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+"Normally, when we consider a mathematical rule such as addition, we think of ourselves as guided in our application of it to each new instance. Just this is the difference between someone who computes new values of a function and someone who calls out numbers at random. Given my past intentions regarding the symbol '+', one and only one answer is dictated as the one appropriate to '68+57'. On the other hand, although an intelligence tester may suppose that there is only one possible continuation to the sequence 2, 4, 6, 8,…, mathematical and philosophical sophisticates know that an indefinite number of rules (even rules stated in terms of mathematical functions as conventional as ordinary polynomials) are compatible with any such finite initial segment. So if the tester urges me to respond, after 2, 4, 6, 8, . . ., with the unique appropriate next number, the proper response is that no such unique number exists, nor is there any unique (rule determined) infinite sequence that continues the given one. The problem can then be put this way: Did I myself, in the directions for the future that I gave myself regarding ~~plus~~ '+', really differ from the intelligence tester? True, I may not merely stipulate that ~~plus~~ '+' is to be a function instantiated by a finite number of computations. In addition, I may give myself directions for the further computation of ~~plus~~ '+', stated in terms of other functions and rules. In turn, I may give myself directions for the further computation of these functions and rules, and so on. Eventually, however, the process must stop, with 'ultimate' functions and rules that I have stipulated for myself only by a finite number of examples, just as in the intelligence test. If so, is not my procedure as arbitrary as that of the man who guesses the continuation of the intelligence test? In what sense is my actual computation procedure, following an algorithm that yields '125', more justified by my past instructions than an alternative procedure that would have resulted in '5'? Am I not simply following an unjustifiable impulse?" Of course, these problems apply throughout language and are not confined to mathematical examples, though it is with mathematical examples that they can be most smoothly brought out. I think that I have learned the term 'table' in such a way that it will ~~to~~ apply to indefinitely many future items. So I can apply the term to a new situation, say when I enter the Eiffel Tower for the first time and see a table at the base. Can I answer a sceptic who supposes that by `table' in the past I meant tabair, where a 'tabair' is anything that is a table not found at the base of the Eiffel Tower, or a chair found there? Did I think explicitly of the Eiffel Tower when I first `grasped the concept of' a table, gave myself directions for what I meant by `table'? And even if I did think of the Tower, cannot any directions I gave myself mentioning it be reinterpreted compatibly with the sceptic's hypothesis? Most important for the 'private language' argument, the point of course applies to predicates o f sensations, visual impressions, and the like, as well: "How do I know that in working out the series -f 2 I must write " 20,004, 20,006" and not " 20,004, 20,008" ? - (The question: "How do I know that this color is-'red'?" is similar.)" (Remarks on the Foundations ofMathematics, I, §3.) The passage strikingly illustrates a central thesis of this essay: that Wittgenstein regards the fundamental problems o f the philo sophy of mathematics and of the 'private language argument' - the problem of sensation language ~~ as at root identical, stemming from his paradox. The whole o f §3 is a succinct and beautiful statement o f the Wittgensteinian paradox; indeed the whole initial section of part I of Remarks' on the Foundations of Mathematics is a development o f the problem with special reference to mathematics and logical inference. It has been supposed that all Ineed to do to determine my use ofthe word 'green' is to have an image, a sample, of green that I bring to mind whenever I apply the word in the future. When I use this tojustify my application of'green'to anew object, should not the sceptical problem be obvious to any reader of Goodman?14 Perhaps by 'green', in the past I meant grue,15 and the color image, which indeed was grue, was meant to direct me to apply the word 'green' tourne objects always. If the blue object before me now is grue, then it falls in the extension o f'green', as I meant it in the past. It is no help to suppose that in the past I stipulated that 'green' was to apply to all and only those things 'of the same color as' the sample. The sceptic can reinterpret 'same color' as same schmolor, l 6 where things have the same schmolor if . . ." (1hhhhhhhhhhhhhhhhhhhhhhh7-20) |