## Representational Crankery: the New Reals and the Dark Number

There's one kind of crank that I haven't really paid much attention to on this blog, and that's the real number cranks. I've touched on real number crankery in my little encounter with John Gabriel, and back in the old 0.999...=1 post, but I've never really given them the attention that they deserve.

There are a huge number of people who hate the logical implications of our definitions real numbers, and who insist that those unpleasant complications mean that our concept of real numbers is based on a faulty definition, or even that the whole concept of real numbers is ill-defined.

This is an underlying theme of a lot of Cantor crankery, but it goes well beyond that. And the basic problem underlies a lot of bad mathematical arguments. The root of this particular problem comes from a confusion between the representation of a number, and that number itself. "" isn't a number: it's a notation that we understand refers to the number that you get by dividing one by two.

There's a similar form of looniness that you get from people who dislike the set-theoretic construction of numbers. In classic set theory, you can construct the set of integers by starting with the empty set, which is used as the representation of 0. Then the set containing the empty set is the value 1 - so 1 is represented as { 0 }. Then 2 is represented as { 1, 0 }; 3 as { 2, 1, 0}; and so on. (There are several variations of this, but this is the basic idea.) You'll see arguments from people who dislike this saying things like "This isn't a construction of the natural numbers, because you can take the intersection of 8 and 3, and set intersection is meaningless on numbers." The problem with that is the same as the problem with the notational crankery: the set theoretic construction doesn't say "the empty set is the value 0", it says "in a set theoretic construction, the empty set can be used as a representation of the number 0.

The particular version of this crankery that I'm going to focus on today is somewhat related to the inverse-19 loonies. If you recall their monument, the plaque talks about how their work was praised by a math professor by the name of Edgar Escultura. Well, it turns out that Escultura himself is a bit of a crank.

The specify manifestation of his crankery is this representational issue. But the root of it is really related to the discomfort that many people feel at some of the conclusions of modern math.

A lot of what we learned about math has turned out to be non-intuitive. There's Cantor, and Gödel, of course: there are lots of different sizes of infinities; and there are mathematical statements that are neither true nor false. And there are all sorts of related things - for example, the whole ideaof undescribable numbers. Undescribable numbers drive people nuts. An undescribable number is a number which has the property that there's absolutely no way that you can write it down, ever. Not that you can't write it in, say, base-10 decimals, but that you can't ever write down anything, in any form that uniquely describes it. And, it turns out, that the vast majority of numbers are undescribable.

This leads to the representational issue. Many people insist that if you can't represent a number, that number doesn't really exist. It's nothing but an artifact of an flawed definition. Therefore, by this argument, those numbers don't exist; the only reason that we think that they do is because the real numbers are ill-defined.

This kind of crackpottery isn't limited to stupid people. Professor Escultura isn't a moron - but he is a crackpot. What he's done is take the representational argument, and run with it. According to him, the only real numbers are numbers that are representable. What he proposes is very nearly a theory of computable numbers - but he tangles it up in the representational issue. And in a fascinatingly ironic turn-around, he takes the artifacts of representational limitations, and insists that they represent real mathematical phenomena - resulting in an ill-defined number theory as a way of correcting what he alleges is an ill-defined number theory.

His system is called the New Real Numbers.

In the New Real Numbers, which he notates as , the decimal notation is fundamental. The set of new real numbers consists exactly of the set of numbers with finite representations in decimal form. This leads to some astonishingly bizarre things. From his paper:

3) Then the inverse operation to multiplication called division; the result of dividing a decimal by another if it exists is called quotient provided the divisor is not zero. Only when the integral part of the devisor is not prime other than 2 or 5 is the quotient well defined. For example, 2/7 is ill defined because the quotient is not a terminating decimal (we interpret a fraction as division).

So 2/7ths is not a new real number: it's ill-defined. 1/3 isn't a real number: it's ill-defined.

4) Since a decimal is determined or well-defined by its digits, nonterminating decimals are ambiguous or ill-defined. Consequently, the notion irrational is ill-defined since we cannot cheeckd all its digits and verify if the digits of a nonterminaing decimal are periodic or nonperiodic.

After that last one, this isn't too surprising. But it's still absolutely amazing. The square root of two? Ill-defined: it doesn't really exist. e? Ill-defined, it doesn't exist. ? Ill-defined, it doesn't really exist. All of those triangles, circles, everything that depends on e? They're all bullshit according to Escultura. Because if he can't write them down in a piece of paper in decimal notation in a finite amount of time, they don't exist.

Of course, this is entirely too ridiculous, so he backtracks a bit, and defines a non-terminating decimal number. His definition is quite peculiar. I can't say that I really follow it. I think this may be a language issue - Escultura isn't a native english speaker. I'm not sure which parts of this are crackpottery, which are linguistic struggles, and which are notational difficulties in reading math rendered as plain text.

5) Consider the sequence of decimals,

(d)^na_1a_2...a_k, n = 1, 2, ..., (1)

where d is any of the decimals, 0.1, 0.2, 0.3, ..., 0.9, a_1, ..., a_k, basic integers (not all 0 simultaneously). We call the nonstandard sequence (1) d-sequence and its nth term nth d-term. For fixed combination of d and the a_j's, j = 1, ..., k, in (1) the nth term is a terminating decimal and as n increases indefinitely it traces the tail digits of some nonterminating decimal and becomes smaller and smaller until we cannot see it anymore and indistinguishable from the tail digits of the other decimals (note that the nth d-term recedes to the right with increasing n by one decimal digit at a time). The sequence (1) is called nonstandard d-sequence since the nth term is not standard g-term; while it has standard limit (in the standard norm) which is 0 it is not a g-limit since it is not a decimal but it exists because it is well-defined by its nonstandard d-sequence. We call its nonstandard g-limit dark number and denote by d. Then we call its norm d-norm (standard distance from 0) which is d > 0. Moreover, while the nth term becomes smaller and smaller with indefinitely increasing n it is greater than 0 no matter how large n is so that if x is a decimal, 0 < d < x.

I think that what he's trying to say there is that a non-terminating decimal is a sequence of finite representations that approach a limit. So there's still no real infinite representations - instead, you've got an infinite sequence of finite representations, where each finite representation in the sequence can be generated from the previous one. This bit is why I said that this is nearly a theory of the computable numbers. Obviously, undescribable numbers can't exist in this theory, because you can't generate this sequence.

Where this really goes totally off the rails is that throughout this, he's working on the assumption that there's a one-to-one relationship between representations and numbers. That's what that "dark number" stuff is about. You see, in Escultura's system, 0.999999... is not equal to one. It's not a representational artifact. In Escultura's system, there are no representational artifacts: the representations are the numbers. The "dark number", which he notates as , is (1-0.99999999...) and . In fact, is the smallest number greater than 0. And you can generate a complete ordered enumeration of all of the new real numbers, .

Reading Escultura, every once in a while, you might think he's joking. For example, he claims to have disproven Fermat's last theorem. Fermat's theorem says that for n>2, there are no integer solutions for the equation . Escultura says he's disproven this:

The exact solutions of Fermat's equation, which are the counterexamples to FLT, are given by the triples (x,y,z) = ((0.99...)10^T,d*,10^T), T = 1, 2, ..., that clearly satisfies Fermat's equation,

x^n + y^n = z^n, (4)

for n = NT > 2. Moreover, for k = 1, 2, ..., the triple (kx,ky,kz) also satisfies Fermat's equation. They are the countably infinite counterexamples to FLT that prove the conjecture false. One counterexample is, of course, sufficient to disprove a conjecture.

Even if you accept the reality of the notational artifact , this makes no sense: the point of Fermat's last theorem is that there are no integer solutions; is not an integer; is not an integer. Surely he's not that stupid. Surely he can't possibly believe that he's disproven Fermat using non-integer solutions? I mean, how is this different from just claiming that you can use (2, 3, 351/3) as a counterexample for n=3?

But... he's serious. He's serious enough that he's published published a real paper making the claim (albeit in crackpot journals, which are the only places that would accept this rubbish).

Anyway, jumping back for a moment... You can create a theory of numbers around this rubbish. The problem is, it's not a particularly useful theory. Why? Because it breaks some of the fundamental properties that we expect numbers to have. The real numbers define a structure called a field, and a huge amount of what we really do with numbers is built on the fundamental properties of the field structure. One of the necessary properties of a field is that it has unique identity elements for addition and multiplication. If you don't have unique identities, then everything collapses.

So... Take . That's the multiplicative inverse of 9. So, by definition, - the multiplicative identity.

In Escultura's theory, is a shorthand for the number that has a representation of 0.1111.... So, . So is also a multiplicative identity. By a similar process, you can show that itself must be the additive identity. So either , or else you've lost the field structure, and with it, pretty much all of real number theory.

## It's MathematicS, not Mathematic

As you may have noticed, the crank behind the "Inverse 19" rubbish in my Loony Toony Tangents post has shown up in the comments. And of course, he's also peppering me with private mail.

Anyway... I don't want to belabor his lunacy, but there is one thing that I realized that I didn't mention in the original post, and which is a common error among cranks. Let me focus on a particular quote. From his original email (with punctuation and spacing corrected; it's too hard to preserve his idiosyncratic lunacy in HTML), focus on the part that I've highlighted in italics:

I feel that with our -1 tangent mathematics, and the -1 tangent configuration, with proper computer language it will be possible to detect even the tiniest leak of nuclear energy from space because this mathematics has two planes. I can show you the -1 configuration, it is a inverse curve

Or from his latest missive:

thus there are two planes in mathematics , one divergent at value 4 and one convergent at value 3 both at -1 tangent(3:4 equalization). So when you see our prime numbers , they are the first in history to be segregated by divergence in one plane , and convergence in the other plane. A circle is the convergence of an open square at 8 points, 4/3 at 8Pi

One of the things that crackpots commonly believe is that all of mathematics is one thing. That there's one theory of numbers, one geometry, one unified concept of these things that underlies all of mathematics. As he says repeatedly, what makes his math correct where our math is wrong is that there are two planes for his numbers, where there's one for ours.

The fundamental error in there is the assumption that there is just one math. That all of math is euclidian geometry, or that all of math is real number theory, or that real number theory and euclidian geometry are really one and the same thing.

That's wrong.

## Loony Toony Tangents

As I've mentioned before, one of the pitfalls of writing this blog is that I get a lot of mail from crazy christians. I'm not sure why it's just the christian crazies that come after me, but that's the way it is.

Anyway, yesterday, I got a fresh one which is really quite bizarre. I can't figure out what the heck the dumbass is trying to get at, so I thought I'd share.

It all starts yesterday at 2:30 or so, when I got the following, under the title "To Marcus From a Christian Physician Mathematician". I put it in a pre-format region, in order to give you the full experience. This is exactly how it appeared in my inbox. I've done my best to preserve the exact formatting, so that you get the full sense of looniness.

 Mark ,

The Lord has helped me and all I need from you is to
help write a manuscript in math language. I have
developed a new mathematics -1 tangent, very hard to
communicate and very difficult , but by a simple
computer program we have placed and sieved all
prime numbers ( Please examine our site )

. Basically the mathematics creates a tangent over the
original primordial universe , tangents used are 1/6
and 5/ 6 at Inverse 19  and if you can solve
this simple equation a help from my lord from my
lord then work with me. No PHDs
have been able to solve this, and no one has been
able to understand the
mathematics. My papers were accepted as
assignment by the worlds top Physics
journal , and they said they have a hard time
understanding the tangents, write it better

X- 0.5=(0.5X)/10        ( The lord parted the waters)
What is the rational value of X ( Least whole number ratio)

You may  write  these papers with us in the grace of
our Lord Jesus Christ. We
will give you that site if you
acknowledge our Lord Jesus Christ that stilled
the waters.


My first thought was the usual annoyance at being pestered by one of these twits. My second was "perhaps the reason why no one has been able to understand the mathematics is because you're making absolutely no sense at all. And my third was to be really annoyed, because the moron sent me a request to "look at his site", without even bothering to give me a URL!

So... I responded. I know that I shouldn't have, but I can't resist a good crank. A couple of quick and mostly nonsensical exchanges occured, which just aren't worth the effort of copying. But one thing that I did say to him was:

If you send me your stuff, I'll take a look at it. But you should understand that if, as I suspect, it turns out to be nothing but garbage, then I'm going to post it on my blog, with an appropriate amount of mockery of you and your work.

There were a couple of stupid back and forths... including his explaining that the reason that he sent this to me was because I've written about christian mathematicians on by blog. From this, I conclude that the guy's reading comprehension is about as good as his writing, because the only times that I've mentioned the religion of anyone (except myself) that I've written about, it's to mock them. (Like, for example, I've frequently mocked Dembski's, and the way that he substitutes christian apologetics for actual math.)

Anyway, the first thing with any actual substance to it in our exchange, was this:

What ever , is fine, my Paper as I say was accepted as assigned by the AIP Journals and it is not accepted for publication because it is sloppy and poorly written in different mathematics. The reason I will only give you the prime number placement and the Computer program because you will not understand -1 tangent or some of the mathematical statements like " A divisor of Space must be 2* a tangent ( a tangent always has a midline. Divisor 19 is exactly 1:3 (1/6+1/6) . That is it , do you solve or understand X-0.5= (0.5X)/10

Attached is a very tiny snippet slow prime number sieve/placement Program that no one has seen or understood yet but "each prime number is connected to each prime number and is continuous program" so unlike all the yobos prime number sieve out there , this one is different . It does not need a proof . We have done a billion and it is already copywrited to our site . Attached is the source code and the sample prime numbers by gaps and placement. It is my gift to you, and if you understand this then I will show you the rest of the mathematics, and why I can help you and you me .

Dont you dare call it Garbage, because then I can do the same to you , what is garbage is current mathematics understanding of prime numbers etc. I have a Phd education too, so it do not matter, I am a fellow of the royal college of Surgeons . See only the rest at your risk , you will not get it, because it is -1 tangent mathematics. It is copy righted ten times over.

V.C.

Along with this, he included a PDF file that had Fortran-77 source code superimposed on background images...

Now, I have no idea of just what this twit is trying to get at. But he did at least send me a link to his website. He's created his own "research institute" called hope research. And it's an absolute gem of almost time-cube caliber insanity. He's got a picture of a file of rocks, with a metal plaque on a pole above them, reading:

BY THE GRACE OF OUR LORD JESUS CHRIST, A FEW ORDINARY
ATHENS RESIDENTS DISCOVERED "INVERSE 19 MATHEMATICS",
AT TAN 19, 1:3 (1/6 + 1/6), WITH DIVERGENCE/CONVERGENCE
AT 1 AND MINUS 1(0.999.), IN 2009-2010

WORLD RENOWNED MATHEMATICIAN PROFESSOR EDGAR
IN WRITING THE BASIC PREMISE OF TAN 19, AND ALSO IN
WRITING HAS SAID "THAT IS IS A NEW NUMBER SYSTEM
AND A NEW GEOMETRIC PLANE". THE FUTURE WILL HOLD THE
REST

THEOREM OF INVERSE 19- "A CIRCLE IS A SPACE CONSTRUCTED
BY THE INVERSE OF PERFECT CONVERGENCE/DIVERGENCE
OF 1 AND -1. 1=MUN 1(0.999.) AT NATURAL 1/3(1/6 + 1/6)
DIVERGENCE OF NUMBERS. WHEN A CIRCLE IS COLLAPSED
IT COLLAPSES NOT TO A NULL ZERO AS PER PRESENT
MATHEMATICAL THEORY, BUT INTO A TANGENT CURVED
SPACE 1,-1 AND 0.000166666667 (1/6X1/1000)


Try to make sense out of that, eh?

Looking at his web-page a bit, it's an amazing jumble of incoherent rubbish. Most of it is just pure incoherence. But, as near as I can figure it out... the nugget, the basic idea at the center of it all, is:

CURRENT MATHEMATICS THEORY is wrong because it is based on a single square plane with a squared center, "a circle can never be squared”, vice versa, by a single mathematical plane, the mistake of Riemann, Euclid, Archimedes, and Einstein.

In somewhat more coherent terms: he believes that our number system is fundamentally defined by a square plane, and that all sorts of errors come from the fact that we always analyze things in terms of a "square space". He believes that there are actually two overlapping spaces - one square, and one circular.

The "circle can never be squared" bit is really quite interesting, because it's something that cranks constantly bring up, without ever bothering to understand what it means.

There's an old traditional of geometry dating back to the ancient Greeks, which looks at things you can do using nothing but a straight-edge and a compass. You can do a lot of interesting things; for example, you can construct a perfect square without needing to measure any lengths or angles. Below is an animation of the process, from wikipedia.

Squaring a circle is a straight-edge and compass problem: if I give you a circle, can you draw a square which has the same area as that circle using nothing but a straight-edge and a compass? And the answer is: No, you can't. When someone talks about "squaring a circle", that's all that they're talking about: you can't draw a square and a circle with the same area using nothing but a straight-edge and a compass.

People like our incoherent friend here believe that it means something much, much stronger: that you can never convert between circles and squares; that things that are round, and things that have right angles are completely, fundamentally incompatible. This is utter nonsense.

In fact, given a plane, we can identify points in the plane in two different ways: by picking a line and an arbitrary 0 point, we can then measure its distance from the origin in two directions (the rectangular coordinate), or we can measure its angle and distance from the origin and baseline (the polar or circular coordinate). And we can freely convert back and forth between those two representations.

He doesn't understand that at all. He believes that the cartesian plane is actually rectangular, and believes he's made some brilliant discovery by inventing a circular form of a plane. (A plane isn't rectangular or circular. It's a plane.)

As far as his prime number stuff goes... I can't make head or tail out of it. He seems to be using the word "tangent" in a novel way, and I can't figure out what his definition of the word is. Without that, there's no hope of rendering his babble into anything meaningful.

But for your entertainment... He claims that he's got this program which somehow demonstrates his prime discovery. For you, my loyal readers, I have actually copied it out of his PDF file. This appears to some version of BASIC.. it's amusing; his programming is just as incoherent as his english. I mean, look at it: there's no way that this program can work. None. Nil. Zero.

I doubt that it's even valid syntax. I can't say that for certain, because there are so many different variants of BASIC, and so many of them are so wacky. But even if the syntax, by some miracle, is actually valid in some version of basic, it doesn't work.

How can I say that? Just look at the program - you don't need to look very far. Look at the line with line number 10: 10 IF PRIME(X)=0 THEN GOTO 1009. There is no line 1009. There are jumps to line 1014; there is no line 1014. There are statements that jump to line 2002; there is no line 2002.

DIM PRIME(100000)' HOW FAR DO YOU WANT TO GO
DIM RIME(100000)' HOW FAR DO YOU WANT TO GO
PRIME(X)=7
RIME(Y)=5
X=0
Y=0
5 A = A + 1
7 AA=0

[BB]
D=D+1
X=0
Y=0
10 IF PRIME(X)=0 THEN GOTO 1009
IF RIME(Y)=0 THEN GOTO 1009

20 IF BB/RIME(Y)=1 THEN GOTO 1999 ELSE IF BB/RIME(Y)=0
OR BB/RIME(Y)=INT(BB/RIME(Y)) THEN GOTO 2000
30 IF BB/PRIME(X)=1 THEN GOTO 1999 ELSE IF BB/PRIME(X)=0
OR BB/PRIME(X)=INT(BB/PRIME(X)) THEN GOTO 2000
40 IF INT(BB/2)<RIME(Y) AND INT(BB/2)<PRIME(X) GOTO 1999

X=X+1
Y=Y+1

GOTO 10

1999 AA=AA +1
2000 X=0
Y=0
[BA]

IF PRIME(X)=0 THEN GOTO 1009
IF RIME(Y)=0 THEN GOTO 1014

IF BA/RIME(Y)=1 THEN GOTO 2002 ELSE IF BA/RIME(Y)=0
OR BA/RIME(Y)=INT(BA/RIME(Y)) THEN GOTO 2001
IF BA/PRIME(X)=1 THEN GOTO 2002 ELSE IF BA/PRIME(X)=0
OR BA/PRIME(X)=INT(BA/PRIME(X)) THEN GOTO 2001
IF INT(BA/2)<RIME(Y) AND INT(BA/2)<PRIME(X) GOTO 2002

X=X+1
Y=Y+1
GOTO [BA]
2001  GOTO 2003
2002

AA=AA+2
2003 IF AA=1 THEN PRINT TAB(32); BB
IF AA=2 THEN PRINT BA
IF AA=3 THEN PRINT BA; TAB(32); BB

BB =BB +6
BA =BA +6
X=A
Y=A

LET PRIME(X)=BB 'BB
E=X
LET RIME(Y)=BA  'BA
F=Y

IF A = 100000 THEN GOTO [MEM]  ' HOW FAR DO YOU WANT TO GO
IF D= 71 THEN GOTO [PAGE] ELSE GOTO 5

[PAGE]
D=0
B=B+1 'PAGE NUMBERS

PRINT TAB(45);"PAGE " ;B

C=C+A              'NUMBER OF LOOPS

' START NEXT LOOP

IF B=200 THEN GOTO [QUIT]

GOTO 5

[QUIT]

open "LOOP" for text as #1
print #1, "NUMBER OF LOOPS "; C
PRINT #1, "X ";E;" Y ";F

confirm "DO YOU WISH TO CONTINUE?"; answer$if answer$ = "no" then [END]

GOTO [CONTINUE]

[CONTINUE]
CLOSE #1
GOTO 5

[END]
CLOSE #1

END

[MEM]
PRINT "OUT OF ALOTTED MEMORY   A"

END


I wanted to give you folks a version of this that actually ran... to at least see if this was, in any way, shape, or form a prime number generator. I tried to translate it into Python... But I can't make any kind of sense out of it. Even with all of the obscure and deliberately pathological languages I've learned, I can't make this make sense. For example, BA and [BB] seem to branch targets. But they also seem to somehow be used as variable prefixes. I'm not sure what, if anything, that's supposed to mean.

If you know the variant of BASIC that this is written for, and you can explain it to me, I'll be glad to make another stab at rewriting it into a runnable program in Python.

To conclude.. Why should I bother to do this? According to my loony correspondent:

I feel that with our -1 tangent mathematics, and the -1 tangent configuration , with proper computer language it will be possible to detect even the tiniest leak of nuclear energy from space because this mathematics has two planes. I can show you the -1 configuration, it is a inverse curve

I reluctantly give you the raw very primitive site of the mathematics without the calculus , it is not written in modern math language,but we are sure of it the mathematics and the numbers placement. DO NOT ridicule us, and if you can help find a partner to write this mathematics with us , let us know, we will teach you the calculus

Yes, we'll be able to detect the tiniest leak of nuclear energy using his prime number sieve! (Which, in so far as I can understand it, isn't even a sieve.) And I'd better not ridicule him. Oops, too late.

Oh, and according to him, π is exactly 22/7.

## Grandiose Crackpottery Proves Pi=4

Someone recently sent me a link to a really terrific crank. This guy really takes the cake. Seriously, no joke, this guy is the most grandiose crank that I've ever seen, and I doubt that it's possible to top him. He claims, among other things, to have:

1. Demonstrated that every mathematician since (and including) Euclid was wrong;
2. Corrected the problems with relativity;
3. Turned relativity into a unification theory by proving that magnetism is part of the relativistic gravitational field;
4. Shown that all of gravitational/orbital dynamics is completely, utterly wrong; and, last but not least:
5. proved that the one true correct value of is exactly 4.

I'm going to focus on the last one - because it's the simplest illustration of both his own comical insanity, of of the fundamental error underlying all of his rubbish.

## Obfuscatory Vaccination Math

Over at my friend Pal's blog, in a discussion about vaccination, a commenter came up with the following in an argument against the value of vaccination:

Mathematical formula:

100% – % of population who are not/cannot be vaccinated – % of population who have been vaccinated but are not immune (1-effective rate)-% of population who have been vaccinated but immunity has waned – % of population who have become immune compromised-(any other variables an immunologist would know that I may not)

What vaccine preventable illnesses have the result of that formula above the necessary threshold to maintain herd immunity?

I don’t know if the population is still immune to Smallpox, but I would hope that that is just a science fiction question. Smallpox was eradicated, but that vaccine did have the highest number of adverse reaction (I’m sure PAL will correct me if that statement is wrong)

It's a classic example of what I call obfuscatory mathematics: that is, it's an attempt to use fake math in an attempt to intimidate people into believing that there's a real argument, when in fact they're just hiding behind the appearance of mathematics in order to avoid having to really make their argument. It's a classic technique, frequently used by crackpots of all stripes.

It's largely illegible, due to notation, punctuation, and general babble. That's typical of obfuscatory math: the point isn't to use math to be comprehensible, or to use formal reasoning; it's to create an appearance of credibility. So let's take that, and try to make it sort of readable.

What he wants to do is to take each group of people who, supposedly, aren't protected by vaccines, and try to put together an argument about how it's unlikely that vaccines can possibly create a large enough group of protected people to really provide herd immunity.

So, let's consider the population of people. Per Chuck's argument, we can consider the following subgroups:

• is the percentage of the population that does not get vaccinated, for whatever reason.
• is the percentage of people who got vaccinated; obviously equal to .
• is the percentage of people who were vaccinated, but who didn't gain any immunity from their vaccination.
• is the percentage of people who were vaccinated, but whose immunity from the vaccine has worn off.
• is the percentage of people who were vaccinated, but who have for some reason become immune-compromised, and thus gain no immunity from the vaccine.

He's arguing then, that the percentage of effectively vaccinated people is . And he implies that there are other groups. Since herd immunity requires a very large part of the population to be immune to a disease, and there are so many groups of people who can't be part of the immune population, then with so many people excluded, what's the chance that we really have effective herd immunity to any disease?

There's a whole lot wrong with this, ranging from the trivial to the moderately interesting. We'll start with the trivial, and move on to the more interesting.

## Too Crazy to Be Fun: Pi Crackpottery

I always appreciate it when readers send me links to good crackpottery. But one of the big problems with a lot of the links that I get is that a lot of them are just too crazy. When you've got someone going off on a time-cube style rant, there's just no good way to make fun of them - the stuff just doesn't make enough sense to make fun of.

For example, someone sent me a really... interesting link recently, to a book by a guy who claims to have proved that . Let me quote the beginning of his book, to give you an idea of what I mean. I've attempted to reproduce the formatting as well as I can, but it's frankly worse that I can figure out how to reproduce with HTML.

CONCEPTIONS OF π

One conception of π is the value 3.141... that is used for calculations, involving geometrical figures containing circles.

Another conception is that the number 3.141... is only an approximation. I interpret

π in this book as the relationship between a circle and its diameter, and not as the irrational number 3.141...

I have attempted to find a value that will result in exact calculations of circles.

SQUARING

The word “squaring” is used for the following:

A. The square with side of 4 u.l. so-called square squaring form

B. A circle with the diameter of 4 u.l., the circle squaring form

C. The only cylinder that has been produced by a square and two circles, from which come the cylinder squaring form

I identify the characteristics found in figures that I call square squaring, circle squaring and cylinder squaring and the principles behind these figures. I refer to three figures:

1. Square

2. Circle

3. Cylinder

It's not particularly easy to make fun of that, because it's so utterly and bizarrely nonsensical.

It's pretty hard to get through his drek... But he's got this way of characterizing different kinds of squares, and then different kinds of circles based on the different kinds of squares. The ways of characterizing the squares are based on screwing up units. There are three kinds of squares: squares where the number of length units in the perimeter are larger than the number of area units in the area; squares where the number of length units in the perimeter are smaller than the number of area units in the area; and squares where they're equal.

That last group contains only one element: the square who's sides have length 4. He concludes that this is a profoundly important square, and says that a square whose side-length is four of some unit is the "square squaring form" of the square. This is a really important idea to him: he goes out of his way to write a special note in extra large font:

N.B.

Squares with sides of 4 u.l. have a perimeter of 16 u.l. and an area of 16 u.a. Perimeter = 16 u.l. and area = 16 u.a. What I immediately observed was the common number for the perimeter and the area.

As you can see, we're dealing with a real genius here.

From there, he launches into a description of circles. According to him, every circle is defined by a square, where the circle is inscribed in the square. It makes no sense at all; this section, I can't even attempt to mock. It's just so damned incoherent that it's not even funny. The conclusion is that for magical reasons to be explained later, the circle with diameter 4 is special.

Then we get to the heart of the matter: what he calls "the circle squaring form". This continues to make no sense. But it's got some interesting typography. It starts with:

ln

of

the logarithm e

For no apparent reason. Then he goes on to start presenting the notation he's going to use... And to call it insane is kind. In includes two distinct definitions: "Logarithm e = log e" and "Logarithm ln of e = log ln". I have no clue what this is supposed to mean.

From there, he goes through a bunch of definitions, leading up to a set of purported equations describing the special circle related to the special square whose sides are 4 units long. What are the equations going to show us?

The formulae will define a circle that shows relation to;

• Its diameter to its circumference and area.
• Circles relation to its square.
• Its relation of the shaded area that is not covered by the
circle.
• Finally, how many per cent a circle cover its square’s
area and perimeter.
• Also relations to the cylinder.

So he gets to the equations, which are defined in terms of "ln of logarithm e". His first equation, presented without explanation, is:

What in the hell that's supposed to mean, I don't know. He doesn't define Q. s is the length of the side of a square. Where eln s comes from, I have no idea... but he gets rid of it, replacing it with s. Apparently, this is supposed to be a meaningful step - we're supposed to learn something really important from it! He goes through a bunch of steps, ending up with "Relevant Formula: ⇒ ", which supposedly defines "the relationship between area, circumference, and diameter of a circle".

I'll stop here. I think by now you can see my problem. How can you make fun of this in an entertaining way? There's just nothing that I can say about this stuff beyond "huh? what in the bloody hell is he trying to say here?"

He offers a cash prize to anyone who can prove him wrong. I think he's pretty safe in not needing to worry about paying that prize out; you can't prove that something nonsensical is wrong. Yeah, sure, π=3.2 or whatever in his universe: after all, for any statement S, . Hell, , therefore the moon is made of green cheese!

What kills me about this is how utterly, insanely, ridiculously wrong it is... My daughter, who is in fifth grade, did experiments last year in math class where they roll a circle along a piece of paper to get its diameter, and then compare that to its length. A bunch of fourth graders can easily do this accurately enough to show that the ratio of the circumference to the diameter is around 22/7. Any attempt to actually verify his number totally fails. But it would seem that in his world, when reality conflicts with theory, reality is the one that's wrong.

## Return of a Classic: The Electromagnetic Gravity Revolution!

Between work, trying to finish my AppEngine book, and doing all of the technical work getting Scientopia running smoothly on the new hosting service, I haven’t had a lot of time for writing new blog posts. So, once again, I'm recycling some old stuff.

It's that time again - yes, we have yet another wacko reinvention of physics that pretends to have math on its side. This time, it's "The Electro-Magnetic Radiation Pressure Gravity Theory", by "Engineer Xavier Borg". (Yes, he signs all of his papers that way - it's always with the title "Engineer".) This one is as wacky as Neal Adams and his PMPs, except that the author seems to be less clueless.

At first I wondered if this were a hoax - I mean, "Engineer Borg"? It seems like a deliberately goofy name for someone with a crackpot theory of physics... But on reading through his web-pages, the quantity and depth of his writing has me leaning towards believing that this stuff is legit. (And as several commenters pointed out the first time I posted this, in Germany, you need a special license to be an engineer, and as a result, "Engineer" is actually really used as a title. Still seems pompous to me - I mean, technically, I'm entitled to go around calling myself Dr. Mark Chu-Carroll, PhD., but I don't generally do that.)

Between work, trying to finish my AppEngine book, and doing all of the technical work getting Scientopia running smoothly on the new hosting service, I haven't had a lot of time for writing new blog posts.

But in the process of doing my technical work around here, I was browsing through some archives, and seeing some of my old posts that I'd forgotten about. And odds are, if I forgot about it, then there are a lot of readers who've never seen it. So I'm going to bring back some of the classic old material.

For example, Neal Adams. Comic book fans will know about Neal: he's a comic book artist who worked on some of the most famous comics in the 1970s: he drew Batman, Superman, Deadman, Green Lantern, the Spectre, the X-men. More recently, he's done a lot of work in general commercial art - for example, he did the animated nasonex bee commercials a few years ago.

But he's not just an artist. No, he's so much more than that! He's also a brilliant scientist. He's much smarter than all of those eggheads with college degrees. They're struggling to build giant particle accelerators to help understand things like mass. But Neal - he's got them beat. He's figured out exactly how things work!

According to Neal, there is no such thing as gravity - it's all just pressure. People trying to figure out stuff about how gravity works are just wasting time. The earth (and all other planets) is actually a matter factory - matter is constantly created in the hollow center of the earth, and the pressure of all the new matter forces the earth to constantly expand. The constant expansion creates pressure on the surface as things expand - and that constant expansion is what creates gravity! You're standing on a point on the surface of the earth. And the earth is expanding - the ground is pushing up on you because of that expansion. You're not being pulled down towards the earth: the earth is pushing up on you.

And according to Neal, the best part is the math works!. In the original version of this post, I had a link to Neal's page with his explanation of how the math works - but he has, since then, moved most of his science stuff behind a paywall - you now need to pay Neal \$20 to get to see his material, so I can't provide a direct link. But it's in a video <a href="http://www.continuitystudios.net/prime.html"here, and you can see the original using the Wayback Machine.

## Turing Crackpottery!

One of the long-time cranks who's commented on this blog is a bozo who goes by the name "Vorlath". Vorlath is a hard-core Cantor crank, and so he usually shows up to rant whenever the subject of Cantor comes up. But last week, while I was busy dealing with the Scientopia site hosting trouble, a reader sent me a link to a piece Vorlath wrote about the Halting problem. Apparently, he doesn't like that proof either.

Personally, the proof that the halting problem is unsolvable is one of my all-time favorite mathematical proofs. It's incredibly simple - just a couple of steps. It's very concrete - the key to the proof is a program that you can actually write, easily, in a couple of lines of code in a scripting language. And best of all, it's incredibly profound - it proves something very similar to Gödel's incompleteness theorem. It's wonderful.

To show you how simple it is, I'm going to walk you through it - in all of its technical details.