# Why we need formality in mathematics

The comment thread from my last Cantor crankery post has continued in a way that demonstrates a common issue when dealing with bad math, so I thought it was worth taking the discussion and promoting it to a proper top-level post.

The defender of the Cantor crankery tried to show what he alleged to be the problem with Cantor, by presenting a simple proof:

If we have a unit line, then this line will have an infinite number of points in it. Some of these points will be an irrational distance away from the origin and some will be a rational distance away from the origin.

Premise 1.

To have more irrational points on this line than rational points (plus 1), it is necessary to have at least two irrational points on the line so that there exists no rational point between them.

Premise 2.

It is not possible to have two irrational points on a line so that no rational point exists between them.

Conclusion.

It is not possible to have more irrational points on a line than rational points (plus 1).

This contradicts Cantor’s conclusion, so Cantor must have made a mistake in his reasoning.

(I’ve done a bit of formatting of this to make it look cleaner, but I have not changed any of the content.)

This is not a valid proof. It looks nice on the surface – it intuitively feels right. But it’s not. Why?

Because math isn’t intuition. Math is a formal system. When we’re talking about Cantor’s diagonalization, we’re working in the formal system of set theory. In most modern math, we’re specifically working in the formal system of Zermelo-Fraenkel (ZF) set theory. And that “proof” relies on two premises, which are not correct in ZF set theory. I pointed this out in verbose detail, to which the commenter responded:

I can understand your desire for a proof to be in ZFC, Peano arithmetic and FOPL, it is a good methodology but not the only one, and I am certain that it is not a perfect one. You are not doing yourself any favors if you let any methodology trump understanding. For me it is far more important to understand a proof, than to just know it “works” under some methodology that simply manipulates symbols.

This is the point I really wanted to get to here. It’s a form of complaint that I’ve seen over and over again – not just in the Cantor crankery, but in nearly all of the math posts.

There’s a common belief among crackpots of various sorts that scientists and mathematicians use symbols and formalisms just because we like them, or because we want to obscure things and make simple things seem complicated, so that we’ll look smart.

That’s just not the case. We use formalisms and notation because they are absolutely essential. We can’t do math without the formalisms; we could do it without the notation, but the notation makes things clearer than natural language prose.

The reason for all of that is because we want to be correct.

If we’re working with a definition that contains any vagueness – even the most subtle unintentional kind (or, actually, especially the most subtle unintentional kind!) – then we can easily produce nonsense. There’s a simple “proof” that we’ve discussed before that shows that 0 is equal to 1. It looks correct when you read it. But it contains a subtle error. If we weren’t being careful and formal, that kind of mistake can easily creep in – and once you allow one, single, innocuous looking error into a proof, the entire proof falls apart. The reason for all the formalism and all the notation is to give us a way to unambiguously, precisely state exactly what we mean. The reason that we insist of detailed logical step-by-step proofs is because that’s the only way to make sure that we aren’t introducing errors.

We can’t rely on intuition, because our intuition is frequently not correct. That’s why we use logic. We can’t rely on informal statements, because informal statements lack precision: they can mean many different things, some of which are true, and some of which are not.

In the case of Cantor’s diagonalization, when we’re being carefully precise, we’re not talking about the size of things: we’re talking about the cardinality of sets. That’s an important distinction, because “size” can mean many different things. Cardinality means one, very precise thing.

Similarly, we’re talking about the cardinality of the set of real numbers compared to the cardinality of the set of natural numbers. When I say that, I’m not just hand-waving the real numbers: the real numbers means something very specific: it’s the unique complete totally ordered field $(R, +, *, <)$ up to isomorphism. To understand that, we’re implicitly referencing the formal definition of a field (with all of its sub-definitions) and the formal definitions of the addition, multiplication, and ordering operations.

I’m not just saying that to be pedantic. I’m saying that because we need to know exactly what we’re talking about. It’s very easy to put together an informal definition of the real numbers that’s different from the proper mathematical set of real numbers. For example, you can define a number system consisting of the set of all numbers that can be generated by a finite, non-terminating computer program. Intuitively, it might seem like that’s just another way of describing the real numbers – but it describes a very different set.

Beyond just definitions, we insist on using formal symbolic logic for a similar reason. If we can reduce the argument to symbolic reasoning, then we’ve abstracted away anything that could bias or deceive us. The symbolic logic makes every statement absolutely precise, and every reasoning step pure, precise, and unbiased.

So what’s wrong with the “proof” above? It’s got two premises. Let’s just look at the first one: “To have more irrational points on this line than rational points (plus 1), it is necessary to have at least two irrational points on the line so that there exists no rational point between them.”.

If this statement is true, then Cantor’s proof must be wrong. But is this statement true? The commenter’s argument is that it’s obviously intuitively true.

If we weren’t doing math, that might be OK. But this is math. We can’t just rely on our intuition, because we know that our intuition is often wrong. So we need to ask: can you prove that that’s true?

And how do you prove something like that? Well, you start with the basic rules of your proof system. In a discussion of a set theory proof, that means ZF set theory and first order predicate logic. Then you add in the definitions you need to talk about the objects you’re interested in: so Peano arithmetic, rational numbers, real number theory, and the definition of irrational numbers in real number theory. That gives you a formal system that you can use to talk about the sets of real numbers, rational numbers, and natural numbers.

The problem for our commenter is that you can’t prove that premise using ZF logic, FOPL, and real number theory. It’s not true. It’s based on a faulty understanding of the behavior of infinite sets. It’s taking an assumption that comes from our intuition, which seems reasonable, but which isn’t actually true within the formal system o mathematics.

In particular, it’s trying to say that in set theory, the cardinality of the set of real numbers is equal to the cardinality of the set of natural numbers – but doing so by saying “Ah, Why are you worrying about that set theory nonsense? Sure, it would be nice to prove this statement about set theory using set theory, but you’re just being picky on insisting that.”

Once you really see it in these terms, it’s an absurd statement. It’s equivalent to something as ridiculous as saying that you don’t need to modify verbs by conjugating them when you speak english, because in Chinese, the spoken words don’t change for conjugation.

# UD Creationists and Proof

A reader sent me a link to a comment on one of my least favorite major creationist websites, Uncommon Descent (No link, I refuse to link to UD). It’s dumb enough that it really deserves a good mocking.

Barry Arrington, June 10, 2016 at 2:45 pm

daveS:
“That 2 + 3 = 5 is true by definition can be verified in a purely mechanical, absolutely certain way.”

This may be counter intuitive to you dave, but your statement is false. There is no way to verify that statement. It is either accepted as self-evidently true, or not. Think about it. What more basic steps of reasoning would you employ to verify the equation? That’s right; there are none. You can say the same thing in different ways such as || + ||| = ||||| or “a set with a cardinality of two added to a set with cardinality of three results in a set with a cardinality of five.” But they all amount to the same statement.

That is another feature of a self-evident truth. It does not depend upon (indeed cannot be) “verified” (as you say) by a process of “precept upon precept” reasoning. As WJM has been trying to tell you, a self-evident truth is, by definition, a truth that is accepted because rejection would be upon pain of patent absurdity.

2+3=5 cannot be verified. It is accepted as self-evidently true because any denial would come at the price of affirming an absurdity.

It’s absolutely possible to verify the statement “2 + 3 = 5”. It’s also absolutely possible to prove that statement. In fact, both of those are more than possible: they’re downright easy, provided you accept the standard definitions of arithmetic. And frankly, only a total idiot who has absolutely no concept of what verification or proof mean would ever claim otherwise.

Verification is the process of testing a hypothesis to determine if it correctly predicts the outcome. Here’s how you verify that 2+3=5:

1. Get two pennies, and put them in a pile.
2. Get three pennies, and put them in a pile.
3. Put the pile of 2 pennies on top of the pile of 3 pennies.
4. Count the resulting pile of pennies.
5. If there are 5 pennies, then you have verified that 2+3=5.

Verification isn’t perfect. It’s the result of a single test that confirms what you expect. But verification is repeatable: you can repeat that experiment as many times as you want, and you’ll always get the same result: the resulting pile will always have 5 pennies.

Proof is something different. Proof is a process of using a formal system to demonstrate that within that formal system, a given statement necessarily follows from a set of premises. If the formal system has a valid model, and you accept the premises, then the proof shows that the conclusion must be true.

In formal terms, a proof operates within a formal system called a logic. The logic consists of:

1. A collection of rules (called syntax rules or formation rules) that define how to construct a valid statement are in the logical language.
2. )

3. A collection of rules (called inference rules) that define how to use true statements to determine other true statements.
4. A collection of foundational true statements called axioms.

Note that “validity”, as mentioned in the syntax rules, is a very different thing from “truth”. Validity means that the statement has the correct structural form. A statement can be valid, and yet be completely meaningless. “The moon is made of green cheese” is a valid sentence, which can easily be rendered in valid logical form, but it’s not true. The classic example of a meaningless statement is “Colorless green ideas sleep furiously”, which is syntactically valid, but utterly meaningless.

Most of the time, when we’re talking about logic and proofs, we’re using a system of logic called first order predicate logic, and a foundational system of axioms called ZFC set theory. Built on those, we define numbers using a collection of definitions called Peano arithmetic.

In Peano arithmetic, we define the natural numbers (that is, the set of non-negative integers) by defining 0 (the cardinality of the empty set), and then defining the other natural numbers using the successor function. In this system, the number zero can be written as $z$; one is $s(z)$ (the successor of $z$); two is the successor of 1: $s(1) = s(s(z))$. And so on.

Using Peano arithmetic, addition is defined recursively:

1. For any number $x$, $x + 0 = x$.
2. For any number numbers x and y: $s(x)+y=x+s(y)$.

So, using peano arithmetic, here’s how we can prove that $2+3=5$:

1. In Peano arithemetic form, $2+3$ means $s(s(z)) + s(s(s(z)))$.
2. From rule 2 of addition, we can infer that $s(s(z)) + s(s(s(z)))$ is the same as $s(z) + s(s(s(s(z))))$. (In numerical syntax, 2+3 is the same as 1+4.)
3. Using rule 2 of addition again, we can infer that $s(z) + s(s(s(s(z)))) = z + s(s(s(s(s(z)))))$ (1+4=0+5); and so, by transitivity, that 2+3=0+5.
4. Using rule 1 of addition, we can then infer that $0+5=5$; and so, by transitivity, 2+3=5.

You can get around this by pointing out that it’s certainly not a proof from first principles. But I’d argue that if you’re talking about the statement “2+3=5” in the terms of the quoted discussion, that you’re clearly already living in the world of FOPL with some axioms that support peano arithmetic: if you weren’t, then the statement “2+3=5” wouldn’t have any meaning at all. For you to be able to argue that it’s true but unprovable, you must be living in a world in which arithmetic works, and that means that the statement is both verifiable and provable.

If you want to play games and argue about axioms, then I’ll point at the Principia Mathematica. The Principia was an ultimately misguided effort to put mathematics on a perfect, sound foundation. It started with a minimal form of predicate logic and a tiny set of inarguably true axioms, and attempted to derive all of mathematics from nothing but those absolute, unquestionable first principles. It took them a ton of work, but using that foundation, you can derive all of number theory – and that’s what they did. It took them 378 pages of dense logic, but they ultimately build a rock-solid model of the natural numbers, and used that to demonstrate the validity of Peano arithmetic, and then in turn used that to prove, once and for all, that 1+1=2. Using the same proof technique, you can show from first principles, that 2+3=5.

But in a world in which we don’t play semantic games, and we accept the basic principle of Peano arithmetic as a given, it’s a simple proof. It’s a simple proof that can be found in almost any textbook on foundational mathematics or logic. But note how Arrington responds to it: by playing word-games, rephrasing the question in a couple of different ways to show off how much he knows, while completely avoiding the point.

What does it take to conclude that you can’t verify or prove something like 2+3=5? Profound, utter ignorance. Anyone who’s spent any time learning math should know better.

But over at Uncommon Descent? They don’t think they need to actually learn stuff. They don’t need to refer to ungodly things like textbook. They’ve got their God, and that’s all they need to know.

To be clear, I’m not anti-religion. I’m a religious Jew. Uncommon Descent and their rubbish don’t annoy me because I have a grudge against theism. I have a grudge against ignorance. And UD is a huge promoter of arrogant, dishonest ignorance.

# Elon Musk’s Techno-Religion

A couple of people have written to me asking me to say something about Elon Musk’s simulation argument.

Unfortunately, I haven’t been able to find a verbatim quote from Musk about his argument, and I’ve seen a couple of slightly different arguments presented as being what Musk said. So I’m not really going to focus so much on Musk, but instead, just going to try to take the basic simulation argument, and talk about what’s wrong with it from a mathematical perspective.

The argument isn’t really all that new. I’ve found a couple of sources that attribute it to a paper published in 2003. That 2003 paper may have been the first academic publication, and it might have been the first to present the argument in formal terms, but I definitely remember discussing this in one of my philophy classes in college in the late 1980s.

Here’s the argument:

1. Any advanced technological civilization is going to develop massive computational capabilities.
2. With immense computational capabilities, they’ll run very detailed simulations of their own ancestors in order to understand where they came from.
3. Once it is possible to run simulations, they will run many of them to explore how different parameters will affect the simulated universe.
4. That means that advanced technological civilization will run many simulations of universes where their ancestors evolved.
5. Therefore the number of simulated universes with intelligent life will be dramatically larger than the number of original non-simulated civilizations.

If you follow that reasoning, then the odds are, for any given form of intelligent life, it’s more likely that they are living in a simulation than in an actual non-simulated universe.

As an argument, it’s pretty much the kind of crap you’d expect from a bunch of half drunk college kids in a middle-of-the-night bullshit session.

Let’s look at a couple of simple problems with it.

The biggest one is a question of size and storage. The heart of this argument is the assumption that for an advanced civilization, nearly infinite computational capability will effectively become free. If you actually try to look at that assumption in detail, it’s not reasonable.

The problem is, we live in a quantum universe. That is, we live in a universe made up of discrete entities. You can take an object, and cut it in half only a finite number of times, before you get to something that can’t be cut into smaller parts. It doesn’t matter how advanced your technology gets; it’s got to be made of the basic particles – and that means that there’s a limit to how small it can get.

Again, it doesn’t matter how advanced your computers get; it’s going to take more than one particle in the real universe to simulate the behavior of a particle. To simulate a universe, you’d need a computer bigger than the universe you want to simulate. There’s really no way around that: you need to maintain state information about every particle in the universe. You need to store information about everything in the universe, and you need to also have some amount of hardware to actually do the simulation with the state information. So even with the most advanced technology that you can possible imagine, you can’t possible to better than one particle in the real universe containing all of the state information about a particle in the simulated universe. If you did, then you’d be guaranteeing that your simulated universe wasn’t realistic, because its particles would have less state than particles in the real universe.

This means that to simulate something in full detail, you effectively need something bigger than the thing you’re simulating.

That might sound silly: we do lots of things with tiny computers. I’ve got an iPad in my computer bag with a couple of hundred books on it: it’s much smaller than the books it simulates, right?

The “in full detail” is the catch. When my iPad simulates a book, it’s not capturing all the detail. It doesn’t simulate the individual pages, much less the individual molecules that make up those pages, the individual atoms that make up those molecules, etc.

But when you’re talking about perfectly simulating a system well enough to make it possible for an intelligent being to be self-aware, you need that kind of detail. We know, from our own observations of ourselves, that the way our cells operates is dependent on incredibly fine-grained sub-molecular interactions. To make our bodies work correctly, you need to simulate things on that level.

You can’t simulate the full detail of a universe bigger that the computer that simulates it. Because the computer is made of the same things as the universe that it’s simulating.

There’s a lot of handwaving you can do about what things you can omit from your model. But at the end of the day, you’re looking at an incredibly massive problem, and you’re stuck with the simple fact that you’re talking, at least, about building a computer that can simulate an entire planet and its environs. And you’re trying to do it in a universe just like the one you’re simulating.

But OK, we don’t actually need to simulate the whole universe, right? I mean, you’re really interested in developing a single species like yourself, so you only care about one planet.

But to make that planet behave absolutely correctly, you need to be able to correctly simulate everything observable from that planet. Its solar system, you need to simulate pretty precisely. The galaxy around it needs less precision, but it still needs a lot of work. Even getting very far away, you’ve got an awful lot of stuff to simulate, because your simulated intelligences, from their little planet, are going to be able to observe an awful lot.

To simulate a planet and its environment with enough precision to get life and intelligence and civilization, and to do it at a reasonable speed, you pretty much need to have a computer bigger than the planet. You can cheat a little bit, and maybe abstract parts of the planet; but you’ve got to do pretty good simulations of lots of stuff outside the planet.

It’s possible, but it’s not particularly useful. Because you need to run that simulation. And since it’s made up of the same particles as the things it’s simulating, it can’t move faster than the universe it simulates. To get useful results, you’d need to build it to be massively parallel. And that means that your computer needs to be even larger – something like a million times bigger.

If technology were to get good enough, you could, in theory, do that. But it’s not going to be something you do a lot of: no matter how advanced technology gets, building a computer that can simulate an entire planet and its people in full detail is going to be a truly massive undertaking. You’re not going to run large numbers of simulations.

You can certainly wave you hands and say that the “real” people live in a universe without the kind of quantum limit that we live with. But if you do, you’re throwing other assumptions out the window. You’re not talking about ancestor simulation any more. And you’re pretending that you can make predictions based on our technology about the technology of people living in a universe with dramatically different properties.

This just doesn’t make any sense. It’s really just techno-religion. It’s based on the belief that technology is going to continue to develop computational capability without limit. That the fundamental structure of the universe won’t limit technology and computation. Essentially, it’s saying that technology is omnipotent. Technology is God, and just as in any other religion, it’s adherents believe that you can’t place any limits on it.

Rubbish.

# One plus one equals Two?

My friend Dr24hours sent me a link via twitter to a new piece of mathematical crackpottery. It’s the sort of thing that’s so trivial that I might lust ignore it – but it’s also a good example of something that someone commented on in my previous post.

This comes from, of all places, Rolling Stone magazine, in a puff-piece about an actor named Terrence Howard. When he’s not acting, Mr. Howard believes that he’s a mathematical genius who’s caught on to the greatest mathematical error of all time. According to Mr. Howard, the product of one times one is not one, it’s two.

After high school, he attended Pratt Institute in Brooklyn, studying chemical engineering, until he got into an argument with a professor about what one times one equals. “How can it equal one?” he said. “If one times one equals one that means that two is of no value because one times itself has no effect. One times one equals two because the square root of four is two, so what’s the square root of two? Should be one, but we’re told it’s two, and that cannot be.” This did not go over well, he says, and he soon left school. “I mean, you can’t conform when you know innately that something is wrong.”

I don’t want to harp on Mr. Howard too much. He’s clueless, but sadly, he’s a not too atypical student of american schools. I’ll take a couple of minutes to talk about what’s wrong with his stuff, but in context of a discussion of where I think this kind of stuff comes from.

In American schools, math is taught largely by rote. When I was a kid, set theory came into vogue, but by and large math teachers didn’t understand it – so they’d draw a few meaningless Venn diagrams, and then switch into pure procedure.

An example of this from my own life involves my older brother. My brother is not a dummy – he’s a very smart guy. He’s at least as smart as I am, but he’s interested in very different things, and math was never one of his interests.

I barely ever learned math in school. My father noticed pretty early on that I really enjoyed math, and so he did math with me for fun. He taught me stuff – not as any kind of “they’re not going to teach it right in school”, but just purely as something fun to do with a kid who was interested. So I learned a lot of math – almost everything up through calculus – from him, not from school. My brother didn’t – because he didn’t enjoy math, and so my dad did other things with him.

When we were in high school, my brother got a job at a local fast food joint. At the end of the year, he had to do his taxes, and my dad insisted that he do it himself. When he needed to figure out how much tax he owed on his income, he needed to compute a percentage. I don’t know the numbers, but for the sake of the discussion, let’s say that he made $5482 that summer, and the tax rate on that was 18%. He wrote down a pair of ratios: $\frac{18}{100} = \frac{x}{5482}$ And then he cross-multiplied, getting: $18 \times 5482 = 100 \times x$ $98676 = 100 \times x$ and so $x = 986.76$. My dad was shocked by this – it’s such a laborious way of doing it. So he started pressing at my brother. He asked him, if you went to a store, and they told you there was a 20% off sale on a pair of jeans that cost$18, how much of a discount would you get? He didn’t know. The only way he knew to figure it out was to do the whole ratios, cross-multiply, and solve. If you told him that 20% off of $18 was$5, he would have believed you. Because percentages just didn’t mean anything to him.

Now, as I said: my brother isn’t a dummy. But none of his math teachers had every taught him what percentages meant. He had no concept of their meaning: he knew a procedure for getting the value, but it was a completely blind procedure, devoid of meaning. And that’s what everything he’d learned about math was like: meaningless procedures performed by rote, without any comprehension.

That’s where nonsense like Terence Howard’s stuff comes from: math education that never bothered to teach students what anything means. If anyone had attempted to teach any form of meaning for arithmetic, the ridiculous of Mr. Howard’s supposed mathematics would be obvious.

For understanding basic arithmetic, I like to look at a geometric model of numbers.

Put a dot on a piece of paper. Label it “0”. Draw a line starting at zero, and put tick-marks on the line separated by equal distances. Starting at the first mark after 0, label the tick-marks 1, 2, 3, 4, 5, ….

In this model, the number one is the distance from 0 (the start of the line) to 1. The number two is the distance from 0 to 2. And so on.

Addition is just stacking lines, one after the other. Suppose you wanted to add 3 + 2. You draw a line that’s 3 tick-marks long. Then, starting from the end of that line, you draw a second line that’s 2 tick-marks long. 3 + 2 is the length of the resulting line: by putting it next to the original number-line, we can see that it’s five tick-marks long, so 3 + 2 = 5.

Multiplication is a different process. In multiplication, you’re not putting lines tip-to-tail: you’re building rectangles. If you want to multiply 3 * 2, what you do is draw a rectangle who’s width is 3 tick-marks long, and whose height is 2 tick-marks long. Now divide that into squares that are 1 tick-mark by one tick-mark. How many squares can you fit into that rectangle? 6. So 3*2 = 6.

Why does 1 times 1 equal 1? Because if you draw a rectangle that’s one hash-mark wide, and one hash-mark high, it forms exactly one 1×1 square. 1 times 1 can’t be two: it forms one square, not two.

If you think about the repercussions of the idea that 1*1=2, as long as you’re clear about meanings, it’s pretty obvious that 1*1=2 has a disastrously dramatic impact on math: it turns all of math into a pile of gibberish.

What’s 1*2? 2. 1*1=2 and 1*2=2, therefore 1=2. If 1=2, then 2=3, 3=4, 4=5: all integers are equal. If that’s true, then… well, numbers are, quite literally, meaningless. Which is quite a serious problem, unless you already believe that numbers are meaningless anyway.

In my last post, someone asked why I was so upset about the error in a math textbook. This is a good example of why. The new common core math curriculum, for all its flaws, does a better job of teaching understanding of math. But when the book teaches “facts” that are wrong, what they’re doing becomes the opposite. It doesn’t make sense – if you actually try to understand it, you just get more confused.

That teaches you one of two things. Either it teaches you that understanding this stuff is futile: that all you can do is just learn to blindly reproduce the procedures that you were taught, without understanding why. Or it teaches you that no one really understands any of it, and that therefore nothing that anyone tells you can possibly be trusted.

# The Bad Logic of Good People Can’t be Sexists

One of my constant off-topic rants around here is about racism and sexism. This is going to be a nice little post that straddles the line. It’s one of those off-topic-ish rants about sexism in our society, but it’s built around a core of bad logic – so there is a tiny little bit of on-topicness.

We live in a culture that embodies a basic conflict. On one hand, racism and sexism are a deeply integrated part of our worldview. But on the other wand, we’ve come to believe that racism and sexism are bad. This puts us into an awkward situation. We don’t want to admit to saying or doing racist things. But there’s so much of it embedded in every facet of our society that it takes a lot of effort and awareness to even begin to avoid saying and doing racist things.

The problem there is that we can’t stop being racist/sexist until we admit that we are. We can’t stop doing sexist and racist things until we admit that we do sexist and racist things.

And here’s where we hit the logic piece. The problem is easiest to explain by looking at it in formal logical terms. We’ll look at it from the viewpoint of sexism, but the same argument applies for racism.

1. We’ll say $\text{Sexist}(x)$ to mean that “x” is sexist.
2. We’ll say $\text{Bad}(x)$ to mean that x is bad, and $\text{Good}(x)$ to mean that x is good.
3. We’ll have an axiom that bad and good are logical opposites: $\text{Bad}(x) \Leftrightarrow \lnot \text{Good}(x)$.
4. We’ll have another axiom that sexism is bad: $\forall x: \text{Sexist}(x) \Rightarrow \text{Bad}(x)$.
5. We’ll say $\text{Does}(p, x)$ means that person $p$ does an action $x$.

The key statement that I want to get to is: We believe that people who do bad things are bad people: $\forall p, x: \text{Does}(p, x) \land \text{Bad}(x) \Rightarrow \text{Bad}(p)$.

That means that if you do something sexist, you are a bad person:

• $s$ is a sexist action: $\text{Sexist}(s)$.
• I do something sexist: $\text{Does}(\textbf{markcc}, s)$.
• By rule 5 above, that means that I am sexist.
• If I am sexist, then by rule 4 above, I am bad.

We know that we aren’t bad people: I’m a good person, right? So we reject that conclusion. I’m not bad; therefore, I can’t be sexist, therefore whatever I did couldn’t have been sexist.

This looks shallow and silly on the surface. Surely mature adults, mature educated adults couldn’t be quite that foolish!

If his crime was to use the phrase “boys with toys”, and that is your threshold for sexism worthy of some of the abusive responses above, then ok – stop reading now.

My problem is that I have known Shri for many years, and I don’t believe that he’s even remotely sexist. But in 2015 can one defend someone who’s been labeled sexist without a social media storm?

Are people open to the possibility that actually Kulkarni might be very honourable in his dealings with women?

In an interview a week or so ago, Professor Shri Kulkarni said something stupid and sexist. The author of that piece believes that Professor Kulkarni couldn’t have said something sexist, because he knows him, and he knows that he’s not sexist, because he’s a good guy who treats women well.

The thing is, that doesn’t matter. He messed up, and said something sexist. It’s not a big deal; we all do stupid things from time to time. He’s not a bad person because he said something sexist. He just messed up. People are, correctly, pointing out that he messed up: you can’t fix a problem until you acknowledge that it exists! When you say something stupid, you should expect to get called on it, and when you do, you should accept it, apologize, and move on with your life, using that experience to improve yourself, and not make the mistake again.

The thing about racism and sexism is that we’re immersed in it, day in and day out. It’s part of the background of our lives – it’s inescapable. Living in that society means means that we’ve all absorbed a lot of racism and exism without meaning to. We don’t have to like that, but it’s true. In order to make things better, we need to first acklowledge the reality of the world that we live in, and the influence that it has on us.

In mathematical terms, the problem is that good and bad, sexist and not sexist, are absolutes. When we render them into pure two-valued logic, we’re taking shades of gray, and turning them into black and white.

There are people who are profoundly sexist or racist, and that makes them bad people. Just look at the MRAs involved in Gamergate: they’re utterly disgusting human beings, and the thing that makes them so despicably awful is the awfulness of their sexism. Look at a KKKer, and you find a terrible person, and the thing that makes them so terrible is their racism.

But most people aren’t that extreme. We’ve just absorbed a whole lot of racism and sexism from the world we’ve lived our lives in, and that influences us. We’re a little bit racist, and that makes us a little bit bad – we have room for improvement. But we’re still, mostly, good people. The two-valued logic creates an apparent conflict where none really exists.

Where do these sexist/racist attitudes come from? Picture a scientist. What do you see in your minds eye? It’s almost certainly a white guy. It is for me. Why is that?

1. In school, from the time that I got into a grade where we had a dedicated science teacher, every science teacher that I had was a white guy. I can still name ’em: Mr. Fidele, Mr. Schwartz, Mr. Remoli, Mr. Laurie, Dr. Braun, Mr. Hicken, etc.
2. On into college, in my undergrad days, where I took a ton of physics and chemistry (I started out as an EE major), every science professor that I had was a white guy.
3. My brother and I used to watch a ton of trashy sci-fi movie some free movie apps from the internet. In those movies, every time there was a character who was a scientist, he was a white guy.
4. My father was a physicist working in semiconductor manufacturing for satellites and military applications. From the time I was a little kid until they day he retired, he had exactly one coworker who wasn’t a white man. (And everyone on his team complained bitterly that the black guy wasn’t any good, that he only got and kept the job because he was black, and if they tried to fire him, he’d sue them. I really don’t believe that my dad was a terrible racist person; I think he was a wonderful guy, the person who is a role model for so much of my life. But looking back at this? He didn’t mean to be racist, but I think that he was.)

In short, in all of my exposure to science, from kindergarten to graduate school, scientists were white men. (For some reason, I encountered a lot of women in math and comp sci, but not in the traditional sciences.) So when I picture a scientist, it’s just natural that I picture a man. There’s a similar story for most of us who’ve grown up in the current American culture.

When you consider that, it’s both an explanation of why we’ve got such a deeply embedded sexist sense about who can be a scientist, and an explanation how, despite the fact that we’re not deliberately being sexist, our subconscious sexism has a real impact.

I’ve told this story a thousand times, but during the time I worked at IBM, I ran the intership program for my department one summer. We had a deparmental quota of how many interns each department could pay for. But we had a second program that paid for interns that were women or minority – so they didn’t count against the quota. The first choice intern candidate of everyone in the department was a guy. When we ran out of slots, the guy across the hall from me ranted and raved about how unfair it was. We were discriminating against male candidates! It was reverse sexism! On and on. But the budget was what the budget was. Two days later, he showed up with a resume for a young woman, all excited – he’d found a candidate who was a woman, and she was even better than the guy he’d originally wanted to hire. We hired her, and she was brilliant, and did a great job that summer.

The question that I asked my office-neighbor afterwards was: Why didn’t he find the woman the first time through the resumes? He went through the resumes of all of the candidates before picking the original guy. The woman that he eventually hired had a resume that was clearly better than the guy. Why’d he pass her resume to settle on the guy? He didn’t know.

That little story demonstrates two things. One, it demonstrates the kind of subconscious bias we have. We don’t have to be mustache-twirling black-hatted villains to be sexists or racists. We just have to be human. Two, it demonstrates the way that these low-level biases actually harm people. Without our secondary budget for women/minority hires, that brilliant young woman would never have gotten an internship at IBM; without that internship, she probably wouldn’t have gotten a permanent job at IBM after graduation.

Professor Kulkarni said something silly. He knew he was saying something he shouldn’t have, but he went ahead and did it anyway, because it was normal and funny and harmless.

It’s not harmless. It reinforces that constant flood of experience that says that all scientists are men. If we want to change the culture of science to get rid of the sexism, we have to start with changing the deep attitudes that we aren’t even really aware of, but that influence our thoughts and decisions. That means that when someone says we did something sexist or racist, we need to be called on it. And when we get called on it, we need to admit that we did something wrong, apologize, and try not to make the same mistake again.

We can’t let the black and white reasoning blind us. Good people can be sexists or racists. Good people can do bad things without meaning to. We can’t allow our belief in our essential goodness prevent us from recognizing it when we do wrong, and making the choices that will allow us to become better people.

# Genius Continuum Crackpottery

This post was revised on June 25, 2014. Mr. Wince has been threatening to sue me for libel. I don’t think that that’s right, but one thing that he’s complained about is correct. I called him a high school dropout. In his article, Wince refers to “when he dropped out of high school”, but in the same sentence, he goes on to say that he dropped out to attend community college. Calling him a dropout is a cheap shot, which I shouldn’t have included, and for that, I apologize. I’ve removed the line from the post. I still think that his math is laughably wrong, but I shouldn’t have called him a dropout.

There’s a lot of mathematical crackpottery out there. Most of it is just pointless and dull. People making the same stupid mistakes over and over again, like the endless repetitions of the same-old supposed refutations of Cantor’s diagonalization.

After you eliminate that, you get reams of insanity – stuff which
is simply so incoherent that it doesn’t make any sense. This kind of thing is usually word salad – words strung together in ways that don’t make sense.

After you eliminate that, sometimes, if you’re really lucky, you’ll come accross something truly special. Crackpottery as utter genius. Not genius in a good way, like they’re an outsider genius who discovered something amazing, but genius in the worst possible way, where someone has created something so bizarre, so overwrought, so utterly ridiculous that it’s a masterpiece of insane, delusional foolishness.

Today, we have an example of that: Existics!. This is a body of work by a guy named Gavin Wince with truly immense delusions of grandeur. Pomposity on a truly epic scale!

I’ll walk you through just a tiny sample of Mr. Wince’s genius. You can go look at his site to get more, and develop a true appreciation for this. He doesn’t limit himself to mere mathematics: math, physics, biology, cosmology – you name it, Mr. Wince has mastered it and written about it!

The best of his mathematical crackpottery is something called C3: the Canonized Cardinal Continuum. Mr. Wince has created an algebraic solution to the continuum hypothesis, and along the way, has revolutionized number theory, algebra, calculus, real analysis, and god only knows what else!

Since Mr. Wince believes that he has solved the continuum hypothesis. Let me remind you of what that is:

1. If you use Cantor’s set theory to explore numbers, you get to the uncomfortable result that there are different sizes of infinity.
2. The smallest infinite cardinal number is called ℵ0,
and it’s the size of the set of natural numbers.
3. There are cardinal numbers larger than ℵ0. The first
one larger than ℵ0 is ℵ1.
4. We know that the set of real numbers is the size of the powerset
of the natural numbers – 20 – is larger than the set of the naturals.
5. The question that the continuum hypothesis tries to answer is: is the size
of the set of real numbers equal to ℵ1? That is, is there
a cardinal number between ℵ0 and |20|?

The continuum hypothesis was “solved” in 1963. In 1940, Gödel showed that you couldn’t disprove the continuum hypothesis using ZFC. In 1963,
another mathematician named Paul Cohen, showed that it couldn’t be proven using ZFC. So – a hypothesis which is about set theory can be neither proven nor disproven using set theory. It’s independent of the axioms of set theory. You can choose to take the continuum hypothesis as an axiom, or you can choose to take the negation of the continuum hypothesis as an axiom: either choice is consistent and valid!

It’s not a happy solution. But it’s solved in the sense that we’ve got a solid proof that you can’t prove it’s true, and another solid proof that you can’t prove it’s false. That means that given ZFC set theory as a basis, there is no proof either way that doesn’t set it as an axiom.

But… Mr. Wince knows better.

The set of errors that Wince makes is really astonishing. This is really seriously epic crackpottery.

He makes it through one page without saying anything egregious. But then he makes up for it on page 2, by making multiple errors.

First, he pulls an Escultura:

x1 = 1/21 = 1/2 = 0.5
x2 = 1/21 + 1/22 = 1/2 + 1/4 = 0.75
x3 = 1/21 + 1/22 + 1/23 = 1/2 + 1/4 + 1/8 = 0.875

At the end or limit of the infinite sequence, the final term of the sequence is 1.0

In this example we can see that as the number of finite sums of the sequence approaches the limit infinity, the last term of the sequence equals one.
xn = 1.0
If we are going to assume that the last term of the sequence equals one, it can be deduced that, prior to the last term in the sequence, some finite sum in the series occurs where:
xn-1 = 0.999…
xn-1 = 1/21 + 1/22 + 1/23 + 1/24 + … + 1/2n-1 = 0.999…
Therefore, at the limit, the last term of the series of the last term of the sequence would be the term, which, when added to the sum 0.999… equals 1.0.

There is no such thing as the last term of an infinite sequence. Even if there were, the number 0.999…. is exactly the same as 1. It’s a notational artifact, not a distinct number.

But this is the least of his errors. For example, the first paragraph on the next page:

The set of all countable numbers, or natural numbers, is a subset of the continuum. Since the set of all natural numbers is a subset of the continuum, it is reasonable to assume that the set of all natural numbers is less in degree of infinity than the set containing the continuum.

We didn’t need to go through the difficult of Cantor’s diagonalization! We could have just blindly asserted that it’s obvious!

or actually… The fact that there are multiple degrees of infinity is anything but obvious. I don’t know anyone who wasn’t surprised the first time they saw Cantor’s proof. It’s a really strange idea that there’s something bigger than infinity.

Moving on… the real heart of his stuff is built around some extremely strange notions about infinite and infinitessimal values.

Before we even look at what he says, there’s an important error here
which is worth mentioning. What Mr. Wince is trying to do is talk about the
continuum hypothesis. The continuum hypothesis is a question about the cardinality of the set of real numbers and the set of natural numbers.
Neither infinites nor infinitessimals are part of either set.

Infinite values come into play in Cantor’s work: the cardinality of the natural numbers and the cardinality of the reals are clearly infinite cardinal numbers. But ℵ0, the smallest infinite cardinal, is not a member of either set.

Infinitessimals are fascinating. You can reconstruct differential and integral calculus without using limits by building in terms of infinitessimals. There’s some great stuff in surreal numbers playing with infinitessimals. But infinitessimals are not real numbers. You can’t reason about them as if they were members of the set of real numbers, because they aren’t.

Many of his mistakes are based on this idea.

For example, he’s got a very strange idea that infinites and infinitessimals don’t have fixed values, but that their values cover a range. The way that he gets to that idea is by asserting the existence
of infinity as a specific, numeric value, and then using it in algebraic manipulations, like taking the “infinityth root” of a real number.

For example, on his way to “proving” that infinitessimals have this range property that he calls “perambulation”, he defines a value that he calls κ:

$sqrt[infty]{infty} = 1 + kappa$

In terms of the theory of numbers, this is nonsense. There is no such thing as an infinityth root. You can define an Nth root, where N is a real number, just like you can define an Nth power – exponents and roots are mirror images of the same concept. But roots and exponents aren’t defined for infinity, because infinity isn’t a number. There is no infinityth root.

You could, if you really wanted to, come up with a definition of exponents that that allowed you to define an infinityth root. But it wouldn’t be very interesting. If you followed the usual pattern for these things, it would be a limit: $sqrt[infty]{x} lim_{nrightarrowinfty} sqrt[n]{x}$. That’s clearly 1. Not 1 plus something: just exactly 1.

But Mr. Cringe doesn’t let himself be limited by silly notions of consistency. No, he defines things his own way, and runs with it. As a result, he gets a notion that he calls perambulation. How?

Take the definition of κ:

$sqrt[infty]{infty} = 1 + kappa$

Now, you can, obviously, raise both sides to the power of infinity:

$infty = (1 + kappa)^{infty}$

Now, you can substitute ℵ0 for $infty$. (Why? Don’t ask why. You just can.) Then you can factor it. His factoring makes no rational sense, so I won’t even try to explain it. But he concludes that:

• Factored and simplified one way, you end up with (κ+1) = 1 + x, where x is some infinitessimal number larger than κ. (Why? Why the heck not?)
• Factored and simplified another way, you end up with (κ+1) = ℵ
• If you take the mean of of all of the possible factorings and reductions, you get a third result, that (κ+1) = 2.

He goes on, and on, and on like this. From perambulation to perambulating reciprocals, to subambulation, to ambulation. Then un-ordinals, un-sets… this is really an absolute masterwork of utter insane crackpottery.

# Sloppy Dualism Denies Free Will?

When I was an undergrad in college, I was a philosophy minor. I spent countless hours debating ideas about things like free will. My final paper was a 60 page rebuttal to what I thought was a sloppy argument against free will. Now, it’s been more years since I wrote that than I care to admit – and I still keep seeing the same kind of sloppy arguments, that I argue are ultimately circular, because they’re hiding their conclusion in their premises.

There’s an argument against free will that I find pretty compelling. I don’t agree with it, but I do think that it’s a solid argument:

Everything in our experience of the universe ultimately comes down to physics. Every phenomenon that we can observe is, ultimately, the result of particles interacting according to basic physical laws. Thermodynamics is the ultimate, fundamental ruler of the universe: everything that we observe is a result of a thermodynamic process. There are no exceptions to that.

Our brain is just another physical device. It’s another complex system made of an astonishing number of tiny particles, interacting in amazingly complicated ways. But ultimately, it’s particles interacting the way that particles interact. Our behavior is an emergent phenomenon, but ultimately, we don’t have any ability to make choice, because there’s no mechanism that allows us free choice. Our choice is determined by the physical interactions, and our consciousness of those results is just a side-effect of that.

If you want to argue that free will doesn’t exist, that argument is rock solid.

But for some reason, people constantly come up with other arguments – in fact, much weaker arguments that come from what I call sloppy dualism. Dualism is the philosophical position that says that a conscious being has two different parts: a physical part, and a non-physical part. In classical terms, you’ve got a body which is physical, and a mind/soul which is non-physical.

In this kind of argument, you rely on that implicit assumption of dualism, essentially asserting that whatever physical process we can observe isn’t really you, and that therefore by observing any physical process of decision-making, you infer that you didn’t really make the decision.

For example…

And indeed, this is starting to happen. As the early results of scientific brain experiments are showing, our minds appear to be making decisions before we’re actually aware of them — and at times by a significant degree. It’s a disturbing observation that has led some neuroscientists to conclude that we’re less in control of our choices than we think — at least as far as some basic movements and tasks are concerned.

This is something that I’ve seen a lot lately: when you do things like functional MRI, you can find that our brains settled on a decision before we consciously became aware of making the choice.

Why do I call it sloppy dualism? Because it’s based on the idea that somehow the piece of our brain that makes the decision is different from the part of our brain that is our consciousness.

If our brain is our mind, then everything that’s going on in our brain is part of our mind. Taking a piece of our brain, saying “Whoops, that piece of your brain isn’t you, so when it made the decision, it was deciding for you instead of it being you deciding.

By starting with the assumption that the physical process of decision-making we can observe is something different from your conscious choice of the decision, this kind of argument is building the conclusion into the premises.

If you don’t start with the assumption of sloppy dualism, then this whole argument says nothing. If we don’t separate our brain from our mind, then this whole experiment says nothing about the question of free will. It says a lot of very interesting things about how our brain works: it shows that there are multiple levels to our minds, and that we can observe those different levels in how our brains function. That’s a fascinating thing to know! But does it say anything about whether we can really make choices? No.

# For every natural number N, there's a Cantor Crank C(n)

More crankery? of course! What kind? What else? Cantor crankery!

It’s amazing that so many people are so obsessed with Cantor. Cantor just gets under peoples’ skin, because it feels wrong. How can there be more than one infinity? How can it possibly make sense?

As usual in math, it all comes down to the axioms. In most math, we’re working from a form of set theory – and the result of the axioms of set theory are quite clear: the way that we define numbers, the way that we define sizes, this is the way it is.

Today’s crackpot doesn’t understand this. But interestingly, the focus of his problem with Cantor isn’t the diagonalization. He thinks Cantor went wrong way before that: Cantor showed that the set of even natural numbers and the set of all natural numbers are the same size!

Unfortunately, his original piece is written in Portuguese, and I don’t speak Portuguese, so I’m going from a translation, here.

The Brazilian philosopher Olavo de Carvalho has written a philosophical “refutation” of Cantor’s theorem in his book “O Jardim das Aflições” (“The Garden of Afflictions”). Since the book has only been published in Portuguese, I’m translating the main points here. The enunciation of his thesis is:

Georg Cantor believed to have been able to refute Euclid’s fifth common notion (that the whole is greater than its parts). To achieve this, he uses the argument that the set of even numbers can be arranged in biunivocal correspondence with the set of integers, so that both sets would have the same number of elements and, thus, the part would be equal to the whole.

And his main arguments are:

It is true that if we represent the integers each by a different sign (or figure), we will have a (infinite) set of signs; and if, in that set, we wish to highlight with special signs, the numbers that represent evens, then we will have a “second” set that will be part of the first; and, being infinite, both sets will have the same number of elements, confirming Cantor’s argument. But he is confusing numbers with their mere signs, making an unjustifiable abstraction of mathematical properties that define and differentiate the numbers from each other.

The series of even numbers is composed of evens only because it is counted in twos, i.e., skipping one unit every two numbers; if that series were not counted this way, the numbers would not be considered even. It is hopeless here to appeal to the artifice of saying that Cantor is just referring to the “set” and not to the “ordered series”; for the set of even numbers would not be comprised of evens if its elements could not be ordered in twos in an increasing series that progresses by increments of 2, never of 1; and no number would be considered even if it could be freely swapped in the series of integeres.

He makes two arguments, but they both ultimately come down to: “Cantor contradicts Euclid, and his argument just can’t possibly make sense, so it must be wrong”.

The problem here is: Euclid, in “The Elements”, wrote severaldifferent collections of axioms as a part of his axioms. One of them was the following five rules:

1. Things which are equal to the same thing are also equal to one another.
2. If equals be added to equals, the wholes are equal.
3. If equals be subtracted from equals, the remainders are equal.
4. Things which coincide with one another are equal to one another.
5. The whole is greater that the part.

The problem that our subject has is that Euclid’s axiom isn’t an axiom of mathematics. Euclid proposed it, but it doesn’t work in number theory as we formulate it. When we do math, the axioms that we start with do not include this axiom of Euclid.

In fact, Euclid’s axioms aren’t what modern math considers axioms at all. These aren’t really primitive ground statements. Most of them are statements that are provable from the actual axioms of math. For example, the second and third axioms are provable using the axioms of Peano arithmetic. The fourth one doesn’t appear to be a statement about numbers at all; it’s a statement about geometry. And in modern terms, the fifth one is either a statement about geometry, or a statement about measure theory.

The first argument is based on some strange notion of signs distinct from numbers. I can’t help but wonder if this is an error in translation, because the argument is so ridiculously shallow. Basically, it concedes that Cantor is right if we’re considering the representations of numbers, but then goes on to draw a distinction between representations (“signs”) and the numbers themselves, and argues that for the numbers, the argument doesn’t work. That’s the beginning of an interesting argument: numbers and the representations of numbers are different things. It’s definitely possible to make profound mistakes by confusing the two. You can prove things about representations of numbers that aren’t true about the numbers themselves. Only he doesn’t actually bother to make an argument beyond simply asserting that Cantor’s proof only works for the representations.

That’s particularly silly because Cantor’s proof that the even naturals and the naturals have the same cardinality doesn’t talk about representation at all. It shows that there’s a 1 to 1 mapping between the even naturals and the naturals. Period. No “signs”, no representations.

The second argument is, if anything, even worse. It’s almost the rhetorical equivalent of sticking his fingers in his ears and shouting “la la la la la”. Basically – he says that when you’re producing the set of even naturals, you’re skipping things. And if you’re skipping things, those things can’t possible be in the set that doesn’t include the skipped things. And if there are things that got skipped and left out, well that means that it’s ridiculous to say that the set that included the left out stuff is the same size as the set that omitted the left out stuff, because, well, stuff got left out!!!.

Here’s the point. Math isn’t about intuition. The properties of infinitely large sets don’t make intuitive sense. That doesn’t mean that they’re wrong. Things in math are about formal reasoning: starting with a valid inference system and a set of axioms, and then using the inference to reason. If we look at set theory, we use the axioms of ZFC. And using the axioms of ZFC, we define the size (or, technically, the cardinality) of sets. Using that definition, two sets have the same cardinality if and only if there is a one-to-one mapping between the elements of the two sets. If there is, then they’re the same size. Period. End of discussion. That’s what the math says.

Cantor showed, quite simply, that there is such a mapping:

${ (i rightarrow itimes 2) | i in N }$

There it is. It exists. It’s simple. It works, by the axioms of Peano arithmetic and the axiom of comprehension from ZFC. It doesn’t matter whether it fits your notion of “the whole is greater than the part”. The entire proof is that set comprehension. It exists. Therefore the two sets have the same size.

# 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. “$\frac{1}{2}$” 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 $R^*$, 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. $\pi$? 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 $d^*$, is (1-0.99999999…) and is the smallest number greater than 0. And you can generate a complete ordered enumeration of all of the new real numbers, ${0, d^*, 2d^*, 3d^*, ..., n-2d^*, n-d^*, n, n+d^*, ...}$.

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 $x^n + y^n = z^n$. 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 $d^*$, this makes no sense: the point of Fermat’s last theorem is that there are no integer solutions; $d^*$ is not an integer; $(1-d^*)10$ 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 $d^*$ 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 $\frac{1}{9}$. That’s the multiplicative inverse of 9. So, by definition, $\frac{1}{9}*9 = 1$ – the multiplicative identity.

In Escultura’s theory, $\frac{1}{9}$ is a shorthand for the number that has a representation of 0.1111…. So, $\frac{1}{9}*9 = 0.1111....*9 = 0.9999... = (1-d^*)$. So $(1-d^*)$ is also a multiplicative identity. By a similar process, you can show that $d^*$ itself must be the additive identity. So either $d^* == 0$, or else you’ve lost the field structure, and with it, pretty much all of real number theory.

# 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 $pi$ 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.