Insulting people's intelligence

There is a movement among some academics to celebrate the bad reviews - the ones that led to rejected papers - because the good reviews simply confirm us, but the bad ones make us think.

I got a bad review for one of my eBooks - The Art of DSP  which is in fact one of my best books (in my own opinion). A really angry review, actually. Apart from taking issue with a typo (which I have since corrected) the reviewer took issue with my saying:

"we need to admit how little mathematics we have that is practically useful"

which I admit is a deliberately challenging statement.

So the reviewer said:

I am stunned by that statement and offended at this arrogant insult to my intelligence

Fair enough. We are all entitled to our opinion, and this is not the first time in my career that I have been told in no uncertain times that I am wrong.

It is interesting, by the way, how much passion math can invoke. I admit to being sometimes 'stunned' - but I am long past being offended by mathematics or by any statements about it - and my intelligence, I have learnt through many painful lessons, is probably beneath insult. To some extent this is why I drafted my statement as I did: it is a deliberate challenge to what I think is a growing 'faith' in mathematics and science - its promotion to something more akin to a faith or religion, something about which one can be offended or insulted, rather than about which one enjoys a dispassionate discussion and thought. Beyond the philosophical implications, faith in mathematics can I think be risky: skepticism, checking, verification of the implicit assumptions, challenging of the model behind it, can all too easily be neglected when one has faith.

But I want to explore the statement, and what I meant by it.

I could have phrased it better, and in time I will, but in it I am trying to say something that I think is really important.

I was taught as a scientist, and later as a mathematician, to question my own knowledge and especially to respect its limits. My professor of Physics, Daphne Jackson, used to say: "It's only a model". By which she meant, you have only made a limited, probably simplified, most likely deficient mathematical or physical model of something that actually neither you nor I fully understand - so you had better not fall into the trap of believing that your model IS the reality.

Let me take an example. My book was about DSP - Digital Signal Processing. One of the most basic tools - and the fundamental basis for most theories and practical applications - in DSP is the Fourier Transform. The Fourier Transform models all signals as if they are composed of what in school we learnt to call 'Simple Harmonic Motion'. You see simple harmonic motion in all sorts of systems: waves, pendulums, loaded springs, all obey the laws of simple harmonic motion. Basically simple harmonic motion says that the more you stretch or displace a thing, the more it tends to pull back to where it was: the force to restore it to where it was is 'proportional' to the stretch or displacement. The 'harmonic' comes from the way sound waves resonate in a wind instrument, and the way strings vibrate in a stringed instrument : these simple systems result in preferred waves whose wave length either matches, or is a simple whole number multiple of, the length of the pipe or string - the 'harmonics' that make the musical tone sound rich but sweet.

Simple Harmonic Motion comes from things that 'cycle' - that go in a circle like a wheel - and are measured in units such as 'cycles per second'. In the Fourier Transform the movements actually are circles - two dimensional shapes - but often we are modelling simpler, one dimensional movements (like the stretching of a spring). In these cases we imagine looking at the cycling wheel from one edge, plotting a graph of how a mark on the wheel's rim moves with time and we get what in school we learnt to call a 'sine' wave: a smooth regularly undulating curve that plots out the mathematical 'sine' of an angle - that angle being the angle between the line from our point of view to the wheel's axle, and the line from the axle to the mark on the rim. These 'cyclic' or 'sine' movements are our models - mathematical functions that are called the 'basis functions' for the Fourier Transform.

These are special, simple functions. In fact they are functions from trigonometry - which is itself the mathematical equational representation of geometry. Our base model is a perfect circle. And, we have solutions to trigonometry: the Ancient Greeks solved those equations for us. Hipparchus, Menelaus, Pythagoras, Euclid - these Ancient Greeks wrote the equations, and solved them, which is why we spend so much time at school learning them. Not all that much more has been solved since - math is hard.

Of course neither pipes nor strings nor many real physical systems, nor many signals, do actually model as perfect cyclic functions. If we photograph the shape of a vibrating piano string, or a flicked skipping rope, the shape we see is often very far from the smooth regularly undulating sine wave. In fact most of the time the shape we see is unique and - crucially - does not match to any known mathematical function. Let me go through this point again so I can try to make myself clear. There is no known mathematical equation that describes the arbitrary shape of a plucked string, a flicked skipping rope the sound wave in a blown wind instrument, or just about any real signal or physical system other than the most carefully constrained, limited, simplified systems.

And here is the problem: if you can't write the equation for the system, you can't solve it - that is, you can't work out things about it, like what it is going to do in future, or what it tells you about what is inside the system, or how the system works. Again, to be clear, what I mean by 'solve' is, you have a mathematical equation to describe the system (given) and from that you can work out another mathematical equation that describes the system at a different time or under differing conditions, or that describes some inner working of the system or the driving forces behind the system. From equation, to equation.

So this is what I meant. Most real systems, physical phenomena, signals, cannot even be described by a mathematical equation - let alone solved to another mathematical equation. This is the limit of our mathematics.

Now, many - even most - systems and signals can be described by some form of equation. Take a system where motions very closely do match perfect circles - the solar system, where the planets orbit cyclically around the sun, and moons around their planets. Thanks to Newton we have equations for  gravitational attraction - Newton's Law of Gravity - and for motion - Newton's Laws of Motion. Each of these are really simple equations. The Laws of Motion let us write equations for the force operating between sun and planets, and the Laws of Motion let us write equations for how that force moves the planets. But here is the problem: those laws tell us how the planets move, but not where they are: they are simultaneous differential equations (remember those from school?). To tell where the planets are at any time, we need to solve those simultaneous differential equations. And guess what? We can't. Nobody ever yet solved the equations of motion and gravity to write down equations for where the planets were, are or will be. There is a special case: where we limit ourselves to only two bodies - Earth and Moon alone, or Sun and Earth alone - we can solve the equations. But that neglects the effects - however weak - of all the other planets, and galaxies, and dark matter and dust and...

We can't solve the equations of gravity and motion for the solar system.

This has a name: it is called the 'Three Body Problem'. If you try with just three bodies, you can't solve the equations. More, and you can solve even less.

So, suppose we want to know where the Moon will be, so we can forecast, through its gravitational effect on the seas, the tides? We can't write an equation, into which we could plug the desired date and time. We can do a computational thing called 'iteration': from where it is now, work out - using the equations of gravity and motion - what forces are acting on it, hence how it is moving now, from that work out where it will be in a moment, then start again from there (iterate). This is great. But each time, we have a little error because we are jumping in little leaps, not continuously as the Moon really moves - and those errors can build, so that after a short while we really have no idea at all.

Or we can do the Fourier thing.

Kepler showed that the planets move in ellipses - ovals if you like. We could write equations for the ellipses - ellipsoidal functions - but those are quite hard, and lack the elegantly simple geometry of perfect circles. But an ellipse is almost a circle. So what if we modeled the motion as perfect circles? Then, we could apply all our simple geometry and trigonometry - to which  we do have solutions, thanks to the Ancient Greeks. Now, there is a difference - an error if you like - between our circle and the actual ellipse. And here is the Fourier thing. Let's see if we can model that difference also as a perfect circle - whose motion we simply add to the first circle. Of course there is still a bit of error - hopefully small - but we can model that too as a perfect circle, and so on and on until the error is small enough that we can ignore it as insignificant compared to the accuracy we want.

This is Fourier Analysis - the most common basis for DSP. It is also, if you remember, the Ptolemaic model of planetary motions as perfect circles - that we all learnt to laugh at when we were taught about Copernicus and Kepler and the sun-centered solar system. Ptolemy had the planets fixed to crystal spheres, whose movement generated music - the Music of the Crystal Spheres - harmonics, if you like. So the Ptolemaic model is a Fourier model: both models 'work', to a degree, so long as we remember their limitations - and both fail, to a degree, if we think they ARE the system rather than simplified, flawed models of it. If we could deal with the motions as they are, and could solve the equations, then we could do better - but we can't, so we fall back on Fourier or some other model.

In fact, in forecasting planetary positions, the Fourier model is quite often used. Ptolemy would approve. And in forecasting the tides - which then need to include the movements of the 'tidal waves' and other actual ebbs and flows - the motions of the seas are also often modeled using the Fourier model - cycles within cycles.

So, if this works so well, why did I make a fuss about it?

Well, mainly because it doesn't. For example, my current work is in microwave medical imaging.  Microwaves are electromagnetic waves - actual waves, sine waves, perfect electromagnetic cycles: so ideal for Fourier analysis. And in fact I can quite easily, with just some basic wave physics and a few lines of equations, show that the medical image is the 3-dimensional Fourier Transform of the measured Scattering Parameters. Except that it isn't. Maxwell's Equations, like Newton's, are simple, elegant - and differential. And, like Newton's, nobody yet solved them. There are special cases - scattering from a tiny point, from a perfect sphere, from an infinitely long cylinder - but no solution for the general case of the internal body organs of a medical imaging subject. Which is why we use Fourier. But Fourier fails, as it does with planetary motions and tides, because microwaves are not quite cyclic when they pass through a human body. Their speed changes, they are attenuated, bent, distorted: all of which we can indeed take into account by modifying our Fourier model to take into account ever more corrections, compensations and adjustments - but all that leads us further and further from the perfect crystal spheres of the Fourier model, and further and further into the gritty muddy cloudiness of the real world.

Which is what I meant: "It's just a model".