A Perspective on Science 1 -- Science and mathematics

DrRocket · May 3, 2008

A Perspective on Science 1 -- Science and mathematicsFor the purpose of further discussion I propose the following definitions, with credit to TheShadow for the bulk of the definition of “theory”.   It is most important that it be recognized that a theory is much more than a conjecture or a hypothesis.  Science – the pursuit of explanations for the phenomena observed in the natural world.  The ultimate goal of a field of science is the development of a theory based on a minimal number of principles with predictive power that is consistent will all verifiable observations. Theory – a set of statements or principles devised to explain a group of facts or phenomena, especially one that has been repeatedly tested or is widely accepted and can be used to make predictions about natural phenomena.  Most successful theories involve a description using a mathematical model.Mathematics – the study of any kind of order that the human mind can recognize based on a small number of ideas, called axioms, that are assumed to be true.  All other mathematical assertions must be consequences of the fundamental axioms and any relevant outside definitions. Mathematics is not a science.  It differs from a science in two respects.  First, its objective is not specifically to explain phenomena from the natural world but rather to study order and logical connections among concepts.  Second, while the method by which mathematical “truths” are determined is often via close examination of representative examples, the method by which those ideas are verified is by deductive logic alone, based on a small set of axioms and definitions constructed to define the issue at hand and make use of the axioms in a strictly logical argument.  No attempt is made to verify the “truth” of those axioms, though they may often seem to be intuitively obvious. With apologies to formal logicians, I will concentrate on the foundations as used by  most other working mathematicians.  The axioms on which mathematics is based are basically those embodied in what you may have learned in school as set theory, to which one adds the Peano postulates, which essentially define the natural numbers and the axiom of choice.  The axiom of choice basically admits the possibility of “choosing” an element from each member of an arbitrary family of non-empty sets.  This seems quite obvious, but has some rather non-intuitive consequences.  I will not go into any of the finer points of logic or set theory except to note that there still remain some difficulties at the root of these subjects, including a paradox or two that can be obtained when one deals with very large families.  Basically there are some things that are too big to be sets, and in particular the notion of the set of all sets gives rise to Russell’s paradox.  For the mathematics necessary for science we need not be overly concerned about such issues.The point of the previous paragraph is that mathematics has been successfully developed using a very small set of basic assumptions and rules.  From those assumptions one can actually construct the real number system, show that it is topologically complete and from there go on to develop calculus and more advanced mathematics.  Literally all of mathematics can be traced back to these roots.  Not only can it be done in principle, it is quite often done in fact during the education of a research mathematician.  For those who would like to see the development of the real and complex numbers from the basics, I recommend the book “Foundations of Analysis” by Landau.  It is short, and his style is dry and telegraphic, but he gets the job done quickly and cleanly.  For those of you who have a life, you can take my word for it.Physics is a science.  It attempts to explain, with as few rules as possible, the workings of the natural world.  That is a bit more of a challenge than the one given to mathematicians, since mathematicians get to choose that which they attempt to organize and explain.  Physicists have their problems thrust upon them by natures, and often in the form of experimental data which must be understood and some of which may be found to be flawed and in need of discard.  Knowing what to keep and what to discard is a challenge. Those physicists who are pursuing the “theory of everything” are in fact trying to emulate the mathematical foundations of set theory and logic to find a framework from which the rest of science can flow.  This has proved to be a daunting task, worthy of the best minds on the planet.Ernest Rutherford once said “All science is either physics or stamp collecting.”  I find this a very perceptive statement if interpreted properly. That interpretation is not intended to denigrate other areas of science.I believe that Rutherford’s point is that the objective of science must be to develop quantitative models with predictive power, as is the obvious case in the study of physics.  Physics thus stands as a model for science in its ability to exhibit the connection between theory and experiment.  In a perfect world all branches of science would be based on fundamental physics and derivable from those basic laws.  Would that we were actually smart enough to accomplish that task.Science does not proceed solely by deductive means.  The defining issues of science come from observations of the natural world, and the recognition of some order and regularity in the observed phenomena.  Before modern genetics could be developed there first needed to be a scheme for classification of flora and fauna – i.e, some judicious stamp collecting.   Mendel’s laws of inheritance are another example of judicious and productive stamp collecting.   Before modern chemistry could arise there needed to be the development of the periodic table – again judicious stamp collecting, on the part of Mendeleev.  There is a role for observation and correlation, but that task marks only the beginning of the development of a scientific discipline.  The task of developing quantitative predictive models remains the end goal, but one that increases in difficulty as the systems studies by a branch of science become more complex.  The more complex the system being studied, the more important is the role of judicious “stamp collecting” in providing an organization of the data so that quantitative predictive theories can eventually be developed.For the chemists in the audience, and for the purpose at hand, I consider chemistry to be physics – in fact one of the most useful branches of applied quantum electrodynamics.Before ending this section, there is one other role of mathematics in the sciences, and in physics in particular, that is worth discussing.  To start out this discussion I would like to quote Richard Feynman from The Character of Physical Law (a book that I highly recommend):            “To summarize, I would use the words of Jeans, who said that ‘the Great Architect seems to be a mathematician.  To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature.  C.P Snow talked about two cultures.  I really think that those two cultures separate people who have and people who have not had this experience of understanding mathematics well enough to appreciate nature once.” One will find similar sentiments expressed by Weinberg, Dirac and Einstein.  Dirac used the beauty of physical laws expressed in mathematical terms as a guiding principle of his research.  Einstein expressed interest only in beautiful theories.  Eugene Wigner wrote an essay entitled “ The Unreasonable Effectiveness of Mathematics in the Natural Sciences”, which I highly recommend.  The major point here is that mathematical beauty is an aspect of the development of physical law that has inspired some of the most productive scientists in the history of mankind and is an important factor in their work.  It is not something that is usually discussed as a part of the “scientific method”, but it is something that has had a great influence on the development of scientific theories.  It has been proved empirically to be an effective tool in the evaluation of the likelihood that a theory will stand the test of time and of experiment.  Ignore it at your peril. One might rightly take the position of agreeing with me regarding the role of mathematical beauty and then challenge me to explain how to recognize and evaluate it.  I readily admit that I am not up to that challenge.  I think that I can recognize it myself, but am at a loss to explain it to another or to teach another to see it.  I find the notion readily accepted by most mathematicians and physicists, but not amenable to objective discussion.  I suspect that artists might understand the notion, but be unable to participate directly since they are attuned to a different sort of beauty.   I think I can say to the category 2 population that may have read this far that if you continue to study science you may eventually develop an understanding of this aspect of physical law.  I hope that is so.   Your scientific experiences will be the better for it. <div class="Discussion_UserSignature"> </div>

nimbus · May 3, 2008

n/t <div class="Discussion_UserSignature"> </div>

nimbus · May 3, 2008

I hope I didn't miss the formating you had intended by too far.  If you come back and edit your post to match what you meant it to look like, I'll clear my above post.My comment on that sense of beauty: I think it's what crackpots let themselves be lured into without caring whether it matches reality or not. They're only after the beauty they think they've recognized in nature, and just get drunk with it. That's pretty close to insanity, in my opinion. Try and slap em out of it, and they'll get very upset, because you're taking away their source of happiness.. However naive and/or nonsensical it is. "Stuck on aesthetic", as it were. <div class="Discussion_UserSignature"> </div>

DrRocket · May 10, 2008

Replying to: <DIV CLASS='Discussion_PostQuote'>I hope I didn't miss the formating you had intended by too far.  If you come back and edit your post to match what you meant it to look like, I'll clear my above post.My comment on that sense of beauty: I think it's what crackpots let themselves be lured into without caring whether it matches reality or not. They're only after the beauty they think they've recognized in nature, and just get drunk with it. That's pretty close to insanity, in my opinion.Try and slap em out of it, and they'll get very upset, because you're taking away their source of happiness.. However naive and/or nonsensical it is."Stuck on aesthetic", as it were. Posted by nimbus</DIV>You did a great job on the formatting.  I have no idea how you managed to do so well, since in the compressed state even I had a hard time separating paragraphs,  and I could look in the back of the book.I think the crackpots see something quite different from the beauty that I was tryiing to convey.  That beauty is related to simplicity and tight logic, the ability to convey deep meaning with a set of simple principles from which the remainder of the subject can be deduced.  Volumes have been written on the consequences of F=ma. It is something that is difficult to see without a firm foundation in mathematics.  I have not yet run across a crackpot who is proficient with mathematics (or with physics), and I can't really appreciate what it is they are hallucinatiing.  I think they are fixated, but that fixation lies in something other than what a real physicist or mathematician means by beauty.  Maybe they are fixated on "pretty pictures" much as an infant can be with wallpaper patterns or with colorful moving toys.  Or maybe they are fascinated in the manner of a someone on an acid trip.  I don't know. The beauty of logical reasoning and mathematical elegance certainly doesn't make an impression on them.  Neither does the intricacy and interrelationship of know physical laws, in which there is, in my opinion, deep beauty.Let me try give an example, to ilustrate and perhaps to spark some useful discussion.  This is more that a wee bit speculative, but bear with me.  I am going to assume for the purpose of this example that the idea put forth by Roger Penrose that the indeterminacy of quantum mechanics  in some accounts for what we perceive as consciousness and free will.  I don't know if this true, but it does seem plausible.  So here goes.On the one hand we have the macroscopic world which seems to be deterministic and extremely predictable.  Newton's mechanics allows very precise predictions of the motion of bodies and Maxwell's equations for electrodynamics cover the remainder of things in our everyday macroscopic experience.  We can precisely predict the motion of satellites and planets.  We can design radios and telephones, and iPods that work reliably.  We have electric power at our beck and call.  It appears that if we only had enough initial data we could in principle predict everything in the future.  But that would be really boring and we ourselves might be nothiing more than wind-up dolls going through the motions. On the other hand we have the quantum world of the atom.  Nothing is for certain, and the best that can be done is to predict probabilities.  Maybe a particle will decay, and maybe it won't.  If it decays we don't know when it will do so.  Maybe and electron is stopped by the potential wall, and maybe it tunnels through.  Maybe the photon went through that slot and maybe it didn't and the answer may depend on whether and how its passage was recorded.  Nothing is for certain.  And perhaps this is why we seem to not be just responding to a previously written program.But Quantum mechanics also applies, in principle in the macroscopic world.  There is a non-zero probability that each atom in the sun could suddenly appear in my back yard.  That would be most unfortunate, but it is theoretically possible.  But it doesn't happen.  Somehow, chance and indeterminancy at the atomic level succumb to the law of large numbers at the macroscopic level and predictability is restored.  Were it not so we probably could not survive (certainly not if the sun popped up willy nilly).  So, somewhere in the grand scheme of physical law there is room for at least perceived free willl and still a predictiable place in which to exercise it.  In the laws of physics and the disparate scales of the elementary particles, the atom, the world, the solar system, the galaxy and the universe is an elegance that is awe-inspiring.You can leave out the speculation linking quantum mechanics to free will and the picture is still pretty awesom, but I think if you accept that piece of speculation at least momentarily it gets across the idea of beauty a bit more clearly. <div class="Discussion_UserSignature"> </div>

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A Perspective on Science 1 -- Science and mathematics

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