Friday, March 25, 2011

A Return To The Invisible College In Science 4a

Will the crisis in reductionism in Science lead us to a new scientific method?

The origins of science were mixed with the mystical from early on.  In the ancient world, the greatest attention was paid to the movement of the stars and planets not for its own sake, but for how they would affect the events and actions of kings and nations.  Later, on in the 1600's western european men of letters were alchemists and astrologers.

Over time, science moved further away from these two studies and chose instead to concentrate on "practical" considerations such as agriculture and husbandry.  These men adopted what has been called the Baconian method, named after Sir Francis Bacon.  This despite the strong evidence that Bacon was an alchemist and perhaps even practiced astrology.

Science became more rational and under the period of the Enlightenment, it finally abandoned any notion of magic, metaphysics and eventually married itself to mathematics.  This union of the two disciplines accomplished great breakthroughs in understanding the universe and nature.  Western science then discovered the mathematical nature of nature.  This realization led to an attempt to quantize every phenomena observed in nature.  This almost exclusive obsession with mathematical quantification has also become the very soul of science.

There are however certain voices that are beginning to question this blind allegiance to mathematics in the sciences.  The former Jacob Schwartz, professor of Mathematics and Computer Science and founder of the Computer Science department at New York University wrote an article entitled, The Pernicious Influence of Mathematics on Science.  We will quote this article extensively because of its relevance.  He stated:
Jacob Schwartz
science tries to deal with reality that even the most precise sciences normally work with more or less ill-understood approximations toward which the scientist must maintain an appropriate skepticism. Thus, for instance, it may come as a shock to the mathematician to learn that the Schrodinger equation for the hydrogen atom, which he is able to solve only after a considerable effort of functional analysis and special function theory, is not a literally correct description of this atom, but only an approximation to a somewhat more correct equation taking account of spin, magnetic dipole, and relativistic effects; that this corrected equation is itself only an ill-understood approximation to an infinite set of quantum field-theoretical equations; and finally hat the quantum field theory, besides diverging, neglects a myriad of strange-particle interactions whose strength and form are largely unknown. The physicist, looking at the original Schrodinger equation, learns to sense in it the presence of many invisible terms, integral, integrodifferential, perhaps even more complicated types of operators, in addition to the differential terms visible, and this sense inspires an entirely appropriate disregard for the purely technical features of the equation which he sees. This very healthy self-skepticism is foreign to the mathematical approach.
Schwartz then laid bare one of the critical problems mathematics faces when applied to science:
Mathematics must deal with well-defined situations. Thus, in its relations with science mathematics depends on an intellectual effort outside of mathematics for the crucial specification of the approximation which mathematics is to take literally. Give a mathematician a situation which is the least bit ill-defined—he will first of all make it well defined. Perhaps appropriately, bit perhaps also inappropriately. The hydrogen atom illustrates this process nicely. The physicist asks: "What are the eigenfunctions of such-and-such a differential operator?" The mathematician replies: "The question as put is not well defined. First you must specify the linear space in which you wish to operate, then the precise domain of the operator as a subspace. Carrying all this out in the simplest way, we find the following result..." Whereupon the physicist may answer, much to the mathematician's chagrin: "Incidentally, I am not so much interested in the operator you have just analyzed as in the following operator, which has four or five additional small terms—how different is the analysis of this modified problem?" In the case just cited, one may perhaps consider that nothing much is lost, nothing at any rate but the vigor and wide sweep of the physicist's less formal attack. But, in other cases, the mathematician's habit of making definite his literal-mindedness may have more unfortunate consequences. The mathematician turns the scientist's theoretical assumptions, i.e., convenient points of analytical emphasis, into axioms, and then takes these axioms literally. This brings with it the danger that he may also persuade the scientist to take these axioms literally. The question, central to the scientific investigation but intensely disturbing in the mathematical context—what happens to all this if the axioms are relaxed?—is thereby put into shadow.
This article is so devastating to the common way mathematics is used in the sciences that one see a crisis approaching when the mathematical models will dramatically fail, thus perhaps bringing the entire mathematical/science edifice into partial disrepute.
The literal-mindedness of mathematics thus makes it essential, if mathematics is to be appropriately used in science, that the assumptions upon which mathematics is to elaborate be correctly chosen from a larger point of view, invisible to mathematics itself. The single-mindedness of mathematics reinforces this conclusion. Mathematics is able to deal successfully only with the simplest of situations, more precisely, with a complex situation only to the extent that rare good fortune makes this complex situation hinge upon a few dominant simple factors. Beyond the well-traversed path, mathematics loses its bearings in a jungle of unnamed special functions and impenetrable combinatorial particularities. Thus, the mathematical technique can only reach far if it starts from a point close to the simple essentials of a problem which has simple essentials. That form of wisdom which is the opposite of single-mindedness, the ability to keep many threads in hand, to draw for an argument from many disparate sources, is quite foreign to mathematics. This inability accounts for much of the difficulty which mathematics experiences in attempting to penetrate the social sciences. We may perhaps attempt a mathematical economics—but how difficult would be a mathematical history! Mathematics adjusts only with reluctance to the external, and vitally necessary, approximating of the scientists,
 and shudders each time a batch of small terms is cavalierly erased. Only with difficulty does it find its way to the scientist's ready grasp of the relative importance of may factors. Quite typically, science leaps ahead and mathematics plods behind.
Finally, at the risk of quoting the entire article, Schwartz unmasked what many of us have suspected all along about the way mathematics is being used in the sciences.
Related to this deficiency of mathematics, and perhaps more productive or rueful consequence, is the simple-mindedness of mathematics—its willingness, like that of a computing machine, to elaborate upon any idea, however absurd; to dress scientific brilliancies and scientific absurdities alike in the impressive uniform of formulae and theorems. Unfortunately however, an absurdity in uniform is far more persuasive than an absurdity unclad. The very fact that a theory appears in mathematical form, that, for instance, a theory has provided the occasion for the application of a fixed-point theorem, or of a result about difference equations, somehow makes us more ready to take it seriously. And the mathematical-intellectual effort of applying the theorem fixes in us the particular point of view of the theory with which we deal, making us blind to whatever appears neither as a dependent nore as an independent parameter in its mathematical formulation. The result, perhaps most common in the social sciences, is bad theory with a mathematical passport.
We present to you a documentary named Dangerous Knowledge on the slow realization on the part of mathematicians in the late 19th and early 20th century on weakening pillars supporting mathematics as sure fire way to understand the structure of the universe.  If you cannot see the embedded video here is the link:

In our final part 4b of this article we will consider the radical new direction Science might choose to go to escape the dilemma it finds itself in.  


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Sean said...

Yes he did indeed discuss the physical sciences as apologies. As for mathematics needing to simplify all that it models, this is what everyone does all the time. Is the author suggesting that there is a human out there who does not simplify/model their reality? We have to remember that if we investigate something as a human we are perceiving. What is perception? It is a biased abstraction of the mind. In other words we can only guess what are senses tell us about our world. If I show you my car and you tell me that car is blue...are you correct? What is blue? Is it kind of blue? Bluish-green? If I ask you for the time and you tell me it's 5:00pm..what does that mean? Close to 5:00? Exactly 5:00? But there was a bit of time that passed while you were telling me it was 5:00pm. So was it when you first moved your lips? Or after?

As humans we MUST simplify/model our world. An exact description will always be unattainable as there will ALWAYS be too many variables to include.


"it may come as a shock to the mathematician to learn that the Schrodinger equation for the hydrogen atom... is not a literally correct description of this atom"

Seriously??? Are there mathematicians out there this naive? It would surprise me that a mathematician would actually believe his/her formula could provide an actual description of anything. This simply isn't the case...ever. No system, no matter how "simple" can EVER be exactly described. system can EVER be exactly known by any human in any way. We ALL model our reality. If we didn't, then we would be able to define reality. Our greatest philosophers cannot do this.

The idea of simplifying and modeling is inherent to being human...whether you apply some mathematical formalism or use your human is modeled. Your perceptions....your ideas....your beliefs....your anything intellectual is a model. Everything you an approximation. All concepts in science, or any form of reality, is an approximation. To suggest otherwise is to invite a severe amount of naïveté into our understanding of the world. There is simply NO equation that does NOT neglect some (and more likely many) variables. Impossible.

Finally, yes individuals were vehemently attacked and not taken seriously. In fact this is usually a sign of a good scientist. If everyone accepts your theory initially there is no paradigm shift and you are not "thinking outside the box." Bohr hated Feynman's diagrams and his approach to QED is one the most successful descriptions (modeled of course) of our world. This is simply healthy skepticism and, in my view, is evolution working in the enterprise of science. The initial rejection from other scientists is the "environment" that provides the stress in order for only the fittest to survive. "Bad" mathematics and/or theory needs to have this stress applied just as "good" mathematics and/or theory must have. I am saying that the stress of rejection ensures that only the math that describes our reality to a decent degree survives.

I love your site and your comments. Keep up the great work.

Guillermo Santamaria said...

Sean these are wonderful comments. You took a very balanced approach to what you said. The only thing I would not agree with is that Schwartz's comments were limited to the use of mathematics in the social sciences. If you read this statements he mentions the "hard sciences" as well. I think his biggest problem is that mathematics by its nature needs to simplify all it models. Looks at this statement, "it may come as a shock to the mathematician to learn that the Schrodinger equation for the hydrogen atom, which he is able to solve only after a considerable effort of functional analysis and special function theory, is not a literally correct description of this atom, but only an approximation to a somewhat more correct equation taking account of spin, magnetic dipole, and relativistic effects; that this corrected equation is itself only an ill-understood approximation to an infinite set of quantum field-theoretical equations; and finally hat the quantum field theory, besides diverging, neglects a myriad of strange-particle interactions whose strength and form are largely unknown." As for being taken seriously by the "real" scientific community, science is full of people who were not taken seriously and who's theories were vehemently attacked by the scientific establishment. This coupled with the serious problems in the current peer review process, render the idea that science is like evolution in that only the fit will survive a bit optimistic. Nevertheless, I truly thank you for your thoughtful comments.

Sean said...

Definitely an interesting article. Mathematics should of course be taken with a grain of salt when applying its constructions to the real world...whatever that is. But far more important is the realization that in the absence of mathematics science is merely a collection of ideas and findings with little ability to predict or systematize in any useful manner. Without a mathematical description of our world we are left with doing what "feels right." This would of course undermine the edifice of quantum theory whose foundation is almost entirely counter-intuitive. Can bad theory propagate via a "mathematical passport" ? To a certain extent. Would such a theory have a far-reaching influence on other scientists? Unlikely. Almost nobody accepted relativity until it was both verified experimentally (Principe Island) and mathematically solidified by many many others. Quantum theory is THE most experimentally verified scientific theory of all time and would have made little to no progress in the absence of mathematics. This author is focusing more on the social sciences and so the question is what role will math play in this area. If the social studies want to attach the word science to what they do then math MUST play a significant and inherent role. Will social theories propagate on a mathematical passport of erroneous arguments? Unlikely. Regardless of any apparent crisis in reductionism, (what aspect of human endeavor isn't in some form of crisis?) any social theory MUST be compatible with existing hardcore sciences (physics and chemistry) otherwise it is destined to remain in the annals of social science and not taken seriously by the REAL scientific community. Science is as evolution...humans do not throw away the simply doesn't survive. Science advances via the mass contribution of individuals following the mindless powers of chaos theory. The fit survives....the rest dies...whether you like it or not. Mathematics, GOOD mathematics, that DOES say something about our reality is the only type that survives into the REAL enterprise we call science. Math isn't going away, and any "lies" that are put forth in its name will simply die as fast as an untrained swimmer sporting a lifeguard's uniform.

Guillermo Santamaria said...

Great comments Jan. I will look at those links.

Jan Henderson said...

I’m enjoying this series. In high school I became fascinated with the question of how pure mathematical theories (e.g., infinite-dimensional Hilbert spaces) turn out to describe the “real” world. Hilary Putnam writes on this. I majored in math as an undergrad, but switched to the “softer” history of science and medicine in grad school. Reductionism is a much underappreciated problem in Western (not Eastern) medicine, and of course medicine is not really a science, just a wanna be. This is getting some attention lately, thanks to the work of John Ioannidis (“Why Most Published Research Findings Are False”).

You might enjoy – if you’re not familiar with it already – the work of Steven L. Goldman, especially Science Wars: What Scientists Know and How They Know It (there’s an informative review at I think Goldman is brilliant.

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