Encore #10: Parsons on intuition

Just yesterday, Brian Leiter posted the results of one of his entertaining/instructive online polls, this time on the “Best Anglophone and German Kant scholars  since 1945“. Not really my scene at all. Though I did, back in the day, really love Bennett’s Kant’s Analytic (as philosophy this is surely brilliant, whatever its standing as “scholarship”). I note that in comments after his post, Leiter expresses regret for not having listed Charles Parsons in his list of contributors to Kant scholarship to be voted on. Well, true enough, Parsons has battled over the years to try to make sense of/rescue something from Kantian thoughts about ‘intuition’ as grounding e.g. arithmetical knowledge. But with what success, I do wonder? I found the passages about intuition in Mathematical Thought and Its Objects rather baffling. Here I put together some thoughts from 2008 blog posts.

Is any of our arithmetical knowledge intuitive knowledge, grounded on intuitions of mathematical objects? Parsons writes, “It is hard to see what could make a cognitive relation to objects count as intuition if not some analogy with perception” (p. 144). But how is such an analogy to be developed?

Parsons tries to soften us up for the idea that we can have intuitions of abstracta (where these intuitions are somehow quasi-perceptual) by considering the putative case of perceptions – or are they intuitions? – of abstract types such as letters. The claim is that “the talk of perception of types is something normal and everyday” (p. 159).

But it is of course not enough to remark that we talk of e.g. seeing types: we need to argue that we can take our talk here as indeed reporting a (quasi)-perceptual relation to types. Well, here I am, looking at a squiggle on paper: I immediately see it as being, for example, a Greek letter phi. And we might well say: I see the letter phi written here. But, in this case, it might well be said, ‘perception of the type’ is surely a matter of perceiving the squiggle as a token of the type, i.e. perceiving the squiggle and taking it as a phi.

Now, it would be wrong to suppose that – at an experiential level – ‘seeing as’ just factors into a perception and the superadded exercise of a concept or of a recognitional ability. When the aspect changes, and I see the lines in a drawing as a picture of a duck rather than a rabbit, at some level the content of my conscious perception itself, the way it is articulated, changes. Still, in seeing the lines as a duck, it isn’t that there is more epistemic input than is given by sight (visual engagement with a humdrum object, the lines) together with the exercise of a concept or of a recognitional ability. Similarly, seeing the squiggle as a token of the Greek letter phi again doesn’t require me to have some epistemic source over and above ordinary sight and conceptual/recognitional abilities. There is no need, it seems, to postulate something further going on, i.e. quasi-perceptual ‘intuition’ of the type.

The deflationist idea, then, is that seeing the type phi instantiated on the page is a matter of seeing the written squiggle as a phi, and this involves bring to bear the concept of an instance of phi. And, the suggestion continues, having such a concept is not to be explained in terms of a quasi-perceptual cognitive relation with an abstract object, the type. If anything it goes the other way about: ‘intuitive knowledge of types’ is to be explained in terms of our conceptual capacities, and is not a further epistemic source. (But note, the deflationist who resists the stronger idea of intuition as a distinctive epistemic source isn’t barred from taking Parsons’s permissive line on objects, and can still allow the introduction of talk via abstraction principles of abstract objects such as types. He needn’t have a nominalist horror of talk of abstracta.)

Let’s be clear here. It may well be that, as a matter of the workings of our cognitive psychology, we recognize a squiggle as a token phi by comparing it with some stored template. But that of course does not imply that we need be able, at the personal level, to bring the template to consciousness: and even if we were to have some quasi-perceptual access to the template itself, it wouldn’t follow that we have quasi-perceptual access to the type. Templates are mental representations, not the abstracta represented.

Parsons, however, explicitly rejects the sketched deflationary story about our intuition of types when he turns to consider the particular case of the perception of expressions from a very simple ‘language’, containing just one primitive symbol ‘|’ (call it ‘stroke’), which can be concatenated. The deflationary reading

does not accurately render our perceptual consciousness of strokes. It would make what I want to call intuition of a string an instance of seeing a certain inscription as  of a type …. But in actual cases, the identification of the type will be firmer and more explicit that the identification of any physical inscriptions that is an instance of the type. That the inscriptions are real physical objects with definite physical properties plays no role in the mathematical treatment of the language, which is what concerns us. An illusory presentation of a string, provided it is sufficiently clear, will do as well to illustrate a mathematical notion as a real one. (p. 161)

There seem to be two points here, neither of which will really trouble the deflationist.

The first point is that the identification of a squiggle’s type may be “firmer and more explicit” than our determination of its physical properties as a token (which I suppose means that a somewhat blurry shape may still definitely be a letter phi). But so what? Suppose we have some discrete conceptual pigeon-holes, and have reason to take what we see as belonging in one pigeonhole or another (as when we are reading Greek script, primed with the thought that what we are seeing will be a sequence of letters from forty eight upper and lower case possibilities). Then fuzzy tokens can get sharply pigeonholed. But there’s nothing here that the deflationist about seeing types can’t accommodate.

The second point is that, for certain illustrative purposes, illusory strings are as good as physical strings. But again, so what? Why shouldn’t seeing an illusory strokes as a string be a matter of our tricked perceptual apparatus engaging with our conceptual and/or /recognitional abilities? Again, there is certainly no need to postulate some further cognitive achievement, ‘intuition of a type’.

Oddly, Parsons himself, when wrestling with issues about vagueness, comes close to making these very points. For you might initially worry that intuitions which are founded in perceptions and imaginings will inherit the vagueness of those perceptions or imaginings – and how would that then square the idea that mathematical intuition latches onto sharply delineated objects? But Parsons moves to block the worry, using the example of seeing letters again. His thought now seems to be the one above, that we have some discrete conceptual pigeon-holes, and in seeing squiggles as a phi or a psi (say), we are pigeon-holing them. And the fact that some squiggles might be borderline candidates for putting in this or that pigeon-hole doesn’t (so to speak) make the pigeon-holes less sharply delineated. Well, fair enough. But thinking in these terms surely does not sustain the idea that we need some basic notion of the intuition of the type phi to explain our pigeon-holing capacities.

So, I’m unpersuaded that we actually need (or indeed can make much sense of) any notion of the quasi-perceptual ‘intuition of types’ – and in particular, any notion of the intuition of types of stroke-strings – that resists a deflationary reading. But let’s suppose for a moment that we follow Parsons and think we can make sense of such a notion. Then what use does he want to make of the idea of intuiting strokes and stroke-strings?

Parsons writes

What is distinctive of intuitions of types [here, types of stroke-strings] is that the perceptions and imaginings that found them play a paradigmatic role. It is through this that intuition of a type can give rise to propositional knowledge about the type, an instance of intuition that. I will in these cases use the term ‘intuitive knowledge’. A simple case is singular propositions about types, such as that ||| is the successor of ||. We see this to be true on the basis of a single intuition, but of course in its implications for tokens it is a general proposition. (p. 162)

This passage raises a couple of issues worth commenting on.

One issue concerns the claim that there is a ‘single intuition’ here on basis of which we see that that ||| is the successor of ||. Well, I can think of a few cognitive situations which we might agree to describe as grounding quasi-perceptual knowledge that ||| is the successor of || (even if some of us would want to give a deflationary construal of the cases, one which doesn’t appeal to intuition of abstracta). For example,

  1. We perceive two stroke-strings
           ||
           |||
    and aligning the two, we judge one to be the successor or the other.
  2. We perceive a single sequence of three strokes ||| and flip to and fro between seeing it as a threesome and as a block of two followed by an extra stroke.

But, even going along with Parsons on intuition, neither of those cases seems aptly described as seeing something to be true on the basis of a single intuition. In the first case, don’t we have an intuition of ||| and a separate intuition of || plus a recognition of the relation between them? In the second case, don’t we have successive intuitions, and again a recognition of the relation between them? It seems that our knowledge that ||| is the successor of || is in either case grounded on intuitions, plural, plus a judgement about their relation. And now the suspicion is that it is the thoughts about the relations that really do the essential grounding of knowledge here (thoughts that could as well be engaging with perceived real tokens or with imagined tokens, rather than with putative Parsonian intuitions that, as it were, reach past the real or imagined inscriptions to the abstracta).

The other issue raised by the quoted passage concerns the way that the notion of ‘intuitive knowledge’ is introduced here, as the notion of propositional knowledge that arises in a very direct and non-inferential way from intuition(s) of the objects the knowledge is about: “an item of intuitive knowledge would be something that can be ‘seen’ to be true on the basis of intuiting objects that it is about” (p. 171). Such a notion looks very restrictive – on the face of it, there won’t be much intuitive knowledge to be had.

But Parsons later wants to extend the notion in two ways. First

Evidently, at least some simple, general propositions about strings can be seen to be true. I will argue that in at least some important cases of this kind, the correct description involves imagining arbitrary strings. Thus, that will be included in ‘intuiting objects that a proposition is about’. (p. 171)

But even if we now allow intuition of ‘arbitrary objects’, that still would seem to leave intuitive knowledge essentially non-inferential. However,

I do not wish to argue that the term ‘intuitive knowledge’ should not be used in that [restrictive] way. Our sense, following that of the Hilbert School, is a more extended one that allows that certain inferences preserve intuitive knowledge, so that there can actually be a developed body of mathematics that counts as intuitively known. This seems to me a more interesting conception, in addition to its historical significance. Once one has adopted this conception, one has to consider case by case what inferences preserve intuitive knowledge. (p. 172)

Two comments about his. Take the second proposed extension first. The obvious question to ask is: what will constrain our case-by-case considerations of which kinds of inference preserve intuitive knowledge? To repeat, the concept of intuitive knowledge was introduced by reference to an example of knowledge seemingly non-inferentially obtained. So how are we supposed to ‘carry on’, applying the concept now to inferential cases? It seems that nothing in our original way of introducing the concept tells us which such further applications are legitimate, and which aren’t. But there must be some constraints here if our case-by-case examinations are not just to involve arbitrary decisions. So what are these constraints? I struggle to find any clear explanation in Parsons.

And what about intuiting ‘arbitrary’ strings? How does this ground, for example, the knowledge that every string has a successor? Well, supposedly, (1) “If we imagine any [particular] string of strokes, it is immediately apparent that a new stroke can be added.” (p. 173) (2) But we can “leave inexplicit its articulation into single strokes” (p. 173), so we are imagining an arbitrary string, and it is evident that a new stroke can be added to this too. (3) “However, …it is clear that the kind of thought experiments I have been describing can be taken as intuitive verifications of such statements as that any string of strokes can be extended only if one carries them out on the basis of specific concepts, such as that of a string of strokes. If that were not so, they would not confer any generality.” (p. 174) (4) “Although intuition yields one essential element of the idea that there are, at least potentially, infinitely many strings …more is involved in the idea, in particular that the operation of adding an additional stroke can be indefinitely iterated. The sense, if any, in which intuition tells us that is not obvious.” (p. 176) But (5) “Once one has seen that every string can be extended, it is still another question whether the string resulting by adding another symbol is a different string from the original one. For this it must be of a different type, and it is not obvious why this must be the case. … Although it will follow from considerations advanced in Chapter 7 that it is intuitively known that every string can be extended by one of a different type, ideas connected with induction are needed to see it” (p. 178).

There’s a lot to be said about all that, though (4) and (5) already indicate that Parsons thinks that, by itself, ‘intuition’ of stroke-strings might not take us terribly far. But does it take us even as far as Parsons says? For surely it is not the case that imagining/intuiting adding a stroke to an inexplicitly articulated string, together with the exercise of the concept of a string of strokes, suffices to give us the idea that any string can be extended. For we can surely conceive of a particularist reasoner, who has the concept of a string, can bring various arrays (more or less explicitly articulated) under that concept, and given a string can recognize that this one can be extended – but who can’t advance to frame the thought that they can all be extended? The generalizing move surely requires a further thought, not given in intuition.

Indeed, we might now wonder quite what the notion of intuition is doing here at all. For note that (1) and (2) are a claims about what is imaginable. But if we can get to general results about extensibility by imagining particular strings (or at any rate, imagining strings “leaving inexplicit their articulation into single strokes”, thus perhaps |||| with a blurry filling) and then bring them under concepts and generalizing, why do we also need to think in terms of having cognitive access to something else which is intrinsically general, i.e. stroke-string types? It seems that Parsonian intuitions actually drop out of the picture. What gives them an essential role in the story?

Finally, note Parsons’s pointer forward to the claim that ideas “connected with induction” can still be involved in what is ‘intuitively known”. We might well wonder again as we did before: what integrity is left to the notion of intuitive knowledge once it is no longer tightly coupled with the idea of some quasi-perceptual source and allows inference, now even non-logical inference, to preserve intuitive knowledge? I can’t wrestle with this issue further here: but Parsons ensuing discussion of these matters left me puzzled and unpersuaded.

Again, as with the last post, that’s how things seemed to be more than seven years ago. If other readers have a better of sense of what a Parsonian line on intuition might come to, comments are open!

This entry was posted in Math. Thought and Its Objects, Phil. of maths. Bookmark the permalink.

Leave a Reply

Your email address will not be published. Required fields are marked *