by Scott Aaronson

*Note:* I must have been in a finitistic mood when I wrote this piece three years ago. I don't know how much I agree with it. -SA (2001)

Since so little is known about consciousness, it often seems that one can
make any assertion about it that one wishes without fear of being proven wrong.
For example, one could claim that the true seat of consciousness is an invisible
cashew residing in the pancreas, and challenge scientists to find a better
explanation. Or one could speculate that consciousness arises from
as-yet-undiscovered noncomputable laws of quantum gravity operating within brain
structures called microtubules, as Sir Roger Penrose did in his 1994 book
Now we know that the
2. Finiteness Perhaps you have the sensation of being able to do infinitely many things
with your computer. You can visit web sites dealing with medieval weapons or the
flammability of Pop Tarts; you can play Minesweeper or Quake, or make the mouse
pointer dance across the screen, or write a program or an article about
consciousness -- surely this variety has no end? But on closer thought, your
computer is a finite object. If it can store N bits of information in memory
(where maybe N >> 500 million), then it has at most
2 Furthermore, time for your computer is broken up into discrete units, so that
viewed as an FSA, it might only have, say, 200 million opportunities per second
to change state. So clearly there's some finite upper bound M on the number of
state transitions your computer can make before it breaks down. To be
conservative, we could set M equal to a billion billion billion billion billion
(10 Here's a disconcerting implication: if you have friends with whom you've only
interacted online, then the whole history of your interactions can be described
by one of those 2 "When you plead with your lover over the telephone, every nuance, every catch in the voice, every passionate sigh and yearning timbre is carried along the wire solely in the form of numbers. You can be moved to tears by numbers -- provided they are encoded and decoded fast enough." To which I'd add that yearning timbres can be encoded not only by numbers, but by numbers of bounded size -- that is, by finitely many of them. These considerations apply not only to conversations, but to any information
that you could store on your computer. The complete works of Shakespeare
downloaded from Project Gutenberg, the Mona Lisa stored as a high-resolution
JPEG image, and Beethoven's 5 For now, let's ask how far our 'finitizing' of human experience can go. We've
seen that there are only finitely many phone conversations that could be carried
over a digital switch. But what about conversations over an analog switch? Or
face-to-face conversations? First kisses? Walks through the park on an autumn
day? I argue that there are only finitely many possibilities for each of these
things, and indeed for all of human experience. This requires only that a
computer could, in principle, simulate a human's experience of the world such
that it would be impossible for the human to tell the difference from the real
thing -- in other words, that 'total-immersion virtual reality' is theoretically
possible. First, note that there's some finite upper bound T on how long any
human could live. Again, to be conservative, we can set T equal to
10
3. The Fallacy of the Virtually Infinite You might object that, even if there are a finite number of phone
conversations that could be had, paintings that could be painted, and human
lives that could be lived, the numbers are so astronomical that it would make no
difference if they were infinite. This objection illustrates what might be
called the 'fallacy of the virtually infinite': the conflation of the distinct
concepts of 'arbitrarily large' and 'infinite.' That we humans regularly commit
this fallacy is understandable: the ability to distinguish between, say, eight
goats and ten goats undoubtedly carried an evolutionary advantage, but a
prehistoric human who pondered the difference between 10 Mathematicians denote the cardinality (or size) of the set of whole numbers, which is usually what we mean by 'infinity,' as (pronounced aleph-null). is not a number, nor is there any sense in which it can be considered the 'largest quantity.' In the 1880's, Georg Cantor showed that given any set S (which might be infinite), one can form a larger set by taking the set of subsets of S. The study of higher orders of infinity led to the amazing theorem of Kurt Gödel (1938) and Paul Cohen (1963) that whether there are orders of infinity between and the cardinality of real numbers is undecidable within the usual axioms of set theory [Coh66], but I digress. What's relevant for us is that it's easy to prove that is the lowest order of infinity -- that is, that there are no sets straddling the fence between finite and infinite cardinality. This means that there's a sharp distinction between sets of size N, where N could be an arbitrarily large integer, and infinite sets. These two classes of sets have very different properties: an infinite set can be placed in one-to-one correspondence with a proper subset of itself (think of the whole numbers and the even whole numbers), but this isn't the case for any finite set, no matter how large. The ancient Greeks were suspicious of infinity because of 'paradoxes' related to the Fallacy of the Virtually Infinite, and because of their suspicion humanity had to wait two millennia for Isaac Newton and Gottfried Leibniz to discover differential calculus. But you can avoid the fallacy by remembering this simple rule: that for every whole number N, there are infinitely many whole numbers larger than N. This rule implies that 2, 17, and the number of possible human life experiences are all equally distant from infinity.
4. Implications for Penrose's Argument When Penrose asserts that consciousness arises from noncomputable processes
in the brain, he means that these processes can't be simulated by a Turing
machine. The Turing machine is a model of computation proposed by the English
mathematician Alan Turing in 1936. At first it seems bizarre: it involves a tape
head moving back and forth, reading and writing symbols, on an infinitely long
paper tape divided into squares. But the Turing machine can simulate all other
models of computation that have ever been proposed, leading to the Church-Turing
Thesis, that 'computable by a Turing machine' is what we The incompleteness theorem says roughly that given any formal proof system F
that allows reasoning about numbers and that's consistent (i.e., doesn't allow
falsehoods to be proved), there's a statement of F, called G(F), which is true
for F and yet unprovable within F. Gödel constructed G(F) by starting with the
statement "This statement doesn't have a proof in F," which we can easily see is
both true and unprovable in F given that F is consistent. He then showed how to
express this statement in the language of F, by encoding the concepts of
'statement' and 'proof' as numbers. Gödel's result is a cornerstone of
mathematical logic, but Penrose argues that it's relevant for consciousness as
well. His reasoning is that, while a computer operating within the fixed formal
system F can't prove G(F), a human can This argument isn't new (it goes back at least to John Lucas in 1961), and
logicians and computer scientists have pointed out a major flaw in it. This is
that human mathematicians don't use any consistent formal system such as F: they
rely on intuition, and they frequently make mistakes. If we grant a computer
this same liberty to make mistakes, then it need not operate strictly within F,
and there's nothing paradoxical about it being able to 'see' the truth of G(F).
Even without this consideration, that a computer is algorithmic doesn't imply
that it must or should use a consistent formal system: if we program it to print
'1+1=3,' then it will oblige. Penrose is aware of this flaw, and he tries at
great length in But refuting Penrose's argument is like a refuting a proposed method for
squaring the circle: although finding the specific flaw can be instructive, we
can decide before even looking at the argument that there must - All thinking is computation; in particular, feelings of conscious awareness are evoked merely by the carrying out of appropriate computations.
- Awareness is a feature of the brain's physical action; and whereas any physical action can be simulated computationally, computational simulation cannot by itself evoke awareness.
- Appropriate physical action of the brain evokes awareness, but this physical action cannot even be properly simulated computationally.
- Awareness cannot be explained by physical, computational, or any other scientific terms.
Views A and B, I think, are the ones compatible with the knowledge that consciousness is finite. Penrose states, unsurprisingly, that view C "is the one which I believe myself to be closest to the truth" [Pen94, p. 15]. (He states at the outset that his focus is on explanations for consciousness that at least attempt to be scientific, thus ruling out view D.) The closest Penrose comes to addressing the objection that consciousness is finite is in his 'Q7' (one of twenty objections he raises against his theory, together with his responses). Though Q8 also deals with the fact that computers and brains are finite, it involves mathematical issues that are less relevant to us here. So let's look at Q7 [Pen94, p. 82-83]: The total output of all the mathematicians who have ever lived, together with
the output of all the human mathematicians of the next (say) thousand years is
finite and could be contained in the memory banks of an appropriate computer.
Surely this particular computer To which Penrose responds, in part: … One could equally well envisage computers that contain nothing but lists of
totally false mathematical 'theorems', or lists containing random jumbles of
truths and falsehoods. How are we to tell which computer to trust? The arguments
that I am trying to make here do not say that an effective simulation of the
output of conscious human activity (here mathematics) is impossible, since
purely by chance the computer might 'happen' to get it right -- even without any
understanding whatsoever. But the odds against this are absurdly enormous, and
the issues that are being addressed here, namely how one decides This sounds like view B, directly contradicting Penrose's stated belief in view C. Penrose might respond by emphasizing the word 'properly' in view C, and arguing that simulating a mind by simply listing each of its finitely many contingencies, together with its chosen responses, isn't a 'proper' simulation. But in that case, why does he even distinguish between views B and C? (Penrose further blurs his stated position in a fantasy dialogue [Pen94, p. 179-190] between a human and a robot. The robot is driven insane when the human challenges it to prove a statement corresponding to G(F), but that the robot can hold an articulate conversation at all would seem to indicate Penrose's agreement with views A or B.) Penrose may not have sufficiently considered the impact that the finiteness of our minds has on his theory.
5. Why Finiteness Isn't So Bad We've argued that, "Since the time of Greek philosophy, scholars have prided themselves on their ability to understand something about infinity; and it has become traditional in some circles to regard finite things as essentially trivial, too limited to be of any interest. It is hard to debunk such a notion, since there are no accepted standards for demonstrating that something is interesting, especially when something finite is compared with something transcendent. Yet I believe that the climate of thought is changing, since finite processes are proving to be such fascinating objects of study." So how did finite objects, dismissed as insignificant less than a century
ago, come to take their proper place at the mathematical table? Part of the
explanation might lie with Paul Erdös, a giant of 20 The greatest impetus, though, has been the computer, which has fueled the
creation of whole new branches of finite mathematics. One of these branches,
called complexity theory, deals with how quickly the time and memory required to
solve a problem grows as a function of the problem's size. For example, in the
Maximum Clique problem, we're given a list of N people, together with a list of
who's friends with whom, and are asked to find the size of the largest group of
people who are all friends with one another. We can solve any instance of
Maximum Clique by examining finitely many groups of people (there are
2 This last claim might seem surprising, especially given Penrose's guess that
"the issues of complexity theory are not quite the central ones in relation to
mental phenomena" [Pen89, p. 145]. So let's elaborate. Recall that in Section 2,
we asked how a work of art could be 'creative' if it's just a selection among
finitely many pre-existing possibilities. The answer most people would give, I
think, is that if a set of possibilities is That our minds are finite helps to shed light on certain philosophical arguments, such as Penrose's. But as we've seen, it doesn't render consciousness trivial, nor does it diminish the role of creativity. So I don't mind that my entire life can be modeled by the choosing of a single element from a finite set, and I hope you don't mind that yours can be so modeled either. The finiteness of our minds may even be cause for optimism, because it makes our ability to contemplate the infinite even more astounding.
References [Coh66] Cohen, Paul. [Daw95] Dawkins, Richard. [Hay95] Hayes, Brian. "Debugging Myself," [Knu76] Knuth, Donald. "Mathematics and Computer Science: Coping With
Finiteness," in [McD95] McDermott, Drew. "[STAR] Penrose is wrong," [Pen94] Penrose, Roger. [Pen89] Penrose, Roger. |