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Re: [Phys-l] Science, verification and proving



On 03/08/2011 12:42 AM, Savinainen Antti wrote:
I had recently a discussion on science and physics with a teacher who
apparently had no background in science. He said that you should
never say "science has proven that...". I proposed that one could
talk about verification of predictions of a theory. In physics this
means that an experimental result matches with the theoretical
prediction within uncertainty limits. However, one might argue that
the theory itself was not proven whereas one could say that theory is
validated by experimental evidence. The concept of proof would in
this scenario be reserved to mathematics and logic.

I quite often hear another claim: science changes all the time so who
knows, perhaps everything is different in future. My response is
that, yes, there will be better theories in future. Then again,
technology based on science we have now does not cease to work and
well checked empirical results are not likely to be false. We have
not abandoned Newtonian mechanics even though we know that relativity
theory is a more accurate description of nature. The point is to know
the limits of validity of a theory with respect to the accuracy of
measurements we want to make. This is not to say that a conceptual
framework in which the empiria is interpreted can be very different.

So...what do you think about science, verification and proof?

That's a tricky issue. There are at least two ways of approaching it.

If you insist on absolute proof, beyond any doubt whatsoever, then
nothing will ever be proved. Absolute certainty is not found in
the science lab or in the courtroom. Sometimes religion claims
absolute certainty, but that is a matter of faith, not a matter of
logical proof.

A weaker standard is certainty _for all practical purposes_. For
example, I don't know everything about the moon, and I don't know
everything about cheese, but I am certain for all practical purposes
that the moon is not made of green cheese.

As for the example of Newton's laws of motion: We know that they
are not exact in principle, but when analyzing a game of billiards
the inexactitude is insignificant compared to the uncertainties in
mass, shape, position, initial velocity, et cetera. So for all
billiards purposes, a model based on the uncorrected Newtonian laws
is just as good as a model that includes relativistic corrections.

Einstein said the the most incomprehensible thing about the universe
is that it is comprehensible. I would have said /partly/ comprehensible.
That is, much of what we do involves building models. It is astounding
that our models work at all, but some of them do work quite well.

In formal computational learning theory, one speaks of PAC models.
That stands for Probably Almost Correct. That means there are two
parameters, epsilon and delta, and under appropriate conditions it
can be shown that a certain model will have a probability (1-epsilon)
of getting the right answer within plus-or-minus delta. So the model
claims neither complete confidence nor complete exactitude ... but
the model is Probably Almost Correct. You can /prove/ it is PAC,
quite formally. The formulas for epsilon and delta are nontrivial.

This formalism allows us to determine such things as the amount of
data needed to pin down this-or-that fitting parameter, and how
many times we need to replicate a result before we begin to trust
it. Or, as the lawyers would say, how much circumstantial evidence
constitutes a proof.

I'm not sure this solves the fundamental philosophical metaphysical
question, because the PAC-learning theorems require certain conditions,
and it's not clear how to ensure those conditions hold in real-world
situations. So in some sense we are back where we started: we have
a formalism that is good enough for a wide range of practical purposes.

All in all, science is not so much different from buying shoes. You're
never going to find a shoe that is exactly the right size. Even if
the shoe is exactly size 12, your foot isn't, so you buy something that
is close enough, and/or has a suitable range of adjustment. In science,
mathematical proof is often helpful when building a model, but the
resulting mathematically perfect model applies only imperfectly to the
real world.

The sad fact is that the folks who claim absolute certainty are wrong
more often than the folks who don't. Specifically, I am thinking of
the various religions that hold incompatible views, all with vehement
certainty. Also there are the people who told us in August 2002
"Simply stated, there is no doubt that Saddam Hussein now has weapons
of mass destruction."

Science does not give us certainty. Rather, it gives us a good way
to get things done in a somewhat-uncertain world.

People who do not understand this often make the mistake of treating
all approximations as equally bad or equally good. For example:
a1) When the situation calls for making an approximation, they may
be paralyzed by the lack of certainty. I see this in student
pilots all the time. We are flying northeast, and it is time to
turn to a certain westerly course, but they haven't calculated
the exact heading, so they don't turn at all. I suggest that it
would be better to turn to some approximately westerly heading
and hold that heading while figuring out a more exact answer.
a2) And then there is climate change. I wonder about the folks
who say we must not act until we are certain. Are they really
that clueless? Or are they just being dishonest, and hoping we
are too stupid to notice?
b) At the opposite extreme, sometimes when people are forced to
make an approximation, they latch onto the first approximation
that comes to mind, even though vastly better approximations are
readily available. Choosing a suitable approximation sometimes
requires judgment and skill.