2015 |
If a straight line be cut at random, the square of the whole is equal to the squares on the segments and twice the rectangle contained by the segments.Now, given such a problem, Euclid or any of those following in his tradition would draw a diagram and proceed to prove from the axioms of plane geometry the correctness of the statement. But it isn't obvious how to apply this identity to other problems, or how it illustrates the behaviour of general numbers. Today, we'd express the problem and proceed as follows:
Once again, faced with the word problem, it's difficult to know where to begin, but once expressed in symbolic form, it can be solved by applying rules of algebra which many master before reaching high school. Indeed, the process of simplifying such an equation is so mechanical that computer tools are readily available to do so. Or consider the following brain-twister posed in the 7th century A.D. about the Greek mathematician and father of algebra Diophantus: how many years did he live?
“Here lies Diophantus,” the wonder behold.Oh, go ahead, give it a try before reading on! Today, we'd read through the problem and write a system of two simultaneous equations, where x is the age of Diophantus at his death and y the number of years his son lived. Then:
Through art algebraic, the stone tells how old;
“God gave him his boyhood one-sixth of his life,
One twelfth more as youth while whiskers grew rife;
And then one-seventh ere marriage begun;
In five years there came a bounding new son.
Alas, the dear child of master and sage
After attaining half the measure of his father's life chill fate took him.
After consoling his fate by the science of numbers for four years, he ended his life.”
Plug the second equation into the first, do a little algebraic symbol twiddling, and the answer, 84, pops right out. Note that not only are the rules for solving this equation the same as for any other, with a little practice it is easy to read the word problem and write down the equations ready to solve. Go back and re-read the original problem and the equations and you'll see how straightforwardly they follow. Once you have transformed a mass of words into symbols, they invite you to discover new ways in which they apply. What is the solution of the equation x+4=0? In antiquity many would have said the equation is meaningless: there is no number you can add to four to get zero. But that's because their conception of number was too limited: negative numbers such as −4 are completely valid and obey all the laws of algebra. By admitting them, we discovered we'd overlooked half of the real numbers. What about the solution to the equation x² + 4 = 0? This was again considered ill-formed, or imaginary, since the square of any real number, positive or negative, is positive. Another leap of imagination, admitting the square root of minus one to the family of numbers, expanded the number line into the complex plane, yielding the answer 2i as we'd now express it, and extending our concept of number into one which is now fundamental not only in abstract mathematics but also science and engineering. And in recognising negative and complex numbers, we'd come closer to unifying algebra and geometry by bringing rotation into the family of numbers. This book explores the groping over centuries toward a symbolic representation of mathematics which hid the specifics while revealing the commonality underlying them. As one who learned mathematics during the height of the “new math” craze, I can't recall a time when I didn't think of mathematics as a game of symbolic transformation of expressions which may or may not have any connection with the real world. But what one discovers in reading this book is that while this is a concept very easy to brainwash into a 7th grader, it was extraordinarily difficult for even some of the most brilliant humans ever to have lived to grasp in the first place. When Newton invented calculus, for example, he always expressed his “fluxions” as derivatives of time, and did not write of the general derivative of a function of arbitrary variables. Also, notation is important. Writing something in a more expressive and easily manipulated way can reveal new insights about it. We benefit not just from the discoveries of those in the past, but from those who created the symbolic language in which we now express them. This book is a treasure chest of information about how the language of science came to be. We encounter a host of characters along the way, not just great mathematicians and scientists, but scoundrels, master forgers, chauvinists, those who preserved precious manuscripts and those who burned them, all leading to the symbolic language in which we so effortlessly write and do mathematics today.
Assume now a group of people aware of the reality of interpersonal conflicts and in search of a way out of this predicament. And assume that I then propose the following as a solution: In every case of conflict, including conflicts in which I myself am involved, I will have the last and final word. I will be the ultimate judge as to who owns what and when and who is accordingly right or wrong in any dispute regarding scarce resources. This way, all conflicts can be avoided or smoothly resolved. What would be my chances of finding your or anyone else's agreement to this proposal? My guess is that my chances would be virtually zero, nil. In fact, you and most people will think of this proposal as ridiculous and likely consider me crazy, a case for psychiatric treatment. For you will immediately realize that under this proposal you must literally fear for your life and property. Because this solution would allow me to cause or provoke a conflict with you and then decide this conflict in my own favor. Indeed, under this proposal you would essentially give up your right to life and property or even any pretense to such a right. You have a right to life and property only insofar as I grant you such a right, i.e., as long as I decide to let you live and keep whatever you consider yours. Ultimately, only I have a right to life and I am the owner of all goods. And yet—and here is the puzzle—this obviously crazy solution is the reality. Wherever you look, it has been put into effect in the form of the institution of a State. The State is the ultimate judge in every case of conflict. There is no appeal beyond its verdicts. If you get into conflicts with the State, with its agents, it is the State and its agents who decide who is right and who is wrong. The State has the right to tax you. Thereby, it is the State that makes the decision how much of your property you are allowed to keep—that is, your property is only “fiat” property. And the State can make laws, legislate—that is, your entire life is at the mercy of the State. It can even order that you be killed—not in defense of your own life and property but in the defense of the State or whatever the State considers “defense” of its “state-property.”This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International License and may be redistributed pursuant to the terms of that license. In addition to the paperback and Kindle editions available from Amazon The book may be downloaded for free from the Library of the Mises Institute in PDF or EPUB formats, or read on-line in an HTML edition.
You are at war with us? Then we are at war with you. A condition of war has existed, and will continue to exist, until you surrender without condition, or until every drug judge, including you, … and every drug prosecutor, and every drug cop is dead. So have I said it. So shall it be.Shortly after the sentencing, Windsor Annesley's younger brother, Worthington (“Worthy”) meets with Matthew and the bookstore crew (including, of course, the feline contingent) to discuss a rumoured H. P. Lovecraft notebook, “The Miskatonic Manuscript”, which Lovecraft alluded to in correspondence but which has never been found. At the time, Lovecraft was visiting Worthy's great-uncle, Henry Annesley, who was conducting curious experiments aimed at seeing things beyond the range of human perception. It was right after this period that Lovecraft wrote his breakthrough story “From Beyond”. Worthy suspects that the story was based upon Henry Annesley's experiments, which may have opened a technological path to the other worlds described in Lovecraft's fiction and explored by Church of Cthulhu members through their sacraments. After discussing the odd career of Lovecraft, Worthy offers a handsome finder's fee to Matthew for the notebook. Matthew accepts. The game, on the leisurely time scale of the rare book world, is afoot. And finally, the manuscript is located. And now things start to get weird—very weird—Lovecraft weird. A mysterious gadget arrives with instructions to plug it into a computer. Impossible crimes. Glowing orbs. Secret laboratories. Native American shamans. Vortices. Big hungry things with sharp teeth. Matthew and Chantal find themselves on an adventure as risky and lurid as those on the Golden Age pulp science fiction shelves of the bookstore. Along with the adventure (in which a hero cat, Tabbyhunter, plays a key part), there are insightful quotes about the millennia humans have explored alternative realities through the use of plants placed on the Earth for that purpose by Nature's God, and the folly of those who would try to criminalise that human right through a coercive War on Drugs. The book concludes with a teaser for the next adventure, which I eagerly await. The full text of H. P. Lovecraft's “From Beyond” is included; if you've read the story before, you'll look at it an another light after reading this superb novel. End notes provide citations to items you might think fictional until you discover the extent to which we're living in the Crazy Years. Drug warriors, law 'n order fundamentalists, prudes, and those whose consciousness has never dared to broach the terrifying “what if” there's something more than we usually see out there may find this novel offensive or even dangerous. Libertarians, the adventurous, and lovers of a great yarn will delight in it. The cover art is racy, even by the standards of pulp, but completely faithful to the story. The link above is to the Kindle edition, which is available from Amazon. The hardcover, in a limited edition of 650 copies, numbered and signed by the author, is available from the publisher via AbeBooks.