So where exactly does the buck stop?
Another reason why Lucas and Penrose should not be faulted for continuing to believe in their well-known Gödelian arguments against computationalism lies in the lack of an adequate consensus on the concept of `effective computability’.
For instance, Boolos, Burgess and Jeffrey (2003: Computability and Logic, 4th ed.~CUP, p37) define a diagonal halting function, , any value of which can be computed effectively, although there is no single algorithm that can effectively compute .
“According to Turing’s Thesis, since is not Turing-computable, cannot be effectively computable. Why not? After all, although no Turing machine computes the function , we were able to compute at least its first few values, For since, as we have noted, the empty function we have . And it may seem that we can actually compute for any positive integer —if we don’t run out of time.”
… ibid. 2003. p37.
Now, the straightforward way of expressing this phenomenon should be to say that there are well-defined real numbers that are instantiationally computable, but not algorithmically computable.
Yet, following Church and Turing, such functions are labeled as effectively uncomputable!
The issue here seems to be that, when using language to express the abstract objects of our individual, and common, mental `concept spaces’, we use the word `exists’ loosely in three senses, without making explicit distinctions between them.
First, we may mean that an individually conceivable object exists, within a language , if it lies within the range of the variables of . The existence of such objects is necessarily derived from the grammar, and rules of construction, of the appropriate constant terms of the language—generally finitary in recursively defined languages—and can be termed as constructive in by definition.
Second, we may mean that an individually conceivable object exists, under a formal interpretation of in another formal language, say ′, if it lies within the range of a variable of under the interpretation.
Again, the existence of such an object in ′ is necessarily derivable from the grammar, and rules of construction, of the appropriate constant terms of ′, and can be termed as constructive in ′ by definition.
Third, we may mean that an individually conceivable object exists, in an interpretation of , if it lies within the range of an interpreted variable of , where is a Platonic interpretation of in an individual’s subjective mental conception (in Brouwer’s sense).
Clearly, the debatable issue is the third case.
So the question is whether we can—and, if so, how we may—correspond the Platonically conceivable objects of various individual interpretations of , say , ′, ′′, …, unambiguously to the mathematical objects that are definable as the constant terms of .
If we can achieve this, we can then attempt to relate to a common external world and try to communicate effectively about our individual mental concepts of the world that we accept as lying, by consensus, in a common, Platonic, `concept-space’.
For mathematical languages, such a common `concept-space’ is implicitly accepted as the collection of individual intuitive, Platonically conceivable, perceptions—, ′, ′′, …,—of the standard intuitive interpretation, say , of Dedekind’s axiomatic formulation of the Peano Postulates.
Reasonably, if we intend a language or a set of languages to be adequate, first, for the expression of the abstract concepts of collective individual consciousnesses, and, second, for the unambiguous and effective communication of those of such concepts that we can accept as lying within our common concept-space, then we need to give effective guidelines for determining the Platonically conceivable mathematical objects of an individual perception of that we can agree upon, by common consensus, as corresponding to the constants (mathematical objects) definable within the language.
Now, in the case of mathematical languages in standard expositions of classical theory, this role is sought to be filled by the Church-Turing Thesis (CT). Its standard formulation postulates that every number-theoretic function (or relation, treated as a Boolean function) of , which can intuitively be termed as effectively computable, is partial recursive / Turing-computable.
However, CT does not succeed in its objective completely.
Thus, even if we accept CT, we still cannot conclude that we have specified explicitly that the domain of consists of only constructive mathematical objects that can be represented in the most basic of our formal mathematical languages, namely, first-order Peano Arithmetic (PA) and Recursive Arithmetic (RA).
The reason seems to be that CT is postulated as a strong identity, which, prima facie, goes beyond the minimum requirements for the correspondence between the Platonically conceivable mathematical objects of and those of PA and RA.
“We now define the notion, already discussed, of an effectively calculable function of positive integers by identifying it with the notion of a recursive function of positive integers.”
… Church 1936: An unsolvable problem of elementary number theory, Am.~J.~Math., Vol.~58, pp.~345–363.
“The theorem that all effectively calculable sequences are computable and its converse are proved below in outline.
… Turing 1936: On computable numbers, with an application to the Entscheidungsproblem, Proceedings of the London Mathematical Society, ser.~2.~vol.~42 (1936–7), pp.~230–265.
This violation of the principle of Occam’s Razor is highlighted if we note (e.g., Gödel 1931: On undecidable propositions of Principia Mathematica and related systems I, Theorem VII) that, pedantically, every recursive function (or relation) is not shown as identical to a unique arithmetical function (or relation), but (see the comment following Lemma 9 of this paper) only as instantiationally equivalent to an infinity of arithmetical functions (or relations).
Now, the standard form of CT only postulates algorithmically computable number-theoretic functions of as effectively computable.
It overlooks the possibility that there may be number-theoretic functions and relations which are effectively computable / decidable instantiationally in a Tarskian sense, but not algorithmically.
BBJ03 George S. Boolos, John P. Burgess, Richard C. Jeffrey. 2003. Computability and Logic. (4th ed). Cambridge University Press, Cambridge.
Go31 Kurt Gödel. 1931. On formally undecidable propositions of Principia Mathematica and related systems I. Translated by Elliott Mendelson. In M. Davis (ed.). 1965. The Undecidable. Raven Press, New York. pp.5-38.
Lu61 John Randolph Lucas. 1961. Minds, Machines and Gödel. In Philosophy. Vol. 36, No. 137 (Apr. – Jul., 1961), pp. 112-127, Cambridge University Press.
Lu03 John Randolph Lucas. 2003. The Gödelian Argument: Turn Over the Page. In Etica & Politica / Ethics & Politics, 2003, 1.
Lu06 John Randolph Lucas. 2006. Reason and Reality. Edited by Charles Tandy. Ria University Press, Palo Alto, California.
Me64 Elliott Mendelson. 1964. Introduction to Mathematical Logic. Van Norstrand. pp.145-146.
Pe90 Roger Penrose. 1990. The Emperor’s New Mind: Concerning Computers, Minds and the Laws of Physics. 1990, Vintage edition. Oxford University Press.
Pe94 Roger Penrose. 1994. Shadows of the Mind: A Search for the Missing Science of Consciousness. Oxford University Press.
Sc67 Joseph R. Schoenfield. 1967. Mathematical Logic. Reprinted 2001. A. K. Peters Ltd., Massachusetts.
Ta33 Alfred Tarski. 1933. The concept of truth in the languages of the deductive sciences. In Logic, Semantics, Metamathematics, papers from 1923 to 1938. (p152-278). ed. John Corcoran. 1983. Hackett Publishing Company, Indianapolis.
Wa63 Hao Wang. 1963. A survey of Mathematical Logic. North Holland Publishing Company, Amsterdam.
An12 … 2012. Evidence-Based Interpretations of PA. In Proceedings of the Symposium on Computational Philosophy at the AISB/IACAP World Congress 2012-Alan Turing 2012, 2-6 July 2012, University of Birmingham, Birmingham, UK.