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CHAPTER FOUR: FOURS
IV. 3. EXCURSUS: DEMONSTRATION
AND PROOF
ARTIFICIAL INTELLIGENCE -- NUMBER-CRUNCHING
THE SOURCE OF CONFUSION
THE QUEST FOR BIG N
IV. 3. EXCURSUS:
DEMONSTRATION AND PROOF
Let us take an example from number theory
sometimes called "slicing a pancake." The problem is to get the
maximum number of pieces of pancake with the minimum number of slices. One
slice can only yield two pieces. Two parallel slices only produce three
pieces; but if they cross, they can yield four pieces. None of this requires
belief: with demonstrable certainty we know that the maximum number of pancake
pieces we can get with a single slice is two; with two slices of the pancake,
four; and with four slices, eleven. If anyone does not believe the consequences
of our slicing the pancake--no matter how many times we may slice it, there
is little enough to be done about it except to begin again with a new pancake
and to proceed by repeating the demonstration, step by step, slice by slice,
counting the pieces as we cut. There is no way that any one can ever get
five or more pieces of a pancake with only two (straight) slices; nor can
one get more pieces than eleven, with slices four.
This so-called "pancake" is an
abstract two-dimensional circular figure, and not really a four-dimensional
pancake, hot from the griddle, crowned with a pad of melting butter and
anointed with maple syrup. So obviously we can't split the "pancake"
with a slice parallel to our plate, nor fold it over, since it is supposed
to be a plane, two-dimensional, discoidal figure; just regular slices are
allowed. Proceeding thusly, such truths can be illustrated certainly and
convincingly on one's morning breakfast plate. Three slices yield a maximum
of seven pieces; and inexorably, the maximum number of pieces obtainable
with four slices of the pancake is eleven. These numbers are not a matter
of anyone's beliefs or opinions. They are necessarily true, provided (and
this is important) that our demonstration follows all of the rules. And
the truth so established has the same rigor and verity as a demonstrated
consequence of ordinary school algebra with x's and y's, after which we
are permitted to write Q.E.D., standing for quod erat demonstrandum,
or "this has been demonstrated."
[ See, N.J.A. Sloane, A Handbook of Integer
Sequences, p. 20, and sequence 391 on page 59. The sequence, also called
"central polygonal numbers" has the general formula N (N - 1)
/ 2 + 1; or, 1/2 n (n + 1) + 1. The sequence continues: 5 slices yield
16, then 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, and 137 pieces from
16 slices, and so forth.
THE TANTRIC PANCAKE
The word "demonstration," if
it is to be understood with the same force and clarity as in the technical
or mathematical sense, must be differentiated from those approximations
suggested by current, casual uses of the word. We may gain some appreciation
of the complete and utter totality possible for a demonstrable experience--
to complement the "breakfast plate" view of things being cut up
into pieces--by reflecting upon the meaning of a "demonstration"
from the wholistic viewpoint of certain high Tantric teachings according
to the Vajrayana tradition of Tibetan Buddhism. In the practicing tradition
of Tantric Buddhism, at the level of the ninth yana--the final stage
of the path, which is called maha ati, or ati yoga--one is
in a sense at both the beginning and the end, where the view is characterized
by a panoramic, global perspective with enormous space and total openness.
Of course, in maha ati there is warmth, there
is openness, there is penetration--all those things are there. But if we
begin to divide the dharma [Sanskrit = truth], cutting it into little pieces
as we would cut a side of beef into sirloin steaks, hamburger, and chuck,
with certain cuts of beef more expensive than others, then the dharma is
being marketed....The maha ati level is necessary in order to save the
dharma from being parcelled and marketed; that is, it is necessary to preserve
the wholesomeness of the whole path....
There is a children's story about the sky falling,
but we do not actually believe that such a thing could happen. The sky
turns into a blue pancake and drops on our head--nobody believes that.
But in maha ati experience, it actually does happen. There is a new dimension
of shock, a new dimension of logic. It is as though we were furiously calculating
a mathematical problem in our notebook, and suddenly a new approach altogether
dawned on us, stopping us in our tracks. Our perspective becomes completely
different....
Our ordinary approach to reality and truth is
so poverty stricken that we don't realize that the truth is not one truth,
but all truth....There are all sorts of philosophical, psychological, religious,
and emotional tactics that we use to motivate ourselves, which say that
we can do something but nobody else can. Since we think we are the only
one who can do something, we crank up our machine and we do it. And if
it turns out that someone else has done it already, we begin to feel jealous
and resentful. In fact, the dharma has been marketed or auctioned in that
way. But from the point of view of ati, there is "all" dharma
rather than "the" dharma. The notion of "one and only"
does not apply anymore. If the gigantic pancake falls on our head, it falls
on everybody's head.
In some sense it is both a big joke and a big
message. You cannot even run to your next-door neighbor saying, "I
had a little pancake fall on my head. What can I do? I want to wash my
hair." You have nowhere to go. It is a cosmic pancake that falls everywhere
on the face of the earth. You cannot escape-- that is the basic point.
From that point of view, both the problem and the promise are cosmic.
[ Trungpa, Journey Without Goal: The Tantric
Wisdom of the Buddha {Dharma Ocean Series}, Prajna Press, Boulder and
London (1981), pp. 135-137. ]
SPENCER BROWN / KEYS
For our interests, the crucial distinction
between demonstration and proof may be elaborated by citing an eloquent
discourse on the subject by G. Spencer Brown. We offer below a slightly-edited
version of remarks-- heretofore only privately published--that Spencer Brown
presented to an extraordinary audience in 1973. The group included: dolphin
researchers John Lilly and his late wife Toni, cybernetician Heinz von Foerster,
mapper of the brain Karl Pribram, electronic publisher Clifford Barney,
psychologists of altered states Charles Tart and Ram Dass (who altered his
state from being Dr. Richard Alpert, to which he--at last report--has once
again returned), psychiatrist George Gallagher, the Gestalt-Sufi-Buddhist
Claudio Naranjo, mathematician-yogis Douglas Kelly and Ted Guinn, cataloguer
of the Whole Earth Stewart Brand, the late ecologist of the mind Gregory
Bateson and the late (but Perennial) Philosopher, Alan Watts, and distinguished
others. The Esalen Institute setting was appropriately spectacular: a luxuriant
green niche of nature, with healthy food and hot baths, above the brontobooming
Big Sur surf, under a full moon at the vernal equinox.
The difference between demonstration and proof
is that a demonstration is always done by the rules. A computer can do
a demonstration....Whether we are giving a demonstration or a computer
is demonstrating, we just follow the rules within the calculus. But where
we have to prove something, we always find that we cannot find with the
rules within the calculus. In other words, no computer will compute a proof.
Proof is quite different. Proof can never be demonstrated.
I will give an example of proof--one which is familiar to us all, an illustration
not in the Boolean form but in the common school form of the arithmetic
of numbers--a very beautiful theorem and a very beautiful proof by Euclid.
The question asked: Is the number of primes infinite?
The prime numbers, as we see--and it is obvious
when you think of it--as they go on, they get sparser. It is very obvious
that they will, if you consider it, because every time we have a new one,
we have a new divisor which is likely to hit one of the numbers we're looking
for to see if it is prime. If it hits it--if it divides into it-- then
it won't be prime (a strange sort of statement: the science of certainty
taken in probability terms) because the more primes there are that could
divide into it. So for fairly obvious reasons, as we continue in the number
series, the primes get, in general, further and further apart. There are
fewer and fewer of them. And what Euclid asked was, do they get so thinly
scattered that in the end they stop altogether? Or does this never happen?
This is an example, now, of a mathematical theorem.
To make it into a theorem, you actually give the answer, you actually state
the proposition: "The number of primes is endless." You may not
be certain whether it is true or not. You may still be asking the question,
do they come to an end, or do they go on?
Well, to illustrate the difference between mathematical
art--because it now needs an art to do the theorem, where it only used
a techique, a mechanical application, to demonstrate something (and we
don't need to do that ourselves as computers can do it so much better)--we
will now do something that a computer can never do. Because what we are
going to do is find the answer to this question, do the primes go on forever
or not? We are going to find this answer quite definitely, and we are not
going to find it by computation, because it cannot be found by computation.
But it can be found like this. This is the way
Euclid found it. He said--supposing they come to a stop--all right, if
they come to a stop then we know they are going to go on for a long time
until we come to big primes, but, if they do come to a stop there will
be some largest prime, call it Big N. That's it. That is the last prime,
the biggest of the lot. If they come to a stop, there must be such a prime.
Now, if there is such a prime--and there it is, up there--let us construct
another number which looks like this: all primes, every single one of them,
up to and including Big N. Right. We have made this new number by multiplying
all the primes together. Now, Big N being the largest prime, this new number
is made up of all the primes there are. There isn't another prime, because
we have assumed that Big N is the largest.
On the hypothesis that Big N is the largest, this
new number is all of the primes multiplied together. And having done this
multiplication and getting the answer, we'll call the (new) answer Big
M. We will take this new number, Big M, and we will add one. Now we will
examine the properties of Big M Plus One.
You see, this is why arithmetic is so lovely:
it's about individuals. Here is our number Big M, as an individual, and
here is Big M Plus One. It is a hypothetical number. Actually, it is a
nonexistent number. And this is why we can't properly speak of number as
existing or as not existing, because some of them do and some of them don't.
Big M Plus One, let's examine its properties.
Well, it is obviously not divisible by any other prime, including Big N,
because we know that they all divide Big M. Therefore, every single prime
leaves a remainder when we attempt to divide it into Big M Plus One. So
Big M Plus One must either be prime, because it is not divisible by any
existing prime, or if it ain't prime, then it must be divisible by a prime
which is larger than Big N. Therefore, by assuming that there is a biggest
prime, which we have called Big N, we have ineluctably shown that this
assumption leads, absolutely without any doubt, to the construction of
a larger prime which is either Big M Plus One, or some other prime larger
than Big N which divides Big M Plus One.
And that is how Euclid did it. There are many
other, later proofs, of course. But that is still one of the simplest and
most beautiful. And the answer is absolutely certain that there is no largest
prime, that they do go on forever. This cannot be done by a computer. Currently
there is no computer that has done that.
We can do the prime factorials. Let us multiply
the first three primes: 2 X 3 X 5 = 30, and then add one. Right, we get
31. 31 is prime, but if you go out far enough, you will find that you get
one that isn't prime. But it will be divisible by a prime bigger than the
largest prime you have used. Let's see if we can find one. Here, wait a
minute, 211 is prime, isn't it? I'm just thinking of the prime factorial
plus one: at seven, it's (2 X 3 X 5 X 7 = 210, + 1 = 211), that's prime
factorial plus one. And 211 is prime as far as I know. We want a table
of primes here. We multiply the next one, 11 and add one, and it comes
out 2311 (2 X 3 X 5 X 7 X 11 + 1 = 2311). Is that prime?
[ It is.]
Anyway, I do assure you that if you go
on long enough, getting the final factorial, adding one, you will find
one that is not prime. But that doesn't matter (so far as the implications
for Euclid's proof are concerned), because it will be divisible by a prime
that is bigger than the biggest prime you have used to produce it.
[ March 19-20, 1973. The AUM Conference Transcript--documenting
what several of the participants felt was a truly astounding performance
by G. Spencer Brown/James Keys--was recorded, edited, and privately published
by the present author, together with Clifford Barney. It is available on
the World Wide Web at http://members.aol.com/lawsofform/.]
211 and 2311 are, of course, only two of
the many aliases of Big M Plus One; and each of these is in fact prime,
that is, not divisible by any other number but itself and one. In its next
manifestation, however, as 30,031 (the prime factorial for 13 plus one),
our "non-existent" messenger (or angel?) from the eternal realms,
Big M Plus One, is composite or non-prime, thus confirming the Spencer Brown
/ Keys assurances. For in this case 30,031 is divisible by two primes, 59
and 509, both of them larger than the prime used to produce the number,
which was 13.
If it were always prime, you would have immediately
a means, a formula, for producing primes, and this we haven't got. There
is no formula for producing primes except going about it the hard way and
seeing that they don't divide by anything.
[ James Keys, AUM Conference, Transcript
pp. 50-56. ]
ARTIFICIAL INTELLIGENCE
-- NUMBER-CRUNCHING
These days people use the computer to approach
such problems as determining whether or not a number is prime, in the anxious
hope of avoiding what we now see is the only truly effective method: employing
extensive and tedious computations in order to see if it divides by anything.
We refer to such wishful- thinking ruses as number-crunching: an activity
altogether different from conceiving or designing a proof; and so we do
not know if this activity REALLY leads to the True.
The confusion Spencer Brown referred to
has been endemic since the beginnings of modern AI (artificial intelligence)
research. The team of Allen Newell and Herbert Simon, then at the Rand Corporation,
in their first program called Logic Theorist, attempted to show that the
new electronic computers
were more than merely "number crunchers"
and...actually prove theorems taken from Whithead and Russell's Principia....And
in August 1956, the Logic Theorist program actually produced on Rand's
Johnniac computer the first complete proof of a theorem (Russell and Whitehead's
theorem 2.01....The demonstration [sic!] that the Logical Theorist could
prove [sic!] theorems was itself remarkable....However, the Journal
of Symbolic Logic declined to publish an article co-authored by the
Logic Theorist in which this proof was reported.
[ Howard Gardner, The Mind's New Science: A
History of the Cognitive Revolution, Basic Books, New York (1987),
p. 145. P. McCorduck, Machines Who Think, W. H. Freeman, San Francisco,
p. 142, is cited in reference to the Journal's rejection notice
and their demonstrable lack of any sense of humor. ]
This incident is noteworthy because of
deliberate attempts to create--with their symbol manipulation, including
list processing and detatchment methods--analogs of human thought processes,
based upon the symbolic logic of Russell and Whitehead's Principia.
As Howard Gardner recounts the story, supplying his own emphasis:
Newell, Simon, and their colleague [at Rand Corporation]
Cliff Shaw stressed that they were demonstrating not merely thinking of
a generic sort but, rather, thinking of the kind in which humans engage.
[ Gardner, The Mind's New Science, p. 147.
]
This claim was met with serious challenges
and, as it transpired, the history of AI and programming in general was
strongly marked by Newell and Simon's "dry" approach based on
notions of the "physical symbol system" and "production systems"
developed in their subsequent General Problem Solver program (1972). Even
so, many in AI and related fields remain philosophically confused because
of an underlying problem with the mathematical terminology itself. Further
along in the transcript quoted at length above, Spencer Brown/Keys pointed
to the venerable roots of this problem:
I know, as an engineer, the computer boys have
vastly oversold their products by saying that they can do anything that
the human mind can do and this is not so.
Newell, Simon and Shaw did not, themselves,
make an explicit claim to have designed a thinking machine, but they went
to great lengths to emphasize the parallels between human and machine problem
solving. Enthusiasm for their earlier achievements has promoted allusions
to "insight" in reference to "theorems."
For example, they reported certain moments of
apparent insight as well as a reliance on executive process that coordinates
the elementary operations of Logical Theorist...and selects the subproblem
and theorems upon which the methods operate.
[Gardner, The Mind's New Science, p. 147
f. ]
This realm of theory, however, is pervaded
by shadows, and misty areas of secrecy in which the understanding of just
what computer programs are actually doing, or what they soon will be able
to do, is frankly not very clear. Some reasons for this is obvious: the
security surrounding research in the field of Artificial Intelligence and
the fact that so much of this work is funded by entities not committed to
the free dissemination of knowledge. Added to this are the intrinsic difficulties
of the subject, in part because of its newness. But within a decade of Spencer
Brown's memorious presentation at Esalen, an astounding advance toward theory-formation
by machine has been accomplished by Douglas Lenat with his EURISKO. This
work has profound implications for epistemology and for the study of inductive
reasoning that so deeply characterize our quest for the True. Mr. Lenat's
program, working at the task of "discovering and modifying useful new
heuristics" (running for ten thousand hours) presented evidence of
machine-generated theorems with some apparent qualities of "insight."
[ Douglas B. Lenat, "Theory Formation and
Heuristic Search. The Nature of Heuristics II: Background and Examples,"
Artificial Intelligence: An International Journal, Volume 21 (1983),
pp. 31-59; quote, p. 57. ]
Just a year after Lenat's publication,
Paul Levinson helped to put these issues into perspective, while emphasizing
the distinction between two types of Artificial Intelligence:
"auxiliary" or "augmentative"
intelligence (as in mainframes extending and augmenting the social epistemological
enterprise of science, and micros extending and augmenting thinking and
communication on the individual level), and "autonomous" intelligence,
or claims that computers/robots can or will function as self-operating
entities, in independence of humans after the initial human programming.
The difference between these two types of AI is akin to the difference
between eyeglasses and eyes.
"Expert systems" and "human meat
machines" claims for autonomous intelligence in machines will be examined
and found wanting....The problem with current attempts at autonomous intelligence
is that the machines in which they are situated are not alive, or no not
have enough of the characteristics necessary for the sustenance of the
"living" label. Put otherwise, the conclusion will be: in order
to have artificial intelligence (the autonomous kind), we first must have
artificial life; or: when we indeed have created artificial intelligence
which everyone agrees are truly intelligent and autonomous, we'll look
at theese "machines" and say: My God (or whatever)! They're alive!
[ Paul Levinson, "Artificial Intelligence
and Real Life." Abstract of a talk given at the New School for Social
Research, as part of the Colloquium on Philosophy and Technology,
sponsored by the Polytechnic Institute of New York and the New School,
(November 12, 1984). ]
THE SOURCE OF
CONFUSION
Returning once again to the comments of
G. Spencer Brown, who diligently and patiently strives to unravel the knotted
twine of logic and inference the still binds much mathematical and programming
thought in confusion about the real potential of computers:
They cannot do the most elementary things that
the human mind can do. And I blame Russell and Whitehead for totally mixing
up proof and demonstration. If you go through the Principia, there
is not a single theorem, not one theorem...because what they call theorems
are consequences. Now this is totally confused: the idea of the difference
between demonstration and proof in mathematics. In fact Russell, you see,
in suggesting it, completely confused them; and people have done so ever
since. What he called theorems are in fact consequences; they are algebraic
consequences which can be, in fact demonstrated [as shown-- demonstrated--by
the Logic Theorist]. And indeed, Russell says, "These theorems"--he
calls them theorems, they are consequences--"can be proved."
And then he does the demonstration, and then he calls it "Dem."
"Dem." is short for "demonstration."
The two words are used interchangeably, and wrongly.
There is a difference, and what can be demonstrated is done within the
system and can be done by computer. And what cannot be demon-strated, but
may be proved, cannot be done by computer. It must have a person to do
it. No computer can prove it, because it is not proved by computation.
The steps of this proof, Euclid's proof, were not computational steps.
The computer cannot do it because it is not computation.
[However], they had a precedent in that Euclid
himself already rightly called this a theorem [that "the number of
primes is endless," and rightly] calls it algebraic. His geometric
consequences he called theorems; they are not. So the confusion developed
right at the beginning with Euclid, who called his geometric consequences,
which can be computed, "theorems." Wrongly. Euclid was
the first offender. And from him: it just shows how we have copied; we
have copied his error through hundreds of years.
I may be wrong, you see. My Latin--I have little
of it--perhaps he was O.K. Euclid said quod erat demonstrandum,
"this has been demonstrated." It is O.K. after a demonstration;
it is misleading after a proof. And maybe he did not make this error, but
we have. We have called them theorems when we should call them consequences.
And this has been responsible for a vast system of error [which] has grown
up, because a computer has been found to be able to demonstrate consequences--and
all you need is the calculating facility to do this.
And consequently, the demonstration of consequences--in
other words, calculations--has been confused with the proof of theorems,
which is another matter altogether. Because of this confusion it has been
thought that a computer, therefore, can do practically all that a man's
mind can do. But it can't, because only the most minor function of a man's
mind...is to compute. And we have, in fact, this tremendous emphasis--because
of the confusion in mathematics, the difference between computation and
actual mathematical thinking--which has lead us to believe that computers
have minds and can do what we can do....Even here, what a computer can
do, a man can do better if he gives himself to the problem, because he
has the capacity of seeing in a way the computer never can.
[ Keys, AUM Conference Transcript, pp.
56-60. ]
Since the book which Russell and Whitehead
chose to call the Principia Mathematica resides menacingly in the
distant gloom of this historical confusion, it may be worthwhile tracking
the mystery to the early years of this century, noting the possible influence
of biographical factors in circumstances at Cambridge around 1910- 1913.
Whitehead was a man of mathematics. Russell knew
the forms, but he actually had no instinctual ability in mathematics. Whitehead
actually had. But Russell, being a stronger character, was able to program
Whitehead, and you will see this if you examine the last mathematical work
Whitehead wrote, which is called the Treatise on Universal Algebra with
Applications, volume one. I asked Bertrand Russell...I said I had never
been able to get volume two, and Russell said, "Oh, he never wrote
it." So it's all sort of a mystery.
But the mathematical principles of algebra, in
the usual complicated way are set out, including the Boolean algebras,
in this volume produced in 1898 as an only edition. By that time, Russell,
who was the stronger of the two characters, had got together with
Whitehead to do Principia Mathematica, which nobody was ever going
to digest. It was a very ostentatious title, because they had chosen the
title which Newton had used for his greatest work. Incidentally--it is
an extraordinary thing in the academic world, people are very silent about
these things but--it was a very, very presumptuous title, I think, to take
for this work.
[ Keys, AUM Conference Transcript, p.20
f. ]
Late in life, Lord Russell himself paid
tribute to Laws of Form-- first published in Great Britain by George
Allen and Unwin (1969), and in the United States by the Julian Press (1972)--in
comments printed on the dust jacket of the U.S. edition:
In this book G. Spencer Brown has succeeded in
doing what, in mathematics, is very rare indeed. He has revealed a new
calculus, of great power and simplicity...that particular calculus which
lets us see deeper into the nature of mathematics. Indeed, I still consider,
on re-examining this book after a two-year interval, that it is a work
of genius.
So too, in its way, is the American
Heritage Dictionary a work of genius, particularly for its rich network
of references compiled of probable (attested) and hypothetical (unattested)
Indo-European roots for American English words. But unfortunately, it does
not provide much help in distinguishing and clarifying either the words
or the concepts of DEMONSTRATION and PROOF. The first of these (through
the Latin demonstrare) is formed from DE "completely" +
monstrare "to show," from monstrum "divine
portent," from monere, "to warn," which words (with
cognates at both MONSTRANCE and MONSTER) are from the Indo-European root
men-(1), "to think." For PROOF (or the verb to PROVE) the
link is made back to the Latin verb probare, "to test, demonstrate
as good, from probus, good, virtuous." Here we note two points
of interest: 1. that the word DEMONSTRATE is used in the definitions related
to PROOF, and 2. that the idea of PROOF which we have introduced into the
discussion of the True (and the False) is etymologically related to the
Good. The Indo-European root cited is per-(1), the base for many
prepositions and other lexical elelments meaning "through," "forward,"
"at" and "around." Coming at it the other way, this
Dictionary gives, as its first definition for the word DEMONSTRATION,
"To prove or make manifest by reasoning or evidence." At this
level we are going around in circles; and a useful distinction between the
two words depends upon following the conventions of a more specialized context,
such as the way these words might be employed for articulating formal methodologies
in the discipline of mathematics.
THE QUEST FOR
BIG N
Obviously, a set of operations is very
simply programmed to conduct a test of divisibility, although for a large
number it may go on for a while, even when run on the largest and fastest
of today'scomputational machines. One modern candidate for Big N (as it
were, prime to the largest) was produced in 1984 by Harry Nelson and David
Slowinsky who devised an operational shortcut for a program which they ran
on a CRAY supercomputer, generating the number: 2 raised to the power of
132,049, minus one--which they say is prime. Like all of the Mersenne primes
(following Euclid in Book IX of the Elements) it leads to an even
perfect number if, in this case, it is multiplied by 2 raised to the power
of 132,048 (i.e., two to the n, minus one, times two to the n
minus one). The general form two to the n, minus one, is usually
written as a Mersenne number: M sub n, or here M sub 132,409. The
notation is named after the form developed by Pre Marin Mersenne,
a natural philosopher, theologian, mathematician
and musical theorist, and a moving spirit of one of the most important
French scientific groups of the early seventeenth century. He was a friend
of Descartes, with whom he studied at Jesuit college... Fermat...and the
Pascals...to whom he proposed problems concerning perfect numbers and related
ideas.
The perfect numbers correspond one for one with
the Mersenne primes....As long as only hand calculation was available,
the discovery of Mersenne primes depended on human labor in actually making
the necessary calculations, and subtle theorems that showed that only possible
divisors of a certain type need be tried. The labor for large numbers was
immense. Mersenne himself stated that all eternity would not be sufficient
to decide if a 15- or 20-digit number were prime.
In 1814 Peter Barlow in an article in A New
Mathematical and Philosophical Dictionary wrote, "Euler ascertained
that 2 raised to the power of 31 minus one = 2,147,483,647 is a prime number;
and this is the greatest at present known to be such, and, consequently,
the last of the above perfect numbers which depends upon this, [i.e., 2
raised to the power of 30, times 2 raised to the power of 31 minus one]
is the greatest perfect number known at present, and probably the greatest
that ever will be discovered; for as they are merely curious without being
useful, it is not likely that any person will attempt to find one beyond
it." Barlow underestimated the fascination of record-breaking for
mathematicians, and he could not forsee the electronic computer.
By allowing millions of calculations per second,
the computer opened up vast reaches of numbers that had previously been
inaccessible and allowed mathematicians to make effective use of much more
powerful tests for primality. These tests decide whether n is prime
by analysing the factors of either n - 1 or
n + 1.Because
of their special form, Mersenne and Fermat numbers are easier to test for
primality than any other forms, and all the recent record-breaking primes
have been Mersenne numbers, and have automatically led to a new perfect
number [ David Wells, The Penguin Dictionary of Curious and Interesting
Numbers, Penguin Books (1986), p. 107- 110, where appears a list of
perfect numbers, including the four known to ancient Greeks: 6, 28, 496,
and 8128; see also p. 137 f. for the Mersenne numbers. ]
The number-crunchers Harry Nelson and David
Slowinski have since had their big prime number exceeded by a newer and
bigger candidate for Big N, 2 raised to the power of 216,091, minus one--and
hence for the largest perfect number: 2 raised to the power of 216,090 times
M sub 216,091 (i.e., times 2 raised to the power of 216,091, minus one)--generated
by Chevron Geosciences in 1985. Even so, the French mathematician Ren Thom,
featured on one of the Nova TV programs, clearly stated the standard
methodological objection to computer processing: "But is it a proof?"
Actually, this was in reference to a set of configurations drawn by a computer
in 1976 in order to attack the Four Color Problem. But after all, as Thom
protested, the computer could have made a mistake, and how would anyone
know?
This underscores the need for distinctions
between demonstration and proof drawn with such painstaking clarity by G.
Spencer Brown/James Keys at the AUM Conference. Referring to Euclid's classic
example of a mathematical approach to the True by inventing a theorem and
conducting a proof, this methodological point is driven home:
No one could do it on a computer because we were
not doing computation. Computation is counting in either direction, no
more, no less. There is nothing more to computation than that, nothing
more. Let's go through the steps again. Where's the computing? We compute
nowhere. There is no computation in this proof. Not a single computation
can be made, not one. The whole process is a proof. In the whole process
of a proof, there is not one single computation, nothing that a computer
could do. We were imagining doing a computation of a particular kind--but
we weren't actually doing it, because there were no numbers to put in the
places. In fact, there only could have been a computation if our number,
Big N, being prime to the largest, happened to exist. Yes. If it happened
to exist and if we knew what it was, then we could do this whole thing
on a computer. But it doesn't happen to exist. Yet, in order to find that
it doesn't happen to exist, we had to go through the imaginary steps of
computing in this particular way. We were divining the answer. We were
divining what had to be done by making certain deductions and seeing what
they led to. This was an artistic process, not a mechanical one.
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