The mathematically provable idea that something exists but is unknowable has clear philosophical and theological implications. Here’s a look
Here is why Chaitin’s number is mind-blowing: There are many problems in mathematics that can be disproved by one counterexample. Consider the theorem that all skunks are black and white. Show me a pink skunk and the theorem is disproved. Chaitin’s number—one number—can be used to prove or disprove almost every known math conjecture that can be disproven by a single counterexample… It is an astonishing fact that, given a lot of computing time and power, any list of problems that can be disproven by a single counterexample could be proved or disproved if we knew Chaitin’s number to finite precision. Think of it. A list of the most perplexing math problems not yet solved by the world’s top mathematicians can be answered with one single number. And not only does Chaitin’s number exist, we know it lies between zero and one. More.
He goes on to explain why we cannot know the number.
Robert J. Marks is one of the authors of Introduction to Evolutionary Informatics
Also by Robert J. Marks: Random Thoughts on Recent AI Headlines: There is usually a story under those layers of hype but not always the one you thought
When Thomas Sowell was writing his syndicated column on economics, I always looked forward to his sporadically appearing “Random Thoughts on the Passing Scene.” Reminding readers that imitation is the sincerest form of flattery, I offer my own “Random Thoughts on Recent AI Headlines.”
From the linked essay:
This equating of “noncomputable” with “unknowable” is interesting to me, in view of the fact that some people here insist that the human mind is very much not like a computer. If this is true, then the human mind might be capable of performing tasks that a computer cannot, for example “knowing” a noncomputable number.
DS
The human mind (as early as 1931) has already performed a task that a computer cannot. Read Penrose’s “Shadows of the mind” where Penrose debates an AI and employs Gödel’s 2nd Incompleteness Theorem to good effect.
On the other hand, it is highly unlikely that we’ll ever be able to prove that the billionth binary bit of Chaitin’s constant is 1. (Pick a particular computing framework first, so that the constant is unique, change billion to trillion if needed.)
This is not the case for pi or other computable real numbers, where our only constraints are time, memory, and energy. We can in principle compute any previously specified decimal place for pi.
Math Guy,
Thanks for the reference to the Penrose book. I will put it on my list. It looks like it has generated some criticism as well, which I’ll try and look at .
What’s your argument for this? The best I can come up with is that while the mind is not merely a computer, its power is limited, and the solution to this problem lies beyond those limits.
Math Guy,
For reference, here’s Solomon Feferman’s critique of Penrose’s book (apparently Penrose cites Feferman in Shadows of the Mind).
Feferman also includes a reference to Martin Davis’ criticism of The Emperor’s New Mind (here).
Apparently neither Feferman nor Davis were terribly impressed with Penrose’s argument.
Of related interest:
of supplemental note:
Godel’s incompleteness theorem has now been extended to physics and now undermines the entire reductive materialistic framework that undergirds Darwinian evolution:
In the following article entitled ‘Quantum physics problem proved unsolvable’, which studied the derivation of macroscopic properties from a complete microscopic description, the researchers remark that even a perfect and complete description of the microscopic properties of a material is not enough to predict its macroscopic behaviour.,,, The researchers further commented that their findings challenge the reductionists’ point of view, as the insurmountable difficulty lies precisely in the derivation of macroscopic properties from a microscopic description.”
DS @ 4
In his critique of Penrose, Fefferman writes:
“The main question, though, is whether these errors undermine the conclusions that Penrose wishes to draw from the Gödelian argument. I don’t think that they do, at least not by themselves. That is, I think that the extended case he makes from sec. 2.6 on through the end of Ch. 3 would be unaffected if he put the logical facts right; but the merits of that case itself are another matter.”
So Fefferman caught some non-critical gaps or miss-statements that don’t affect Penrose’s basic argument. Fefferman is righteously indignant for being quoted out of context or mistakenly in a best-selling book written by a 20th century genius. He also disagrees with the philosophical interpretation put forward by Penrose.
The section of Penrose’s book that I mentioned are not the deep stuff to which Fefferman has issue with. The section is a sort of Socratic dialogue between Penrose and an advanced AI. It is easy reading (unlike other sections) and rather amusing.
Math Guy,
Thanks, I have now read the dialogue. I’ll post some background info here FTR.
For reference, some responses by Penrose to his critics.
Penrose singles out Chalmers as about the only one who discusses the argument which is summarized by the dialogue.
Chalmers’ summary of this argument: