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Re: What Fun it is to Learn
|It is one thing to document something mathematically. It is quite another to prove something mathematically. It is even less favorable to try and grasp the reality of a phenomena soley through mathematics. After awhile, mathematics turns into a huge shell game.|
I'll start off ignoring the condition just to make a point.
Mathematics is the language of physics. Newton could not even begin to communicate his ideas about physics until he had invented calculus. You need mathematics to think about and to communicate with others about physics.
On the language aspect. In high school algebra class, I had difficulties with word problems. The main point was to set up the worded problem, but I could never figure out how. So I started college majoring in what I could do best: German, plus branching out into other languages. Then one day on PBS I watched a series of lectures Leonard Bernstein gave on a pet idea of his, musical linguistics (1973 Norton Lectures entitled, "The Unanswered Question" after a Charles Ives piece that violated standard melodic phrasing of call-and-response). At one point he presented a linguistical spectrum of degree of metaphor with poetry ranking higher than prose and music being the most metaphoric linguistical expression. I immediately went to the opposite pole of that spectrum, to the most concrete form of expression, and decided that that was where mathematics must lie. Referring to analytic geometry (graphing functions and non-functions as lines, curves, and conic sections), I saw that the equation of a line was an extremely concise and concrete description of each and every point on that line. The same with a circle and ellipses and parabolae and hyperbolae and all manner of curves. And word problems! Solving them must be simply a matter of translating from English to algebra and if there was one thing I had really learned, it was translating from one language to another.
If you cannot speak the language, how can you possibly communicate? Or even think? In 1984, the purpose of Newspeak was to limit citizens' ability to think; if they have no language to express an idea or ideal, then they cannot express it or even think it. In Stranger in a Strange Land, with knowledge of the Martian language came the ability to tap into enormous mental powers. If you don't know the language of physics, you cannot even think of physics.
But, true, there is the ability to imagine, to picture how things work. Indeed, that is what Darwin said of the evolution of complex organs, such as the eye, though he is frequently misquoted about that. What Darwin actually said was that the problem in trying to imagine how the eye could have evolved is due to limitations in our ability to imagine (meaning our ability to picture it in our minds). Then, as creationists consistently fail to inform you, he went on for another two pages naming one example after another in nature of intermediate forms of eyes that serve their possessors quite well, thank you very much! In his The Blind Watchmaker, Dawkins revisited the question by answering a common false creationist retort, "What good is half an eye?", with "Far better than a quarter of an eye, or none at all." (strictly from memory), whereupon he also launched into the intermediate steps for the evolution of the eye showing that it is not only functional throughout, but also that each step is an improvement over the previous.
Now, being able to picture in your mind is not only very satisfactory, but also very useful. Being able to do that in school, I could remember concepts much better and could even work out problems primarily through mental imagery (eg, during a Scout campout where we wanted to direct the boys through some star gazing, by knowing the orbital period of the moon, I was able to calculate in my head how much time after sunset we had on the last night before the moon rose and would spoil the view).
But at the same time, what we imagine is not true. I once read three decades ago an article about scientific illiteracy in which a physics problem would be present as an image and the respondents were to choose between different answer images. The one I remember was of a steel ball rolling down a spiral track that suddenly terminated. The majority of respondents chose the answer image of the ball falling in a spiral trajectory, whereas the real answer was the image of the ball going straight out of the track and falling in a downward curve due to gravity. Common sense is not common! The way that we think things should work is not always how they actually work. Drop two objects, both weighing differently, from a height, such as from the Tower of Pisa. Common sense tells us that the heavier object should hit the ground first, but in reality they both fall at the same rate. Why isn't that "common knowledge"? Because falling objects are also subject to aerodynamic drag. Drop a penny and a feather from the same height at the same time. The penny hits the ground before the feather does, each and every time. Now do it from the surface of the moon, which lacks a measurable atmosphere and hence eliminates aerodynamic drag. They both hit the ground at the same time, as demonstrated by one of the Apollo teams.
The appeal to ignore mathematics can be alluring, especially for creationists and other pseudo-scientists who want to ignore the math and the real evidence in favor of a lot of hand-waving. You can do a lot just with a lot of hand-waving, but only if you can ignore the math. For example, creationist idol "Dr" Kent Hovind presented this revelation about the rate at which the sun is losing mass "as it burns its fuel" (from his seminar video tape downloaded from his drdino.com site circa 2003, from the audio of video #7, "Questions and Answers", from 37 minutes 40 seconds to 39 minutes 54 second):
All you got to do is step outside and look up. Obviously the sun is burning. It's losing 5 million tons every second. You can't just keep losing 5 million tons a second, pretty soon you start to lose weight. And so the sun is losing this mass -- 5 million tons every second -- which means it used to be larger. And it used to be more massive. If you increase the mass of the sun, going backwards in time for several billion years, you start to create a problem with the gravitational balance between the earth and the sun. It's going to suck the earth in and destroy everything.
He's waving his hands here and throwing (quite literally!) astronomical figures out at you. 5 million tons per second. for about 5 billion years! How much mass is that being lost at that rate for that much time? Something to the order of 1024 tons! That is something to the order of a thousand earths! You, average Joe Christian or even some everyday rube off the street, look at that and think, "Wow! He must be right!"
Well, that is what we have when physics is reduced to nothing but mere hand-waving. Something to the order of 1024 tons lost over 5 billion years. OK, how does that compare to the total mass of the sun? Uh, the total mass of the sun is about a thousand times greater than the total mass lost. So then in 5 billion years the sun has lost a few hundredths of one percent of its total mass? Uh, yeah. So then because the sun's gravitational field, being directly proportional to its mass, used to be only a few hundredths of a percent more intense, had "sucked the earth in" by only about 60,000 miles? Uh ... uh ... oh! Look at that shiny object over there!
Allow them to do nothing but hand-wave and they will convince you of all kinds of false bullshit. But make them do the math ...
Please refer to my work-in-progress page at http://cre-ev.dwise1.net/solar_mass_loss/index.html. By the way, when I contacted Kent Hovind for more information on his solar-mass-loss claim and his sources for it, he tried everything he could to avoid discussing it, including twice trying to pick a fight with me over my AOL screenname (which I use here; it's really a very mundane story should you ever wish to hear it).
Now to return to what you had actually said (my emphasis added):
|It is even less favorable to try and grasp the reality of a phenomena soley through mathematics|
You do have a point there. Returning to Bernstein's Metaphor Spectrum, most of us operate in the middle, with prose. Poets (die Künstler (artists) as opposed to die Bürger (the middle-class bourgeoisie)) may prefer to operate more towards the metaphoric pole, while musicians don't even realize that they're making a linguistic choice as they just groove up there. Music and poetry sounds to enriching, while most regard mathematics as dry and sterile.
And yet ... I once found perhaps the most beautiful expression in mathematics. Conic sections are all the same except for the angle at which they intersect a cone. If you look at the analytic geometry formulae for circles, ellipses, parabolae, and hyperbolae, they're all different and increasingly messy. But those formulae are for rectangular 2-space (X-Y Cartesian coordinate system; des Carte had supposedly thought of it while imprisoned and looking at the bars in his cell window). There is also the polar coordinate system consisting of angles and radii. In polar coordinates, there is one and only one formula for all conic sections. The only difference is in the eccentricity (e=0 is a circle, e>0 and 1 is a hyperbola). One formula for all possible conic sections. All orbits being conic sections or composed of conic sections, all of which can be expressed by a single function ... now that is true beauty, more beautiful than we can imagine!
But there was a related event supportive of your (and most everybody else's) emotional needs as expressed by you in the quote box. In 1967, there was a conference at the Wistar Institute. One statement in those proceedings has been grossly misquoted in the creationist community, by the late Luther Sunderland, to the effect that a scientist had pronounced Darwinism's "survival of the fittest" to be a tautology. Here is what was actually said (I personally looked up that document) -- the parts that Sunderland had selectively quoted (thus pulling it out of context) are capitalized:
Dr. Waddington: I AM A BELIEVER THAT SOME OF THE BASIC STATEMENTS OF NEO-DARWINISM ARE VACUOUS; and I think there is a confusion here, possibly, about whether we are talking about Darwinism or neo-Darwinism. Dr. Medawar mentioned this phrase, 'the survival of the fittest,' and it is a very elementary, old-fashioned, long outdated concept; but, of course, this is what Darwin was talking about. By 'fittest,' he meant best able to carry out the functions of life, best adapted to some environmental situation and some way of life. By a fit horse, he meant a horse that could gallop fastest and escape best from wolves, or whatever it might be. That is a real theory which is perfectly capable of refutation.
What has happened to it since, in the process of turning this into a lot of mathematics, is that 'fitness' has been redefined, leaving out anything to do with way of life, simply in terms of leaving offspring. SO THE THEORY OF NEO-DARWINISM IS A THEORY OF THE EVOLUTION OF THE CHANGING OF THE POPULATION IN
RESPECT TO LEAVING OFFSPRING AND NOT IN RESPECT TO ANYTHING ELSE. NOTHING ELSE IS MENTIONED IN THE MATHEMATICAL THEORY OF NEO-DARWINISM. IT IS SMUGGLED IN AND EVERYDOGY HAS IN THE BACK OF HIS MIND THAT THE ANIMALS THAT LEAVE THE LARGEST NUMBER OF OFFSPRING ARE GOING TO BE THOSE BEST ADAPTED ALSO FOR EATING PECULIAR
VEGETATION, OR SOMETHING OF THIS SORT; BUT THIS IS NOT EXPLICIT IN THE THEORY. ALL THAT IS EXPLICIT IN THE THEORY IS THAT THEY WILL LEAVE MORE OFFSPRING.
THERE, YOU DO COME TO WHAT IS, IN EFFECT, A VACUOUS STATEMENT: NATURAL SELECTION IS THAT SOME THINGS LEAVE MORE OFFSPRING THAN OTHERS; AND YOU ASK, WHICH LEAVE MORE OFFSPRING THAN OTHERS; AND IT IS THOSE THAT LEAVE MORE OFFSPRING; AND THERE IS NOTHING MORE TO IT THAN THAT.
THE WHOLE REAL GUTS OF EVOLUTION-WHICH IS, HOW DO YOU COME TO HAVE HORSES AND TIGERS, AND THINGS- IS OUTSIDE THE MATHEMATICAL THEORY. The whole real guts of evolution -- which is, how do you come to have horses and tigers, and things -- is outside the mathematical theory. So when people say that a thing is vacuous, I think they may be thinking of this part of it, this type of statement. The sheer mathematical statement is largely vacuous. The actual way this is applied, not by the mathematical theorists but by the biologists working with the subject, is not vacuous at all.
(P.S.Moorehead, and M.M.Kaplan, Eds., Mathematical Challenges to the Neo-Darwinian Interpretation of Evolution, The Wistar Institute Symposim Monograph N. 5 (Philadelphia: Wistar Institute Press, 1967), pp.13,14)
Now, perhaps you do not understand what neo-Darwinism is. Basically, Darwinism is what Darwin had proposed a little over 150 years ago. One of the problems that Darwin was faced with and which he never could solve was how new traits were inherited and how they could establish themselves. His imaginary model was that of mixing pigments together in paint, but no new "pigment" could possibly survive that. He ended up coming up with pangenetics, a way in which new traits could be acquire through use and disuse, basically a retreat into a form of neo-Lamarckism. All the while on his bookshelf sat the answer to his dilemma, a monograph by an Augustinian friar, Gregor Mendel, which he had apparently never gotten around to reading.
Come the turn of the century (ie, 19th to 20th centuries), the other scientists had done what Darwin had apparently not done: read Mendel's monograph. They started to map out as best as they could the gene maps of lab animals, most notably fruit flies because of their short lifetimes. Genetic mutation became the scientific rage. And all their work directly contradicted Darwin's ideas about pangenetics, which they did not hesitate to point out. A continuing legacy of that time are all the "quotations of prominent scientists that Darwinism had been disproven", whereas in reality they were not disproving Darwinism and evolution, but rather only Darwin's faulty pangenetic ideas.
Then came the Modern evolutionary synthesis, in which scientists (eg, Julian Huxley (inventor of the term), R. A. Fisher, Theodosius Dobzhansky, J. B. S. Haldane, Sewall Wright, E. B. Ford, Ernst Mayr, Bernhard Rensch, Sergei Chetverikov, George Gaylord Simpson, and G. Ledyard Stebbins) had come to realize that instead of refuting Darwinism, Mendelian genetics actually complemented it and filled in the missing pieces. This modern synthesis produced neo-Darwinism and the mathematical discipline of population genetics. That population genetics approach was completely necessary, but at the same time it had to abstract several things, such as fitness.
So now we come to Waddington's concern that in neo-Darwinism fitness has been abstracted away such that we are left without a feeling for why a particular trait was more fit than another.
So where then does that leave us? Mathematics alone feels sterile, and yet if we were to abandon mathematics then we leave ourselves open to being deceived by all kinds of pseudo-scientific bullshit. There is a balance that must be maintained there.