Fun with the ISO 216 Standard

ISO 216 is a lovely standard. It's the one that defines the A4 paper size, and all its friends. Bengt likes papers, not just for writing on, but also for other things, like making paper airplanes. He claims that the A4 size is particularly good for making paper planes, but that would most likely be a bit of a mess to try to prove, so we'll leave this out of this problem.

One other thing they can be used for is to measure lengths. See, the meter isn't really a very convenient unit of measurement - it's historically based on the circumference of the Earth, which to be honest isn't something most people have an intuitive grasp on. The foot is better in that sense, but since people's feet aren't all the same length, that's also not quite ideal. In fact, there may not be any obvious things in nature to base a length unit on, because most things in everyday life vary in size. But a standardised paper size - well, it may not be "in nature", but papers sure are pretty much everywhere these days, and they're a convenient size too. And we could potentially set the definition of the paper size so that it fits with some scientific constant - say, a nanolightsecond; that way, the definition would be convenient for both scientists and normal people.

But there is a slight problem with the A4 as it stands: It's based on area, not length. A0 has the area 1 m², and each subsequent one has half that area. It would probably be better to define them by saying that, for example, the short side of the first one would be one meter. It's a little known fact that the same ISO standard actually defines that as well - it's called the B series. So, for example, the B4 is a quarter of a meter wide.

Still, there's another slight problem: Every other size will have half the length, but the metric system is based on powers of ten, not two. So once you get down to B10, the width is 31.25 mm - or rather, it's 31 mm, since the standard requires that they be rounded off to the nearest millimeter, which is clearly an abomination.

If you ask Bengt, this is further proof that the decimal system is inherently flawed, and we should use base sixteen instead - in that case, the area of an A4 would be exactly one d(m²), which is admittedly a confusing unit, as it's easily confused with the much smaller (dm)², but anyway. And the B8 would be one dm wide, so we could for example give all children a little notepad in that size, and use that as our unit of measurement, so we would all grow up with a remarkably accurate intuition for length units, and there would be no need to fight over it, which would probably lead to world peace.

Now, just to make fun of the decimal system and all that, let's try to construct a paper size which does the same thing but with each subsequent area being a tenth of the previous area, rather than half. It should have such proportions that if you fold the short side in half, and the long side in fifths, you get the same proportions again. It would be awfully tricky to fold anything like that, but that's sort of the idea.

1. What is the width of an A0?
2. On a side note - how long is a nanolightsecond, in, say, feet?
3. What proportions would the "decimal" system have? Suppose the short side is 1 m, how long would the long one be?
4. Is there a simple similar system where both the areas and lengths have integer ratios?
5. How far does the A series go, before they get rounded off to nothing?

The Problem with Bathtubs

Bengt is in the bath. It's very hot, because Bengt thinks real men bathe in at least 50°C, but that has nothing to do with the problem. Suddenly he notices a little bit of lint in the water. Yuck! He pulls out the plug to let the water out and get rid of the lint. We can assume that the lint is in a random location and has the same density as the water; it behaves essentially like a randomly selected water molecule. But Bengt is annoyed at how slowly the water is flowing, so he considers whether it might possibly help to add more water. That way the pressure and therefore the rate of flow would increase, right?

1. Is there any way adding water could decrease the expected time for the lint to disappear? Assume that the added water appears uniformly; that is, we can think of it in an entirely statistical way, with water molecules being added and removed at random. The only physics involved is the increased rate of flow from the pressure.
2. What if we involve a little more physics? Presumably Bengt has the sense to add the new water at the opposite end of the bathtub from the drain. Could it reasonably be expected to be a good idea?

Bengt vs. Peano, part s(0)

In the last problem, we took arithmetic down to what seems like a very fundamental level, seemingly without any preconceived ideas or assumptions. But Bengt is not satisfied.

Now, you might say, there's that definition symbol that we haven't defined, right? Peano himself famously asked "How do you define a definition?" But Bengt thinks this is a stupid question, because obviously you don't. The definition symbol is not part of the system, it's outside the system, so it's fine.

We've also sort of skipped over the issue of precedence - unless of course you decided to deal with that in your answer to the last problem, in which case, good on you - but that's not a problem either, because we could just express "x+y" as "+(x, y)" instead, that is, use only prefix operators.

No, what Bengt has a problem with is all those variables. What is x? Well, it varies, obviously, that's kind of the point of variables, but Bengt is not happy with that. Also we're assuming that there are functions, which takes arguments, and all that. And what about all those definitions of numbers? That's a great many definitions, isn't it? So either we have to make an infinite number of definitions, or we're stuck writing numbers as s(s(s(s(s(...

Bengt would like to sort out all that stuff and make everything simple. In the new system, every rule has to contain an exact sequence of symbols on each side.

Decimal numbers would be a little cumbersome to define, so we might resort to binary - once we've got that down, it's easy enough to do the same with any other base. Alternatively, we could use unary numbers, which would probably make things even easier to define, but is a pretty messy way of writing numbers. Also, to make sure we know where a number starts and ends, we'll enclose them in brackets. So eleven is written as [1011] in binary, or [•••••••••••] in unary.

Furthermore, it's probably a good idea to write the operators prefix instead of infix, so we don't have to worry about precedence and parentheses and all that. So 1+1=2 could be written (in binary) as =+[1][1][10].

As a warm-up, we could start with logical expressions. We might define true and false as... oh, it doesn't matter, really, let's just say T and F. We can always define those as something else later. If we let for example the & symbol stand for "and" (which makes sense, since that's what it means) we could write

&TT:T
&TF:F
&FT:F
&FF:F

The trickiness appears of course when we want to deal with numbers, since we can't just spell out all possibilities. But that doesn't mean it can't be done. If we take our old pal the successor function, for example, it could be written (for unary) as
s[:[•

Now all that remains is to go from there to a complete arithmetic system, not using any variables or anything else that isn't a literal sequence of symbols.

1. Define the "or" operator.
2. Define addition for unary numbers. There are various ways, with or without using the successor function - feel free to try out more than one. Note that this function (and the others) should be able to handle any natural numbers.
3. And multiplication.
4. Define the successor function for binary numbers.
5. Define addition for binary numbers.
6. And multiplication.
7. Define binary numbers in terms of unary. This obviously solves (e) and (f) automatically, so it's nice if you've got some other solution to those as well.
8. What can't you do in this type of system that you can do in Peano's? You might want to look up "Peano's axioms" if you're not familiar with them.

Bengt vs. Peano, part 0

Bengt likes maths, and he likes starting from the very basics. So that's what we're doing today.

The foundations of arithmetic are often expressed by starting from one number, zero, and one function, the successor function. We denote those by 0 and s. (The successor function of a given number represents the following number, so s(x) is what you would normally call x+1.) From those, we can easily define things like the other natural numbers:
1 : s(0)
2 : s(1)
3 : s(2)
...

We can also define the basic operators. For example, addition can be described like so:
x + 0 : x
x + s(y) : s(x) + y

Just to horrify mathematicians, Bengt decides to include infinity in the natural numbers. Just like 0 and s, it is not defined as anything in particular. Then we need a couple more lines to define addition:
x + ∞ : ∞
∞ + x : ∞

To include things like the equals operator, which is essentially a function from two numbers to a truth value, we also need to define true and false. But Bengt decides to cheat a little by reusing the numbers, like in some programming languages. Let's be a little bit original and say that 0 denotes true, and ∞ denotes false.

1. Define multiplication.
2. Define subtraction. Since we're only dealing with natural numbers, we'll use a slightly unusual definition of subtraction; the smaller number is always subtracted from the larger one. That is, what we want is actually the absolute value of subtraction. Infinity minus infinity should be zero.
3. Define equality. Remember that you can use the things you have defined in (a) and (b) - or haven't, if you skipped those.
4. Define the => operator, that is, equal-or-greater-than.
5. Define the other => operator, that is, logical implication.
6. What about logical negation? Will that be interesting somehow?
7. Finally, put those operators to use by proving that 1+1=2.