Mathematical
Platonism takes numbers and other mathematical objects to exist in a third
realm distinct from the material and mental worlds, after the fashion of the
Forms of Plato’s famous theory.  A common
objection to this view, associated with philosophers like Paul Benacerraf, is
epistemological.  In order for us to have
knowledge of something, say these philosophers, we must be in causal contact
with it.  But if numbers are abstract
objects outside of space and time, then we cannot be in any such contact with
them, because they would be causally inert and inaccessible to perception.  So, if Platonism were true, we couldn’t have
knowledge of them.  Yet we do have such
knowledge, which (the argument concludes) implies that Platonism is false.  This is known as the “access problem” for
mathematical Platonism.

Brown’s defense

Is this a serious
problem?  No and yes.  On the one hand, the way the problem is often
framed is too underdeveloped and question-begging to worry a sophisticated
Platonist.  The idea seems to be that,
when we know a chair, for example, that is because light travels from the chair
to our eyes, resulting in retinal stimulation, which in turn generates neural
activity that brings about a conscious perception of the chair.  But nothing like this is possible where
Platonic objects are concerned.

But there
are several problems with leaving it at that. 
First, as James Robert Brown
points
out
, the objection presupposes that we have a clear and
uncontroversial account of how neural processes generate conscious perceptual
experiences.  But of course, we don’t,
which is why there is such a thing as a mind-body problem.  Now, with occasional exceptions (such as
Berkeley), philosophers tend not to take the mind-body problem to be a reason
to doubt the existence of the material world. 
Though there is no agreement about how conscious experiences can be
caused by material objects and processes, they don’t take that to be a reason
for us to doubt that there really are material objects and processes, that our
experiences are in causal contact with them, and that those experiences
therefore give us knowledge of them.  But
in that case, Brown quite reasonably concludes, such philosophers can hardly
take the absence of an account of how we can get in causal contact with
abstract mathematical objects to be a reason to doubt that there are such
objects.

Brown also
suggests (less plausibly, I think) that quantum mechanics gives us reason to
doubt that a causal connection with what is known is really a necessary condition
for our knowing it.  He has in mind J. S.
Bell’s famous nonlocality result. 
Consider an EPR scenario in which two photons arrive at different locations,
B and C, from a common source A.  When
the photons arrive at their destinations, measurements will show that one has
the property spin-up and the other
the property spin-down, though
nothing about what is happening at A could tell us which photon will have which
property.  Furthermore, supposing that B
and C are outside of each other’s light cones, information about what is
happening at one of these locations cannot get to the other.  Nevertheless, if I know, for example, that
the photon that arrives at B has the property spin-up, then I can know that the one that arrives at C will have
the property spin-down.  But nothing about any causal relation between
A on the one hand and B and C on the other, or between B and C, will have told
me this.  And that, Brown says, refutes the
assumption that a causal connection is necessary for knowledge. 

But this
seems to me not quite right.  After all,
if the photons had never been emitted from A, they would not have arrived at B
and C, and had I not been there to take the measurement at B, I would not have
been able to infer from it what was going on at C.  And these are causal facts.  So, the right conclusion to draw from Bell’s
result is not that there are no causal connections at all involved in my knowing what I know, but rather that the
causal connections are very weird.  This
raises many questions, of course, but I don’t see that they need to be
addressed in order to make the narrow point that Bell’s result doesn’t provide
a compelling way to respond to the access problem.

Plato’s defense

Another
problem with the way the access problem is usually framed is that it rather
shamelessly begs the question against Plato himself.  After all, it is hardly as if Plato was
unaware of the difficulty of modeling our knowledge of Platonic abstract objects
on perceptual knowledge of physical objects. 
Indeed, Plato himself insists that knowledge of the Forms cannot work
that way.  That’s why he thinks that we
must have come to know them prior to embodiment in this life, and why he thinks
the soul must be unlike perceptual organs in being immaterial.

In short,
Plato is well aware that there is an access problem, but thinks he’s solved
it.  Contemporary naturalists don’t like
the solution, but part of Plato’s point is that the reality of Platonic objects,
and of our knowledge of them, give us reason to reject naturalism.  To object to mathematical Platonism on the
grounds that it is hard to square with naturalism is simply to assume, without argument, precisely what
is at issue.

Plato would
also reject the naturalist’s assumption that explanation is at bottom a matter
of identifying relations of efficient causation between material objects.  For Plato, the participation relation that he
thinks holds between particular things and the Forms provides another mode of
explanation, and teleology yet another. 
Of course, the Platonist would have to spell out exactly how Plato’s
richer account of explanation can be deployed to solve the access problem.  But the point is that, by simply assuming, without
argument, a broadly naturalistic metaphysics and epistemology, the usual way of
presenting the access problem does not constitute as powerful an objection as
is often supposed, because it simply begs the question against Plato.

Aristotle’s critique

But that doesn’t
mean that the mathematical Platonist is out of the woods.  We Aristotelians also reject Platonism, for
several reasons, and some of these are relevant to the access problem.  In particular, Aristotle too is critical of the
idea that an entity like a Platonic Form could be an efficient cause.  In Metaphysics,
Book XII,
Part 6
, he writes:

But if there is something which is
capable of moving things or acting on them, but is not actually doing so, there
will not necessarily be movement; for that which has a potency need not
exercise it.  Nothing, then, is gained
even if we suppose eternal substances, as the believers in the Forms do, unless
there is to be in them some principle which can cause change; nay, even this is
not enough, nor is another substance besides the Forms enough; for if it is not
to act, there will be no movement
.

Aristotle’s
point here is, first, that something can function as an efficient cause only if
it both has an active causal power (which is what a “potency” is in this
context), and that power is actually exercised on some particular occasion.  For example, I can cause the pen in front of
me to move just by touching it, but I cannot cause it to dissolve just by
touching it.  For I have an active causal
power of the first sort, but not a power of the second sort.  But in addition to my having the first power,
I have to exercise it in order for the pen actually to move.  If I don’t decide to touch the pen, it will
just sit there motionless, despite my having the power to move it.

Similarly,
Aristotle says, in order for a Platonic Form (or a mathematical object
conceived of on the model of a Form) to function as an efficient cause, it
would have to have the active causal power to do so, and it would have to be
actually exercising that power on some particular occasion.  And Aristotle’s implication is that these
conditions don’t hold in the case of the Forms. 
They don’t function as efficient causes. 
But why not?

Well, think
about what, from an Aristotelian point of view, is true of me that makes it the case that I can function as an efficient cause
of the movement of the pen.  I am part of
a larger system of substances with their own causal powers, the exercise of
which contributes to triggering the operation of my own.  For example, the phone rings, which leads me
to pick it up, which is followed by somebody on the other end of the line
telling me something I want to write down, which leads me in turn to exercise
my power to pick up the pen.  All of this
unfolds in time and involves my being changed in various ways by the substances
I interact with, leading me in turn to bring about changes in them.

These
circumstances do not hold of the Forms. 
The Forms (and mathematical objects conceived of on the model of the
Forms) are eternal and unchanging.  So, nothing could happen to them to trigger the operation of their
causal powers, if they have any.  Now,
you might respond that God, in Aristotelian-Thomistic theology, is eternal and
unchanging, yet he is still said to be an efficient cause.  So why couldn’t the same thing be said of the
Forms?

But there is
a crucial difference.  There is in God
something analogous to intellect and will, but that is not true of the Forms,
which are impersonal.  The reason this
matters is evident from a point Aristotle makes in On Generation and Corruption, Book II, Part 9,
where he writes:

Some… thought the nature of ‘the Forms’
was adequate to account for coming-to-be.  Thus Socrates in the
Phaedo first blames everybody else for having given no explanation; and then
lays it down that ‘some things are Forms, others Participants in the Forms’,
and that ‘while a thing is said to “be” in virtue of the Form, it is said to “come-to-be”
qua sharing in, to “pass-away” qua “losing,” the ‘Form’.  Hence he thinks that ‘assuming the truth of
these theses, the Forms must be causes both of coming-to-be and of passing-away’…

[But] if the Forms are causes, why is
their generating activity intermittent instead of perpetual and continuous – since
there always are Participants as well as Forms?

The idea, as
I read Aristotle here, is this. 
Consider, for example, the Form of Triangle.  It never comes into being or passes away, nor
does it change in any other respect.  So,
if it is functioning as an efficient cause, its effects – particular triangles –
should be similarly temporally unbounded. 
They should simply always exist, past, present, and future.  But they don’t – they come into being and
pass away.  The point even more obviously
holds of living things like Tyrannosaurus Rex, which came into existence at
some point and have now gone extinct – even though the Platonic Form of Tyrannosaurus Rex, like every
other Form, is eternal.

Now, if we were
to attribute something like rationality and free choice to the Forms – as we
can to God – we could find a way to make sense of how an eternal cause could
have a temporally limited effect.  All we
need is the idea there is some reason
why the cause saw fit to produce an effect that is temporally bounded in just
the way it is.  We don’t need to know what the reason is; the mere fact that
there could be one is sufficient to make intelligible the possibility of an
eternal cause having such an effect.  (Readers
familiar with William Lane Craig’s work might recognize this as among the
arguments he gives for the claim that the cause of the beginning of the
universe must be personal rather than impersonal.)

But we can’t
do this with numbers and other Platonic objects, because, again, they are
impersonal.  Hence that way of answering
Aristotle’s criticism is not open to the Platonist. 

Aristotelianizing Plato

The problem,
to sum up, is that if a thing really has active causal powers, then there has
to be something that accounts for how those powers get triggered in the ways
they do.  Now, we have such accounts in
the case of physical substances (in terms of their relations to other physical substances)
and in the case of immaterial mental substances (in terms of their rationality and
free choice).  But there is no account
available in the case of the purported occupants of Plato’s “third realm” –
immaterial but impersonal entities. 

Factor in
the Scholastic principle agere sequitur
esse
(“action follows being”) – that the way a thing acts reflects what it
is – and we have the ingredients for an argument to the effect that Platonic
Forms would have to be causally inert.  For
if there is no way in principle that the causal powers of such Forms could ever
be exercised, in what sense would
they even have such powers in the
first place?

Now, the
passages from Aristotle I cited do not address the access problem,
specifically, but their relevance to it should by now be obvious.  If mathematical objects conceived of on the
model of Platonic Forms would have to be causally inert, then they cannot be
what causes us to have knowledge of them. 
But then, how do we have
knowledge of them?  (Notice that it won’t
do to posit some third thing – call it X – that has access to the Forms and
then in turn imparts knowledge of them to us. 
For that just kicks the problem back a stage.  How could X
gain knowledge of the Forms if they are causally inert, and thus cannot be
what causes X to know about them?)

Notice that
the problem does not arise for the Augustinian position that the Forms (and
numbers and other mathematical objects) are to be identified with ideas in the
divine intellect.  For then it wouldn’t
be the Forms per se that directly act
on the world, but rather God, who is not causally inert. 

This
position, adopted by later Scholastics like Aquinas and thus sometimes labeled “Scholastic
realism” (as opposed to Platonic realism and Aristotelian realism) can be
interpreted as a kind of Aristotelianizing of Plato.  Plato posits three realms, the material, the
mental, and the Platonic third realm. 
Aristotle holds that only the first two are real.  Scholastic realism agrees with Aristotle that
there is no third mode of being apart from the material realm and the mental
realm.  But it agrees with Plato that
truths about mathematical objects and other Forms can’t be grounded in truths
about material substances or even in truths about finite mental
substances.  Hence it takes the infinite, divine mind to be their ultimate ground.

Exactly how our knowledge of these objects works
is another question.  Augustine says it
is by illumination, but there
are problems with that account
.  Whatever
the right answer, though, it needn’t be saddled with the difficulties facing
Platonism.

Related
reading:

Review of
Craig’s God Over All: Divine Aseity and
the Challenge of Platonism

Frege
on what mathematics isn’t

Rucker’s
Mindscape

David
Foster Wallace on abstraction

Augustine
on divine illumination

Plato’s
affinity argument

Five
Proofs of the Existence of God
, chapter 3

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