Robert J. Marks: Things Exist That Are Unknowable

The mathematically provable idea that something exists but is unknowable has clear philosophical and theological implications. Here’s a look atChaitin’s number:

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.”

7 Replies to “Robert J. Marks: Things Exist That Are Unknowable”

1. 1
daveS says:

Second, knowing Chaitin’s number involves a cumulative sum of values obtained by a Turing Halting Oracle. The Turing Halting Oracle is noncomputable. Consequently, Chaitin’s number is unknowable.

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.

2. 2
math guy says:

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.

3. 3
daveS says:

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 .

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.)

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.

4. 4
daveS says:

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.

5. 5
bornagain77 says:

Of related interest:

A Mono-Theism Theorem: Gödelian Consistency in the Hierarchy of Inference – Winston Ewert and Robert J. Marks II – June 2014
Abstract: Logic is foundational in the assessment of philosophy and the validation of theology. In 1931 Kurt Gödel derailed Russell and Whitehead’s Principia Mathematica by showing logically that any set of consistent axioms will eventually yield unknowable propositions. Gödel did so by showing that, otherwise, the formal system would be inconsistent. Turing, in the first celebrated application of Gödelian ideas, demonstrated the impossibility of writing a computer program capable of examining another arbitrary program and announcing whether or not that program would halt or run forever. He did so by showing that the existence of a halting program can lead to self-refuting propositions. We propose that, through application of Gödelian reasoning, there can be, at most, one being in the universe omniscient over all other beings. This Supreme Being must by necessity exist or have existed outside of time and space. The conclusion results simply from the requirement of a logical consistency of one being having the ability to answer questions about another. The existence of any question that generates a self refuting response is assumed to invalidate the ability of a being to be all-knowing about the being who was the subject of the question.
http://robertmarks.org/REPRINT.....heorem.pdf

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.”

Quantum physics problem proved unsolvable: Gödel and Turing enter quantum physics – December 9, 2015
Excerpt: A mathematical problem underlying fundamental questions in particle and quantum physics is provably unsolvable,,,
It is the first major problem in physics for which such a fundamental limitation could be proven. The findings are important because they show that even a perfect and complete description of the microscopic properties of a material is not enough to predict its macroscopic behaviour.,,,
“We knew about the possibility of problems that are undecidable in principle since the works of Turing and Gödel in the 1930s,” added Co-author Professor Michael Wolf from Technical University of Munich. “So far, however, this only concerned the very abstract corners of theoretical computer science and mathematical logic. No one had seriously contemplated this as a possibility right in the heart of theoretical physics before. But our results change this picture. From a more philosophical perspective, they also challenge the reductionists’ point of view, as the insurmountable difficulty lies precisely in the derivation of macroscopic properties from a microscopic description.”
http://phys.org/news/2015-12-q.....godel.html

6. 6
math guy says:

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.

7. 7
daveS says:

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:

(1) Assume my reasoning powers are captured by some formal system F (to put this more briefly, “I am F”). Consider the class of statements I can know to be true, given this assumption.

(2) Given that I know that I am F, I know that F is sound (as I know that I am sound). Indeed, I know that the larger system F’ is sound, where F’ is F supplemented by the further assumption “I am F”. (Supplementing a sound system with a true statement yields a sound system.)

(3) So I know that G(F’) is true, where this is the Gödel sentence of the system F’.

(4) But F’ could not see that G(F’) is true (by Gödel’s theorem).

(5) By assumption, however, I am now effectively equivalent to F’. After all, I am F supplemented by the knowledge that I am F.

(6) This is a contradiction, so the initial assumption must be false, and F must not have captured my powers of reasoning after all.

(7) The conclusion generalizes: my reasoning powers cannot be captured by any formal system.