Darwin was wrong.

This headline, or something like it, is a staple of lazy sensationalist journalists. It seems that every time something interesting about biology or evolution (I dont think there's a difference between the two but that's by the by) crops up in some study there's a semi-literate hack waiting in the wings to say that this proves Darwin wrong. Normally this just reflects a lack of understanding of the complexity of living things (which is of course evidence that evolution is true, no other mechanism can produce organic complexity). However sometimes it reflects a real lack of understanding of how science works.

We see the same bad reasoning whenever someone says something like “Einstein proved Newton wrong” or that “scientists are always changing their minds, one day they say one thing, the next they might say the opposite”. I think this reflects a real misunderstanding of the nature of the word wrong and does not reflect the historical reality of scientific progress. I dont think a scientific Theory is simply right or wrong as an absolute, nor do I think that future Theories are totally unpredictable.

I'm a mathematician and as such I think about everything as a mathematical analogy. To me science is a line of best fit. We are presented with data and observations. Given that we are required to predict what will happen in the future. All of science works like this, even history or other apparently non-predictive subjects work like this. When we say that there was a iron-aged fort at location X, we can interpret that as a prediction: “if more artefacts are uncovered from this site they will follow the distribution we expect from a fort of this size”. This is a prediction about future observations. Without future observations science is a meaningless task. Science is about constructing Theories, a scientific Theory is a (generally mathematical) model into which we can feed an experiment and from which we get a prediction. Any other sort of claim about the world is metaphysics, which bores me rigid so I wont talk about it.

So, given that, let us analyse what happened with gravity. Everyone knows things fall, the ancients said that the elements of earth and water tried to return to the ground and to the sea respectively. This is a scientific Theory. It makes predictions, and most of the time those predictions are pretty good. Rivers do flow towards the sea not away from it, rocks fall towards the earth not the sky. But then along came the Renaissance, then it was claimed that things fall towards the centre of the spherical earth at a constant rate of acceleration and that fire etc rise due to buoyancy. This is a very accurate prediction, it gives you much more detail than the ancient theory, if you've done any kind of mechanics with gravity in school, this is the theory you used. Newton then came along, and said that the force of gravity on earth was the same inverse square law that was known to rule celestial mechanics. This meant that very accurate predictions could be made for the force of gravity almost anywhere in the universe. And finally, Einstein shows up and yacked about bent space for a bit and allowed us to predict the precession of Mercury better than Newton.

What has happened here? I'm going to take some daft numbers for sake of example but they're not important. Let us say that the Ancient theory gives predictions that are as accurate as our instruments 50% of the time, this is very generous. Then let us suppose that the Renaissance model is as accurate as we want 90% of the time. What has happened to the ancient predictions? They're still just as valid. They're still easy to do and intuitive. They're still right with exactly the same accuracy. The Renaissance did not change the accuracy of the ancients. What it did do is provide a new theory that is more accurate. Anyone who stuck with the old theory ends up with worse results and is more likely to have things blow up in their face.

I claim that Einstein didn't prove Newton wrong. Before 1915 it was well known that Newton was accurate to within (made up figure) 1%, but it was also well know that this 1% error wasn't going away. Anyone who claimed that Newton's laws were perfect was either hopelessly optimistic or really out of touch with experimental results. After 1915 it was well known that Newton was accurate to within 1% and that whilst this is far from perfect it was used in the Apollo missions and almost all space-flight without too much worry. Einstein did not change this fact. One thing he did do that was important was knock down any notions that Newton was a god or had direct dealings with a god. This is very healthy. We need to remember that Einstein's Theory is more accurate than Newton's only quantitatively, it is accurate to within 0.1% not 1%. This is not the same thing as being right in an absolute sense. Relativity makes predictions about black holes and about the subatomic universe that just aren't right. All we say for Einstein is that these cases are rare so he's very accurate on average.

My point here is we need to get away from the notion that a Theory exists in one of two states, right or wrong. A simplistic version of the principle of falsification runs like this: we propose a Theory, provisionally accept it, then run tests, and if it fails them then it is wrong. I hate to disillusion you, but by that definition all Theories are equally wrong. Not one scientific Theory ever proposed has made 100% accurate predictions. Relativity goes all to hell near black holes, quantum theory is great, gets predictions right down to the 12th decimal place, but the 13th? Totally wrong.

As someone who's excited by physics I'm quite glad that a 100% accurate Theory Of Everything doesn't exist. A world where there was nothing left to do but measure some constants ever more accurately (as was thought by some at the start of the last century) would be a very boring one to live in.

Another suggestion that annoys me is that Theories change, so tomorrow a Theory may be discovered that is completely the opposite of the current one and that might be accepted. Remember the analogy of the line of best fit? I've got an extension of the idea. To me, Theories of science are a Taylor expansion of reality. A Taylor expansion is a sequence of curves of best fit that get closer and closer to the true value of some curve as time goes on.




Here we have a sequence of Theories, each one is an approximation of the real observation, each one is very close to right in a small area, and very far from right as you go away from there. I want to claim that these sequences of Theories have a very important property. As you go forward in time I want to claim that the numerical predictions made by a Theory will tend to get ever closer together before ending up very near some point.


I want to suggest that if you look at the history of mainstream Theories in one area of science you will see that over time the Theories become ever closer in predictions, until (as with current Quantum Theory) it is very very hard indeed to do an experiment where the difference between an old Theory and a new one is measurable. The red dots represent different Theories that are the most accurate available at different times. They fluctuate wildly at the start, then gradually settle down towards a definite value. What is that value? No way to tell, the best approximation we have is the most accurate Theory we have.

I suggest that if we observe one Theory to be very accurate then any more accurate future Theory will fall roughly within the error bar of the first. In short, I suggest that the green Theory could never be more reliable and accurate than the cluster of reds. Simply because if the green Theory is accurate you have a lot of work to do to explain why it is that all the red Theories work so well.

The Darwin example: Darwin wasn't a god. Of course he made predictions that have turned out to be false. And of course the core Theory of biology hasn't been left lying around for the last 150 years. We have new Theories of evolution all the time. Darwin's was simplistic, not very quantitative. It was an improvement on Lemark. It wasn't as accurate as the Neo-Darwinian synthesis. In the same way Richard Dawkins didn't show that evolution was wrong when he proposed his gene-centric version was more quantifiable and more accurate than group-selection theories. In the same way the latest improvement or refinement of the theory does not show that evolution isn't true. It is. Just now we have a better understanding of it.

It is of course always logically possible that anything could happen and any new theory could in principle be successful. But anything that seriously contradicts previous theories has an awful lot of explaining to do about why all the evidence of the past showed something different. I wouldn't dare suggest that it is impossible that evolution or some other well tested Theory will ever be replaced by a Theory that is substantially different. But it is a good prediction that this wont happen. I'm defiantly more confident than 1 in 2 million and would thus happily stake my life on it being true. (And so would you, you've taken a 1 in 2 million bet with your life if you've ever been on a plane for more than an hour).

Please note here I'm talking only about the predictive element of the theory. The metaphysical and philosophical differences between theories can be immense. The shift to a non-absolute time was enormous and totally contradicted all previous theories. But the change in predictions was very very small. The metaphysical difference between absolute time and relative time is vast, the predictive difference is billionths of seconds in any reasonable case.

Jupiter and beyond the infinite - Part 5

Solving Zeno's paradox

Adam: Ok, we have seen that different sizes of infinite sets can exist by thinking about size as pairing things up. We've seen that adding one to an infinite set makes it the same size and today I'd like to try and deal today with some of the problems raised by Zeno. Would you like to explain the paradox of the arrow so I can see how best to respond?

Plato: Ok, suppose an arrow is fired at a target. Before it can hit the target it has to get half way there, before it can get half way there it has to get a quarter of the way there and so on. So it needs to do an infinite number of things before it can hit the target. Doing an infinite number of things takes an infinite amount of time. So the arrow will never hit the target.

Jupiter and beyond the infinite - Part 4

Different infinities.

Adam: So last time we worked out that we measure size by pairing things up. And from that we worked out that the set of evens is the same size as the set of all whole numbers, that the set of square numbers is the same size as the set of whole numbers, etc.

Jupiter and beyond the infinite - Part 3

Counting the infinite.

Adam: Hello Plato, sorry I was busy in Friday, care for another chat about infinity?

Plato: Definitely, let's see. Last time we decided that a set is a collection of some number of objects, that can be thought of as a box. We also said it makes sense to say that an infinite set cannot be given a number as its size. But I said that we had missed something and wanted to know how we could say the size of something infinite.

Jupiter and beyond the infinite - Part 2

defining infinity, sets and counting

Plato: So lets recap what we worked out last time. If I remember right we realised that if our definitions aren't sorted out we can end up confusing and contradicting ourselves. And as we dont find definitions set in stone, we have to make them to suit our needs. (Though I'm not sure I agree with you about that).

Adam: Yeh that's right, basically today I'm going to suggest that the idea of a set is a useful one for discussing infinity. … hang on a second … why am I talking to you?

Plato: Oh it's called a Socratic dialogue. You talk to fictional student,in this case me, then in arguing with me you find a solution to the problem.

A lie told often enough.

To suppose that the eye with all its inimitable contrivances for adjusting the focus to different distances, for admitting different amounts of light, and for the correction of spherical and chromatic aberration, could have been formed by natural selection, seems, I freely confess, absurd in the highest degree." ~ Charles Darwin

Just a quick note to say that evolution is all a lie and even Charles Darwin, the god of atheism, thought so. And also to say a few things.

Dont trust authority, just because a respected scientist says something doesn't make it right.

Respect evidence, dont assume your opinion is valid just because it makes you feel happy or you have money riding on it. If you've not got reasons you've got nothing.

Read the whole paragraph. Sometimes quotes are misleading.

The quote above can be found in 46,300 websites. The next sentence, 7,670. This happens all over the place, not just science, but in politics, friendships, art, everywhere. It's a common trick that is used too often, and one should always be on guard for this kind of deception, as with any other kind.

With this in mind, here's the context to the above quote.

To suppose that the eye with all its inimitable contrivances for adjusting the focus to different distances, for admitting different amounts of light, and for the correction of spherical and chromatic aberration, could have been formed by natural selection, seems, I freely confess, absurd in the highest degree.
When it was first said that the sun stood still and the world turned round, the common sense of mankind declared the doctrine false; but the old saying of Vox populi, vox Dei, as every philosopher knows, cannot be trusted in science. Reason tells me, that if numerous gradations from a simple and imperfect eye to one complex and perfect can be shown to exist, each grade being useful to its possessor, as is certainly the case; if further, the eye ever varies and the variations be inherited, as is likewise certainly the case and if such variations should be useful to any animal under changing conditions of life, then the difficulty of believing that a perfect and complex eye could be formed by natural selection, though insuperable by our imagination, should not be considered as subversive of the theory. How a nerve comes to be sensitive to light, hardly concerns us more than how life itself originated; but I may remark that, as some of the lowest organisms, in which nerves cannot be detected, are capable of perceiving light, it does not seem impossible that certain sensitive elements in their sarcode should become aggregated and developed into nerves, endowed with this special sensibility.

In searching for the gradations through which an orgain in any species has been perfected, we ought to look exclusively to its lineal progenitors; but this is scarcely ever possible, and we are forced to look to other species and genera of the same group, that is to the collateral descendants from the same parent-form, in order to see what gradations are possible, and for the chance of some gradations having been transmitted in an unaltered or little altered condition. But the state of the same organ in distinct classes may incidentally throw light on the steps by which it has been perfected.

The simplest organ which can be called an eye consists of an optic nerve, surrounded by pigment-cells, and covered by translucent skin, but without any lens or other refractive body. We may, however, according to M. Jourdain, descend even a step lower and find aggregates of pigment-cells, apparently serving as organs of vision, without any nerves, and resting merely on sarcodic tissue. Eyes of the above simple nature are not capable of distinct vision, and serve only to distinguish light from darkness. In certain star-fishes, small depressions in the layer of pigment which surrounds the nerve are filled, as described by the author just quoted, with transparent gelatinous matter, projecting with a convex surface, like the cornea in the higher animals. He suggests that this serves not to form an image, but only to concentrate the luminous rays and render their perception more easy. In this concentration of the rays we gain the first and by far the most important step towards the formation of a true, picture-forming eye; for we have only to place the naked extremity of the optic nerve, which in some of the lower animals lies deeply buried in the body, and in some near the surface, at the right distance from the concentrating apparatus, and an image will be formed on it.

In the great class of the Articulata, we may start from an optic nerve simply coated with pigment, the latter sometimes forming a sort of pupil, but destitute of a lens or other optical contrivance. With insects it is now known that the numerous facets on the cornea of their great compound eyes form true lenses, and that the cones include curiously modified nervous filaments. But these organs in the Articulata are so much diversified that Muller formerly made three main classes with seven subdivisions, besides a fourth main class of aggregated simple eyes.

When we reflect on these facts, here given much too briefly, with respect to the wide, diversified, and graduated range of structure in the eyes of the lower animals; and when we bear in mind how small the number of all living forms must be in comparison with those which have become extinct, the difficulty ceases to be very great in believing that natural selection may have converted the simple apparatus of an optic nerve, coated with pigment and invested by transparent membrane, into an optical instrument as perfect as is possessed by any member of the articulate class.

He who will go thus far, ought not to hesitate to go one step further, if he finds on finishing this volume that large bodies of facts, otherwise inexplicable, can be explained by the theory of modification through natural selection; he ought to admit that a structure even as perfect as an eagle's eye might thus be formed, although in this case he does not know the transitional states. It has been objected that in order to modify the eye and still preserve it as a perfect instrument, many changes would have to be effected simultaneously, which, it is assumed, could not be done through natural selection; but as I have attempted to show in my work on the variation of domestic animals, it is not necessary to suppose that the modifications were all simultaneous, if they were extremely slight and gradual. Different kinds of modification would, also, serve for the same general purpose: as Mr. Wallace has remarked, "if a lens has too short or too long a focus, it may be amended either by an alteration of curvature, or an alteration of density; if the curvature be irregular, and the rays do not converge to a point, then any increased regularity of curvature will be an improvement. So the contraction of the iris and the muscular movements of the eye are neither of them essential to vision, but only improvements which might have been added and perfected at any stage of the construction of the instrument." Within the highest division of the animal kingdom, namely, the Vertebrata, we can start from an eye so simple, that it consists, as in the lancelet, of a little sack of transparent skin, furnished with a nerve and lined with pigment, but destitute of any other apparatus. In fishes and reptiles, as Owen has remarked, "the range of gradations of dioptric structures is very great." It is a significant fact that even in man, according to the high authority of Virchow, the beautiful crystalline lens is formed in the embryo by an accumulation of epidermic cells, lying in a sack-like fold of the skin; and the vitreous body is formed from embryonic sub-cutaneous tissue. To arrive, however, at a just conclusion regarding the formation of the eye, with all its marvellous yet not absolutely perfect characters, it is indispensable that the reason should conquer the imagination; but I have felt the difficulty far too keenly to be surprised at others hesitating to extend the principle of natural selection to so startling a length.

It is scarcely possible to avoid comparing the eye with a telescope. We know that this instrument has been perfected by the long-continued efforts of the highest human intellects; and we naturally infer that the eye has been formed by a somewhat analogous process. But may not this inference be presumptuous? Have we any right to assume that the Creator works by intellectual powers like those of man? If we must compare the eye to an optical instrument, we ought in imagination to take a thick layer of transparent tissue, with spaces filled with fluid, and with a nerve sensitive to light beneath, and then suppose every part of this layer to be continually changing slowly in density, so as to separate into layers of different densities and thicknesses, placed at different distances from each other, and with the surfaces of each layer slowly changing in form. Further we must suppose that there is a power, represented by natural selection or the survival of the fittest, always intently watching each slight alteration in the transparent layers; and carefully preserving each which, under varied circumstances, in any way or in any degree, tends to produce a distincter image. We must suppose each new state of the instrument to be multiplied by the million; each to be preserved until a better one is produced, and then the old ones to be all destroyed. In living bodies, variation will cause the slight alterations, generation will multiply them almost infinitely, and natural selection will pick out with unerring skill each improvement. Let this process go on for millions of years; and during each year on millions of individuals of many kinds; and may we not believe that a living optical instrument might thus be formed as superior to one of glass, as the works of the Creator are to those of man?

Jupiter and beyond the infinite

Infinity is a very interesting subject, and one that I need to understand better. I'm going to make this post a regular (weekly?) thing. I hope that by going slowly over what infinity is and what we can say about it that I can a) ensure my understanding of it is clear b) interest people who might want to know about it and c) show how mathematical ideas are generated and created. I hope you enjoy this. Feedback, as ever, is appreciated.