If the Only Tool You Allow Yourself is a Hammer...

...you'll have a hard time painting your ceramics. 

Ed Feser, over on his philosophy blog, makes the following useful observation:

[T]he problem [of consciousness] is a problem in principle, and ...it isn’t going to be solved by further application of existing methods precisely because the problem is generated by the application of existing methods.  ... [T]he problem of consciousness is a result of the move Galileo, Descartes, and Co. made of taking color, sound, heat, cold, and other sensory qualities out of the material world and relocating them in the mind’s experience of that world.  Having thus made matter essentially devoid of qualia and qualia essentially immaterial, there is no way on this picture of things you are ever going to “naturalize” qualia.  You are stuck either with a Cartesian-style dualism ..., or with an incoherent eliminativism.  (Incoherent because the qualitative experience whose existence you will be denying in the name of science forms the evidential base of science -- a problem Democritus and Schrödinger saw but Dennett does not, pushing ... his usual line that consciousness is an “illusion.”)  The only true solution to the problem is to see that the post-Cartesian conception of matter does not capture the entirety of its real nature in the first place, but is merely a useful simplification.

Or to put it another way, materialism does not even completely account for matter. 

There's Another Cow Off the Ice

One of the tactics often used for solving intractable mathematical and physical problems is to simplify it until it does become solvable. Thus, we might assume an infinite surface rather than one with boundaries for the simple, if inconvenient, fact that physical systems grow irritable at the boundaries.  Or we might suppose a frictionless surface, an ideal gas, a perfectly elastic collision.  Think of the phrase "all else being equal."  They never are, but by assuming they are we might gain some insight into the more complex reality.  Ptolemy (and Copernicus) solved the orbits of each planet as a separate problem, and wound up with each planet orbiting a different center.

The Systems of the World.
The Copernican model (right) is not notably simpler than the Ptolemaic model (left). 
It uses more epicycles; the Sun– like Ptolemy’s Earth – is off-center;
and each planet orbits a different center.  Note also the double epicycle for the Copernican Moon

Newton, of course, solved the 2-body problem, not the 7-body problem. Ivar Eklund noted in Mathematics and the Unexpected that a billiards player attempting to calculate the motion of a seven-ball cannon would have to take account of the gravitational attraction of the spectators distributed about the table.  The further out a prediction runs, the more such things must be taken into account. In calculating orbits for satellites and Moon trips, NASA still uses geocentric math, not because it is factual, but because it is true within the times and distances involved. That is, it produces results that approximate reality "close enough."   

There is No Such Thing as a Problem in General

There are only problems in particular.  This was a favorite aphorism coined by TOF during his storied career as a quality consultant.  Quotidian problems like "too much scrap in the factory" are often intractable because it is not actually a problem.  It is many problems lumped under the same name, and each one of them has its own cause or causal nexus.  The problem-solver's first step is often to break the generic "problem" down into specific problems, a process called by Juran "Pareto Analysis."

Pareto Analysis: breaking big unsolvable problems into small solvable ones.

For example (above), "excessive steam usage" on a plant site making vitamins proceeded by first breaking steam usage down by building (66% of usage was in the building making product A), then by process step (36% of the steam in building A was used in Step 2), then by particular equipments (20% of the steam in Step 2 was used in Evaporator #2).  This provided a problem precisely defined around which one could wrap one's head.  Excessive steam usage on Evaporator #2 could be traced to specific physical factors, equipment could be inspected, leaking seals identified, damages repaired, and so on.  Excessive steam usage in general can only be ascribed to vague generic factors, much as in sociopolitical discourse, where all ills are ascribed to the vague and malignant spirit of G.W.Bush.  

Naturally, one continues in this fashion, eating the big problem one bite at a time. 

Never Eat Anything Bigger Than Your Head

“All models are wrong, but some are useful.”
– George E. P. Box,
"Robustness in the Strategy of Scientific Model Building"

One of the problems the medievals had was that they tried to solve too much at once.  For them, motion was kinesis, all actualization of potentials.  So an orbiting planet, a ripening apple, and a maturing puppy dog were all in motion.  Try digging a theory of motion out of that, monkey boy. 

So the moderns broke it all up into separate problems.  They reduced motion to local motion: to change only of location from change in general.  The morphogenesis of a puppy dog became a different problem, addressed be it noted with different tools.  The tools of physics are not entirely suited to the puppy dog.   This way they could make stunning progress within a limited domain. 

But while this made specific problems more easily solvable to an approximation useful for the range of normal life, it glossed over the genuine complexity of the Ugly Reality™.  The model is elegant in its simplicity.  The world is not.  And just as the fish stinks from the head, the model begins to stink at the extremes, at the boundary conditions.  In an example using 153 consecutive heats of steel of a certain grade, the model predictions of the percentage to be expected above 90 ppm N2 accorded well with actual counts; but the predictions above 104 ppm diverged considerably.  A Normal Distribution goes off to infinity in both directions.  Nitrogen content of actual empirical physical heats does not. 

Ugly Reality™ versus Platonic Ideal
Left: Nitrogen content of 153 heats of a certain grade of steel.
Right: Idealized model of the data on the left.
(Nitrogen content above 90 ppm makes the steel too brittle for its application.)

 A Close Shave With Billy's Razor

I have been saying that modern science broke down the barriers that separated the heavens
and the earth, and that it united and unified the universe. And that is true. But, as I have
said, too, it did this by substituting for our world of quality and sense perception, the world
in which we live, and love, and die, another world—the world of quantity, of reified
geometry, a world in which, though there is place for everything, there is no place for man.

— Alexandre Koyré, Newtonian Studies

The problem comes when we forget that we made the simplification, and go on as if reality were as simple as their model, that what the world looks like depends largely on the instruments we use to view it.  Newtonian mechanics breaks down when things become very small or very fast.  But even today there are those who try to apply 19th cent. mechanical models to quantum events.  Or for that matter to maturing puppy dogs and the human mind.  But while 99% of the motions in the universe might be grasped by mechanical models that does not mean that

1. the universe really is mechanical; or
2. the other 1% might not be rather important.

After all, scientific progress has come largely from noting the importance of very small exceptions around the margins.  

The Old Inceptor

A curious thing is that oft-times a person disdainful of philosophy will cite "Ockham's Razor" as if it were a scientific truth rather than an methodological choice.  The simplest explanation is not necessarily the correct one -- and William of Ockham said this flat out.  He was making an epistemological point, not an ontological one.  What he said was that we ought to prefer simpler explanations because we can grasp them more easily.  In modern language, he said Don't have too many terms in your models or else you won't understand your own models.  In this age of Big Data and computer-generated regression on orthogonal factor analysis, this advice is well taken.  In many cases, the variables in the resultant model do not even correspond to variables in the original data!

We are right to laugh at the legendary philosophers with a predilection for the abstract who,
out of their loyalty to obsolete theories, refused to look at the world through a telescope.
One hopes the day might arrive when we will find equally amusing the scientific type
who refuses to remember what the world looks like without one.
-- Michael Augros, "Reconciling Science with Natural Philosophy"

For several hundred years, modern science has insisted on shoving things like color, sound, etc. into the closet of the mind, where it was comfortably labeled "subjective" and ignored in favor of bouncing photons, compression waves, and the like.  This was necessary because the methodology of modern science focused on only those properties of the physical world that were metrical and controllable.  Once this methodological choice was forgotten, we began to think that the World was metrical and controllable, rather than only the aspects of it that were visible using this instrument.  This does not mean that the view was wrong, only that it is and always was a partial view.  The world, it turns out, is neither geocentric nor heliocentric.  It is egocentric.  Everything revolves around my interests and specialties. 

It does no good to solve the problem of steam usage on Evaporator #2 unless one goes on to address other high-usage equipments, so that the aggregate sum of the savings adds up to a respectable amount.  If one eats an elephant one bite at a time, one should not stop with the first bite. 

In the same fashion, if one has developed a scientific law for a specified set of conditions, one should be aware of what those conditions are, and where the boundaries might lie.  By experimentation, we can discover a relationship between the length of a stretched rubber band and the weight hung on it.  But that does not mean that heavier weights will always result in longer stretches.  There comes a point where a domain boundary is reached - the tensile strength - and the rubber band breaks rather than stretches further.  Particle theories, wave theories, mechanistic theories, and so forth will likewise stretch only so far, even within the domain of their own competencies.  When they cross a domain boundary...  Well, Not all local motion is explained by Newtonian gravitation.  Consider magnetism as a source of motion.  And it is not scientifically unreasonable that there are some evolutions not explained by natural selection.

References

Michael Augros.  "Reconciling Science with Natural Philosophy," The Thomist 68 (2004): 105-41
Edward Feser.  "Naturalism in the news"  June 5, 2013
Michael F. Flynn.  "Spanking Bad Data Won't Make Them Behave."  Analog (to appear). 
David Weinberger.  "To Know, but Not Understand."  The Atlantic, Jan 3, 2012