This is a critical commentary on Adrian Moore’s A
History
of the Infinite, broadcast in ten episodes by the BBC (on Radio 4) across
two weeks in late September/early October 2016. The first episode was broadcast
on the 19
th September. Episodes are discussed in order of
broadcast.
I was in conversation with Adrian Moore from halfway through the
series, and this conversation continued into early 2018. Moore wrote a book on the philosophical and
mathematical aspects of the infinite in 1990, which was a volume in the series
‘Problems of Philosophy’. This radio series gave its arguments much more
exposure than they ever had before.
I have spent many years studying Greek philosophy, and as a
result I found both Moore’s arguments and his narrative concerning the idea of
the infinite to be oddly structured. There is a gaping hole at the start, since
Plato is scarcely mentioned, and none of his arguments appear in the narrative
(sometimes voiced in the dialogues by his master Socrates). He does
discuss the ideas of Pythagoras, but in such a way that it is hard to recognise
him, and the many parallels which exist in Plato’s writing. As a result, this
history of the infinite is not a complete history, tracing the discussion of
the idea from the earliest period possible, but a history with a strong point
of view, which begins at a point which is convenient for the arguments
which follow (Moore’s book on the infinite has a much broader compass).
Part of my purpose here is to outline Plato’s engagement
with the idea of the infinite, and to place it before Moore’s chosen point of departure.
Understanding what Plato said concerning the unlimited and unbounded
necessarily changes the interpretation of Aristotle’s views and arguments, with
which Moore begins. Simply writing Plato out of the narrative not only creates
something of a fictitious narrative, but also creates difficulties that
otherwise would not exist.
Oddly for an account of man’s engagement with the infinite,
the first of the series of programmes is titled ‘Horror of the Infinite’. Moore
quotes the mathematician David Hilbert:
The infinite has always stirred the emotions of
mankind more deeply than any other question; the infinite has stimulated and
fertilized reason as few other ideas have; but also the infinite, more than
other notion, is in need of clarification.
Moore accepts Hilbert’s characterisation of the idea of the
infinite. He begins by saying that
ever since people have been able to reflect, they’ve
been captivated and puzzled by the infinite, in its many varied guises; by the
endlessness of space and time; by the thought that between any two points in
space, however close, there is always another; by the fact that numbers go on
forever; and by the idea of an all-knowing, all powerful god. People have been
by turns attracted, fascinated, perplexed, and disturbed, by these various
different forms of infinity.
Indeed yes. But Moore’s account appears to start at
‘disturbed’, rather than ‘attracted’.
Is God the Infinite, and Reality
itself? Moore does not much concern himself with this question in this sequence
of programmes, at least not in the terms in which the Greeks understood the
question. The following is an extract from The
Sacred History of Being (2015):
The Greeks did
not contemplate the idea that the ‘existence’ of God, or the supremely perfect
Being, was subject to proof. This would have been anathema to them, for the
reason that they understood the very concept of the divine is inevitably beyond
the capacity of the human mind to understand, or to frame. It is also beyond
space and time. It is possible to say something about the divine, but that is
all. Saying that the supreme perfect Being has a property ‘perfection’ is fine,
but the meaning of this perfection is strictly limited in its human
understandability. To attribute the property of secular ‘existence’ to this
Being would have been regarded as absurd.
Yet it would be granted that one could argue that,
without the property of existence, the perfection, or the completeness of God,
was compromised. But for it to be in the world of change and corruption would
also be understood as compromising the perfection of the supreme Being. At
least in terms of public discussion. Thus the Greek view of reality and the
Divine was that there was a paradox at the root of reality and the gods, and
that it was not possible to define the nature of the Divine without
exposing that definition to contradiction. The enlightened enquirer into
the nature of the divine therefore is spared further pointless argument about
the nature and the very existence of God. Both are conceivably true. But the
true nature of the Divine, being a paradox, rises beyond our capacity to argue
about that nature. It remains a matter of conjecture.
Our human experience tells us we live in a world in
which change is possible, and inevitable. The definition of the Divine on the
other hand, tells us, the divine reality beyond this world of appearances is a
place of eternal invariance. It suggests that at the apex of reality, it is not
possible for the divine to act in any way, or to participate in the world of
change. Again there is a difficulty if we hold that the greatest and most
perfect Being can do nothing without contravening its essential nature. A whole
range of properties would clearly be missing from the divine nature.
It would seem that the Greek solution to this problem
was to argue, as Plato and the neoplatonists did, that the world of reality was
in fact invariable, as the theory requires. And it did not at any time change.
But a copy was made. As a copy it was less than perfect, and this imperfection
created the possibility of change, action, and corruption. This copy is
eternally partnered by the original, which stands behind it, unchanging and
unchanged by anything which happens in the copy of the original divine model.
As a copy it is the same, but as a copy it is different.
This however, is a solution which Plato labelled as a
likelihood. Which is code for: ‘this is not the answer to the problem’.
One of the properties of the supremely perfect Being
would be that he was one and not two. In the creation of a copy, the
invariability of the divine has been breached, and the divine is now two, not
one. Two, not one, would seem to be a fatal objection. Firstly the copy is a
representation of the original, and not the original itself. Secondly, the copy
is imperfect, and through the act of representation, it has become different.
The original continues complete in its original nature, with its original
properties and characteristics. Plato hints at territory beyond this
contradiction, but does not venture into it overtly.
This is the key mystery of ancient thought. To
understand the full significance of this problem, and its implications for
ancient models of reality, we need to look closely, as they would, at what a
copy of Being actually means. There can be no copy, at least not in
an objective sense. And if there is no objective copy, then the world which
moves and which has existence, must be a subjective view of
Being.
Apart from anything else, if the world is a wholly
subjective experience, occurring (if we dare to use that word) within Being
itself, then the change and motion which is apparent to us, and which
contradistinguishes the world of existence from Being, which is itself and only
itself, must be illusory. The illusion may be convincing, but ultimately it
remains as an illusion, however persuasive it is to us, that there is an
objective reality which is subject to change and movement.
This is the correct answer to the problem. Our
experience in the world is of finite things, which are finite representations
of things which are infinite. But this world is also infinite, and at the same
time. It is therefore a matter of apprehension, understanding, and will, if man
is to engage with infinity, and reality itself.
Hence Plato’s discussion of the ascent to The Good via
the Forms, to that infinite place where all knowledge is to be had, and to
descend again with divine knowledge, again entirely via the Forms, to the world
of sensibles. What he is actually talking about is a formal process and
discipline by which the finite human mind can engage with infinity.
Pythagoras was much closer to Plato in terms of
doctrine than scholars normally allow. I can demonstrate this by quoting the
Neoplatonist Porphyry who wrote about Pythagoras many centuries after his
lifetime. Porphyry’s account tells us that:
He cultivated philosophy, the scope of which is to
free the mind implanted within us from the impediments and fetters within which
it is confined; without whose freedom none can learn anything sound or true, or
perceive the unsoundedness in the operation of sense. Pythagoras thought that
mind alone sees and hears, while all the rest are blind and deaf. The purified
mind should be applied to the discovery of beneficial things, which can be
effected by, certain artificial ways, which by degrees induce it to the
contemplation of eternal and incorporeal things, which never vary. This
orderliness of perception should begin from consideration of the most minute
things, lest by any change the mind should be jarred and withdraw itself,
through the failure of continuousness in its subject-matter.
That is exactly the doctrine of the ascent and descent
via the Forms which is described by Plato. The definition of transcendent
reality in Plato (articulated by Socrates) is that it is a place beyond shape,
form, size, etc., and occupies no place on earth. It is however the place where
knowledge has its reality (the ‘eternal and incorporeal things’ mentioned by
Pythagoras). Connection with transcendent reality is possible by the likenesses
to the transcendent which have existence on earth, such as things which are
complete and whole, which therefore participate in the completeness and
wholeness of the transcendent reality. Completeness and wholeness require (in the
world of the mundane) delineation and limits, and so the limits and the
extremes of things are also things which participate in transcendent reality.
The principle of ascent to the ‘eternal and
incorporeal things’ is entirely a mental process, which does not involve any of
the senses. It proceeds via chains of similitudes, both up and down, as a
sequence of orderly perceptions. The goal is a form of communion with that
which never varies, and which is always one and unchanging, as Plato tells us
in the Sophist. The return from the communion with the Good
delivers beneficial things, because the Good is the source of all knowledge.
What is transmitted to us via the writings of the
Platonists, is something of the basis of both their understanding of what the Divine
actually is (the Infinite, the Limitless, and Reality itself), and how man may
have commerce with the Divine, through sacred rather than profane practices, in
a world which has a double nature, and in which man has a choice.
Looked at in this way, rather than being a history of
infinity, Moore’s argument is about the idea of the infinite from the point of
view of finitude. This is the way Aristotle chose to deal with the infinite, by
dividing the concept into the actual infinite, and a potential infinite, and
dealing with the latter. Moore has said elsewhere that the way he treats the
infinite is generally in terms of an Aristotelian Finitism.
We might pause here and consider what the implications might
be of the identification of the Infinite and the Divine, which seems to be
implicit in the views of a number of ancient philosophers. If they did so
identify these concepts, then much of Greek religious thought and
practice was based on a philosophical understanding of the infinite. In
which case, Moore’s history is a history of what happens when the actual
importance of the infinite in the life of man is forgotten, misunderstood, and
eventually no longer noticed for what it is. Much of Moore’s argument is shaped
by his Aristotelian Finitism.
In the first programme, Moore argues that the Pythagoreans
thought finite things were good, and that infinite things were bad (this
information comes to us via Aristotle), and that they thought they had evidence
that the finite had some kind of control over what was infinite. And that the
usefulness of rational numbers showed that this was the case. This is clearly a
garbling of Pythagorean thought from a distant age, if Pythagoras thought that
ascent to eternal and incorporeal things was important, as I’ve suggested. There
is also discussion of musical ratios, and the Pythagorean discovery that
different string lengths with simple ratios are more consonant to the ears than
those which involve large values. Their ‘discovery’ of irrational numbers,
which can be found using the theorem of Pythagoras, is said to have filled the
Pythagoreans with horror, and the story of one of their number being drowned at
sea after revealing their existence, is referenced. Rather than revealing their
horror of irrational numbers, this is a story which points to their interest in
whole numbers. The idea that they once had no idea about the existence of
irrational numbers is absurd.
The programme moves on to consider whether other ancient
Greeks had the same resistance to the infinite. The views of Anaxagoras on
infinite divisibility are discussed. Anaxagoras was relatively comfortable
about these ideas. Zeno’s paradoxes in connection with infinite divisibility
are also discussed, including his paradox of travelling by an infinite number of
half distances, which seems to imply that movement is impossible. The similar
paradox of Achilles and the Tortoise is also referenced. Observation and
reflection thus seem to contradict each other. Zeno distrusted observation to
the point that he believed that movement was impossible. Parmenides was Zeno’s
teacher, and taught the universe to be a simple unity. So, only the appearance
of motion is possible. Otherwise the universe would have to have infinite
complexity. Moore winds up the episode by suggesting that because of these
paradoxes, and the existence of irrational numbers, that there is some truth in
the suggestion that the Greeks had a horror of the infinite.
Looking at the content of this episode in the light of the
added preamble about ideas of the infinite held by Plato and Pythagoras, we can
see that something old and valuable is contained in the writings of some
earlier philosophers, transformed into more or less secularised accounts of the
arguments the Greeks used to illustrate the paradoxical nature of the infinite
aspects of the world, as they manifest in the world of the finite.
We get many clues about the Greek understanding of the
infinite and the unlimited from a number of Plato’s dialogues, including The
Timaeus, The Sophist, The Republic, The
Theaetetus, The Laws, and The Parmenides. In
skipping Plato, the first reference to Parmenides and his notion of the
universe as simply one and one alone, is as an introduction in the first
episode to his pupil Zeno of Elea, and his response to paradox. There is no
discussion of Plato’s demolition of Parmenides arguments, no discussion of the
Platonic forms, no discussion of the relationship of the forms to the form of
the Good, which is another way of talking about what is infinite, and no discussion
of what amounts to a different logical modality in the pages of Plato (where he
discusses things passing into one another by means of their similitude), which
is a way of understanding the relationship of finite things to the
infinite.
Essentially Aristotle’s rapprochement, which Moore
characterises as an attempt to make the concept of the infinite more palatable
to the Greeks, involved dividing the idea of the infinite into two. As already
mentioned, one of these was the potential infinite, and the second was the
actual infinite. As outlined in the first episode, Zeno’s paradoxes depended on
the idea of an infinite divisibility, which seemed to make the idea of any kind
of movement impossible, since that would require a universe of infinite
complexity. Zeno therefore regarded all forms of movement as illusion. Since in
order to travel a certain distance, you would have to travel half the distance
to your destination, and then half of the distance remaining, and then half of
that, and half of what still remained, and so on. Which would result in an
infinite number of steps. Which would be impossible.
Aristotle’s response was that though the various stages of
the journey could be understood in such a way, the stages were not marked, and
did not have to be considered in making a journey. The idea of limit is however
a crucial point. What Aristotle was saying is that there are two ways of
looking at the idea of what a limit is. Essentially there is limitation
which is defined by what a thing is, and there is limitation which is not. In
the first case the limit of a thing cannot be transcended without the nature of
that thing turning into something else.
The essence of this argument is that there are forms of
limit which can be ignored. One of which is the actual infinite: instead we
should deal with the potential infinite. The actual infinite, by its nature, is
always there. But we cannot deal with it. The potential infinite we can work
with, since it is not always there, and spread infinitely through reality. So
we can count numbers without ever arriving at infinity, or ever being in danger
of arriving there. Moore mentioned that this conception of infinity more or
less became an orthodoxy after Aristotle, though not everyone accepted that his
argument against actual infinity was solid. Which is something of an
understatement. Aristotle’s distinction between the potential infinite and the
actual infinite is between what is, in practical terms, something we can treat
as finite, and what is actually infinite.
It might seem surprising that Moore’s first port of call in
part three is the philosopher Plotinus, who was writing in the third century C.E.,
some five centuries after Aristotle. The reason that he has jumped to Plotinus
is because he argues that Plotinus claimed not only that the divine was
infinite, but that the divine was the infinite. Thus
conflating the ideas of divinity and infinity in a way that – he says – no one
had done before. Or, to be more precise, he declared the identity of the divine
and the infinite in a way no-one had done before.
Well no. As I’ve argued at the beginning of this essay,
Plato’s principal interest was in a transcendent reality, which it would be
hard to distinguish from the infinite, except in hair-splitting terms. He
refers to the necessity of ‘looking to the one thing’, and that the ‘one thing’
is something which is found nowhere on earth. In one of his dialogues, he has
Socrates describe that transcendent realm as something which possesses ‘no
form, shape or colour.’ It is clearly without definition and limitation, with
no finite properties and attributes, which means it is unlimited, and infinite.
It is also the ultimate source of all knowledge. So it also seems to possess
the properties and attributes which are associated with the divine. Plotinus’
supposed innovation is therefore no such thing. Anaximander’s understanding of
the ‘apeiron’ (the unlimited) as the cause of all things is just such an
equation of the divine with the infinite, which means the idea was around in the
sixth century B.C.E.
Moore tells us that Plotinus’ idea that the divine was
infinity itself was taken up by Thomas Aquinas, Aquinas was schooled in
Aristotle (there was a renaissance in the study of Aristotle at this time,
which was not mentioned), and he was aware of the distinction which Aristotle
had made between a potential infinite, and the actual infinite. He also became
an authority in the Catholic Church, and wanted to incorporate the philosophy
of Aristotle into Christian thought, but with an emphasis on the idea that the
divine had reality in the world in terms of a potential infinite, rather than
as actual infinity. Aquinas essentially divided infinity into a mathematical
conception, and a metaphysical or theological conception.
Aquinas’ ideas went on to dominate later Christian thought,
but not everyone agreed with what became the doctrinal position. Much later
again (Moore’s argument makes large temporal jumps), the arrest of the scholar
Giordano Bruno in the late sixteenth century, showed how dangerous the idea of
the Infinite could be. He was imprisoned and tortured by the Inquisition for
having argued that the universe was infinite, and that there existed an
infinite number of worlds. He was burned at the stake in 1600. Moore suggests
that this marked the end of the renaissance. Galileo Galilei was lucky
the same thing did not happen to him. His interest in the number of numbers in
existence was discussed in this episode, and these and the numbers of their
squares and roots, seemed to him to be infinite. This represented the birth of
a whole new set of paradoxes with later repercussions in mathematics.
The fourth programme discusses the views of Rene Descartes
in the sixteenth century, and also the views of philosophers from the
eighteenth-century Enlightenment. I haven’t added up the number of centuries of
thought which have not been discussed at all, but so far argument has been
drawn from the sixth century B.C.E. (Pythagoras) fourth century B.C.E.
(Aristotle, Zeno), the third century C.E. (Plotinus), the 13th century
C.E. (Aquinas), and the 16th century C.E. (Bruno). Which is a
journey of around twenty centuries.
It isn’t that there is nothing to say about the idea of
infinity during those long centuries, but that where Moore is going determined
his selection of evidence and argument. He wants to talk mainly about the role
and history of infinity in mathematics and in physics, and the fascinating
paradoxes and problems which later investigation has thrown up. And a little
about religious faith and the infinite. The first episodes are therefore a
necessary introduction to set the scene.
As he puts it in the text introduction to this episode, 'we
have arrived at a time where people think about these things as we now do.' A
telling statement, which hints at the richness and strangeness of the
unexplored territory between the sixth century B.C.E. and the sixteenth century
C.E., and that most of it is best skipped over as quickly as possible. It also
lets us know that he has a normative view of human thought, and that what he
thinks is rational and reasonable is mostly to be found in modern times. His is
the Enlightenment agenda, which he mentions during this episode.
Descartes famous ‘Cogito Ergo Sum’ (‘I think therefore I am’)
is mentioned in the context of Descartes massive reduction of all the ideas and
beliefs which he could accept unequivocally as true. He engaged in this
reduction in order not to rely on tradition and authority, but on the
intellectual resources available to the finite human mind. The question of
whether the infinite can be grasped at all by the human mind is discussed,
since we cannot see it or touch it. It is hard for us to know it, because it is
the infinite. Descartes is quoted as saying that you cannot put your arms around
a mountain as you can around a tree. So our knowledge of the infinite is
necessarily less intimate than our knowledge of finite things.
In the next part, the relationship between Descartes
confidence in his own existence and capacity to think (expressed in the
‘cogito’) and his understanding of the infinite nature of God, is less than
clear. It is true that Descartes suggested that he might have an idea of an
infinitely perfect, infinitely powerful God because God put that idea into his
mind. That might be the case. Alternatively, it may be that you as a finite
being do not have to have an intimate acquaintance with the infinite in order
to understand what you are talking about.
Moore does not use the expression which Descartes employed
to explain why it was not necessary to have intimate knowledge of something in
order to have a useful and intelligible idea of what it is. He used ‘clear and
distinct’ idea to indicate when he had such a useful and intelligible notion of
what he was talking about. Later, Bertrand Russell would reformulate the
distinction between knowledge by acquaintance and knowledge by description (in
his Problems of Philosophy). So, by ‘clear and distinct ideas’
about God Descartes is relying on a description of what is, which means that he
could be sure what he meant, and that his idea of God was a rational
idea.
In fact, Descartes idea of his own finite reality was dependent
on his certainty of the reality of an infinite God. If he could conceive of
such a God clearly and distinctly, then it was likely that such a God was
real.
Moore skips on to the second half of the eighteenth century,
mentioning Berkeley (‘there is no such thing as the 10,000 part of an inch’ is
all that is said), and Hume also, in connection with the indivisibility of
reality (the disappearing inkspot when seen from sufficient distance, which is
a matter of perception and experience rather than indivisibility per se).
Berkeley was an idealist philosopher, who held that the only reason the world
is perceptible is because it is held in the mind of God. He also denied
materiality, at least as a metaphysical concept.
Finally Moore discusses a narrow aspect of Kant’s
understanding of the idea of infinity. This final part of the episode
represents a highly misleading understanding of Kant.
Moore argues that Kant agreed with Descartes that we have a
clear idea of the infinite (the nearest he gets to the Cartesian formulation
‘things which are clear and distinct’). But that our idea is limited to what we
can experience and perhaps what we can invest faith in. Really? I don’t think
it is. Did Kant say that knowledge is confined to the five senses? And if
we don’t understand knowledge this way, we leave solid ground and end up in
metaphysics? That is what seems to be suggested at this point in the series.
One of Kant’s principal interests was metaphysics, and how
we apprehend things and have knowledge of them. Hume’s empiricism was one of
the things which impelled Kant to write some of his most important works (The
Critique of Pure Reason, and The Prolegomena to any Future
Metaphysics which may Present itself as a Science). It isn’t the case
that Kant thought our ideas are limited to what we can experience in terms of
the senses, but instead what is intelligible to us is interpreted through the
categories of our understanding. He sought to understand shape and form without
these things being associated with form possessing scalar values and spatial
angles, which are matters of experience. In that he was very close indeed to Plato’s
understanding of the Platonic forms.
Kant, a figure so important to the concept of reason, is
quoted as saying that ‘I go beyond knowledge to make room for faith’. It is
true that Kant had the idea that rational thought and reason did not have to exclude
a life of faith. It had space in which to exist. But it does not mean that Kant
thought that faith was important to the life of reason. For Kant, like
Pythagoras and Plato, knowledge is not gained through knowledge of sensible
things, but is acquired by the contemplation of things which have a
transcendent reality. This isn’t something which everyone can do, or will ever
be able to do. Since there is an equation between the Divine and the Infinite,
what Kant is doing is leaving space for some sort of understanding of the
Divine for those who will never have a genuine understanding of transcendental
reality and the Infinite. He is not arguing that faith creates a functional
connection with the Infinite.
Karl Lōwith wrote that, in his book Religion within
the Limits of Reason Alone, Kant had
interpreted the whole history of Christianity as a gradual advance from a religion of revelation to a religion of reason…. It is the most advanced expression of the Christian faith for the very reason that it eliminates the irrational presupposition of faith and grace.
Moore then turns to Kant’s conception of the moral law.
Aspects of the life of the mind which put us in contact with the infinite are
about our reason, our rationality. Our reason enables us to grasp the moral
law, which gives us infinite dignity (since we are rational beings). He says
that “the moral law is what ought to direct us in all we do, with infinite
respect granted to fellow rational beings”.
Which explains little. The origin of Kant’s moral law may be
the idea that the life of reason, and rationality itself (as he defined it) is
about connecting with the infinite. If man is truly rational, then he is
connected with the Infinite (the ancient concept of the soul, as discussed by
Plato, is related to this idea). But we need to accept Kant’s understanding of
what reason is, and not distort it by saying knowledge is obtained through the
five senses. Through this distortion, what Moore is left with is the
Calvinist notion of a ‘sensus divinitatis’ (sense of divinity). Which is
a poor substitute for the kind of engagement with divinity which was understood
to be possible in the ancient world. Such engagement was not achieved through
knowledge of the world of the five senses or space and time.
In the episode which follows, the subject is the nature and
development of the calculus. It begins with the observation that to divide zero
by zero, or zero into anything at all, makes no sense. If you know anything
about the calculus, it is clear what is being talked about in this episode, but
the way it is discussed is lacking in the kind of precise description you might
expect.
A train is used as an illustration. Travelling a distance of
sixty miles over an hour means that the train had an average speed of sixty
miles an hour. However, the train might have been travelling at a much higher
speed for half of the journey, and have been delayed by signal failure during
the second half of the journey. So if you measure the distance travelled and
the speed at a particular point in the journey, the result may be misleading.
If the time period measured is very short, say close to zero, and the distance
travelled is close to zero, then you will know nothing useful about how fast
the train is going, and how long it will take to complete its journey.
Calculus enables the accurate measure of quantities which
are subject to change (which is why the inventor of the calculus as we know it
today, Isaac Newton, referred to it as ‘Fluxions’). The episode makes clear how
important the development of the mathematics of change has been ever since, and
that much of the modern world depends on the use of calculus. The term
‘integration’ makes no appearance in this episode.
Much of the rest of the programme discusses the invention of
calculus, and the bitter dispute which arose between Isaac Newton and the
philosopher Leibniz, who developed a similar approach to the mathematics of
change quite independently. Newton appears to have begun to develop the
mathematics for ‘fluxions’ early on – perhaps as early as the 1660s. The
chronology is not clearly established, but Leibniz may have developed his
version some ten years later.
Newton did not publish any information about the mathematics
involved in the calculus until many years later, preferring to share a few
details with his friends and colleagues. Newton was aware of Leibniz and his
work, not least because he too was a member of the Royal Society. Eventually he
wrote to Leibniz with some limited details of the calculus (Moore suggests that
Leibniz could not have understood these details since they were in
code). Newton became aware that Leibniz had developed similar mathematics
to deal with change, and a long dispute ensued, mostly conducted via
intermediaries. Leibniz was often travelling, and so correspondence sometimes
took months to reach him. Newton launched attacks on the integrity of Leibniz,
accusing him of plagiarizing his ideas. Leibniz was bemused by his attacks and
the force with which they were made. But Newton had decided that Leibniz was
his enemy, and that was that.
Eventually it was proposed that a report be prepared by the
Royal Society to establish who had the prior claim to the invention of
calculus. This sounds fair, except that the President of the Royal Society
wrote the report, and the President was Isaac Newton. As Moore says, ‘not
Newton’s finest hour’.
The philosopher George Berkeley makes another more
substantial appearance in this episode, since he wrote a criticism of what he
called ‘the analysts’ (The Analyst: a Discourse addressed to an Infidel
Mathematician (1734)). His criticism was based on the general lack of
rigour with which calculus was often used at the time, and argued that scholars
who attacked religious and theological arguments for lack of rigour were being
similarly careless. The criticism revolves around the limitations of the
technique already mentioned, when the quantities and measures chosen are too
small to produce intelligible results.
The most famous quotation from this book describes infinitesimals
as ‘the ghosts of departed quantities’. The book seems to have been aimed
particularly at the mathematician Edmund Halley, who is reported to have
described the doctrines of Christianity as ‘incomprehensible’, and the religion
itself as an ‘imposture’. Moore references the fact that the technique of the
calculus lacked technical rigour until the early nineteenth century, until the
idea of the limit was introduced (in fact Cauchy, and later Riemann and
Weierstrass redefined both the derivative and the integral using a rigorous
definition of the concept of limit. But that is another story).
Moore concludes the episode by saying that:
precisely what such precision and rigour show, is that
the calculus can be framed without any reference to infinitely small
quantities. There is certainly no need to divide zero by zero. What then
remains is a branch of mathematics, which is regarded by many, in its beauty,
depth and power, as one of the greatest ever monuments to mathematical
excellence.
The sixth episode 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.
The eighth episode opens with a discussion of the Andromeda
Galaxy, which is the furthest object in the universe which can be seen with the
naked eye. Its light takes two million years to reach us. Yet it is a close
neighbour to our own galaxy. The distance is mind-numbing, but it isn’t
infinite. The central question of this programme is ‘where does the concept of
infinity fit in with our attempts to understand physical reality?’
Looking up at the night sky is evocative of the infinite for
many of us because of the enormous distances involved, but how appropriate is
this? Is the number of stars infinite? Is space infinite? Is anything in nature
infinite? The Greeks contemplated these ideas. Moore quotes Archytas of Tarentum
(4th century B.C.E.) on this question:
If I am at the extremity of the
heaven of the fixed stars, can I stretch outwards my hand or staff? It is
absurd to suppose that I could not; and if I can, what is outside must be
either body or space. We may then in the same way get to the outside of that
again, and so on; and if there is always a new place to which the staff may be
held out, this clearly involves extension without limit.
Aristotle, who wrote a little later, resisted Archytas’
argument, and said that although it is true that the universe can’t be bounded
by anything outside it, nevertheless, it is only spatially finite. This may be
difficult for us to grasp, but it has become a staple of contemporary
cosmology. Moore asks, how do we arbitrate between the views of Archytas and
Aristotle? Moore then quotes the seventeenth century philosopher Thomas Hobbes,
who didn’t think there was anything to arbitrate on, since neither view
(infinite or finite) is based on anything which presents itself to the mind. So
the question (for Hobbes) is meaningless.
The mathematician John Barrow (Cambridge) distinguishes
between speculations about the universe as a whole, and our piece of it, the
part which is visible, about which we can have a scientific understanding. So
though the universe is potentially infinite in size, we can only see a part of
it, which may not be infinite. If someone says that it is infinite, that is a
purely philosophical speculation, and cannot be demonstrated. Moore then asks:
If we confine ourselves to the part which is visible to us, can we ask if that
part of the universe is infinite or finite?
Isaac Newton though that the matter in the universe had to
be infinite, otherwise the cosmos would collapse in on itself. The modern
consensus, Moore suggests, is that the amount of matter in the universe is
finite. And that indeed, space itself is finite. Even if unbounded, as
Aristotle thought. This is an idea which is difficult to grasp – how can the
universe be both finite and unbounded?
It might be possible for the universe to be both finite and
unbounded. The example is the surface of the earth. Whichever direction you
travel, you will always arrive back at the same place (assuming the oceans are
not a barrier of course). There is no limit to the distance you can travel, but
the surface of the earth is finite, and not infinite. Space itself may be like
this, so that there is no limit to the distance you can travel, but that its
extent is finite.
If space were curved like the surface of the earth, how
would that manifest itself? It is possible to distinguish between the
properties of triangles on a flat Euclidean surface, and their properties in a
curved space. You can tell the difference between these surfaces, because the
angles add up differently. It appears the evidence that we have suggests space
itself is not curved. The evidence needs to be drawn from examples which
involve distances of billions of miles [no explanation here of how this might
be done], so that the differences in the angles would show up.
It is still possible that space is finite, and indeed that
distant galaxies are (as they appear to be) moving away from us through the
expansion of space. The evidence for this is the phenomenon of the Doppler
shift. The same evidence allows us to extrapolate backwards. As the galaxies
are moving further and further apart, so we can infer that they were once very
close together, and originated in an enormous explosion of energy which may
have given rise to the universe as we know it today, a finite length of time in
the past.
Moore points out that scientists like to evade the infinite.
If an infinite crops up in equations, there is a feeling that a theory is
incomplete. Moore suggests that this finite point so long ago, doesn’t
eliminate the infinite, it reintroduces it. Mass and energy at the point of the
so called ‘big bang’ would have been infinite, if that apparent beginning was a
real event, there would have been infinite density, and infinite
temperature and, had the mass not been travelling at great speed, there would
have been the danger that the whole universe would have collapsed through
gravitational attraction, and there would be a big crunch of infinite density.
As may happen in the distant future.With infinitudes, you lose the capacity to predict things.
Which is another reason why scientists like to evade the infinite.
Moore concludes by recalling how the series opened, where
the attitudes of the Greeks to the infinite were explored. He repeats his view
that the Greeks had trouble understanding the infinite, and also found it hard
to ignore it. Now we find ourselves in a similar position. Exploration of the
cosmos invites us to reckon with the infinite, even when we are not sure what
the infinite is.
The programme following (number nine in the series) began
with a scene from the opera ‘The Makropolos Case’ by the Czech composer
Janáček. The premise of the opera is simple: more than three hundred years
earlier the heroine of the opera, Elina Makropolos, was given an elixir of life
by her father, the court physician. She is now nearly three hundred and fifty
years old. She has reached a state of utter indifference to everything, and her
life has lost its meaning. In the opera excerpt she sings a lament: ‘Dying or
living it is all one. It is the same thing. In me my life has come to a
standstill. I cannot go on. In the end it is the same. Singing, and silence.’
Makropolos refuses to take the elixir again, and dies.
The opera raises some profound questions, about life, about
death, about purpose, and about our finitude. But how should we understand our
finitude? Human finitude has many facets. We live in a reality, which for the
most part is quite independent of us. We are limited in what we can know, and
in what we can do, but importantly, we also have temporal and spatial limits.
Though it isn’t entirely clear what those actually are. Moore asks, as an
example, if he began to exist when he was born, or if he was himself when he
was still a foetus. Another question concerns how big he is. He gives his
dimensions and weight, but points out that you could cut his hair off, or even
amputate his legs, without destroying him. Some philosophers would argue that
who a person is, is represented principally by the brain of the individual in question.
And other philosophers might argue that we are not physical entities at
all.
In any case, it is clear that human beings are not infinite
in size. And, unless there is an afterlife, there will come a time when
we no longer exist. Is the prospect of our annihilation something we should
fear, deplore, and does it reduce our lives to meaninglessness? The Greek
philosopher Epicurus did not believe in an afterlife. But the Epicurians did
not fear or deplore death. They did not see how they should be affected by
something they would not be around to witness. They were of the view that death
was not an evil to us, since we were not around to witness it. Lucretius, also
an Epicurean, reinforced the point by saying that we didn’t exist before we
were born, and the fact that we won’t exist after we are dead, is just a mirror
image of that. Lucretius asked, ‘is there anything terrible there? Anything
gloomy? It seems more peaceful than sleep.’
The twentieth century philosopher Bernard Williams went even
further. Rather than dwelling on the innocuousness of being dead, he dwelt on
the awfulness of being perpetually alive. He wrote a famous article which took
both its theme and its title from the Makropolos Case. Its subtitle was
‘reflections on the tedium of immortality.’ He argued that a never-ending life
would become what Elina Makropoulous’s life had become – tedious to the point
of unendurability. For Williams, it was about whether or not you could have a
life of your own, if you could live for eternity. If you are going to live for
eternity, it would seem that you would need to keep finding new things to do,
or new ways to be satisfied doing the same things again and again.
Williams’ argument is that you can only talk about such a life as your
own life if you remain reasonably close to how you started out. In other words,
can it still be your life if it goes on for eternity? Williams’ answer was
‘no’.
For some philosophers, it is straightforwardly obvious that
annihilation, followed by nothingness, is a great and uncompensated evil. Moore
quotes the American philosopher Thomas Nagel, who writes that being given the
alternatives of living for another week or dying in five minutes would always
(all things being equal) opt for living another week. If there were no other
catastrophe which could be averted by his death. Which Nagel interprets as
being tantamount to wanting to live for ever. He wrote that ‘there is little to
be said for death: it is a great curse. If we truly face it, nothing can make
it palatable.’ Moore suggests that the opposing points of view of Williams and
Nagel may be the consequence of a temperamental difference, as much as an
intellectual one. Nagel also suggested the possibility that Williams
might have been more easily bored than he is. Moore says this might have been
the case.
Moore feels that choosing the option of living for another
week is not tantamount to wanting to live forever, and that he might choose the
option while being appalled at the prospect of living forever. Where does this
leave us? Some philosophers celebrate our finite nature, and others lament it.
But there is consensus on one point, which is that finitude does not deprive
our lives of meaning. Those who celebrated finitude say that it helps to confer
meaning. Those who lament it, do so because, in the fullness of time, it
threatens to take that meaning away. Moore suggests that what this perhaps
shows, is that our finitude is what enables us to reach out and touch the
infinite.
John Cottingham argues that there is something about us,
though we are obviously animals, obviously biological creatures, which is
restless and inclined to reach out. We have transcendent aspirations. What St
Augustine called ‘the restlessness of the human heart’. Our finitude fuels our
aspirations to something more perfect than we are.
In the final programme, Moore returns to the words of the
mathematician David Hilbert, from a lecture in 1925, on the nature of the
infinite, which opened the series:
The infinite has always stirred the emotions of
mankind more deeply than any other question; the infinite has stimulated and
fertilized reason as few other ideas have; but also the infinite, more than
other notion, is in need of clarification.
Moore says that what we have seen in the course of this
series amply bears out Hilbert’s claim. Philosophers have pondered the vastness
of space and time, the limitless of numbers, and the perfection of God. Many
however, aware that the idea of the infinite is associated with paradox and
mystery, have been disturbed by it, and have been wary of it. At the root
of this division of opinion about the infinite, is our knowledge that we
ourselves are finite.
There is something more fundamental to this than the fact
that we are small and ephemeral. We find ourselves limited and constrained by
what is utterly independent of us. This gives us a contrasting sense of what is
unlimited, unconstrained, and self-sufficient. If scientific investigation
should show at some point that space and time are themselves finite, still we
would have this sense of the infinite. But it is a sense of something which
eludes us. We might say that it is a sense of something which we cannot have
any sense of. We are unsure of what to make of the infinite, and consequently
unsure of what to make of ourselves.
Like the Greeks two and a half thousand years ago, we find
ourselves on the one hand troubled by the infinite, and on the other hand,
unable to ignore it. Rene Descartes took issue with the fact that we begin with
the finite, and just think of the opposite of it; on the contrary he said that
we begin with the infinite, and think of the finite as the opposite of that.
And this led him to argue that there must be something in reality, something
infinite, from which we get the very idea. Descartes had in mind God. There is
a non-theological version of this argument which has survived into contemporary
philosophy. Thomas Nagel is quoted on the idea of infinity in connection with
numbers:
To get the idea of infinity, we must understand that
the numbers we use to count things, are merely the first part of a series that
never ends. Though our direct acquaintance with specific numbers is extremely
limited, we cannot make sense of it, except by putting them and ourselves, in
the context of something larger. Something whose existence is independent of
our fragmentary experience of it. When we think about the finite activity
of counting, we come to realise it can only be understood as part of something
infinite.
Thomas Nagel finds it striking that we have evolved in a
complex way, in such a way that we have a grasp of very powerful objective
truths – of logic and mathematics for example, and also certain objective
truths of morality, and, according to Nagel, this means that our minds are what
he calls ‘instruments of transcendence’. Though we are finite biological
creatures, our minds nevertheless reach out to grasp these objective truths,
and that is a remarkable fact about us which Nagel thinks can’t be fully
explained by the biological processes of mutation and natural selection.
So an atheist such as Nagel nevertheless thinks we are in
touch with something transcendent. The topic of this series is the infinite –
what does the infinite have to do with what is transcendent? Perhaps these are
two quite different ideas. Perhaps the infinite isn’t anything grand at all. It
might well be argued, as we have seen in previous programmes, it has been
argued, that any time you move from A to B, you do infinitely many
things.
Ludwig Wittgenstein tried to take some of the mystique out
of the infinite. We naturally think that the infinite, if it exists at all,
must be something awesome, and utterly beyond our comprehension. His view was
that to understand the infinite, we need to understand phrases, like, ‘and so
on’. Simple phrases, which use simple grammatical rules, which finite creatures
like ourselves, can easily grasp. “The expression ‘and so on’ is nothing
but the expression ‘and so on’. Nothing, that is, but a sign which can’t have
meaning but by the rules which have hold of it.”
Moore suggests that, if we accept Wittgenstein’s debunking,
it allows us to separate completely the idea of the infinite from the idea of
the transcendent. It allows us to treat the infinite as something
quite tame, and unremarkable. Moore should realise this only applies to
the potential infinite, and if the idea of an actual infinite, which has
connections with the finite world, is baseless.
He continues by suggesting that, whether or not we side with
Wittgenstein and admit that there is nothing to the infinite, beyond our finite
linguistic resources and the rules governing their use, we must acknowledge our
urge to think there is more to the infinite than that. We must concede that we
think that there is an infinite with a capital 'I'. The unconditioned,
self-sufficient, and transcendent.
Where does this urge come from? How does it manifest
itself? In conversation with Moore, John Cottingham suggests that there are
three categories in our awareness of the beauties of the natural
world which seem to have something in them which isn’t just a mere ability
to produce pleasure in us – a special quality which is sometimes called a
sacred, or even a numinous aspect. Another area is the requirements of
morality, which I recognise as incumbent on me to observe, even though I may
not want to. A third area is what used to be called ‘the
eternal truths’ of logic and mathematics, which I recognise as necessary,
universal and unalterable. So in all these areas I seem to recognise something
which transcends, which goes beyond the ordinary contingent flux of biological
and physical circumstances. In so far as I can recognise those areas, I seem to
be reaching forward to what might be called 'the dimension of the
transcendent'.
Moore comments that we have a sense of the transcendent
then, and a sense of the infinite, with the capital ‘I’. This is something akin
to a religious experience. Or an experience of the sacred. It suggests
something infinitely greater than us, which can give meaning to our lives. But
how might such experiences be located in a secular world view? John Cottingham
says that these might be understood in terms of weird, funny states we can get
into, via psychedelic drugs, or by fasting, which would be a reductionist or deflationary
interpretation of the sacred. If we don’t want to go down that route, then we
have to say that there is something about reality which calls forth such
experiences, and they are not just a private ‘trip’, but a response to
something which is real.
The weakest parts of this series tend to be when religion
and ideas of what is transcendent are being discussed. The secular and finitist
approach is a ball and chain when it comes to talking about the actual
infinite.
Moore begins the windup to the series, saying this brings us
right back to Descartes, back to the idea that the best explanation for these
responses, maybe that they are genuine responses to something which is
genuinely infinite, with a capital ‘I’.
There is however another approach which Moore suggests we
can take however, and he gives a striking example of this alternative
approach, expressed by the British novelist and philosopher Iris Murdoch.
Whereas Descartes had said that our idea of God as something perfect and great,
could only have been implanted in us by something real which is in fact so
perfect and great. Iris Murdoch however turned this around, and said our
idea of god was so great that nothing in reality could match up. God does not
exist, and cannot exist, because something so great could not be confined by
mere existence. But what implants the idea in our minds, does exist. And is
constantly experienced and pictured. Incarnate in work, knowledge, and
love.
Moore’s judgement of the worth of Murdoch’s formulation is
sound. It might have been better to have started the series with this view of
the infinite, which does not prioritise existence as a property which is
necessary to what is real. Instead, it makes more sense to understand
existence as something which is generated by the Infinite, rather than trying
to locate the Infinite within what exists. Which is the principal difference
between ancient and modern philosophy, when considering the infinite, and what
is real. We look at important questions upside down, and are tempted to
re-interpret what came before, in terms of how we think now. Much of our
inherited history of ideas is bent out of shape, and sometimes unintelligible
to us, having been deformed by many attempts to understand difficult ideas over
many centuries.
This particular history of the infinite represents a more or
less consensus view of an important aspect of the history of ideas. A consensus
in philosophy however, usually means that a serious line of argument has
run out of steam, and can tell us nothing new. I think that this is the case
here. Moore's account is full of fascinating detail, but the overall narrative
is open to question in a number of respects, and it is clear that the story of
man's engagement with the infinite could be written in a quite different way.
