Quality assessment:
On the 5-point scale current in Estonia, and surely in nearby nations, and familiar to observers of the academic arrangements of the late, unlamented, Union of Soviet Socialist Republics (applying the easy and lax standards Kmo deploys in his grubby imaginary "Aleksandr Stepanovitsh Popovi nimeline sangarliku raadio instituut" (the "Alexandr Stepanovitch Popov Institute of Heroic Radio") and his grubby imaginary "Nikolai Ivanovitsh Lobatshevski nimeline sotsalitsliku matemaatika instituut" (the "Nicolai Ivanovich Lobachevsky Institute of Socialist Mathematics") - where, on the lax and easy grading philosophy of the twin Institutes, 1/5 is "epic fail", 2/5 is "failure not so disastrous as to be epic", 3'5 is "mediocre pass", 4.5 is "good", and 5/5 is "excellent"): 4/5. Justification: There was enough time to develop the requisite points to some reasonable length.
Revision history:
- UTC=20160609T1521Z/version1 1.3.0: Kmo repaired deficiencies in his earlier discussion of the trigronometry of the capping surface, and related points (indicating within the text, to the extent necessary, what changes he had made, and why he had made them). - He retained the right to make further tiny cosmetic tweaks, over the coming week, as here-undocumented uploads 1.3.1, 1.3.2, 1.3.3, ... .
- UTC=20160607T1455Z.version 1.2.0: Kmo added some minor details on the smooth loop-capping surface; and added some potentially useful, less minor, detail on effective Illumination (with remarks on cosine of angle between impinging field and normal-to-surface; he had previously used only sine of angle between impinging field and surface); and added some potentially useful, less minor, detail on radio waves. - He retained the right to make further tiny cosmetic tweaks, over the entire coming week, as here-undocumented uploads 1.2.1, 1.2.2, 1.2.3, ... .
- UTC=20160607T0237Z/version 1.1.0: Kmo corrected what seemed to be a rather bad error, or potential error, in his description of the magnetic field induced by a changing electric field. We do not want to imagine tiny magnets in the tubing being urged into motion around the tubing. Instead, we want to imagine simply a set of magnets being torqued, so that the sequence of magnets traces out a magnetic field running around the loop. - Kmo retained the right to make further tiny cosmetic tweaks, over coming hours and over the entire coming week, as here-undocumented uploads 1.1.1, 1.1.2, 1.1.3, ... .
- UTC=20160607T0001Z/version 1.0.0: Kmo uploaded a moderately polished base version, while reserving the right to make tiny cosmetic tweaks over the ensuing four or so hours, as here-undocumented uploads 1.0.1, 1.0.2, 1.0.3, ... .
[CAUTION: A bug in the blogger software has shown a propensity to insert inappropriate whitespace at some late points in some of my posted essays. If a screen seems to end in empty space, keep scrolling down. The end of the posting is not reached until the usual blogger "Posted by Toomas (Tom) Karmo at" appears.]
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Cantor's is one recent revolution. Another was sparked in geometry through the separate, but synergistic, researches of Nikolai Ivanovitch Lobachevsky (1792-1856) and Georg Friedrich Bernhard Riemann (1826-1866). A third was sparked in mathematical logic (created, as a development itself revolutionary, by Gottlob Frege (1848-1925)) through unavoidable-incompleteness-of-axiomatizations results from Alan Turing (1912-1954), Alonzo Church (1903-1995), and Kurt Gödel (1906-1978). Taken as a whole, the record exhibits mathematics as liable to unpredictable upheavals, occurring independently of technology.
Indeed the experience of a mathematical revolution goes back to classical antiquity.
At the dawn of Greek mathematics, research was in the hands of the "Pythagoreans", who had religious views regarding the cosmic significance of ratios. Their views seemed confirmed by the observation that vibrating harp strings in simple ratios of lengths sound out pleasant musical intervals. (If, for instance, a string of length k sounds one particular note, then a string of length 2k sounds a note exactly one octave lower.)
The Pythagoreans were surprised to find, their encouraging musical investigations notwithstanding, that not all pairs of lengths can be compared as ratios. In particular, they were surprised to discover a proof that there is no ratio p:q (as we would put it, no "pair of positive integers p, q") such that the ratio p:q exactly equals the ratio of the length of a diagonal of a square to the length of a side in the same square.
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Mathematical insight, then, is subject to revolutions. I have already briefly suggested that it is mathematical insight, rather than technology, that is the fundamental driver of physical science. Although we cannot discount technology, we must be on guard against a tendency to overrate it. The tendency is of course specially tempting in our own age, when we in the world's (temporarily?) more affluent countries find ourselves (temporarily?) awash in a tide of Web servers and motor vehicles.
I now develop this suggestion further.
Productive though the Greek mathematicians were (and brave though they were to face the Pythagorean revolution at almost the outset of their mathematical culture), they did not - the lone figure of Archimedes aside - progress strongly in mathematical physics.
One problem was their failure to develop laboratory disciplines with the workshop technology available to them. Had they thought of it, they could have started taking measurements of distances, durations, speeds, accelerations, masses, and forces with things available to them - rulers, swinging plumb lines, protractors, and balances. These tools must have been developed to a high level, in an engineering culture which could even attempt to compensate tall temple columns against an optical illusion-of-curvature.
At a more fundamental level, the Greek mathematicians, from the Pythagoreans at the perplexing dawn right out to the Byzantines at sunset, were handicapped by a failure to develop much algebra.
I gather that simple pairs of simultaneous linear equations were solved, as in the following puzzle (discussed at http://mathworld.wolfram.com/DiophantussRiddle.html): Diophantus's youth lasted for one-sixth of his life. After a further one-twelfth of his life, he grew a beard. After a further one-seventh of his life, he married. He had a son five years later. The son lived exactly half as long as Diophantus himself, and Diophantus died four years after his son's death. How long did the son live, and how long did Diophantus live?
And I gather, without knowing details, that in Roman-epoch Alexandria, at the hands of a worker actually called Diophantus, there was some study of polynomial equations, with integer solutions sought.
But the very fact that the spotlight is put on the special case of integer solutions points to a failure of imagination. It is a far step from doing algebra with integers to grasping the full algebraic structure, as an ordered "field" with a Dedekind completeness property, of the real number line.
Most crucially, the Greeks failed to develop the idea of coordinate systems as algebraizations of geometry.
Perhaps the most fundamental of their deficiencies in algebra, and so in imagination, was their failure to investigate zero. It is as simple as Cantor: we must not only be able to count the Canadian provincial capitals; we must additionally be able to count the Belgian dragons.
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With the concept of zero to hand (I suggest, admittedly without arguing it rigorously here), it becomes easier to build - to develop, next, the concept of a number line, supplementing zero and the positive integers with the negative integers; and then to identify the Pythagorean "ratios", or "rationals", with points on that line; and then to add a concept of "irrationals" (including the troublesome quantities revealed to the Greeks by their so-early Pythagorean revolution) as points also on that line. These various constructions were worked out for the most part not in the Graeco-Roman-Byzantine world, but in India, in Islam, and in post-mediaeval Europe.
Once the constructions are in place, it at last becomes possible to amplify an old Greek method called "exhaustion", and I think used by the Greeks for the computation of the areas under some curves and the volumes under some curving surfaces. Upon due pondering on "exhaustion", we have integral calculus. I like to think of this as the "Study of Accumulations". With a bit of further work, we have differential calculus, which I like to think of as the "Study of Rates". And we additionally have a bridge between the two - if one proposition is at the heart of physics and engineering, it is this proposition - the "Fundamental Theorem of Calculus", showing how to compute Accumulations from given Rates. We thereupon find ourselves able to solve, in just a few efficient lines of writing, anchored in rigorous reasoning and yielding an exact final answer, problems like the Nasty Spigot.
The Nasty Spigot is a problem in Accumulations. It is the one and only half-sensible problem that I have so far succeeded in seeing in bed, while asleep and dreaming. I do not know if I will ever learn to dream in a properly productive way.
The Spigot is Nasty because it has, perversely, a screw thread of smoothly increasing pitch. An ordinary screw thread, for instance from the home-hardware retailer "Canadian Tire", has a thread of constant pitch. (In the familiar Canadian Tire situation, in other words, the thread has some unvarying number n of turns in its thread for every centimetre of shaft. Inspect the first centimetre of shaft in that long carriage bolt, and you will find n turns. Inspect the third centimetre, and you will find the same. Inspect the eighth centimetre, and you will again find n nurns. So regular, so prosaic, so unwearyingly Canadian Tire.) This more exotic, smoothly and continuously broadening pitch (it might be called "nonlinear") is configured in such a way that as the handle is steadily turned, the screw advances at a rate by no means steady. From Canadian Tire, one might expect, as it were, a steady 0.00001-metres-of-advance-for-each-and-every-degree-of-rotation. With this more exotic screw, by contrast, the advance becomes greater and greater for each successive degree, even for each successive tenth-of-a-degree, even for each successive hundredth-of-a-degree, ... .
At one second past noon - just before someone initiates an unvarying, so-many-degrees-in-each-and-every-second, rotation of the handle, the Nasty Spigot's instantaneous flow is a mere litre per hour. At two seconds past noon, the instantaneous flow rate is four litres per hour. At three seconds past noon (for at this point on the treacherous screw the pitch is already decidedly more coarse than when the rotation began), the flow rate is nine litres per hour. At ten seconds past noon, the flow rate is one hundred litres per hour - by now no negligible trickle, but an insistent stream.
The Problem of the Nasty Spigot is to calculate the accumulated flow (how many litres, total, have been sent down into the storm sewer?) from the period which starts at 12:00:01 and ends at 12:00:31.
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I will develop a further example to bring out the position of mathematical advances, as undergirding advances in physical science.
Before Einstein formulated Special Relativity from a consideration of moving charged bodies, it was necessary to have a correct mathematical representation of two things: on the one hand, of the electric field associated with a charged body (such as the cloth-stroked glass rod, or the hair-stroked plastic comb), and on the other hand the "magnetic fields" surrounding those charges which, in a given choice of reference frame, happen to be in motion. (It is, it should perhaps be added, hard to avoid being in the presence of charges which in one's chosen frame of reference happen to be in motion: if there are no electric currents at the macroscopic level, there may at any rate be things happening rapidly at the atomic level, as with the electrons in their various orbitals within the refrigerator magnet - or happening, at any rate, in some perhaps conceptually fuzzy sense, to be explored in quantum mechanics. In quantum mechanics, it is not quite right to say that electrons literally whiz around nuclei, like planets orbiting their stars. Here I begin to be out of my depth, having achieved the appalling grade of C- in 1990s third-year Quantum Mechanics, and not yet having had time to return to more than the first-year or second-year aspects of the subject.)
This mathematical representation was achieved by Scottish worker James Clerk Maxwell (1831-1879). Maxwell's analysis, although based on laboratory physics, was itself purely mathematical.
The American physicist Feynman discusses Maxwell in the second volume of his physics Lectures. I am reliably informed that these Lectures were so well regarded in the Soviet physics establishment as to be available even in ordinary bookstores, to ordinary pupils, in translation, in even a remote part of the erstwhile Union. So here we have an authority of an altogether different order from the National Geographic and the BBC. Feyman, as a Nobel laureate, says that Maxwell's work will be remembered and appreciated in the remote future, when such things as the USA Civil War are forgotten.
I must now substantiate, in my own modest way, Prof. Feynman's favourable assessment.
Although Maxwell's electromagnetism work as a whole illustrates the special role of mathematics, one particular point in it - the point at which he introduces the idea of a time-varying electric field as pairing up with an "induced" magnetic field - is of special vividness. I will have to develop this particular point only after expending many paragraphs on unavoidable preliminaries. The preliminaries, however, while less deep than our destination, are already in their own terms interesting.
Take a closed, non-self-intersecting, loop, such as can be formed with wire or glass tubing. The loop could be a mere "plane curve", for example a circle or a non-circular ellipse. It could, alternatively, be a "space curve which is not a plane curve" - i.e., a closed curve not confined to a plane, and so preventing the wire or the glass from lying fully flat when the loop is placed onto a flat table.
Take this closed loop as the boundary of some imaginary "capping surface", of some finite area, in space.
If the loop is a mere circle, then the chosen "capping surface" could be that bounded planar surface which is a disk. But it could, even in the case where the loop is so politely behaved as to be a circle, be something less politely behaved - for instance, a surface of constant nonzero concavity (a hemisphere, or some part of a sphere smaller than a hemisphere, or some part of a sphere larger than a hemisphere). Again, it could be an almost-disk deformed in three local regions into bumps and in seventeen other local regions into little dimples, and in five other local regions into five cunning combinations of short ridges and short valleys.
One rather special case is the "capping surface" of minimal area, obtained by stretching a soap film across the loop.
In versions of this essay up to and including 1.2.0, 1.2.1, 1.2.2, ... I tried to keep things easily visualized by imposing a requirement not required by the mathematics - namely, that the capping surface be unamenable to multiple piercings. By this I meant that if you were to take that particular capping surface which is of minimum area - the rather special capping surface obtained by putting a soap film across the loop - and if you were to stick lots and lots of toothpicks out from the film, each toothpick being locally perpendicular to the film at the base of the toothpick, then no such toothpick would pierce the the actual, possibly not-of-minimum-area, capping surface more than in one single point. On more mature consideration, however, it seems to me best not to impose this requirement. Admittedly, we do now get into tedious formal questions, as when we say, "Suppose the capping surface to be shocking outré - being shaped like the skin of a sausage, fastened primly around the loop, but with the sausage remarkably long in proportion to its prim diameter, and spiralling fifty times." But it now seems to me best to make the discussion duly general.
One rather special case is the "capping surface" of minimal area, obtained by stretching a soap film across the loop.
In versions of this essay up to and including 1.2.0, 1.2.1, 1.2.2, ... I tried to keep things easily visualized by imposing a requirement not required by the mathematics - namely, that the capping surface be unamenable to multiple piercings. By this I meant that if you were to take that particular capping surface which is of minimum area - the rather special capping surface obtained by putting a soap film across the loop - and if you were to stick lots and lots of toothpicks out from the film, each toothpick being locally perpendicular to the film at the base of the toothpick, then no such toothpick would pierce the the actual, possibly not-of-minimum-area, capping surface more than in one single point. On more mature consideration, however, it seems to me best not to impose this requirement. Admittedly, we do now get into tedious formal questions, as when we say, "Suppose the capping surface to be shocking outré - being shaped like the skin of a sausage, fastened primly around the loop, but with the sausage remarkably long in proportion to its prim diameter, and spiralling fifty times." But it now seems to me best to make the discussion duly general.
To avoid largely irrelevant complications, of a character which I am in any case not quite able to discuss with mathematical competence, I do assume the capping surface to be smooth, lacking creases or wrinkles. (In properly professional terms, let this be the assumption that throughout the capping surface, all directional derivatives exist - I say in my ignorance, to be very safe indeed, all directional derivatives, even of arbitrarily high orders. So although it is fine, for example, for the imaginary capping surface to have a little local trough in it, somewhere, the bottom of the trough has to be like a U, not like a V.)
We now imagine the capping surface, with all its possible local (smooth) divagations, to be immersed in some possibly non-uniform magnetic field, as when a refrigerator magnet happens to be near the loop.
Suppose for the moment that the magnetic field, although varying in space, is constant in time. This will be the case if, for instance, the field is due to a refrigerator magnet, and the relative positions of the loop and magnet are fixed, as by embedding loop and magnet in Lucite.
I will now write of the "Illumination", by a magnetic field, of a tiny planar surface of some fixed area - say, for instance, of a tiny disk, one nanometre in radius.
Readers of physics will recognize that I am talking about "flux", with the Weber (the product of the Tesla and the square metre; one Tesla equals 10,000 Gauss) as its unit in the Système International d’Unités (its "SI Unit"). But "Illumination" is here more vivid, since this term lets me soon develop an analogy with something familiar in households, namely the orientation in sunlight of a solar photovoltaic panel.
A magnetic field, such as surrounds a refrigerator magnet, has at every point in space a certain strength, and additionally has at every point in space a certain direction. The direction can be ascertained, at least in principle, with a tiny compass needle, mounted in gimbals permitting rotation about pretty much any axis, in analogy with Euler's 1776 sphere (discussed earlier in this section, in the blog posting of 2016-05-31).
If the field is of some particular strength, and is uniform throughout the space in the neighbourhood of the surface, and the surface is planar, and the field is at all points directed at right angles to the surface, we will here say that some particular "Magnetic Illumination" of the surface has been achieved. (The textbooks use instead the term "flux". For purposes of the present exposition, however, "Illumination" is more vivid.)
If the field becomes weaker - this would be the case if, for instance, the magnet got damaged, by getting severely warmed, or alternatively if the magnet got pulled farther away from the loop - the "Magnetic Illumination" of the little surface shall be said to diminish.
If the field remains of constant strength, but the little nanometre-disk surface is tilted, the "Magnetic Illumination" must somehow be said to diminish. But what is the right bit of trigonometry here? One might be tempted to write (and fool that I am, I did in version of this essay up to 1.2.0, 1.2.1, 1.2.2, ... write) "in accordance with the sine of the angle the surface makes with the impinging magnetic field". The folly here lies in the fact that there is no such thing as "the" angle. Consider a plane surface, say a disk, and consider a straight line in space, not in the plane of the disk, passing through the centre of the disk. Different diameters of the disk will make different angles with this line, except in the special case where the line and the disk are perpendicular.
So we really do have to do what all the textbooks do.
Label (in the spirit of all the textbooks) one side of the capping surface as the "inside", the other as the "outside" - say, by imagining one side painted indigo, the other side painted orange. This can always be done - even if the surface, far from being a minimal-area soap film, is the skin of some outré spiral sausage. Always there are sure to be two sides to a finite capping surface. In particular, since we have here a mere finite capping surface - a finite surface bounded by a mere non-intersecting closed curve - the surface cannot possibly be that terrifying single-sided, orange-indigo-paint-defying, thing which is a Moebius strip.
Next, imagine the surface adorned with toothpicks, each of which is locally perpendicular to the surface at its base, and each of which points in the direction of Outside. (The toothpicks are, in the language of all the textbooks, "outward normals".) A flea perched on the point of each toothpick, and looking toward the toothpick base, will see orange, not indigo.
We are now ready to introduce, in conformity with all the textbooks, the right bit of trigonometry. Use not sines, but cosines. Consider, at each point on the capping surface, the angle, in the range from 0 degrees to 180 degrees, between the outward normal and the local direction of magnetic field. If the field is locally in the same direction as the outward normal, then the relevant angle is 0 degrees - an angle whose cosine is 1. If the field is moderately oblique to the outward normal, making an angle of 30 degrees with it, then the local Illumination falls off, by a factor of cos 30 degrees. (This is, to three significant figures, 0.866.) If the field is decidedly oblique to the outward normal, making an angle of 60 degrees with it, the local Illumination falls off still more, by a factor of 0.5. (The cosine of 60 degrees is exactly 0.5). If the field direction is perpendicular to the toothpick, the Illumination at the base of the toothpick falls to zero (reflecting the fact that the cosine of 90 degrees is zero).
This "Magnetic Illumination" is analogous in its geometry to the illumination of a solar panel by sunlight. To get the maximum power output at any one instant, a solar panel must be made perpendicular to the incoming rays - being a little after sunrise at vernal or autumnal equinox, for instance, at any non-equatorial point on Earth where the sun clears the horizon, aligned almost-but-not-quite north-south in its yawing aspect, and being made almost-but-not-quite vertical. As the morning progresses, the yaw needs adjusting away from the north-south direction, and the panel needs additionally to become more horizontal. At exact local noon at equinox on the equator, the panel in fact must for maximum power output be exactly horizontal, when for one magical instant the equinoctial shadows of all exactly vertical poles at the equator vanish. At exact local noon at sunlit places off the equator, the panel has, even at equinox, for maximum output to be tilted a bit off the horizontal - and very far indeed off the horizontal if we are generating solar power in a high global latitude, with even the noon Sun correspondingly low in our local sky.
So we really do have to do what all the textbooks do.
Label (in the spirit of all the textbooks) one side of the capping surface as the "inside", the other as the "outside" - say, by imagining one side painted indigo, the other side painted orange. This can always be done - even if the surface, far from being a minimal-area soap film, is the skin of some outré spiral sausage. Always there are sure to be two sides to a finite capping surface. In particular, since we have here a mere finite capping surface - a finite surface bounded by a mere non-intersecting closed curve - the surface cannot possibly be that terrifying single-sided, orange-indigo-paint-defying, thing which is a Moebius strip.
Next, imagine the surface adorned with toothpicks, each of which is locally perpendicular to the surface at its base, and each of which points in the direction of Outside. (The toothpicks are, in the language of all the textbooks, "outward normals".) A flea perched on the point of each toothpick, and looking toward the toothpick base, will see orange, not indigo.
We are now ready to introduce, in conformity with all the textbooks, the right bit of trigonometry. Use not sines, but cosines. Consider, at each point on the capping surface, the angle, in the range from 0 degrees to 180 degrees, between the outward normal and the local direction of magnetic field. If the field is locally in the same direction as the outward normal, then the relevant angle is 0 degrees - an angle whose cosine is 1. If the field is moderately oblique to the outward normal, making an angle of 30 degrees with it, then the local Illumination falls off, by a factor of cos 30 degrees. (This is, to three significant figures, 0.866.) If the field is decidedly oblique to the outward normal, making an angle of 60 degrees with it, the local Illumination falls off still more, by a factor of 0.5. (The cosine of 60 degrees is exactly 0.5). If the field direction is perpendicular to the toothpick, the Illumination at the base of the toothpick falls to zero (reflecting the fact that the cosine of 90 degrees is zero).
This "Magnetic Illumination" is analogous in its geometry to the illumination of a solar panel by sunlight. To get the maximum power output at any one instant, a solar panel must be made perpendicular to the incoming rays - being a little after sunrise at vernal or autumnal equinox, for instance, at any non-equatorial point on Earth where the sun clears the horizon, aligned almost-but-not-quite north-south in its yawing aspect, and being made almost-but-not-quite vertical. As the morning progresses, the yaw needs adjusting away from the north-south direction, and the panel needs additionally to become more horizontal. At exact local noon at equinox on the equator, the panel in fact must for maximum power output be exactly horizontal, when for one magical instant the equinoctial shadows of all exactly vertical poles at the equator vanish. At exact local noon at sunlit places off the equator, the panel has, even at equinox, for maximum output to be tilted a bit off the horizontal - and very far indeed off the horizontal if we are generating solar power in a high global latitude, with even the noon Sun correspondingly low in our local sky.
As the solar panel is more and more severely tilted away from the optimal orientation (in other words, with its "normal", i.e., its own perpendicular, becoming less and less parallel to the rays), the power output sinks further and further, under the cosine-of-angle law. The output falls to zero when the panel is finally parallel to the sunlight, so that the incoming rays merely skim its surface, with the angle between ray and normal finally a right angle. In terms of a toothpick perpendicular to the panel: we achieve optimal orientation when the toothpick casts no shadow, being parallel to the rays, and our orientation becomes worse and worse as the toothpick shadow lengthens.
The tilting of a flat solar panel, of some given area, more and more severely away from its optimal orientation is equivalent to keeping the panel correctly broadside to the Sun, but reducing its area, in proportion to the cosine of the angle between sunlight and toothpick.
There is one point, admittedly, at which the analogy between sunlit solar panel and magnetically "Illuminated" capping surface fails. In our case, we could have the magnetic field pointing in some places to the indigo-painted side, in other places to the orange-painted side. Consider again the spiralling sausage. If we follow the magnetic field, starting from some one point in space, and continually flying forward in the direction indicated from instant to instant by a gimbal-mounted compass needle, we will trace some curve through space. (If the magnetic field comes from a bar magnet, and we start from a point at which the field is pointing more or less away from one end of the bar, then we will eventually loop round until we find ourselves, in our long flight, returning to the bar, steadily approaching its other end.) It may well be that if our curve is rather gentle, and the encountered capping surface rather outré, then we will in our flight pierce the surface multiple times - we will hit, perhaps, some Indigo first, and exit duly through the Orange (making, perhaps, some pleasantly small angle with the direction in which the local toothpick points), only to find that ahead of us is lies another bit of Indigo, to be pierced in turn with an exit through an Orange landscape. To describe everything tersely, we are forced to distinguish "positive Illumination" (when the magnetic field and toothpick are respectively pointing in directions that make an acute angle; the cosine of an acute angle is positive) and "negative Illumination" (when the magnetic field and toothpick are respectively pointing in directions that make an obtuse angle; the cosine of an obtuse angle is negative).
We now re-introduce (but this time for space) the ideas of Accumulation and Rate, initially introduced (not for space, but for time) with the Nasty Spigot. The whole capping surface in the magnetic-field setup may tilt locally in many different directions. Further, the impinging magnetic field may itself be nonuniform in direction and strength. Nevertheless, we can form an idea of the "strictly local effective Illumination", at any one point on the surface, as a spatial analogue to the temporal idea of the current strictly instantaneous flow rate at the Spigot.
Briefly: consider, at any one point on the surface, smaller and smaller disks - not even nanometres in radius, but now picometres, femtometres, whatever. Extrapolate from these tiny disks, by analogy with the extrapolation we perform when considering the instantaneous flow rate at a spigot.
What do we mean when we say that at, e.g., 12:00:07, the instantaneous flow rate is 49 litres per hour? "Vot meenz?" - as I like to think people saying by the samovar, over glasses of tea-with-raspberry-jam, in my imaginary "Aleksandr Stepanovitch Popov Institute of Heroic Radio", on the scarier side of the Urals. (Or, better still, in my imaginary "Nikolai Ivanovitch Lobachevsky Institute of Socialist Mathematics", across that muddy Siberian street. This is better still, because radio is itself erected on a foundation of maths.)
We mean merely (closely enough for present purposes, where we do not resort to epsilon-delta-definition rigour) that if the tiny aggregate flow over some tiny interval around 12:00:07 were - counterfactually and hypothetically - extrapolated to a whole hour, an accumulated total of 49 litres would pass.
Analogously, then, when we say that the strictly local effective Magnetic Illumination at point P on the capping surface is such-and-such-per-square-metre, we mean (short, admittedly, of full epsilon-delta-definition rigour) that if the tiny aggregate Magnetic Illumination on some tiny flat region around P were, counterfactually and hypothetically, extrapolated to a flat square-metre region, of the same toothpick-measured tilt as we locally have, and under the same conditions of magnetic field strength and direction as we locally have, then the total Magnetic Illumination over that square metre would be such-and-such.
So, to reiterate: we can, by pondering Rates, form a spatial idea of "strictly local effective Illumination", just as we can in the case of the Spigot form a temporal idea of "strictly instantaneous rate-of-flow".
Having formed this idea, we can proceed to a notion of Spatially Accumulated Illumination, over the entire capping surface, analogous to our idea of total temporally accumulated volume-of-water over half a minute under the Nasty Spigot.
Conveniently enough, for a given magnetic field impinging on a given loop, the Spatially Accumulated Illumination over any one finite capping surface is the same as the Spatially Accumulated Illumination over any other. Start with the cap of minimal area, the simple soap-film cap. Proceed to enlarge this cap, by so-to-speak blowing on the film. Enlarge the film into something grotesque, such as the spiralling sausage. Can the Illumination be changed by this manoeuvre? No. (To take one of the two possibilities, can it be increased? No. The capping surface may get bigger and bigger, but as it does so it takes on more and more severe tilts with respect to the locally impinging magnetic field - perhaps even, in some localities, suffering negative Illumination, by locally exposing to the impinging field not its Indigo but to its Orange side. The increase in area is exactly offset by the introduction of more severe tilts, keeping the Spatially Accumulated Illumination constant.)
Conveniently enough, for a given magnetic field impinging on a given loop, the Spatially Accumulated Illumination over any one finite capping surface is the same as the Spatially Accumulated Illumination over any other. Start with the cap of minimal area, the simple soap-film cap. Proceed to enlarge this cap, by so-to-speak blowing on the film. Enlarge the film into something grotesque, such as the spiralling sausage. Can the Illumination be changed by this manoeuvre? No. (To take one of the two possibilities, can it be increased? No. The capping surface may get bigger and bigger, but as it does so it takes on more and more severe tilts with respect to the locally impinging magnetic field - perhaps even, in some localities, suffering negative Illumination, by locally exposing to the impinging field not its Indigo but to its Orange side. The increase in area is exactly offset by the introduction of more severe tilts, keeping the Spatially Accumulated Illumination constant.)
Maxwell asked, in effect (now combining, as we have so far not done, spatial considerations with temporal ones): what happens when the total Spatially Accumulated Magnetic Illumination of the entire capping surface changes as time passes?
Suppose, for instance, that the refrigerator magnet is pulled steadily away from the wire loop, so that the total Spatially Accumulated Magnetic Illumination at 15:44:02 is a little less than the Total Spatially Accumulated Magnetic Illumination at 15:44:01, and with the Total Spatially Accumulated Magnetic Illumination at 15:44:03 in turn a little less than it was at 15:44:02. What happens?
The answer was well known in Maxwell's time, from lab work, ultimately from Faraday's "Induction" experiments. (We have not yet, despite our rather heavy exertions, come to the part of Maxwell that is a distinctive vindication of mathematics. We are still in the part which is a preliminary, interesting also in its own terms.) The answer is something that I kept demonstrating to tour groups at the David Dunlap Observatory one summer, using perhaps 10 or 20 dollars' worth of gear, largely picked up from Active Surplus on Queen Street in Toronto - wire, and a magnet, and a swinging-needle microammeter.
As the total Spatially Accumulated Magnetic Illumination changes over time, a voltage is developed around the loop, like the voltage that can be developed by a battery. This is a so-called "electromotive force" - although it is "field", rather than "force" , which is here the formally correct term. This force - or rather (to be formally correct), this field - will drive any available free-to-move charged particle, such as an electron, around and around the loop.
In copper, there are lots of free-to-move charges. Solid copper is in reality a lattice of positive copper ions penetrated by a gas of ever-so-mobile electrons. (In a typical electrical application such as lab wiring, each electron has, superimposed on its mad to-and-fro thermally driven particle-of-a-gas random dashings, a slow and systematic "drift", from the negative terminal of the battery around to the positive terminal, at a "drift speed" of around one millimetre per second.)
Let the loop, then, be made of copper wire. A current of electrons in this case flows, with each individual electron proceeding, on average, along the loop axis, over and above whatever it may be doing in its random rapid to-and-fro gaseous dashings, at that stately "drift speed". It is this current which is demonstrated by the deflection of the lecture-room microammeter pointer, as the magnet is pulled away from the wire.
A second example may be helpful. If the loop is made of glass tubing, enclosing not electrons but the positively charged ions of some gas previously ionized (as through high heat), then again a current will flow - this time a current of ions, in fact a modest little wind, circulating around and around in the glass, and in a sense opposite to the sense of circulation that would under those same conditions of magnetic change beset electrons in a wire.
For completeness, I may as well add that if the receding refrigerator magnet had been placed in the opposite orientation (North pole where we had in fact placed South, South where we had in fact placed North), then the electron current would be reversed in its sense - as it were, from clockwise-as-viewed-from-magnet to anticlockwise-as-viewed-from-magnet, and that the positive-ion current would likewise be reversed in its sense - as it were, from anticlockwise to clockwise. And I also add that in each of these scenarios, if the magnetic field strength were not decreased but increased, the direction of circulation would again be reversed.
We now, at last, come to the part of Maxwell's work which distinctively vindicates mathematics.
In working out the maths, Maxwell realized that his equations would assume a specially tidy form if he introduced an assumption hitherto untested in lab work. Here, then, was a brave man.
We have so far been imagining magnetic fields, as in the vicinity of a refrigerator magnet. What about electric fields, as in the vicinity of a rubbed plastic comb? Maxwell found his mathematics to become specially tidy once he assumed, in effect, the following: Consider a closed non-self-intersecting loop L, with a smooth capping surface S, as already discussed in connection with magnetism. Consider many tiny magnets positioned within the loop, each free to swing around into any orientation. (Perhaps the loop is made of glass tubing filled with oil, in which freely float tiny gas bladders, each held at neutral buoyancy by the weight of an attached tiny steel magnet.) If the total Spatially Accumulated Electric Illumination changes over time (as when the plastic comb is pulled steadily farther and farther away from the loop), then a "magnet-twisting field" is developed around the loop. (So, in particular, the suspended magnets will feel a torquing force. Let the geomagnetically north-seeking end of each magnet be marked, following the usual convention, "N", and the other end be marked "S". Then each magnet will be urged to swing, like a compass needle. If all other magnetic fields, such as the Earth's own field, are neutralized - we can achieve this in a lab with a nulling pair of direct-current Helmholtz coils, oriented in opposition to the geomagnetic field as it is oriented in our particular place on Earth - then the various magnets around the loop will swing either in such a way that as we move along the loop in some fixed sense of circulation, we encounter first the N end, then the S end, of each successive magnet, or else the various magnets will swing in such a way that on that same peregrination around the loop we encounter in each case first the S end of each given magnet, and then its N end.)
Maxwell introduced, in other words, the following assumption: As changing the Magnetic Illumination upon a capping surface induces an electric field along the surface-bounding loop, so also does changing the Electric Illumination upon such a capping surface induce a magnetic field along the surface-bounding loop.
For completeness, I may as well add that the direction of the induced magnetic field depends on the polarity of the charge creating the Electric Illumination. Rubbery things stroked with fur are negatively charged. Glass stroked with silk is positively charged. If the diminishing Electric Illumination is due to the steady withdrawal of a charged rubbery object, and we find each of the magnets in our peregrination around the loop to be oriented N-then-S, we will find each of the magnets to be oriented S-then-N when the diminishing Electric Illumination is produced instead by the steady withdrawal of a charged glassy object.
I also add, for completeness, that just as in the Magnetic Illumination case, if the electric field is made to strengthen over time, rather than to diminish, the direction of the looping magnet field is in each case reversed - from, so to speak, clockwise to anticlockwise around the loop, or from anticlockwise around the loop to clockwise.
I reiterate and stress that there was no experimental consideration, at the time of Maxwell's writing (his textbook came out in 1873) either in favour of his assumption or against it. The assumption cannot be confirmed except with equipment subtler than the twenty dollars of Active Surplus gear that I used to lug into the David Dunlap Observatory auditorium on public-tour nights, for demonstrating that easy thing which is the mere production of an electric field through the mere varying of a magnetic field. If I correctly recall what I have briefly read, the laboratory confirmation came at some point in the 1920s, and I know not with what delicate and expensive setup.
I also add, for completeness, that just as in the Magnetic Illumination case, if the electric field is made to strengthen over time, rather than to diminish, the direction of the looping magnet field is in each case reversed - from, so to speak, clockwise to anticlockwise around the loop, or from anticlockwise around the loop to clockwise.
I reiterate and stress that there was no experimental consideration, at the time of Maxwell's writing (his textbook came out in 1873) either in favour of his assumption or against it. The assumption cannot be confirmed except with equipment subtler than the twenty dollars of Active Surplus gear that I used to lug into the David Dunlap Observatory auditorium on public-tour nights, for demonstrating that easy thing which is the mere production of an electric field through the mere varying of a magnetic field. If I correctly recall what I have briefly read, the laboratory confirmation came at some point in the 1920s, and I know not with what delicate and expensive setup.
Yet the truth of Maxwell's assumption was suggested, in an informal way, by a prediction entailed by his overall theory. With electric and magnetic fields coupled in the tidily symmetric way Maxwell claimed them to be, an answer was right away available to the seemingly almost philosophical, seemingly almost metaphysical, question "What is light?" If Maxwell's mathematics was correct, light could be identified with a propagating wave disturbance, comprising coupled, rapidly and regularly fluctuating, electric and magnetic fields.
In particular, Maxwell's mathematics correctly entailed the propagation speed of light, as measured in lab work.
Additionally, his mathematics predicted the existence of light-like radiations ("radio waves") from oscillating electric charges. The predicted radiations would be in waves of a length dependent on the frequency of the oscillator, and in typical laboratory cases much longer than the wavelength of light: not, now, 300 nanometres as with borderline-ultraviolet light, or 656.3 nanometres as with the much-studied red light of a low-pressure hydrogen-discharge tube, but to a wavelength conveniently measured in centimetres or metres. The existence of these radiations was confirmed rapidly, not long after his death, in the centimetre (the modern microwave-oven) regime. (In this case, the oscillator has to be driven somewhere in the gigahertz frequencies. If the regime of hundreds of thousands of gigahertz were by some electronics-engineering miracle achieved, the antenna would emit light instead of radio waves.)
It is pleasant to recall that of the many Victorian lecturers demonstrating radio waves to their spellbound audiences was the eventual 1935 David Dunlap Observatory founder, the then young Prof. Clarence Augustus Chant. He did his demonstration, surely with a spark-gap oscillator sending a weak signal from one end of an auditorium to the other, in 1890 or 1895 or 1899 or so - at any rate, I am quite sure, before 1900.
It is pleasant to recall that of the many Victorian lecturers demonstrating radio waves to their spellbound audiences was the eventual 1935 David Dunlap Observatory founder, the then young Prof. Clarence Augustus Chant. He did his demonstration, surely with a spark-gap oscillator sending a weak signal from one end of an auditorium to the other, in 1890 or 1895 or 1899 or so - at any rate, I am quite sure, before 1900.
I cannot leave Maxwell without commenting further, on a subsidiary mathematical point.
I have not read Maxwell's 1873 textbook, and these days few students have. What everyone uses nowadays are not Maxwell's "quaternions". (Quaternions are, one gathers, a tedious fourfold elaboration of the reals, analogous to that twofold elaboration of the reals into a non-ordered, and yet in some suitably extended sense - how on earth do the profs do this? - Dedekind-cut-complete, field which is the complex plane.) Rather, what everyone uses nowadays are "vectors" - further elaborated, indeed, into "tensors" in the superstructure that Einstein puts onto Maxwell.
On the strength of vectors, Maxwell's 24-or-thereabouts equations for electromagnetic theory can be condensed into just four equations. I keep the four emblazoned in white paint on the side of my British-Army-surplus duffel bag, to render mine instantly distinguishable from all probable others at the railway baggage offices. In duffel-bag terms, Maxwell's bold conjecture shows up as a special term in the sum which is the right-hand side of the longest equation. The "closed line integral of the magnetic field" (the left hand side) is none other than the sum of two right-hand-side terms: not just a constant times the "current enclosed by the loop" (this is banal labwork, long antedating Maxwell), but also a constant times the "rate-of-change, with respect to time, of the electrix flux through any finite-area surface smoothly capping the loop" (this is the modern formal expression of Maxwell's experimentally untested mathematical conjecture).
The possibility of condensing Maxwell's 24-or-thereabouts equations into a compact foursome is a discovery of Oliver Heaviside (1850-1925), who also introduced into electromagnetic theory the universally employed terms "impedance" and "inductance", along with other useful terms. Heaviside is also one of the people, or perhaps the person, to whom we owe the now ubiquitous talk of "unit vectors".
Heaviside is, along with Einstein and Maxwell, one of my main personal heroes - not only for his mathematical fortitude, but also for his contributions to the English tradition of social nonconformity. He was in all but his earliest adult years unemployed, and was self-trained in mathematics and physics; he got into trouble with important people (notably with a head, under Queen Vicky, of the Post Office telegraphy-and-telephony service); and in late life he painted his finernails pink (or was it black?), and signed his letters "W.O.R.M.", and replaced the furniture in his house with granite blocks. As it has rightly been said, "There'll always be an England."
The subsidiary mathematical point, at any rate, is that a theory can be presented in many ways, and that what looks prolix when first published can in a radically imaginative, simple, formalism undergo a correspondingly radical simplification.
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Enough has now been said to indicate the irrationality of proceeding, from the admitted realities of the Law of Diminishing Returns, to predicting the demise of physical science itself. I do not know if any authors have actually fallen into that trap, but the danger is to be noted. I have held up Einstein as a votary of Truth. I have stressed the mathematical, as opposed to experimental, character of Einstein's contribution. Parallel points are now seen to hold for Maxwell-Heaviside.
We cannot know what the future of physics holds. But if points as fundamental as Einstein's and Maxwell's have emerged from mathematical work, the same may happen again. The next time around, it may perhaps happen in something still scarier than "the true nature of so-called magnetic forces" (as with Special Relativity) or "the true nature of gravitational forces" (as with General Relativity) or "the true nature of light" (as with Maxwell). The next time round, perhaps (I am guessing, as everyone must guess), there might be some insight into the relationship of quantum mechanics to conscious observers, resolving twentieth-century conceptual controversies, and answering some questions which seem at present to belong more to philosophy or metaphysics than to sober science.
We have at any rate to keep reminding ourselves that there is more to physics than the discovery of yet further subatomic particles, at accelerators of increasing beam power, coupled to supercomputing clusters of increasing aggregate floating-point-operations-per-second, in a research culture whose more familiar paradigms are anchored in the comparatively tame world of engineering.
[To be continued, and with just fifty-percent-probability concluded, in the upload of UTC=20160614T0001Z/20160614T0401Z. That impending upload might possibly contain points regarding the miseries and joys - the concrete, three-colours-of-pencil, practicalities - of private mathematics study. It is pretty much expected that this essay will end (whenever it does finally end, whether on 2016-06-14 or later) with a moral on joy and suffering drawn from Catholic-praxis author Dorothy Day.]
