martes, 12 de octubre de 2010

The Unreasonable Effectiveness of Maths in Natural Sciences

AND what if...:

our Maths are actualy the result of our own intuitions, constructed from our own sensorial experience?

NOTE THAT ex perience expell something out, out of use?

IF NOW WE WENT TO DISCOVER THAT EXPERIENCE IS THE MINOR CONTRIBUTOR TO SCIENCE, UNTIL RECENT PHENOMENOLOGICAL REVOLUTION...

THROUGH THAT EXPERIENTIAL EXPERIENCE SENSORIALY EQUIPED HUMANS SCANER THER TERRITORRIES, WITH DIFERENT TYPES OF ATTENTION. OK.

WE SCANNER EVERYDAY OUR TERRITORIES.
EACH OF US CAN DESCRIBE MANY OF THE DYNAMISC OF THEIR BIORREGION, HOUSES, OR SURROUNDINGS...

iN THAT WAY WE EVERY DAY ARE DEVOURING RELITY IN TONNES...

WITHIN THOSE TONNES OF REALITY WE ENGULF BY OUR ALWAYS HUNGER MKPUTOMKO,

WE INPUT ALSO INSIDE OUR BRAIN-AND-SENSES-ANDBODIES.

MANY TONNES OF EXPONENTIAL, DIFERENTIALS, AND TOUT THE CREME OF THE CREME OF THE PERCEPTIONS WE CONSTRUCT IN OUR EVERY DAY LIFE ZIRKUS.

nO PROBLEM IF WE ARE, OR WE WEREN'T NOT AWARE OF THAT.

the question is, as with socialdarwinism which actualy were at the basis of Darwin Cultural Thoughts, in our present case is not wich magicaly matematical formulas will connect splebndously with reality, physical, as we live with her every day,
not,

The question is that every day our conviviality with complex reality, changing, dynamical, chaotics, irregular, anarquist, buuuuuuuuuuuuuuuuur ciclic.

All that information we receive and interact with her,

is all iformation from that actual reality, wich of course can be related with the structural geometry of anyway of explainining physical world.

¿As if we scientists were su abstracticaly out of physical reality that now, we could one day to ask what that person asked in this ARTICLE"!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

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The Unreasonable Effectiveness of Mathematics in the Natural Sciences is the title of an article published in 1960 by the physicist Eugene Wigner^[1]. In it, he observed that the mathematical structure of a physics theory often points the way to further advances in that theory and even to empirical predictions, and argued that this is not just a coincidence and therefore must reflect some larger and deeper truth about both mathematics and physics.

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[edit] The miracle of mathematics in the natural sciences

Wigner begins his paper with the belief, common to all those familiar with mathematics, that mathematical concepts have applicability far beyond the context in which they were originally developed. Based on his experience, he says "it is important to point out that the mathematical formulation of the physicist’s often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena." He then invokes the fundamental law of gravitation as an example. Originally used to model freely falling bodies on the surface of the earth, this law was extended on the basis of what Wigner terms "very scanty observations" to describe the motion of the planets, where it "has proved accurate beyond all reasonable expectations."

Another oft-cited example is Maxwell's equations, derived to model the elementary electrical and magnetic phenomena known as of the mid 19th century. These equations also describe radio waves, discovered by Heinrich Hertz in 1887 a few years after Maxwell's death. Wigner sums up his argument by saying that "the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it." He concludes his paper with the same question with which he began:

The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.

[edit] The deep connection between science and mathematics

Wigner's work provided a fresh insight into both physics and the philosophy of mathematics, and has been fairly often cited in the academic literature on the philosophy of physics and of mathematics. Wigner speculated on the relationship between the philosophy of science and the foundations of mathematics as follows:

It is difficult to avoid the impression that a miracle confronts us here, quite comparable in its striking nature to the miracle that the human mind can string a thousand arguments together without getting itself into contradictions, or to the two miracles of laws of nature and of the human mind's capacity to divine them.

Later, Hilary Putnam (1975) explained these "two miracles" as being necessary consequences of a realist (but not Platonist) view of the philosophy of mathematics^[2]. However, in a passage discussing cognitive bias Wigner cautiously labeled as "not reliable," he went further:

The writer is convinced that it is useful, in epistemological discussions, to abandon the idealization that the level of human intelligence has a singular position on an absolute scale. In some cases it may even be useful to consider the attainment which is possible at the level of the intelligence of some other species.

Whether humans checking the results of humans can be considered an objective basis for observation of the known (to humans) universe is an interesting question, one followed up in both cosmology and the philosophy of mathematics.

Wigner also laid out the challenge of a cognitive approach to integrating the sciences:

A much more difficult and confusing situation would arise if we could, some day, establish a theory of the phenomena of consciousness, or of biology, which would be as coherent and convincing as our present theories of the inanimate world.

He further proposed that arguments could be found that might...

...put a heavy strain on our faith in our theories and on our belief in the reality of the concepts which we form. It would give us a deep sense of frustration in our search for what I called 'the ultimate truth'. The reason that such a situation is conceivable is that, fundamentally, we do not know why our theories work so well. Hence, their accuracy may not prove their truth and consistency. Indeed, it is this writer's belief that something rather akin to the situation which was described above exists if the present laws of heredity and of physics are confronted.

Some believe that this conflict exists in string theory, where very abstract models are impossible to test given existent experimental apparatus. While this remains the case, the "string" must be thought of either as real but untestable, or simply as an illusion or artifact of either mathematics or cognition.

[edit] Hamming's follow-on to Wigner

Richard Hamming, an applied mathematician and a founder of computer science, reflected on and extended Wigner's Unreasonable Effectiveness in 1980, mulling over four "partial explanations" for it^[3]. Hamming concluded that the four explanations he gave were unsatisfactory. They were:

1. Humans see what they look for. The belief that science is experimentally grounded is only partially true. Rather, our intellectual apparatus is such that much of what we see comes from the glasses we put on. Eddington went so far as to claim that a sufficiently wise mind could deduce all of physics, illustrating his point with the following joke: "Some men went fishing in the sea with a net, and upon examining what they caught they concluded that there was a minimum size to the fish in the sea."

Hamming gives four examples of nontrivial physical phenomena he believes arose from the mathematical tools employed and not from the intrinsic properties of physical reality.

Hamming proposes that Galileo discovered the law of falling bodies not by experimenting, but by simple but careful thinking. Hamming imagines Galileo as having engaged in the following thought experiment (Hamming calls it "scholastic reasoning"):

Suppose that a falling body broke into two pieces. Of course the two pieces would immediately slow down to their appropriate speeds. But suppose further that one piece happened to touch the other one. Would they now be one piece and both speed up? Suppose I tie the two pieces together. How tightly must I do it to make them one piece? A light string? A rope? Glue? When are two pieces one?"

There is simply no way a falling body can "answer" such hypothetical "questions." Hence Galileo would have concluded that "falling bodies need not know anything if they all fall with the same velocity, unless interfered with by another force." After coming up with this argument, Hamming found a related discussion in Polya (1963: 83-85)^[4]. Hamming's account does not reveal an awareness of the 20th century scholarly debate over just what Galileo did.

The inverse square law of universal gravitation necessarily follows from the conservation of energy and of space having three dimensions. Measuring the exponent in the law of universal gravitation is more a test of whether space is Euclidean than a test of the properties of the gravitational field.

The inequality at the heart of the uncertainty principle of quantum mechanics follows from the properties of Fourier integrals and from assuming time invariance.

Hamming argues that Albert Einstein's pioneering work on special relativity was largely "scholastic" in its approach. He knew from the outset what the theory should look like (although he only knew this because of the Michelson-Morley Experiment), and explored candidate theories with mathematical tools, not actual experiments. Hamming alleges that Einstein was so confident that his relativity theories were correct that the outcomes of observations designed to test them did not much interest him. If the observations were inconsistent with his theories, it would be the observations that were at fault.

2. Humans create and select the mathematics that fit a situation. The mathematics at hand does not always work. For example, when mere scalars proved awkward for understanding forces, first vectors, then tensors, were invented.

3. Mathematics addresses only a part of human experience. Much of human experience does not fall under science or mathematics but under the philosophy of value, including ethics, aesthetics, and political philosophy. To assert that the world can be explained via mathematics amounts to an act of faith.

4. Evolution has primed humans to think mathematically. The earliest lifeforms must have contained the seeds of the human ability to create and follow long chains of close reasoning. Hamming, whose expertise is far from biology, otherwise says little to flesh out this contention.

[edit] Tegmark's response

A different response, advocated by Physicist Max Tegmark in 2007, is that physics is so successfully described by mathematics because the physical world is completely mathematical, isomorphic to a mathematical structure, and that we are simply uncovering this bit by bit^[5]. In this interpretation, the various approximations that constitute our current physics theories are successful because simple mathematical structures can provide good approximations of certain aspects of more complex mathematical structures. In other words, our successful theories are not mathematics approximating physics, but mathematics approximating mathematics.

[edit] Response by Grattan-Guinness

Ivor Grattan-Guinness (2008) finds the effectiveness in question eminently reasonable, and explained in terms of analogy, generalisation, metaphor, and similar techniques.

[edit] Related quotes

The most incomprehensible thing about the universe is that it is comprehensible. — Albert Einstein

How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? — Albert Einstein

There is only one thing which is more unreasonable than the unreasonable effectiveness of mathematics in physics, and this is the unreasonable ineffectiveness of mathematics in biology. — Israel Gelfand^[6]

"... if nature is really structured with a mathematical language and mathematics invented by man can manage to understand it, this demonstrates something extraordinary. The objective structure of the universe and the intellectual structure of the human being coincide." - Pope Benedict XVI^[7]

[edit] See also

[edit] Further reading

Sundar Sarukkai (10 February 2005). "Revisiting the ‘unreasonable effectiveness’ of mathematics". Current Science 88 (3). http://www.ias.ac.in/currsci/feb102005/415.pdf.
Alex Kasman (April 2003). "Unreasonable Effectiveness". Math Horizons magazine: 29–31. http://kasmana.people.cofc.edu/MATHFICT/mfview.php?callnumber=mf334. , a piece of "mathematical fiction".
Arthur Lesk (2000). "The Unreasonable Effectiveness of Mathematics in Molecular Biology". The Mathematical Intelligencer 22 (2): 28–36. doi:10.1007/BF03025372.
Ivor Grattan-Guinness (2008). "Solving Wigner’s mystery: The reasonable (though perhaps limited) effectiveness of mathematics in the natural sciences". The Mathematical Intelligencer 30 (3): 7–17. doi:10.1007/BF02985373.