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Mathematical Elegance: An Approachable Guide to Understanding Basic Concepts
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.
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.
- GenresMathematics
123 pages, Kindle Edition
First published December 31, 2014
2 people are currently reading
32 people want to read
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Displaying 1 - 4 of 4 reviews
January 3, 2018
A really cool guide for people sort of far from maths. Prime numbers, rings, fields, etc.
Q:
Primes are the crème de la crème of numbers because they are the mathematical atoms, the building blocks of all other numbers.(c)
Q:
Gödel’s 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. (c)
Q:
Heisenberg’s 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.
Here’s 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. (C)
Q:
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 differently. It considers empirical truths (“facts”) and generalizes about them. Where mathematics is certain, science is always tentative.
We have excellent reasons for believing that no cow can fly, but we always leave open the possibility that tomorrow we will spot a flying cow and all start carrying umbrellas.(c)
Q:
All that exists or could have existed or could come to exist—in the mind or in potential or in reality—is the set of all sets, which has the same structure as the set of all complex one-dimensional subspaces of a complex infinite-dimensional Hilbert space.(c)
Q:
Primes are the crème de la crème of numbers because they are the mathematical atoms, the building blocks of all other numbers.(c)
Q:
Gödel’s 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. (c)
Q:
Heisenberg’s 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.
Here’s 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. (C)
Q:
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 differently. It considers empirical truths (“facts”) and generalizes about them. Where mathematics is certain, science is always tentative.
We have excellent reasons for believing that no cow can fly, but we always leave open the possibility that tomorrow we will spot a flying cow and all start carrying umbrellas.(c)
Q:
All that exists or could have existed or could come to exist—in the mind or in potential or in reality—is the set of all sets, which has the same structure as the set of all complex one-dimensional subspaces of a complex infinite-dimensional Hilbert space.(c)
December 26, 2014
Moderately interesting. Some stories were repeated twice so an editor was sorely needed, and the author chose to point out ideas and then said they didn't understand the reasoning and couldn't explain it. Disappointing to say the least.
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February 27, 2015510 G6185 2015
March 2, 2016
A book so well done it reignited my passion for mathematics and what the science has to show us. Very entertaining read for those who are unsure of math.
Displaying 1 - 4 of 4 reviews