The Fabric of Reality David Deutch

10
 
The Nature of Mathematics
 
The ‘fabric of reality’ that I have been describing so far has been the fabric
of 
physical reality. Yet I have also referred freely to entities that are nowhere
to be found in the physical world — abstractions such as numbers and
infinite sets of computer programs. Nor are the laws of physics themselves
physical entities in the sense that rocks and planets are. As I have said,
Galileo’s ‘Book of Nature’ is only a metaphor. And then there are the fictions
of virtual reality, the non-existent environments whose laws differ from the
real laws of physics. Beyond those are what I have called the ‘Cantgotu’
environments, which cannot even be rendered in virtual reality. I have said
that there exist infinitely many of those for every environment that can be
rendered. But what does it mean to say that such environments ‘exist’? If
they do not exist in reality, or even in virtual reality, where do they exist?
Do abstract, non-physical entities exist? Are they part of the fabric of reality?
I am not interested here in issues of mere word usage. It is obvious that
numbers, the laws of physics, and so on do ‘exist’ in some senses and not in
others. The substantive question is this: how are we to understand such
entities? Which of them are merely convenient forms of words, referring
ultimately only to ordinary, physical reality? Which are merely ephemeral
features of our culture? Which are arbitrary, like the rules of a trivial game
that we need only look up? And which, if any, can be explained only in a way
that attributes an independent existence to 
them? Things of this last type
must be part of the fabric of reality as defined in this book, because one
would have to understand them in order to understand everything that is
understood.
This suggests that we ought to apply Dr Johnson’s criterion again. If we want
to know whether a given abstraction really exists, we should ask whether it
‘kicks back’ in a complex, autonomous way. For example, mathematicians
characterize the ‘natural numbers’ 1, 2, 3,… in the first instance through a
precise definition such as:
1 is a natural number.
Each natural number has precisely one successor, which is also a natural
number.
1 is not the successor of any natural number.
Two natural numbers with the same successor are the same.
Such definitions are attempts to express abstractly the intuitive 
physical
notion of successive amounts of a discrete quantity. (More precisely, as I
explained in the previous chapter, that notion is really quantum-mechanical.)
The operations of arithmetic, such as multiplication and addition, and further
concepts such as that of 1 prime number, are then defined with reference to
the ‘natural numbers’. But having created abstract ‘natural numbers’ through
that definition, and having understood them through that intuition, we find
that there is a lot more that we still do not understand about them. The
definition of a prime number fixes once and for all which numbers are primes
and which are not. But the 
understanding of which numbers are prime — for
instance, how prime numbers are distributed on very large scales, how
clumped they are, how ‘random’ they are, and why — involves a wealth of

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new insights and new explanations. Indeed, it turns out that number theory is
a whole world (the term is often used) in itself. To understand numbers more
fully we have to define many new classes of abstract entities, and postulate
many new structures and connections among those structures. We find that
some of these abstract structures are related to other intuitions that we
already had but which, on the face of it, had nothing to do with numbers —
such as 
symmetry, rotation, the continuum, sets, infinity, and many more.
Thus, abstract mathematical entities we think we are familiar with can
nevertheless surprise or disappoint us. They can pop up unexpectedly in
new guises, or disguises. They can be inexplicable, and then later conform
to a new explanation. So they are complex and autonomous, and therefore
by Dr Johnson’s criterion we must conclude that they are real. Since we
cannot understand them either as being part of ourselves or as being part of
something else that we already understand, but we 
can understand them as
independent entities, we must conclude that they 
are real, independent
entities.
Nevertheless, abstract entities are intangible. They do not kick back
physically in the sense that a stone does, so experiment and observation
cannot play quite the same role in mathematics as they do in science. In
mathematics, 
proof plays that role. Dr Johnson’s stone kicked back by
making his foot rebound. Prime numbers kick back when we prove
something unexpected about them especially if we can go on to explain it
too. In the traditional view, the crucial difference between proof and
experiment is that a proof makes no reference to the physical world. We can
perform a proof in the privacy of our own minds, or we can perform a proof
trapped inside a virtual-reality generator rendering the wrong physics.
Provided only that we follow the rules of mathematical inference, we should
come up with the same answer as anyone else. And again, the prevailing
view is that, apart from the possibility of making blunders, when we have
proved something we know with 
absolute certainty that it is true.
Mathematicians are rather proud of this absolute certainty, and scientists
tend to be a little envious of it. For in science there is no way of being certain
of any proposition. However well one’s theories explain existing
observations, at any moment someone may make a new, inexplicable
observation that casts doubt on the whole of the current explanatory
structure. Worse, someone may reach a better understanding that explains
not only all existing observations but also why the previous explanations
seemed to work but are nevertheless quite wrong. Galileo, for instance,
found new explanation of the age-old observation that the ground beneath
our feet is at rest, an explanation that involved the ground actually moving.
Virtual reality — which can make one environment seem to be another —
underlines the fact that when observation is the ultimate arbiter between
theories, there can never be any certainty that an existing explanation,
however obvious, is even remotely true. But when proof is the arbiter, it is
supposed, there can be certainty.
It is said that the rules of logic were first formulated in the hope that they
would provide an impartial and infallible method of resolving all disputes.
This hope can never be fulfilled. The study of logic itself revealed that the
scope of logical deduction as a means of discovering the truth is severely
limited. Given substantive assumptions about the world, one can deduce

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conclusions; but the conclusions are no more secure than the assumptions.
The only repositions that logic can prove without recourse to assumptions
are tautologies — statements such as ‘all planets are planets’, which assert
nothing. In particular, all substantive questions of science lie outside the
domain in which logic alone can settle disputes. But mathematics, it is
supposed, lies 
inside that domain. Thus mathematicians seek absolute but
abstract truth, while scientists console themselves with the thought that they
can gain substantive and useful knowledge of the physical world. But they
must accept that this knowledge comes without guarantees. It is forever
tentative, forever fallible. The idea that science is characterized by
‘induction’, a method of justification which is supposed to be a slightly fallible
analogue of logical deduction, is an attempt to make the best of this
perceived second-class status of scientific knowledge. Instead of deductively
justified certainties, perhaps we can make do with inductively justified near-
certainties.
As I have said, there is no such method of justification as ‘induction’. The
idea of reasoning one’s way to ‘near-certainty’ in science is myth. How could
I prove with ‘near-certainty’ that a wonderful new theory of physics,
overturning my most unquestioned assumptions about reality, will not be
published tomorrow? Or that I am in not inside a virtual-reality generator?
But all this is not to say that scientific knowledge is indeed ‘second-class’.
For the idea that mathematics yields certainties is 
a myth too.
Since ancient times, the idea that mathematical knowledge has a privileged
status has often been associated with the idea that some abstract entities, at
least, are not merely part of the fabric of reality but are even more real than
the physical world. Pythagoras believed that regularities in nature are the
expression of mathematical relationships between natural numbers. ‘All
things are numbers’ was the slogan. This was not meant quite literally, but
Plato went further and effectively denied that the physical world is real at all.
He regarded our apparent experiences of it as worthless or misleading, and
argued that the physical objects and phenomena we perceive are merely
‘shadows’ or imperfect imitations of their ideal essences (‘Forms’, or ‘Ideas’)
which exist in a separate realm that is the true reality. In that realm there
exist, among other things, the Forms of pure numbers such as 1,
z, 3, …,
and the Forms of mathematical operations such as addition and
multiplication. We may perceive some shadows of these Forms, as when we
place an apple on the table, and then another apple, and then see that there
are two apples. But the apples exhibit ‘one-ness’ and ‘two-ness’ (and, for
that matter, ‘apple-ness’) only imperfectly. They are not perfectly identical, so
there are never really 
two of anything on the table. It might be objected that
the number two could also be represented by there being two 
different things
on the table. But such a representation is still imperfect because we must
then admit that there are cells that have fallen from the apples, and dust,
and air, on the table as well. Unlike Pythagoras, Plato had no particular axe
to grind about the natural numbers. His reality contained the Forms of all
concepts. For example, it contained the Form of a perfect circle. The ‘circles’
we experience are never really circles. They are not perfectly round, nor
perfectly planar; they have a finite thickness; and so on. All of them are
imperfect.

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Plato then pointed out a problem. Given all this Earthly imperfection (and, he
could have added, given our imperfect sensory access even to Earthly
circles), how can we possibly know what we know about real, perfect
circles? Evidently we do know about them, but how? Where did Euclid obtain
the knowledge of geometry which he expressed in his famous axioms, when
no genuine circles, points or straight lines were available to him? Where
does the certainty of a mathematical proof come from, if no one can perceive
the abstract entities that the proof refers to? Plato’s answer was that we do
not obtain our knowledge of such things from this world of shadow and
illusion. Instead, we obtain it directly from the real world of Forms itself. We
have perfect inborn knowledge of that world which is, he suggests, forgotten
at birth, and then obscured by layers of errors caused by trusting our
senses. But reality can be remembered through the diligent application of
‘reason’, which then yields the absolute certainty that experience can never
provide.
I wonder whether anyone has ever believed this rather rickety fantasy
(including Plato himself, who was after all a very competent philosopher who
believed in telling ennobling lies to the public). However, the problem he
posed — of how we can possibly have knowledge, let alone certainty, of
abstract entities — is real enough, and some elements of his proposed
solution have been part of the prevailing theory of knowledge ever since. In