The n-Category Café

The link for this book is
here
There is also an online version available.

Perhaps you can explain why this book should be compared to the one in the main post? It’s not immediately obvious to me what is so shocking about the things that it claims to show, though perhaps the application of information theory is slightly unexpected?

Just for the record, the central argument in Farming Human Pathogens, in chapters 2-4, is based on a long peer reviewed article:

R. Wallace and D. Wallace (2008), Punctuated equilibrium in statistical models of generalized coevolutionary resilinece…, Transactions on Computational Systems Biology IX, 23-85.

Chapter 5 and most of 6 are also based on published peer reviewed articles.

I don’t work anywhere close to mathematical biology, but I’ve been to a lot of talks on that field and at a superficial glance I don’t see anything that looks shocking or controversial about the Wallace book. (In particular there’s nothing unexpected about applying information theory. Groupoids are slightly unexpected, but applied mathematics can be surprisingly omnivorous.) The treatment of mathematics is mostly not up to pure mathematicians’ standards, but that’s to be expected.

In any case, as far as I can see they aren’t making any big, hard-to-believe claims.

If the mathematical ideas are reasonable (and other posters have suggested that they are, although I wouldn't be able to tell you without reading the book or other such material), then this actually looks interesting.

While some editor should have corrected this, the author might be forgiven this mistake as it is a logical mistake to make if you’re a native Dutch speaker (as this person seems to be). However, the author claims a PhD from Waterloo University, undermining my point.

The author seems to be some sort of electrical engineer, and also a frequent poster on sci.math. Looking at his webpage, one wonders what his crackpot score would be…

“How can Springer possibly justify such a price, especially when they appear to have put little or no effort into the book?”

Well, if they behave just like Elsevier, then all the high rating and rave reviews their books recieve on the internet are perhaps something to do with it:

http://news.bbc.co.uk/1/hi/magazine/8118577.stm

Of course, now the guy’s claims are in print, they can be added back into Wikipedia along with a citation, making them satisfy the rules for not being deleted.

Hi GS: It did cross my mind, having such a powerful tool as residue-and-carry (viz. applying semigroup structure to elementary arithmetic problems, and NOT forgetting about the carry - which was normally done by number theorists since Hensel… who’s extension lemma was a major block against progress - only a CS engineer familiar with the pain/importance of carries in arithmetic was apparently necessary to see that the Hensel-lift can be broken in special cases, one of which are the cubic roots of 1 mod p^k, the extension of FST that Fermat probably discovered soon after his FST mod p in 1637).

However, Riemann’s Hypothesis works in the complex plain, that is: on real-pairs, which is ‘too far from my bed’. Moreover, in industry (where I worked in the digital VLSI domain as an EE for 32 years) one has to show practical engineering relevance, which despite mathematicians stating the opposite, does hold for FLT (and maybe for Goldbach, which was so close to FLT that I couldn’t resist). – NB

PS: You’re welcome to try and apply my residue-and-carry method to Riemann. Let me know the result. In Holland we say (in Dutch of course): “you never know how a cow catches a hare”. In other words: keep it simple, you may be pleasantly surprised ;-)

I hope people from outside mathematics don’t get the impression that Benschop’s not getting a fair hearing for his proofs. At least for the FLT proof, several mathematicians (Alf van der Poorten and Robin Chapman) have debunked Benschop’s work. At least as of a few years ago, when he first managed to get it into print (in Acta Mathematica Univ. Bratislava, which I’ve never heard of in any other context), the proof was not only wrong but showed no progress or even promise of future progress. I’d be shocked if the proof in his new book is any better, but I have no intention of paying $25 for a 16-page chapter to find out.

In any case, the real issue here isn’t whether his proof is valid. Rather, it’s why a reputable scientific publisher would publish a technical book seemingly without any substantive review of its contents. (Acta Mathematica Univ. Bratislava has no reputation to protect, and may be run by crackpots for all I know, but Springer has a good reputation.) Books are often not refereed as carefully as journal articles, but there’s no excuse for publishing proofs of FLT and Goldbach that have not been endorsed by any mainstream mathematician whatsoever.

The trouble is that unless you are willing and able to provide free access to the relevant material, people here may not be willing to spend 139 US dollars just for the purpose of trying to refute the claims.

Can you provide names of professional mathematicians who have examined your claims and are willing to vouch for their validity?

There is at least one very fine mathematician, Robin Chapman, who has examined claims you’ve made in the past (contained in various drafts of papers you’ve put on the arXiv and in sci.math postings) and considers them seriously wanting. Your book would seem to be about very similar material; given the professional standing of Dr. Chapman, some degree of skepticism doesn’t seem all that out of place.

But in the end, this post is not about you, it’s about the practices of Springer Verlag. A big question is whether Springer made any effort to ask mathematicians to review the dramatic, eyebrow-raising mathematical claims made in the book. Can you address this?

Nico wrote:

PS: would you care to read the proof first?

Sorry, I don’t have time. I didn’t claim your proofs are wrong — though I’m willing to bet they are, purely as a matter of gambling. And my criticism was not directed at you. It was directed at Springer Verlag, for not adequately refereeing your book.

How do I know it was not adequately refereed? Simple:

If your proofs are wrong, Springer should have caught this. Why? Not because publishers should catch every error, but because Goldbach’s Conjecture and Fermat’s Last Theorem are extremely famous. If your proofs are wrong, it will be very embarrassing for Springer to have published them!

On the other hand, if your proofs are correct, Springer should have announced this loudly to the whole world. News of your work should be on the front page of every newspaper! It would be a huge triumph for Springer.

So, either way, Springer did the wrong thing by letting proofs of these results appear quietly, without having a team of experts carefully check them and then formally announce to the world that an earth-shaking mathematical event has occurred.

How could they have made such a huge mistake? That’s the interesting question.

I should make clear that “the rest of your work in academia would hold you in good stead”, if it is the case that the work in question is of merit. However I should make it clear that since it is mostly in electrical engineering and related topics, I have nothing to say about it. That would be up to the experts to pronounce.

Dear GS and Todd,

First regarding Todds question on: “A big question is whether Springer made any effort to ask mathematicians to review the dramatic, eyebrow-raising mathematical claims made in the book. Can you address this?” - Sure I can: I know as much of the reviewers Springer engaged as you do about those that read any paper you submitted to a journal, thus ZIP.

Next point: *why* do you want another authority than yourself to say it is OK? Think of Wiles’ >150 page proof (indirect via Tanyama-Shimura conjecture about elliptic curves and modular forms having a 1-1 mapping). It is said that only a handfull of specialists could verify it, and when the white smoke curled to heaven it was announced OK. So are *you* happy now. Q: why were you interested in FLT in the first place? Is it just like Mt.Everest (why climb it? Because it is there;-) Would not you rather verify a proof yourself, if it’s important to you?
For a Wiles type of proof: forget it.

Let me repeat a quote from Carver Mead (at CIT if I’m not mistaken), which I found so to-the-point that it is among the six quotes on the dedication page:
“It always worked out that when I understood something, it turned out to be simple. Take the connection between the quantum stuff and electrodynamics in my book. It took me thirty years to figure out, and in the end it was almost trivial. It’s so simple that any freshman could read it and understand it. But it was hard for me to get there, with all this historical junk in the way.”

I feel with him: important stuff you must be able to check yourself! I’m an EE with formal self-education in discrete math (associative algebra’s of various kinds, for my research in digital VLSI design methods). I’ll tell you something surprising: I did not look for a proof of FLT, I just stumbled over a property of three p-th powers mod p^k summing to zero (k-digit number representation base prime p) to find a better way of multiplying numbers than by a binary array-multiplier of n x n Full Adders, which is *much* to powerful for its purpose. A log-arithmetic scheme seemed useful, with a multiplicative generator, in math called a primitive root (see chap.7, I got a US-patent out of that within a year!). The group of units mod p^k (any k>0) is known to be cyclic (1 generator) and Fermat knew this! Its period equals (p-1)p^{k-1} because there are p^{k-1} multiples of p (hence not units). If 3 divides p-1 (p=1 mod 6) then there is a 3-cycle as subgroup, and any subgroup sums to zero (mod p^k). So there you have it: 3 p-th powers adding to 0 mod p^k (k>1), with the extra property that Exponent Distributes over a Sum (EDS property): (x+y)^p == x+y == x^p +y^p (mod p^k). That, in January 1994, reminded me of Fermat, and I had to make a decision whether I would spend company- (and private-) time to “go for FLT” or not.
With the EDS property it should be possible to show the impossibility of extending such cubic root of 1 mod p^k type solution to integers. It took a while but you can, without high-math verify it for yourself (both cases 1 and 2 of FLT).

These are solutions for p=1 mod 6, but there are also another type of (more general) solutions for some primes -1 mod 6, starting with p=59 (more complex, but cubic roots are a special case), of which I’m sure Fermat had no clue. Possibly he realized that to present his cubic-root proof he would have to show they are the *only* solutions, which he could not… so he left it at rest (not divulging what he knew already). For p=7 and arithmetic mod 7^2 the first case of FLT case1 occurs, easily done and found by hand.

Now I was then 55 years old, a senior researcher (having parttime tought 6 years at Delft University) so I thought: I’ll give it a half year upto a year (between other research). And by december that year I submitted my paper to the internal screening board, with a math prof. of Eindhoven Univ. as advisor. He immediately blocked it, even though I showed the engineering relevance for a log-arithmetic scheme based on (p+1^)* generating a cycle of period p^{k-1}, namely all k-digit numbers that are 1 mod p, implying that each odd binary coded number equals a unique power of 3 (mod p^k). He said: if *that* were true we would have known this years ago (the proof is half a page!) So then my ‘paper polishing’ phase began, which lasted some 10 years… until publication by the experts in semigroups applied to elementary arithmetic: Univ. of Bratislava! (re: prof Stefan Schwarz).

The clue for Goldbach came also by chance: I had only to change the mudulus to \prod(first k primes) and first solve Goldbach-for-residues (mod m_k): each even residue equals the sum of two units,
whic took me just a few weeks, and another polishing phase started,in 1996,
until last year Springer (first Germany, then Holland: Kluwer Academic publishers before 2004) accepted in august my manuscript. So if you’re interested: chapter 9 has all the introductory semigroup / lattice theory you need for the proof: no other books necessary.

It was nice talking to you both as civilized people do. – NB

Dear GS, you write “Would you be able to give a source where I can read it without paying money?” – Sure, the moderator of sci.math.research gave a link to my Goldbach paper in arxiv.org :
http://arxiv.org/abs/math/0103091 where you find an older version (2001), and links to various intros on my homepage. That is an early version, only in non-essential details differing from the present book version. If you search for ‘Benschop’ there, another 7 papers appear, including that for FLT. BTW: I derive much more than FLT, namely the full additive structure of the group of units mod p^k, characterised by a threefold successor function I call ‘triplet’: (a+1)b == (b+1)c == (c+1)a == -1 with abc==1 mod p^k. Of which the cubic roots form a special case, when a==b==c follows: a+1=-1/a (with 1/a == a^2). If you have any questions, please let me know. – NB

To GS. You say “However, in your work, I failed to see any connections of astonishing and surprising depth, which one sees in the work of Wiles, and the people who laid the groundwork for him (… on the shoulders of giants…;). One has difficulty in believing that the profound methods can be replaced by half a page of computational considerations.”

NB: It’s not ‘half a page’ but 16 pages, yet including the full additive structure of the units group in Z(.) mod p^k.
The key in my work is to treat the two symmeteries of residue arithmetic as *functions* : complement C(n) = -n, and inverse I(n) = 1/n, together with the successor function S(n) = n+1.
Then by inspection you can verify that I and C commute: IC = CI, leaving four possible combinations of these three functions, namely SCI, ICS, ISC, CSI which *all* have a period of three :
denote the identity function E(n) = n, then for instance (SCI)(SCI)(SCI)=E,
where working left-to-right: SCI = 1-(1/n), repeatedly applied to itself (substitute n by 1-(1/n) three times). From this treatment of the two symmetries of residue arithmetic I derive the ‘triplet’ structure (see my other msg today) as the only possible additive structure of the units group. The integers come in by carry extension *and* the EDS property of such additive solution (including FLT mod p^k type solution, which *must* hold also for any integer solution) makes preservation of equivalence for large enough modulus mod p^{pk} impossible. Notice the p-th power of a k-digit number x smaller than p^k requires at most pk digits (base p).

And regarding a complex problem reduced to a much simpler one: think of the 77 circle/epicircle model of Ptolemeus for the orbits of the known five planets. Just by making the basic circle into an ellipse Kepler needed only one elliptic orbit per planet, enabeling Newton to reduce that further to a simple inverse quadratic law of dynamic motion (his gravity law). In this FLT case: arithmetic is associative and commutative, and it requires the one level more powerful algebra of function composition, which is not commutative, to solve hard additive number theory problems
(here: S does not commute with I nor with C, where successor function S is characteristic for addition). In other words: “You need steel to cut wood, and you need a diamnant to cut steel” (where Boolean logic = wood, arithmetic = steel, and diamant = function-composition (semigroups). – NB

You refer to, as it says, * one version* of the Hensel lemma. The simple version that I refer to (Hensel’s extension lemma, or shortly: Hensel-lift) does not require a derivative. It says that if an equivalence F(n) == G(n) mod p^2 (prime p) holds, then this can be extended to equivalence mod p^t for any t>2. For his p-adic number theory Hensel extends to infinite precision (inf t).
This has been a block to looking at carry extensions to (finite) integers, because any precision t can be made to hold. However, in the case of an equivalence (x+y)^p == x+y == x^p + y^p mod p^k (k>1) where the exponent distributes over a sum (EDS property), taking the p-th powers of a k-digit residue solution yields already inequivalence mod p^{3k+1}, which you could call a ‘bootstrap’ effect: for any prcision k of a FLT mod p^k solution, its p-th powers (taken as integers) cannot preserve the quivalence beyond triple precision 3k+1.

And re the ‘structure’ of the units group G of mod p^k, I mean its *additive* structure, not its multiplicative structure, which is cyclic of order (p-1)p^{k-1. Hence G is the direct product of two cyclic components G = A.B where A is the ‘core’ of order p-1 with n^p == n for all p-1 in core A (FST extension to mod p^k), while the ‘extension component’ B = (p+1)* is generated by p+1, yielding p^{k-1} distinct powers of p+1. So if A == g* (primitive root of core A) then each unit residue n = g^i.(p+1)^j for unique pair of naturals i \lt p-1, j \lt p^{k-1}. But that is its * multiplicative* structure, not the additive one. – NB

These discussions with Robin Chapman took place some 10 years ago on the sci.math newsgroup. He did indeed find errors, mainly due to misunderstanding my intent, which was not expressed accuratly enough. Such errors were readily corrected.

For instance (Ch.7 on the extension of FST mod p to mod p^3, removed from Ch.10 that was published, and where this was a sideline subject): in search of primitive roots of 1 mod p I considered divisors of p-1 resp. p+1, since these are powerfull generators in the units group G mod p. This I thought to compress into: divisors of p^2-1, which is in error because such divisor can contain prime divisors of p-1 and p+1 about which product I had proven nothing. So p-1 and p+1 should be considered separately. Trivial, and easily corrected, but essential. A second point relating to this was my claim that different divisors r of p-1 would have distinct r^{p-1} mod p^3. In fact this should be reduced to: divisors with different prime-divisor sets, hence generating distinct idempotents in Z(.) mod p-1.

PS: the unpleasent memories I have from those discussions was his denigrating tone towards me (as most mathematicians still do): the total lack of civilized discussion style. But I got used to that, and thanked him for his efforts. – NB

Some mathematicians are still working on it for a second opinion (among others Todd Trimble in this column). A small correction, upon his remark, is made in thm(9.2)- in the sentence starting with “Now assume GC to fail for some 2n ….”. This version can (temporarily) be found on my homepage
http://home.claranet.nl/users/benschop/ngb0203.pdf – NB

Whether Mr. Benschop’s alleged proof of the Goldbach Conjecture has been invalidated may depend on who you ask. If you were to ask Robin Chapman, then I believe it is likely that he would reply that he has invalidated the proof put forward in the arXiv article, which Nico has said does not differ essentially from the proof given in his book. (See Chapman’s statement following the sci.math.research posting where the publication of Benschop’s book was announced.)

For some reason, some people seem to be looking to me to make a pronouncement. Unfortunately, I haven’t found enough time to examine the paper in detail. So I can’t point yet to a specific statement that I think is demonstrably false, but this doesn’t mean that I believe that Nico’s arguments are rigorous and/or free of gaps. Indeed, I personally find many of his arguments quite difficult to follow. In some such cases I’ve checked to see whether I could at least construct proofs for myself of certain assertions, and this has met with partial, but needless to say not complete success.

I won’t make any promises whether or when I’ll get around to forming an opinion on this paper which I think I can back up rigorously (although I’ll report back here if and when I do). As I’ve said before, life is short, and alleged elementary proofs of Fermat’s Last Theorem, Goldbach’s Conjecture, and so on are pretty far removed from my current interests. Seeing whether I can spot a flaw in this attempt would have to be considered a diversion for me, with no fixed time frame.

I assume Nico will not mind my saying that according to him, a “well-known Dutch number theorist” is also looking at his paper, for a second opinion.

The first 171 pages of the book contain no proofs (I scanned the pdf file for the word “proof”). The book is not wrong! :)

Even more interestingly, searching for Benschop or the book title or the isbn on the Springer website gives no result. Any trace of the book has vanished.

A nice Stalinist touch! Must have been the Stasi sleeper cells at Springer that just woke up.

Amazon still lists it.

Buy it while you still can! It may become a collector’s item.

Or maybe Springer will continue to sell it while trying to minimize its visibility. They do have a contract with Benschop, after all…

… though authors usually forget to check whether the contract requires the publisher to keep the book in print.

The image of the book cover that graces this blog entry is still available on Springer’s website. Any bets on how long it takes for Springer to make it disappear, after someone there reads this?

WoW, the voice of Authority who knows it all. Do you include, next to the present version of Goldbach’s Conjecture proof (chapter 9) the other 10 chapters of the book - which you apparently claim to have seen? Six of which are published in reputable journals and/or conferences: from state machine structure (the 5 basic components and the ways to couple them), Boolean Logic implementation (including fault-tolerant logic design) and Arithmetic (incl. a European Esprit project for a 32 bit logarithmetic unit on silicon). Maybe it’s useful for you to check the arXiv at
http://de.arxiv.org/abs/math/0103091 (v5)
to see the Goldbach Conjecture proof. Don’t be embarrassed to report if it improved (possibly even due to your suggestions;-)

Best greetings and wishes for the New Year.

By the way: the elementary proof of FLT that you mention as being ‘complete nonsense’ is published in the Nov.2005 issue of Acta Mathematica of Univ-Bratislava: http://pc2.iam.fmph.uniba.sk/amuc/_vol74n2.html