Schneier on Security
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February 25, 2009
The Doghouse: Singularics
This is priceless:
Our advances in Prime Number Theory have led to a new branch of mathematics called Neutronics. Neutronic functions make possible for the first time the ability to analyze regions of mathematics commonly thought to be undefined, such as the point where one is divided by zero. In short, we have developed a new way to analyze the undefined point at the singularity which appears throughout higher mathematics.
This new analytic technique has given us profound insight into the way that prime numbers are distributed throughout the integers. According to RSA's website, there are over 1 billion licensed instances of RSA public-key encryption in use in the world today. Each of these instances of the prime number based RSA algorithm can now be deciphered using Neutronic analysis. Unlike RSA, Neutronic Encryption is not based on two large prime numbers but rather on the Neutronic forces that govern the distribution of the primes themselves. The encryption that results from Singularic's Neutronic public-key algorithm is theoretically impossible to break.
You'd think that anyone who claims to be able to decrypt RSA at the key lengths in use today would, maybe, um, demonstrate that at least once. Otherwise, this can all be safely ignored as snake oil.
The founder and CTO also claims to have proved the Riemann Hypothesis, if you care to wade through the 63-page paper.
EDITED TO ADD (3/30): The CTO has responded to me.
Posted on February 25, 2009 at 2:00 PM
• 108 Comments
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Do they neutronic you in the doghouse?
Reading through other things on their web site, it appears that they're trolling for someone to buy their company. The disappointing thing is that a lot of executives looking for small innovators to buy will probably fall for it, its like something from the late '90s.
This is absurd!
I've spent the last 20 years studying mathematics and I can safely say that this company's research is completely bogus!
Through the introduction of what I like to call "non-modulo" arithmetic and "improper" fractions, I have discovered that prime numbers do not, in fact, exist at all!
Take, for example, the number 7. A prime number can only be divided by one and itself, but if I use fractions, I can divide 7 by 3, or 2, or even 6!
I've used my unique insight to create a truly marvelous crypto-system, but this comment box is too small to contain it.
You know, I doubt the Clay mathematics institute intended it, but in a way their Millennium prizes have an effect similar to James Randi's million-dollar prize for proving paranormal claims. If these guys did have a proof of the Reimann hypothesis, they could earn a million dollars, plus a ridiculous amount of publicity for their company. That they haven't done this is the best layman's indictment of their claim to a proof.
Link to the hypothesis is broken.
I've heard of "pseudo-science". This is "pseudo-math".
Sorry to be a little petty, but isn't it Riemann?
Yeah, he proved Riemann's hypothesis, but didn't collect the millenium prize ($1,000,000) and his Fields Medal to boot. He'd rather just try to sell us something.
@Damon Loved the Fermat reference.
You would also think that someone who is claiming to have broken all RSA keys to know better than to claim "The encryption ... is theoretically impossible to break" of their own system.
I would love to see some REAL math about the distribution of prime numbers. Sadly this is not it let alone factual information about secure encryption methods.
An internet search for "jnoelcook" will also reveal some inventive contributions to YouTube and various discussion forums, including claims to have created synthetic life.
Maybe a beginners course in Algebra for these incompetents? Divinding by zero is either defined or undefined. You cannot fix this, it is by definition.
@Doug Napoleone: Look here:
Conjectured in 1796, proved in 1896.
I believe that company is run by Underpants Gnomes. It seems their business plan is:
Phase 1: Float Web 3.0 vapors.
Phase 2: ?
Phase 3: Profit
When I saw divide by zero, I couldn't help but think that these guys also have built aircraft that propel themselves by pulling on lines of longitude and latitude. Are you sure this isn't some something from The Onion?
What's with all the negative criticism -- shouldn't we at least evaluate their claims thoroughly before passing judgment? The first math on their site I saw was perfectly valid. Euler's Identity, to be precise, in the banner image of the "math" page. Anyone with that level of a grasp of higher math surely is worth taking seriously. Also, they successfully used commas in large numbers on the "investments" page.
Hmm. This "Singularic Power" the Singularics people have also looks interesting. A Perpetuum Mobile maybe?
Wow! Neutronic forces...ooooh. Straight from the droppings in the bat cave.
Ouch. I'm a former mathematics student, and that's definitely a level of kookery not usually found even in the Doghouse.
Next thing you'll know they'll tell us that time is really a cube.
From the proof: A limes for 1->infinity in (11). Priceless.
I'm not sure if I am to take what you said as sarcasm or not.
Assuming it was not sarcasm, consider this: The NSA employs the most mathematicians in the world. If there were any legitimacy to this company at all, the web site would have vanished and their top thinkers employed at the NSA to update military cryptography before this bombshell undermines some of the best cryptography in the world.
Not only that, such a revelation would be a potent weapon in information warfare. Individuals who are using high grade commercial cryptography probably have more interesting things to say than those of us who normally have conversations in the clear.
The web site didn't vanish and they weren't absorbed by the NSA overnight, so I'm inclined to doubt their legitimacy without having given any time whatsoever to evaluating their claims.
Perhaps this should be moved from the "Doghouse" to the "Outhouse"?
@Ward S. Denker:
I think "they successfully used commas in large numbers" is a definite clue...
I think this is the math that enabled the Steorn perpetual motion discovery from 2 years ago.
@Ward: Their website has already been redacted. The nutjobbery is the NSA's way of making us think there's nothing there---but maybe there *was*! dun-dun-dun!
I have to imagine that they'll shortly join companies like JAWS and Non-Elephant encryption as ghosts in the Google cache . . .
My sarcasm detector is a little out of whack lately. I've taken to arguing with people who employ very contorted logic to "prove" their points (I guess you could call it "intellectual slumming"). I am absolutely certain that they're convinced there is nothing at all wrong with the way they've argued their points, otherwise they've been employing a very efficient conspiracy to drive me crazy.
I've had to step back from it all. Hopefully that rights my sarcasm detector again. I'm trying to calibrate it by having conversations with intelligent humans who employ feasible, logical arguments.
I doubt whether these guys have anything at all. I don't think they are even kidding themselves. They are simply con men trying to scam potential investors.
Note that in addition to their "Neutronic Encryption" they are also flogging something called "Singularic Power" which is supposedly a green power solution with zero emissions:
"This technique is based on the physics at the singularity which we have been able to explore using Neutronic analysis and quantum physics' nonlinear wave mechanics."
Just don't get so wrapped up in proving the RH that you ignore your thesis project. See http://abstrusegoose.com/14 (and all other episodes of this wonderfully geeky comic). ;-)
"Great scientific discoveries mark the milestones of human history. Such are the accomplishments achieved by the men and women of Singularics. Standing on the shoulders of giants such as Albert Einstein and Bernhard Riemann, we have advanced human knowledge which has resulted in 8 new inventions (patents pending). "
I searched the US Pat. Office, no hits on Alex Petty, Singularics or any of the other buzzwords.
Yours for only $20M:
The company has advanced the state of art in cryptography by developing a new theoretically unbreakable public-key algorithm called Neutronic Encryption. In so doing, well known fixed prime based algorithms such as RSA have been greatly weakened.
REQUIRED AMOUNT: $20,000,000
MINIMUM INVESTMENT AMOUNT: $1,000,000
REGION: District of Columbia
INDUSTRY 1: Products
INDUSTRY 2: Computer Hardware/Software
INVESTMENT REASON: Working Capital
PREMIUM SERVICES: Yes
LAST POSTED: January 12, 2009
Sheesh. Scientists cobble together Frankenwords from Greek and Latin spare parts. Mathematicians use perfectly ordinary English words and define them within an inch of their lives. "Neutronics"? Harrumph.
This is vintage doghouse - of a sort we haven't seen for years.
Neutronic. Singularic. Undocumented forces.
Sounds like this guy dropped acid while attending a planetarium lecture on black holes.
And let's not forget a neutron star is ULTRA DENSE. I think that sums it up quite nicely.
DEAR MR SCHNEIER,
I AM MUGU MUJOTO, SON OF THE INFORMATION PRIME NUMBER MINISTER OF NIGERIA. BEFORE MY FATHER WAS BRUTALLY ASSASSINATED, HE HID A LARGE AMOUNT OF PRIME NUMBERS IN A SECRET BANK ACCOUNT, WHICH I WOULD LIKE YOUR HELP IN MOVING OUT OF MY COUNTRY. HIS ADVANCES IN PRIME NUMBER THEORY HAVE LED TO A NEW BRANCH OF MATHEMATICS CALLED NEUTRONICS. THIS TRANSACTION IS RISK-FREE AND ALL MODALITIES HAVE BEEN ASSURED.
PLEASE SEND ME YOUR BANK ACCOUNT INFORMATION, YOUR PIN NUMBERS, 5 MILLION DOLLARS, AND A PONY SO WE MAY CONCLUDE THIS MUTUALLY BENEFICIAL TRANSACTION.
The only poster that got it right was 'kermit the blog' http://www.schneier.com/blog/archives/2009/02/...
They're obviously targeting mentally ill people to invest in their company. It's the same principle as: "Bad credit? No problem!" = "We're looking for people who can't handle money."
> I'm trying to calibrate it by having conversations with intelligent
> humans who employ feasible, logical arguments.
Not sure the Interweb is going to help you there, although the signal to noise ratio is admittedly higher at this blog than many other elsewheres :)
Note to self: reload before typing message. A lot can happen in 45 minutes.
Apologies to all the other posters, who got it right.
I was looking over the RH proof that was linked above. I was interested in it because I recently read "Prime Obsession" by John Derbyshire which deals with this very subject. The proof that J Noel Cook provides is rather compelling. Did anyone else actually read it? In the proof it appears that its author has provided not one, but two different ways to show that RH is true. I am not qualified to discern if this work is mathematically sound, but if it is a fake, it sure is stated in a manner contrary to what one might expect of a fake.
Then what's with all the b.s. about neutronics and non-existent singularities? You had to invent a new kind of math to prove/disprove the RH?
"And ever since I meet this man,
My life is not the same
And Nikolai Ivanovich Lobachevsky is his name!"
If they could crack RSA, they would have either done so and made of fortune from intercepted messages or sold the technique to someone who would. Either way they wouldn't need a nice round $20M of additional capital.
I dont know the answer Nick, but perhaps a new kind of math *is* needed to proove this question. According to Derbyshire's book, no one has made any progress on the RH problem since it was introduced about 150 years ago, and some really smart people have been working on it over the years.
As for singularities, wikipedia says:
It seems that the paper provides a way look at "undefined points" which are, by the definition given at wikipedia, called singularities.
I wish I could tell if this was real or not - but it feels to me like there may be something to it. Are there any mathematicians out there who can dig deeper on this?
Also on the name Neutronics, which at first struck me as a really cheesy sci-fi name, the author says in his paper:
"The term I gave the mathematics used to prove the Riemann Hypothesis, even before it
had been discovered, was Neutronics, as it was a system first envisioned to provide the
means to define the seemingly random growth of patterns throughout nature. The word
has later come to me to mean the neutralization of any periodic mathematical pattern or
function in relation with linear growth, as well as exponential growth, which are both also
common patterns found throughout nature that can be directly linked to this function."
I dont know what to think about it. If its real, it could be pretty amazing.
Doing a bit of Google searching on this guy, particularly by following references to his self-published novel, turns up the following MySpace page:
Reading the bits on this page, I'm increasingly suspicious that this is some form of personal entertainment hobby that exists only in amusing web pages. Note for instance that he says on his MySpace page that he's a college dropout who was majoring in entertainment business.
What'll happen now is he starts hitting up investors with "Mentioned on Bruce Schneier's blog".
I have discovered a truly marvellous proof of the Riemann hypothesis, but unfortunately I encrypted it using Twofish, and I have forgotten the password!
Nope, still not buying it. He talks about it starting as a definition of 'seemingly random growth' and then jumps to 'periodic mathematical pattern or function in relation with LINEAR growth as well as EXPONENTIAL growth.' It's a fancy description that comes off as total b.s. of the kind that we see in descriptions of New Age phenomena.
"Neutronic" is a word meaning 'related to neutrons' - so he's inventing stuff out of whole cloth. If he really means the characteristic of neutralization, the word is 'neutralistic'.
I think it's fair to say that no matter how you slice this, it comes out baloney.
Joe Trader: he starts out by defining some bizarre nonstandard notation for functions (namely, that $f_x$ will mean what most people would write $f(x)$). Along the way there's a lot of chatty commentary on how all other math papers are excessively complicated and use unclear notation, and how cool it would be if his claims were true.
The first thing in the paper that's claimed to actually be a proof is where he shows a table of the first dozen or so values of a function, all of which obey the pattern he's claiming to prove, and then says that it's clear the pattern holds all the way up to infinity, QED.
That's as far as I cared to read it.
@Damon heh heh. Fermat would be proud.
Wow, just reading through http://www.jeffreyncook.com/ made my brain hurt. It's like some weird blender of B- work on a Trig test in high school, new age 'soul' mumbo jumbo, and that 'ghost hunters' show that my father in law is obsessed with... oh, and he himself looks like a software salesman who just got back from a convention in Vegas (I happen to be one of those myself, I recognize the tan and booze droopy eyes anywhere!)
"The encryption that results from Singularic's Neutronic public-key algorithm is theoretically impossible to break."
Of course, ANY encryption algorithm is 'theoretically impossible to break', given a sufficiently poor grasp of the relevant theory...
I presume they also have a secret proof that P==NP (in exchange for only a few million dollars, they'll reveal the secret behind their proof, that N=1; they had an alternative solution, P=0, but that wasn't compatible with 'Neutronics' so it had to be suppressed).
At least here in good old Germany we don't need this high tech Neutronics stuff to encrypt our files. Here we have the best and most secure crypto application ever built by mankind: the famous KRYPTO application developed by Kryptochef:
If you can read German or can decipher his "german-english" (you are crypto experts, aren't you?) you will get much more in-depth understanding of crypto technology as a whole ;-) .
But the user will still superglue their token to the laptop, and use their birthday as the passcode
The paper reads like something generated by SciGen ;-)
Let's see if we can get their Google Group taken off line. These people are engaged in fraud, and that's illegal. So, report the group at
Top notch gobbledegook. What are the options here? Either (a) it's a spoof to wind us up or (b) the authors actually believe this stuff, in which case they are pathologically out of touch with reality, or (c) it's a scam (as Mugu Mujoto tells us).
(a) means there are a couple of quite clever people with a *lot* of spare time on their hands. Possible, but sad if that's how they get enjoyment from life.
(b) wouldn't be the first time some very clever people came off the rails.
(c) .. but surely they would be trying to scam people who are themselves very bright and probably very cynical? There must be easier marks.
Hmm. (b) then.
@wernerd: Even Kryptochef's German leaves a lot of room for improvement... but funny site, nevertheless. :)
A lot of people say, "I'm not qualified to know if its legit but it looks like it is."
Notice the contradiction. If you can't tell if its a proof, how can you tell its not star trek speak?
There are quite a few of the these crackpots. They can factor numbers in p time. But can't actually give an example etc etc etc.
No data, No examples, No dice.
Its not our time that should be on the line for every crackpot. We have to do some real science and work out some real math and get some real funding.
@Damon: your small comment box is icing on my morning cake of reading this whole thing. Keep up the good work, I am looking forward to your next insight.
Multiply both sides by zero.
There - I've developed Supersonic Encryption™. Someone send me a cookie, preferably chocolate chip.
Okay, the blurb Bruce provided was good. But if you have a few minutes, here is the Neutronics "paper" that explains this fantastic new breakthrough:
Here is a snippet to whet your appetite:
While I do not feel it necessary to show all my math, such as reductions, rearrangements or the like, not even with Complex Arithmetic, I do describe as best I possibly can to describe the meaning of nearly all my equations is at least a short word or two in plain English. So many papers today leave too much room for the reader to make assumptions, which I do feel is completely abhorrent. With that said, I do still make this mistake myself from time to time as well; it seems to me to be an unruly habit of many a modern mathematician. However, I have tried with all my might to keep the reader informed with every step I make throughout; but I clearly own the fault when the reader does not follow and offer my apologies herein if such comes about. With all that said, I shall now divulge my purported proof of the Riemann Hypothesis...
Wow, you would think that an organization with enough on the ball to prove Riemann's and to decrypt RSA would be able to avoid misspelling "privatley" and Issac" .
I once tried to prove the Riemann Hypothesis. I decided I needed a list of every prime number before I could begin. As you might imagine forming such a list is taking much longer than expected. I use PGP full disk encryption on my laptop to prevent anyone else getting access to my list. I think I might try and sell the list to the Neutronics folks.
The PDF file was created on April 12, 2007.
Is two years long enough for Clay Mathematics Institute or any other mathematicians to study and validate (or not) this proof ?
Or is he misunderstood ?
@ all posters (except Joe Trader):
I agree with Joe Trader: Can anyone come up with a reasoned, *mathematical* rebuke to this work, amidst all this wry and masturbatory clamor? How can a poster's ability to evaluate the substance of this work be assured, when his or her own arguments are simply hollow, and at least a little snide?
That this is such an obvious masturbatory prank akin more to performance art than fraud, is hilarious.
That a blog-full of Bruce Schneier followers is even typing one char in an effort to try to prove / convince others that it is utter nonsense is priceless.
It couldn't have been more obvious if he inserted 'poopy fart cakes / cherry pie = the root of all evil + the ascent into heaven' somewhere in his proof.
Some stay at home nerd's mom cut off his WoW account, and he got bored. Did a pretty good job of fooling some of you guys though, haha.
Thanks, I needed that! ;)
Wait a minute. Apparently Gingrich, along with Singularics and the Institute of Multidimensional Medicine, PLLC and anyone else can get that address from Intelligent Office of Washington DC:
(Despite my choice of name here I am not affiliated with them: I didn't find the IO site until after my previous post. A coincidence worthy of crackpot theorists.)
I stopped perusing at page 21, but I think I get the gist of it...
Twice by that point, at least, he has basically stated "I can make a model of system X, which is validated against a subset of the legal inputs for system X. Since it was validated, I can now state what system X would do using my model."
There's one great section around page 20 where he deals with a particle which is speeding up asymptotically. He "solves" this by applying a curve fit to each side, and seeing where the two curves fit. That intersection is CLEARLY the "speed" at the time where the speed is undefined.
In addition to this classic modeling mistake, it seems that his method for proving the Riemann hypothesis is to take a weaker form of it, then prove that he can write a function which is, within error bounds, identical to the Riemann Zeta function, then argue that since his function (which is close to correct) has proven intercepts only at 1/2, then the Riemann Zeta function must also follow suit.
I didn't read the last part, but it really does look like that's where its going (again, I made it to page 21). Classic application of models beyond the scope where the model has meaning. I expect him to continue making said assumption.
And I will continue to rely on RSA!
No formal proof or vaildation for (87), (95), and (103) are given. But if this assumption is false the whole proof falls appart.
Function (86) can be shortened to x*log(c_1*x)+c_2 for large x, where c_1 and c_2 are constants. But this nothing more than a simple, well known, prime number approximation.
From the Summary:
"analyze regions of mathematics commonly thought to be undefined, such as the point where one is divided by zero."
Division is a simple operation, 9 divided by 3 asks the question: How many 3's do you need to add together to get 9? So 1 divided by 0 asks the question: How many 0's do you need to add together to get 1? There is no "new math" you can invent that will answer this question. 1/0 is undefined due to the definition of the division operator.
Surely you've heard the old standard, 'burden of proof is on the claimant'?
That is, if this is a valid solution to the RH, then it's up to Dr. Neutronic and his Amazing Singularic Decoder Ring to prove such. The fact that his text includes 'any failure to understand my hypotheses are the fault of the reader and not mine' should be a tip-off that this is all nonsense.
Cook is truly the Renaissance man. Beyond mastering esoteric maths and music, he also paid PublishAmerica (a well known vanity press) to print his novel in 2007.
Unfortunately, it's out of stock. :(
Equation #18 - are you kidding me? "lim bo(x) = ..." - sure, bought it.
A friend alerted me to the Singularics site after noticing that they'd stolen a photo of mine for use on their site. But after reviewing the utter nonsense on the site, I'm quite convinced that this is *not* the work of a con man, but rather the work of a schizophrenic. It's amazingly consistent with the thought processes of those patients I encountered in my psych rotation in medical school. Sadly, he really believes he's proved the Riemann hypothesis, can divide by zero, and can produce perfectly clean energy, oh...yeah...and even crack RSA while provding uncrackable encryption himself. Last I heard, the Vernam cipher...well, one-time pad...was the only uncrackable algorithm.
I flipped through the paper, actually; didn't have the patience to try to follow up on all of it. One particularly interesting point is equation #108, which states:
This he deduces from the identity that exp(2πi)=1 and then taking the log of both sides. I wonder why he doesn't take the next obvious step and divide by πi and thus prove that 2=0. Once you accept 2=0, anything becomes provable.
@agniyo - a useful saying in this context is "That's not right. It's not even wrong".
And I'd agree with fpsurgeon - it's either a very well (too well, really) constructed hoax, or someone who is delusional.
To begin, this Singularics thing is wonderfully funny. I've been a loyal reader of Bruce's blog for years, and this is the best yet.
As soon as I looked at this guy's web page, I felt that I knew him... because he so strongly resembles a beloved relation of mine.
My relative is wonderfully optimistic, charming, breezy, funny, and has a really shaky grasp of reality. He gets enthused by high-flown concepts, which he rarely connects to anything concrete. What he's imagining is so vivid for him, and he speaks of it with such conviction, that it's easy to believe it along with him... at least, for a little while.
It's hard to be angry with him, because when he's robbing you, he also makes you feel good.
He's a fountain of entertainment with his jokes and stories, and there seems to be no sharp dividing line between the jokes and stories (on the one hand) and what he believes to be true (on the other)... it's as though everything is a variant of theatre.
So if I have it right, and Mr. Cook IS like my relative, what he's doing isn't exactly a hoax (because he partly believes it), and it isn't exactly sincere (because he's having his joke). It's his own crazy way of sending love to the world. It sure made my day!
The trickster archetype is as old as our discovery of language. Maybe this guy is just messing with us, just like the people who make crop circles mess with the UFO believers.
Of course, if they are really asking for money then it crosses over into fraud. Carlos Castaneda comes to mind as someone who went to the dark side.
For those looking for a mathematical argument, just look at his definition of Submission.
Item 2 is wrong.
I'd expect any student of Algebra to be able to inform me why it's wrong (why? because it doesn't account for the case where x=0 in item 2).
Unfortunately, this problem seems to be part of the foundation for his "branch of mathematics".
It just gets worse... For example, he doesn't like limits (nor does he understand Newtonian Physics, complex numbers, nor vectors), so he invents "functions" (that he never properly defines) to get rid of them.
It just gets worse, and worse, and worse.
Mr. Cook's mathematical work is about as interesting as a "proof", using a standard algebra, that 1=0. Yes, if you ignore the problems with 0 being a denominator, it's real easy to do. It's also really useless.
And yes, it is possible to create an algebra which has some of the features that Mr. Cook wants (but dosen't bother to define) in his algebra. Dr. Anderson has made one such interesting attempt, by re-defining division:
The only problems with Dr. Anderson's attempt are:
1) his algebra is not a field.
2) his algebra is self-contradictory (i.e. you can proove anything you like).
Mr. Cook didn't even bother giving definitions, but it appears that he's doing something similar, but far less rigorously or consistently.
I just looked at his physics work. Mind you, I'm far less educated in physics (high school) than mathematics (graduate school), but his physics work looks even worse than is mathematics work.
He doesn't understand Newtonian physics, and his problems mount as he tries anything else (and he does).
I like that his "Cook Coil" 1) creates antimatter, then 2) shoots it up (away from the direction he wants to move).
It seems that if you could do that, 1) not creating antimatter, or 2) at least shooting it in the direction you want to move would help his device immensely.
But I'm guessing that he threw out Einstein's "Does the inertia of a body depend upon its energy-content?" Around the time he threw out Newton's "Philosophiæ Naturalis Principia Mathematica" and algebra.
Well, at least his work is mildly amusing.
I notice a defect in my last post. His movement of antimatter is in the wrong direction, but my statement is also inverted.
He wants his device to move up, but his device is moving antimatter up (and presumably, the same amount of matter down, at the same velocity). As the antimatter (but not the matter) will tend to annihilate with matter, producing a lot of energy, thus pressing against anything nearby, the antimatter will tend to press his device down (not up, as he apparently wants).
On the other hand, I'm likely spending far too much time on someone's work who deliberately conflates mass and weight.
I don't see any reason to continue past "regions of mathematics commonly thought to be undefined, such as the point where one is divided by zero."
Now, real mathematicians do analyze the limit of f(x)/x as x approaches zero. While 0/0 is undefined, if f(0)=0 and f(x) approaches 0 fast enough as x approaches zero, the limit may be defined. Calculus is based on such analysis: dy/dx is the limit as dx approaches zero of
(y(x+dx) - y(x))/dx
which is the slope of y(x) and is defined if y(x) doesn't jump at the point in question. If you want to be a mathematician, physicist, or engineer, you'd better be able to learn this mathematical legerdemain at the beginning of your freshman year.
But it's not called "neutronics", and this guy's work sounds like a schizophrenic garbling of freshman calculus and physics. If it's not a joke hoax - and the only way to test that is to inquire about investing and see if he takes your money - it's a scam or a schizophrenic. (The last two cases are not mutually exclusive...)
Don't dismiss him because he defines a whole new territory. Taking the square root of a negative number was "thought to be undefined" until someone decided to invent the imaginary axis. They even had the gutso to call it imaginary!
Turned out that imaginary axis was pretty darn useful, and consistent. Nowdays its part of any engineer or mathematician's toolkit.
Don't get me wrong, he makes plenty of other mistakes worth dismissing him over, but the simple fact that he tried to give a definition to something historically undefined isn't one.
One key detail of the complex number system is that the result turned out to be useful. The resulting number system followed many very useful patterns which lead to drastic simplification of things previously defined only with trig functions. The mathematical operations were also defined rigorously, instead of just assuming that they operate as a field.
Dont dismiss his proof simply because he devised a new math. Before complex numbers, sqrt(-1) was "commonly thought to be undefined" until someone built a whole theory aroudn it.
The acid test for new theories such as that is usefulness. Complex numbers proved to describe a whole host of things in very consise notation, and maintained field logic.
Don't get me wrong, he makes PLENTY of other hangable mistakes... but inventing a new theory isn't one of them.
Consider the really important question here. A sane individual does not author a 63 page proof of the RH while claiming new math via division by zero (presumably). As JimFive points out: What does 1 divided by zero ask? It asks: How many 0s need to be added together to give 1? (Notice he doesn't ask further: And what is 0? What is 1?). Furthermore, are con artists sane? Most are presumably aware they are sane and that they are conning people. On the other hand, this individual does not appear to be sane - as fpsurgeon points out (paraphrasing): this man's behavior adds up to and is reminiscent of mental illness. The real question therefore: what is it specifically about this mental illness that makes an individual unable to evaluate: What is 0? vs. What is 1? To put it another way: What is a number, from the neurophysiological perspective and how does it relate to reality? (A question first and foremost for biological science).
His paper entertained me, and the discussion here has been reasonable, but a few folks are still a bit overimpressed with his attempts to define 1 divided by zero. This is actually a highly useful concept in Complex Analysis, and by "folding up" the complex plane so that all lines going towards infinity meet up at a point called the "point at infinity", then infinity becomes a point that is as convenient and any other. Certain classes of functions, called meromorphic functions, such as f(x) = 1/x, then are well behaved on the entire plane, including the point infinity. e.g. f(0) = infinity, f(infinity) = 0.
Here's the relevant wikipedia article if this sounds pseudomathematical to you:
This picture of the complex plane can also lead to beautiful computer visualizations, see: http://www.youtube.com/watch?v=JX3VmDgiFnY
Note that if you're talking about a field, 1 and 0 are defined (and the multiplicative and additive identity). Of course, if you ignore division by 0, you're probably not in a field. (This may be provable, but I have little interest in trying at the moment).
Complex analysis also uses information from singularities - for example, you can perform integrals around them. Once again, Cook's apparent complete cluelessness as to mathematics that was developed 40 years ago is a large warning sign.
One of his "big advances" is re-writing exponents in his own nomenclature "submission". Unfortunately, during his describing of this format, he makes fundamental errors in mathematics that I'd hope that anyone passing an Algebra class wouldn't make (specifically, he pretends that there are no problems with 0 being in a denominator).
His other "big advance" is similar, is - he re-writes exponents (again) as "Annarithms", and in steps 10 and 11 in his "paper" makes other fundamental math errors (in his paper, e^(-Pi*i) = Pi*i, and e^(-inf) = 0).
@Divide By 0-
A few of his problems could be explained away if he were using a Riemann sphere (which is plausible). Unfortunately, in both of his mathematics papers that I read, his math elsewhere assumes that he's not using a Riemann sphere, and in his text of his "proof", he basically states that he doesn't understand that division by 0 is a problem, or even why. Later on, he expresses extreme ignorance in what a proof is (you know, what he's alleging to be doing) and invents a "solution" to division by 0 via an approximation function (of course, omitting data which would quickly show his approximation function to be wildly inaccurate).
didn't mean to give the impression I thought any of his ideas were anything other than total nonsense. Quite the opposite; some people were giving him some credit for "creating new theory" when these ideas have been well developed in real mathematics, and I was hoping to inform on that topic.
This is a clearly autistic guy, living in his own dreamworld. No need to ruin it for him or take him seriously.
@Divide by 0-
I created a "new branch of mathematics" (an extension of machine theory) back in grad school. I'm sure you've never heard of it. Why?
Because I could never actually get a useful result from using it. More specifically, I was able to using a different branch of machine theory to do everything that my "new branch of mathematics" could do (that had any practical use). For that matter, the existing mathematics was simpler than my "new branch of mathematics", and wasn't prone to fall into rather nasty traps (where any physical representation of the machine would be required to be infinite in size to function).
I'd submit that the vast majority of attempts at "new mathematics" are much like that. The better mathematicians don't bother to publish, realizing the problems prior to publication; the worse mathematicians publish, not realizing that they're either making a horrible mistake or that there's an existing branch of mathematics that solves their problems as well, or better than their "new branch of mathematics".
Submission and Annarithms are just a particularly bad example of the later - there are both horrible mistakes and any problems they can solve are solved at least as well using powers.
These days, due to the wide variety of branches of mathematics, I'm highly skeptical when someone claims to have created "new mathematics" without a degree in mathematics - it's far too likely that something as good or better already exists (that they are unaware of), even if there are no horrible mistakes.
I just encountered this thread. I must say, some of the arguments made here:
are actually fairly compelling from a standpoint of pure logic.
I was particularly impressed with the logic in the "dog example" mid way down the page.
Leaderboard, you say you "just encountered this thread," yet you have the same IP address as Joe Trader. Care to explain what gives you such an interest in Singularics that you're willing to defend them by sockpuppeting?
Your soft-sell, concern-trolling approach was an interesting one, but counterproductive, since it only led people to spend more time pointing out that the emperor has no clothes. Coming back to try again after most people had lost interest in the thread--and on a weekend, when fewer people read blogs--was actually smart: if you'd managed to sneak in the last word, it might have raised some doubts for readers who don't understand any of the math in the thread. But "Leaderboard's" tone, approach, and use of language were much too similar to "Joe Trader"'s to pass unnoticed, and -- not to give away too many secrets -- when people go out of their way to point out that they just happened by, it almost always means they didn't.
Your worst mistake was trying to sockpuppet on a security blog without using a proxy. That's just embarrassing. Overall, I give your efforts a D minus.
I just re-encountered this thread. I must say, some of the arguments made here:
are actually fairly compelling from a standpoint of pure debunking.
I was particularly impressed with the logic in the "doghouse example" that starts at the top of the page.
(Moderator: don't make me put sarc tags on that.)
Fred P -
Fascinating. I guess I'll stick to reality, at least what I understand it to be (at this moment in time). Though, I must say, I would hope to see math as an exploration of sanity and other corners of the mind be married to biomedical research.
To answer your question, I'm unimpressed. According to the site:
Okay, but for x / 0 to be defined don't you need to change the mathematical system that you are using?
Then, in the answer just above the "dog" example, it describes (very poorly) a change to the mathematical system in use. Just in case one missed it, a (possibly separate) change to the mathematical system is made when "equality" is re-defined, shortly after the "dog" example.
By this point, the mathematics in question aren't a field, they aren't in a Reimann sphere, and they're not well defined.
In such a case, you can no longer steal things from field theory without being extremely careful not to trip over the variant mathematics in use - but that's done all over the "alleged proof".
This is either a deliberate attempt to confuse those with little mathematical background, or the symptom of a self-proclaimed "mathematician" who needs to take a remedial course in proofs.
Hi, friends, from the Doghouse. Jeff Cook here. Been reading up on the comments circulating the Internet regarding my work, starting particularly at this thread. While most of the derogatives are quite funny to say the least, some of you probably ought to speak with someone. I don't know...
In any case, I welcome a mathematical debate or discussion with anyone at any level over my paper and/or my credibility. Probably not here though, as I think Bruce would rather this thread simply die away. But I will respond here if you prefer. Else, feel free to raise any questions or comments or the like at my blogged response to Mr. Schneier's accusations of fraud and my work: http://blog.singularics.com/ . Here's a snippet:
"Thus, being co-founder and CTO of Singularics, I feel perhaps in a small part the responsible party for best explaining the factual value of our technology and the validity of the mathematics underlying it, being its primary engineer. This post is an honest response to the inaccurate criticism..."
I read the "response" by the fake company. Loved it! I like it when snake oil salesmen dance -- which is, perhaps, the whole point of exposing them. The entertainment value.
The tone of conspiracy oppressing the insights of spectacular thinkers who are unrecognized is particularly entertaining.
Perhaps the gentleman would be so kind as to *demonstrate* is exciting new hyperdrive (or whatever it is) so that the great unwashed, unbelieving masses might observe the phenomena.
I'd love to see it.
My opinions only, as always, and only my opinions.
Um, in retrospect after actually picking through some of the amusing nonsense on the web site, I must conclude that it's a joke -- or more accurately a spoof.
Fooled me for a while. }:-}
There is no joke...if anything it's on me. The company is no more, that part is true. The propaganda against my work has taken its toll on Singularics and my personal reputation. Not claiming any conspiracy, just the evidence in the blogs on me over the Internet. Just fact...all I ever work with.
But it has been no joke...never was on my part. I love the math and continue to work on securing the Internet.
Sorry about any confusion.
Jeffrey N. Cook
Schneier.com is a personal website. Opinions expressed are not necessarily those of Co3 Systems, Inc.