Penrose's Triangles: The Large,
The Small, and the Human Mind
by
Arkady Plotnitsky
© 1997
----------------------------------------------------------
Roger Penrose, The Large, the Small, and the Human Mind
(with Abner Shimony, Nancy Cartwright, and Steven
Hawking), Cambridge: Cambridge University Press, 1997;
with a glance back at The Emperor's New Mind, Shadows of
the Mind, and The Nature of Space and Time.
1. At 4 p.m. on May 11, 1997, "the truly impossible
occurred," as Newsweek reported (May 26, 1997, 84). The
computer Deep Blue defeated the world chess champion and
one of the greatest chess players of all time, Garry
Kasparov. In the process, his confidence, bolstered by
his impressive previous victories over computers, was
shattered along with the confidence and hopes of much of
the chess world. Indeed, the defeat was taken by some,
including by Kasparov, as a humiliation. It appears to
have been especially humiliating because it was inflicted
on a great chess mind, capable of the most complex
tactical and strategic thinking, by the raw power of
computation. A great chess mind was defeated by crude
number-crunching--a much inferior form of thinking or, in
this case, not even thinking. Such minds have always been
seen as able to circumvent and transcend protracted
computational routines, and as entities whose own
workings are themselves inaccessible to computational
analysis.
2. In the last installment of his ongoing argument against
the possibility of artificial intelligence, The Large,
the Small, and the Human Mind, Roger Penrose uses chess
to illustrate the difference between the computational
approach used by digital computers and non-computational
thinking, which, according to him, fundamentally defines
the human mind and gives it ultimate superiority over
computers or computer-like computational intelligence
(103-5). Penrose gives examples of two chess problems
that are easily solved by even mediocre human players but
have defeated a computer (in this case Deep Thought, the
forerunner of Deep Blue). Penrose's lectures (given in
1995 and published earlier this year) preceded the latest
chapter of the story of computer chess just described,
and he may even have been inspired in part by Kasparov's
previous decisive victories over computers and by the
seemingly unquestionable ultimate superiority of chess
players or, one might say, chess thinkers over chess
computers.
3. This latest episode of this, by now long, history does
not prove the opposite. Nor does it prove the ultimate
superiority of any one form of thinking over another, or
of (human) thinking over (computer-like or other)
non-thinking. That is, assuming that human thinking is
indeed non-computational--a claim that, however appealing
or likely, remains hypothetical, as Penrose admits.
Everyone acknowledges that Deep Blue cannot
think--leaving aside here the complexity, if not
impossibility, of the latter concept itself. Nor does the
episode prove that artificial intelligence is any more
likely now than it was before. What it does prove is that
computational thinking or (even more humiliatingly for
its opponents) computation without thinking,
number-crunching, can be taken to a level high enough to
supersede non-computational thinking in certain specific
cases. The fact that this is possible in some cases is in
part (there are more general conceptual reasons) what
drives the idea--which is more or less the idea of
artificial intelligence--that it may be possible in any
given case. Much else, it is further extrapolated, would
be possible as well for computational devices. It may,
for example, be possible for machines to perform any
given task that human thinking can perform, or
conceivably even to think or have consciousness or even
to feel, just as humans do. Or--a rarely, if ever,
discussed but interesting and radical possibility--it may
be possible to perform even the most complex tasks better
than humans without thinking and, thus, in a certain
sense without intelligence. Or, as I said, it might also
be possible to assume that at bottom the human mind is
itself only a computational device, a number-crunching
machine. These are the types of possibilities that are
pursued in the field of artificial intelligence and
related endeavors and that are argued against by, among
others, Penrose.
4. At a certain level, at stake here is still the power of
creative thinking (seen as ultimately non-computational)
against computational or calculating thinking (seen as
ultimately uncreative) or against the nonthinking of
machines. Along and interactively with other classical
oppositions and hierarchies of that type, this opposition
and this hierarchy has, from Plato (or even the
pre-Socratics) to the present, defined the history of
thinking about human thinking and its superiority over
other animals, on the one hand, and machines, on the
other.
5. This theme is among several that link Penrose's books to
what have become known as "postmodern" problematics
(using this term with caution here), within which the
nature and structure, or deconstruction, of both of these
oppositions--the human and the animal, and the human and
the machine--and their interactions have been explored at
great length. The general question of artificial
intelligence is, of course, another such theme, although
it is indissociable from the oppositions just described.
This question is fundamentally connected by Penrose to a
number of key questions raised by such revolutionary
developments in twentieth century science and technology
as Einsteinian relativity and quantum physics,
post-Gödelian mathematical logic, modern biology, and
computer technology; and these questions have in turn
been central to postmodernist thinking. Such issues as
the nature of causality and reality of the physical
world, of truth and certainty in mathematics and
mathematical logic, the nature of scientific explanations
necessary for understanding the human brain (or indeed
mind), all considered by Penrose, are part of the
postmodernist intellectual scene and have been hotly
debated recently, or, again, throughout modern (or
earlier) intellectual history.
6. Penrose's ideas have themselves been the subject of
considerable debate. The Large, the Small and the Human
Mind is the last installment of his "trilogy" on
"computers, minds, and the laws of physics," initiated by
The Emperor's New Mind: Concerning Computers, Minds and
the Laws of Physics (1989), and continued, in part in
reply to questions posed and debates provoked by the
first volume, with Shadows of the Mind: A Search for the
Missing Science of Consciousness (1994). There is,
moreover, another companion (technical) volume,
containing a debate with Stephen Hawking, The Nature of
Space and Time (1996), which is more specialized and
concerned more exclusively with physics, but with many
echoes of and significant connections to Penrose's books.
7. Beyond and as part of an investigation into the question
of the human mind and artificial intelligence, Penrose's
trilogy contains major semi-popular or, more accurately,
semi-technical expositions of key revolutionary
mathematical and scientific theories defining
twentieth-century science--post-Gödelian mathematical
logic, relativity, and quantum physics--and his
controversial forays into the biology of the brain.
Derived from a 1995 series of lectures, The Large, the
Small, and the Human Mind is a summary presentation (with
some updating) of the two previous volumes and a sample
of the debate provoked by them. It includes contributions
by Abner Shimony, Nancy Cartwright, and Stephen Hawking,
and Penrose's responses to their arguments. Both earlier
volumes contain elegant extended expositions of the
mathematical and scientific theories just mentioned.
While, in principle, these expositions do not require a
specialized knowledge of mathematics and science, it is
doubtful that, in practice, they are sufficiently
accessible to general readers to be read in depth by most
of them. By contrast, the latest volume can be read as a
more general introduction to Penrose's main ideas and is
more readable for general readers than the two previous
volumes or, especially, The Nature of Space and Time. The
latter work, however, contains arguably the most
interesting discussions of general relativity and
cosmology, parts of which can be read and productively
assimilated by non-specialists, albeit with considerable
(and, in my view, well-deserving) effort. Some of
Penrose's elegant mathematical thinking and presentation
is found in the last volume as well, in particular in his
discussions of relativity and non-Euclidean geometry in
Chapter 1, and the reader will be especially rewarded
here. Quantum mechanics, mathematical logic, and biology
are, as will be seen, more complex cases in this respect.
8. Penrose's overall argument can be summarized as follows.
He wants to argue definitively for the impossibility for
artificial intelligence (at least that based on
computation, such as that carried on by digital
computers) to reach the level of human intelligence and,
more generally, consciousness (both based, he believes,
on non-computational processes and themselves effects of
physical systems whose design is non-computable). This
definitive argument is to be rigorously grounded in
mathematically and scientifically ascertainable facts
about (physical) nature and (mathematical) mind. Or, more
accurately--and this nuance is crucial--Penrose contends
that there is, at this point, enough scientific data to
hope that one will be able to offer a definitive,
rigorous argument of that type in the future (it is,
Penrose acknowledges, speculative at present) from the
new physics of the world and/as the new physics of the
brain. Penrose also offers a specific argument for, or
more accurately, a vision of what kind of physics it
would and should be. Penrose's argument, however, is
fundamentally based on his interpretations of these
theories and data, in particular quantum physics and
post-Gödelian mathematical logic, two crucial ingredients
of his overall argument. These interpretations are not
uncontestable (and have been contested), as Penrose
admits. This point is especially significant precisely
because Penrose offers a vision of specific future
theories of non-computational physical processes
modifying existing theories in physics (and a program for
developing such theories), and his case against
artificial intelligence is fundamentally tied to the
possibility and plausibility of such theories. Most
especially at stake is a particular form of "quantum
gravity" theory, a theory that joins quantum physics and
general relativity, the Einsteinian theory of
gravitation. Theories envisioned by Penrose will also
account for a non-computable physics of the human brain
and the non-computational mathematics of the human mind,
and specifically for the phenomenon of consciousness.
That is, these will be theories of physics that will lead
to the physics of the brain that makes thinking and
consciousness possible, along the way bridging the
classical (macro) level--"the Large"--and the quantum
(micro) level--"the Small"--so far resisting such a
bridging in physical theory. In Penrose's own words,
We look for the non-computability in physics which
bridges the quantum and the classical levels. This
is quite a tall order. I am saying that not only do
we need new physics, but we also need new physics
which is relevant to the action of the brain. (103)
One needs the non-computability in physics, which will,
first, bridge the hitherto unbridged classical and
quantum physics, and then will bridge two other hitherto
unbridged territories--the physics and biology of the
human brain. A tall order indeed, and Penrose
acknowledges that his vision is speculative and that it
reflects certain particular "prejudices" of a
philosophical nature (97). There are, as will be seen,
also "prejudices" that Penrose does not quite see, either
by (overtly) seeing some of them as non-prejudicial
arguments or by (unconsciously) missing, being blind to,
the presence of others in his argument.
9. Before commenting on the nature or, one might say, the
structure of Penrose's speculation, I would like to
discuss Penrose's presentation of key scientific theories
involved. As I have indicated, Chapter 1, "Space-Time and
Cosmology," which deals with "the Large," may well be the
most rewarding chapter of the book. It also offers a
discussion of the kind of theory (modelled on and
extending Einstein's general relativity) that
conceptually grounds Penrose's vision of what a physical
theory should be, whether as a theory of the large, the
small, or of the human brain/mind, or, as is Penrose's
ultimate desideratum, of all three together. One finds in
this chapter a beautiful, highly informative, and
reasonably accessible exposition of, among other things,
special and general relativity, relativistic cosmology,
and non-Euclidean geometry (the mathematical basis of
general relativity). The overall exposition is itself
mostly geometrical, which, as will be seen, also signals
a significant philosophical point, as an example of
something that is more likely to be non-computable.
Penrose's readers, especially non-scientists, will learn
much about the conceptual richness of modern mathematics
and science, and might be motivated to read or reread
Penrose's discussion of these subjects in his earlier
books, or even in the more technical exposition of The
Nature of Space and Time, his debate with Hawking.
10. While the arguments of that debate may appear only
tangential to the main ostensible concern ("the human
mind") of Penrose's project, these arguments are in fact
crucial, as is clear from the extension of this debate in
The Large, The Small, and the Human Mind. The Nature of
Space and Time provides a more comprehensive and more
sustained picture of the experimental evidence (or at
least testability) and theoretical argument as to what
kind of theory "quantum gravity" could or should be.
Penrose and Hawking disagree on that issue, as well as in
their overall philosophical positions--the Platonism of
Penrose versus the positivism of Hawking, or what they
see as such.
11. The specific shape of the evidence and arguments for (or
against) a theory of quantum gravity is, as I said,
crucial to Penrose's argument concerning the human mind
as something that, in contrast to computational digital
computers, is based on non-computational thinking,
mathematical or other. The debate with Hawking shows,
however, how deeply speculative and at times problematic
(at least at certain points and along certain lines) are
Penrose's ideas concerning quantum gravity even on
scientific grounds. These complexities would remain,
regardless of whether one agrees or disagrees with
Hawking's own vision of this theory, which is not without
some speculation either; or regardless of how one can use
currently available theories in approaching some of the
problems at issue (Hawking's primary concern).
12. Obviously, such complexities in themselves offer no
grounds for a criticism of Penrose's ideas concerning
modern physics, especially relativity, which have
elicited much admiration, as have Hawking's ideas. One
might want to be more cautious in evaluating certain
aspects of Penrose's presentation of his argument as
limiting the scope of the debate in modern physics
concerning key scientific and philosophical questions at
issue, on which I shall further comment below. My point
at the moment is double. First, general readers of
Penrose's work should be aware of the complexities of the
scientific (rather than only philosophical) nature of
Penrose's argument concerning relativity and cosmology,
and even more so of other scientific theories and
questions he considers. Second, The Nature of Space and
Time is especially indicative of this situation and is
especially significant in exposing such complexities,
more than is The Large, the Small, and the Human Mind.
13. For this and other reasons, of all the books mentioned
here, The Nature of Space and Time may well be the most
interesting and exciting one for scientists and, I would
even argue, for general readers as well. The latter
argument can, I think, be made in spite of the more
technical and difficult, and at points prohibitive,
nature of the book, but, to a degree, also because of
that nature. A nonscientist may be unrecoverably
frustrated even by such (by the book's standards) benign
elaborations as: "The No-Boundary Proposal (Hartle and
Hawking): The path integral for quantum gravity should be
taken over all compact Euclidean metrics. One can
paraphrase this as 'The Boundary Condition of the
Universe Is That It Has No Boundary'" (The Nature of
Space and Time, 79). The primary interest for general
readers may be the very scene of the debate between two
great scientific minds, rather than the scientific or
even philosophical substance at stake, and the book is
framed (not altogether justifiably) as a continuation of
the Bohr-Einstein debate concerning quantum mechanics. An
attempt to penetrate more deeply into the philosophical
substance of the debate may, however, bring considerable
rewards. Michael Atiyah (a great mathematician in his own
right) describes the situation well in his foreword:
Although some of the presentation requires a
technical understanding of the mathematics and
physics, much of the argument is conducted at a
higher (or deeper) level that will interest a
broader audience. The reader will at least get an
indication of the scope and subtlety of the ideas
being discussed and of the enormous challenge of
producing a coherent picture of the universe that
takes full account of both gravitation and quantum
physics. (viii)
First, then, by reading The Nature of Space and Time one
gets an indication of the scope and subtlety of the ideas
being discussed, and, I would add, of their conceptual
and metaphorical richness. This richness is one of the
great philosophical or, one might even say, poetic
achievements of modern mathematics and science. Penrose's
and Hawking's ideas and their ways of thinking offer some
superb examples here. I am thinking in particular of
Hawking's ideas concerning the "gluing" of spaces of
different non-Euclidean geometries and curvatures in
constructing his model or rather theory of the universe,
or his ideas concerning more radical (than even in
conventional quantum physics) non-causalities in quantum
gravity (59-60, 103), or of Penrose's own rich
geometrical ideas, which inform and shape his books, and
in many ways define his mathematical imagination.
14. It is, one could argue, this richness that connects
modern science and modern--and postmodern--discourses in
the humanities in the most interesting and significant
ways. It is easy to understand from this perspective why,
for example, Gilles Deleuze appeals to Riemann's
mathematical ideas at key junctures of his work. We are,
however, far from having really approached what Riemann's
work and subsequent mathematics and science have to offer
us by way of new concepts, metaphors, ways of thinking,
and so forth. Arguably the most interesting and important
challenge for the interdisciplinary studies involving
science is to convey or indeed present this richness, on
the one hand, and meaningfully to absorb as much of this
richness as possible, on the other. Obviously, this
traffic can proceed in both directions, and Penrose's
philosophical arguments could benefit from absorbing
certain key recent developments in the humanities. One
also could and should, at least at certain points, expect
in the theoretical discourses in the humanities a level
of complexity and even a certain technical specificity,
and hence also difficulty, comparable to those of
mathematics and science. Philosophical (rather than
mathematical and scientific) complexity is sometimes
lacking in Penrose's books. Their main value in this
context is instead in the possibilities they offer for
this traffic by their presentation of mathematical and
scientific ideas.
15. Equally important is Atiyah's remark concerning "the
enormous challenge of producing a coherent picture of the
universe that takes full account of both gravitation and
quantum physics." First of all, this remark is, again,
indicative of the speculative and tentative nature of
Penrose's ideas (and other ideas he considers) even as
concerns relativity and, even more so, quantum physics,
or, to a still greater degree, those concerning the
biology of the brain, even in the scientific context.
These scientific ideas, moreover, are the subject of much
controversy and debate in these fields themselves.
Secondly, equally significantly, not only key scientific
but key philosophical ideas--such as those concerning
physical reality and its mathematical nature,
determinism, the question of truth in mathematics and
mathematical logic, and so forth--are as much part of the
debate within the scientific community itself as of the
debate in the humanities (or the social sciences), or of
the debate between both communities.
16. As we progress with Penrose from the classical world,
including relativity, the world of the large (although it
is no longer quite clear how classical this world really
is), to quantum physics, to Gödel's theorem, to the
biology of the brain, or (all the more so) as we travel
between them, the level of complexity, ambiguity,
speculation, debate, and so forth increases. This is all
the more so because, as I indicated at the outset,
Penrose's arguments, including his ultimate argument for
the non-computability of the human mind and against
artificial intelligence, depend not only on complex
aspects of mathematical and physical theories themselves
but on his interpretations of these theories. These
interpretations are far from being broadly accepted
(which Penrose acknowledges) and some of their aspects
are rather idiosyncratic, and sometimes problematic.
Penrose's non-computability argument, however,
irreducibly depends on these interpretations. While, to
his credit, Penrose acknowledges such complexities and
complications and considers some of them in his books,
these books, I would argue, do not fully reveal to their
readers the extent of these complexities and
complications, and of the debates concerning them. There
are, that is, levels of complexity surrounding Penrose's
ideas that are hidden (I am not saying deliberately) from
an unprepared reader, and the debates incorporated in The
Large, the Small, and the Human Mind or even in The
Nature of Space and Time help only partially in this
regard. Science itself appears to provide no conclusive
evidence for most, if any, of Penrose's key ideas, and
indeed certain of his claims concerning scientific
theories are questionable. One can argue that modern
mathematics and science, especially quantum physics or
post-Gödelian mathematical logic, provide more evidence
against classical or traditional thinking (based on the
concepts of reality, determinism, truth, knowledge, and
so forth) than for it. Nor is there indeed any measurable
consensus of opinion on these issues among the scientific
or philosophical communities themselves. Whether one
speaks of mathematics, physics, or biology (or indeed of
consciousness and the mind), classical ideas and ideals
are put into question by science itself, including even
by what is seen as classical science. Modern science
questions some of the same ideas as does some of the most
radical postmodernist work in the humanities, and
questions them just as radically.
17. To illustrate the argument just offered, I would like to
consider one of Penrose's key comments on quantum
mechanics in The Large, the Small, and the Human Mind:
One of the things which people say about quantum
mechanics is that it is fuzzy and indeterministic,
but this is not true. So long as we remain at this
level [of the quantum, small-scale behavior],
quantum theory is deterministic and precise. In its
most familiar form, quantum mechanics involves use
of the equation known as Schrödinger's Equation
which governs the behavior of the physical state of
a quantum system--called its quantum state--and this
is a deterministic equation.... Indeterminacy in
quantum mechanics only arises when you perform what
is called "making a measurement" and that involves
magnifying an event from the quantum level to the
classical level. (8)
One can indeed say that there is nothing fuzzy or
imprecise about quantum mechanics in the sense that it is
as precise and effective as any mathematical theory in
the history of physics. The claim that it is
deterministic is far more complicated, however, and may
indeed be unacceptable, at least in this strong
form--"this is not true." There is certainly more
disagreement with the view advocated by Penrose than this
statement or Penrose's overall treatment of the subject
would suggest, even though he, again, indicates that his
overall view of quantum physics is not widely accepted.
One might argue that there is a degree of consensus that
Schrödinger's equation itself is mathematically
deterministic. There is, however, hardly any consensus at
all as to what, if anything, it is deterministic about.
At best it may be deterministic about indeterminism--that
is, in gauging the distribution of the randomness in
quantum behavior, which behavior, it is true, is manifest
only at the macro level of measurement. One cannot,
however, infer from this fact, in the way Penrose appears
to do, that the micro--quantum--behavior is physically
deterministic on the basis of Schrödinger's equation
alone. This is a (metaphysical) assumption, not a logical
inference on the basis of the available data of quantum
physics. In Max Born's elegant formulation: "The motion
of particles follows the probability law but the
probability itself propagates according to the law of
causality" (cited in Pais, 258). Probabilities can be
gauged in a reasonably deterministic manner, for example,
by using Schrödinger's equation. The process itself,
however, is never fully predictable, and is constrained
by Heisenberg's uncertainty relations, which are inherent
in Schrödinger's equation as well. Indeed in any given
case just about anything can happen. In this, quantum
physics is very much like life, or chess. To cite
Hawking's comments in The Nature of Space and Time:
"Einstein was wrong when he said, 'God does not play
dice.' Consideration of black holes suggests, not only
that God does play dice, but that he sometimes confuses
us by throwing them where they can't be seen" (26), and
speaking of further indeterminacy that gravity may
introduce: "It means the end of the hope of scientific
determinism, that we could predict the future with
certainty. It seems God still has a few tricks up his
sleeve" (60).
18. A number of other examples of the kind just considered
can be given here, in particular (still in his discussion
of the quantum world) certain aspects of his
interpretation of the Einstein-Podolsky-Rosen argument
and Bell's theorem, or some aspects of his interpretation
of Gödel's and Turing's findings in mathematical logic.
As with Penrose's claim concerning quantum determinism,
these examples are not random. They occur at crucial
junctures of his overall argument concerning the human
mind and artificial intelligence. I mention these
examples even though my space does not allow me to
consider them in the detail necessary to offer a fully
rigorous critical argument. My aim, however, is not so
much to criticize Penrose, but to indicate the broader
(than Penrose himself suggests) scope of the hypothetical
and the "prejudicial" in the landscape surveyed by his
books.
19. I borrow the characterization "prejudicial" from Penrose
himself, but give it a broader philosophical rather than
negative meaning, as Penrose perhaps does as well.
Penrose organizes his key philosophical
"prejudices"--"that the entire physical world can in
principle be described in terms of mathematics"; "that
there are not mental objects floating around out there
that are not based in physicality"; and "that, in our
understanding of mathematics, in principle at least, any
individual item in the Platonic world is accessible to
our mentality, in some sense"--into a Penrose triangle of
the Platonic, Physical, and Mental Worlds (96-97,
137-39). The Penrose triangle is arguably the most famous
object which can be drawn so as to appear physically
possible, but which cannot actually exist, and as such it
was an inspiration for Escher's famous drawings, which
are often in turn used by Penrose. One finds a picture of
the Penrose triangle in The Large, the Small, and The
Human Mind (138). The title itself suggests (I think
deliberately) a triangle and a Penrose triangle, similar
to that of Platonic, Physical, and Mental Worlds. Both
these triangles are in fact multiply connected into a
kind of complex and perhaps ultimately impossible
network. As I have pointed out, one of the main questions
of the book (of all the books at issue here) is that of
the possibility of bridging the hitherto unbridgeable;
and the Penrose triangle is of course a very fitting
figure in this context. While Penrose ultimately aims at,
at least, some bridging, the metaphor itself inevitably
suggests that at best one can only achieve an illusion of
bridging, but can never actually implement it. Penrose is
obviously aware of this, but I think that the broader
space of the hypothetical and the prejudicial, as here
considered, not only makes the figure of the Penrose
triangle even more pertinent and poignant here, but also
suggests a different implication of its use by Penrose.
It suggests that each of the entities Penrose wants to
bridge--whether large or small, human or inhuman--are
themselves networks of real and Penrose triangles, or of
much more complex figures or unfigures and networks of
that type. This irreducible multiplicity might give us a
better sense of the figures and of the unfigurability of
the large, the small, and the human mind, and of the
possible or impossible interconnections between them.
20. By the same token, however, this richer and more complex
conceptual geometry also suggests that we may connect
things that Penrose (perhaps) wants to separate, for
example, the computable and the noncomputable, or the
human and the nonhuman. As I have pointed out earlier,
the "prejudice" against computational thinking has a very
long history which extends from the pre-Socratics to
Heidegger and beyond. I can only consider here one early
event in this history, in which it is, fittingly,
geometry that (as against both arithmetic and logic)
appears to have been especially associated with
non-computational and/as creative thinking--the thinking
of mathematical and perhaps (at least for Plato) all
philosophical discovery. It appears that ultimately
Penrose takes a similar view as well, although he does,
of course, argue for the ultimate non-computability of
arithmetic as well in view of Gödel's findings. My
example is all the more fitting here since it has to do
with the diagonal of the square. Just as it was the
square where numerical computation was defeated by the
Greeks, it was the square--now that of the chess
board--where the latest defeat of the non-computational,
the mind of Garry Kasparov, took place.
21. The diagonal of the square was both a great glory and a
great problem, almost a scandal, in Greek mathematics and
philosophy. For the diagonal and the side of a square
were proved to be incommensurable, a discovery often
attributed to Plato's student Theaetetus. Their "ratio"
is irrational, that is, it cannot be represented as a
ratio of two whole numbers, and hence is not a rational
number. This was the first example of such a number--what
we now call the square root of, for example, two--a
number that was proved to be unrepresentable as a ratio
of two positive integers. It was an extraordinary and, at
the time, shocking discovery, which was in part
responsible for a crucial shift from arithmetics to
geometry in mathematics and philosophy, since the
diagonal is well within the limits of geometrical
representation but outside those of arithmetical
representation--as the Greeks conceived of it. To cite
Maurice Blanchot:
The Greek experience, as we reconstitute it, accords
special value to the "limit" and reemphasizes the
long-recognized scandalousness of the irrational:
the indecency of that which, in measurement, is
immeasurable. (He who first discovered the
incommensurability of the diagonal of the square
perished; he drowned in a shipwreck, for he had met
with a strange and utterly foreign death, in the
nonplace bounded by absent frontiers). (103)
The Greeks, then, might have been more ambivalent about
the relationships between geometrical and arithmetical,
or logical, thinking (and their relation to computation
and the non-computable) than is commonly thought, even
though Plato or Socrates might have seen geometry as the
greatest model of mathematical or even philosophical
discovery. In closing his book Penrose relates (a bit too
loosely) his philosophical triangle to the so-called
cohomology theory, which is part of the field of
algebraic topology:
You may ask, "Where is the impossibility [of the
Penrose triangle]?" Can you locate it?....You cannot
say that the impossibility is at any specific place
in the picture--the impossibility is a feature of
the whole structure. Nevertheless, there are precise
mathematical ways in which you can talk about such
things. This can be done in terms of breaking it
apart, glueing it together and extracting certain
abstract mathematical ideas from the detailed total
pattern of glueings. The notion of cohomology is the
appropriate notion in this case. This notion
provides us with a means of calculating the degree
of impossibility of this figure. (137-39, emphasis
added)
The appeal to calculation in the end of a book that
celebrates the incalculable and the non-computable could
delight an early deconstructionist a couple of decades
ago, and one finds the deconstruction of oppositions of
that very type in the works of Derrida and de Man, among
others. Penrose's comment, however, can hardly be
conceived as unselfconscious here. We must of course also
be aware of the difference between calculation and
computability. (Penrose, it should be noted, does not
deny the significance of either). My point is that, by
associating algebraic structures with topological ones,
cohomology theory connects the often incalculable or even
inconceivable geometry and topology (or indeed
inconceivable algebra) to arithmetical and algebraic
calculations and makes it possible to know something
about the noncomputable and the (geometrically or
otherwise) inconceivable. Mathematics may suggest to us a
better model than we might be able to offer to
mathematics. This model may be simultaneously both that
of computation and that of noncomputability, or even of
that which is neither one nor the other, and a sign of
intelligence that is neither artificial (or otherwise
inhuman) nor human, nor divine.
Literature Program
Duke University
aplotnit@acpub.duke.edu
Copyright © 1997 Arkady Plotnitsky
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Hawking, Stephen and Roger Penrose. The Nature of Space
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Pais, Abraham. Inward Bound: Of Matter and Forces in
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Penrose, Roger. The Emperor's New Mind: Concerning Minds
and the Laws of Physics. Oxford: Oxford UP, 1989.
---. The Large, the Small and the Human Mind. Cambridge
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---. Shadows of the Mind: A Search for the Science of
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