a reply to professor abelson

A Reply to Professor AbelsonAuthor(s): Michael LockwoodSource: Philosophical Studies: An International Journal for Philosophy in the AnalyticTradition, Vol. 24, No. 2 (Mar., 1973), pp. 133-135Published by: SpringerStable URL: http://www.jstor.org/stable/4318774 .

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MICHAEL LOCKWOOD

A REPLY TO PROFESSOR ABELSON

(Received 4 November, 1971)

In a recent article in Philosophical Studies, Raziel Abelson claims to have refuted the identity thesis, according to which every mental state is identical with some state of the brain. His argument rests on the premise that a human being has the capacity to think of literally any natural number, no matter how large. It seems to me that Professor Abelson has failed to establish this premise. I shall argue that there are no sufficient grounds for supposing that we do have the capacity to think of numbers of unlimited magnitude - in any sense, at least, that is incompatible with the truth of the identity thesis.

The essence of my objection is as follows. Abelson would no doubt admit' that, with respect to any specific notation, there will be numbers which are simply too large for one's mind to handle. His claim that one can think of absolutely any number therefore rests upon the possibility of switching to successively more compact notations, which will render manageable numbers that, otherwise represented, are simply beyond our intellectual grasp. What Abelson seems not to have noticed is that one's grasp of successively more compact notations itself makes progressively greater demands upon one's intellect. For each makes use of progressively more complex mathematical functions.2 And it is arguable that if a per- son does not genuinely understand the mathematical principle upon which a given notation is based, he cannot genuinely think of numbers with its aid. All he can do is contemplate symbols that have no meaning for him. So my question is this: Why could there not, in principle, be numbers so large that the simplest notation capable of representing them with the requisite degree of compactness was itself based upon a mathematical principle of such staggering complexity that one's intellect was simply incapable of grasping it? I am temped to argue that there must be such numbers, although I shall not press this stronger claim here.

It is, admittedly, somewhat unclear just exactly what Abelson means by

Philosophical Studies 24 (1973) 133-135. All Rights Reserved Copyright 0 1973 by D. Reidel Publishing Company, Dordrecht-Holland



134 MICHAEL LOCKWOOD

"thinking of a number". And it might be argued that, in the line of rea- soning just presented, I have been construing this notion too narrowly. There are, as far as I can see, two weaker senses of 'thinking of a number' which might be held relevant to the present discussion.

First, one might think of a number serially. That is to say, one might build it up, in stages, by a chain of connected cognitive acts. Thus, we might say to ourselves: 'Three billion, multiplied by a trillion, raised to the power sixteen million, etc. etc.' Even with a notation which enabled us, at each stage of this process, to achieve the most that was compatible with knowing what we were doing, the intervention of death would of course ultimately prevent our defining numbers of unlimited magnitude. But this might be counted irrelevant. Note, however, that failing unlimited power to remember what we had already done, we should not have a gestalt with respect to any very large number thus 'thought of'.

If what I have just envisaged were to be counted as a case of thinking of a number, then, given sufficient time, perhaps we could, in this sense, think of literally any number. But this is no longer inconsistent with the supposition that our minds are capable of assuming only a finite number of distinct possible states. For what is now involved is a chain of states or events. And since there is no limit to the length of the chain (except that imposed by our mortality), it does not matter that at each stage there is only a finite number of possible states - or transitions from one state of another - that can occur. The situation is roughly analogous to that of a language. It may have a finite vocubulary. But, provided there is no upper limit on sentence length, there will be an infinity of distinct possible sentences which belong to that language.

The second possible weaker sense of thinking of a number, which bears consideration, is this. One might think of a number under some wholly or partially non-mathematical description. For example, one might think of 'the number written down on this piece of paper' or 'the number just generated by Columbia University's new computer, raised to the power six billion'. With respect to some given number, too large to grasp under any purely arithmatic characterization, there might well be some description or other under which, in this weak sense, one could think of the number. (Sheer numerical magnitude is not at issue here quite as it was previously.) But there is still no reason to suppose that, at any given time, there is for any arbitrarily chosen number a description simple enough



A REPLY TO PROFESSOR ABELSON 135

for one to grasp which uniquely identifies the number in question. Of course, the point of appealing to non-mathematical descriptions

might be differently conceived. Think, for example, of possible computer outputs. Any number whatever might be, say, the largest number yet generated by a computer, or the largest number yet generated by a computer raised to the power six billion. And if so, one could obviously think of it as such. But clearly it can be true, in this latter sense, that any number is a possible object of thought, without our minds having the capacity to assume any of an infinity of possible states. For what is being envisaged is the possibility that any of an infinity of different numbers might satisfy some single given description containing token-reflexive elements. And when one thinks of an object under a description, one's state of mind is a function of the description itself, not of the object which happens to satisfy it.

In conclusion, it seems to me that Professor Abelson has not established that we really do possess the capacity to think of numbers of unlimited magnitude 3, in any sense that is inconsistent with supposing that the workings of our mind are identical with the operations of a Turing machine with only a finite number of possible states (such as the human brain is frequently posulated to be). What his argument does show is that i the human mind can be thus regarded, then with respect to any given human being there are intellectual tasks which are, in principle, beyond his capacity. The consequent of this conditional may be false. But Professor Abelson has not, I think, demonstrated its falsity. And until he does, he cannot justifiably claim to have refuted the identity thesis.

New York University

NOTES

He has admitted this in conversation. 2 Or, if not exactly more complex, involving numbers which themselves would ultimately become unmanageably immense. I am thinking, for example, of progressively larger integral bases. 3 I have been thinking, here, merely in terms of the natural numbers. Similar arguments could be used to cast doubt on supposition that we could ,for example, think of any of the infinity of rational or real numbers between, say, zero and one.



a reply to professor abelson

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