Dummett's Backward Road to Frege and to Intuitionism - Tripod
Dummett's Backward Road to Frege and to Intuitionism - Tripod Dummett's Backward Road to Frege and to Intuitionism - Tripod
the identity relation. Here the leap is between not wholly distinct entities in the strong sense that primitive references cannot fail to exist if the senses reflecting their nature exist, entailing symmetric ontological dependencies. This is the later Frege’s implicit atomistic version of the Parmenidean thesis that the rational is the real and the real is the rational, though Frege has a more general and somewhat different version in his thesis that every true or false sentence must have a referring logical subject (1970f: 62). It is in any case the later Frege’s implicit version of his own earlier thesis that even some complex logical entities, namely numbers, are transparent to the reason (Grundlagen, § 105). 34 7. The consequences for intuitionism By parity of reason, there is no backward road from truth to propositional sense. 35 This is fatal to intuitionism, and to strong verificationism. It does not matter whether a propositional sense describes a truth-condition, or whether it is logically tied to a truth-condition at all. What matters is that the truth- value, truth, is propositionally presented in indefinitely many ways in the order of cognition. Thus there is no backward road from truth to propositional sense in the order of cognition, any more than there is from truth-conditions. We cannot squeeze an intension in sense (2) out of all the extensions in sense (2) in the world, even including assignments of truth-values among the extensions. My point is so general, it does not even matter if the assignments of truth-values are based on proofs or not. We may say that the truth of a statement underdetermines the thought it expresses. A second problem of intensionality in sense (2) arises more specifically for intuitionism in that there can be many different proofs of the same thesis, so that there is no backward road from the truth of a thesis to any one proof as the sense the thesis expresses. As Frege says, “Frequently several routes for a proof are open” (Grundgesetze, p. 3; see Dummett 1981: 634). Thus we may say the truth of a thesis underdetermines its proof. 62
We cannot need to be guaranteed to discover an object in order to understand the sense its name expresses, since we cannot need to single out that object in order to understand that sense. For an objectual inquiry to be genuine, it must be at least zetetically, i.e., investigationally, possible for the object to turn out to have properties other than those we think it has, or not to exist at all. By parity of reason, we cannot need to know how to prove that a statement is true in order to understand its sense, because we cannot need to have the truth in order to understand its sense. For the inquiry to be genuine, it must be at least zetetically possible for the statement to turn out to be false. These two points coalesce for Frege, for whom truth is an object. I need not add that the order of investigation is a form of the order of cognition. This is the problem of Plato’s Meno. How can it make sense to inquire about anything if the object of inquiry must already visibly contain the answer? What would we be discovering? The problem of how informative inquiry is possible is more generally the problem of how informative identity statements are possible. Thus the Meno problem is more generally Frege’s puzzle. And for Frege, for a statement “to be informative—to have ‘cognitive value’—it is necessary that a mere knowledge of its sense be insufficient to guarantee a recognition of its truth” (Dummett 1981: 228). The burden is on the intuitionist to explain why the Meno problem and Frege’s puzzle are not genuine puzzles. Can an intuitionist find any sense of discovery in moving from having an effective means for obtaining a proof (“informal proof” or “demonstration”) to obtaining a proof (“canonical proof”) (Dummett 2000: 270–71)? No, this is illegitimate on intuitionism’s own showing. For the expressions “an effective means” and “a proof” are existential generalizations. And according to intuitionism, the only way to prove an existential generalization is to prove that there is an instance. 36 Thus the only way to prove “There is a proof,” “There is an effective means to a proof,” or even “This is an effective means to some proof,” so that these assertions have meaning, is to produce an actual proof. Thus there 63
- Page 11 and 12: eferences in a given sentence. The
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We cannot need <strong>to</strong> be guaranteed <strong>to</strong> discover an object in order <strong>to</strong> underst<strong>and</strong> the sense its name<br />
expresses, since we cannot need <strong>to</strong> single out that object in order <strong>to</strong> underst<strong>and</strong> that sense. For an<br />
objectual inquiry <strong>to</strong> be genuine, it must be at least zetetically, i.e., investigationally, possible for the<br />
object <strong>to</strong> turn out <strong>to</strong> have properties other than those we think it has, or not <strong>to</strong> exist at all. By parity of<br />
reason, we cannot need <strong>to</strong> know how <strong>to</strong> prove that a statement is true in order <strong>to</strong> underst<strong>and</strong> its sense,<br />
because we cannot need <strong>to</strong> have the truth in order <strong>to</strong> underst<strong>and</strong> its sense. For the inquiry <strong>to</strong> be genuine,<br />
it must be at least zetetically possible for the statement <strong>to</strong> turn out <strong>to</strong> be false. These two points<br />
coalesce for <strong>Frege</strong>, for whom truth is an object. I need not add that the order of investigation is a form<br />
of the order of cognition.<br />
This is the problem of Pla<strong>to</strong>’s Meno. How can it make sense <strong>to</strong> inquire about anything if the<br />
object of inquiry must already visibly contain the answer? What would we be discovering?<br />
The problem of how informative inquiry is possible is more generally the problem of how<br />
informative identity statements are possible. Thus the Meno problem is more generally <strong>Frege</strong>’s puzzle.<br />
And for <strong>Frege</strong>, for a statement “<strong>to</strong> be informative—<strong>to</strong> have ‘cognitive value’—it is necessary that a<br />
mere knowledge of its sense be insufficient <strong>to</strong> guarantee a recognition of its truth” (Dummett 1981:<br />
228). The burden is on the intuitionist <strong>to</strong> explain why the Meno problem <strong>and</strong> <strong>Frege</strong>’s puzzle are not<br />
genuine puzzles.<br />
Can an intuitionist find any sense of discovery in moving from having an effective means for<br />
obtaining a proof (“informal proof” or “demonstration”) <strong>to</strong> obtaining a proof (“canonical proof”)<br />
(Dummett 2000: 270–71)? No, this is illegitimate on intuitionism’s own showing. For the expressions<br />
“an effective means” <strong>and</strong> “a proof” are existential generalizations. And according <strong>to</strong> intuitionism, the<br />
only way <strong>to</strong> prove an existential generalization is <strong>to</strong> prove that there is an instance. 36 Thus the only way<br />
<strong>to</strong> prove “There is a proof,” “There is an effective means <strong>to</strong> a proof,” or even “This is an effective<br />
means <strong>to</strong> some proof,” so that these assertions have meaning, is <strong>to</strong> produce an actual proof. Thus there<br />
63
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