Chaitin on why mathematicians know nothing, but it is good news!

From a recent paper* by Gregory Chaitin, here is some rigour to support my previous self-indulgent speculation, when I suggested that however much human knowledge progresses we get no closer to understanding the universe.

Chaitin discusses an algorithmic information-theoretic understanding of Gödel incompleteness and concludes:

The bottom line is that these metatheorems show that the world of pure mathematics has infinite conceptual complexity, infinite information content, but any mathematical theory has finite conceptual complexity, and can therefore capture only an infinitesimal part of mathematical truth!

And furthermore:

The psychological defense mechanisms of the mathematics community have rejected incompleteness, they have decided that incompleteness is insignificant. They prefer to continue to believe, that they, not physicists, are the unique possessors of absolute truth. Incompleteness is too dreadful, let's forget about it, let's suppress it.

But Chaitin himself has a more optimistic interpretation:

But in the past few years, my work on biology has led me to a radical reinterpretation of incompleteness. Incompleteness is good, not bad! Incompleteness means, as Post stated a long time ago, but which it has taken me many years to understand, that mathematics is creative. Math is an open, not a closed system. I now see the incompleteness theorems as the first steps in the direction of a mathematical theory of creativity, and as providing a bridge between pure mathematics and biology. I think that mathematical creativity and biological creativity are related.

*Is God a computer programmer, not a mathematician? Building the world out of information and computation. Accessed from academia.edu

Human knowledge and tricks with percentages

A bit of a self-indulgent post today.  (OK, even more self-indulgent than usual.)

Suppose your destination is 100 miles away and you've travelled 20 miles. One day you travel another 10 miles, so you have travelled half as much again: you've increased the distance you have travelled by 50%.  Or, in that day, you have gone from having covered 20% of your total distance to having travelled 30%, so you have got 10% closer to your destination.

On another journey your destination is 1000 miles away and you've travelled 20 miles. Again, one day you travel another 10 miles, so you have travelled half as much again: you've increased the distance you have travelled by 50%.  But in this case, in that day, you have gone from having covered 2% of your total distance to having travelled 3%, so you have got 1% closer to your destination.

Then, destination 1000,000 miles away, still increase the distance you've travelled by 50%, but only get 0.001% closer.

But what if your destination is infinitely far away?  Do you get any closer however far you travel? So are you really travelling at all?

Is that what human knowledge is like?  Certainly we know lots more about the universe than we did 1000 years ago, but are we any closer to understanding the universe?

Update: see later post

A couple of snippets from Martin Bean's presentation at the OU Teaching Awards

A few things I wanted to keep from the presentation of Martin Bean, OU Vice Chancellor, at the OU Teaching Awards yesterday.

"Transforming information into meaningful knowledge"

In … a world (in which information is everywhere), sense-making and the ability to assess the credibility of information are paramount.

Mentoring and preparing students for the world in which they live, the central role of the university when it achieved its modern form in the 14th Century, is again at the forefront

Horizon Report 2010 [http://wp.nmc.org/horizon2010/chapters/trends/]

As long as the centuries continue to unfold, the number of books will grow continually, and one can predict that a time will come when it will be almost as difficult to learn anything from books as from the direct study of the whole universe. It will be almost as convenient to search for some bit of truth concealed in nature as it will be to find it hidden away in an immense multitude of bound volumes”

Denis Diderot, "Encyclopédie" (1755) [Online, I could only find this in wikipedia]

Conclusion: Teaching Matters

“..it is the role of us teachers, to make sense of the information that’s out there. To think differently, to act differently, to solve problems, to confront society in different ways”

Dretske on 'what we see'

Long-time readers of this blog (as if!) will know I admire Dretske's 'Knowledge and the flow of information'. It turns out that as well as an excellent writer he's a great speaker, as I found from his 2007 Howison Lecture at UC Berkeley, on 'What we see' - embedded below.

He's arguing that we DO see the rich complexity of things, even though we don't realise it. One aspect of his line of argument is that you don't have to think about something actively to know it.

Eg:

Q: Were there any giraffe's in your bedroom when you got up this morning?

A: You most likely know there weren't even though you didn't actively think about it.

This is interesting, and some other examples he gives - with some nice illustrations using complicated images - probe it further, but I'm not 100% convinced.



In the questions at the end, someone asks about the gorilla in the famous video:



(An earlier version of this idea here: http://viscog.beckman.illinois.edu/flashmovie/15.php)

Dretske argues that those of us who think we didn't see the gorilla, probably did know something about it, which could be revealed if we were asked the right question. Well, maybe, but surely if we'd seen it we would have been actively aware of it, something as out of place as that? It seems a bit of 'clincher' to me!

PS: Here's another one in the same vein:

Wise words of Solomonoff

I've been reading some of the work of Ray Solomonoff, having come across him in the context of Kolmogorov complexity/algorithmic information, and I like this (from The Discovery of Algorithmic Probability, Journal of Computer and System Sciences 55, 73-88 (1997)):

From Freud I got the idea of the unconscious mind: that
there were things going on in one's brain that one did not
have direct access to. Poincaré, made it clear that much of
his serious problem solving occurred in his subconscious,
and I felt this was very common in problem solving of all
kinds in the sciences and the arts.

This view was one important reason for my later rejection
of "expert systems" as a significant step toward artificial
intelligence. Expert systems were (at best) expressions of
peoples' conscious thought, which was, I felt, a very small
fraction of human problem solving activity.

Other implications: Memory is what you invent to explain
the things that you find in your head. Over the years, the
"facts" in this paper will be gradually revised as I reread my
research notes.

Explanations that people give for their own behavior are
not to be taken too seriously - including discussions in this
paper.

Dretske's definition of knowledge: and atoms

Dretske (Knowledge and the flow of information, CLSI 1999) page 86

K knows that s is F = K's belief that s is F is caused (or causally sustained) by the information that s is F.

Dretske argues that this is a better definition of knowledge than the 'traditional'

Knowledge is justified true belief

You have to take Dretske's definition together with his definition of information.

Consider the atomic theory of matter. Now, in what follows, you have to accept all sorts of 'suspension of skepticism', and it really needs reams of footnotes and caveats, but I think it does provide a valuable insight.

Did Democritus know that matter is structured into atoms?

Democritus did NOT know, on either definition. Although the belief was true, it was not justified because he didn't have any (modern, scientific) evidence. Similarly, Democritus's belief was not caused by the information that matter is structured into atoms

Did Rutherford know that matter is structured into atoms?

Yes, on both definitions. The belief was true, and it was justified by the experimental evidence - which was the information from his experiment.

Did a student of Democritus know that matter is structured into atoms

This I think is the key one, because I think on the 'justified true belief' a student of Democritus did know. That is, if we are going to allow anyone to learn anything 'second hand', from a teacher, then we must allow the possibility that a student of Democtritus is justified in believing his teacher. But, he does not know anything of the sort in Dretske's definition, because he did not receive the information that matter is structured into atoms.

Do I know that matter is structured into atoms?

Well, yes, I think I do know that, on both definitions. Except, of course, that there's a bit of the skeptic in me (especially, dogmatic scientific statement), but I did say above that there were all sorts of caveats and footnotes needed, and Dretske does discuss the position of the skeptic