The sixth episode of Adrian Moore’s radio ‘A History of the Infinite’ is
concerned with the infinitely big, considered not in terms of physical size,
but in the context of mathematics. It focuses on the work of the German mathematician,
Georg Cantor, who devised a way of distinguishing between different infinite
sizes, and of calculating with infinite numbers. Cantor was the first to do
such a thing. One of the most interesting developments in modern
mathematics, and as Moore says, his work was ‘utterly revolutionary.’
Everyone knows there is no such thing as the biggest number.
No matter how far you travel along a sequence of numbers, you can always count
further. Even Aristotle, who Moore suggested in an earlier episode was an
arch-sceptic about the infinitely big, accepted the reality of the infinite
only in terms of processes and sequences which were destined to go on for
ever.
This might be a little tendentious, since as Moore has
already pointed out in the episode ‘Aristotle’s Rapprochment’, he divided the
concept of the infinite into two things: the actual infinite, and a potential
infinite. The world of numbers and calculation exists in the context of the
potential infinite, in which change happens in space and time. The actual
infinite, for the purposes of mathematics, is simply ignored, since it is
(apparently) not possible to work with it. I make this point since there is
much about Aristotle’s wider philosophical work which points to a strong
concern with the actual infinite. He isn’t sceptical about the reality of the
infinite.
Aristotle’s view prevailed for over two thousand years, and
during that period there was hostility to the idea that the infinite itself
could be the subject of mathematical study in its own right. This orthodoxy was
not challenged until the late nineteenth century, when Cantor presented a
systematic, rigorous, formal theory of the infinite. Moore is interested in
what drove him, and at what cost.
Cantor had a very hard time in trying to have his ideas
accepted by the mathematical community, partly because of the perception that
there was a religious component to his work. Henri Poincaré said of his work
that: ‘it was a disease, and there would be a cure.’ His teacher Leopold
Kronecker, who might have been expected to support his pupil, was hostile to
his work, and made it difficult for him to publish. Kronecker said ‘God made
the integers, all the rest is the work of man’. Cantor suffered several nervous
breakdowns, possibly because of the sheer perplexity of his work, and died in
an asylum.
Moore now considers set theory. How do you count without
actually counting, and know if a set or collection is the same size as another?
You can assemble pairs of things, such as male and female, cats, dogs, etc. If
they are paired, and there are no extra males, females, cats or dogs left over,
then you know that they are the same size without counting the individuals in
the sets.
Does this apply to the infinite? Cantor asked why not? But
here things get a little weird. The set of what Moore refers to as ‘the
counting numbers’ (positive integers) appears to be the same size as the set of
the even numbers. Even though the first set includes all the numbers in the set
of even numbers, plus all the odd numbers. If we want to show the number of
counting numbers is the same as the number of even numbers, we can do this
fairly easily by pairing the counting numbers with the even numbers which
result from doubling them. There will be nothing left over, so we can say that
these two sets are the same size as each other. Moore says that it is tempting
to say that comparisons of size just don’t make sense in the infinite case. But
Cantor accepted that they were the same size, despite the fact that the first
set contained everything in the second set, plus more besides.
Can we use this technique to show that all infinite sets are
the same size, which might not be a counter-intuitive conclusion? In fact, some
infinite sets are bigger than others, as Cantor discovered. Even if you start
with an infinite set, it will always have more subsets than it does have
members. You cannot pair numbers with the subsets: there will always be a
subset left over. So there are different infinite sizes. Moore does not draw
the conclusion that it is the unbounded nature of the infinite which makes the
differently sized infinities true. What is infinite contains all things which
are possible. It is not just something which is extremely large.
Cantor’s work polarized opinion in his lifetime, and it has
continued to polarize opinion ever since. The mathematician David Hilbert
famously said ‘No one shall be able to drive us from the paradise which Cantor
has created for us’. To which Wittgenstein responded: 'I wouldn’t dream of
trying to drive anyone from this paradise: I would do something quite different
– I would try to show you that it is not a paradise, so that you leave of your
own accord’
Moore concludes with a question: “Is Cantor’s work of any
significance outside mathematics? Some would say that it is not. It certainly
made its mark by creating as many problems as it solved.”
It can however be argued that many difficult questions are
difficult for us as the result of an important concept dropping out of western
philosophy, which is the concept of the plenum. This concept is not discussed
by Moore in this series of programmes. The idea of the plenum is that reality
itself is undifferentiated possibility, something which does not exist in time
and space, but contains every possible aspect of time and space, and everything
which might be contained in it as potential, as something which might be
generated within physical reality. With the idea of such a transcendent
reality, almost anything which can be imagined to exist, can have existence. But
such things will inevitably point back to the nature of the initial plenum in
some way, and be full of puzzles and paradoxes. In rejecting this view of
infinity, and treating it as if it had no bearing on sensible reality,
Aristotle and those who followed afterwards, effectively closed off the
possibility of understanding why such paradoxes exist in the physical universe.
In the seventh episode there is a brief introductory recap,
reminding us that Georg Cantor created a formal theory of the infinite in the
late nineteenth century. The impact of his work on mathematics was large, and
led to a period of unprecedented crisis and uncertainty. Subjecting the
infinite to formal scrutiny, led to mathematicians confronting puzzles at the
heart of their discipline. These puzzles indicate some basic limits to human
knowledge.
Moore invites us to consider the issue of sets of sets. How
can there be more sets of sets, than there are sets? He suggests at this point
that our heads may begin to reel. But why shouldn’t we have, say the set of
sets which have seven members? Enter Bertrand Russell, who, in trying to come
to terms with some of these issues, arrived at what is known as Russell’s
Paradox. He argued that once we have accepted that there are sets of sets, we
can acknowledge sets which belong to themselves, and those which don’t. A set
of apples is not a member of itself, for example, since it is not an apple.
The paradox arises in connection with the set of all sets
which are not members of themselves. On the face of it, there should be such a
set, but there is not. For the same reason that there cannot be a nun in a
convent who prays for all those nuns in that convent who do not pray for
themselves. This is a matter of logical rules. She is going to pray for herself,
only if she does not pray for herself, which is impossible. Russell’s paradox
seemed to indicate a crisis at the heart of mathematics, where sets play a
pivotal role.
Russell communicated
his paradox to the German mathematician Gottlob Frege, which is a
well-rehearsed incident in the history of philosophy and mathematics. Frege had
been trying to put these mathematical issues on a sound footing in a
three-volume work, which was two thirds completed. Russell’s paradox came like
a bolt from the blue. Frege replied saying he was ‘thunderstruck’, since the
paradox undermined his attempt to give a sure foundation to arithmetic, while
he was engaged in writing and publishing his life’s work. Frege died
embittered.
Returning to Cantor, Moore discusses his work with the
problem of the ‘counting numbers’, (1,2,3,4, etc), which constitutes a smaller
group than the group of possible sets of the counting numbers. The question
arose of how much smaller the first group was. Cantor’s hypothesis was that it
was just one size smaller, and that there were no sets of intermediate size.
But he was unable to confirm that this was the case, or to refute the idea. So
he was in a state of uncertainty for a long time, and this exascerbated his
lifelong problem with depression. This question was listed by David Hilbert as
one of the 23 most important questions in mathematics to be addressed in the
ensuing century.
The matter is not settled, even now. Is this the result of
mathematicians not being assiduous enough? Moore says that it has been shown
that it is impossible, using all of the tools available to mathematicians, to
resolve the issue. It looks as though we are stuck with an unanswerable
question. Perhaps not completely unanswerable, but it is with the toolkit
of mathematical principles which are currently available. No new principle has
been discovered in the decades since, so it looks as though we have stumbled on
an inherent limitation on mathematical knowledge.
The logician Kurt Gōdel showed that this limitation was in a
sense unavoidable, in that, with a limited set of mathematical principles,
there will always be truths which lie beyond their reach.
So there are many questions about the foundations of
mathematics, and their security, or insecurity. Russell’s paradox of the set of
all sets which don’t contain themselves, had revealed an inconsistency in the
principles mathematician’s had been working with up to then. David Hilbert had
said “how do we know there isn’t another inconsistency elsewhere in mathematics
generating the problem?” He devised a programme to map mathematics with a
limited but very precise set of principles, in order to discover if this was
the case. Gōdel’s work however, made it unlikely that this programme would be a
success. Is there a crisis in modern mathematics? It was suggested that
modified versions of the Hilbert programme have proved that there are no other
inconsistencies in basic mathematical principles. And that consequently the
rest of mathematics is essentially reliable and consistent. Moore concludes
that mathematical work on the infinite has left us acutely aware of what we do
not know, and indeed what we cannot know
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