The Intuitively Given and the Logically Derived
The View from Oregon — 311: Friday 18 October 2024
In newsletter 303, in a discussion of intuitive formalisms, I quoted Ian Stewart and David Tall in The Foundations of Mathematics:
“…our aim is to develop the formal approach as a natural outgrowth of the underlying pattern of ideas. A sixth-form student has a broad grasp of many mathematical principles, and our aim is to make use of this, honing his mathematical intuition into a razor-sharp tool which will cut to the heart of a problem. Our point of view is diametrically opposed to that where (all too often) the student is told ‘Forget all you’ve learned up till now, it’s wrong, we’ll begin again from scratch, only this time we’ll get it right’. Not only is such a statement damaging to a student’s confidence: it is also untrue. Further, it is grossly misleading: a student who really did forget all he had learned so far would find himself in a very sorry position.”
After quoting this I mentioned that there was another quote that I couldn’t find, which I have since located, which is only a few pages further along from the above quote:
“The expert can look at a logical construction written in a book and say, ‘I guess that thing there is meant to be “zero”, so that thing is “one”, and that’s “two”, … this load of junk must be the integers, … what’s that? Oh, I think I see: it must be “addition” …’ The non-expert is faced with an indecipherable mass of symbols.” (Stewart and Tall, The Foundations of Mathematics, p. 6)
My point in referring to this passage is that even when we rationally reconstruct our knowledge, we continue to interpret it in a way that makes sense to us, and the way that makes sense to us is the way that appeals to our native intuitions. Philosophers use “intuition” in a peculiar way. In ordinary language, intuition is a subtle and often unconscious process of reasoning by means of which we reach conclusions that might easily be missed, and which are indeed missed by most, but picked up by a few — those with superior intuition.
In philosophy, intuition isn’t a reasoning faculty of any kind, consciousness or unconscious; it is, on the contrary, the given and not the derived. Kant made a distinction that has come down to us as the distinction between sensual intuition and intellectual intuition: that which is given to the senses and that which is given to the mind. In this way, that which is given to the mind, such as a grasp of the concept of number, is entirely parallel to that which is given by the senses, such as the taste of pineapple. Both are forms of givenness, though one is given by way of the mind and the other by way of the body. But there is yet another sense of intuition described by Ian Stewart in another book, Concepts of Modern Mathematics:
“Logic can be used to solve problems, but it cannot suggest which problems to try. No one has ever formalized significance. To recognize what is significant you need a certain amount of experience, plus that elusive quality: intuition. I cannot define what I mean by ‘intuition.’ It is simply what makes mathematicians (or physicists, or engineers, or poets) tick. It gives them a ‘feel’ for the subject; with it they can see that a theorem is true, without giving a formal proof, and on the basis of their vision produce a proof that works.” (Concepts of Modern Mathematics, p. 4)
One could argue that what Stewart here describes is either a subtle faculty of reasoning or a form of givenness; Stewart’s position occupies a kind of halfway point between understanding that is derived and understanding that is given. It is an extraordinarily elusive idea, and it is understandable that he says he can’t define it. However, some mathematicians have tried to explain what they mean by significance: G. H. Hardy’s A Mathematician’s Apology has a remarkable discussion of cognitive significance (one might even call it a phenomenology of mathematical significance), but Hardy would not have said that he formalized the idea. In a maddening reflexivity, we could say that Hardy’s is an intuitive presentation of mathematical significance. (I made use of Hardy’s approach to cognitive significance in my paper “Space Philosophy: The Symmetry Hypothesis.”)
Given the philosopher’s conception of intuition, we might well ask whether the distinction between sensual intuition and intellectual intuition is misleading, since that which is revealed by the senses, i.e., that which is given to us, varies according to the senses by which it is given. Sensual intuition, then, may be of many kinds. In the classical Aristotelian analysis from On the Soul, there are five senses, and each of the five senses could be said to represent a distinct form of givenness. Our contemporary understanding considerably expands upon the five senses, recognizing kinesthetic sensations (or proprioception) and visceroception, both of which are forms of interoception. Already here we have a wide range of sensory intuitions, different enough from each other that we might want to put them in separate classes.
The same can be done with intellectual intuitions. Here we fall far short of the language we need to describe the different ways in which intellectual intuitions are given to us, but the many kinds of ideas we have suggest that they are at least as internally different as the many kinds of sensual intuitions we experience. (It says something about our cognitive architecture that our conceptual framework for the exposition of sensory experience is so much more sophisticated than our conceptual framework for the exposition of thought.) There is a famous passage from Gödel (Collected Works, Vol. II, p. 268) where he says that we have intellectual intuition of numbers (“…we do have something like a perception also of the objects of set theory…”), but he follows this with the observation that there is nothing in a sensory experience that gives us, “…the idea of object itself…” so the idea of an object must derive from our intellectual intuition (although Gödel does not use “intellectual intuition”).
We could argue that it is only a degree of abstraction that separates the intellectual perception of the idea of object and the intellectual perception of numbers as objects (the latter a position especially associated with Frege), but there is no reason we should deny that a taxonomy of abstract objects is based on their degree of abstractness, and, with a hierarchy of abstract objects given in intellectual intuition, we then have a range of intellectual intuitions as I previously suggested that we have a range of sensual intuitions, i.e., the internal diversity of the two classes of intuition is parallel. Of course, we might also have qualitatively distinct intellectual intuitions and not only a hierarchy of greater or lesser abstraction, which only makes intellectual intuition more variegated.
However many forms of intuition we possess, they are all equally “basic” in the sense of being given to us immediately, whether by the senses or by the mind. From these basic units of intuition, we build up whatever thoughts we have. As our thinking becomes more sophisticated, we eventually see the need to bring order to it, and this brings us to realize that our while our intuitions are all equally basic to us, there are relations among them of a different order than immediate givenness. Our intuitions are assimilated to concepts, and concepts stand in definite logical relationships to each other. As we puzzle out the logical relationships among concepts (a process that has taken us more than two thousand years), the equality of intuitions given to consciousness gives way to a logical order of derivation, and the logical order of derivation gives us a different meaning of what it is to be basic — logical simplicity. To be logically simple is to be first in the order of derivation, but the order of derivation is constructed ex post facto from basic intuitions.
What is logically simple is not necessarily basic to consciousness, and what is basic to consciousness, i.e., intuition, is not necessarily logically simple. Thus we work backward through our intuitions and we try to reconstruct them in the logical order. It is this rational reconstruction of concepts that gives us a logical order that is distinct from the intuitive order. It is also this rational reconstruction that gives us the sophisticated logical expression of intuitive ideas as in the quote above from Stewart and Tall. The mind that naturally thinks in terms of its intuitions comes to see the reasonableness of thinking in logical terms, but these logical terms never possess the conceptual naturalness of basic intuitions. To understand their relationship is itself an accomplishment.
Ultimately there is, then a relationship between the sense of intuition as a subtle form of derivation and intuition as that which is immediately given to consciousness. Intuitions exhibit conceptual naturalness; the order of derivation exhibits logical simplicity, which lacks conceptual naturalness. But there is, moreover, an intuition (is this a second-order intuition?) in recognizing our intuitive ideas within a logical framework distantly derived from them. We “see” the relationship between logical simples and conceptual naturalness (if we see it at all, which most do not, as this is an experience reserved for the few communicants of logic) and, if we are sufficiently sophisticated in our thought, we may experience this relationship as exhibiting a kind of conceptual naturalness. If we’ve made it to that point, we see that this relationship between intuitive simples and logical simples is itself a subtle form of derivation in which the two are seen to be related to each other in an elusive way that escapes most of us, which is nothing but the ordinary language sense of intuition that I gave above.
