[FOM] CH and mathematics

[Neil Tennant]

Joe Shipman wrote (Sat, 19 Jan 2008 17:37:07 -0500):

> I don't see why our inability to know something should cast any doubt 
> on its definiteness. That's epistemological arrogance.

The anti-realist position that all truths are in principle knowable often
attracts this kind of criticism.

But the anti-realist would say that there's another kind of arrogance at
work behind the realist position (which is Shipman's position here). 
It is the *semantic* arrogance of thinking that one has a grasp of such
truth-conditions (or meaning) as make it possible for the
(allegedly determinate) truth-value of a sentence forever to elude
detection.

There appears to be a stalemate:

(1) The realist claims epistemic humility, but displays semantic
arrogance.

(2) The anti-realist claims semantic humility, but displays epistemic
arrogance.

The stalemate is best broken, I think, by arguing that the alleged
epistemic arrogance in (2) is only apparent. If there really are any
truth-value gaps---or "holes in the world"---then even an omnisicent
God would not be in a position to assert or deny the sentence in question. 
For omniscience is knowledge of *all truths*; and neither the sentence in
question, nor its negation, would be true.

Saying that mere mortals cannot do what even an omniscient God would not
be able to do hardly strikes one as arrogant. 

Nor need the anti-realist be lured into claiming, of any particular
sentence (such as CH) that it is an example of a sentence that is neither
true nor false. Indeed, the anti-realist will avoid asserting even the
bare existential, that there is such a sentence. For that would lead to
contradiction. The anti-realist denies the claim "Every sentence is
(determinately) either true or false", without asserting its (strictly
classical) consequence "Some sentence is neither true nor false".

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Neil W. Tennant

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