Sidelights on Number:  The Alliance between Mathematics and Science

M. C. Ma

Preface:  I am indebted to colleagues and friends for their invaluable suggestions and support, especially Tolen Parall, and Eleanor Schorkbsky for the revision of the manuscript.  Although there are instances in which I have decided to not follow their sound advice, I have done so only with the utmost consideration.  It seemed necessary to leave some sections as they were originally written in the first m.s.  As I do not  yet know if this is the more prudent of the two choices we shall have to wait and see.  If in the future I should realize that such a decision was a mistake, the reader can expect a few words from the present author in the preface to the second edition.  No doubt he will offer countless justificatons for his lack of foresight at that time!  Under these circumstances I acknowledge sole responsibility for any errors, large or small.  If this grand experiment should in fact generate in our youth some interest, perhaps even a consideration of independent and personal study, if this and nothing else, the present author will be fully satisfied.  I also find myself unable to deny several counts against me on the grounds that I have treated the subject matter with excessive zeal.  "It's just mathematics!" says the little girl with a thousand hairpins.  So it is for you and another to which I dedicate this work, even against your protests.

I am under the assumption that those topics which have interested me, will stand a chance of interesting you, my reader.  I take this time also to bring to the reader's attention that there is no particular order in which any of this should be read. 

M.C. Ma
August 2005

To Erik-John, an old friend, without which, no such work would have been possible and
To Abby, who needs no excuse

If "Number rules the universe" as Pythagoras asserted, Number is merely our delegate to the throne, for we rule Number.

E. T. Bell

Contents

Introduction

        0-1    The Need for Education Reform:  Why should the non-mathematician concern himself with geometry, trigonometry, and calculus when he can devote this time instead to the study of practical disciplines- basic electronics, plumbing, and the like?  One may reply, "mathematics for mathematics sake is reason enough."  What if a 12-year curriculum included an in-depth study of adobe brick houses?  Should we argue the study of adobe bricks for their own sake?  The proper training of our youth is as important to the wellfare of Science and Mathematics as are the current theories being created today, for who will create them tomorrow?

        0-2    The Role of Science in Mathematical Creation:  How is mathematics created?  Certainly not deductively as the textbooks seem to indicate.  How then?

Chapter 1: Mathematics, Queen and Servant of Science

        1-1   Zeno's Paradoxes: The most convincing arguments against Zeno’s conclusions are those contradictions we can observe readily –every time we walk from the living room to the study, whenever we see Lucky Mistress overtake Yesterday’s Lunch at the track.  One needs only to recall that the strongest evidence against the generality of Newtonian mechanics was the plain and visible.  Any scientist that looked through a particle accelerator was convinced, more so than any proof would render him to believe otherwise.  But the present author shall appease the reader with mathematical arguments. 

        1-2    Calculus:  We shall see that the invention of the Calculus was motivated by a desire to uncover Nature's secret scheme, that the subject is not as dry as you would be lead to believe.  The environment leading up to Newton and Leibniz's differential calculus was one of intellectual fervor, in which the average educated man, no longer satisfied with mere myths and superstitions, set out with great strides to conquer his own ignorance. He sought physical explanations, and in desiring to understand planetary motion and the behavior of light, among other things, mathematics most fruitful branch was born.  Let us not forget that those looking through the telescopes were themselves the mathematicians.  What the average textbook has to offer are the shells of palaces, for mathematics is alive.  So let us not work ourselves up needlessly in remembering details and techniques which the non-mathematician will never need.  Let us instead look at the beast, not counting his hairs, but instead see if we can by some other means experience the same excitement as those that first discovered and those that later tamed him.  

Chapter 2: Applied Science and Engineering

        2-1    Matlab:  Matlab is like an abacus with a keyboard.  Not satisfied with the analogy?  Let us just say that an engineer that can only work with a hammer usually sees only nails before him.  Stop asking me to define what Matlab is.  Just use it and see!

        2-2   Torque and Horsepower: 

Chapter 3: Selected Topics in Physics

        3-1    A Very Brief introduction to Vectors:  Fundamental to all of Physics is the idealization of a vector.  What is all this talk about vectors?  It will be shown that a vector is quite intuitive after all.

Chapter 4: Mind-candy for Abby

        4-0    Who is Abby?  Self-Explanatory, no?

        4-1    Syllogisms for Abby:  So you mean to say that what is said cannot mean what is to be meant to be said?  Kid, you can't throw me off so easily.  Watch, I'm going to take your head for a spin.

       4-2    Lessons for Abby:  "What about a circle?"  "Abby, what do you mean what about a circle?"  "No, I'm asking you about the circle.  Why are you asking me about the circle?"  "I'm asking you what the what of the circle is referring to."  "To the circle."  "I know the circle but what of the circle."  So goes a usual lecture of ours.  These are lessons I have written for Abby.  I have only made minor changes to my original set of notes so they still read much like the informal lectures they were intended to be.

         4-3     Puzzles for Abby:    Searching for words in those scrambled matrices are utterly pointless, like searching for a number through a whole phonebook.  A good teasers should work the brain and have you scratching your head up until the very end.

        4-4     Abby in Relativity Land:    What's all this talk about relativity?  It's been exactly 100 years since the publication of Einstein's special theory and his manuscript on Brownian motion, and since our number system is base 10 we celebrate. 

         4-5    Stories and Links:  I have designed this section for the class of little girls that like to collect stickers.  If you are not a member of this aggregate I refer you to the Calculus or Engineering Section.  Leave me to my vices.

Chapter 5: Essays and Selected Works

      5-1    Speculative Fiction:  A term which I use in place of science fiction. However we choose to label the literature, without exception, we must note that it is often essential, for the consistency of a story, that a SF writer employ some assumptions unacceptable to the average scientist.  Nonetheless we must accept them as givens if we are to follow along.  It is no surprise that some readers balk when asked to accept this nonexistent camel, one which, furthermore, is said to be found in a nonexistent desert.  So in asking that our readers accept conclusions derived from those premises we cannot deny that we are asking them to swallow this camel whole.  Let us however note that these facts have not been disproved by physics, just deemed highly unlikely.  For instance, many stories depend upon the premise that light-speed travel is possible and even feasible (at one time rocketing to the moon was a feat that seemed wholly impossible, not just unlikely).  Therefore we should expect that these stories should take place sometime in the future, and it must, or else structurally the work would be contrary to the scientific facts of today. A good SF writer, however doesn't guess blindly. With a knowledge of where we've been and what we're doing, he is able to extrapolate new data and predict future trends; that is, he writes of a future that might take place, not one that must. The key elements are science and mathematics, without which you have just brain-eaters, brain-eaters that happen to find us interesting enough to want to subjugate the race.

        5-2    Free Verse:  A favorite pastime of every mechanical engineer is writing free verse.  Absurd you say?  But where does he keep them?  Check his lowest drawer and you will be sure to find some.

        5-3    Clarity:  A Comedy on Manners:  This is an unfinished play.  Don't wait up, I have no intention on finishing it.

        5-4    Problems of Philosophy:  Philosophy: Searching for a black cat in a dark room that isn't there.

Chapter 6: Essays and Selected Works of Others

       6-1    To Learners:  An excerpt from Lewis Carroll's Symbolic Logic. For anyone who habitually thinks about thinking, the difference between learning to play a violin, enjoying Keat's Ode to a Grecian Urn, and studying a treatise on wave mechanics is a matter of specifiying the parameter; the formula remains the same.  Once you master the technical details the rest follows naturally.  In the first you master the scales and harmonics, the second, the meter and rhythm, the last the symbolics and invariant relations, which are common to all mathematical sciences.  Time spent learning about learning is not time wasted.

Recommended Sites

I never realized how useful a bibliography was until I had read Count Korzybski's Science and Sanity.  In place of a bibliography are some recommended links.  This way the reader can find additional sources without havng to leave his terminal. 

Nature: All that is needed to understand a majority of the content here is a basic working knowledge of introductory calculus and physics.  Nature allows a nonscientist to keep up with current advances without burdening him with details.  This is not a watered-down popular site, however, as there are links to the original manuscripts.

NASA: When I was a squirt I had read my fair share of  science fiction.  Rocketship Galileo, anyone?  I am now an incurable idealist. I can't really think of myself as a young man anymore, and seeing how old men have their foolish dreams, I try not to ask for much.  I just want to stick around long enough to see spacesuits sold at Wal-Mart: "Discount on Junior Class 2 Pressure Suits, Aisle 7."  I could live a thousand  lives and this still shouldn't concern me, but something tells me that this hairless mammal we call homo sapien has been around too long to let one star determine his destiny.  We have to go back to the moon.  In colonizing earth's only satellite we take our first step as a race in space, not just a race on one planet.  When I see what NASA is doing I think, we won't go out like the T-rex, we won't go out at all.  As individuals we must die, but as a race we can live on forever.  Spread the word, with your help we can have the public fall in love with space again.

Heinlein Society:  What the Nobel Prize award does for Science in motivating men to create new theories is what the Heinlein Society does for the space program.  It offers a cash award to those individuals that advance private commercial space flight.

Addresses to the graduating class, Annapolis Naval Academy:  It had always been my recommendation to any man feeling that the world has treated him unfairly to immediately join the military to see for himself first hand what fairness this world has to offer.  One only has to recall the Budapest Massacre or the subjugation of Lithuania by the Soviets to see that this world is anything but fair.  I do not think it is mere politics when I say this, that few things are as sacred as an individual's sense of personal-honor, and you can always tell the difference between the man who stays up late thinking about his and the scoundrel who has no such conception.

Project Gutenberg:  What happens when a copyright for a book expires?  It becomes free.  You can download Huckleberry Finn, Macbeth, or the complete sonnets of Keats.  How do these online books become available?  It's simple.  Ordinary people volunteer to type books as textfiles.  Given enough volunteers, the result is this free electronic library- no late fees.