I have not yet found the mathematics book I seek. Age Of Genius by Michael J. Bradley was not it, and yet I did gleam a few things from it– though mostly things I already suspected or half-realized.
It would help perhaps if I could better describe to myself the book I seek. Basically, I want to trace the historical development of mathematics, learning the concepts and procedures used in different periods of history, so that by the time I arrive at modern math, I have a pretty clear concept of why math looks the way it does.
I am, perforce, slowly writing the very book I seek to find, but I will never complete it, for Math is not my primary area, and I have many other interests that turn my head like so many pretty girls (and by God, the world is full of pretty girls!).
Mathematics is not PURE Reason, as I have sometimes been taught. There be ART in Mathematics, and artifice, too. What Mathematics turns out to be is a self-contained system logically and internally consistent as long as the rules laid down for its game are followed. I think it is possible that OTHER Mathematics could have been invented, equally logical, equally self-contained. Of course, the present system has a head start of a few thousand years and several million thoughtful mathematicians- -and since it seems to be performing pretty well for us– I doubt we’ll see the emergence of a completely new math anytime soon.
Mathematics is not as perfect or eternal as once I thought. In fact, the rules by which Mathematicians play were largely made-up along the way. Mathematicians had to learn, by trial and error, rules they could apply to their cipherin’ that would facilitate the continued smooth functioning of their game– a game which basically boils down to forms of addition.
Gaps here and there were discovered in Mathematics and had to been filled-in ad hoc. Places where the merely logical gaped or pinched had to be adjusted. Stopgaps were invented over the centuries in order to save this valuable game. Some of the patches added to Arithmetic over time include:
* the value of zero (how can something which doesn’t exist have a value?!)
* negative values (how can something be less-there than not-there?!)
* assigning the zero-power exponent a value (is there any intrinsic reason why something NOT times itself should equal one?!)
* granting existence to the square roots of negative numbers (come now, you don’t really believe in imaginary numbers do you?!)
* and of course, Mathematicians also had to settle upon the fundamental rules of playing their game– its operational order (upon seeing an equation, calculations must be performed in a certain order or else the whole game breaks down).
All the devices and assumptions and rules used in Mathematics are indeed logical– but they are not PURELY logically determined. As I said, some art has been used along the way.
Actually, all that stuff I’ve typed above has nothing particularly to do with The Age Of Genius, but the book set me to thinking about such things. Slowly, as I read books like Age Of Genius, the lineage of mathematics becomes clearer to me. For example, Astronomy (or astrology, or the study of the sky) is intricately bound-up in the DNA of today’s living mathematics. The Age Of Genius does not go into this heritage, but allow me some digressive progression here…
It was the study of the stars and Sun that often drove thinkers to come up with the new ideas that would today be called Mathematics. The Ancient Egyptians, as they did in so much, led the way. They realized that the seasonal flooding of the Nile ran parallel to events in the sky, all of which appeared to move in cycles. This heavenly connection to cycles and life-giving waters is one of the reasons I think circles have always had a bit of mystical aura surrounding them.
Measurement is half of Math. Two of the most important measurements when it comes to heavenly circles are the “Day” and the “Year.” The Egyptians figured the relationship between the Day and the Year was about 360-to-1. In other words, each year contained about 360 days. In some way of which I’m not certain (and as far as I know, no one is), this 360 number led to the number 60 becoming the base-unit of the emergent Mathematics of Egyptland. 360 is obviously divisible by 60, and so that is the reason cited as to why 60 was chosen as their base number.
Yet, why 60 instead of, say, 30? Thirty was much closer to the cycle of the moon, and it divides just as easily into 360. Or why not 12? Twelve divides 360, and also accounts for the 12 astrological houses (or are they 13? not my area). Or why did they not choose 45? …Or, for that matter, 3 or 9? I have to wonder if the number 10 did not in some way figure into the base-60 choice– after all, it is the number of “digits” (on our hands) and arguably the most natural way to count. Also, it might have felt aesthetically pleasing to the first Mathematicians to have six 10s make a 60, and six 60s make a 360.
Now… “make a 360.” That can imply turning a circle, can’t it? This is no coincidence. To the Ancient Egyptians, the heavens clearly described a circle… a circle which MOVED… and it moved a certain steady distance every night. This distance of night’s traverse equals 1/360ths of the full, 360-day circle of the heavenly sphere. This is precisely why today’s circles are composed of 360 “degrees“– we could just as well call degrees “nights” (though, admittedly it might lead to some confusion nowadays).
Birthed from the heavens, mathematics has thus been tied to the stars and to the circle (and to the base-60 system) for ages. [By the way, the base-60 system still comes up sometimes… the measurements we use for telling time still utilize the ancient base-60 method… 60 seconds in a minute, 60 minutes an hour; you can thank an Egyptian for that]. Much of the impetus for next few millennia behind mathematical innovation would come from the field of astronomy.
The way Math is taught in school, it is easy to forget its origins in circles. Of course, the clues are there if your eyes are open. For example, the circle-drawing compass has long been a symbol of mathematics and of the mathematician (as well as of the architect or mason). Also, I’m beginning to suspect that all this Sine and Cosine business that I used in Trigonometry (and hated using– though I love Geometry in general) comes not so much from the study of triangles as from the study of the chords and radii of circles. In fact, I’m starting to believe that all of Trigonometry is but a special branch of the study of the circle.
For instance, Bradley writes that al-Kashi [al KAH shee] (1380-1429) independently discovered a special case of the Law Of Cosines while attempting to find chord-lengths of a circle.
The Law Of Cosines, for the the record, is…
c^2 = a^2 + b^2 – 2ab * cos of angle C
Also, consider how significant “Pi” is in the realm of Mathematics. It is almost a holy number. Pi is simply the value you get whenever you divide a circle’s Circumference by its Diameter. No matter how big or small the circle, if it is perfect, the ratio will be 3.14. And it only adds to the mystique of Pi that it is and forever will be a rounded number– you can divide Diameter into Circumference to the 999th decimal place, and you still will have a reminder that you could keep on dividing.
Cones also hold a special place in the history of Mathematics. All those parabolas and hyperbolas and ellipses we studied in our Math courses were for centuries viewed by Mathematicians first and foremost as Conic Sections gotten by slicing a Cone at an angle (or straight across for a circle). In this way, even with the Cone, the Circle, or at least the Curve, crops up again.
One of the ironies of Mathematics is that, though it stems from observations of the qualities and quantities of a self-seeking curve (the Circle), the area beneath a curve was the last area of a normalish geometric shape to be solved. Newton and, independently, Leibniz, were still working on “the Calculus” in the 1600s.
Calculus… inside that perfect book on Mathematics I seek would be a combination of chapters which would elucidate the philosophy behind Calculus. Sure, I learned the steps and calculations used for completing Calculus problems in school, but I never truly deeply understood what I was doing. [addendum: months after writing this essay, I begin to get a better understanding. See: Hammering Shield’s Essay On Mathematics )
However, I have learned this much… That before Calculus, Mathematicians guesstimated the area beneath a curve by dividing it into very narrow rectangles and summing the area of all those rectangles. Quite commonsensical really. And this method could get you a pretty darn close estimate. But of course, we’re talking about Math here… exact answers are preferred. One of the things I hope to learn during this coming year is exactly how Calculus works under the hood.