life contain memories of
past successes and failures. Instructions of what to do and what no to do.
D.C.: Yes. I can understand that being a large part of a definition of life.
Muse: So, a first attempt has us with metabolism, memory, and re
production. How does
that sound to you?
D.C.: Not sure. After all, a whole lot of things have been left out.
Muse: Such as?
D.C.: Breathing, eating, sleeping, and so on, for examples.
Muse: I would consider each of those as a part of metabolism. We breath,
eat, and sleep in
order to maintain our metabolism in some way. See what I mean. If you’re going to
make the definition manageable, you’re going to have to make it possible for many
of the things we do as life forms fit within the terms we choose.
D.C.: I
do see that. But ho
w about things such as learning?
And I don’t necessarily mean
learning as we humans do, but learning in i
ts basic sense. Trail and error?
Muse: Wouldn’t that fall under memory? Memory is what makes learning possible. We use
trail and er
ror to figure things out, yes, but in the end the only way it matters is by
remembering what we’ve achieved.
D.C.: I see where you’re going
with this
.
Muse: That everything about life can be included within one or more of these three basic
premises of li
fe?
D.C.: Yes.
Muse: Well, then, let’s run with this as see where it takes us.
D.C.: One last question.
Muse: Yes?
D.C.: What about the notion of carbon as being the root of all life? Where does that fit
, given
that with electricity we have only electrons
?
41
Muse: Carl Sagan argued for years against what he called ‘carbon
chauvinism
,’ the notion
that just because we have a carbon

water based chemistry does not in any way
preclude other forms of life.
Astrophysicist Victor Stenger writes that:
There is no
good reason to assume that there's only one kind of
life possible

we know far too little about life in our own
universe, let alone ‘other’ universes, to reach such a conclusion.
It is ‘carbon chauvinism’ to assume that life requires carbon;
other chemica
l elements, such as silicon, can also form
molecules of considerable complexity. Indeed, it is ‘molecular
chauvinism’ to assume that molecules are required at all; in a
universe with different properties, atomic nuclei or other
structures might assemble in
totally unfamiliar ways.
And I subscribe to these notions completely.
It seems to me the height of
arrogance to base our definition on the only form of life we know
when the universe is full of other possibilities. Maybe some of them
exist right here on
Earth without us knowing about them.
D.C.: Why wouldn’t we have discovered them by now?
Muse: I don’t know. Maybe they live lives so short that we cannot perceive
them. Maybe we perceive them but just don’t consider them life forms.
I just don’t know. But
I have an open mind.
D.C.: I do as well. However this is sounding more like science fiction than
science to me.
Muse: I can understand your feelings and, surprisingly I’m shocked that you
haven’t expressed such a feeling long before now. Much of what we’
ve
been discussing, from the idea that mathematics represents reality
and that nature a mere reflection of that reality, borders on sci

fi.
D.C.: I suppose I didn’t mention it because up until now these things seemed
philosophical. Now, with this machine y
ou have here, you’ve brought
them right down to the possibility that you think they may be real and
42
that we can experience them. That makes the whole thing take on a
whole different perspective.
Muse: I see. Would you rather stop now?
D.C.: Absolutely not
. I’ve come this far, I want to see what’s left. To make my
own decision rather then let what I’ve been programmed to think
make up my mind for me.
Muse: Glad to hear you say it in that way.
After all, many of us take what
we’ve learned at face value and a
s we get older shut out many
possibilities that could be true. A shame.
D.C.: I agree,
Muse: Then let’s us continue with our definition.
D.C.: I thought we’d finished.
Muse: We have, but only to a degree. We have to see what our three terms,
metabolism, r
eproduction, and memory have left out. What kinds of
things we can proclaim are not life
. To make sure we can really
recognize it when we see it.
D.C.: When?
Muse: Yes, when.
D.C.: You mean you have seen it?
Muse: I’m looking at it right now. You.
D.C.: No
, I mean these alternate life forms that you’ve been referring to.
Muse:
I’m not sure I’m not looking at one right now.
D.C.: Me?
Muse: No. But everything else I can see besides you. The computer behind
you, the air between us, the walls, the table, and so
on.
D.C.: But we know those are alive with bacteria, molds, and so on. Even the
desk was once alive since it’s wood.
Muse: Then what you’re saying is that most everything is alive or has once
been alive.
D.C.: Not really. Obviously the Earth hosts a grea
t deal of life, but rocks aren’t
alive.
Nor, I suppose, are all of the various attempts to create artificial
43
life on computers such as cellular automata, genetic algorithms, and
so on.
Muse:
No, but don’t discount their importance. Many of these types of
e
xperiments that have been going on for the last thirty or more years
have provided significant insights into life as we know it.
But, you’re
right, no one pretends they are actually alive. Hence the name,
artificial life. A

Life some call it. Many of my ow
n ideas come from that
research.
D.C.: Interesti
ng.
Muse: We’ve now come to an interesting point in my description of the
transcendent machine.
It may be a good time to take a break. Next
time we’ll discuss
how we’re going to search for alternative forms
of
life and then beyond that.
D.C.: Beyond that?
Muse: Yes. How do we prove it beyond just fitting o
ur ad hoc definition of
what lif
e is.
44
Chapter 7.
D.C. I gather our topic for today is going to be a continuation of last time with
an emphasis on how we
recognize life

like behavior in attractors.
Muse: Seems incredible doesn’t it?
D.C.: Yes.
Muse: Especially since, I remind you, we’re not talking about life

like
behavior here, we’re talking about the behavior or life. I know that
sounds like nit

picking,
but we’ve come a long way in order to
understand that what’s occurring in this current that’s flowing out of
the analog computer is the actual thing and not a simulation of it. If we
were to discover life amidst those electrons, it would be the real
McCoy
and not a representation of it.
D.C.: I understand and am duly reprimanded.
Muse: Not meant to be a reprimand. Only a reminder.
D.C.: Good.
Muse: So, how do you recommend proceeding.
D.C.: I’m getting the idea that as we continue I’m playing an ever mo
re
important role in the process.
Muse: You are right about that. I’m hoping that you’ll be as eager and as
knowledgeable about all this by the time we finish as I am.
And ready
to carry on further experiments as we go.
D.C.: Okay, then, I suggest we fin
d an attractor and see if any aspects of it
appear to have metabolism, reproduction, and/or memory.
Muse: Sounds good. But how will we recognize those things?
D.C.: By looking at the attractor in the oscilloscope I imagine.
Muse: What does a metabolism look like?
D.C.: Not sure. Shouldn’t we just take a look first?
Mu
se: I think we should discuss it initially before we do that. Otherwise, we
might not recognize it when we see it.
45
D.C.: Okay.
Then my guess for observing metab
olism in an attractor would be
something moving, changing, that while keeping a general
intact
form
it would grow, change, or at least develop in some way.
Muse:
Are you sure you just want to use your eyes?
D.C.: What other choice do we have?
Muse: Rememb
er, this is electricity. You can do lots of stuff with it. We can
hear it, for example. With a bit of amplification, we can listen to it
through loudspeakers and get a feel for it that way. We could place
some salt on top of a laid flat plate of iron and t
hen use the electrons
from the analog computer to make the plate move and watch the salt
move around, find resonance spots, and so on.
Or we could simply
hold the same plate and feel the vibrations the electrons make. Lots of
ways to make the attractors or
even the chaos come alive.
D.C.: Hadn’t given any of that a thought, but I guess you’re right. Lots of
different ways to perceive what we’re looking for, no reason to stick
to the oscilloscope.
Muse:
Maybe you had the right idea, though, at least to begi
n with. Use the
oscilloscope screen until we find something interesting and then
transfer it to the other mediums to see if what we think we’ve found
sounds right and feels right as well as looks right.
D.C.: Excellent.
Muse: Before we do that, however, l
et’s make sure we know what we’re
looking for. You said that we’re looking for something that keeps a
general intact form but also grow, change, or at least develop in some
way.
That seems to me to be vague. Lots of things do just that, but we
wouldn’t cal
l them alive. For example,
water tumbling over a falls
keeps its general shape but at the same time constant changes within
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