Rawlings and Hypercomplex Numbers: Over a hundred years late

Yoo Hoo Mr. Rawlings.

One of the first thing a person doing mathematical research does is -check the literature-. That is to make sure he is not reinventing the wheel.

Please, please look at this

http://en.wikipedia.org/wiki/Hypercomplex_numbers

and follow the references to the literature.

The quaternions and octonians have been been around over a hundred years. The Cayley-Dickson construction shows how to get all of the hypergeometric algebras.

You have not invented anything new and it is doubtful the quality of your work is up to modern standards of mathematical rigor.

But don't believe me. Try getting your stuff published in a refereed mathematical journal of good standing. See what the rejection notice says.

Bob Kolker

Post 1

Sunday, June 29, 2008 - 10:23am

Mr. Kolker,

I believe it is necessary for me to say a few words in Mr. Rawlings’s defense.

I, more than most people, am aware of how long the hypercomplex numbers have been known and studied – as I have formally studied them myself. Mr. Rawlings never claimed to have made an unprecedented discovery; he simply arrived at an idea that was already known independently and using a method different from the conventional one.

The Chinese invented gunpowder and the printing press centuries before anyone in the West – but that does not detract from the merits of Roger Bacon and Johannes Gutenberg. They used their rational minds, without any external aid, to arrive at these discoveries and so should be esteemed accordingly.

I do not think Mr. Rawlings intends to publish his work in an official mathematical journal (though I may be mistaken with regard to his intentions). Rather, he simply wishes to make it publicly available to illustrate an approach that might help even those without extensive formal mathematical training arrive at some of the impressive higher concepts.

When I was ten years old, I independently discovered that (a + b)(a – b) = a² – b², before I ever took an algebra course. I did this by a quasi-inductive approach, after playing around with the products of a few numbers. For instance, I noticed that 16*16 = 256, 17*15 = 255, 18*14 = 252, 19*13 = 247, etc. I started to see a pattern and then formalized it using variables. Even though I found out a few minutes later that this property had been known for over two millennia, I was still justifiably proud of arriving at it on my own – which illustrated my mathematical understanding as well as my ability and willingness to think.

May I also inquire, Mr. Kolker, what the motivations were behind your prior post? I do not mean to condemn or criticize you; I am merely curious.

Sincerely,
Gennady Stolyarov II

Editor-in-Chief, The Rational Argumentator: http://rationalargumentator.com

Writer, Associated Content: http://www.associatedcontent.com/user/46796/g_stolyarov_ii.html

Author, The Best Self-Help is Free: http://rationalargumentator.com/selfhelpfree.html

Author, The Progress of Liberty Blog: http://progressofliberty.today.com/

Post 2

Sunday, June 29, 2008 - 4:43pm

Excellent, excellent post, Mr. Stolyarov.

Post 3

Monday, June 30, 2008 - 4:06am

"

May I also inquire, Mr. Kolker, what the motivations were behind your prior post? I do not mean to condemn or criticize you; I am merely curious.

"

You sure may. Mr. Rawlings is selling his work at $2.50 a download. Since his is asking a price (and I have no objection to that) it seems to me he ought to indicate what value there is in his work that is worth two and a half bucks a pop.

Does he claim his work is new. No. In fact he says it is not.

Does he show how his approach to deriving the hyper complex numbers adds to the mathematical art or ties in with the rest of mathematics? He does not.

Does he show how his approach might be of value to scientific or other mathematical applications? He does not.

What virtue does he claim?

1. It is the work of an amateur (Mr. Rawlings himself).
2. It is somehow derived from Ayn Rand's thinking.

Is that sufficient reason for -buying- his stuff? Not for me it isn't so I am not buying and I am saying why I am not buying.

Mr. Rawlings has neglected to tell us what the cash value of his work is, other than it is -somehow- derived from Rand's thought.

Having seen Rand's comments on mathematics I have low expectations. Her plot McGuffin in -Atlas Shrugged- is a perputual motion machine (for Christ's sake!). I am justified in having my doubts about Rand's mathematical and and scientific insight.

The way mathematics developed in the Real World was -first- from practical applications and -then- by abstract extension using the axiomatic hypothetical logical approach which the Greeks first invented. The way mathematics developed in Europe in the last 300 years with increasing scope, generality and abstraction is how we got what we have mathematically. Mathematics and Physics have synergically fed and nurtured each other. And I assure you as a professional mathematician (I did applied math in my youth and got paid good bucks for it) mathematics was not derived primarily from philosophical considerations. Logical considerations, yes; issues raised in physics, yes; philosophical considerations very little.

Gauss once said of the metaphysicians pronouncements on mathematics: "Where they are correct they are not original. Where they original they are not correct". It sounds like that just might apply to what Mr. Rawlings has to offer.

Look. If you want to spend $2.50 of your own money, go ahead. Es su dinero. Me. I am going with known quality and technique and it does not cost me a cent. Why should I spend $2.50 on an amateur work without a decent abstract, when I can get the best of Hamilton, Cayley, Dickson and others for free.

I close on a positive note. If you want to see an alternative development of vectors that really have some good applications in physics dig into the Hestenes mother lode. Google David Hestenes. There are PDF and PS files of some of his papers on line. It will only cost you the time to download it. Get the good stuff.

Bob Kolker

Post 4

Monday, June 30, 2008 - 7:21pm

Mr. Kolker,

Thank you for your response. It is true that Mr. Rawlings is selling his work – but there is nothing wrong with this, in my view. He is simply offering it on terms which he believes would fairly compensate him and duly reward him for the effort of creating it. How each of us judges the rewards we believe are due to us as creators will differ among individuals. For some people, the psychic satisfaction of making available a discovery is enough. For others, money is the benefit sought.

The fact is, Mr. Rawlings’s work is more original than that of, say, popular authors of mathematics textbooks that sell their books at $100 apiece, even though the books merely contain ideas discovered by the now dead great mathematicians of the past. These authors do not even contribute a new approach to existent discoveries; they simply present existing approaches (and often in a not too accessible way; it takes other skills beside mathematics aptitude to be able to teach mathematics well)– and yet they charge much more than Mr. Rawlings, and you do not seem to object.

I think both Mr. Rawlings and the textbook authors are offering a useful service by getting different kinds of people introduced to and involved in mathematical ideas such as the hypercomplex numbers. The conventional approaches for explaining and teaching mathematics may work well for some people, but certainly others could be brought to appreciate mathematical ideas with different methods of arriving at these ideas.

Of course, you are fully entitled to your preferences as a consumer – and I will not question your decisions regarding how you spend your money. I appreciate your positive recommendations, and I am sure that others will benefit from also looking at the sources you have cited.

If you wish to criticize Mr. Rawlings’s work, though, I would encourage you to read it first and then write a review of whatever tenor you desire. But to try to ward people off from a work without oneself having examined it seems to me overly hasty. Might there not be some other value in that work – from your own perspective – that you would not have expected or known of unless you had read it? Informing other consumers of one’s opinion of a quality of a particular work is helpful, but only when one has actually been fully exposed to the work in question.

Sincerely,
Gennady Stolyarov II

Editor-in-Chief, The Rational Argumentator: http://rationalargumentator.com

Writer, Associated Content: http://www.associatedcontent.com/user/46796/g_stolyarov_ii.html

Author, The Best Self-Help is Free: http://rationalargumentator.com/selfhelpfree.html

Author, The Progress of Liberty Blog: http://progressofliberty.today.com/

Post 5

Tuesday, July 1, 2008 - 2:43am

Stolyarov:

Thank you for your response. It is true that Mr. Rawlings is selling his work – but there is nothing wrong with this, in my view. He is simply offering it on terms which he believes would fairly compensate him and duly reward him for the effort of creating it. How each of us judges the rewards we believe are due to us as creators will differ among individuals. For some people, the psychic satisfaction of making available a discovery is enough. For others, money is the benefit sought.

..................................................

If you wish to criticize Mr. Rawlings’s work, though, I would encourage you to read it first and then write a review of whatever tenor you desire. But to try to ward people off from a work without oneself having examined it seems to me overly hasty. Might there not be some other value in that work – from your own perspective – that you would not have expected or known of unless you had read it? Informing other consumers of one’s opinion of a quality of a particular work is helpful, but only when one has actually been fully exposed to the work in question.

Me:

In the first five sentences I made plain that I have no problem with people selling the product of their labor. I do. That is how I make my living. What I -also- said is that I expected a better abstract, an indication of why his work is worth 2.50 a pop. People who sell their wares should advertise them adequately.

I am the last person in the cosmos to object to selling one's goodies and making a profit at it. I am one of the last of the red-hot pro capitalists in a society that has come to consider enterprise as somewhat anti-social if not downright criminal. As the late great Robert Heinlein once wrote: There ain't no such thing as a free lunch.

.........................................................

As to purchasing a work purely for the purpose of reviewing it, that is not how the world works. If I were a professional reviewer (which I am not) and the author wished me to review his work (he probably does not) he would provide me with a free copy for that purpose. That is how it is done in the real world.

I am not interested in chiseling a free copy for a "mathematical" work based on (oh goodness me) the work of Ayn Rand who has exhibited little or no grasp of either mathematics or physics. As I indicated my opinions of Rand's grasp of mathematics and physics are on the low side. In fact the author's claim to basing a -mathematical work- on Rand's thinking is a disincentive (for me anyway) to spending a cent on it.

I don't need a caricature of mathematics when for absolutely free I can go straight to the masters and get the -real stuff-. And it does not cost me a cent. Nada, zip, zero.

Actually I think it is silly (at least for me) to pay for what one can get for free.

Since you seem inclined to buy the thing then buy it. Then compare it to the real thing (which is readily available at no cost) and YOU review it.

I also mention that in 2006 the author published some of his findings (on this very forum, by the way) on hyper complex numbers. Actually he produced three or four postings shot through with errors that he himself corrected. In the mathematics business one checks his work carefully. Then he gets his work vetted (checked for error) by a qualified pro, preferably not in public and then he shows his stuff to the world. And even then, that is no guarantee of error free work. When Andrew Wiles (after seven years of hard labor) presented his first proof of Fermat's Last Theorem (so-called) to a gathering of mathematicians, a very subtle error was pointed out. He went at it for another year and then presented his repaired version which has been checked out by the world's leading mathematicians in the field and it appears to be holding up.

I seriously doubt that Mr. Rawlings is of Andrew Wile's caliber and that is all the more reason he should get his work vetted by professionals (simply for mathematical correctness) before publishing. Did he? He never claimed he did, but he still wants 2.50 a pop. No thanks.

In the field of mathematics there are amateurs (i.e. people who do not do or teach mathematics for a living) who publish (and some of it is good stuff). They go through the same vetting process as the professionals. That is how it is done. Do you know why? Because if mathematical works were not properly vetted we would be deluged by papers which claim to have trisected the angle or duplicated the cube (both shown to be impossible) or which derive sacred proportions from the measurements of the Great Pyramid of Cheops. In short, without proper vetting there would be a flood of shit and the good stuff would get drowned in it. And that is why we have professional journals. It is a matter of quality assurance.

If Mr. Rawlings wants to be taken seriously he should (1) produce something original and non-trivial and (2) he should have it properly checked out prior to publication. Has he done either? I don't think so.

Bob Kolker

(Edited by Robert J. Kolker on 7/01, 3:13am)

Post 6

Tuesday, July 1, 2008 - 6:18pm

Mr. Kolker,

Just a few comments.

Having been through the peer review process once myself, I believe that, in our age, it is quite a hindrance to scientific progress, as opposed to a boon to it. Yes, it does provide a mechanism for checking people's work and ensuring that it is accurate. But it also takes an extremely long time to get anything published, simply because the feedback mechanisms are so slow. I wrote an economics paper in eight hours, submitted it to a journal, and got back a response with a tiny number of requested corrections a full year later. It took me two hours to make the corrections, I submitted the paper again, and it got published another half-year later. At that rate, important and perhaps ground-breaking discoveries might be left to fester on the tables of referrees who have zero economic incentives to review them quickly. The journal I submitted my paper to was a good journal also — with no nasty political agendas or political criteria for publication. Most other scientific journals are fraught with these problems in addition to the time delays.

I believe the Internet is an excellent antidote to the peer review process. Anyone can publish a work online, but not everyone will be able to convince people of the value of his work. The good works have a chance of spreading faster than the peer review process will permit, especially if some kind of free-market quality determination system springs up. Granted, this has not yet happened, but that is because academia is notoriously slow to adapt to new technologies and new ways of thinking.

Consider this: a person publishes a paper online. The paper can be read by anyone, but individual scholars who like its contents and believe them to be of value can place their virtual seal of approval on the paper, and then have the paper rise in status as a result. The more seals of approval from scholars or other reputed authors of scientific works a paper has, the higher its quality rating and the more reliable it will tend to be. How much a person's judgments of a particular paper are weighted might depend on the number and quality of his own past contributions.

With regard to Mr. Rawlings, I am pleased that anyone would wish to share ideas outside the rigid constraints of the peer review process. Moreover, there is nothing wrong with publishing works that contain errors, provided that one is intellectually honest and is willing to correct the errors when either they are pointed out to him or he discovers them himself. In that respect, the Internet is a much better form of “peer review” than the traditional kind, because anyone can notice the errors and report them, and there is a much larger pool of people who potentially have access to the work. Moreover, the errors can be rectified right away, instead of the author having to go through the tedious and expensive process of issuing errata.

Galileo, Newton, Lavoisier, and virtually all the pioneers of science and mathematics were not subjected to the constraints of the conventional peer review process. That was a twentieth-century invention, and it is high time we discarded it as a relic of the past.

Sincerely,
Gennady Stolyarov II

Editor-in-Chief, The Rational Argumentator: http://rationalargumentator.com

Writer, Associated Content: http://www.associatedcontent.com/user/46796/g_stolyarov_ii.html

Author, The Best Self-Help is Free: http://rationalargumentator.com/selfhelpfree.html

Author, The Progress of Liberty Blog: http://progressofliberty.today.com/

Post 7

Tuesday, July 1, 2008 - 8:26pm

"With regard to Mr. Rawlings, I am pleased that anyone would wish to share ideas outside the rigid constraints of the peer review process. Moreover, there is nothing wrong with publishing works that contain errors, provided that one is intellectually honest and is willing to correct the errors when either they are pointed out to him or he discovers them himself. In that respect, the Internet is a much better form of “peer review” than the traditional kind, because anyone can notice the errors and report them, and there is a much larger pool of people who potentially have access to the work. Moreover, the errors can be rectified right away, instead of the author having to go through the tedious and expensive process of issuing errata."

I respond:

He isn't sharing. He is -selling-. Do you think it reasonable for someone to pay for the privilege of correcting the author's errors?

You other remarks about the slowness of vetting a paper are well taken. That is why many mathematicians and physicists are putting their preprints on arXiv.org. All the material on arXiv, good, bad or in between can be had for -free-. One need not pay the author. If anything the author should pay the reviewer for his labor or at least offer a free copy to the reviewer who is spending his good time reviewing the work.

I notice Mr. Rawlings has not referred to or cited a place where his stuff has been reviewed.

I am not going to buy a pig in a poke. Are you?

Bob Kolker

Post 8

Thursday, July 3, 2008 - 10:22am