Wednesday, February 26, 2014

Godel's principle and respect for failure

In 1931 Kurt Godel shocked the mathematical world.  Math is the ultimate sanctuary for those who believe that some ultimately Platonic sense there is absolute, universal, unexceptionable--and understandable truth. The facts of geometry and mathematics are cosmically true (the Pythagorean theorem, the sum of angles =180 degrees, etc., 2+2=4, and the fact that the derivative of x-squared is 2x, etc). The way we do math may be a human or cultural convention, but the facts are not.  Our cultural convention of how we do it, or whatever it is, may for cultural or historical reasons simply overlook many equivalent truths, but it is working with at least some set of ultimate truths.  But how is it that Pythagorean theorem is true about right triangles, but there are no actual, perfect right triangles in the world??


At least, at the turn of the 20th century, for those studying such ultimate truths, it was widely thought that the principles of logic and logical reasoning were essentially the same as the principles of mathematics.  Not only were both equally true but logical reasoning could be expressed in the same kinds of terms as those of mathematics.  This would make the world a certain place, in a sense.  Truth is truth. Truth is internally consistent.  And truth is discoverable!

Infinity symbol in various typefaces; Wikipedia

There were some problems.  For example, what do we do with or about the notion of 'infinity'?  Nineteenth century mathematicians, notably Cantor, showed that there are even different levels of infinity.  The whole numbers are one level.  You can match even numbers up one for one with odd numbers.  But you can't match either up like that for the numbers between just 0 and 1.  That's because there are far more of the latter:  if we match 0, 1, and 2 with 0, 0.1, and 0.2 it might seem fine, but then how do we match up 0.001, 0.002, 0.00001, and so on?

The constant π is represented in thismosaic outside the mathematics building at the Technische Universität Berlin. Wikipedia

And then there is the little problem of what 'randomness' means.  For example, I recently learned that the digits in the value of  pi (relating radius to circumference in a circle) are randomly distributed.  Take any sequence, like 7623116, and you'll find it, on average every 10 million 7-digit sequencs in pi.  Or look at Stephan Wolfram's 'cellular automata'; these are simple rules for transitions in a string of (say) black or white boxes if each box produces a 'descendant' box and the rule says what color it is based on the array of colors in the current generation.  The resulting black/white pattern, determined by a simple rule, is, Wolfram says, indistinguishable from random: no pattern can be found along the string at any given time.  Yet 'random' seems such an obvious concept...until you think about them too much, and then they become quite disturbing.

And I have not mentioned the very problematic notion of probability.  We know how a cause can lead to an effect--well, we think we know that--but it is totally unclear how a cause can only lead to the future probabilistically.   Usually, the probabilistic nature of such results is attributed to measurement error, poor theoretical understanding, and so on, in a cosmos that, were we to know everything, would be purely and rigidly law-like.  But how could something cause something else 'with 27% probability'?  If you think about it, it is not at all clear what that means, in terms of actual causation, beyond errors and sampling effects.

And then there is 'chaos' theory.  Even in a purely deterministic, rule-bound process of cause and effect, where there is no uncertainty or probability involved, unless you have 100% measurement accuracy of things at some given time, you cannot predict with any accuracy what things will be like over the future.  Your predictive power, even with perfectly true theory, is zilch.  But how can you know what the underlying reason is?

Such things fly in the face of the views of the cosmos as a law-like place where certainty rules.  We want a knowable universe.  We spend our puny lives as scientists trying to understand it, and assuming it at least exists, even if it's hard to understand!

What we have learned
To general chagrin, what mathematician Kurt Godel showed halfway through the last century, was that even if it were true that the mathematical realm was all-of-perfection, an unknown fraction of it was unknowable.  That is, things that are true can't be proved and, worse, you could never know whether something you thought were true and were trying to prove it, was in reality untrue or just unprovably true.

Given all of this, it is surprising that in so many cases what we have learned is that the universe may not be law-like in the way we'd thought, true probability may or may not exist, not all things can be shown to be true even if they are true, and (in quantum mechanics and relativitiy and gravity at least) there are phenomena that seem truly to be unlike any of the above concepts or, as physicists often say, just are not consistent with 'common sense'....even if they're true.

Even physics and chemistry, not to mention biology and psychology or economics, are often swimming in uncertainty and claims of knowledge that, no matter how confidently asserted, simply don't hold up.

These various incarnations of indeterminacy, like probability, can shake our faith in the idea of a knowable universe whose causal nature we can pin down tight.  In ordinary sciences, and sometimes even in our daily lives, we don't know exactly how we should be viewing, much less approaching, causation.  What we end up doing is designing studies or experiments that we know how to design, using methods we know how to use, and crossing our fingers.  Whether we're being ostriches to our peril, or whether it doesn't matter and we should just carry on regardless, is unclear.

To the young and thoughtful, these provide things to think about, both in the practical sense of actually moving the science forward more than a millimeter at a time.....and in terms of our ultimate hope to understand life as it really is, not just in a statistical analysis.


Holly Dunsworth said...

Thanks for this post Ken!

Given what you wrote today, I'm curious to know what you meant in your title yesterday with "pop culture isn't knowledge."

Ken Weiss said...

Yesterday's post didn't seem to ring many good bells, and seemed to ring some bad ones, so I probably shouldn't dwell on it. But basically that I think popular science has become too much like television entertainment, and too little about facts that aren't babied too much.

But, it may have been ill-advised in general.

Holly Dunsworth said...

Well I wasn't picking at your yesterday's post. I was trying to get at what is knowledge. That tiny thing. That's all :)

Manoj Samanta said...

Well, just because he is telling 'what is not knowledge' does not mean he has to have answer of 'what is knowledge' :)

Ken Weiss said...

What is knowledge? I actually know (but I'm not telling!)