The title of this post is a real headline from the Berkeley Barb (many years ago).
It made quite an impression on me at the time though it’s not clear what it means. I believe it was a discussion of a marijuana arrest at a sexual freedom league meeting. It sums up a lot of what was going on in Berkeley in those days.
What exactly are mathematical objects like π and the unit circle? In what sense do they exist?
In the last post I proposes neofictionalism, the idea that the mathematical universe is a collective fiction (like Lord of the Rings, Star Trek or GOT) but about which statements can be objectively true or false. Thus the ordinal ω is comparable to the Iron Throne and the statement “ω is the set of all natural numbers” is objectively true in the same way that “the Iron Throne is made of swords” is objectively true.
Mathematicians discovered a new [whole] number. It’s between six and seven and is called “bleen”.
-George Carlin
When we’re talking about π, what are we talking about? What kind of object is it?
A number, your text book will tell you. Where is it? On the real line. Where is the real line and how long has it been around? It’s just there, and has been forever …
Your textbook may not answer these questions but philosophers have tried. Needless to say there is no consensus. They’ve come up with several answers, and in my humble opinion, all have something to offer, but all miss the mark.
I have an answer which you’ll be happy to hear I am going to share with you. Briefly, π and its friends live in our collective imagination. To be corny, in our hearts and minds.
I’ve been to Greece often enough that I’ve picked up a bit of (modern) Greek. Like anyone in my situation, I’ve had fun spotting Greek words with Englishs cognates based on Greek roots, popping up with unusual meanings in unusual contexts. Here’s my story of a typical day in a Greek visit, using some of these words. (I also translate some Greek idioms, like “the Bill Wadge”). Based on a true story.
Remember this Wadge’s Law
Whenever you want to do something, there’s always someone who says there’s something else you have to do first.
Call the thing you want to do A, the thing you’ve been told you have to do first B. They’re telling you you have to do B before A.
Continue readingTime for another break from research (at least the normal kind).
I seem to be always discovering fundamental Laws of the Universe, especially about teaching. I’d like to share some of them with you. They are each called “Wadge’s Law” … by me. Maybe the name will catch on. Here they are.
Wadge’s Law (of traffic)
Continue readingNo matter how late you go through an orange light, the guy/gal behind you follows you through.
Nothing says fun like formal power series!
A formal power series is a (usually) infinite polynomial in x. For example
1 + x + x2 + x3 + x4 + …
This is an expression, not a number. If we give a value to x, we may get a number, or the evaluation may run away (diverge) on us (if |x|≥1).
They’re called formal power series because we don’t normally try to evaluate them, we just manipulate them formally and symbolically.
For example, if we multiply the above series by 1-x, we get … 1. So 1-x is the multiplicative inverse of that series, even though the series diverges for most values of x. These are purely formal manipulations.
So where’s the fun? Well recently my colleagues and I produced a Lucid interpreter. It’s written in Python and handles the full language described in the Lucid book, plus it supports an extra space dimension. I’ll tell you about it in another post and make it available through github.
Anyway pyLucid plus formal power series spells fun. Clearly a formal power series is determined by the infinite sequence of its coefficients, and such a sequence can be represented in a straightforward way as a space vector. Thus the power series 1 is 1 sby 0, the series x is 0 sby 1 sby 0, the series x2 is 0 sby 0 sby 1 sby 0 and the power series first given is simply 1.
More precisely 1 sby 0 is the vector
1, 0, 0, 0, …
which represents
1 + 0x + 0x^2+ 0x^3 + …
0 sby 1 sby 0 is the vector
0, 1, 0, 0, 0, …
which represents the series
0 + 1x + 0x^2+ 0x^3 + …
0 sby 0 sby 1 sby 0 represents
0 + 0x + 1x^2+ 0x^3 + …
and 1 is the vector
1, 1, 1, 1, …
which represents the original series above.
What about operations on power series? Addition is just that: if pyLucid vectors P and Q represent two power series, P+Q represents their sum, since addition is coefficientwise.Multiplication is more complex.
(I’m going using ASCII notation instead of proper sub and super scripting. I tried but the cockamamy new wordpress editor ate my html).
Anyway we can break down P to p0+xP’ where P’ is p1+p2x+p3x^2+… . Similarly Q is q0+xQ’ and multiplying them we get
p0q0 + (p0q1+p1q0)x + P’Q’x^2
This defines multiplication of P and Q in terms of multiplication of P’ and Q’. This is recursion but P’ and Q’ are in no sense simpler than P and Q. However we can use it to write pyLucid code because it produces two coefficients before it recurses. It doesn’t deadlock.
The pyLucid definition of product of two series represented as space vectors is
pprod(p,q) = r
where
p0 = init p; q0 = init q;
pp = succ p; qp = succ q;
r0 = p0*q0; r = (r0 sby (p0*qp + q0*pp)) + (0 sby 0 sby pprod(pp,qp));
end;
(Dangnabbit, there doesn’t seem to be a way to indent a block in Gutenberg, WordPress’ newfangled editor.)
We can test pprod by multiplying 1 (our original power series) by itself, and we get
1 + 2x + 3x^2 + 4x^3 + …
which is correct. In other words, pprod(1,1) is
1, 2, 3, 4, …
Division is even trickier, I’ll spare you the explanation, the code is
pdiv(q,w) = t
where
q0 = init q;
w0 = init w;
r = q0/w0;
v = succ(q – r*w);
t = r sby pdiv(v,w);
end;
We can test it by setting one = 1 sby 0 and calculating pdiv(one,1) which gives coefficients
1, -1, 0, 0, 0, …
which is correct since the result is 1-x.
Formal power series can be integrated and differentiated (formally) and that’s where things get interesting. The derivative of
p0 + p1x + p2x^2 + p3x^3 + …
is
p1 + 2p2x + 3p3x^2 + …
and this is easy to code:
pderiv(g) = h
where
gp = succ g;
k = 1 sby k+1;
h = gp*k;
end;
Integration is even simpler, the integral of
p0 + p1x + p2x^2 + p3x^3 + …
is
c + p0x + p1x^2/2 + p2x^3/3 + …
(c is the constant of integration.)
The code is
pinteg(c,s) = d where
i = 1 sby i+1;
d = c sby s/i;
end;
Simple enough … but now let’s think about the power series corresponding to the exponential function e^x. This function is its own derivative and therefore also its own integral. More precisely, e^x is one plus the integral of e^x. In code, we get
ex = pinteg(1,ex)
and this works! Even though it’s recursive, we get all the coefficients:
1.00000 1.00000 0.50000 0.16667 0.04167 0.00833 0.00139 0.00020 …
Let’s compute e! By evaluating ex at x=1. Evaluating doesn’t always work but sometimes we can get approximations. The code
peval(p,a) = v
where
v = init p +(0 sby a * peval(succ p, a));
end;
gives us the sequence of partial sums, which sometimes converges. Sure enough, peval(ex,1) yields
1.00000 2.00000 2.50000 2.66667 2.70833 2.71667 2.71806 2.71825 2.71828 …
In the same way, sin(x) and cos(x) are each others derivatives/integrals and the pyLucid code
sinx = pinteg(0,cosx)
cosx = pinteg(1,sinx)
works. For example, the coefficients of sinx are
0.00000 1.00000 0.00000 -0.16667 0.00000 0.00833 0.00000 -0.00020 …
Now for the fireworks! Arctan(x) is really interesting and its derivative is 1/(1+x^2). We can integrate this in code:
x2 = 0 sby 0 sby 1 sby 0; one = 1 sby 0;
atanx = pinteg(0,pdiv(one,one+x2));
This does the job, giving the coefficients of arctan(x) as
0.00000 1.00000 0.00000 -0.33333 0.00000 0.20000 0.00000 -0.14286 …
Now it so happens that the arctan of 1/2 plus the arctan of 1/3 is pi/4. So the expression
4*(peval(atanx,1/2)+peval(atanx,1/3))
is what we want, and sure enough we get
0.00000 3.33333 3.33333 3.11728 3.11728 3.14558 3.14558 3.14085 3.14085 3.14174 3.14174 3.14156 3.14156 3.14160 3.14160 3.14159 …
The amazing thing about all this is that pi emerges from a relatively simple program (given in full below) in which only small integers appear.
Enough fun for now. I’ll leave you with the complete program and a promise to tell you more about the interpreter and to make it publicly available.
4*(peval(atanx,1/2)+peval(atanx,1/3))
where
x2 = 0 sby 0 sby 1 sby 0;
atanx = pinteg(0,pdiv(one,one+x2));
one = 1 sby 0;
peval(p,a) = v where v = init p +(0 sby a * peval(succ p, a)); end; pdiv(q,w) = t where q0 = init q; w0 = init w; r = q0/w0; v = succ(q - r*w); t = r sby pdiv(v,w); end; pinteg(c,s) = d where i = 1 sby i+1; d = c sby s/i; end; columns = 16; rows = 1; numformat = '%7.5f';
end
(Sorry this is the best I can do with the persnickety editor which even screws up a simple paste. Land o’ Goshen)
