What is the meaning of Gödel's theorem?

Hofstadter's most recent book "I am a strange loop" deals with cognitive science, mathematics and the problem of consciousness and self-consciousness.

By their very nature, these works can only be critically considered in terms of the intersection of disciplines that they involve. I would like to consider a dimension of Hofstadter's work that is sometimes overlooked and that is of fundamental intrinsic value for both science and mathematics.

Hofstadter understood something about the fundamental nature of mathematics that is sometimes overlooked by his readers, who focus on the cognitivistic dimension of his work, an example of this is Deutsch's most recent review of the above cited book (http://physicsworld.com/cws/article/print/30347).

Deutsch believed (as stated in the review) that Hofstadter’s point was that the brain worked like acomputer and that consciousness was an attribute of certain programs, when, in fact, this is not the case, in Hofstadter’s proposal.

Indeed, Hofstadter’s work is actually a deep insight into Gödel’s work, and the significance of Gödel’s work for mathematics.

In order to make this point clearer, and the reason why Hofstadter’s strange loop is fundamentally an insight into the nature of mathematical objects, it is important to understand the argumentative position of Gödel with respect to Hilbert’s project for mathematics. Hilbert believed that all mathematics could be expressed in terms of formal languages, and that any proof could be conducted through a series of mechanical steps, within a formal system, which included that formal language and the rules of deduction of well formed formulas.

If Hilbert was right, then, mathematics could be reduced to a purely syntactical game, more properly, each mathematical object would be, upon formalization, essentially dispossessed of its meaning beyond the pure meaningless syntax. Mathematics could be reduced to logic. Gödel put this into question by showing that mathematical truth and mathematics cannot be reduced to logic. A formal system, simple enough to include only the rules of arithmetic, would, either, be inconsistent, or, incomplete. This had a twofold consequence, mathematics may possess an irreducible semantics and not all mathematical knowledge is accessible to the mathematician.

Not all could be proven using deduction rules within a formal system, this was the common understanding of the meaning Gödel’s work. However, Hofstadter uncovered a second layer of meaning, deeply present within Gödel’s work – the paradox is the necessary point of fracture that triggers a mathematical evolutionary expansion from a formal system to another.

In order to transcend the paradox, one needs to move beyond the formal system, a motion that is made by the mathematician, but that is fundamentally inscribed in the very nature of the formal system itself. It is mathematics that demands the motion, in that sense the motion can be identified as present within mathematics itself, and independent of the mathematician, the mathematician must follow the nature and motion of the mathematical object.

The fundamental limitations of the formal system are simultaneously the threat to that formal system and the opportunity for mathematics to grow. Turning in on itself, the self-referential statement logically precedes the mathematical leap.

Carlos Pedro Gonçalves

Mathematics researcher at UNIDE