Models of Thought and the Limits of the Thinkable

We think, it is sometimes said, by building models of phenomena. It seems therefore natural to conclude that we can understand thought itself by elaborating a model of what it is. Many domains of empirical science can be described as being engaged in precisely this activity (experimental psychology, neurophysiology, robotics and artificial intelligence, to name the main categories), but is there a contribution  to thinking about thought that could still be called philosophical in our age of naturalized epistemology?

There is a long tradition in philosophy of attempting to delineate what is thinkable, of characterizing the limits of thought. Producing a critique of the claims of reason to know certain things, or to be able in principle to come to know them, is associated most obviously with the name of Kant, but it is not difficult to find both antecedents and more recent versions of this very general project : in a way Zeno's paradoxes of motion are a critique of the pretension of thought to conceive of the infinite divisibility of space and time, whereas more recently Wittgenstein's Tractatus or (differently) Carnap's logical positivism embody a philosophical effort at relegating into meaninglessness many of metaphysics' traditional questions and answers.

It is clear that claiming to rule out certain modes of thinking or certain classes of assertions entails having a certain model of what valid thinking is. The aim of this short note is to describe a specific claim to meaninglessness or unintelligibility as well as the explicit or implicit model of what thought is which underlies it, and to try in a cursory way to explain what we think a philosophical reflection on (aspects of ) thought could amount to. The claim we wish to discuss belongs to a family of claims expressing the doubts of mathematicians and philosophers concerning the legitimacy of the notion of mathematical infinity and the clarity of the grasp we have of it.

The notion of infinity has a peculiar dual status: in the hand of mathematicians it has been- especially since the end of the nineteenth century- an object of positive knowledge, yielding theorems and constructions; on the other hand it has been the focus of philosophical doubts and controversy. The philosophical discussions about the uses of infinity in mathematics often concentrate on what we may call an epistemological paradox: how is it possible that our finite minds have a grasp of notions of infinity, and how do we know this grasp is not illusory? This is an old question that resurfaced every time a significant new mathematical way of handling infinity appeared. We shall soon zoom on the specific example we wish to concentrate on, but in order to illustrate the recurring nature of the discussion let us first consider a quote from Bishop Berkeley's criticism of Newton's mathematics:

It must indeed be acknowledged the modern mathematicians do not consider these points as mysteries, but as clearly conceived and mastered by their comprehensive minds. They scruple not to say that by the help of these new analytics they can penetrate into infinity itself: that they can even extend their views beyond infinity: that their art comprehends not only infinite, but infinite of infinite (as they express it), or an infinity of infinites. But, notwithstanding all these assertions and pretensions, it may be justly questioned whether, as other men in other inquiries are often deceived by words or terms, so they likewise are not wonderfully deceived and deluded by their own peculiar signs, symbols or species.

George Berkeley, The Analyst: A discourse addressed to an infidel mathematician (1734)

The objections of Berkeley1 amount to asking for a  rigorous foundation for a mathematical algorithm (the differential calculus and its applications to physics). In a way those objections were taken very seriously by the mathematicians, even if not through Berkeley's influence. They answered them not by philosophizing but simply by producing (in the course of almost two hundred years of collective work) a logically impeccable foundation for the calculus, one in which the consideration of explicit infinities or infinitesimal was avoided. To give a flavor of the kind of analysis involved consider the formulation: "when  x becomes infinite, 1/x becomes zero", this statement refers directly to infinity, however when reworked in the 19-th century style of elucidation it becomes: "for any assigned finite positive non-zero quantity e there exists a finite quantity A such that for any x bigger than A, 1/x is smaller than e", now there is no mention of infinity and the second formulation can be seen as an arithmetic analysis of what was meant in the first. The second formulation is also considerably more cumbersome than the first, but this is only poetic justice, if you insist on getting rid of any mention of infinity this should cost you something!  However the rigorous arithmetization of analysis achieved by mathematicians during the nineteenth century and the correlative increasing independence of the notion of function from explicit means of computing led to Cantor's set theory and to a renewal of mathematical/philosophical polemics about infinity. This last polemics is the one we will concentrate upon.

We shall adopt the presentation proposed by Emile Borel (1871-1956) a French mathematician who made important contributions to the theory of real and complex functions, to measure theory and probability and, as we shall emphasize, to a reflection on foundational aspects of the study of infinity. He was one of the first promoters of an abstract, set theoretical, approach to function theory, and an early adopter of the ideas of Cantor2. On the other hand Borel wrote abundantly about methodological matters, criticizing in particular the unchecked reliance on transfinite methods and on set theory in general. Borel contributed to these discussions both in his mathematical production and in texts presenting methodological/philosophical reflections about the role of infinity in mathematics. In a pair of papers for the general philosophical public,3 he discusses what he takes to be the main problem concerning the new notions of infinity developed by G. Cantor two decades before.

Here, as Borel does in his own papers we shall have to digress briefly into some expository mathematics (nothing that can't be taken in by the mathematically innocent reader!).

Consider the set of natural numbers: 1,2,3,4 … We seem to understand it reasonably well, at least in the sense that we understand the mode of generation of the successive elements (add 1 repeatedly) and that this description itself gives us a clear grasp of the general notion of natural number. To put it in another way, we have an ordered sequence, infinite in the sense that there is no obstruction to continue it indefinitely (add 1, again and again!), and we agree (or should agree) that it is fundamentally right to apply to it the following principle of induction : if some property of natural numbers is true of 1 and if the property is such that if a number has it then that number plus one has it too, then we can safely conclude that every natural number has it.

For some thinkers the above principle of induction is in fact the defining property of the natural numbers. This, so far, is the good old notion of infinity, a structure in which there is a notion of order (5 is smaller than 100), and in which for any given element a bigger one can be found.

The "new infinite" can be described (although this is not Cantor's original formulation) through the following considerations:

Consider now a function f1, from the natural numbers to the natural numbers, that is a way of associating to each natural number another natural number. To start with a very simple example let us put f1(x) = x. In words: f1 is the function that associates to any number this number itself, thus f1(7)=7,   f1 (1789)=1789, etc…

Once we have f1 we can consider f2 defined by   f2 (x) = f1 (x)^. f1 (x), so that           

  f2 (7) = f1 (7)^. f1 (7) = 7^.7 = 49

We can see that for all but finitely many values of x  f1 (x) is smaller than f2 (x) (actually here this happens as soon as x is strictly bigger than 1). We shall say, in such a situation, that f1 is smaller than f2 or that f2 is bigger than f14. It is not difficult to guess what comes next: we can iterate this procedure and get f3

f3 (x) = f1 (x)^. f1 (x). f1 (x)

and similarly  f4, f5 etc…

So we have obtained a sequence of objects f1, f2, f3 … with an order relation (let us denote it by "<") such that f1< f2 < f3 < f4 < … Exactly as in the case of natural numbers for any function fn we can define a bigger function fn+1.  However here something new happens:  we can define a function f^* that is simultaneously bigger than all the fn's. This is done very simply by putting

   f1^*(x) = fx(x), so f^* associates to a number x the value that the function denoted by  fx associates to x. For instance:    

 f1^*(7) =  f7 (7) = f1 (7)^. f1 (7)^. f1 (7)^. f1 (7)^. f1 (7)^. f1 (7)^. f1 (7) = 7^.7^.7^.7^.7^.7^.7 = 7⁷ = 823543. You should be able to convince yourself that f1^* is bigger than any of the fn's: as soon as the number x is bigger than n, f1^* (x) is bigger than fn (x). Why does this introduce a new kind of infinity? Well, as can be seen from our construction, once we have f1^* nothing prevents us from defining

f2^*(x) = f1^*(x)^. f1^*(x)  ,   f3^*(x) = f1^*(x)^. f1^*(x)^. f1^*(x), etc. and while we are at it we can define f1^** which will dominate all the fn^*'s , and continuing that way  f1^{** *} ,f1^****, ...

But if we do that we shall obtain a sequence of functions each of which dominates the preceding one, and once again nothing can prevent us from defining f1^{super *}which will simultaneously dominate all of them. How long can this go on, and why is it disturbing? As can be seen from the structure of the construction of the dominating function, as long as the sequence of functions to be dominated can be indexed by natural numbers, we can apply the construction, exactly in analogy to the fact that as long as a number is finite we can add 1 to it. But if this is so we must pursue the analogy with the natural numbers and conclude that the set of functions that we can obtain through this procedure cannot be enumerated by means of the natural numbers (exactly analogous to the fact that the set of all natural numbers cannot be counted by a natural number). But, this means that some of the functions that will appear in this newly infinite sequence cannot be described by a procedure, indeed by any finite number of intelligible instructions for constructing the function. To see this, notice that the set of all finite sequences of instructions (say in English) can be fully enumerated by the natural numbers: think of the instructions as words in an alphabet that includes empty spaces, punctuation signs etc… and arrange them like words, in the usual sense, are arranged in the dictionary, this is an enumeration. But, in contrast, such an enumeration is impossible with the set of functions since we can always apply our f^* construction to such a purported enumeration, thus getting a new function which was not, after all, included in the enumeration. On the other hand the analogy to the natural numbers runs deep: in fact we can effectively compute only finitely many natural numbers, so our conviction that it is meaningful to speak of all the natural numbers must be founded on some leap of idealization. Following this line of thought we seem to reach the conclusion that in order to be coherent with ourselves we should also accept the leap to this new kind of infinity which cannot be enumerated by the set of all natural numbers. Borel is sharply conscious of the dilemma and summarizes the situation thus:

"Are we justified in introducing a new principle of induction, applying not only to the denumerable sequence of all integers, but to the non-denumerable sequence of all functions of increasing rate of growth?"5

His pragmatic conclusion is that we will gradually get to understand whether we must (and therefore can) admit this new intuition of the infinite in order to account for the objects and reasonings that will eventually appear in the work of mathematicians, and in particular in the work that is pertinent to the pursuit of empirical science (physics). If the thoughts related to the "new infinite" find an organic link to the more traditional parts of mathematics, then we will develop a new intuition, the new infinite will have become intelligible. Borel adds a commentary which will lead us to the philosophical conclusion of this short note:

"If this clarification ever obtains, the study of the means by which our mind will have reached it seems to me extremely interesting; I think there may have never been previous opportunities to observe the functioning of the human mind as it strives to conquer a new abstract notion and create at the same time a new mode of reasoning".

Thought, Borel seems to suggest, is not given in its totality to our introspection, not even at the level of the most fundamental principles, it is always open to the possibility of an extension of our present notion of "the thinkable".  If this is so, then any attempt, philosophical or otherwise, at legislating the boundaries of the conceivable, are bound to fail, and with them the model of thought that underlies the attempt. What task then would be left for a philosophical contribution to the study of thought, one that would neither be "regional" (like the various empirical disciplines I have enumerated at the beginning of this note) nor unduly sure of knowing ahead of times the limits of the thinkable? We believe that Borel's suggestion is very fruitful: philosophy could be a careful analysis of the process whereby (in history or in our own life) the unthinkable becomes thinkable, the most fundamental principles are extended, distorted, sometimes negated.