Well, it’s too late for me to make a post about “St. Patrick’s Day” that is actually *on* “St. Patrick’s Day,” but I won’t let that stop me!

If anyone is interested in the *truth* about Christianity and Ireland, which is far more than the stories associated with Patrick (and which take place much earlier), here’s an article for you: “Behind the Mists of Ireland” by Dr. Douglas Winnail.

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Good. DON’T let it stop you! It was and remains an outstanding article! 🙂

Hello Mr. Smith.

This message is actually a response to your March 20, 2007 “Why Is a Negative Times a Negative a Positive?” I thought it might be of interest to you.

I recently decided to see if Amazon had anything on the negative numbers controversy, and found that they did. I ordered the book “Negative Math How Mathematical Rules Can Be Positively Bent” by Alberto A. Martinez

https://webspace.utexas.edu/aam829/1/m/NegativeMath.html has a book description with some short reviews.

I found it quite fascinating reading. The first hundred pages examined historical controversies surrounding the development of algebra largely from the 1500s and on, which includes material on quaternions and vector algebra. (Negative numbers are, of course, a big issue in this material.)

The rest of the book explores different rules one could introduce to produce different algebras, starting with the negative times negative = negative rule. This apparently is doable, although one would have to do an extensive rejiggering of algebra to come up with an appropriate algebraic system. I like that this eliminates imaginary numbers. There are examples and explanations.

Other rules are explored also. I particularly liked his “distinction” rule. I’ll use $ to symbolize it.

Sums are commutative but subtraction isn’t. Disinction is also commutative.

If $ means “difference” then

+7 $ +3 = 4

+3 $ +7 = 4

-7 $ -3 = 4

-3 $ -7 = 4

Algebraicly, we can now define length simply by l = x1 $ x2

If I had more time, I would look more deeply into these things.

Ed Ewert

Howdy, Mr. Ewert, and thanks for pointing me to that book. It sounds interesting! I’ve been a fan of exploring the true boundaries of math rules since trying to come up with a workable definition of 0/0 when I was in junior high, and there is a lot of good stuff out there to be sure. I do find it interesting that to make sense of much of it we often have to define the new concept in terms of the old ones (e.g., defining your $ operator in terms if absolute value and subtraction: a $ b = |a – b|).

And I would hate to “eliminate” imaginary numbers! After all, they are pretty real… 🙂

Thanks, again! (Maybe I will be able to figure out a way to transfer your comment over to the actual “Neg. times a neg.” post, which continues to this day to receive daily hits on search engines.)

But…but…but…I LIKE imaginary numbers! 🙂

I’d like to think that mathematicians are smart (and wise) enough to come up with the simplest explanation of all the facts, and the system we have seems simple and elegant enough for me. If it ain’t broke, don’t fix it! 😀

Can I have some fun with this one?

Senator John Q. Public supports earmark legislation adding $10 billion to the deficit. After certain ammendments, the earmark is reduced to $9 billion in deficit spending. He then dramatically tell his constituents, “we have SAVED $1 billion dollars!” (-10 $ -9 = 1).

And speaking of imaginary numbers… well, never mind.

I just want to add one additional math related note. I hope my examples didn’t make the book sound lame. I just didn’t want to quote 2 or 3 pages to give the full flavor of each situation.

I’ll just add this simple example. Say you owe the bank $16. (You’re acct stands at $-16.) You are told that you now owe the square root of that. So the square root of -16 = 4i is what you owe. Of course you understand that you owe $4. But the algebra gives a funny result. But where – * – = -, the square root of -16 = -4, a rational result. This is a simple example, but there are complex algebra problems generating both a rational result and a result with an imaginary number which makes no sense. The – * – = – rule causes the irrational result not to be generated (again, I’m oversimplifying).

Thanks, Mr. Ewert! I think I will have to get the book, because the example you provided doesn’t seem to work. If “you owe 16” then the “square root of what you owe” would be the same as “the square root of 16” which is 4, not 4i. If the account value is -$16, then it is a reflection of how much you “have” not how much you owe. (By the way, where can I sign up for one of these “we take the square root of what you owe to get the new amount you owe” credit cards? Sounds great!)

I understand from the (generally positive) Amazon reviews and from the author’s website that he is interested in promulgating a non-commutative arithmetic that better satisfies his personal feelings about what is “intuitive,” but it seems as exercise in replacing one subjective judgment with another. For instance, the language of arithmetic currently in popular use is quite intuitive to most, and it would take an awful lot of convincing on Mr. Martinez’ part to convince me that his formulations would truly be more intuitive.

This isn’t to say that there is not value in exploring new operators (say @, where -2 @ -3 = -6) and new mathematical systems where popular properties, such as the distributive property, are cast aside. Mathematicians already routinely recognize this, though, and new approaches to mathematics are explored all the time. That would mean that Mr. Martinez’ ideas aren’t really revolutionary, but if his book helps to popularize this creative side of mathematical work then he deserves kudos. Looking at his website, he seems a bit of a contrarian to me, but he does use that to say good things about Euler, my favorite mathematician, so I won’t hold that against him. 🙂

I hope Ed doesn’t think my comment was directed at him. Nothing like that.

I’ve been closely following the budgetary process in Washington, and it constantly amazes me how politicians will jump through intellectual hoops to explain how increasing the deficit is actually saving money.

The national debt just keeps racing out of control.

It was Ed’s use of the dollar sign that prompted my comment. Wrong venue, though.

I doubt he would think that, Steve. I’m just sad that no one here is talking about the actual topic of this post. 😦

OK, Mr. Smith, I’ll bite. (Looks like I got caught up in what became quite a tangent.)

This article has so many fascinating things to say that one fears just repeating what is said out of sheer awe. Early on, Dr. Winnail wrote: “The real plot in the drama of Western civilization is the struggle for the soul of Europe. Most assume Apostolic Christianity prevailed. Surprising clues nearly hidden on the Emerald Isle suggest something else!” And then he goes on to document the point as few articles even by him have done.

One might wonder, though, if by the end of days the driving paradigm of interpretation of history will be, “Historical Christianity prevailed, which was an improvement over apostolic Christianity anyway thanks to the latter’s ‘Jewish roots’.” It’s my impression that the Catholics know theirs isn’t “apostolic Christianity” in the sense the article speaks of it, and don’t care; they seek to change times and laws in the name of apostolic authority anyway. It’s the Protestants who’ve traditionally claimed that theirs was a restoration (more or less) of “apostolic Christianity” on the basis of “sola Scriptura”. And either way, the assertion is groundless.

Thanks, Mr. Wheeler! And I agree — they may believe it to be “Apostolic Christianity” in some sense, but I would gather that they clearly understand it is not the Christianity of the apostles.