eBook - ePub
Mathematical Elegance
An Approachable Guide to Understanding Basic Concepts
Steven Goldberg
This is a test
Share book
- 123 pages
- English
- ePUB (mobile friendly)
- Available on iOS & Android
eBook - ePub
Mathematical Elegance
An Approachable Guide to Understanding Basic Concepts
Steven Goldberg
Book details
Book preview
Table of contents
Citations
About This Book
The heart of mathematics is its elegance; the way it all fits together. Unfortunately, its beauty often eludes the vast majority of people who are intimidated by fear of the difficulty of numbers. Mathematical Elegance remedies this. Using hundreds of examples, the author presents a view of the mathematical landscape that is both accessible and fascinating. At a time of concern that American youth are bored by math, there is renewed interest in improving math skills. Mathematical Elegance stimulates students, along with those already experienced in the discipline, to explore some of the unexpected pleasures of quantitative thinking. Invoking mathematical proofs famous for their simplicity and brainteasers that are fun and illuminating, the author leaves readers feeling exuberant-as well as convinced that their IQs have been raised by ten points. A host of anecdotes about well-known mathematicians humanize and provide new insights into their lofty subjects. Recalling such classic works as Lewis Carroll's Introduction to Logic and A Mathematician Reads the Newspaper by John Allen Paulos, Mathematical Elegance will energize and delight a wide audience, ranging from intellectually curious students to the enthusiastic general reader.
Frequently asked questions
How do I cancel my subscription?
Can/how do I download books?
At the moment all of our mobile-responsive ePub books are available to download via the app. Most of our PDFs are also available to download and we're working on making the final remaining ones downloadable now. Learn more here.
What is the difference between the pricing plans?
Both plans give you full access to the library and all of Perlegoâs features. The only differences are the price and subscription period: With the annual plan youâll save around 30% compared to 12 months on the monthly plan.
What is Perlego?
We are an online textbook subscription service, where you can get access to an entire online library for less than the price of a single book per month. With over 1 million books across 1000+ topics, weâve got you covered! Learn more here.
Do you support text-to-speech?
Look out for the read-aloud symbol on your next book to see if you can listen to it. The read-aloud tool reads text aloud for you, highlighting the text as it is being read. You can pause it, speed it up and slow it down. Learn more here.
Is Mathematical Elegance an online PDF/ePUB?
Yes, you can access Mathematical Elegance by Steven Goldberg in PDF and/or ePUB format, as well as other popular books in Mathematics & History & Philosophy of Mathematics. We have over one million books available in our catalogue for you to explore.
Information
1
The Lay of the Land
Well, I Did Not Want to Know That Anyway
Many of the greatest discoveries of the twentieth century are discoveries of the inherent limitations that will forever withhold knowledge from even our smartest descendants.
Gödelâs Theorem: For millennia mathematicians dreamt of discovering an all-encompassing mathematical system in which it was possible to prove every mathematical truth. Half a century ago, however, Kurt Gödel proved that every nontrivial logical and mathematical system will possess truths not provable in that system. (Trust me, you do not want to read here how he proved this.)
We can demonstrate the truths using a more inclusive system, but we cannot prove that a larger system will have truths without going to an even more robust system. (Ad infinitum.)
The dream is dead.
Heisenbergâs Uncertainty Principle: There is a limit beyond which full knowledge is impossible. In other words, there are things we can never, even in principle, know. For example, we can never learn both the position and velocity of a subatomic particle. We can learn either with as much precision as we wish, but the more precisely we know one, the less precisely we can know the other.
Hereâs why. When we measure something, we use particles to see the thing we measure and to make the measurement. For example, we use photons, particles of light, to see and measure the tabletop we wish to measure. The fact that we use these particles makes no practical difference when we are viewing and measuring tabletops, elephants, or even bacteria; the effect of the tiny photons on the thing we measure is essentially nonexistent and has no more effect than a ping-pong ball thrown at a mountain range. However, when we attempt to measure particles on the tiny scale of the particles we use to measure them, the effect of the latter on the former is tremendous and sets limits on what we can ever know about the measured particles.
Unpredictability: Even when a system is determinative, it is often the case that complexity and sensitive dependence on initial conditions renders the system forever unknowable in practice. This is why we will never be any good at predicting weather conditions more than a couple of weeks ahead.
Unsolvable in Practice (Probably): There are a large number of problems that are probably inherently unsolvable in a finite amount of time. Some of these are problems that you would think would be easy pickings for the mathematician armed with a computer. The traveling salesman problem is a good example.
Say you are a traveling salesman who must visit a certain number of cities, and you want to take the shortest route possible. You might think that there is no practical application of mathematics more easily accomplished than this. And, in fact, the problem is easy to solve by trial and error if the number of cities is relatively small.
For any specific number of cities, the number of routes is the number of cities âfactorialâ; take my word for this. The meaning of âfactorialâwhich is indicated by an exclamation pointâis most easily seen by example: 5! means 5 Ă 4 Ă 3 Ă 2 Ă 1, or 120. Thus, there are 120 different routes for visiting five cities one time each.
If you must visit ten cities, the number of routes your trial-and-error methods must compare is over three and a half million. To be sure, you could obviously immediately disregard inefficient routes such as New York to Los Angeles to Boston to Seattle . . . , but there would remain a daunting number of possible routes that would require actual comparison. With the help of a computer you could do this in a reasonable amount of time, but this is for only ten cities.
Perhaps you must visit thirty thousand cities. A mathematician cannot give you the shortest route that reaches all of the cities; he cannot even tell you with certainty whether there is any way other than trial and error for finding the shortest route. There are shortcuts that guarantee a route not more than about 20 percent longer than the shortest route, but no known method other than trial and error that guarantees the shortest route.
Now, you may well say, âWho has to visit thirty thousand cities?â And, of course, you would be right. But there are a great many analogous problems (circuit design, storing and transporting millions of packages, and the like) that have the equivalent of many thousands of âcities.â A saving of a few percent in the length of trucking routes, for example, can save the industry billions of dollars a year.
If a method better than trial and error for guaranteeing the most efficient (shortest, quickest, etc.) answer for even one of these problems is found, it will work for all of them. If, on the other hand, it can be proved that there can be no such method for even one of these problems, then there can be no such method for any of them. Most mathematicians believe that there can be no such method.
But It Looks So Easy
You no doubt remember the Pythagorean theorem:
Pythagoras was considering right triangles. His theorem states that the lengths of the two shorter sides determine the length of the longer side (or the other way around, which is the same thing). Specifically, the square of one short side plus the square of the other short side equals the square of the long side (the hypotenuse). In other words, a2 + b2 = c2. If the two short sides are 3 and 4 inches, the hypotenuse is 5 (i.e., 9 + 16 = 25; the square root of 25 is 5). The lengths need not be integers (nor need they be consecutive numbers).
Now, what about a solution to the equation ax + bx = cx when x (the exponentâthe power) is an integer greater than two? Is there a nontrivial solution? (A trivial solution would be that a, b, and c are all 0 and x is anything you want.)
The question of whether there is a nontrivial solution to the equation ax + bx = cx (x is an integer > 2; a and b and c are positive integers) was the most famous unsolved problem in mathematics. It is referred to as Fermatâs Last Theorem, after Pierre de Fermat (1601â1665), who wrote in the margin of a book that he had âdiscovered a truly marvelousâ proof that there could be no solution.
Fermat did not provide the proof. His margin note added that âdemonstration of this proposition (is one) that this margin is too narrow to contain.â Now, who would actually bother to write those words if he did not plan on someoneâs reading them? It is not inconceivable that Fermat was a practical joker of a particularly insidious typeâafter all, the man wrote in Latinâand that he knew he could flummox three and a half centuries of great mathematicians by claiming he had a proof.
In any case, there are many reasons for believing that Fermat found his âproof,â if he had one, to be faulty. If someone had found just one solution for ax + bx = cx, where x is greater than 2, that exception to Fermatâs claim would, of course, have been sufficient to disprove Fermatâs conjecture and to demonstrate that there could be no valid proof. No one ever found such an exception, but failure to find an exception can never prove that there is none. The exception might be the next number after the one you where you stopped checking. Here we see an asymmetry: failure to find any exception does not prove the conjecture is true, but one exception does prove the claim is untrue.
However, it was proved that there is no solution for exponents less than one million. Thus, do not bother checking to see whether, say, 163 + 233 = 473, or any other combination. It does not. If there were a solution, it would have had to include numbers unimaginably much larger than those necessary to count all the particles in the universe.
Unlike science, which deals in probability-like realities, mathematics recognizes only proofs and solutions. The Fermat question can be answered only by finding a solution to the equation, which would show the theorem to be false, or proving that there could not be one. Incidentally, as Ian Stewart (Game, Set, and Math: Enigmas and Conundrums) has pointed out, there are many âclose callsâ:
The philosopher W. V. Quine has shown that Fermatâs Last Theorem can be stated in terms purely of power, rather than addition and power. This is the first of a few entries in this book that are included simply because they are nifty. An explanation would take us too far afieldâout to that area where I would have no idea what I was talking about.
Fermatâs theorem has finally, after three centuries, been proved, by the British mathematician Sir Andrew John Wiles. There is no solution to ax + bx = cx (x is an integer > 2; a and b and c are positive integers). Do not even ask about what the proof was. It took Professor Wiles over seven years and depended on techniques discovered in areas of mathematics unknown for centuries after Fermat. Of course, it is always possible that Fermat had a different, simple proof. But do not bet on it.
Until recently, most mathematicians merely felt, rather than knew, that there is no solution to the equation. This was certainly plausible (and, as we have seen, turned out to be true). One would think that, if there were a solution, it would show up long before the unimaginably high numbers that are the lowest that could possibly provide a solution.
Some âFermat-likeâ equations do have simple solutions that are easy to find. For example:
However, there are many equations that have very high numbers as first solutions, and this renders inductive thinking in mathematics extremely dangerous.
The Dangers of Induction
Logic and mathematics work by deduction. Each step is logically entailed in the previous step(s). Thus, if (A) Andy is taller than Bob, and (B) Bob is taller than Charles, then (C) Andy is taller than Charles. You do not have to measure Andy and Charles to know that Andy is taller than Charles.
Logic and mathematics do not even care whether A and B are true; their interest is only in the relationship between the premises. That is one of the things that makes math so great: you do not have to know anything. Science, while nearly always making use of logic and mathematics, works differen...