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The Importance of School Mathematics
Guardian columnist Simon Jenkins recently claimed that what education (needs is less maths and more lessons in English grammar. Jenkins has form in this area complaining about our fixation on maths. I am sure you have heard others saying “Who needs to solve quadratic equations?” or “When will I ever use trigonometry?”.
I have nothing against the teaching of English: it should be taught and taught well, but civilisation would not, in fact, grind to a halt if schools produced students who were unable to comply with rules defined by long-dead grammarians, who thought that the rules of English should be more like those in Latin. Furthermore, as psycholinguists such as Stephen Pinker have convincingly demonstrated, the grammar rules one sees in textbooks often have little to do with the way real people construct and interpret the language - and no one has ever prevented the language evolving. Grammatically correct English is undeniably useful, as all communications conventions are useful, but not by any means essential: I am rarely in doubt about the intended meaning of those who choose to speak in a different way. It is, however, all too true that, as Bernard Shaw said, "It is impossible for an Englishman to open his mouth without making some other Englishman hate or despise him."
Our civilisation would, however, be in serious trouble if we failed to deliver sufficient students with an adequate knowledge of mathematics.
Simon Jenkins no doubt typed his articles using a computer and a word processor programme; he also probably submitted his material to the editor via a computer network. After which it had further electronic handling ultimately becoming part of an electronic download to the printing machinery. If he purchases groceries in a supermarket, passing through the tills automatically generates restocking orders, which pass back through just-in-time supply networks, dependent on sophisticated mathematical algorithms that optimise the process to move the maximum amount of goods with the minimum of effort.
If he buys goods from Amazon, the orders were passed to robotic warehouses operated by artificial intelligence which then handed his order to delivery drivers following optimally efficient routes calculated by mathematical algorithms. He may very well have used his mobile phone which itself is a virtuoso exercise in modern physics (relying on mathematics) while the process by which one phone is connected to another depends on sophisticated routing algorithms providing seamless communication even while both parties may be in motion, being passed from one 4G base station to another.
None of this would be possible without computer science - a branch of applied mathematics and physics. Without our mathematical infrastructure the shops would empty and we would quickly starve.
The planes you fly in, the cars you drive, the bridges you cross were designed by engineers who use mathematics as an everyday tool, and apply physical science knowledge that itself could not be acquired and applied without sophisticated use of high-level mathematics. Even the genetic sequencing and RNA synthesis that has figured so prominently in the fight against COVID are only feasible because we can now use complex robots supported by mathematical algorithms.
Our civilisation runs on mathematics, and usually it runs so well that most of us are hardly aware of the complex machinery ticking away under the surface.
That, you may well argue, is all very well for those will make and maintain this infrastructure. Why do the rest need to treat it as other than incomprehensible magic?
All this would, indeed, start to fall apart unless schools produce significant number of students who are are equipped to make further progress in science and engineering. You absolutely cannot do without the few thousand students who graduate each year in STEM subjects, a number already too few to meet the employment demand. We certainly cannot afford to further restrict the potential population from which these professionals will eventually emerge.
What about those who have ambitions to become leaders in industry or politics? Can they be trusted to make well informed decisions if they have little insight into the way the world actually functions? Do they really appreciate the potential for massive disruption if the seamless operation of our highly interconnected society suddenly has to cope with additional frictions? Recent events should give us pause for thought.
We all need some mathematical insight in order to make well-informed decisions. Where do we invest our money? Should we get vaccinated? Is an amniocentesis test during pregnancy likely to inform or mislead? Is that politician asking for our vote simply fabricating his statistical claims? (Do they get away with it so often because so few political commentators have the mathematical confidence to challenge them? Do we have too many politicians and commentators who have no feel for numbers?)
The World is now facing a Climate Crisis that will have profound implications for our children. Crossed fingers and delayed decisions are no longer a sufficiently responsible approach. The numbers told us what was happening and what actions were needed several decades ago: you and I preferred not to look at those numbers and allowed our political leaders to prevaricate. Those numbers now need to finally add up and we need to understand enough to hold the leaders to account for any deficits.
It bothers me that so many are blind to the cultural importance of mathematics. The Greek’s invention of logic and proof was a truly momentous departure in human thought, with profound influence on everything that followed, and in truth it was of far greater moment than their undoubted achievements in literature and art. (My website, however, is of course predicated on an assumption that art and maths are connected.) Those early mathematical discoveries are still as valid today as they were two and half thousand years ago, and the calculus of Leibniz and Newton will still be taught two thousand years from now, and probably long after other aspects of our culture have faded to obscurity. Those who do not have some appreciation of mathematical thinking are as culturally impoverished as those who have never read great literature or seen great art.
Finally, Mr Jenkins, when applying the rules of a language grammar, you are in fact exercising a mathematical algorithm.
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- Category: Maths
Where is the Feeling?
Some time ago (Feb 2017) I gave a talk on "Artful Computing" to the Cheltenham branch of the British Computer Society (of which I am a member). I was listened to very respectfully, but in the question and answer session one of the audience members suggested that my digital art did not appeal to him because it did not express feeling.
I understand where he is coming from, but I think that it is an unnecessarily narrow view of what constitutes valid or worthwhile forms of art.
The emphasis in the modern art world on "self-expression" is after all relatively recent. Artists in the past have had many motivations underlying their work: they have been required to provide decorative surfaces, tell stories, uplift the viewer to a higher moral plane, provide titillation, illustrate ideas, describe the world as it is..... and so on and so on. Of course, other kinds of media now do many of these jobs often more cheaply and perhaps even better than a traditional visual artist, so what is there left for an artist to do? Possibly the only unique thing an artist can now offer is his particular view of the world and what he feels about it - and perhaps what he can make us feel about it.
In this respect it is of course very difficult to be original, and find something to say and feel that has not been done before Unfortunately, it sometimes feels as though nothing less that Alexander Pope's "What oft was thought, but ne'er so well express'd" will make the work of art worth the considerable effort.
Perhaps I am less ambitious, or perhaps I have a wider view. I think that it is OK to produce visual works that intrigue the intellect, without necessarily wringing the heart-strings. I do not think that I am out of step with much of art history. We may now admire the Impressionists for the "feeling" in their paintings, but to a large extent they just believed that they were proving a better way of describing what their eyes experienced. Movements in abstract art in the early 20th Century were not at all concerned with feeling: they sought a language of art based on "meaning free" symbols.
All I claim for the work that I do is that there were abstract ideas that interested me, leading to a visually arresting result, and that it may just possible interest you as well.
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- Category: Principles and Philosophy
Video from Processing
FFMPEG
The essential point is that one can use an open source application called ffmpeg to stitch together a sequence of JPEG images into a video clip. ffmpeg and the libraries that support it are widely used in other open source video applications such as VLC media player - but here I use it as a stand-along application. it is available on all the same platforms for which Processing can be installed. (I have used it on both 32-bint Intel/Window 10 and iMac/OS X hosts.)
Nothing, of course, could be easier than making Processing write a series of JPEG images with file names containing sequentially increasing numerical parts, for example, using saveFrame() each time draw() is called, and ffmpeg is then easily configured to read such a file sequence. The main issue therefore is programming the generation of the desired image sequence.
The Practical Problem
it would certainly be possible to program a loop inside the Processing program that iterated through an algorithmically defined variation of the parameters controlling the image generation. I did it - and found that it became very time consuming to progressively adjust the algorithm to get exactly the timing and sequence of transformations the appealed above all the others. This is because a great deal of computation is required to produce each frame and if one has to wait an hour to see the effect of each adjustment the whole process becomes exceedingly tedious.
The Solution
Fortunately, in order to be able to reproduce successful static transformations (perhaps in higher resolution) I had developed a method of saving image transformation parameters (using the saveTable operation) to a comma-separated-values file. Furthermore, in order to explore the space of transformations I had designed in some interactive features, such that the movements of the mouse and some keyboard events could be used to progressively adjust the transformation. A good deal of experimentation is involved choosing values of the configurable parameters (the vector p) that lead to "interesting" images - and in my programs I use the mouse position and key-presses to allow interactive exploration of the image space. I save only the images that appeal. Note that at this point I am producing only static images (for example, those illustrating the p3m wallpaper page and similar pages for other symmetry experiments).
Since successive image saves write new lines to the same CSV file, at the end of an exploratory session, in which I have saved perhaps ten or so "key frames" that appeal, I have a table with ten lines listing all the control parameters. I can now use this list to generate a much longer list of specifications for perhaps several thousand frames. The process is therefore:
- Whenever I save an image (assumed to be "interesting") from my static exploration program, I also save a comma-separated-values (CSV) file containing the parameter vector associated with the image (using the standard library saveTable() method in which the parameter vector appears as a table row).
- I have written an off-line interpolation script which takes a list of such files, each one associated with a time-value (denoting seconds from the beginning of the clip) at which that key-frame should be positioned in the video clip. (I use simple linear interpolation - though there are arguments in favour of using more complex algorithms - such as spline interpolation - that produce smoother variations in parameter values. An experiment for the future.)
- The output from my script is another CSV in the same format as that originally written for my static images, but this time containing many lines of parameter vectors representing the transformation between the key frames. Assuming that 25 frames a second are required in the final video, the interpolation script generates as many intermediate rows (i.e. parameter vectors) as are required to sit between the time-indexed key-frames.
- My Processing image transformation program can then be started in a "replay" mode, where the CSV interpolated parameter file is read (using the loadTable() method) and then each time draw() is called, successive rows are used to define the parameter vector which control each successive image transformation, which is then saved in the normal way using saveFrame().
It now becomes rather easy to adjust the exact timing in the final video between these key frames (perhaps to synchronise with music - experiments in progress!). Although it is not so easy to get around the need to regenerate thousands of images after adjustments, this is an entirely automatic process with very predicable results.
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- Category: Generative Art Techniques
Can Everyone Learn to Program?
It certainly appears to be the case that some people have much more aptitude than others for music or art (or even maths). I could be wrong about this, because there are psychologists who insist on the important of "nurture" as against "nature" - that all we need is the right education, training and encouragement to excel in difficult skills.
There are, of course, many psychologists who take a opposite view, and emphasise the importance of innate differences. They do not deny the importance of nurture, but nurture has to build on the foundations it is given at birth. (The birth of my own children quickly made me suspect that they were far from being blank slates. My wife and I certainly hoped that we would have a strong influence on their development, but it has always been unclear which of our actions had the intended effect - or indeed any effect at all.) Few people have any problem with the idea that elite athletes have to be gifted with the right genetics - and then do lots and lots of training to realise their potential. It would not surprise me if some of our intellectual functions had a significant genetic component (but would also require the right environment to realise their potential).
I do nevertheless hold that many obvious differences in performance are just due to educational advantage and disadvantage, and I also hold that it is morally correct to look for and attempt to develop unrealised potential as far as possible. I do not deny that there may be real differences in ability, but my own history of growing up in a northern working class environment means that I fully appreciate the deeply embedded biasses that probably prevented many of my contemporaries from achieving their potentials. I suspect that I have little innate musical ability - but I never had the chance of music lessons from an early age. Would it have made any difference? Probably not, but who knows?
I cannot see any reason why the average person (who manages to do quite complicated things like driving a car, or navigating city streets) should not, with reasonably instruction and sufficient motivation, be able to attain a reasonable degree of competence in writing computer programs - perhaps the scale and complexity of programs found on this site. Just as with learning a musical instrument, however, regular practice is required, and professional levels of performance are only attainable after professional training and experience.
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- Category: Programming
Programme or Program?
American spelling and usage differs from British conventions.
In the UK the spelling "programme" is more frequently employed when referring to a TV Programme, or a Programme of Work (on a project for example). UK software engineers have, however, generally adopted the US spelling when they are discussing the programs that they work with. (There is so much material coming from the US in this area that it would be hard to do anything else.)
You may, however, notice that unless you specifically load a UK English dictionary into your work processor, it will try to correct programme to program in all contexts. I have little doubt that over time the US spelling will eventually start to predominate whatever pedantic guardians of English say to the contrary. Do not take it to heart. Learn to live with either usage.
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- Category: Programming
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