The Blooming

Alan
Turing inaugurated the theory of computation
in 1936 with the most humble but powerful
manifesto of all time.

What I'd like to do is to explore the
relevance of the theory of computation to computer
art. Both of those terms, however, need a little
unpacking/explanation before talking about their
relations.
Let's start with computer art.
Dominic Lopes, in A
Philosophy of Computer Art, makes a useful
distinction between digital art and computer
art. Digital art, according to Lopes,
can refer to just about any art that is or was
digitized. Such as scanned paintings, online fiction,
digital art videos, or digital audio recordings. Digital
art is not a single form of art, just as fiction
and painting are different forms of art. To call
something digital art is merely to say that the art's
representation is or was, at some point, digital. It
doesn't imply that computers are necessary or even
desirable to view and appreciate the work.
Whereas the term computer art
is much better to describe art in which the computer is
crucial as medium. What does he mean by "medium"? He
says "a technology is an artistic medium for a work just
in case it's use in the display or making of the work is
relevant to its appreciation" (p.15). We don't need to
see most paintings, texts, videos or audio recordings on
computers to display or appreciate them. The art's being
digital is irrelevant to most digital art. Whereas, in
computer art, the art's being digital is crucial to its
production, display and appreciation.
Lopes also argues that whereas digital
art is simply not a single form of art, computer
art should be thought of as a new form of art. He
thinks of a form of art as being a kind of art with
shared properties such that those properties are
important to the art's appreciation. He defines
interactivity as being such that the user's actions
change the display of the work itself. So far so good.
But he identifies the crucial property that works of
computer art share as being interactivity.
I think all but one of the above ideas
by Lopes are quite useful. The problem is that there are
noninteractive works of computer art. For instance,
generative computer art is often not interactive. It
often is different each time you view it, because it's
generated at the time of viewing, but sometimes it
requires no interaction at all. Such work should be
classified as computer art. The computer is crucial to
its production, display, and appreciation.
Lopes's book is quite useful in a
number of ways. It's the first book by a professional
philosopher toward a philosophy of computer art. It
shows us how a philosophy of computer art might look and
proceed. But it is a philosophy, in the end, of interactive
computer art. Which is a more limited thing than a
philosophy that can also accommodate noninteractive
computer art.
Now, why did a professional philosopher
writing a philosophy of computer art fail to account for
noninteractive computer art in his philosophy? Well, to
write a more comprehensive philosophy requires an
appreciation of the importance of programmability. For
it is programmability, not interactivity, that
distinguishes computer art from other arts. And it is at
this point that we begin to glimpse that some
understanding of the theory of computation might be
relevant to an understanding and appreciation of
computer art.
I'll return to this point in another
section. I've given you some idea of what I mean by
computer art. Now let's have a look at the theory of
computation.
It was inaugurated in the work of the
mathematician and logician
Alan Turing in 1936 with his worldshaking paper
entitled "On
Computable Numbers, with an Application to the
Entscheidungsproblem". This is one of the great
intellectual documents of the twentieth century. In this
paper, Turing invented the modern computer. He
introduced us to what we now call the Turing
machine, which is an abstract idea, a
mathematization, an imaginary object that has all the
theoretical capabilities of modern computers. As we
know, lots of things have changed since then in how we
think about computers. However, the Turing machine has
not changed significantly and it is still the bedrock of
how we think about computers. It is still crucial to our
ability to think about the limits of the capabilities of
computers.
And that is precisely what the theory
of computation addresses: the limits of the capabilities
of computers. Not so much today's limits, but
theoretical limits. The theory of computation shows us
what is theoretically possible with computers
and what is theoretically impossible. "Theoretically
impossible" does not mean "probably won't happen". It
means "will absolutely never ever (not ever) happen as
long as the premises of the theory are true".
Since we are dealing with matters of
art, let's first get a sense of the poetry of the theory
of computation. It's a littleappreciated fact that
Turing devised the Turing machine in order to show that
there are some things it will never be capable of doing.
That is, he devised the modern computer not to usher in
the age of computers and the immense capabilities of
computers, but to show that there are some things that
no computer will ever do. That's beautiful. The theory
of computation arises not so much from attempts to
create behemoths of computation as to understand the
theoretical limits of the capabilities of computing
devices.
If you wish to prove that there are
some things that no computer will ever do, or you
suspect that such things do exist, as did Turing—and he
had good reason for this suspicion because of the
earlier work of Kurt
Gödel—then how would you go about proving it? One
way would be to come up with a computer that can do
anything any conceivable computer can do, and then show
that there are things it can't possibly do. That's
precisely how Turing did it.
Why was he more interested in showing
that there are some things no computer will ever do than
in inventing the theoretical modern computer? Well, in
his paper, he solves one of the most famous mathematical
problems of his day. That was closer to the intellectual
focus of his activities as a mathematician and logician,
which is what he was. The famous problem he solved was
called the Entscheidungsproblem,
or the decision problem. Essentially, the problem, posed
by David
Hilbert in 1928, was to demonstrate the existence
or nonexistence of an algorithm that would decide the
truth or falsity of any mathematical/logical
proposition. More specifically, the problem was to
demonstrate the existence or nonexistence of an
algorithm which, given any formal system and any
proposition in that formal system, determines if the
proposition is true or false. Turing showed that no such
algorithm can exist.
At the time, one of the pressing
problems of the day was basically whether mathematics
was over and done with. If it was theoretically possible
to build a computer that could decide the truth or
falsity of any mathematical/logical proposition, then
mathematics was finished as a serious intellectual
enterprise. Just build the machines and let them do the
work. The only possibly serious intellectual work left
in mathematics would be metamathematical.
However, Turing showed that such an
algorithm simply cannot exist. This result was congruent
with Kurt Godel's earlier 1930 work which demonstrated
the existence in any sufficiently powerful formal system
of true but unprovable propositions, socalled
undecidable propositions. In other words, Godel showed
that there are always going to be propositions that are
true but unprovable. Consequently, after Godel's work,
it seemed likely that no algorithm could possibly exist
which could decide whether any/every wellformed
proposition was true or false.
We glimpse the poetics of the theory of
computation in noting that its historical antecedant was
this work by Gödel on unprovable truths and the
necessary incompleteness of knowledge. The theory of
computation needed the work of Gödel to exist before it
could bloom into the world. And let us be clear about
the nature of the unprovable truths adduced by Godel.
They are not gardenvariety axioms. Gardenvariety
axioms, such as the parallel postulate in geometry, are
independent. That is, we are free to assume the
proposition itself or some form of the negation of the
proposition. The so called undecidable
propositions adduced by Gödel are true propositions. We
are not at liberty to assume their negation as we are in
the case of independent axioms. They are (necessarily)
true but unprovably true. And if we then throw in such a
proposition as an axiom, there will necessarily always
be more of them. Not only do they preclude the
possibility of being able to prove every true
proposition, since they are unprovable, but more of them
cannot be avoided, regardless of how many of them we
throw into the system as axioms.
Sufficiently rich and interesting
formal systems are necessarily, then, incomplete, in the
sense that they are by their nature incapable of ever
being able to determine the truth or falsity of all
propositions that they can express.
So the theory of computation begins
just after humanity becomes capable of accomodating a
deep notion of unprovable truth. Of course, unprovable
truth is by no means a new concept! We have sensed for
millenia that there are unprovable truths. But we have
only recently been able to accommodate them in sensible
epistemologies and mathematical analysis. Unprovable
truths are fundamental to poetry and art, also. We know
all too well that reason has its limits.
The more we know about ourselves, the
more we come to acknowledge and understand our own
limitations. It's really only when we can acknowledge
and understand our own limitations that we can begin to
do something about them. The first design for a
socalled Turingcomplete computer—that is, a computer
that has all the theoretical capabilities of a Turing
machine—predated Turing by a hundred years: Charles
Babbage's Analytical Engine. But Babbage was never able
to create his computer. It was not only a matter of not
being able to manufacture the parts in the way he
wanted, but he lacked the theory of computation that
Turing created. A great theory goes a long way. The
Turing machine, as we shall see, is simplicity itself.
Children can understand it. We think of computers as
intimidatingly complex machines, but their operation
becomes much more understandable as Turing machines.
We could have had computers without the
theory of computation, but we wouldn't have understood
them as deeply as we do, wouldn't have any sense of
their theoretical limitations—concerning both what they
can and can't do. And we wouldn't have been able to
develop the technology as we have, because we simply
wouldn't have understood computers as deeply as we do
now, wouldn't have been able to think about them as
productively as we can with the aide of a comprehensive
theory. Try to put a man on the moon without Newtonian
mechanics. It might be doable, but who would want to be
on that ship? Try to develop the age of computers
without an elegant theory to understand them with? No
thank you. That sounds more like the age of the
perpetual blue screen.
Gödel's incompleteness theorems are not
logically prior to Turing's work. In other words,
Turing's work does not logically depend on Gödel's
work—in fact, incompleteness can be deduced from
Turing's work, as computer scientists sometimes point
out. But it was Gödel's work that inspired Turing's
work. Not only as already noted, but even in its use of
Georg Cantor's diagonal
argument. Gödel's work was the historical
antecedant of Turing's work. Gödel's work established a
world view that had the requisite epistemological
complexity for Turing to launch a theory of computation
whose epistemological capabilities may well encompass
thought itself.
The theory of computation does not
begin as a manifesto declaring the great capabilities of
computers, unlike the beginnings of various art
movements. Instead, it begins by establishing that
computers cannot and simply will never ever solve
certain problems. That is the main news of the
manifesto; it means that mathematics is not over, which
had been a legitimate issue for several years. Were
computers going to take over mathematics, basically?
Well, no. That was very interesting news. You don't
often get such news in the form of mathematical proofs.
News that stays news. The other news in the manifesto is
almost incidental: oh, by the way, here is a
mathematization of all conceivable machines—here is the
universal machine, the machine that can compute anything
that any conceivable machine can compute.
His famous paper layed the foundation
for the theory of computation. He put the idea of the
computer and the algorithm in profound relation with
Gödel's epistemologically significant work. He layed the
philosophical foundation for the theory of computation,
establishing that it does indeed have important
limitations, epistemologically, and he also provided us
with an extrordinarily robust mathematization of the
computer in the form of the Turing Machine.
Turing's paper is significant in the
history of mathematics. We see now that the development
of the computer and the theory of computation occurs
after several hundred years of work on the "crisis of
foundations" in mathematics and represents a significant
harvest or bounty from that research. At least since the
seventeenth century, when bishop Berkeley famously
likened Newton's treatment of some types of numbers in
calculus to "the ghosts of departed quantities", and
especially since the birth pains in the eighteenth
century of nonEuclidean geometry, mathematicians had
understood that the foundations of mathematics were
vaguely informal, possibly contradictory, and needed to
be formalized in order to provide philosophical and
logical justification and logical guidelines in the
development of mathematics from first principles.
There's a straight line from that work
to the work of Frege, Cantor, and Gödel. And thence to
Turing. The theory of computation, it turns out, needed
all that work to have been done before it could bloom.
It needed the philosophical perspective and the tools of
symbolic logic afforded by that work. Because the theory
of computation is not simply a theory of widgets and
dodad machines. At least since the time of Leibniz in
the seventeenth century, the quest to develop computing
devices has been understood as a quest to develop aides
to reason and, more generally, the processes of thought.
The Turing Machine and the theory of
computation provide us with machines that operate, very
likely, at the atomic level of thought and mind. Their
development comes after centurys of work on the
philosophical foundations of mathematics and logic. Not
to say that it's flawless. After all, it's necessarily
incomplete and perhaps only relatively consistent,
rather than absolutely consistent. But it's good enough
to give us the theory of computation and a new age of
computers that begins with a fascinatingly humble but
farreaching paper entitled "On Computable Numbers, with
an Application to the Entscheidungsproblem" by Alan
Turing.
It changes our ideas about who and what
we are. Computer art, without it, would be utterly
different. Just as would the world in so many ways.
Greenberg, Modernism,
Computation and Computer Art
In a short but influential piece of
writing by Clement Greenberg called Modernist
Painting written in 1960—and revised
periodically until 1982—the art critic remarked that
"The essence of Modernism lies, as I see it, in the use
of characteristic methods of a discipline to criticize
the discipline itself, not in order to subvert it but in
order to entrench it more firmly in its area of
competence." Such sweeping generalizations are always
problematical, of course. But I want to use the
Greenberg quote to tell you an equally problematical
story about the birth of the theory of computation and,
thereby, computer art. Humor me. It's Clement Greenberg.
Come on.
The work I've mentioned by Gödel and
Turing happened in the thirties, toward the end of
modernism, which was roughly from 1900 till 1945, the
end of World War II. So it's work of late modernism.
Let's grant Greenberg clemency
concerning his conceit, for the moment, that the
"essence"—itself a word left over from previous eras—of
modernism, of the art and culture of that era, at least
in the west, involved a drive to a kind of productive
selfreferentiality or consciousness of the art itself
within the art itself. What work could possibly be more
exemplary of that inclination than the work by Gödel and
Turing that I've mentioned?
Turing's paper in which he invents the
modern computer and solves the Entscheidungsproblem is
profoundly metamathematical;
the Entscheidungsproblem, as already noted, is a problem
of metamathematics. And its argument also involves
interesting selfreferenciality, as is pointed out in plato.stanford.edu/entries/turing:
Turing's proof can be
recast in many ways, but the core idea depends
on the selfreference involved in a machine
operating on symbols, which is itself
described by symbols and so can operate on its
own description. 
Selfreference is crucial also to Godel's
proof in which, among other things, the
proposition "This proposition is not provable" is shown
to be necessarily true but, yes, unprovable, and
exemplary of a kind of proposition which Godel calls
"undecidable". Selfreference is an implicit mode of
metamathematics because mathematics/logic is used to
inquire into the nature or properties of
mathematics/logic, though that doesn't need to be any
more deeply selfreferential than using language to
inquire into the properties of language; what else are
you gonna use? Your big toe? I don't think so.
So we can see the work of Godel and
Turing as a kind of profound culmination of modernism's
"use of characteristic methods of a discipline to
criticize the discipline itself". Greenberg sees
modernism as involving a meta mode of art and thought,
this growing selfcritical, selfaware tendency in art.
Modernism culminates in the work of Godel and Turing and
the consequent development of the computer, a machine
that operates, as it were, at the atomic level of
thought. The culmination of that selfcritical,
selfaware meta mode of modernism in which the work of
art is aware of itself, in a sense, is the creation of a
type of machine that may indeed quite literally possess
the capability of becoming selfaware.
The computer age and, of course,
computer art commences at the end of the modern era,
signalled by the end of a world war, the invention of
the theory of computation, and the atomic bomb, an
understanding of the fierce chemistry of the atom, the
utterly micro, at the level of the chemistry of the sun,
of Apollo, of Ra, the yay very large. Looking inward and
looking outward.
That knowledge of the sun's chemistry
and its harnessing in the creation of the atomic bomb
does indeed reveal important things about our own
nature, but the whole subject is not centrally about
humans and human capacities. Whereas the theory of
computation is a theory inaugurated by Turing as very
explicitly being about human capacities. Recall that the
Entscheidungsproblem, or the decision problem, was to
demonstrate the existence or nonexistence of an
algorithm that would decide the truth or falsity of any
mathematical/logical proposition. A large part of the
difficulty of this problem was in coming to the best
possible formulation of what we mean by "algorithm". As
we read at plato.stanford.edu/entries/turing:
Turing's purpose was to
embody the most general mechanical process as
carried out by a human being. His
analysis began not with any existing computing
machines, but with the picture of a child's
exercise book marked off in squares. From the
beginning, the Turing machine concept aimed to
capture what the human mind can do when
carrying out a procedure. 
From the start, the whole project of
computing has been about us. And involves abstracting
the process of thought into its constituent atoms, as it
were. Computer art, a new form of art, goes beyond
Greenberg's meta imperative into an art that could
possibly create works in which the objects do actually
think. And do actually create art. The human as the meta
artist; the program as the artist. A situation where,
no, art is not over and the only serious artistic work
left is meta art, but the liveliness and
selfconsciousness of the object envisioned in
Greenberg's vision of modernism is taken to the next
level in the age of computers.
Programmability
I said earlier that it's
programmability, not interactivity (or anything else)
that is the crucial matter to consider in computer art.
I want to explain and explore that claim in this
section.
What makes computer art computer art?
We've seen that there is a great deal of art that
appears on computers that could as well appear on a page
or on a TV, in a canvas or on an album. I'm calling that
art digital art and computers are not crucial to the
display or appreciation of it.
The idea I want to capture in the
notion of 'computer art' is art in which computers are
crucial for the production, display and appreciation of
the art, art which takes advantage of the special
properties of computers, art which cannot be translated
into other media without fundamentally altering the work
into something quite different than what it was on the
computer, art in which the computer is crucial as
medium.
And we've noted that interactivity is
not sufficient to characterize the special properties of
computer art, where interactivity means that the display
itself is actually changed/affected by the actions of
the user/viewer. We can imagine theatre, for instance,
that is responsive to input from the audience.
Interactivity is not unique to computers. But there's no
question it's a very important part of many works of
computer art. Also, although interactivity is not unique
to computers, interactivity is the basis of a whole type
of computer art from games to instrumentlike devices to
communication devices. Whereas interactivity does not
have such a crucial and definitive role in other forms
of (noncomputer) art. The computer's ability to respond
almost instantaneously to increasingly complex
situations with deeply considered, highly conditional
responses is, well, unique. Interactivity itself is not,
per se, unique to computing, but some of the ways
computers can react interactively are unique. Computers
can guide us instantaneously through a whole interactive
virtual world of illusions at every point and maintain
the illusion of that world's existence by constructing
it, moment to moment, based on our navigational
decisions. So there is no question that interactivity is
often the crucial property of computer art. In that it's
the interactivity we often think about most deeply in
our appreciation of the work. Interactivity can be
experiential, immediate, engrossing and, yes, immersive
in ways that art rarely is.
However, as I pointed out, there are
types of computer art that art not interactive at all.
Generative computer art is often not interactive. It's
often different each time you see it. The art is
generated by a computer program. This sort of work needs
to be considered as computer art because the art
requires a computer to be generated at all.
So if interactivity is not sufficient
to characterize the special properties of computer art,
what is?
Well, in both interactive computer art
and generative art, the computer's programmability is
crucial. Interactivity requires programmability.
Interactivity requires conditional responses. That's
fundamental to programming. And, in generative art, the
art is different each time because of conditional
programming.
Programmability is what separates
computers from other types of machines. It's important
to understand this to have any idea about what a
computer really is. Nonprogrammable machines may have
some very slight ability to react conditionally. The
light in a refrigerator comes on only when the door is
opened and goes off only when the door is closed. That's
conditional behavior. When you put a penny in a pop
machine you get no pop but you do if you put in enough
money. That's also conditional behavior.
But computers can be as labyrinthine as
we ourselves are, sometimes, in our conditional
responses to life. Computers can make decisions. What
sort of decisions? Think of the sort of decision
computers can make as being like atoms. They're small
and you don't notice them. But put lots of them together
and you get the sort of decisions humans make, just like
when you put lots of atoms together you get stuff we can
see. Basically, computers can only decide if two numbers
are equal or one number is bigger than another number.
And then do different things depending on the outcome.
But just like everything is made out of atoms, all
decisions can be made of lots of smaller decisions.
It's programmability that allows
computers to exhibit conditional behavior that ranges
from the obviously just plainold mechanical, like a
fridge or a toaster's behavior is mechanical, to
decisions and behavior predicated on millions of lines
of programming that not even the programmers can
anticipate. Deep Blue, the famous chess program, plays a
better game of chess than any of its programmers.
Historically, the drive to create
programmable machines arose from the need to make
machines flexible. When you go to the time, trouble, and
expense of building a machine to perform a task, you
would like it to be capable of related tasks. If the
machine is a loom, for instance, to weave fabric, you
would like the loom to be capable of weaving many
different types of cloth that display many different
types of patterns. Rather than having to have a
different machine for each of them. How do you achieve
the utmost in such flexibility? Well, you make the
action of the machine, at any stage in its operation,
dependent on decisions carried out by a program that
guides the machine's actions toward the completion of
the particular desired task. And this program is what
you change when you want the machine to do something
different. You don't change the machine; you change the
program.
In another section, we'll look more
closely at just what a program is, and what a computer
is, exactly. We'll discuss the idea of the Turing
machine, the universal computer, and precisely how it
works. It's quite simple, actually. Children can
understand it. The Turing machine is an imaginary,
theoretical computer. It is our model of every computer
that has ever existed and may ever exist. In any case,
the Turing machine is the ultimate jackpot of machine
flexibility. In the 75 years that have elapsed since it
was first invented by Turing, nobody has been able to
come up with a similarly imaginary, theoretical
computing device that can do more than the Turing
machine.
The Turing machine is profoundly
flexible, as a machine. It is flexible to the point that
there is no proof, and probably never will be, that
there exist thought processes of which humans are
capable and computers are not. Which is to say that it's
very likely that computers are flexible to the point of
being capable of thought itself.
And this flexibility is completely
predicated on programmability. Programmability, not
interactivity or anything else, is what distinguishes
computers from other machines. Programmability is the
fundamental distinguishing characteristic of computers.
Well so what? Of what importance is
this to computer art?
Recall that I'm defining computer art
to be art in which the computer is crucial as medium.
That is, the computer is crucial to the display and
appreciation of the work. The art can't be displayed
(even if it is sonic) on a device that isn't hooked to a
computer; it can't run properly or be properly
appreciated without it.
If you think about what that means, I
think you have to come to the conclusion that such art
requires a computer for its display and appreciation
because the art relies on something about computers that
the art can't get from any other devices. What could
that be? It could be interactivity. But recall that
there is lots of computer art that isn't interactive in
the slightest. In that case, it has to be
programmability. And even when it is interactivity, as
we've seen, computerized interactivity is completely
predicated on programmability.
So the relation of computer art and
programmability is very strong. Computer art is simply
not computer art without programmability. The
programmability of computers is one of the main
things–or, in some cases, the main thing–that computer
artists use to distinguish their art from every other
type of art.
Evolution and the Universal
Machine
Having recently been trying to be less
a fossil concerning knowledge of evolution, I've watched
all sorts of truly excellent documentaries available
online. In several of them, it was said that Darwin's
idea of evolution through natural selection is the best
idea anyone's ever had. Because it's been so powerfully
explanatory and has all the marks of great ideas in its
simplicity and audacious, unexpected and absolutely
revolutionary character.
Uh huh. Ya it's definitely a good one,
that's for sure. But I'll tell you an idea that I think
is right up there but is nowhere near as widely
understood, perhaps permanently so. It's Turing's idea
of the universal machine. Turing invented the modern
computer. This was not at all an engineering feat. It
was a mathematical and conceptual feat, because Turing's
machine is abstract, it's a mathematization of a
computer, it's a theoretical construction.
What puts it in the Darwin range of
supreme brilliance are several factors. First and
foremost, it shows us what is almost certainly a
sufficient (though not a necessary) model of mind. There
is no proof, and probably never will be, that there
exist thought processes of which humans are capable and
computers are not. This is a source of extreme
consternation for many people–very like Darwin's ideas
were and, in some quarters, still are.
The reason why such proof will likely
never be forthcoming is because it would involve
demonstrating that the brain or the mind is capable of
things that a Turing machine is not–and a Turing machine
is a universal machine in the sense that a Turing
machine can perform any computation that can be thought
of algorithmically, involving finitely many steps.
Turing has given us a theoretical model
not only of all possible computing machines, which
launched the age of computing, but a device capable of
thought at, as it were, the atomic level of thought. I
don't really see that there is any reasonable
alternative to the idea that our brains must function as
information processing machines. The universality of
Turing's machine is what allows it to encompass even our
own brains.
Additionally, another reason to rank
Turing's idea very high is that, mathematically, it is
extrordinarily beautiful, drawing, as it does, on
Godel's marvelous ideas and also those of Georg Cantor.
Turing's ideas are apparently the culmination of some of
the most beautiful mathematics ever devised.
Darwin's ideas place us in the context
of "deep history", that is, within the long history of
the planet. And they put us in familial relation with
every living thing on the planet in a shared tree of
life. And they show how the diversity of life on our
planet can theoretically emerge via evolution and
natural selection.
Darwin's ideas outline a process that
operates in history to generate the tree of life.
Turing's ideas outline a process that can generate all
the levels of cognition in all the critters thought of
and unthought. Darwin gives us the contemporary tree of
life; Turing gives us the contemporary tree of
knowledge.
