Monday, 4 April 2016

Wigner's Unreasonable Assessment

Note: I'm extremely pleased to say that this post has been read by one of my friends, a mathematician and logician whose expertise and understanding I have great confidence in based on a large body of interactions. He's identified some problems with it or rather, in his own words:
They haven't been expressed to me yet, but he's agreed to pen a guest post addressing them within the next few weeks. I'm really excited about this. For those who haven't been following from the beginning, this blog represents my author's notes for a book, a sort of manual for applying logic in the kind of situations we face every day, and particularly about how to go about assessing truth-claims. The reason I've been populating this blog is to get my understanding of the ideas I want to talk about in a place where my large pool of extremely knowledgeable friends and long-time cohorts and siblings-in-arms can see it and help me to ensure that I'm not misrepresenting any of the trickier topics in my attempts to make them more generally palatable.

As I've said in this post and others, I make no pretense at being a mathematician. I can fumble through an equation - if I'm careful - well enough to be able to grasp what's going on in a paper, but pure mathematics requires a degree of rigour that is well beyond my patience for the things I'm interested in (although I'm working on it, as time allows).

I'll leave this post here for now, and I'm going to promote it a bit, for two reasons: The first is that I'm tentatively confident that the general point will hold about the relationship between science and mathematics, which is the real topic of the post. The second is that it might serve to be an instructive example of how rational epistemological processes work. I'm certain that I'll learn something important, because I always do from this guy.

Update: I'm happy to say that the critique of this post is now in place, and can be found here:


A potted history of the foundations of mathematics.

Hi,

There'll be more cosmology stuff to follow, but I wanted to take a break to deal with something that's been distracting me since I reopened these pages. My mind's been afire with ideas, but this one's pretty fundamental to all of what follows in one way or another, and I'm known to be easily distrac... Oh, look! A tractor!

Where was I..?

Oh, yes. I wanted to talk a little bit about what Nobel laureate Eugene Wigner called 'the Unreasonable Effectiveness of Mathematics in the Natural Sciences' (Wigner, 1960).


It's a cracking paper, and I recommend it. I certainly don't want anybody to go away with the impression that I broadly disagree with Wigner, but if all one takes away is the title, one could be misled, in precisely the same way as one could from the title of this post.


There has, however, been a fair bit of wibbling of late about how the universe is, in its essence, a mathematical entity - made of numbers (e.g.Tegmark 2014). This statement is either true and trivial (not interesting) or 'not even wrong'. This is what I want to address, and also to lay the groundwork for how it is that we acquire knowledge, and the role that mathematics plays. 

We live in a universe that contains something which we label 'quantity'.

There, that didn't require an entire book, did it?

Thanks for reading.

Oh, you want more?

Fair enough.

I wish first to note that, in what follows, I'm going to be somewhat loose. As I've already stated, I'm no mathematician, and I'm sure that where what I have to say meets technical concepts from that field I'll probably be less than completely rigorous, but I'm more concerned here with the relationship between mathematics and science than the content of the mathematics itself.

So, we have this thing we call quantity. We can apply the concept of quantity to a great many things, from apples to atoms. The beauty of mathematics is that it's axiomatically grounded. We can build from the simplest of axioms, defining our terms as we go, and be sure that what we're building on has good, solid foundations. Specifically, the axioms of mathematics aren't accepted because they seem to be true, but because they are definitionally true. In other words, we define 1 as the singular integer, and 2 is 1+1. You could say that the definition of '1' is the first axiom of mathematics, upon which all other axioms are built. Thus we can define 2 as as the sum of the integers 1 and 1. And from there we can build another axiom, namely, the addition of two integers gives the sum (we'll make this  explicit  shortly ). Much of the rest is about the relationships between operators, so that we can build up to 16 being 1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1, and that being the same as 8+8, which is the same as 8x2, from which we can build another axiom, namely that the multiplication of two integers gives the product. Then we can build a whole set of relationships around the equivalence of these operations on specific sets of integers to build an axiomatically grounded system, and it is this way because of the fact that the core axioms are necessarily true by definition.

It is for this reason that 'proof' is possible in mathematics because, in that sense, it is axiomatically grounded by definition, having been built on the simplest of premises that are necessarily true. The labels themselves are arbitrary, and conceptual, and you'd absolutely correct to say that without observation they don't actually mean anything, but they do inform us with regard to specific characteristics held by that which we observe, and indeed provide information that we couldn't obtain by any other means, so while they don't constitute knowledge in and of themselves, they do provide knowledge. This is especially true when we begin to deal with the relationships between one numerable property and another. All of theoretical physics is based in this.

To make this concrete, I thought I'd do a post about equations and how they're used in science to lay the groundwork for what's to come, so we don't freak out when equations start appearing in earnest (much to Earnest's chagrin).

To begin, just a little background on what equations are and how they work:

So just what is an equation? Well, at bottom, it's simply a description of a relationship. Let's look at the simplest equation, which we met a moment ago:

\[1+1=2\]

So, this all looks very straightforward, doesn't it? It's fairly simple to see what's happening here. The important bit of that is the = sign, which denotes equality. What is on one side of that sign will always be exactly equal to what is on the other side. They are being equated! So the two sides of that sign are directly equivalent.

But what about the terms? Well, in this simple equation, all the terms are variables. The = sign tells us that where one of the terms on one side is varied, the term on the other side will also vary to compensate. So, for example, if we vary the integers in the input, we will get a different output to reflect this:

\[1+2=3\]

And indeed we can generate another equation that shows that, however the input is varied, the output will always give an equality.

\[a+b=\sum\]

Where a and b are any number and where denotes the sum.

Now a look at another familiar equation. We're all (I hope) familiar with Pythagoras' Theorem. It provides a means for calculating the length of one side of a right-angled triangle in Euclidean geometry given the lengths of two sides as input. Formally stated, it tells us that the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides. In geometric form it looks like this:




In the notation, this would be \[a^2+b^2=c^2\] Thus, if we take a to be 3 and b to be 6, we can carry out the calculation using those two lengths as input:

\[3^2=9\] and \[6^2=36\] so we sum those two figures to obtain \[a^2+b^2=45\] Taking that, we find the square root of 45 \[(\sqrt {45})\] which gives us a length for the hypotenuse of \[6.7\]

We can also carry this out backwards, by taking, for example, b and c as input. If we know that b=2, and that c =8, we can calculate what a will be. \[2^2=4\] and \[8^2=64\] so we can calculate this by subtraction: \[64-4=60\] Now we need the square root of 60 \[(\sqrt {60})\] and we now know that \[a=7.7\]

7.7 what, you might ask? Well, as long as the units of measurement are consistent, this doesn't matter. By consistent, we mean that either they are expressed in the same units both sides of the equation or that suitable conversions have been carried out.

Now, all of the above is working with variable terms. In physics, the really interesting things happen when we introduce the notion of a constant. Much of physics can be identified with the finding of constant quantities, because these tell us the most interesting things in physics, not least because constants are often the conversion rates between pairs of variables. Take the most famous equation in physics as an example, Einstein's famous mass/energy relationship.

\[E=mc^2\]

What this equation tells us is that the energy contained in a given quantity of matter is equal to its mass - more accurately rest mass - multiplied by the square of the speed of light. In order to unpack it, we need to define the terms properly and give the units in which the terms are measured.

E denotes the energy, and this is measured in joules. A joule is defined as the energy expended in applying a force of 1 newton through 1 metre. m denotes the mass, and is measured in kilograms. c denotes the speed of light in a vacuum, and this is measured in metres per second. The actual value of this is \[299 792 458 \dfrac m s\] but for simplicity, we'll take it to be \[300000000 \dfrac m s\] or \[3\times10^8 \dfrac m s\] or \[3\times10^8 ms^{-1}\]

So, let's plug in some numbers (actually, just 1 number, namely the mass). Let's take our mass to be 1 kg for simplicity. First, though, we need to square the speed of light in a vacuum. So, we get \[(3\times10^8)\times(3\times10^8)\] which gives us \[9\times10^{16}\] which we multiply again by 1 kg, which gives us \[9\times10^{16} kg\space m^2s^{-2}\] This last result can be converted directly to joules, as the units are given in metres per second. From this, we obtain the result that the energy contained in a mass of 1 kg, is equal to \[9\times10^{16}j\] or 90,000,000,000,000,000 joules.

This is a huge number. What this tells us is that the energy in a single standard bag of sugar (1 kg in the UK) is roughly equivalent to the output of 21.5 million tons of TNT. Putting that into perspective, Little Boy, the bomb dropped on Hiroshima in 1945, had an energy yield equivalent to 15,000 tons of TNT. 


So, we can draw comparisons between enumerable quantities and, indeed, we can even carry out conversions between quantities that seem unrelated, so that we can, for example, relate distances in space and time, simply by relating both distances to a constant that is measured in distance over time (I'll make this explicit in a future post dealing with special relativity in more detail). Anywhere we find a quantity, we can elucidate relationships.

Of course, what Wigner was actually talking about in his paper is the mystery concerning the fact that we can even come up with the concept of number, much less the numerical concepts we've devised that, when introduced, seemed to bear no relationship to the world around us and, of course, the fact that laws formulated in mathematical terms, with little prior justification beyond experience that this can be fruitful, actually work.

Is the effectiveness of mathematics in the natural sciences unreasonable? Certainly not!

Since I've yet to introduce a movie quote into these meanderings, I'll start here with this one.

"Can you count, suckers? I tell you that the future is ours, if you can count" - Cyrus - The Warriors.