Computation, Logic, Philosophy: A Collection of Essays by Professor Hao Wang (auth.)

By Professor Hao Wang (auth.)

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It seems necessary not only to reject the whole argument as a unit but to pin down exactly which step or steps in the argument caused the trouble. Hence, there are the various attempts to reject one or another of the steps as unwarranted. But why can we not say that although each step is in itself all right, they must not be combined in the particular way that leads to the contradiction? Indeed, we may even use this possibility to justify the attitude of indifference, on the part of many working mathematicians, toward the paradoxes.

Here, in fact, it is not even clear that the methods of proof are consistent. The bounds for proofs in Zil can be obtained from Ackermann 1940 in the following manner. Thus, suppose (x)(Ey) Rxy (R primitive recursive) has been proved in Zw Let g, e. j be respectively the number of 8-matrices, the number of 8-terms, and highest degree of terms in the proof. Let further w, r, p, ). be the functions as defined in Ackermann 1940. Then the bound f3(x) is given by: w{x + j, )[r(j,e,g,l, p(g,e)), g + I]}.

For instance, Heyting's formalization of the intuitionistic view oflogic and mathematics helps quite a bit in conveying Brouwer's ideas to those people who have a radically different orientation. Another example is the gradual formalization of the notion of being a definite property, employed for defining sets in Zermelo's axiomatic treatment of set theory. Perhaps we can compare many of the attempts to formalize with the use of an airplane to visit a friend living in the same town. Unless you simply love the airplane ride and want to use the visit as an excuse for having a good time in the air, the procedure would be quite pointless and extremely inconvenient.

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