In Realistic Rationalism, Jerrold Katz develops a new philosophical position integrating realism and rationalism. Realism here means that the objects of study in mathematics and other formal sciences are abstract; rationalism means that our knowledge of them is not empirical.
The Semantic Challenge to Realism 99 triples of numbers that are the plus function and those that are the quus function, but, as the terms plus and quus have to be nonsynonymous expressions that respectively denote those functions, paradoxically, the functions are semantically distinguishable and the reference of sixty-eight plus fty-seven determinate.
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4.3 The General Form of Indeterminacy Arguments The common pattern of reasoning that we have found in Quines and Kripkes indeterminacy arguments is the general form of such arguments. They begin with an explicit or implicit account of our informal knowledge of some domain, typically embodied in the skeptics representation of the situation we face in relation to theories of the domain. The skeptic assumes the account to be a complete representation of that knowledge at least insofar as issues of indeterminacy are concerned, and, on the basis of this assumption, the skeptic argues for an unbreakable symmetry between the intended interpretation and certain deviant interpretations of the theory which precludes our ruling out the deviant interpretations.
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At this point, indeterminacy arguments go their separate ways. Those that are put forth as skeptical arguments, such as Quines, proceed from there being no way to break the symmetry to the conclusion that no interpretation (e.g., translation) is correct because there is no fact of the matter. Those that are not put forth as skeptical arguments, such as Kripkes, stop with the symmetry puzzle and simply ask what is to be done. Going for the robust skeptical conclusion, however, involves a dubious inferential step. As I (1988; 1990b, 183) argued against Quine, symmetry at best delivers the unknowability of language neutral meanings, so unknowable meanings, like Kants noumena, could still exist. If realism is rightand it hasnt been ruled outit is absurd to think that existence of objects in a realm that is independent of us depends in any way on our cognitive powers. The skeptic must do something to shore up the nal step in the argument, but the available props themselves are in need o
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