By V. H. Klenk (auth.)
Wittgenstein's comments on arithmetic haven't got the recogni tion they deserve; they've got for the main half been both missed, or disregarded as unworthy of the writer of the Tractatus and the I nvestiga tions. this is often unlucky, i think, and never in any respect reasonable, for those comments aren't purely relaxing examining, as even the cruelest critics have con ceded, but additionally a wealthy and actual resource of perception into the character of arithmetic. it's probably the truth that they're extra suggestive than systematic which has placed such a lot of humans off; there's not anything right here of formal derivation and extremely little try out even at sustained and arranged argumentation. The feedback are fragmentary and sometimes imprecise, if one doesn't realize the purpose at which they're directed. however, there's a lot the following that's strong, or even a pretty method atic and coherent account of arithmetic. What i've got attempted to do within the following pages is to reconstruct the procedure at the back of the customarily really disconnected statement, and to teach that after the idea emerges, many of the harsh feedback which has been directed opposed to those re marks is obvious to be with out beginning. this is often intended to be a sym pathetic account of Wittgenstein's perspectives on arithmetic, and that i desire that it'll at the least give a contribution to yet another interpreting and reassessment of his contributions to the philosophy of mathematics.
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In that case you do not get any contradiction. ''' (RFM II 80) Or again, "if a contradiction were now actually found in arithmetic - that would only prove that an arithmetic with such a contradiction in it could render very good service; ... " (RFM V 28) One might even want to produce a contradiction for certain purposes, he suggests; a contradiction could conceivably have some use. (RFM II 81, 82; V 8, 12) In any case, Wittgenstein asks, how much is really gained by the discovery of a consistency proof?
RFM III 24) Very ordinary mathematical procedures depend also on the stability of the physical world; we could not measure if it weren't for the fact that "all rulers are made alike; they don't alter much in length; nor do pieces of wood cut up into inch lengths; ... " (RFM V 2) And of course, if objects simply appeared and disappeared at random, any kind of counting would be impossible. Despite the importance of these empirical regularities, however, Wittgenstein does not consider mathematical propositions to be empirical.
I should like to ask something like: 'Is it usefulness you are out for in your calculus? - In that case you do not get any contradiction. ''' (RFM II 80) Or again, "if a contradiction were now actually found in arithmetic - that would only prove that an arithmetic with such a contradiction in it could render very good service; ... " (RFM V 28) One might even want to produce a contradiction for certain purposes, he suggests; a contradiction could conceivably have some use. (RFM II 81, 82; V 8, 12) In any case, Wittgenstein asks, how much is really gained by the discovery of a consistency proof?