Godel’s Argument[119]
In 1951 Godel held one of the prestigious Gibbs Lectures for the American Mathematical Society. The title of his lecture was Some basic theorems on the foundations of mathematics and their implications.
The theorems in question were precisely those of incompleteness, and the philosophical implications concerned the nature of mathematics and the abilities of the human mind (Godel 1951).[120] This was one of the few official occasions in which Godel expounded his opinion on the philosophical implications of his theorems. Without going into details about Godel’s paper, what is interesting here is the first part, where he derives the thesis of essential incompleteness of mathematics from his famous theorems. Such a thesis was sanctioned by the second theorem. Godel’s idea is that if one perceives with absolute certainty that a certain formal system[121] is correct (sound), s/he will also know the consistency of the system, that is, s/he will know the truth of the statement which establishes the consistency of the system itself. But, by Godel’s second theorem, the formal system considered cannot prove its own assertion of consistency, therefore the system does not capture all arithmetical truths, and for this reason “if one makes such a statement he contradicts himself” (1951: 309). But what does all of this mean? Does it mean perhaps that a well defined system of correct (sound) axioms cannot contain all that is strictly mathematical? Godel believes that such a question has two possible answers:It does, if by mathematics proper is understood the system of all true mathematical propositions; it does not, however if someone understands by it the system of all demonstrable mathematical propositions. [...] Evidently no well-defined system of correct axioms can comprise all [of] objective mathematics, since the proposition which states the consistency of the system is true, but not demonstrable in the system.
However, as to subjective mathematics it is not precluded that there should exist a finite rule producing all its evident axioms. However, if such a rule exists, we with our human understanding could certainly never know it to be such, that is, we could never know with mathematical certainty that all the propositions it produces are correct; or in other terms, we could perceive to be true only one proposition after the other, for any finite number of them. The assertion, however, that they are all true could at most be known with empirical certainty, on the basis of a sufficient number of instances or by other inductive inferences. If it were so, this would mean that the human mind (in the realm of pure mathematics) is equivalent to a finite machine that, however, is unable to understand completely its own functioning. This inability [of man] to understand himself would then wrongly appear to him as its [(the mind’s)] boundlessness or inexhaustibility (Godel 1951: 309-310).Therefore, not only does the previous question pose the problem of the inexhaustibility or incompleteness of mathematics considered as the totality of all true mathematical propositions; but it also raises the question as to whether mathematics is in principle inexhaustible for the human mind, that is to say, whether the human mind’s demonstrative abilities are extensionally equivalent to a certain formal system, or to the Turing Machine (TM) connected to it (the TM which enumerates the set of theorems of the corresponding formal system).
The question, then, requires due consideration precisely of the relation between what Godel calls objective and subjective mathematics. First let T be the set of mathematical truths expressible within first-order arithmetic, and call this ‘objective arithmetic’, or following Godel, ‘objective mathematics’, that is “the body of
those mathematical propositions which hold in an absolute sense, without any further hypothesis”.[122] By Tarski’s theorem T is not definable within the language of arithmetic, hence T is not recursively enumerable.
Let us then define K as the set of arithmetical statements which a human being can know and prove absolutely and with mathematical certainty, that is what one can derive[123] and know to be true. Let us call it ‘subjective arithmetic’ or, following Godel, ‘subjective mathematics’, which “consists of all those theorems whose truth is demonstrable in some well- defined system of axioms all of whose axioms are recognized to be objective truths and whose rules preserve objective truth” (Feferman 2006: 135-136). What is then the relation between K and T?Quoting Feferman we could synthesize Godel’s answer by saying: if K was equal to T “then demonstrations in subjective mathematics [would not be] confined to any one system of axioms and rules, though each piece of mathematics is justified by some such system. If they do not, then there are objective truths that can never be humanly demonstrated, and those constitute absolutely unsolvable problems” (Feferman 2006: 136-137). That is, if the equivalence K = T held, the human mind would not be equivalent to any formal system or TM connected to it. In fact, having established characteristics of T, for each formal system there would be a provable statement by the human mind, but not within the formal system. Hence, the mechanistic thesis would certainly be false: T non-recursive enumerability entails, in fact, the non-existence of any effective deductive system whose theorems are only and all truths of arithmetic.
If, on the contrary, K did not coincide with T, and thus the human mind were equivalent to a given formal system or to the TM related to it, the existence of arithmetical statements humanly undecidable in an absolute sense would follow. In fact, as underlined by Godel, the second incompleteness theorem does allow this conclusion: the proposition expressing the consistency of K, say ConK, is true but is not provable within the system itself; the negation of ConK is false and is not provable in K.
Having established the equivalence between the human mind and a formal system, ConK is not even provable by the human mind. Finally, since ConK can be put in the form of a Diophantine problem,[124] it is an absolutely undecidable problem. Such a proposition is, thus, an unknowable truth.Such questions and arguments lead Godel to the idea that from the incompleteness results one can at the most derive the following disjunction:
Either [subjective] mathematics is incompletable in this sense, that its evident axioms can never be comprised in a finite rule, that is to say, the human mind (even within the realm of pure mathematics) infinitely surpasses the powers of any finite machine, or else there exist absolutely unsolvable diophantine problems of the type specified (where the case that both terms of the disjunction are true is not excluded, so that there are, strictly speaking, three alternatives) (Godel 1951: 310).
So, following Tieszen 2006, and considering the translatability between the concept of a well defined formal system and that of a TM, we can say that Godel’s Incompleteness Theorems show that it could not be true that:
The human mind is a finite machine (a TM) and there are for it no absolutely undecidable Diophantine problems.
The incompleteness theorems show that if we think of the human mind as a TM then there is for each TM some ‘absolutely’ undecidable Diophantine problem. The denial of the conjunction (i) is, in so many words, Godel’s disjunction. In formulating the negation of (i) Godel says that the human mind ‘infinitely surpasses the powers of any finite machine’. One reason for using such language, I suppose, is that there are denumerably many different Turing machines and for each of them there is some absolutely diphantine problem of the type Godel mentions. So Godel’s disjunction, understood in this manner, is presumably a mathematically established fact. It is not possible to reject both disjuncts (Tieszen 2006: 230-231).
So the disjunction leaves open the three following possibilities:
I. human intelligence infinitely surpasses the powers of the finite machine (TM), and there are no absolutely unsolvable Diophantine problems (see Godel 1951: 310).
II. human intelligence infinitely surpasses the powers of the finite machine (TM) and there are absolutely unsolvable Diophantine problems. That is, although human intelligence is not a finite machine, nevertheless there are absolutely irresolvable Diophantine problems for it.
III. human intelligence is representable through a finite machine (TM) and there are absolutely irresolvable Diophantine problems for it.
Godel was convinced that (I) held, but he was also aware that his incompleteness theorems did not make the existence of a mechanic procedure equivalent to human mind impossible.
Godel, however, as I explained, believed that from his theorems it followed that if a similar procedure existed we “with our human understanding could certainly never know it to be such, that is, we could never know with mathematical certainty that all the propositions it produces are correct”.[125] But this established Godel’s idea that “the human mind, in demonstrating mathematical truths, only makes use of evidently true axioms and evidently truth preserving rules of inference at each stage”,[126] and this exactly means that “the human mind (in the realm of pure mathematics) is equivalent to a finite machine that, however, is unable to understand completely its own functioning”.[127] This argument, as it can be noticed, reminds those presented by Benacerraf (1967) and Chihara (1972).[128]
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