Bridges to Infinity: The Human side of Mathematics by Guillen (PDF)

Description

Explains important mathematical concepts, such as probability and statistics, set theory, paradoxes, symmetries, dimensions, game theory, randomness, and irrational numbers

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This book tells me that Georg Cantor’s work in mathematics is ready to be taught in high schools. Earlier books on Cantor are too mathematical, hide truths about life, and hide God from the general public. The author brings life into Cantor. The problem is that mathematicians have removed life out of Cantor’s work by rejecting God and God’s determinate infinity.The author’s discussion of mathematics on p.19 should be given lots of attention. There, he has words of Imre Lakatos and Morris Kline on the danger of logic. My recent book on ‘The First Scientific Proof of God’ shows that some opposing concepts can coexist, but only if God exists. By taking Cantor’s work into the high schools, mathematicians might learn that their work can be unified with life and that God really exists.

⭐Marvelous book…delivered a bit late, but that was fine. The book was in very good condition, and very inexpensive. We’ll be making more use of the Kindle as we get familiar with its functions. Thanks.

⭐Loved it. Bought this copy for a friend

⭐Bridges to Infinity is a series of essays by the author, each one explaining a different area or concept of mathematics such as non-Euclidean geometry, set theory and transfinite numbers, game theory, topology, group theory, etc. All told in a very simplified manner without using any equations. This book is simplified enough that a high-school student could read and comprehend it.Which brings me to my criticism: it’s too simplified. The author gives us a small taste of things mathematical without bringing on the main course (or even the appetizer). I came away from this having learned little, and most of what I did learn dealt with the history of mathematics rather than the math itself.The writing is good, the ideas are presented well.I recommend “Bridges to Infinity” for younger, mathematically-inclined people, to give them an introduction to the larger field of mathematics. Teenagers would benefit most. I don’t recommend it for older, college-educated people who enjoy reading “layman’s” science books.

⭐Dr. Guillen has that rare trait that is so difficult to find in anyone who is trying to explain something to someone else – the ability to make things clear! I’ve known people with seven years of college that couldn’t give me a clear explanation of how to get to a friend’s house less than a mile away. Not so with Dr. Guillen. Although he is an obviously super-smart individual with multiple degrees, he has the ability to come down to the average man’s level and make the complicated simple. Reading this book has prompted me to read others of his works.

⭐A previous reviewer claims this book is “full of errors,” but gives no examples of them. While I can’t guarantee that this book is error free, any such errors are likely to involve technicalities which can be overlooked given the introductory and relatively qualitative nature of this book. On the important points, the book indeed appears to be accurate and clear. Moreover, this book is highly enjoyable, and probably accessible for everyone ranging from bright high school students on up. As such, the book is as an excellent prelude before plunging into a more rigorous study of the subject and, indeed, many readers will probably be motivated to do just that!

⭐The section on transfinite numbers is full of errors. For starters, he says”an aleph0 set … has precisely 2^aleph0 conceivable subsets … It is the first stepping stone beyond infinity, the first transfinite number, which Cantor named aleph1 … A set with aleph1 elements in turn, has precisely 2^aleph1 conceivable subsets. This is the second stepping stone, the second transfinite number aleph2, and so forth.”First off, aleph0 is the first transfinite number. Secondly, he’s assuming GCH without telling you so. Finally, Cantor did not “name” 2^aleph0 “aleph1”, he conjectured that this *equation* was true, this is precisely CH.”To this day we still don’t know exactly how many irrationals there are, although it has been established that the total number cannot be more than aleph1.”We don’t know the aleph number of the irrationals, but that’s because we don’t know the aleph number of the reals. And it’s completely wrong to say it “cannot be more than aleph1”. It can be any aleph(x) where x > 0 has uncountable cofinality. This is about as far from “cannot be more than aleph1” as you can get!!”Cantor himself guessed that the total number of irrationals is exactly aleph1, mainly since aleph1 is the next largest infinity after aleph0 defined by set theory. His guess came to be known as the Continuum Hypothesis. But there is still the uneliminated possibility that the number of irrationals actually lies between aleph0 and aleph1.”Laughable. There are NO cardinals between aleph0 and aleph1, by definition. Duh.

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