Here are some of my musings on the state of modern science. I've been incubating a number of ideas for a while, and whilst they are nowhere near joined-up, let alone conclusive, they are quite interesting and perhaps relevant to the thread (I posted this as a note on Facebook a while back, maybe not the best arena for it.)
Like all those with an education in a physics-related science, I learned Quantum Mechanics and was as disturbed by it as any. "If you're not shocked by it, you haven't understood it". Nevertheless it is the most successful mathematical model humans have ever constructed, and has been experimentally validated to the point of tedium and beyond. But what is it a model OF? The answer is that nobody knows. It may just be a convenient mathematical box of tricks (or should a say a number of boxes, because there are several - equivalent but different - formulations), but that scarcely seems likely. And so physicists have come up with a number of interpretations, all of them at odds in some respect with how we experience the universe. That of course, does not make them wrong. And it's not a matter of finding the "most palatable" interpretation so much as the correct one. Which, I should add, we may not yet have conceived. Till then, there is only the maths.
Ah, but what is maths? Why does it work at all? If the universe is anything, it does not seem mathematical. It seems very real and very solid. And indeed, maths provides a means to *model* the properties of the world. When you construct a model, there is an understanding that the map is not the territory. A model captures the salient features of reality and marginalises the rest. This usually gives us a decent approximation, but if modern science has taught us anything, it is that approximations are not necessarily good enough. A non-linear system can amplify tiny differences in initial conditions until they become massive differences. And indeed, non-linear systems are the bane of mathematicians in general. They are difficult not so much to model, but to do anything useful with the model. Early in our heritage, the tendency was to simply *linearise* difficult models that could not be solved in general. This leads to huge difficulties - for example in studying the solar system it has become impossible to tell, using the same models that has taken probes to the outer planets and beyond, whether the solar system is a stable entity or not. Some equations are just not solveable. Some contain terms which get bigger and bigger over long periods until they dwarf other terms, and for the n-body problem representing the solar system this is a lump that can be pushed under the carpet but never entirely hidden. I often wondered: if an equation modelling a physical system cannot be solved, what does it say about that system? For example the Schroedinger Equation, the fundamental mathematical keystone of QM, for the nucleus an atom such as (say) iron cannot be solved in closed form. Iron stubbornly continues to exist, however. The universe does not seem mathematical in this respect, for sure.
One of the most baffling aspects of QM is the idea that the universe, which looks pretty solid and regular to us, is actually highly irregular. Things pop up at random and vanish when the universe blinks. Yet: in order to understand why QM works at all, we need to comprehend the precise regularity that underpins it. QM offers us two models of entities: a wave model, where the characteristics of the object are smeared over space; and a particle model which places a thing at a single point in space. It remains part of the ongoing debate as to whether one - or both, or neither - represent the real entity. But the wave property lets us say one thing, for sure: the behaviour of the object depends on whether that wave fits nicely into a particular space. For example the quantisation of electron orbits is determined by fitting integer multiples of the wave into the potential well created by the nucleus. How strange then: we see that there is a regularity underpinning irregularity.
But then, the regular and the irregular, the patterned and the random, they have an interesting symbiosis. An example that is simple as it is beautiful as it is deep, is as follows: collect some pebbles of various sizes, from tiny to 1 or 2cm in diameter. Put them in a jar in no particular order, but leave room at the top. The distribution is typically "random". A small stone is as likely to be next to a big one as any other one. Now shake the jar a few times. What happens? Suddenly there is order within the jar! Now the bigger stones are more likely to be next to bigger stones than smaller ones, because the little ones fall between the gaps in the big ones and end up at the bottom. Here we see three things in action: an irregular physical system (a collection of pebbles), an irregular perturbation (shaking) and an organising force - in this case, gravity - combine to manufacture regularity out of (apparently) nothing.
So let's return to that question "why does maths work?" What IS mathematics? I would suggest that mathematics is the *study of regularity*. Maths works because there are regularities in the universe which can be modelled mathematically. Physicists then apply this filter to the universe to understand the relationships between elements in their models. But of course, this necessarily screens out the irregular. Which means...? Well the first thing that occurs to me is that we are living in an age when we are scratching our heads trying to figure out what the vast bulk of our universe is made from. So-called "Dark Matter" is required to glue our galaxies together, and its nature is a mystery. "Dark Energy" is even more enigmatic, and is required to explain the disturbing observation that the expansion of our universe is accelerating. It seems that the vast bulk of the universe is hiding beyond our mathematical models and our telescopes. Of course physicists hope that these strange phenomena will drop out of the equations if and when we find a better model of the universe - presumably one in which we understand gravity, the most potent force in the cosmos. But maybe we're missing a trick here? Suppose, for instance, that the vast bulk of the universe is the irregular stuff - the bits that fall inbetween the cracks in our models? In fact, let's go further: let's say that the universe is a huge, utterly irregular systems under continuous agitation (dark energy?) with one (or several) dumb organising forces (like gravity was in the stone jar). Then all that we see around us is the regularity that this meta-system generates.
Furthermore: what would we see when we examine this regularity in more and more detail? We'd see the models start to break down because we'd eventually get down to the irregularities that, by its nature, our maths can't model. We'd be forced to start modelling things statistically for a while, because in statistics we have means of sacrificing quantitative accuracy for qualitative description. But in doing so, the universe suddenly becomes smeared and uncertain. Therefore, Quantum Mechanics - a statistical theory of Something Or Other - would be exactly the sort of thing you would expect to see. In other words, beautiful mathematical models - as powerful as they are - are only good up to a point. Beyond that point lies irregularity - here be dragons.
Talking of dragons, let's cut to the reason behind the title of this note: edge cases. As a software developer, I am more than familiar with these annoying beasts. Just when you think you have written something beautiful and elegant for handling a particular problem, you find that there is some tricksy little problem that your model can't handle, and the consequence is usually having to metaphorically sellotape some extra code over the elegance. Here's a simple example: writing a function to compute factorials. A factorial is easy enough to explain: it's simply the product of all the whole numbers up to a given number. So factorial(4) = 4 x 3 x 2 x 1 = 24. For shorthand we write 4! = 24. The edge case arises from asking "'what is 0!'?" It looks like it ought to be 0, but in fact, for reasons that go way beyond the scope of the problem, it has to be 1. So here we have a hard-wired definition to "sellotape" into our otherwise elegant model. Ok that's a trivial (and slightly disingenuous) example. But usually a "real" software system that delivers some functionality has all manner of edge-cases to handle. They are rarely elegant by the time they are shipped! Part of the reason for this is that, whilst it is possible to elegantly write a small piece of software, when pieces of software start interacting, all bets are off. In fact it it these interactions that cause most of the issues in software development. If you have N software components, and add another 1, that doesn't mean you have to worry about N+1 components, but up to another N issues because any of the existing N could interact with the new one. This makes software increasingly difficult to test - and write in a bullet-proof fashion. Edge case upon edge case!
Here is the point: real models tend to diverge from idealised models because they do not exist in isolation.So what then the hope for an elegant mathematical model of the universe that explains all the forces and all the particles? Imagine the universe as a vast set of loosely related interoperating systems. There would be many "edge cases" brought about by those interactions. The result: a blend of regularity and irregularity - deviations from expected behaviours based on regularities alone. Perhaps then the lesson is that the universe is rich and complicated. "Turtles all the way down". Instead of operating under totally harmonious principles, there are both regularities and irregularities. The simple regularities interact with the irregularities to produce, like the stones in the jar, an observable universe of order, a thin, habitable crust sitting atop and beholden to a vast, swirling miasma of chaos. And that - reassuringly maybe! - the picture we have of the universe in the 21st Century is essentially correct.
Like all those with an education in a physics-related science, I learned Quantum Mechanics and was as disturbed by it as any. "If you're not shocked by it, you haven't understood it". Nevertheless it is the most successful mathematical model humans have ever constructed, and has been experimentally validated to the point of tedium and beyond. But what is it a model OF? The answer is that nobody knows. It may just be a convenient mathematical box of tricks (or should a say a number of boxes, because there are several - equivalent but different - formulations), but that scarcely seems likely. And so physicists have come up with a number of interpretations, all of them at odds in some respect with how we experience the universe. That of course, does not make them wrong. And it's not a matter of finding the "most palatable" interpretation so much as the correct one. Which, I should add, we may not yet have conceived. Till then, there is only the maths.
Ah, but what is maths? Why does it work at all? If the universe is anything, it does not seem mathematical. It seems very real and very solid. And indeed, maths provides a means to *model* the properties of the world. When you construct a model, there is an understanding that the map is not the territory. A model captures the salient features of reality and marginalises the rest. This usually gives us a decent approximation, but if modern science has taught us anything, it is that approximations are not necessarily good enough. A non-linear system can amplify tiny differences in initial conditions until they become massive differences. And indeed, non-linear systems are the bane of mathematicians in general. They are difficult not so much to model, but to do anything useful with the model. Early in our heritage, the tendency was to simply *linearise* difficult models that could not be solved in general. This leads to huge difficulties - for example in studying the solar system it has become impossible to tell, using the same models that has taken probes to the outer planets and beyond, whether the solar system is a stable entity or not. Some equations are just not solveable. Some contain terms which get bigger and bigger over long periods until they dwarf other terms, and for the n-body problem representing the solar system this is a lump that can be pushed under the carpet but never entirely hidden. I often wondered: if an equation modelling a physical system cannot be solved, what does it say about that system? For example the Schroedinger Equation, the fundamental mathematical keystone of QM, for the nucleus an atom such as (say) iron cannot be solved in closed form. Iron stubbornly continues to exist, however. The universe does not seem mathematical in this respect, for sure.
One of the most baffling aspects of QM is the idea that the universe, which looks pretty solid and regular to us, is actually highly irregular. Things pop up at random and vanish when the universe blinks. Yet: in order to understand why QM works at all, we need to comprehend the precise regularity that underpins it. QM offers us two models of entities: a wave model, where the characteristics of the object are smeared over space; and a particle model which places a thing at a single point in space. It remains part of the ongoing debate as to whether one - or both, or neither - represent the real entity. But the wave property lets us say one thing, for sure: the behaviour of the object depends on whether that wave fits nicely into a particular space. For example the quantisation of electron orbits is determined by fitting integer multiples of the wave into the potential well created by the nucleus. How strange then: we see that there is a regularity underpinning irregularity.
But then, the regular and the irregular, the patterned and the random, they have an interesting symbiosis. An example that is simple as it is beautiful as it is deep, is as follows: collect some pebbles of various sizes, from tiny to 1 or 2cm in diameter. Put them in a jar in no particular order, but leave room at the top. The distribution is typically "random". A small stone is as likely to be next to a big one as any other one. Now shake the jar a few times. What happens? Suddenly there is order within the jar! Now the bigger stones are more likely to be next to bigger stones than smaller ones, because the little ones fall between the gaps in the big ones and end up at the bottom. Here we see three things in action: an irregular physical system (a collection of pebbles), an irregular perturbation (shaking) and an organising force - in this case, gravity - combine to manufacture regularity out of (apparently) nothing.
So let's return to that question "why does maths work?" What IS mathematics? I would suggest that mathematics is the *study of regularity*. Maths works because there are regularities in the universe which can be modelled mathematically. Physicists then apply this filter to the universe to understand the relationships between elements in their models. But of course, this necessarily screens out the irregular. Which means...? Well the first thing that occurs to me is that we are living in an age when we are scratching our heads trying to figure out what the vast bulk of our universe is made from. So-called "Dark Matter" is required to glue our galaxies together, and its nature is a mystery. "Dark Energy" is even more enigmatic, and is required to explain the disturbing observation that the expansion of our universe is accelerating. It seems that the vast bulk of the universe is hiding beyond our mathematical models and our telescopes. Of course physicists hope that these strange phenomena will drop out of the equations if and when we find a better model of the universe - presumably one in which we understand gravity, the most potent force in the cosmos. But maybe we're missing a trick here? Suppose, for instance, that the vast bulk of the universe is the irregular stuff - the bits that fall inbetween the cracks in our models? In fact, let's go further: let's say that the universe is a huge, utterly irregular systems under continuous agitation (dark energy?) with one (or several) dumb organising forces (like gravity was in the stone jar). Then all that we see around us is the regularity that this meta-system generates.
Furthermore: what would we see when we examine this regularity in more and more detail? We'd see the models start to break down because we'd eventually get down to the irregularities that, by its nature, our maths can't model. We'd be forced to start modelling things statistically for a while, because in statistics we have means of sacrificing quantitative accuracy for qualitative description. But in doing so, the universe suddenly becomes smeared and uncertain. Therefore, Quantum Mechanics - a statistical theory of Something Or Other - would be exactly the sort of thing you would expect to see. In other words, beautiful mathematical models - as powerful as they are - are only good up to a point. Beyond that point lies irregularity - here be dragons.
Talking of dragons, let's cut to the reason behind the title of this note: edge cases. As a software developer, I am more than familiar with these annoying beasts. Just when you think you have written something beautiful and elegant for handling a particular problem, you find that there is some tricksy little problem that your model can't handle, and the consequence is usually having to metaphorically sellotape some extra code over the elegance. Here's a simple example: writing a function to compute factorials. A factorial is easy enough to explain: it's simply the product of all the whole numbers up to a given number. So factorial(4) = 4 x 3 x 2 x 1 = 24. For shorthand we write 4! = 24. The edge case arises from asking "'what is 0!'?" It looks like it ought to be 0, but in fact, for reasons that go way beyond the scope of the problem, it has to be 1. So here we have a hard-wired definition to "sellotape" into our otherwise elegant model. Ok that's a trivial (and slightly disingenuous) example. But usually a "real" software system that delivers some functionality has all manner of edge-cases to handle. They are rarely elegant by the time they are shipped! Part of the reason for this is that, whilst it is possible to elegantly write a small piece of software, when pieces of software start interacting, all bets are off. In fact it it these interactions that cause most of the issues in software development. If you have N software components, and add another 1, that doesn't mean you have to worry about N+1 components, but up to another N issues because any of the existing N could interact with the new one. This makes software increasingly difficult to test - and write in a bullet-proof fashion. Edge case upon edge case!
Here is the point: real models tend to diverge from idealised models because they do not exist in isolation.So what then the hope for an elegant mathematical model of the universe that explains all the forces and all the particles? Imagine the universe as a vast set of loosely related interoperating systems. There would be many "edge cases" brought about by those interactions. The result: a blend of regularity and irregularity - deviations from expected behaviours based on regularities alone. Perhaps then the lesson is that the universe is rich and complicated. "Turtles all the way down". Instead of operating under totally harmonious principles, there are both regularities and irregularities. The simple regularities interact with the irregularities to produce, like the stones in the jar, an observable universe of order, a thin, habitable crust sitting atop and beholden to a vast, swirling miasma of chaos. And that - reassuringly maybe! - the picture we have of the universe in the 21st Century is essentially correct.
