We address the question:

Exactly what significance should one attach to the semantic and logical paradoxes?

Now, the practical significance of the semantic and logical paradoxes is, of course, that they illustrate the absurd extent to which languages of common discourse need to tolerate ambiguity; both for ease of expression and for practical—even if not theoretically unambiguous and effective—communication in non-critical cases amongst intelligences capable of a lingua franca.

Such absurdity is highlighted by the universal appreciation of Charles Dickens’ Mr. Bumble’s retort that `The law is an ass’; a quote oft used to refer to the absurdities which sometimes surface in cases when judicial pronouncements attempt to resolve an ambiguity by subjective fiat that appeals to the powers—and duties—bestowed upon the judicial authority for the practical resolution of precisely such an ambiguity, even when the ambiguity may be theoretically irresolvable!

So, perhaps we need to make our question somewhat more specific:

Exactly what significance should one attach to the semantic and logical paradoxes in a mathematical language?

In a thought-provoking Opinion piece, `Desperately Seeking Mathematical Truth’, in the August 2008 Notices of the American Mathematical Society, Melvyn B. Nathanson seeks to highlight the significance (and perils) for the mathematical sciences when similar authority for the resolution of ambiguity is vested by society (albeit tacitly) upon academic `bosses’—a reference, presumably, to the collective of reputed (and undisputably respected) experts in any field of human endeavour:

` … many great and important theorems don’t actually have proofs. They have sketches of proofs, outlines of arguments, hints and intuitions that were obvious to the author (at least, at the time of writing) and that, hopefully, are understood and believed by some part of the mathematical community.

But the community itself is tiny. In most fields of mathematics there are few experts. Indeed, there are very few active research mathematicians in the world, and many important problems, so the ratio of the number of mathematicians to the number of problems is small. In every field, there are `bosses’ who proclaim the correctness or incorrectness of a new result, and its importance or unimportance.

Sometimes they disagree, like gang leaders fighting over turf. In any case, there is a web of semi-proved theorems throughout mathematics. Our knowledge of the truth of a theorem depends on the correctness of its proof and on the correctness of all of the theorems used in its proof. It is a shaky foundation.’

Nathanson’s comments are intriguing, because addressing such ambiguity in critical cases—such as communication between mechanical artefacts, or a putative communication between terrestrial and extra-terrestrial intelligences [1] —is the very raison d’etre of mathematical activity!

Such activity we view,:

1. First, as the construction of richer and richer mathematical languages (such as, for instance, the first order set theory ZFC) that can express those of our abstract concepts which can be subjectively addressed unambiguously; and

2. Thereafter, the study of the ability of the mathematical languages (such as, for instance, the first order Peano Arithmetic PA) to precisely express and objectively communicate these concepts effectively.

Our perspective on the expressions `subjectively address unambiguously’ and `objectively communicate effectively’ is broadly that:

(a) By `subjectively address unambiguously’ we intend in this context that there is essentially a subjective acceptance of identity by us between an `original’ abstract concept in our mind that we intended to express symbolically in a language, and the `interpreted’ abstract concept created in our mind each time we subsequently attempt to understand the import of the symbolic expression.

(b) By `objectively communicate effectively’ we intend in this context that there is essentially:

(i) first, an objective acceptance of identity by another mind between the `original’ abstract concept created in the other mind when first attempting to understand the import of what we have expressed symbolically in a language, and the `interpreted’ abstract concept created in the other mind each time it subsequently attempts to understand the import of the symbolic expression; and

(ii) second, an objective acceptance of functional identity between abstract concepts that can be `objectively communicated effectively’ based on the evidence provided by a commonly accepted `agency’ as, for instance the view—gaining currency today [2] —that a simple functional language can be used for specifying evidence for propositions in a constructive logic.

The question arises:

Do the semantic and logical paradoxes reflect an identifiable ambiguity in our initial expression and subsequent interpretation of a formal mathematical language and, if so, is the ambiguity removable?


Return to 1: I address the criticality of Sagan’s Thesis in this presentation.

Return to 2: For instance see Chetan R. Murthy, 1991, An Evaluation Semantics for Classical Proofs, Proceedings of Sixth IEEE Symposium on Logic in Computer Science, pp. 96-109, (also Cornell TR 91-1213), 1991.

Bhupinder Singh Anand