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3. THE EMPIRICIST CASE
AGAINST PLATONISM
Having set out the arguments supporting
platonism in chapter 2, in this chapter I shall present the main arguments
which, from an empiricist perspective, can be deployed against it.
Reducing Universals
The most urgent task facing an empiricist is
to undermine the case for platonism. This has generally been attempted via a
species of reductionism. The intention has been to show that what appears to be
talk about abstract entities is really talk about something else, most usually
internal relations between ideas in our minds. If one could thus reduce
abstract-object-talk to talk about non-contingent features of the human mind,
it would have been shown that the truths of logic and mathematics, while a
priori, are not genuinely substantive. For they would neither give us knowledge
which is contingent (as is our knowledge of our own current mental states), nor
knowledge which relates to any aspect of reality outside of our minds.
To
adopt this reductionist strategy is to try to show that the subject-matter of
logic and mathematics can be interpreted adequately without reference to
abstract entities. In connection with logic, and with the sort of conceptual
analysis which one finds in philosophy, the task is relatively easy. We can
interpret these disciplines as being concerned with relations between rules,
conceiving of rules as mind-dependent entities, reducible in turn to the
semantic intentions of those who employ those rules. Thus 'Anything red is
coloured' may be understood to be true in virtue of a relation obtaining
between the concepts red and colour, where these in turn are
taken to be rules for classifying items in the world. And the rule of
classification expressed by the word 'red' will itself consist in the manner in
which most users of that word intend to continue to employ it. This is in fact
the standard empiricist line: truths of logic and conceptual analysis are analytic,
being concerned only with relations between concepts, where these in turn are
understood to reduce to facts about the human mind.
(Note
that while I have here identified concepts with rules of classification, most
traditional empiricists have thought of them as being images. But the
weaknesses in the imagist theory of concepts have been long recognised. In fact
no image, or sequence of images, can carry unaided the content of even a simple
proposition such as 'Grass is green', let alone of a complex proposition like
'Life may be discovered on Mars in the next ten or twelve years'.)
Does
any such form of reductionism about concepts have the resources to reply to the
platonist arguments from objective truth and necessity? The natural move would
be to interpret sentences which seem to involve generalisations about abstract
universals as being, covertly, statements about concepts, or ideas in our
minds. We should then regard the content of 'There is something that nothing is
(namely a dragon)' as being given in reality by a sentence such as 'There is
some idea which applies to nothing (namely the idea of a dragon)'. We cannot,
however, take the most obvious option with such statements when they relate to
situations in which there are no human beings. For example, we cannot interpet
'Even if there had never been any human beings, it would still have been true
that there is something that nothing is (namely a dragon)' to mean 'Even if
there had never been any humans, there would still have been some idea which
applies to nothing'. Nor can we interpret '20 million years ago there was
something that grass and leaves had in common' to be saying '20 million years
ago there was some idea which applied to both grass and leaves'. For these
sentences would then imply the existence of ideas in situations in which there
are no intelligent agents to possess them, which would be absurd.
In
fact, however, there is no special problem here. We can rather take the first
sentence to say that there is some idea (namely our - presently existing
- idea of a dragon) which applies to nothing in a world which differs from ours
only in never containing any human beings. And we can take the second sentence
to say that there is some idea which applies to grass 20 million years
ago and to leaves 20 million years ago. All we need for this, is the thesis
that presently existing ideas can apply to things across time (as when
we employ them in thoughts about the remote past or future), or across
worlds (as when we employ them in thoughts about a world in which the human
species never evolved), without having to exist at those times or in
those worlds.
A
similar move enables us to avoid the argument from necessary truth to the
necessary existence of universals. For if this argument were to be valid, we
should have to interpret the necessity of 'No surface is red all over and green
all over at once' as holding in virtue of a truth of the following sort:
For all worlds, w, and all times, t, the
concepts red and green exclude one another in w at t.
This would imply the existence of concepts
(that is, universals) at the times and worlds where the original
sentence is true (that is, at all times in all possible worlds). But in fact
there is nothing to stop us regarding the necessity of that sentence as holding
in virtue of a truth of the following form:
For all ways of thinking of a world, w,
and all ways of thinking of a time, t, the concepts red in world w at
time t and green in world w at time t are (now) mutually exclusive.
The idea here, is that no matter how we think
of possible situations, our concepts (ideas) of red and green exclude one
another, provided that they occur in thoughts about one and the same such
situation. An empiricist can thus maintain that analytic truths, while being
true in virtue of merely presently existing ideas, nevertheless constrain our
talk and thought about all other times and possible worlds. We can thus hold on
to the objectivity of at least some necessary truths (namely, those which are
analytic), while being committed only to the existence of concepts as ideas in
the human mind.
Notice,
however, that in thus explaining away the kind of talk which lends credence to
the existence of abstract universals, we are not thereby prevented from
recognising the existence of immanent universals, as forming part of the
natural world. We can allow that there is a sense of 'Grass and leaves have
something in common' which is not merely about ideas in our minds, but rather
claims that there is something which is partially present in individual
leaves and blades of grass, which explains their common appearance. I can see
no reason why empiricists should object to the existence of immanent
universals. For while such universals will be genuinely mind-independent,
forming part of the natural universe just as much as individual mountains and
trees do, we shall have no a priori knowledge of them. Nor will they exist in
all possible worlds. On the contrary, it will be the business of the natural
sciences to discover what common properties there really are in the actual
world, which explain the causal powers which individual objects have in common
with one another.
Reducing Numbers
Reductionism in mathematics is a more complex
affair. This is partly because we seem to have terms here which purport to
refer to individual things (that is, the individual numbers), and it is not so
obvious how these may be reduced to rules or psychological phenomena.
Nevertheless, many of those who pursued the logicist programme in mathematics
around the turn of this century were attempting to show how the truths of
arithmetic could be reduced to those of logic (with these in their turn being
shown to be analytic). If this programme could be carried through, and if it
could be shown that analytic truths are not about anything outside of the human
mind, then the mind-dependence of numbers, too, would have been established.
Note, however, that it is possible to be a logicist and a platonist at the same
time (as Frege in fact was). For one might claim that the truths of logic, and
analytic truths generally, obtain in virtue of facts about necessarily existing
universals. To reject platonism about numbers one might also need to reject
platonism about universals (though not vice versa).
The
first step in the logicist programme was to realise that statements of number
of the form 'There are n Fs' are best understood as statements about concepts
(or as about the sets of things which those concepts pick out). In the last
chapter we noted, in connection with the sentence 'Jupiter has 4 moons', that
the number 4 is certainly not an attribute of each individual moon. But it
might be replied that this does not yet show that numbers are not ordinary
attributes. For they may be distinctive in being attributes, not of individual
things, but of collections of such things. But in fact there are severe
difficulties with such a view. For one thing, the very same physical collection
can have different numbers applied to it. Thus 1 pack of cards can also be
described as 4 suits, or as 52 cards. Moreover, there are problems with the
number 0. If the number of Mary's children is 0, then there exists no
collection for 0 to be an attribute of.
These difficulties can
easily be resolved if statements of number are statements about a concept. Then
'Jupiter has 4 moons' states of the concept moon of Jupiter that it is
instantiated 4 times, and 'Mary has 0 children' says of the concept child of
Mary that it is not instantiated at all.1 In the same way, what
changes when the pack of cards may variously be described as '1', '4' or '52'
is the concept involved in the statement (pack, suit, and card
respectively). But note that it does not follow from this that numbers
themselves are attributes of concepts, or second-level concepts. For as we
noted in the last chapter, sentences of the form 'There are n Fs' can be
rewritten as identities, saying, in effect, 'The number of things falling under
the concept F is none other than n'. While the whole statement can be
regarded as being about the concept F, the numeral 'n' may still be
treated as a proper name.
The
crucial consideration was then to notice that numerical identity can be defined
as a relation between concepts. Thus to say that the number of Fs is none other
than the number of Gs can be analysed as saying that the instances of the
concept F may be placed in a one-to-one relation to the instances of the
concept G, where the idea of such a relation can be defined without
mentioning numbers. To get the feel of this suggestion, notice that a waiter
can tell that the number of bowls on the table is the same as the number of
plates without having to count, by seeing that every bowl stands on a plate,
that every plate has a bowl standing on it, that all plates on which any given
bowl stands are the very same, and that all bowls standing on any given plate
are the very same.
The
individual numbers can then be introduced recursively, defining 0 as the number
of things that are not identical with themselves, 1 as the number of things
that are identical with 0, 2 as the number of things that are either identical
with 0 or 1, and so on.2 In which case 'Mary has 0 children' may be
analysed as saying that the concept child of Mary is one-to-one
correlated with the concept not identical with itself. And to say that
the Earth has 1 moon may be understood as saying that the concept moon of
the Earth is one-to-one correlated with the concept identical with 0.
And so on. Such an analysis need commit us to nothing besides concepts
(reducible in turn to facts about the human mind, for an empiricist) and the
relations between them.
The
project of logicist reduction in mathematics went out of fashion in the early
decades of this century, for a variety of technical reasons; though it has been
revitalised again recently by Crispin Wright (contrary to his intention).3
Other forms of reduction have also been attempted. Notable amongst these, is
the modal-structural approach of Geoffrey Hellman,4 developing some
ideas first suggested by Paul Benacerraf.5 On this account, mathematics deals
with the class of possible structures which share the properties of the natural
number sequence (infinite extent, each member having a unique successor, and so
on). The structures in question are (when instantiated) concrete ones - for
example, the sequence of inscriptions '*', '**', '***', '****', and so on. So
there is no commitment to abstract objects. Yet because the account is modal,
dealing with all possible structures sharing certain properties, it can
achieve the universality (particularly the applicability to other possible
worlds) of mathematics.
However,
it remains true that it would be a fearsome achievement to carry through to
completion any version of the reductionist project. It might therefore be wise
for empiricists to adopt some alternative strategy for opposing platonism in
mathematics if they can, which will carry less risk of failure on technical
grounds. For if there is a lesson to be learned here from the history of
philosophy, it is that conceptual reductions are remarkably difficult to
complete successfully. This is especially likely to be the case where our discourse
is very highly developed and complex, as is that of mathematics. (Compare the
failure of the phenomenalist reductive programme, which attempts to reduce talk
of physical objects to talk about patterns in experience.)
Mind-dependent Existence
There is, however, another strategy available
to an empiricist, which carries less risk of failure. We can grant that
mathematics is concerned with abstract objects, making no attempt to analyse
away such apparent references to numbers as occurs in '7 is prime'. But we can
deny that these objects have an existence which is independent of the human
mind. We can rather claim that their existence supervenes upon aspects of human
activity - particularly on the rules implicit in the practices of those who
count, add and subtract - in such a way that they would not have existed if
human beings (or other intelligent agents) had never existed. So we may allow
that there are abstract objects, but deny that these objects exist necessarily,
or indeed independently of the human mind. If fact we can adopt for numbers
just the sort of position which we sketched concerning the mode of existence of
sentences in the last chapter.6
But
if numbers have only mind-dependent existence, how are we to explain the fact
that we apply arithmetic to remote regions of time, for example in calculating
the orbit of the Earth many millions of years in the past or future (when, we
may presume, there are no minds for numbers to supervene upon)? Indeed,
what of the point that even if there had never been any human beings, it would
still have been true that 2 apples and 2 pears would constitute 4 pieces of
fruit? How can this be so, if in such a case there would have been no such
things as 2 and 4? It is worth exploring in some detail the way in which an
empiricist might respond to these problems. The general strategy will be as it
was in the discussion of universals above, namely to try to account for the
truths in question in terms of only the present existence of numbers.
How
is one who believes that numbers do genuinely exist as abstract objects, but
mind-dependently, to account for a truth such as the following?
Even if there had been no intelligent agents,
the number of moons of Jupiter would still have been 4.
Suppose we let 'a' designate the actual world. Then we might try expressing the
above truth as follows.
For all possible worlds w, if w differs from a only in containing no intelligent
agents, then the number of moons of Jupiter (in w) is none other than 4 (in a).7
The thought here is that we can avoid
commitment to the existence of the number 4 in any other world besides the
actual. The trouble, however, is that the number 4, in a, surely cannot be identical to (be none other than) the number of
moons of Jupiter in w without itself existing in w. What we really
require is something having the following form.
For all possible worlds w, if w differs from a only in containing no intelligent
agents, then the number (in a) of
moons of Jupiter (in w) is none other than 4 (in a).
Here all references to numbers are confined
to the actual world. The problem is to find a form of account which would have
such a feature. For it is by no means easy to see how a single definite
description (in this case 'the number of moons of Jupiter') can contain references
to two distinct possible worlds at once.
I
take it that all platonists should agree that statements of the form 'There are
n Fs' are really statements of numerical identity, having the form 'The number
of Fs is none other than n'. They should also accept that the criterion of
numerical identity is 1-1 correlation between concepts, in the manner explained
earlier. Then our target statement may be expressed by something having the
following general form.
For all possible worlds w, if w differs from a only in containing no intelligent
agents, then the instances of the concept moon of Jupiter (in w) 1-1
correlate the number of things such that .... (in a).
Here the dots would be filled in by the
appropriate definition of the number 4. In such a case we would be relying upon
relations of 1-1 correspondence across possible worlds (that is, between
w and a) to avoid commitment to the
existence of numbers in any but the actual world.8
But
what of these relations of 1-1 correspondence? How can they both obtain across
possible worlds and be mind-dependent? The answer is that they, in their turn,
are not actual relations but possible ones. For consider what it must mean to
say that there are n Fs, if an empiricist insists that concepts (and relations)
are best understood as rules of classification. Certainly not that there is any
actual rule of correspondence relating the number n to the instances of the
concept F. For in many cases no such rule will have been set up. Rather,
it must mean that it is possible to create such a rule, for example by
counting. In full, then, our account of our target statement would run as
follows.
For all possible worlds w, if w differs from a only in containing no intelligent
agents, then there is another possible world v which differs from w only in
whatever is required for there to be both numbers and a rule of correlation
such that the instances of the concept moon of Jupiter (in v) 1-1
correlate the number of things such that... (in v).9
While this does not quite restrict all
references to numbers to the actual world, it does in effect restrict them to
worlds in which there are intelligent agents, which is all that we need for
numbers to have mind-dependent existence.
We
can deal in similar manner with the necessary status (truth about all possible
worlds) of mathematical statements such as '2 + 2 = 4'. What initially causes
us a problem here is that such truths seem naturally to be represented in the
following form.
For all possible worlds w, it is a truth
about w that 2 + 2 = 4.
For it is hard to see how a statement about
numbers can express a truth about a world, without numbers themselves
existing in that world. But in fact we can express the content of our
new target statement somewhat as follows.
It is a truth about a that 2 + 2 = 4, and for all possible worlds w, the truth (in a) of 2 + 2 = 4 is applicable to
w.
Here the idea of the applicability of a
numerical truth is to be cashed out in a similar manner to that outlined above,
involving relations across worlds between numbers and the instances of
concepts. Thus the applicability of 2 + 2 = 4 to a world in which there are no
intelligent agents would consist in the following sorts of facts. If the
instances of the concept apple under the tree in that world are 1-1
correlated to the number 2 in a, and
if the instances of the concept pear under the tree in that world are
similarly correlated to the number 2 in a,
then the instances of the concept apple or pear under the tree in that
world are 1-1 correlated to the number 4 in a. Here we have preserved the intuitive thought that numerical
equalities are necessary (that is, applicable to all worlds) without commitment
to the existence of numbers in any worlds but those in which there are
intelligent agents.
Can there be Causal Contact?
It appears that an empiricist can adequately
undermine the arguments which seem to support platonism. But does an empiricist
have any direct objection to platonism? We have seen that there is insufficient
reason for believing platonism to be true, but are there any good reasons for
positively believing it to be false? The main argument to this effect, arises
out of the question whether a platonist can give an adequate account of our
supposed knowledge of necessarily existing abstract entities.
As
we noted in the last chapter, many platonists have believed that we possess a
special faculty of intellectual intuition, modelled by analogy with
sense-perception. However, it is doubtful whether this quasi-perceptual model
is really coherent. For how are we to make sense of the idea that a changeless
necessarily existing entity (which is what platonists suppose the number 7 to
be) can have causal powers? The problem here, is that the concept of causality
is at least closely bound up with counterfactual and subjunctive conditionals.
(A counterfactual conditional has the form 'If X had been the case, then
Y would have been the case'. A subjunctive conditional has the form 'If
X were to be the case, then Y would be the case'.) To say that A
caused B is to imply that if A had not been the case, then B would not have;
and also to imply that if in other circumstances something sufficiently similar
to A were to be the case, then something similar to B would be also.
Thus,
if it is true that a spark caused a particular explosion, then it must be true
that if the spark had not occurred, nor would the explosion (other things being
equal). It must also be true that in all other sufficiently similar
circumstances a spark would be followed by an explosion. But where A is
something that is necessarily the case, the supposition 'If A had not been so'
will make no sense. If the fact that 7 is prime obtains at all times in all
possible worlds, as platonists believe, then it will be impossible to suppose
that 7 had not been prime. In which case it cannot be the fact that 7 is prime
which causes our belief that it is; indeed such a fact cannot be a cause at
all.10
Although
the quasi-perceptual model is incoherent, it is not the only possible way in
which platonists might conceive of causal access to the abstract realm. An
alternative model would be a gravitational one. Just as it is the fact that an
asteroid enters the gravitational field of the Earth rather than of Mars (which
may for these purposes be thought of as changeless) which explains its
subsequent orbit, so it may be the fact that one intuits the nature of 7 rather
than of some other number which explains our subsequent belief that 7 is prime.
We may thus think of abstract objects as having associated with them permanent
(indeed necessarily existing) 'intuition fields', such that when the mind
stands in the intuition-relation to one abstract object rather than another, it
is caused to have one belief rather than another. This gives us appropriate
counterfactual and subjunctive conditionals. We may truly say that if someone
had not intuited the nature of 7 at that moment, then they would not have come
to believe that 7 is prime; and if in other similar circumstances they were to
intuit the nature of 7, they would then believe that 7 is prime.
This
account enables us to make some headway in explaining how platonic entities can
be causes. But it, too, faces severe problems. For even if abstract objects are
associated with permanent 'intuition fields', it still has to be possible to
explain why any given object is associated with one such field rather than
another. Yet such explanations will return us to the sort of unintelligible
counterfactuals met with earlier. Consider the gravitational case again. Even
if it is the bare fact that an asteroid enters the gravitational field of the
Earth rather than of Mars which explains its subsequent pattern of movement, it
must surely be possible to explain why the Earth has the gravitational
field that it has. And indeed we can provide just such an explanation, in terms
of the Earth's mass.
The
general moral is that relational causal powers need to be explicable in terms
of the categorical properties of the things which possess those powers. That is
to say, if something has the power to affect in a given way things which come
to stand in a certain relation to it, then this must ultimately be explicable
in terms of the non-relational properties which that object possesses. So if
the number 7 has the power to cause those who intuit it to believe that 7 is
prime, this must be causally explicable in terms of the properties of the
number 7. But then this will commit us once again to statements such as 'If 7
had not been prime it would not have had the powers which it has' - which are, as
we saw above, unintelligible (given that 7 is a necessary existent, which
possesses its attributes necessarily).
A
further problem with the idea that platonic objects can have effects on the
human mind (in either of the above versions), is that it requires us to
recognise a whole new species of causality, hitherto unknown to science. For we
would have to believe that an abstract object (non-physical and changeless) can
have effects within the natural (physical and changing) world. The difficulty
faced here by platonists is similar to that faced by dualists in the philosophy
of mind. Those who believe that the mind is a non-physical entity, or who
believe that mental events such as thoughts, decisions and intentions are
non-physical ones, are then faced with the problem of explaining how there can
be causal contact between non-physical events and physical ones. For we do
normally suppose that our thoughts and decisions cause our bodies to
move in certain ways. Now I doubt whether there is any objection in principle
to the idea of such causation. But it does conflict with a well-established
working hypothesis of science, namely that all causes are physical ones.
Science
has progressed by ignoring the possibility of causation by spirits or other
non-physical entities, and by looking wherever possible for physical
mechanisms. So the immense explanatory success of science gives us reason to be
doubtful, at least, of either the quasi-perceptual or gravitational models
employed by platonists to explain our knowledge of logic and mathematics. If we
are impressed by the success of science, we shall wish to insist that such
knowledge should be explained naturalistically, in terms of the
operation of laws and processes which science gives us some reason to believe
in. We shall return to consider this idea in some detail in chapter 9.
Platonism and Scepticism
Yet another objection to platonism, is that
it seems inevitably to lead to scepticism with respect to our knowledge of
logic and mathematics. For if platonism were true, what reason could we have
for thinking that intellectual intuition is generally reliable? It might be
claimed that we have just the same reasons here as we have for thinking
sense-perception to be reliable, namely that the hypothesis of reliability
provides the best over-all explanation of the existence and coherence of our
beliefs. But in fact our belief in perceptual reliability fits into an
explanatory network which is wholly lacking in the case of intuition of
abstract objects. For we have beliefs about our perceptual apparatus itself and
its mode of operation, which mesh closely with our beliefs about the physical
world which we perceive. And these in turn provide the best available
explanation of the course of our experience. (These ideas will be developed
more fully in chapters 11 and 12.)
In
contrast, we have no beliefs about the structure of our faculty of intuition or
its mode of operation. Indeed, the existence of such a faculty remains a bare
hypothesis. Nor can the hypothesis of its reliability play any real explanatory
role. In particular, we cannot use it to explain our success in building
bridges and aeroplanes (in which mathematics plays a significant part), unless
we can suppose that the way things are in the abstract realm can make a
difference to what happens in the physical realm - which brings us back to the
incoherent idea of abstract objects as causes once again. In fact all we need
to suppose, is that mathematical calculation gives us access to whatever
structural features of the physical world underlie our success in action.
The
only other option available to a platonist, in explaining our knowledge of the
abstract realm, is to claim that such knowledge is innate, being brought to
consciousness by the activity of thinking and reasoning. There are two subtly
different versions of such an account. Either it can be claimed that beliefs
about the abstract realm are already latent in us prior to thought and
reasoning. Or it can be claimed that the faculty of reason itself has a constituent
structure, which is such that its operation gives rise to beliefs about the
abstract realm. But the differences between these two forms of account need not
detain us at this stage. For in either case the idea is that there is something
innate in the human mind which mirrors or depicts the nature of the abstract
realm.
Such
an account enables a platonist to do without the supposition of causal contact
between an abstract object and the human mind, and this is a mark in its
favour. But we are still left with the same problem of explaining what reason
we have for supposing that the mirroring or depicting is generally accurate.
For suppose we allow that we have innate beliefs about logic and mathematics.
If we grant the platonic conception of the subject-matter of these disciplines,
then what reason would we have for thinking that the beliefs in question are
generally true? Since we have no idea how it is that these beliefs might come
to be innate, nor any account of why the process which leads them to be innate
should ensure that they mirror the nature of the abstract realm, we lose
nothing in explanatory power if we suppose that our logical and mathematical
beliefs are generally false. (We shall return to this point in greater detail
in chapter 10.) For again, we cannot appeal to the practical success of our
applications of logic and mathematics as a reason for thinking that our innate
beliefs are reliable, unless we can also explain how truth about the abstract
realm could make a difference to the course of events in the physical world.
But this we are debarred from doing, given that the idea of causal influence of
the abstract on the physical is incoherent.
It
would appear then, that neither the hypothesis of a faculty of intuition, nor
the hypothesis of innateness, are capable of explaining how we have knowledge
of the abstract realm (given the platonist's conception of it). We then face a
choice, of either denying that we may have reasonable beliefs about logic and
mathematics, or of finding some alternative (non-platonic) account of their
subject-matter.
In conclusion: there are insufficient reasons for supposing platonism to be true. Yet there are very good reasons for supposing it to be false. For our acceptance of platonism would mean both: that we could give no account of the manner in which we might obtain knowledge of the abstract realm; and that we could no longer hold reasonable beliefs about logic and mathematics, which would purportedly concern such a realm. The empiricist case against this form of substantive a priori knowledge (arguably the most plausible form, as we saw at the beginning of the last chapter) is therefore a powerful one.