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2. KNOWLEDGE OUT OF REASON:
PLATONISM
In this second chapter I shall consider
arguments supporting one of the major forms of rationalism, namely platonism
with respect to the truths of mathematics, logic, and conceptual analysis.
Rationalism and Platonism
Rationalists are united in believing that it
is possible for us to acquire substantial knowledge of the world a priori.
Platonism is (almost always)1 a form of rationalism, since
platonists believe that we may obtain a priori knowledge of a realm of abstract
objects, including numbers and universals. This knowledge is genuinely
substantive, since such objects are held to exist independently of the human
mind. But platonism is not the only form of rationalism. Many rationalists have
also believed that we can know a priori of the existence and immortality of the
soul, of the existence of an all-powerful good God, and of the freedom of the
will. Indeed, in Locke's own day many believed that we could know a priori the
basic constitution of the natural world around us. This was one form of
Medieval-Aristotelian conception of science. It was believed that the various
natural sciences formed deductive systems (somewhat like geometry), the axioms
of which were self-evidently true as well as innately known. (This was the form
of rationalism which Locke himself was most concerned to attack directly.)
While
platonism is only one form of rationalist belief, it nevertheless holds a
central position in the debate with empiricism. This is because logic and
mathematics represent the only examples of a priori knowable truths which are
relatively uncontroversial. Almost everyone (whether empiricist or rationalist)
agrees that it is true that 2 + 2 = 4, and that we may know this a priori, by a
process of reasoning alone. So if it could be shown that this truth is
genuinely substantive, it would follow that rationalism is vindicated - it
would follow that it is possible for us to acquire substantial knowledge
through reason alone. It would then be hard to see why there should be any
objection in principle to the use of reason discovering for us other
substantial truths, such as the existence of the soul or the freedom of the
will. In contrast, in all other domains where rationalists have sought
substantial a priori knowledge - for example, attempting to construct proofs of
the existence of the soul, or of the existence of God - empiricists have
believed (in most cases rightly) that they could detect errors in the reasoning
used which were quite independent of the empiricist/rationalist debate. Such
arguments thus do not constitute a problem for empiricism, in the way that the
truths of logic and mathematics manifestly do. Nor do they amount to very
powerful evidence of the truth of rationalism.
I
therefore propose to concentrate upon platonism in this chapter. This is not
only because of its crucial position in the overall debate, but also because
this is the point at which the issues between empiricists and rationalists can
be presented in their sharpest focus. In chapter 10 I shall return again to
consider rationalists' claims to other forms of substantive a priori knowledge.
Platonists
maintain that alongside the physical world there also exists a realm of
abstract (non-physical and changeless) entities, peopled by universals (such as
greenness, friendship and beauty), and by mathematical objects (including
numbers and geometrical figures). They believe that it is these entities that
the propositions of logic, mathematics, and conceptual analysis are about.
They hold that these entities constitute the subject-matter of our logical and
mathematical propositions. Platonists also generally believe that abstract
entities have a mode of existence which is necessary, believing that
numbers, for example, exist at all times and in all conceivable circumstances
(in all possible worlds). They believe that no matter how distant the past or
future one considers, and no matter how different the natural world might have
been from the way it actually is, it would still have been the case that numbers
exist, and each number would still have had the very same mathematical
properties.
Some
platonists have held that abstract entities have an existence which is timeless
rather than necessary. They have held that, so far from existing at all times
in all possible worlds, it makes no sense to say of an abstract object
that it exists at a particular time, or at all times - just as it makes no
sense to say of an abstract object that it exists in a particular place, or in
all places. But in my view this thesis lacks intuitive plausibility. For
sentences like 'The number 7 always has and always will exist' just do not have
the obvious kind of senselessness possessed by such sentences as 'The number 7
is everywhere'. (Everyone will concede that abstract objects are spaceless.)
On the contrary, such sentences seem perfectly intelligible, if not obviously
true. However, nothing of any great significance turns on this issue for our
present purposes. In what follows I shall confine attention to that version of
platonism which claims necessary existence for abstract entities. For it is
easier to express platonist doctrines, and to present arguments in their
support, if platonic objects are supposed to have necessary rather than
timeless existence.
When
it comes to explaining our knowledge of abstract entities, many platonists have
believed that we possess a special faculty of intellectual intuition, modelled
by analogy with sense-perception. The purpose of this faculty is to make
available to us facts about the abstract realm, just as our senses serve to
make available to us facts about our immediate physical environment. The idea
is that when we come to believe simple conceptual or mathematical truths, we intuit
an aspect of the abstract realm - we 'see it with our mind's eye'. So the fact
that the abstract realm contains the number 7, which has the property of being
prime, is to explain my belief that 7 is prime in something like the manner in
which the fact that there is a cup on the desk in front of me explains my
perceptual belief that there is a cup on the desk. In both cases the state of
affairs in question is supposed to exert an influence upon my mind, via
faculties of intuition and perception respectively.
Does
this then mean that the belief that 7 is prime is not really a priori? Should
we think of intellectual intuition as a way of acquiring experiential
(empirical) knowledge of the abstract realm? I think not. There are two reasons
for this. The first is that, in contrast with genuine sense-experience, there
are no sensations characteristic of acquiring a new mathematical belief. When I
see that there is a cup on the desk, my experience has its own distinctive
phenomenological character. But when I 'see' that 7 is prime, all that happens
is that I come to have a new belief. Phenomenologically speaking, it remains
the case that the only process involved in the acquisition of new mathematical
beliefs is thought. The second difference between intellectual intuition and
genuine perception, is that the supposed faculty of intellectual intuition
remains a bare hypothesis, lacking independent confirmation. In connection with
genuine sense-experience, in contrast, we have well-grounded beliefs about the
structures, locations, and modes of operation of the various senses.2
What
are the main arguments which can be given in support of platonism? I shall
begin by considering the arguments for believing that there are abstract
entities at all, taking the case of numbers first (it is here that the
arguments are strongest). I shall then discuss the existence of universals.
Finally, I shall consider the arguments for believing, not only that there are
abstract entities, but that such entities have mind-independent, indeed
necessary, existence.
Numbers as Objects
A powerful case can be made for saying that
numbers exist as abstract individual objects. It has been presented most
clearly by the 19th century platonist and logician, Gottlob Frege; though many
of the points were implicit in the writings of Plato.3 The argument
begins by noting that in much of our thought and speech we treat numbers as
individual objects. Indeed, in most of their occurrences in sentences numerals
(number-words) appear to function just like proper names. It is worth spelling
this out in some detail, distinguishing six different respects in which
language seems to treat numbers as individual objects.
First,
we predicate things of numbers, just as we predicate things of physical
objects. Compare the sentence 'David is a small man' with the sentences '7 is a
small number', or '7 is prime'. Just as in the first sentence the use of the
word 'David' serves to refer to a particular human, of whom we then predicate
the property of being small, so too in the other sentences it appears that the
numeral '7' is referring to an individual object, of which we then predicate
the property of being small, or the property of being prime. In short: the
sentence '7 is prime' is of subject-predicate form, with the word '7' serving
to introduce its subject, just as a proper name does.
Secondly,
numbers can be referred to by means of definite descriptions (phrases of the
form 'The such-and-such'), which apparently serve to pick out one individual
thing from others. Thus compare the two sentences 'The tallest living person is
the oldest' and 'The successor of 6 is prime'. In the first sentence the
definite description 'The tallest living person' serves to pick out and
distinguish one from the rest of us (even if we do not know whom), who is then
claimed to be the oldest. Similarly in the second sentence, the phrase 'The
successor of 6' seems to pick out and refer to a particular number (in fact the
number 7), which is then said to be prime.
Thirdly,
we quantify over numbers (making 'some' and 'all' statements about them). This suggests
that they form a domain of individual things, just as people form a domain of
individual things. Thus we may say 'Some number between 6 and 10 is prime' just
as we might say 'Some person in the room has fair hair'. The most natural way
to understand the truth-conditions of such sentences, is that they are true if
and only if there is some individual - nameable - thing in the domain having
the property in question. Then it seems that numbers, like people, must be
individual nameable things.
Fourthly,
we take statements about individual numbers to imply existence-statements, just
as we do statements about people. Thus, in the same way that 'David is in the
room' implies 'Someone is in the room' (in virtue of our intention that the
name 'David' should refer to an existing individual), so too does '7 is prime'
imply 'Some number is prime'. This suggests that we must be intending the
numeral '7' to refer to an existing individual thing, in such a way that
statements involving that numeral can only be true if the intended referent
really does exist.
Fifthly,
we make statements of identity concerning numbers, just as we do concerning
individual physical objects. Thus just as we might say 'Thatcher is none other
than Margaret' (or 'Thatcher and Margaret are one and the same person'), so too
we may say '2 + 2 and 4 are one and the same number', or more simply '2 + 2 =
4'. (In order to provide a context for the first of these statements, suppose
you know that Mrs Thatcher was the 1989 Prime Minister of the UK without
knowing what she looks like, and are then introduced to someone at a reception
as 'Margaret' without realising that she was the 1989 Prime Minister, only
later discovering that they are one and the same person.) In this respect, too,
language treats numbers as individual objects, which can either be identical or
distinct from one another.
Finally,
it is a general truth about individual objects that they do not have
contradictories, whereas properties do. And neither do numbers have
contradictories (which are also numbers).4 All properties have
contradictories, in the sense that for any property, there is another property
which is true of something if and only if the first is not true of it. For any
property F, there is the property of being not F. The corresponding principle
does not hold for objects. It is not the case that for any object there is
another object which possesses a property if and only if that property is not
possessed by the first object. Rather, almost any two objects will have something
in common. Similarly, any two numbers will have something in common. For
example, both 7 and 640 are larger than 6. We certainly cannot guarantee that
for any number, there is another number which lacks just the properties which
the former possesses, and vice versa.
While
there are many respects in which our language treats numbers as objects, with
numerals functioning in sentences just like proper names, this is not
universally the case. In statements of number such as 'Jupiter has 4 moons' the
numeral '4' seems rather to occur as an adjective. It appears to be qualifying
the noun 'moons', in something like the way that the adjective 'large' does in
the sentence 'Jupiter has large moons'. Yet there is a difference. For from
'Jupiter has large moons' we may infer 'Each of the moons of Jupiter is large';
whereas from 'Jupiter has 4 moons' we certainly cannot infer 'Each of the moons
of Jupiter is 4'. So if '4' occurs here as an adjective, it is by no means an
ordinary adjective.
In
fact the quasi-adjectival use of numerals can easily be rendered consistent
with their otherwise name-like use, if we notice that the statements of number
in which they figure can be rewritten as identities without any loss of
content. The sentence 'Jupiter has 4 moons' says the same as 'The number of
moons of Jupiter is none other than 4', in which '4' does explicitly occur in
the role of a proper name. (Note, in contrast, that we cannot easily say 'The
size of the moons of Jupiter is none other than large'. This seems nonsensical.)
Since
we use numerals in the manner of proper names, our language treats numbers
themselves as individual objects. Then if any statements of arithmetic are
true, it seems that numbers must really exist. For compare: the statement
'David is a small man' cannot be true unless the name 'David' does succeed in
referring to someone. That is: unless the person David really does exist.
Similarly then, '7 is prime' cannot be true unless the number 7 really exists.
For this is a statement about the intended referent of the name '7', just as
'David is a small man' is a statement about the intended referent of the name
'David'. We then apparently face a stark choice: either to accept that numbers
are genuinely existing objects, or to reject all statements of arithmetic as
false. If the latter is too high a price to pay, then we must accept that
numbers exist as individual things.
From
the fact that numbers are individual objects, it does not immediately follow
that they are abstract (non-physical and changeless) objects. But this
further thesis may be established with very little additional argument. For it
is obvious that numbers are not physical things, like tables and chairs. Thus
it really does seem to be nonsensical to ask where the number 7 might
be, as we noted above. Even those who maintain that numbers are really sets of
sets of things (for example, identifying the number three with the set of all
trios of things) are forced to concede this. For sets of physical things (let
alone sets of such sets) are not themselves physical things. Even if each
fair-haired person is physical, the set of such people is not.5
Moreover,
since the truths of arithmetic are necessary, holding with respect to all times
and all possible worlds, it is clear that numbers must be changeless things. We
certainly have no use for the idea that there are circumstances in which 7
would not have been prime, nor for the idea that at some future time it might
cease to be so. What has to be acknowledged, however, is that numbers can undergo
change in their relational properties; that is to say, in the relations in
which they stand to changing physical things. For suppose Jupiter loses one of
its moons. Then at one time the number of its moons is four and at a later time
it is three. But this is best viewed as a change in the relations obtaining
between numbers and the things in the world satisfying a certain description,
rather than a change in the numbers themselves.
Is any Arithmetic True?
At this point we should consider how
plausible it would be to try resisting the above argument for platonism about
numbers by rejecting all mathematical statements as false. This is the option
defended by Hartry Field. He agrees with the platonist interpretation of the
subject-matter of arithmetic, accepting that '2 + 2 = 4' purports to state a
fact about abstract objects. But because he does not accept the existence of
such objects, he does not accept that arithmetic contains any truths - just as
someone who denies the existence of David would deny that 'David is a small
man' can express a truth. Yet he hopes to explain how arithmetic can be useful
in its applications, in terms of its consistency. In Field's view
mathematical statements might have been true (had there existed any numbers),
although they do not happen to be so; and this is claimed to be sufficient to
explain the practical usefulness of such statements.
An
initial objection to Field is that his position cannot be adequately motivated.
This is because the main difficulty with platonism, as we shall see in chapter
3, is that it cannot provide an acceptable explanation of our knowledge of
mathematics. Thus in order to argue that the platonist interpretation of
mathematics is false, we shall suppose that mathematics itself is largely true,
and indeed is largely known to be true. Such an argument cannot be pursued by
those, such as Field, who wish to deny that mathematics consists of truths.
Field's attempt to deny platonism by rejecting mathematics as false, would then
appear to be self-defeating.
In
fact Field can reply to this objection. His view is that our theory of the
world - our total science - is a better theory if it excludes platonic objects
than if it includes them. And one of the main reasons why belief in platonic
objects would make our overall theory worse, is that we should be left
incapable of explaining how we could have knowledge of those objects. So the
argument against platonism, that it cannot adequately explain how we could have
knowledge of mathematics, can be presented without presupposing that any of our
mathematical statements are true. On the contrary, the reason why our theory of
the world is better for denying that any mathematics is true, is that we could
not explain how we could have knowledge of it if it were.
However,
it is clear that Field's strategy for rejecting platonism is one of very high
risk. For it needs to be shown that a science without any mathematical truths is
a better total theory than a science inclusive of such truths. To begin with,
it is a matter of considerable technical complexity to show that science can
be presented in such a way as not to presuppose that any mathematical
statements are true.6 But even then, Field's task is not complete.
For explanatory completeness is only one good-making feature of a theory.
Another is simplicity. So if, as seems likely, scientific theories will be more
complex when reformulated so as not to presuppose mathematical truth, it will
still not be settled that we should reject mathematics as false. It may be that
the best overall option will prove to be the platonist one, even accepting that
we cannot then explain the genesis of mathematical knowledge.
Perhaps
the main objection to Field's position though, is simply that it is hugely
counterintuitive. It is very hard indeed to induce oneself to believe that '2 +
2 = 4' might in fact be false. As a result, the position contains a crucial
dialectical weakness. This is because our pre-theoretical belief in the truths
of mathematics is much more firmly grounded then any merely philosophical
argument for the platonist interpretation of the subject-matter of those
truths, such as was sketched in the previous section. Since it requires an
argument to convince us that '2 + 2 = 4' is really about abstract objects, it
will always be more reasonable for us to think that something has gone amiss
with the philosopher's argument, than to give up our mathematical beliefs.
Field
also faces a related problem. For mathematics is very naturally taken to
consist of statements which, if true, are true necessarily - holding good with
respect to all possible worlds. But then if those statements are false, they
will be necessarily false - which conflicts with Field's claim that they are
consistent (true with respect to some possible world). In fact it is almost
equally as hard to see how '2 + 2 = 4' might be contingent (true of some worlds
but not others), as it is to see how it might be false.
Now
the objection here is not that if Field were right, it would then be wholly
inexplicable why there should happen to be no abstract objects.7 For
Field may reply that the non-existence of numbers is a brute contingency,
comparable in status to the claim that there are no physical objects, made by
those who have become convinced of the non-existence of matter. (It is obvious
that such a person cannot explain why there are no physical objects,
except perhaps theologically, by saying that God failed to create any. Why then
should Field be under any obligation to explain why there happen to be no
numbers, otherwise than by saying that God never made any?) Rather, the
objection parallels the one made above. It is that our pre-theoretical
conviction that mathematical statements are necessary (as opposed to
contingent) is likely to be more firmly grounded than any philosophical
argument for a platonist interpretation of the subject-matter of mathematics.
Field's
position would seem to be one of last resort for an empiricist. Indeed, it is
so barely believable that if it were the only option available, then this might
be a powerful reason for embracing rationalism. But in fact there are other
possibilities open to us, as we shall see in the next chapter. The conclusion
thus far, is that the existence of numbers as abstract individual objects can
at least be supported by powerful arguments, if not proved outright. At this
point it appears that an empiricist who wishes to oppose platonism faces a
formidable task.
The Existence of Universals
The case for believing in abstract universals
has to be made somewhat differently. For universals are supposed to be what the
predicative expressions of natural language (such as 'wise' or 'small') refer
to. And we cannot claim that such expressions function in sentences as proper
names (as we claimed above that numerals do) without destroying altogether the
distinction between name and predicate. It is true that most predicates admit
of nominalisations, which can then figure in the subject-position in a
sentence. Thus as well as sentences like 'David is wise' we have sentences such
as 'Wisdom is hard to acquire'. But it is doubtful whether anything of much
significance depends on this. For it is generally easy to see how the content
of sentences containing such nominalisations may be expressed without loss by
sentences containing only the corresponding predicate. Thus what 'Wisdom is
hard to acquire' really says, is that it is hard to become wise.
What
does seem to require us to recognise the existence of universals is this. We
accept existence-claims which are not claims about the existence of individual
objects (as is 'There is something which is human'), but are rather claims
about the existence of the properties those objects possess. Thus consider the
sentence 'There is (there exists) something that we all are (namely human)'.
This surely expresses a truth. It seems undeniable that there is something
common to us all. In which case it appears that we must not only accept the
existence of individual humans, but also of what they have in common - the
universal feature humanness. It is also true that there is (there exists)
something that grass and leaves have in common, namely greenness; and so on. If
we accept such truths, we appear to be accepting the existence of universals.
From
the fact that universals exist, it does not immediately follow that they are
abstract. Nothing has as yet ruled out the idea that they may rather be
immanent in the physical world (as Aristotle maintained). Thus it may be that
universals such as greenness are, in part, physically present in the individual
things which instantiate them (that is, in the individual leaves and blades of
grass which share the property of being green). This sort of view has been
defended recently by David Armstrong.8 He argues that we need to
postulate identical natures in things (immanent universals) if we are to give
an adequate explanation of the fact that different things can share similar
causal powers, and of what laws of nature are. He thinks that the best explanation
for the fact that all water behaves similarly in similar circumstances -
freezing at zero centigrade, dissolving sugar, and so on - is that all water
shares an identical inner constitution which necessitates that it behave as it
does. This inner constitution is a repeatable feature of the world, being
identically present in many different items of water, and is hence a universal.
But it is, on this view, a universal which is present in the physical
items which instantiate it.
What
apparently forces us to accept the abstractness of universals, is that we
accept existence-claims with respect to them even when they are not
instantiated in the physical world. For example, it appears to be true that
there is (there exists) something that nothing is (namely, a unicorn, or
a dragon). If this is accepted, then it seems to follow that universals, in
existing independently of their instances (where they have instances), are not
physically present in the natural world. Their changelessness may then be
established using similar considerations to those deployed in the case of
numbers. For we have no use for the idea that the universals dragonness or
greenness themselves undergo change. Although a withering blade of grass is at
one time green and at a later time not, this would best be viewed as a change
in the relation in which it stands to the universal greenness, rather than a
change in that universal itself.
We
may conclude that the existence of universals as abstract entities can also be
supported by powerful considerations, just as can the existence of numbers as abstract
individuals. Then a priori conceptual truths such as 'Anything red is coloured'
and 'No surface can be red all over and green all over at once', in being
concerned with internal relations between universals, will give us a priori
knowledge of a realm of abstract entities. But this is not yet to say that such
knowledge will be genuinely substantive. For it is important to note that we
have not yet established that universals (or indeed numbers) have
mind-independent existence, let alone that their existence is necessary (that
they exist at all times in all possible worlds).
In
order to see that there is a gap here waiting to be bridged, consider what
might naturally be said about the existence of sentences as abstract
types. Given that both the vocabulary and the rules of sentence-formation for a
language have been fixed, it is natural to think of the sentences of that
language as already existing, independently of their being tokened in speech or
writing. And this is just what we say: there are many sentences of English
which no one has ever expressed, or perhaps ever will. Those sentences may
remain unexpressed because they are too complicated ever to be uttered, or too
silly; or simply because no one ever happens to think of them. We also have no
use for the idea that a sentence-type might undergo change. The sentence 'David
is wise' is as it is; it cannot change without becoming another
sentence. Yet we are unlikely to be tempted to think of sentences as existing
independently of the rules implicit in the practices of those who speak the
language. On the contrary, if those rules had never arisen in the way that they
did (for example, in the case of English, if the Normans had never invaded and
conquered England), then the sentences in question would never have existed.
It
is thus natural to think of sentences as abstract but mind-dependent entities,
depending for their existence upon the rules implicit in the practices of the
native speakers of a language. A similar position is then possible in connection
with numbers and universals. One might concede that they are abstract entities
(being non-physical and changeless throughout the time of their existence), but
maintain that their existence supervenes upon the existence, and properties, of
the human mind. Note, however, that I do not take myself to have shown
that sentences have mind-dependent existence. My point has only been to
establish that such an idea makes sense, and hence to show that more needs to
be done if we are to establish full-blown platonism with respect to numbers and
universals. Some have maintained that sentences, too, have necessary existence,
and I do not claim to have answered them here.9
Objective Truth and
Objective Necessity
Are there any reasons for believing that
numbers and universals are abstract entities which have an existence
independent of the human mind? The main argument for such a claim is premised
upon our intuitive belief in the objectivity of truths about such entities.
Thus we are inclined to think that the truth of 'There is something that
nothing is (namely a dragon)' is wholly independent of our beliefs about the
matter. Even if we were all to believe that there are dragons, it would still
be true that there is something that nothing is (namely a dragon). Indeed, not
only are truths about universals independent of our beliefs, but we are
strongly inclined to think that they are independent of our very existence as
well. Even before there were any human beings, it was still true that there was
something that nothing is (namely a dragon). Equally, even if there never had
been any human beings, it would still have been true that there is something
that nothing is (namely a dragon). In which case it would seem that universals
themselves (the entities which such truths are about) must exist independently
of the human mind.
Similar
arguments can be deployed in support of the mind-independent existence of
numbers. Indeed, in this case the impression of objectivity is, if anything,
even stronger. The truth of '7 is prime' is, we think, quite independent of
anything we might believe about the matter. Even if we all believed otherwise -
either through mistake or some sort of collective madness - it would still be
true that 7 is prime. Moreover, we are inclined to think that the truths of
mathematics are independent of our very existence. Even before there were any
human beings, or even if there had never been any human beings, it would still
have been the case that 2 + 2 = 4. For example, if there were two apples and
two pears lying beneath a particular tree on a day exactly 20 million years
ago, then it must still have been the case that there were four pieces of fruit
under the tree, even though there were no intelligent agents in existence to
recognise the fact.
It
appears that the truths of mathematics are mind-independent, obtaining
independently of facts about, or even the very existence of, the human mind.
Yet such truths involve reference to individual abstract objects (the numbers),
as we argued above. Then how could those objects, in turn, fail to exist
independently of the human mind? Compare: it would still have been the case
that Everest is the tallest mountain in the world, even if there had never been
any human beings in existence to appreciate the fact. But this surely could not
be so, unless Everest itself would still have existed in the absence of human
beings. Mind-independent truth seems to require the mind-independent existence
of the objects which figure in those truths. Then so, too, in the case of
mathematics: if the truth of '2 + 2 = 4' is independent of the human mind, then
the objects with which it deals (namely, the numbers 2 and 4) must have a mode
of existence which is independent of the human mind.
It
appears that there is a strong case for saying that abstract entities have
mind-independent existence. In which case our a priori knowledge of the
properties of, and relations between, such entities (of the kind which
platonists maintain is available to us through mathematical proof and
conceptual analysis) will be genuinely substantive. But it does not yet follow
that abstract entities have necessary existence, existing at all times in all
possible worlds. There are of course many things which exist independently of
our minds (Everest, for example) which do not have necessary existence. Are
there, then, any arguments which support the stronger conclusion of necessary
existence for numbers and universals?
There
is indeed such an argument, premised upon the claim that the truths of
mathematics, and many truths about universals, are necessary - being
true with respect to all times in all possible worlds. Thus no matter how
different the world might have been from the way it actually is, it would still
have been the case that 2 + 2 = 4, and it would still have been the case that
no surface could be red all over and green all over at once. We are strongly
inclined to believe that truths such as these hold come what may, no matter
what else may or may not be true of the world, and no matter how different the world
might possibly have been. Then how could the numbers 2 and 4 not exist in all
possible worlds, given that '2 + 2 = 4' is both true with respect to every
possible world, and about the numbers 2 and 4? And since the truth 'No
surface can be red all over and green all over at the same time' is about the
universals redness and greenness (according to the arguments presented
earlier), how could those entities, too, not exist in all possible worlds? It
seems that truths which are necessary must require the necessary existence of
the entities which those truths concern.
The conclusion of this chapter is that there is a seemingly-powerful case supporting platonism with respect to numbers and universals. There appear to be convincing arguments for believing that there are abstract entities which not only exist independently of the human mind, but which exist at all times in all possible worlds. In the next chapter we shall consider what an empiricist can do to undermine these arguments. We shall also consider whether an empiricist has any positive arguments for thinking platonism to be false.