Efficiency is indeed an epistemological problem of science see for example John L. casti's point of view in his paper "Confronting Science's Logical Limits " that one can also find in his book "Would be world "
http://search.barnesandnoble.com/booksearch/isbnInquiry.asp?userid=35IMKTANQA&isbn=0471196932&itm=4
Confronting Science's Logical Limits
by John L.Casti (professor at the Technical University of Vienna and at the Santa Fe Institute like J. Doyne farmer).
The mathematical models now used in many scientific fields may be fundamentally unable to answer certain questions about the real world. Yet there may be ways around these problems.
To anyone infected with the idea that the human mind is unlimited in its capacity to answer questions, a tour of 20th-century mathematics must be rather disturbing. In 1931 Kurt Gödel set forth his incompleteness theorem, which established that no system of deductive inference can answer all questions about numbers. A few years later Alan M. Turing proved an equivalent assertion about computer programs, which states that there is no systematic way to determine whether a given program will ever halt when processing a set of data. More recently, Gregory J. Chaitin of IBM has found arithmetic propositions whose truth can never be established by following any deductive rules.
These findings proscribe our ability to know in the world of mathematics and logic. Are there similar limits to our ability to answer questions about natural and human affairs? The first and perhaps most vexing task in confronting this issue is to settle what we mean by "scientific knowledge." To cut through this philosophical Gordian knot, let me adopt the perhaps moderately controversial position that a scientific way of answering a question takes the form of a set of rules, or program. We simply feed the question into the rules as input, turn the crank of logical deduction and wait for the answer to appear.
Thinking of scientific knowledge as being generated by what amounts to a computer program raises the issue of computational intractability. The difficulty of solving the celebrated travelling salesman problem, which involves finding the shortest route connecting a large number of cities, is widely believed to increase exponentially as the number of destinations rises. For example pinpointing the best itinerary for a, salesman visiting 100 cities would require examining 100 x 99 x 98 x 97 x...x 1 possibilities-a task that would take even the fastest computer billions of years to complete.
But such a computation is possible-at least in principle. Our focus is on questions for which there exists no program at all that can produce an answer. What would be needed for the world of physical phenomena to display the kind of logical unanswerability seen mathematics ? I contend that nature would have to be either inconsistent or incomplete, in the following senses. Consistency means that there are no true paradoxes in nature. In general, when we encounter what appears to be such a paradox-such as jets of gas that seemed to be ejected from quasars at faster than light speeds-subsequent investigation has provided a resolution. (The " super- luminal" jets turned out to be an optical illusion stemming from relativistic effects.)
PROTEIN FOLDING PROBLEM considers how amino acids (left) folds up instantaneously into extraordinarily complex, three-dimensional protein (right) Biologists are no trying to unravel the biochemical "rules" that proteins follow in accomplishing this feat.
Completeness of nature implies that a physical state cannot arise for no reason whatsoever; in short, there is a cause for every effect. Some analysts might object that quantum theory contradicts the claim that nature is consistent and complete. Actually, the equation governing the wave function of a quantum phenomenon provides a causal explanation for every observation (completeness) and is well defined at each instant in time (consistency ) . The notorious "paradoxes" of quantum mechanics arise because we insist on thinking of the quantum object as a classical one.
A Triad of Riddles
It is my belief that nature is both consistent and complete. On the other hand, science's dependence on mathematics and deduction hampers our ability to answer certain questions about the natural world. To bring this issue into sharper focus, let us look at three well-known problems from the areas of physics, biology and economics.
Stability of the solar system. The most famous question of classical mechanics is the N-body problem. Broadly speaking, this problem looks at the behaviour of a number, N, of point-size masses moving in accordance with Newton's law of gravitational attraction. One version of the problem addresses whether two or more of these bodies will collide or whether we will acquire an arbitrarily high velocity in a finite time.In his 1988 doctoral dissertation,Zhihong ( Jeff ) Xia of Northwestern University showed how a single body moving back and forth between two binary systems (for a total of five masses) could approach an arbitrarily high velocity and be expelled from the system. This result, which was based on a special geometric configuration of the bodies, says nothing about the specific case of our solar system. But it does suggest that perhaps the solar system might not be stable .More important, the finding offers new tools with which to investigate the matter.
Protein folding. The proteins making up every living organism formed as sequences of a large number of amino acids, strung out like beads on a necklace. Once the beads are put in the right sequence, the protein folds up rapidly to a highly specific three-dimensional structure that determines its function in the organism. It has been estimated that a supercomputer applying plausible rules for protein folding would need 10127 years to find the final folded form for even a very short sequence consisting of just 100 amino acids. In fact in 1993 Aviezri S. Fraenkel of the University of Pennsylvania showed that the mathematical formulation of the protein-folding problem is computationally "hard" in the same way that the travelling-salesman problem is hard. How does nature do it?
Market efficiency. One of the pillars on which the classical academic theory of finance rests is the idea that markets are "efficient." That is,the market immediately processes all information affecting the price of a stock or commodity and incorporates it into the current price of the security. Consequently prices should move in an unpredictable,essentially random fashion discounting the effect of inflation. This, in turn, means that trading schemes based on any publicly available information, such as price histories, should be useless; there can be no scheme that performs better than the market as a whole over a significant interval. But actual markets do not seem to pay much attention to academic theory. The finance literature is filled with such market "anomalies" as the low price-earnings ratio effect, which states that the stocks of firms whose prices are low relative to their earnings consistently outperform the market overall.
The Unreality of Mathematics
Our examination of the three questions posed above has yielded what appear to be three answers: the solar system may not be stable, protein folding is computationally hard, and financial markets are probably not completely efficient. But what each of these putative "answers" has in common is that it involves a mathematical representation of the real-world question, not the question itself.
