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Date: Thu, 13 Nov 1997 09:56:30 +0200
Reply-To: Discussion List on the History of Mathematics
,
"Antreas P. Hatzipolakis"
From: "Antreas P. Hatzipolakis"
Subject: Surreal Numbers: Quotting Gian-Carlo once more.....
To: MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG
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On 11 Nov 1997, JHC wrote elsewhere:
[...]
> I'm actually someone who discovered "a very large infinity of
>new numbers", the so-called `Surreal numbers'. Despite that, I'm not
>at all convinced of the existence even of infinitely many integers.
>
> John Conway
Surreal numbers are an invention of the great John Conway. They will go
down in history as one of the great inventions of the century.
We thought Dedekind cuts had been given a decent burial, but now
we realize that there is a lot more to them than meets the eye.
A new theory of games lurks behind these innocent cuts. Thanks to Conway's
discovery, we have a new concept of number. We will wait another
fifty years before philosophers get around to telling us what surreal
numbers "really" are.
-- Gian - Carlo Rota, Indiscrete Thoughts.
Birkhaeuser, Boston-Basel-Berlin 1997, pp. 217-218
aph
From: Richard Morris
Subject: Infinity
Date: Mon, 12 May 1997 03:31:17 GMT
Organization: Netcom
The final (summing-up) chapter of Achilles in the Quantum Universe: The
Definitive History of Infinity (just published by Holt):
Chapter 11
is, of course, the mathematical symbol for infinity. It is a
picture of a mathematical curve called the lemniscate, and it was first
used to symbolize infinity in John Wallis’ Arithmetic Infintorum in 1656.
The usage quickly became popular, and the symbol is universally used by
scientists and mathematicians today.
The Greek word for “infinity” was apeiron, which means
“unbounded.” In the ancient Greekworld, apeiron often took on negative
connotations. It could mean “totally disordered,” for example.Apeiron
could be used to describe the chaos out of which the world was formed, or
a crooked line. According to Aristotle, the quality of being infinite was
“a privation, not a perfection.” In the world ofAristotle, as in the
worlds of his predecessors Pythagoras and Plato, there was no place for
infinity.
Aristotle realized that there were many things in the world that
seemed to be infinite. Space and time might go on forever, for example,
and a line might be composed of an infinite number of points. In order to
avoid the disorder that he thought to be associated with the concept,
Aristotle denied that the truly infinite really existed. He developed a
theory of the potentially infinite, and denied that an infinite void
existed outside the celestial spheres.
Aristotle and his predecessor Zeno agreed that infinity could be
a very problematical thing. It was Zeno who first laid bare the
problematical nature of the infinite. He believed that, by doing this,
he could show that the everyday world of common sense was not the true
reality. Though few people today would accept the doctrine of Zeno’s
teacher Parmenides that the world is an unchanging unity, the paradoxes
that Zeno created still fascinate us today. In fact, modern philosophers
have elaborated upon Zeno’s ideas to create paradoxes of their own. One of
the simpler ones concerns a light bulb that is turned on and off an
infinite number of times. As in Zeno’s paradoxes, this involves performing
an infinite number of acts. Logic tells us that it will be both on and
off in its final state. If the light bulb is turned on, and then off an
infinite number of times, then it will be off at the end. But if it is
first turned on, and then is alternately turned off and on an infinite
number of times, the final state will be “on.” This can be represented by
the mathematical series:
1 - 1 + 1 - 1 + 1 - 1 + ....
This series seems to have two different sums according to how we group
the numbers. It can be either
(1 - 1) + (1 - 1) + (1 - 1) + ...
which is equal to zero, or
1 + (-1 +1) + (-1 + 1) + (-1 + 1) + ...
which is equal to 1. Such series present no problem in mathematics.
Mathematicians call them divergent series and deny that they have any real
sums.* But in the real world a paradox does seem to result if one
represents the act of turning the light bulb on by the number “1” and the
act of turning it off by the number “-1.” One can then show that, in the
end, the light bulb is both in the state 1 (on) and in the state 0 (off).
As the example of the ancient Stoic philosophers shows, infinity
presents no problems when it is viewed uncritically. The Stoics believed
that the world was surrounded by an infinite void, and that time repeated
itself endlessly in an infinite series of cycles. They did not think to
ask why the world occupied one particular position in the void, and not
some other. If they had, they might have realized that to speak of
“position” in such a context is not even meaningfulAfter all, if there
were a void, if the world were displaced some distance from its present
position, nothing would have changed. There would still be an infinity of
empty space around it. They did not ask how cyclical time began, or
how it was possible that there had already been an infinite number of
cycles. If they had, they might have realized that infinity was not as
simple an idea as they supposed it to be.
Galileo realized that there was something baffling about the idea
of an infinite universe. His successors found it easy to contemplate this
idea because they didn’t engage in much reflection about the implications.
Because he had apparently never thought very much about the infinitely
large, Newton fell into error when he tried to add up all the
gravitational forces--an infinite number of them-that would act on an
individual star. He mistakenly thought that he could show that they would
balance one another out, and that an infinite universe would be stable.
Newton did ponder the infinitely small, but found himself baffled.
Newton’s infinitesimals presented mathematical problems, and these
problems were eventually solved. In mathematics, problems associated with
infinity are usually tractable, however difficult they may initially seem.
In physics, on the other hand, the appearance of infinite quantities in a
theory is usually a sign that something is terribly wrong. Since modern
physics attempts to probe so deeply into the nature of reality, scientists
have had to attempt to come to terms with the infinite on a number of
different occasions. In some cases, they have had only partial success.
For example, the theory known as quantum electrodynamics is an
extraordinarily successful theory, but no one knows whether or not it is
mathematically consistent. QED views the electron as a point particle.
Because the electron’s charge is seen as being concentrated in a
mathematical point, the “bare” electron turnsout to have an infinite
charge and an infinite mass that are shielded from our view by infinite
numbers of virtual particles. This bizarre picture of the electron is
accepted because the consequences of ascribing a finite size to the
particle are even worse. When calculations were first done using QED,
the quantities that the physicists were trying to find turned out to be
infinite. So physicists “subtractout” the infinities by making use of the
procedure called renormalization. However, renormalization is
a mathematically questionable procedure. Thus we have a situation where
science uses dubious mathematical techniques in QED and in the theories
such as quantum chromodynamics (QCD) that are modeled on it. This seems to
work, and work very well, but it does raise questions as to how well the
fundamental components of nature are really understood. Many theoretical
physicists hope that superstring theory will eventually resolve the
problems. However superstring theory has been a topic of intense debate.
Some scientists think that it will eventually lead physics to a kind of
theoretical nirvana, while others believe that it will eventually prove to
be a dead end.
The difficulties encountered by modern physicists show us that the
infinite is still as much a mystery as it was in the time of Zeno.
However, to my mind, it is the infinities that are encountered in
the fields of astrophysics and cosmology that are the most fascinating.
If we are to believe what Einstein’s general theory of relativity tells
us, then matter is compressed to infinite density in the interior of black
holes. It is agreed that Einstein’s theory should breakdown before this
point is reached. But this means only that the true nature of a black hole
singularity is unknown. No one is sure whether a singularity is a place
where all the known laws of physics break down and where space and time
come to an end, of whether a theory of quantum gravity would reveal
something totally unexpected.
At present, there is no theory of quantum gravity. Gravity is a
more complex force than the other forces of nature. Mass, energy, pressure
and the gravitational fields themselves all give rise to gravitational
forces. It has been shown that attempts to renormalize a quantum version
of general relativity must fail. The infinities that are encountered are
much worse than the infinities in the theories that are used to describe
the behavior of electrons, quarks and other particles. Because such a
quantum theory of gravity does not exist, scientists are not sure what
character space and time might have on a submicroscopic level. Physicist
John Wheeler has suggested that, in the region of the very small, space
and time might cease to be continuous, and that there might exist a kind
of quantum “foam,” that spacetime might be full of a lot of submicroscopic
bridges and holes. Other scientists have suggested that the very concepts
of “space” and “time” might ceato have meaning in this region. Thus, if
the density of matter and gravitational forces do not become infinite in
the interiors of black holes, one has every right to expect that something
very bizarre must be happening.
Most of the time, physicists see infinity as a problem that must
somehow be eliminated before further progress can be made. But, just as
Bruno did in his time, scientists working in the field of quantum
cosmology have embraced the infinite, suggesting that there really might
be an infinite number of quantum universes. In quantum mechanics, we can
often speak only of probabilities. As soon as we begin to apply the theory
to the universe as a whole, we are confronted with an infinite ensemble
of universes with different probabilities of existence.
When we play a game such as roulette, we can assign a probability
to the likelihood that any given number will appear. This probability is
2.63 percent in American roulette, and 2.70 percent in casinos as the one
in Monte Carlo, which use wheels containing only a single zero (as
compared to American wheels which have both zero and double zero). When
the wheel is spun, we know that only one of these probabilities will
become real. Two different numbers cannot be the result of a single spin.
But quantum mechanics views the world in a different way. Here,
probability is the fundamental concept, and probabilities have a kind of
reality that they don’t possess in the everyday world. When one assigns
probabilities to the different positions that an electron might have, then
there is a sense in which the electron occupies all of those positions
simultaneously. Thus when physicists consider the behavior of electrons in
atoms, they picture an electron cloud that surrounds the nucleus.
Similarly, if an electron can be in some arbitrary number of different
energy states, then there is a sense in which it occupies all of them.
These are not just theoretical models. The existence of such probabilities
has been confirmed by experiment. For example, there have been experiments
in which a single neutron has been made to simultaneously follow two
different paths, after which it was observed to interact with itself.
Thus if quantum cosmology envisions an infinite ensemble of
universes, it is necessary to consider the possibility that there really
are other quantum universes, and that many of them might be peopled by
beings very much like us. Some scientists, such as Murray Gell-Mann, have
even begun to speculate as to whether it might become possible to
communicate with these other universes some day.
There is also reason to believe that the number of universes may
be infinite in yet another sense. The big bang could be something that has
happened, not just once, but an infinite number of times. Thus, where some
of the pre-Socratic Greek philosophers, and later Bruno, spoke of an
infinite number of worlds, we now find respected scientists considering
the possibility that there may be an infinity--or perhaps even an infinity
of infinities--of universes.
It must be emphasized that there is no empirical evidence which
would indicate that these universes really exist. Thus it is hard to say
whether such ideas should really considered to be “science” or whether
they are a kind of metaphysical speculation that is framed in the language
of mathematical physics. Nevertheless, the possible existence of
innumerable “alternate” universes givesrise to some sobering thoughts.
Could it be that there are an infinite number of copies of you and me
in others universes? Could they be living lives that differ from ours in
an infinite number of, sometimes important, sometimes trivial ways. If
universes are created endlessly, does this mean that we--or individuals
indistinguishable from us--will live again an infinite number of times?
When I ask such things, I am not even engaging in speculation. On
the contrary, I am simply relating some of the thoughts that have arisen
in a mind that found itself confronted with the infinite, a mind that
feels itself very much in sympathy with Pascal’s famous confession:
“The eternal silence of these infinite spaces frightens me.”