Wednesday, January 12, 2011

Getting Gold When Buying Iron

"The theory of Induction is the despair of philosophy--and yet all our activities are based upon it."
-- Alfred North Whitehead, Science and the Modern World

Jonah Lehrer writes, in the New Yorker,
[A]ll sorts of well-established, multiply confirmed findings have started to look increasingly uncertain. It’s as if our facts were losing their truth: claims that have been enshrined in textbooks are suddenly unprovable. This phenomenon doesn’t yet have an official name, but it’s occurring across a wide range of fields, from psychology to ecology. In the field of medicine, the phenomenon seems extremely widespread, affecting not only antipsychotics but also therapies ranging from cardiac stents to Vitamin E and antidepressants: Davis has a forthcoming analysis demonstrating that the efficacy of antidepressants has gone down as much as threefold in recent decades.
For many scientists, the effect is especially troubling because of what it exposes about the scientific process. If replication is what separates the rigor of science from the squishiness of pseudoscience, where do we put all these rigorously validated findings that can no longer be proved? Which results should we believe?
The answers Lehrer finds for this phenomenon range from regression to the mean to publication bias.  But he shies away from asking the obvious question that rears its ugly head: what rational basis do we have to expect the scientific method to work at all?  Have we have been bred to believe the universe owes us something that we have no logical reason to believe?


Let's step back.  Science has two main stages: discovery and justification.  Scientists come up with hypotheses and then they test them.  The former is a creative act, something that defies description.  The latter is what we think of as the scientific method: the gathering of empirical data to test hypotheses.

In most cases all of the data can not be collected, only a sample.  Einstein postulated as part of his theory of relativity that gravity would bend light rays. The empirical observation during the solar eclipse of May 29, 1919 that light rays were indeed bent by gravity is seen as powerful evidence that his theory is correct.  But not all light rays have been observed, only a few.  Making a few observations and then generalizing these few measurements to cover all the measurements that have not been made is called induction.  Induction is the centerpiece of scientific reasoning.  If the sun rose yesterday, and the day before and the day before that and it also rose today, we have good reason to think it will rise tomorrow.  Induction.

But do we really have good reason?  On what grounds should we believe that induction is a valid way to reason?  Why should we believe that just because something keeps happening, it will continue to happen?  Why should we believe that just because some light rays are bent by gravity, all light rays are bent by gravity?

The answer is, obviously, we should believe this because it works.  This is what has always happened.  Every time we have correctly used induction in the past, it has led to valid conclusions.

But wait, do you see the problem?  We have just proved induction by... using induction.  There is, in fact, no other way we know of to support it.

Frank Ramsey draws the distinction between deductive reasoning--which is supported by formal logic--and inductive reasoning--which is not--in his Truth and Probability (1926, pdf):
The conclusion of a formally valid argument is contained in its premisses; that to deny the conclusion while accepting the premisses would be self-contradictory; that a formal deduction does not increase our knowledge, but only brings out clearly what we already know in another form; and that we are bound to accept its validity on pain of being inconsistent with ourselves. The logical relation which justifies the inference is that the sense or import of the conclusion is contained in that of the premisses.
But in the case of an inductive argument this does not happen in the least; it is impossible to represent it as resembling a deductive argument and merely weaker in degree; it is absurd to say that the sense of the conclusion is partially contained in that of the premisses. We could accept the premisses and utterly reject the conclusion without any sort of inconsistency or contradiction.
This echoes an earlier argument by David Hume, in his Treatise on Human Nature (I.III.VI)*:
[T]here can be no demonstrative arguments to prove, that those instances of which we have had no experience resemble those of which we have had experience. We can at least conceive a change in the course of nature; which sufficiently proves that such a change is not absolutely impossible. To form a clear idea of any thing is an undeniable argument for its possibility, and is alone a refutation of any pretended demonstration against it.
Hume says there can be no proof using induction.  And, if you think about it, there can't even always be belief: probabilistic reasoning, though it seems to support some limited subset of inductions, is not always--or even usually--good enough.  Of the total population of light rays, what would the sample size need to be to give you statistical comfort in the theory of relativity?

Hume goes further.  His rejection of induction as rationally supported is just the inevitable fallout from his rejection of causation altogether (I.III.XIV):
I am sensible that of all the paradoxes which I have had or shall hereafter have occasion to advance in the course of this treatise, the present one is the most violent... any two objects or actions, however related, can never give us any idea of power or of a connexion betwixt them: that this idea arises from the repetition of their union: that the repetition neither discovers nor causes any thing in the objects, but has an influence only on the mind by that customary transition it produces... 
A cause and an effect are two things that we associate simply because the two events always seem to happen together; but there is no way we can discover a connection between the two that guarantees this "causation" to continue.

Our belief in induction and causation come from our belief in the uniformity of nature.  If two events always happen together, or if an event happens the same way over and over, then we believe there must be some unchanging underlying mechanism.  If this light ray does this, then that one does too.  If the earth is turning, it will keep turning.  Why wouldn't it?  Why would anything have changed?

Well, on the contrary, why would it keep turning?  Why would anything stay the same?  The actual unanswerability of these very simple questions makes our entire quest for knowledge seem very fragile.


But in the end Hume was a skeptic who realized that skepticism was not a realistic option, nor a productive one.  Even if the idea of causation itself is suspect--and thus induction--their use goes constructively on.  The scientific method, statistics, these things seem to work.  People who deny induction get hit by cars, their children catch diseases other peoples' children have been vaccinated for.  We have been formed by evolution to believe in induction: babies do not reason about cause and effect, they simply recognize it.  We have been bred to believe that the universe is uniform and unchanging.

Back to Lehrer's problem.  The subtitle of his article is "Is there something wrong with the scientific method?"  He may just as well have asked "Is there something wrong with reality?"  Induction is central to all of our thinking; we do not have any alternative way to reason.  Ramsey again:
We are all convinced by inductive arguments, and our conviction is reasonable because the world is so constituted that inductive arguments lead on the whole to true opinions. We are not, therefore, able to help trusting induction, nor if we could help it do we see any reason why we should, because we believe it to be a reliable process. It is true that if any one has not the habit of induction, we cannot prove to him that he is wrong; but there is nothing peculiar in that. If a man doubts his memory or his perception we cannot prove to him that they are trustworthy; to ask for such a thing to be proved is to cry for the moon, and the same is true of induction. It is one of the ultimate sources of knowledge just as memory is: no one regards it as a scandal to philosophy that there is no proof that the world did not begin two minutes ago and that all our memories are not illusory.

We all agree that a man who did not make inductions would be unreasonable: the question is only what this means. In my view it does not mean that the man would in any way sin against formal logic or formal probability; but that he had not got a very useful habit, without which he would be very much worse off, in the sense of being much less likely to have true opinions.

This is a kind of pragmatism: we judge mental habits by whether they work, i.e. whether the opinions they lead to are for the most part true, or more often true than those which alternative habits would lead to.
Just as with Gödel's incompleteness, or the stack of turtles holding up the universe, we continue ignoring the logical problem of relying on the things they show are inconsistent.  We relegate to the back of our minds the question of why the heck anything seems to work at all.  Because how would we exist in a world where that concern was everpresent?

The philosopher Nāgārjuna said something that Hume would have understood:
Not from itself, not from another, not from both, not without cause,
Never in any way is there any existing thing that has arisen.
This, in the inimitable method of Indian philosophers, is not meant negatively, it is meant as an aid to understanding the ineffable. The Roman Skeptic Sextus Empiricus came to the same conclusion, even if it does not seem that way.  He said that without any way to be sure in our knowledge, we need to suspend judgement over the truth or falsity of our beliefs and simply exist in the present evidence of our senses.  He thought the only truly rational way to live was to rely simply on what is, not on what we think will be.


I've always been the kind of person who wants to know.  After learning how to program, I needed to know how the compiler worked, then the operating system.  Then I needed to know how the microprocessor worked, then integrated circuits, then the logic gate, then the transistor.  How transistors were made, then where silicon comes from.  It is hard for me to accept, in trying to understand, that there may be levels below which I can't pass.

In Hui-k'ai's Gateless Gate, Wumen (Mumon) tells this story:
The temple flag was flapping in the wind.  Two monks argued: one said the flag was moving, the other said the wind was moving.  They argued, but could not agree.  The Sixth Patriarch said "It is not the wind that moves, it is not the flag that moves, it is your mind that moves."
Mumon retorts "It is not the wind that moves, it is not the flag that moves, it is not the mind that moves."  For him.  For me, it is the mind that moves.  Perhaps this is the easier problem to solve.

* I had to take out most of the freakin commas from Hume, they were very distracting.

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