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A friend of mine asked me yesterday about Santilli's isonumbers. I told him that it was quackery. As I based my answer only on the general reputation of the guy and had no knowledge of the subject, I decided to ask this question here.

Question: What is the isonumber? Did any serious mathematician spend any time looking at isomnumbers? What is the conclusion? References would be useful.

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I have no idea of what you are talking about, but a little research with google says, that seems something like a monograph. – Ricky Jul 26 '10 at 3:19
«Despite the achievement of the above epoch making generalizations of all pre-existing mathematical and physical theories [...]» That alone surely will get you a high Baez index... – Mariano Suárez-Alvarez Jul 26 '10 at 3:36
Good to know: see also – Will Jagy Jul 26 '10 at 3:46
See before answering this question – Richard Borcherds Jul 26 '10 at 4:52
@ Richard Good Warning! Although he seems to be more keen on his physical theories... – Bugs Bunny Jul 26 '10 at 5:43

I looked at Jiang's monograph for a little while last night. Here is what I could get from it (I am now quoting from memory, so my terminology and notation may not be exactly the same). If $F$ is a field (of "numbers"), then the field $\overline{F}$ of "isodual numbers" has the same underlying set and addition operation, but multiplication is replaced by the operation $x \ \overline{\bullet}\ y := - (xy)$. The new multiplicative identity is $-1$.

This is mathematically valid, of course: i.e., $\overline{F}$ really is a field. Moreover it is isomorphic to $F$ via the map $x \mapsto -x$, although I couldn't find a clear statement of that. (But somewhat later on I saw references to the isotopy $F \rightarrow \overline{F}$.) Physically speaking, the isodual numbers are supposed to bear the same relation to the ordinary numbers as antimatter does to matter. (I don't know what that means, but I am not a physicist and so am not even going to worry about it.)

Jiang defines a new function $J_2(\omega)$, which is supposed to be some sort of repaired version of the Riemann zeta function. In one of his published works, he claims that the Riemann hypothesis is false -- in fact, he says, the zeta function has no zeros in the critical strip. [Logically speaking, wouldn't that make the Riemann Hypothesis true? Never mind.] From this definition, he immediately deduces proofs of Goldbach, twin primes, primes of the form $n^2+1$, and several other outstanding number theoretic conjectures -- literally immediately, in that I could find no argumentation for them. First these results are stated for "isonumbers" but later on they are stated for the usual integers.

That's about as far as I got. I also noticed, though, that many of the results described in this monograph were first published as papers by the journal Algebra, Groups and Geometries (founding editor: R.M. Santilli). These papers appear on MathSciNet but are not (going to be) reviewed.

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I have read Jiang's book today and this is a definite quackery. I have managed to google He seems to be some kind of "mathematician of the people" in China:-)) I wish I could get hold of Santilli's 1993 paper in Algebras, Groups and Geometries... – Bugs Bunny Jul 26 '10 at 19:23
I found his papers with a couple of friends when I was an undergrad and we had some fun browsing through them. If I interpret his "disproof" of the Riemann hypothesis correctly, he starts using the definition $\zeta(s)=\Prod(1-\frac{1}{p^s})$ as if it converged for $\Re(s)<1$, makes a lot of estimates, and concludes that $|\zeta(s)|>C$ for a nonzero constant $C$ --- but, of course, he gets that because the series he is using as a definition is not convergent. – Federico Poloni Nov 28 '10 at 15:58
Here's a recent, thorough, formal introduction: Ganfornina, Raúl M. Falcón, and Juan Núñez Valdés. “Mathematical Foundations of Santilli Isotopies.” Translated by Alan Aversa. Algebras, Groups, and Geometries 32 (2015): 135–308. – Geremia Jun 17 '15 at 1:56

Looks like total crackpottery.

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Did you actually understand the definition so that you can say what they are in normal language and why they are of no interest? – Bugs Bunny Jul 26 '10 at 5:49
I read enough of his overview paper to come to that determination. The initial abstract algebra is fine, but it doesn't do what it's claimed to do. – Charles Jul 26 '10 at 17:06
Did you read Santilli's paper or Jiang's book? The latter is indeed useless but I cannot access the former one for now... – Bugs Bunny Jul 26 '10 at 19:24
With all due respect, it seems to me that this doesn't answer the question. – Todd Trimble Jan 21 '15 at 17:10
@ToddTrimble: True enough -- it addresses only the third question, not the first two. But at the time it seemed like there might not be any other answers. – Charles Jan 21 '15 at 18:13

protected by Todd Trimble Jan 21 '15 at 17:07

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