Actually, some mathematicians and computer scientists suggest that the undecidable status of the Continuum Hypothesis may be good news, indicating that the universe has more complexity than what can be captured in algorithms. For example, the weirdness of quantum physics, displayed in the Uncertainty Principle of Heisenberg, suggests there is a great deal of randomness in the universe that may not be compressible. Gregory Chaitin, one of the founders of complexity theory, argues that there are things in nature that are so random as to be absolutely incompressible or contained as consequences of a finite set of axioms. Pursuing an argument similar to that presented by Brouwer and others he asks, “Why should I believe in a real number [a number like π with a non-repeating infinite decimal expansion] if I can’t calculate it, if I can’t prove what its bits are, and if I can’t even refer to it?” He suggests that Gödel’s Theorem is good news, because it asserts that nature is not compressible into a small set of axioms and is therefore endowed with the kind of complexity that maximizes diversity. Arguing against the hope that computer-proving software will eventually certify all mathematical proofs, he stated in 2005:
Formal axiomatic systems are a failure! Theorem-proving algorithms do not work [bold lettering, his]. One can publish papers about them, but they only prove trivial theorems.
Of course, most mathematicians would not agree that our current theorems are “trivial.” The kind of complexity that Chaitin is talking about is evident in problems such as the traveling salesman’s problem, which are solvable in exponential time; the time required to derive a solution growing exponentially with the amount of input data. This is a prohibitive restriction on even the most powerful computers because they are deterministic, which implies that they must perform the operations sequentially. (It remains an open question whether quantum computing, which allows parallel processing, will alleviate this problem.)
Roger Penrose argues that consciousness is distinct from algorithmic processing and that human thought has a non-algorithmic dimension not accessible by computers of any given processing power:
There is as much mystery and beauty as one might wish in the precise Platonic mathematical world, and most of this mystery resides with concepts that lie outside the comparatively limited part of it where algorithms and computations reside.
Yet, as ChatGPT and other AI applications appear to close the gap between human and machine thinking, many fear that computers, armed with algorithmic thinking, may surpass human cognition and perhaps, take control of our world. This might prompt someone to ask, “If algorithmic thinking is limited, why do would we fear it?”
Undecidable propositions merely indicate that another axiom may be needed to make a particular theorem decidable, but, as Gödel’s Theorem’s warns, the new axiom may introduce inconsistencies. Meanwhile, mathematicians will continue using deduction and algorithm to develop theorems and continue to look for any additional axioms that may be needed and tested. Computer scientists will continue to explore the limits of algorithmic thinking, and we will all stand in awe as we confront these ideas in our relentless quest to decode nature’s secrets.
