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Sunday, August 25. 2019

A book: Absolutely Elementary Mathematics


What are numbers?  How does addition work? David Berlinski's One, Two, Three: Absolutely Elementary Mathematics


Things our teachers never told us, and maybe never understood. What is zero? Is it a number?

Posted by The News Junkie in The Culture, "Culture," Pop Culture and Recreation at 14:06 | Comments (16) | Trackbacks (0)

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I can recommend another book, too. It's called "The Man Who Loved Only Numbers." It's an interesting book because it contends that most of Einstein's work was not original. His theories were more of a comprehensive review of what had been done before. So why do they make such a big deal out of Einstein? Because he was a Patent Agent. You see, by promoting Einstein as a "Genius" (He was, in fact, really good at math. But in almost every other area, he was a complete moron) they are trying to imply that Patent Agents in general are geniuses. After all, Patents are "Real." And Patent Agents are "Smart". So the Patents are "Valid"! It's kind of a screwy proposition, but what the hell.

Like I said, Einstein was a moron. And he was also a communist. But he served his propaganda purpose. He created the impression that Patent Agents are good people.
So the more patents that he issued for his employer, the Patent Office, the better society will become. But we all know that's not true. There are millions of invalid patents on the books.
In fact, we might have to cancel all of them. Personally, I do not believe that any type of process patent is valid. Software patents, algorithm patents, business method patents, DNA patents, flavor patents; these are all nonsense. And let me tell you something: There are thousands of transactional analysis patents that claim to "own" permutations of up-strokes and down-strokes. Conversations have been patented. And those people will come after you for illegally telling your wife that you love her. That costs fifty cents.
#1 Ron B Liebermann on 2019-08-25 23:57 (Reply)
I have never heard a whisper of an argument that anyone wanted to plump up how important patent agents are and so made an all-star out of Einstein. It seems a singularly ineffective conspiracy, that resulted in all physicists thinking that Einstein is pretty smart, both in his day and unto his posterity, but no one elevating patent agents to the level of general genius.
#1.1 Assistant Village Idiot on 2019-08-26 18:07 (Reply)
I had a math professor, and he was likely quoting someone else, who made the statement along the lines of "there are only three things you needed to understand if you want to understand math: zero, one, and infinity. I think I have a handle on one, and a pretty good understanding of infinity, but I'm not sure anyone really has a full understanding of zero."

For instance, there is a quantity A for which A+X=X for all values of X. There is a quantity B for which B*X=B for all values of B. What is it about the universe that makes A not only equal B, but be indistinguishable from it?
#2 Another Guy Named Dan on 2019-08-26 07:12 (Reply)
typo in above."For instance, there is a quantity A for which A+X=X for all values of X. There is a quantity B for which B*X=B for all values of B. What is it about the universe that makes A not only equal B, but be indistinguishable from it?" should read

For instance, there is a quantity A for which A+X=X for all values of X. There is a quantity B for which B*X=B for all values of X. What is it about the universe that makes A not only equal B, but be indistinguishable from it?
#2.1 aniother guy named Dan on 2019-08-26 10:06 (Reply)
Another Guy Named Dan: For instance, there is a quantity A for which A+X=X for all values of X. There is a quantity B for which B*X=B for all values of X. What is it about the universe that makes A not only equal B, but be indistinguishable from it?

It's not a property of the universe, but a consequence of the axioms of ordinary arithmetic (Peano postulates).
#2.2 Zachriel on 2019-08-26 10:30 (Reply)
Math is but another human language used to describe the Universe. It is not a property of the Universe but only a Map of the known Universe.
#2.2.1 Blick on 2019-08-26 10:42 (Reply)
That's just an acknowledgement that we don't understand it. "Axiomatic" is more or less just a way of saying "we don't understand it, but unless it's true, we don't understand anything else."
Look at Euclid's axioms. For over 2000 years people tried to prove the parallel postulate (given a plane containing a line and a point not on the line, there is only one lone in the plane passing through the given point that is parallel to the given line) in terms of the other axioms. It wasn't until the 1800s that a few people decided to throw it away and see what happens. It turned out that these non-Euclidian geometries were necessary to describe the universe in terms of General Relativity.
The fact that the same quantity shows up in what should be irreducible statements is the mystery.
#2.2.2 Another Guy Named Dan on 2019-08-26 12:45 (Reply)
Another Guy Named Dan: That's just an acknowledgement that we don't understand it.

Postulates are just shared assumptions. People can accept or reject the assumptions, but given the assumptions, certain theorems follow.

Another Guy Named Dan: "Axiomatic" is more or less just a way of saying "we don't understand it, but unless it's true, we don't understand anything else."

If that were true, then there would only be one form of arithmetic and one form of geometry. But that's not the case. We can understand mathematics without Peano arithmetic or Euclidean geometry.

Another Guy Named Dan: For over 2000 years people tried to prove the parallel postulate (given a plane containing a line and a point not on the line, there is only one lone in the plane passing through the given point that is parallel to the given line) in terms of the other axioms.

You can't. Non-parallel postulate geometries are equally consistent.

Another Guy Named Dan: The fact that the same quantity shows up in what should be irreducible statements is the mystery.

It's no more a mystery than the parallel postulate. For Dan living on a flat surface, the parallel postulate appears obvious, but it's not obvious to alter-Dan living on a curved surface, and his ordinary geometry would not have the parallel postulate. Alter-Dan would find the "no parallel lines postulate" to be mysterious, and unless it's true, we don't understand anything else.
#2.2.2.1 Zachriel on 2019-08-26 13:47 (Reply)
Are you trying to say I'm right or I'm wrong. I can't tell.

Descartes's analysis and Newton's laws of motion work great in, and indeed even depend on, a Euclidian universe. But there were things that these laws and ways of thinking did not explain, and it turned out that once you got rid of the rigid space and time of Newton, Galileo, and Euclid, the math started matching up to what people actually were seeing.

The meta point is that the axioms of any system should not be reducible: if you can define axiom 2 in terms of axiom 1, then you have only one axiom and a consequence. Additive identity and one of the multiplicative identities both contain the quantity zero. It then seems that you should then be able to define one of the zero-containing axioms in terms of the other, but no one has been able to do it yet.

So if there is a link between the two axioms, we don't entirely understand what it is. If there is no link, we don't know why the same zero shows up in both. It's something on a philosophical level that is not yet and may never be understood.
#2.2.2.1.1 Another Guy named Dan on 2019-08-26 15:29 (Reply)
Another Guy named Dan: So if there is a link between the two axioms, we don't entirely understand what it is.

One (1) is denoted as the multiplicative identity.
Zero (0) is denoted as the additive identity.

The multiplicative property of zero is a theorem that follows from the basic ring axioms.

(0+0)*a = 0*a, additive identity
0*a + 0*a = 0*a, distributive property
0*a = 0, subtracting 0*a from both sides
#2.2.2.1.1.1 Zachriel on 2019-08-26 16:19 (Reply)
Prove addition of 0 is distributive. You're just introducing another axiomatic statement.
#2.2.2.1.1.1.1 Another guy named Dan on 2019-08-27 11:30 (Reply)
Another guy named Dan: Prove addition of 0 is distributive. You're just introducing another axiomatic statement.

You suggested that there was more than one multiplicative identity when you said, "Additive identity and one of the multiplicative identities both contain the quantity zero." There is only a single multiplicative identity in ordinary arithmetic, and that is the number one.

The distributive property is a ring axiom concerning two operations, multiplication over addition, and applies to all operands. The statement a*0 = 0 is not an axiom, as you had said, but a theorem. Yes, there are axioms in ordinary arithmetic, but a*0 = 0 is not one of them.
#2.2.2.1.1.1.1.1 Zachriel on 2019-08-27 14:41 (Reply)
Another guy named Dan: What is it about the universe that makes A not only equal B, but be indistinguishable from it?

We're not actually far apart. You can assume the multiplicative property of zero and derive the distributive property; or you can assume the distributive property and derive the multiplicative property of zero. You have to make some assumptions for mathematical rigour, if that is your point.

Our question concerned your original statement. Our position is that what makes mathematical 'truths' distinctive are the peculiarities of human cognition, in particular, the perception of distinct objects. A cloud person may not see distinct objects and may develop an entirely different form of mathematics, just as someone living on a curved surface might develop a different form of geometry. We can make up all sorts of consistent axiomatic systems, and this would seemingly contradict your point that "'Axiomatic' is more or less just a way of saying 'we don't understand it, but unless it's true, we don't understand anything else.'"
#2.2.2.1.1.1.1.2 Zachriel on 2019-08-28 10:31 (Reply)
Zero is not Nothing. Zero is the absence of Something. Nothing is, well, Nothing.
#3 Blick on 2019-08-26 08:52 (Reply)
Blick: Zero is not Nothing. Zero is the absence of Something. Nothing is, well, Nothing.

Quite.

{0} ≠ {}
#3.1 Zachriel on 2019-08-26 10:31 (Reply)
I have only been able to get a concept of zero into my head by working from different directions and regarding each as a partial answer. Our intuitive sense of what zero is is probably the most powerful factor, though it does rather break down when we try and and make fine distinctions. It becomes something of a game of whack-a-mole when you get in close, however.

We come in next from thought experiments: Is having an apple, then taking it away, the same as having no space in which there was ever an apple? No, it isn't quite, so Zero and Nothing have similarities, but are not identical. We have to move back out into the idea of what the space is in which the imagining is occurring. And that gets very strange, because space reveals itself as an object in itself, and spaces are not all the same. Objects change spaces.

Whether imagined objects also change spaces is harder to picture, but you can make the math work all day. I go back and forth whether that makes it real or not. I want badly to say yes, but I suspect that is just a prejudice on my part.
#4 Assistant Village Idiot on 2019-08-26 18:25 (Reply)

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