By Professor Hao Wang (auth.)
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Extra resources for Computation, Logic, Philosophy: A Collection of Essays
The usual procedure is to assume given the totality of computable functions and then "diagonalize" to get a function that is not computable. We need not go into the familiar argument, as noncomputable functions got in this manner are clearly of no direct assistance to the task of determining the totality of computable functions. We are left with the alternative of analysing given computable functions. Since addition and multiplication are the most familiar computable functions, it is natural to study first the rules governing the algorithms.
Indeed, we may even use this possibility to justify the attitude of indifference, on the part of many working mathematicians, toward the paradoxes. It is only when we come to constructing a formal system to embody our arguments that this procedure proves awkward. In a logistic system, we break up proofs and arguments into isolated steps so that if a step is valid at all, it is valid no matter where it occurs. In other words, certain combinations of shapes are taken as axioms so that they can be asserted as valid no matter where they occur; and certain (finite) sequences of combinations of shapes are taken as justified by the rules of inference so that any such sequence, wherever it occurs, is taken as determining valid steps.
Similarly, there are many different chairs which can all be employed to support buttocks. In this way formalization, closely tied up with abstraction, produces useful tools. On the other hand, it is often hard to characterize adequately our intuition through the use of formal structures. , of a particular chair. Peano's axioms are thought to be capable of characterizing completely our notion of positive integers. Yet, as Russell observed long ago, Peano's axioms are satisfied by all progressions such as the odd positive integers, the negative integers.
Computation, Logic, Philosophy: A Collection of Essays by Professor Hao Wang (auth.)