The access problem for mathematical Platonism

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