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AlbertJayNock

Unknown Quantity by John Derbyshire: A semi-review

I recently finished reading Unknown Quantity: A Real And Imaginary History Of Algebra by John Derbyshire. Like a lot of mathematical history books (it's a broader genre than you might think) it is an interesting combination of the origins of mathematical ideas with the more general historical background in which the ideas first occurred.

Many of these ideas originated out of necessity. For instance, most people find complex numbers completely counter intuitive. But in the 16th century Gerolamo Cardano realized that you can't solve certain problems without them. (The problem in question was "find two numbers whose sum is 10 and whose product is 40). There is also the story of Rafael Bombelli using complex numbers to derive a real solution for the equation x3=15x+4. Counterintuitive and abstract as complex numbers may be, they did not come into being just because some professor had to publish something

There is not a lot about in the book about mathematical logic in the book, but among the little coverage there are some gems. Derbyshire writes of Augustus DeMorgan, most famous for discovering DeMorgan's theorem which says that

not (A or B) = (not A ) and (not B)

and

not (A and B) = (not A) or (not B)

NOTE: If you are ever in a computer science, math, or logic class and this theorem shows up, do not attempt to debate it with the professor. DeMorgan's Law is not subject to negotiation, and you will look foolish if you appear to think it is.


DeMorgan had the rare distinction of being x years old in the year x2, which is true of people only born on certain years. (The only people reading this for whom it will be true were born in 1980. They will be 45 in the year 2025). x in the case of DeMorgan was equal to 43.
He was born in 1806 and thus was 43 in 1849, or 432. I mentioned in a previous post the personal significance the number 43 has for me: the 43rd president was in office when I turned 43, and my father and I were both 43 when our fathers died. I feel honored that I have something in common with a great mathematician like DeMorgan.

George Boole was also briefly discussed. He invented boolean algebra, a very important area of mathematics and theoretical computer science. It is not too much of an exaggeration to say that you and I would probably not be blogging right now if not for his contribution. The best and wittiest bit of writing concerns, of all things, the manner of Boole's death. He had caught a chill as a result of being out in the rain.
As Derbyshire tells the story,

"His wife believe that a disease should be treated by methods resembling the cause, so she put George in bed and threw buckets of icy water over him. The result, as mathematicians say, followed."

It was an enjoyable book. It entertained me, stimulated my thinking, and perhaps most importantly, motivated me to learn how to do superscripts in HTML so I could write this review.
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Comments

Yeah, that's not up for negotiation and in fact a building block of modern electronics.
See in C:

!(A && B) == (!A || !B)

!(A || B) == (!A && !B)

And here it is done by inputs in logic gates:


ABA op B
001
011
101
110



ABA (op) B
001
011
101
110



ABA (op) B
001
010
100
110



ABA (op) B
001
010
100
110


As long as you look at how it flows, it makes complete sense.

These structures are very special for this reason. Most people call them NAND and NOR gates. As you can see, you can construct nearly anything out of them. Even each other:

!( !( !(A && A) && !(B && B) ) && !( !(A && A) && !(B && B) ) ) == (!A && !B) == !(A || B)


ABA NOR B
001
010
100
110


With enough nand || nor gates, you can do anything, including construct an entire processor...

Though you certainly already knew that, maybe your readers will get a kick out of it. I always enjoyed boolean algebra. Other than being the bread and butter of programmers, it's just... great. I love how huge circuits can just trim away with most of the inputs being 'don't care' cases.
-Joe

Joe, it's a good thing you have a short attention span. I'm pretty sure that if you didn't, you would have either taken over the world or destroyed it by now. Anyway, I miss you and hope OR is treating you well.
Thanks for sharing about the superscripts.
I've never really had any kind of formal instruction on html. Whenever I use it, I've gotten into the habit of putting in the first tag and then I'll say to my self whatever the command is, and then for the last bit I'll say to myself 'quit' whatever the command is. For instance, if I wanted to bold the phrase babydoc3 is brilliant, my internal monologue as I would type would go: "bold- babydoc3 is brilliant- quit bold."
I've done this so much that I refer to a backslash as a 'quit.' h-t-t-p-colon-quit-quit-w-w-w-dot....etc,etc..