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 relevant points to reasonable length.
- UTC=20160622T1709Z/version 1.1.0: Kmo made a string of substantive improvements (remarking on the need to incorporate nuclear-physics assumptions into the mathematical modelling of the solar interior; and anticipating an undergraduate objection to a superficial exposition of Hume; and adding a further example (the Feynman oil slick), which he had this week forgotten, from his recapitulation here of his writing on Mr Greer's circa-2016-02-17 blog; and adding to his discussion of the the Earth mass determination the need to measure gravitational acceleration of falling bodies, as one of the inputs to the eventual mass calculation; and making his ideas about scholastic ossification clearer through a simile or metaphor of terrain-only-two-thirds-covered). He retained the right to make minor, nonsubstantive, improvements over the coming days, without formal documentation in this revision history, as versions 1.1.1, 1.1.2, ... .
- UTC=20160621T0001Z/version 1.0.0: Kmo uploaded base version (and planned to upload in the ensuing four-hour
interval, without formal documentation in this revision history,
nonsubstantive revisions, as versions 1.0.1, 1.0.2, ...) .
We can now see where conservation priorities must lie as we determine, so to speak, how far behind the breached Mannerheim Line our forces shall retreat.
It was strongly argued by John Michael Greer at http://thearchdruidreport.blogspot.com, I think at some point in 2014 or 2015, that a priority in our impending Dark Age must be the conservation of a grasp of scientific method, as opposed to a mere conservation of factual knowledge. To this I assent, with amplification and development.
I have suggested that within the field of method, it is of primary importance to preserve a grasp of those aspects of method that pertain to the enunciation of laws, as opposed to those aspects that pertain to the discovery of manifest contingencies.
We sometimes encounter thin discussions of the "scientific method", in the schema "Make observations, formulate hypotheses, deduce testable predictions, and perform the tests." Many of us will remember sitting through such a sermon in those Places of Scant Learning which are our North American high schools.
The scheme fits some areas of work well enough. It fits double-blind drug trials, as at Servier or SmithGlaxoKline. And (a happier example) it would usefully fit an investigation into the efficacy of mycorrhizal associations - nowadays of agronomic interest as a biosphere-friendly alternative to phosphate fertilizers.
The scheme is, however, of only limited use in guiding the investigation of fundamental principles. The core of the scientific method where it really matters, at the level of principles rather than of individual facts, is hard to capture in a slogan or schema. There is no obvious sense in which Maxwell, in deciding that the line integral, all the way around a non-self-intersecting loop, of magnetic field, had to be sensitive to the rate of time variation in the integrated "Electric Illumination" of the various capping surfaces, was proceeding from "observations". (It was, rather, the correlative principle, that the line integral of electric field is sensitive to the rate of time variation in the integrated "Magnetic Illumination" of the loop's various capping surfaces, that was the well-known observation-backed hypothesis.) The core is, on the other hand, readily demonstrated with examples.
If a slogan or schema is all the same demanded, the following will serve in its bare-bones fashion, given an abundance of corroborating examples: "Mathematics has explanatory power in the physical world."
The very idea of mathematics as possessing explanatory power is somewhat resisted in contemporary culture. People on the Web show some tendency to speak of physical science as the construction of "mathematical models". This is a characterization inviting emendation into a still less flattering phrase - "mere mathematical models".
A Nixon-era USA television programme called "Rowan & Martin's Laugh-In" featured a Black comedian, "Flip Wilson", who at one or more points appeared on the screen in drag. "The devil made me put on this dress," said the loudspeaker voice, in possibly Southern cadences: "Ah said, yew stay away from ME, devil..." A parallel Satanic influence compels me to recount here my little story of mathematical modelling - scant though the value of my anecdote is in illuminating my wider theme, Choice of Tactics for Conserving Appreciation of Scientific Method. Readers concerned to save time can safely skip down as far as the phrase "very nice lady sitting beside me", knowing that they will thereby miss nothing of substance.
The most demanding and frightening, maths course I have taken was not in third year at all, but at second-year level - the University of Toronto 1991-1992 MAT257. MAT257 was a full-year presentation of multivariate real analysis. It culminated, I guess, in a presentation of Green's, Stokes's, and Gauss's theorems as special cases of one single grand theorem involving, I guess with respect to orientable manifolds immersed in real Euclidean spaces of arbitrarily high finite dimension, "differential forms".
This grim rite de passage, in comparison with which rites of adolescent passage in the sub-Saharan interior or in the Papua New Guinea highlands must rank as trifling inconveniences, was in the hands of an expositor of the first rank, the then-young Prof. E. Bierstone. I think professionals will see the gravity of his proceedings as soon as I cite his textbook: it was, alas, Spivak's Calculus on Manifolds.
I propose as a unit for measuring Academic Course Difficulty the "bier", named in my lecturer's honour. MAT257 attained in 1991-1992 a level of exactly 1 bier. Normally, courses are of a difficulty measured in nanobiers, or at worst in tens of millibiers.
Prof. Bierstone, while diligent and exact, was perhaps occasionally a little humourless. At a time at which we were pondering, among other things, the topological concept of an "open covering", Prof. Bierstone found himself having to make one of his infrequent administrative announcements. A competition, he said, was being organized, for such-and-such a venue and date, providing students in the Department the chance to test their skills in "mathematical modelling". We all had the same thought - even middle-aged I, whose hormones can no longer have been raging. Prof. Bierstone asked, in genuine puzzlement, why everyone was smiling or giggling, what everyone was thinking of.
I am in retrospect grateful that I managed to bite my tongue, avoiding the temptation to murmur to the very nice lady sitting beside me, or to say even more loudly - the temptation was there - "open coverings".
Mathematical models, like the observe-hypothesize-deduce-test schema, have their limited place.
In meteorology, it is normal to make numerical models, tiny grid square-or-cube by tiny grid square-or-cube, of the regional or global atmosphere. In such models, the supercomputing cluster - in the case of Canada, I gather the national machine resides in the Montréal suburb of Dorval - steps in some finite-differences way through the tens or hundreds or thousands of coupled differential equations. Such "numerical models" do perhaps in a sense explain something about the weather.
And in a more exalted setting, they do explain something about the inner structure of the Sun. In the solar case, the prof forms a much smaller set of equations, governing spatial variations in temperature and pressure over successively deeper solar layers, and uses her or his computer to work out the temperatures and pressures at great depths from plausible conjectures in nuclear physics, along with a knowledge of the Sun's size, mass, and observable skins - notably, from a knowledge of the Sun's photospheric spectral energy distribution, attainable observationally. - I write here "her or his computer" to mark my suspicion that a lone 1990s Intel 486 DX chip, as opposed to a contemporary Dorval-based Environment Canada supercomputing cluster, would be up to this particular numerical astrophysical task.
These two resorts to explanation through mathematical modelling are rather plebian. There may be some temptation in some quarters to say that this is as good as mathematics ever does get.
A perpetual theme in post-mediaeval philosophy is the recourse to what we might call "Nothing-Buttery".
The supposition of a necessitating causal nexus between events is (it was asserted by 18th-century David Hume) "nothing but" a projection onto the outside world of an inner feeling of confident expectation, this feeling being itself a sort of Pavlovian reflex. - If expounded carefully, the position at any rate escapes the reasonably obvious, undergraduate-level, objection, "Is there a necessitating causal nexus, then, within the mind, necessitating the emergence of the Pavlovian reflex after the projection-prone mind has undergone sufficient observational conditioning?" Even within the realm of the mind, Hume's duly careful expositor will say, putative causal necessities are a mere mythologizing projection.
Mathematics (it was asserted in the 1930s Vienna Circle, contrary to the then-recent adverse experiences of logicians Frege, Russell, and Whitehead) is "nothing but" an edifice of logical necessities.
The thinking familiar to humans is (so the eminent mathematical logician, and wartime codebreaker, Alan Turing (1912-1954) asserted in this 1950 paper, "Computing Machinery and Intelligence") "nothing but" the ability of a physical system, such as a machine, to pass an appropriate simulation test at the hands of human interrogators.
Again, we have in ethics the suggestion that moral duties are "nothing but" arrangements enforced by the positive law of a given society, or alternatively "nothing but" commands residing in the mind of a deity. (A more traditionally Catholic position, opposed to moral Nothing-Buttery in its theological variant, is that if, say, God enjoins some particular form of altruism, then God is enjoining it because it is right. It is not that the thing is right merely because God enjoins it; God is indeed, on the traditional Catholic approach, no more free to say "Evil, be thou my good" than God is free to defy the topology of the Moebius strip. One then proceeds to argue, along lines that are not particularly problematic, that these limitations on God are consistent with divine omnipotence, rightly understood - that omnipotence demands ability to select, as possible feats, only from the domain of feats coherently describable. C.S.Lewis would say that while God "can do anything", there is literally "no such thing" as dissecting the Moebius strip along its median into two separate strips.)
Indeed one might imagine a reductionist, or nothing-but, position in the field of natural essences that I noted here on 2016-06-14. On this possible philosophical position, there are no intrinsically natural ways of carving up the world. To return to my past example: we regard these two maples as belonging to the same "natural kind", taxonomically Acer saccharum, and this third maple as belonging to a different "natural kind", Acer rubrum. But (so the position goes) there is really no right or wrong way to classify things - no deep reason why the second and third trees could not be assigned the same classification, and the first one some different classification. Classifications, it is in a reductionist spirit suggested, are made for reasons of mere practical convenience, as municipal borders might by arbitrary fiat be drawn and redrawn, or currency exchange rates shifted down and up.
Without here delving further into Nothing-Buttery, I will take it that my readers are willing to ponder in a provisional way a philosophical position which opposes some forms of Nothing-Buttery. I invite my readers simply to take this position on provisionally, and to see, in a spirit of open-minded exploration, where it takes us. So let us in particular suppose, provisionally, that mathematics in physical science is not confined to mere "mathematical modelling". If Nothing-Buttery is rejected here, then we can see where our conservationist priorities must lie as our Dark Ages deepen.
A moment's reflection shows how tricky the conservationist task is.
In part, admittedly, inculcating an appreciation of scientific method, on its explanatory-mathematics side, can come from appropriately deep studies of work already done. Here there is little that is tricky. In electromagnetic theory one simply works, at second-year level, from some such author as E.M. Purcell - the clear, inventive expositor who in his earlier life shared with Felix Bloch a Nobel Prize for MRI-related work, and additionally in earlier life detected the 21-centimetre emission in radio astronomy. (It is true that it might prove best to take Purcell's latest, revised, posthumous edition, in which the units are SI rather than c.g.s.. Additionally, it might prove best to use Daniel Fleisch's recent Student's Guide to Maxwell's Equations, or something of this kind, as a lead-in.) And I gather from the pages of Daniel Fleisch and the like that no matter what profs in my early-1990s day may or may not have done at the University of Toronto, it is at any rate eventually normal - if not in third year then either in fourth year or in grad school - to work from J.D. Jackson's Classical Electrodynamics.
There will be lots of problem sets, of some rather standard character, and of course the prof will insist that his or her pupils battle diligently with the problem sets before imploring classmates or Teaching Assistants for hints.
So what is tricky?
What is tricky is that if we stop here, we have covered, so to speak, just two-thirds of the necessary terrain. With the remaining, mission-critical, third left uncovered, we risk falling into a kind of scholasticism. It would be a scholasticism paralleling, at a superficially impressive level of mathematical erudition, the mediaeval-scholastic ossification of Aristotelian biology. We have to add, somehow, the spark of something like original investigation, even as we work within the classics.
The Michelson-Morley experiment, as an educational tool, was much discussed over the week following 2016-02-17 at John Michael Greer's http://thearchdruidreport.blogspot.ca/2016/02/retrotopia-back-to-what-worked.html. I felt uncomfortable at the time, as one of Mr Greer's blog commenters. Here, I felt (though I did not say it too bluntly, or did not say it at all) we have a sterile recitation of authorities, worthy of the Sorbonne of, say, 1395, dealing dogmatically with, say, Aristotle's Historia animalium.
I did for my part try to propose for the classroom some lines of experimentation that would stimulate original conceptual work, as Michelson-Morley in my suspicion does not. In addition to commenting at that blog in the days following 2016-02-17 on Michelson-Morley itself, I suggested for Mr Greer's potential Looming-Dark-Ages classroom the following:
- An astronomical determination of the North direction. (Here I stressed that the superficially convenient bright star Polaris has only an accidental connection with the concept of North, being not in fact exactly at the North celestial pole. I suggested some appropriate questions for pupils who might be working this problem in the day rather than at night: how, for instance, do we get a properly planar and properly horizontal surface as we prepare to start measuring those plumb-line shadows?)
- A rerun of Cavendish's torsion-balance determination of the proportionality constant in the law of universal gravitation. (The law is that the strength of gravitational attraction between point masses is inversely proportional to the square of their separation. With the proportionality constant determined, and with the Earth's radius known, and with the gravitational acceleration of falling bodies measured near the Earth's surface, it is possible - I perhaps neglected to point this out on Mr Greer's blog - to achieve the seemingly miraculous, namely a determination of the mass of the Earth, as a number of, e.g., kilograms. The mass proves greater than the mass of an Earth-sized sphere of mere rock, making this result a potential portal into still further science.)
- An inexpensive classroom or home-lab investigation suggested by Richard Feynman (1918-1988; he originated quantum electrodynamics), on which one determines, with graduated cylinder, medicine dropper, glass tray, and square-ruled paper the thickness of an oil slick. I additionally pointed out some helpful lines of tutorial discussion once this Feynman assignment is coupled with the (also inexpensive) Young double-slit experiment, taken as a method of determining the wavelengths of variously coloured lights.
- An investigation, in the concept of Coriolis pseudo-forces, into the discrepancy between cloud directions of movement and weather-vane-determined wind directions.
The fact that I failed can be inferred from the character of the contemporaneous discussion, duly archived at http://thearchdruidreport.blogspot.ca/2016/02/retrotopia-back-to-what-worked.html. Although the various discussants addressed Mr Greer's ideas, or mine, or both, nobody took my various offerings as an occasion for conceptual probing in physics.
In hindsight, I realize now that I should not, in my battle against impending neomediaeval scientific sterility, have strayed far from Einstein and Maxwell. I should have introduced into Mr Greer's envisaged classroom just one thing - properly anchored, as indeed Michelson-Morley is, in the Maxwell-Einstein classics, and yet adequately subtle. This one thing illustrates, by way of example - I see no good way of working except through example - an approach to the delicate side of the conservation task. It shows what is needed if the ever-so-important last third of the terrain, going beyond conventional problem-set assignments, is to be covered.
To avoid making mistakes, I here adopt a cowardly literary expedient, posing Socratic questions instead of making assertions. But the question is in any case to be preferred in tutorial work to the assertion:
First Question: Is it true that (as stated in popularizing, pre-university, treatments of electromagnetism) "a voltage is induced in a conductor moving so as to cut magnetic lines of flux"? A uniform magnetic field of rather large spatial extent, sustained by (say) a pair of Helmholtz coils of rather generous radius, carrying direct currents, runs parallel to the lab floor. A tiny copper ring, its plane perpendicular to the field, moves downward through the field. Is there some electric-current-driving induced electric field all the way around the ring, or not?
Variant on First Question: Can the presence or, as it might perhaps be, the absence of electric current in this situation be explained in terms of Lorentz forces on individual charge carriers? Let the ring be deformed into a tiny rectangle ABCD, with sides AB and CD parallel to the field, and with sides BC and DA consequently perpendicular to the field. What Lorentz forces (consider directions where appropriate) are experienced by the mobile electrons in the copper in each of these four respective sides?
Rider to Variant on First Question: What, in terms of Maxwell's equations, can be said about - to use my 2016-06-07 language - "Magnetic Illumination" of capping surfaces for the non-self-intersecting loop which is the rectangle ABCD? (Is the "Magnetic Illumination" constant or varying?) What follows regarding presence or absence of an induced electric field-all-the-way-around-the-rectangle?
Second Question: What happens when the rectangle ABCD, while kept perpendicular to the immersing magnetic field, has an axle, parallel to the field, attached to some point on one of its sides, with this axle now caused to spin? (The rectangle in this Gedankenexperiment is moving through the field in the manner of a lawnmower half-blade cutting grass stalks.) What should now be said, in a Maxwell context, regarding variation or failure-to-vary of "Magnetic Illumination" aggregated over capping surface? What is now the case regarding electric fields-all-the-way-around-the-rectangle and the electric currents which they may putatively drive?
Third Question: What happens when the whole surface swept out by the axle-driven rectangle ABCD is replaced by a solid copper disk (with the axle consequently at its centre, driving the disk like the disk sander in a workshop)? What, in terms of current-driving electric fields and consequent electric currents, is observed when conducting brushes are applied, as at Figure One in https://en.wikipedia.org/wiki/Faraday_paradox? Do the various closed non-self-intersecting loops that can be drawn on this disk refute the applicable Maxwell law for electric fields?
Here we reach, through just a short line of questioning, depths which I have not yet mastered, and yet which I and all other students of electromagnetics (even humble radiotelegraph designers in the upcoming Dark Ages, for whose ultimate benefit I hope in future to be writing) will have on pain of disgrace to master.
I bet full mastery will involve close attention to questions regarding choice of reference frame - rest frame of lab? or, rather, rest frame of such-and-such a path segment in the closed non-self-intersecting loop?
I bet, further, that full mastery will involve deploying mathematics of explanatory power, with tensors as opposed to mere vectors, in a duly Einsteinean Special-Relativity setting. Some light is almost certainly shed by the deep Bell Labs thinker (he really is, in real life, called "Dr John Denker") who has published at https://www.av8n.com/physics/faraday-puzzle.htm. But I dare not study his essay before I have done my own "Faraday's Paradox" or "Faraday's Puzzle" thinking.
Finally, I bet, or rather I already know, that full treatment of this "Faraday Paradox" puzzle will involve not merely maths, but supporting lab work, at any rate to confirm the correctness of the various mathematical deductions. To my annoyance, I fail to retreive today, either from my generally well organized Scotland Yard notes or via Google, the relevant Ottawa-based home experimenter in this field. But the abundance of other laboratory material is today suggested by, e.g., a quick Google Images search under the string homopolar generator experiments.
[Coming next week, in the upload of UTC=20160628T0001Z/20160628T0401Z: A discussion of some nuts-and-bolts in conservation of scientific-method knowledge, quite likely with reference to the Vatican Observatory Research Group and other aspects of Catholic scientific life. It is rather likely that next week's upload will conclude this essay. In a separate essay, at some later point, I shall have to discuss the practicalities of work at my own desk, involving such things as the erasable coloured pencils and the just-cited - and just-now-disappointing - "Scotland Yard" files.]