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"): 3/5. Justification: There was enough time to develop some points to some reasonable length, and yet not enough time to open up the number of points required by real thoroughness.
Revision history:
- UTC=20160531T1723Z/version 1.1.1: Kmo belaboured the proof a little more, adding a remark about mappings. He additionally left open the possibility of making very minor here-undocumented tweaks, in the style "1.1.2", "1.1.3", 1.1.4", in the English of this essay as opposed to its maths, over coming hours, days, weeks, ... .
- UTC=20160531T1713Z/version 1.1.0: Kmo, having originally felt it best not to belabour the proof that the set of sets of positive integers is of a higher order of infinity than that simpler thing which is the set of positive integers, changed his mind, and added a few belabouring sentences.
- UTC=20160531T0001Z/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, ... . Version 1.0.0, although substantively correct, was in many details unpolished. Rrepairs were impending over the one or two hours following 20160531T0001Z.
1. Einstein as a votary of Truth
Because I spend easily half my (painfully not-for-pay) working week at my desk, I have developed several of its features with care.
Over the wooden top I have placed a sheet of bevelled glass, custom-cut for me maybe twenty years ago by some Toronto firm.
On this glass, a row of ten-sided white dice, engraved not with dots but with the numerals 0 through 9, indicates the current year, month, and date, in numerical format. At the time I started writing this present essay, the dice were arranged thus: 2, 0, 1, 6; and then a gap, and then 0, 5; and then a second gap; and then 3, 0.
Over the desk is a wooden hutch, comprising twenty-four pigeonholes, elevated by wooden legs a little less than 0.4 metres high. (I had this built for me upon reading the wonderful 1982 book, Tools of the Mind: Techniques and Methods for Intellectual Work, by sometime Phillips engineer Vladimir Stibic.) Each pigeonhole is capable of taking, uncurled, several hundred letter-sized sheets. Or, more realistically, each is capable of taking many tens of such sheets, in half a dozen filing folders, uncurled.
Under the hutch, a forty-watt fluorescent tube, its colour balanced for greenhouse operations (the ordinary fluorescent white is rather too harsh) provides a strong illumination for the writing surface. While avoiding factory or airport fluorescent-tube harshness, the light is admittedly in its own way a bit odd, like high noon on the planet not of a star like our Sun - our Sun is technically in the temperature class "G2" - but in some cooler temperature class. As I loosely imagine it, there is here, if not the deeply alarming red glow of an "M" star, at any rate light from a rosy, subtly disquieting, "K" star. To balance this, however, I have a small twenty-watt desk lamp, with halogen bulb. The specially hot filament of that high-technology bulb emits a light which is indeed close to the more yellow, and to human eyes familiar, G2 stellar class.
Facing me as I look under the hutch is a big, nearly vertical, notice-board panel of cork, or perhaps of some cork ersatz. This panel is around 1.3 metres wide. It runs all the way up from the desk glass to the lowest tier of pigeonholes.
On the notice board, bathed in the K-star glow, are half a dozen inspirational papers, the majority of which I will describe here.
Toward the right edge my drawing pins anchor a postcard, showing the Tartu Observatory tower from which F.G.W. von Struve laid foundations for the cataloguing of binary stars.
Struve taught at Tartu University, or in its Czarist appellation the Universitas Dorpatiensis, from 1813 until his 1839 move out of Estonia to the nascent Pulkovo observatory. I believe Struve's principal contribution in stellar binaries to be a catalogue entitled Stellarum duplicium et multiplicium mensurae micrometricae, comprising micrometric measurements of 2174 systems over the period from 1824 to 1837.
I gather that the "filar micrometer", now superseded by electronic methods, was tedious: as you fought sleep and cold, possibly perched on a ladder, you had to transfix the infuriating, boiling, puddles of light which are stars in a poor atmosphere under high magnification with thin, sinister lines of spider silk. You had to rotate one or two things until everything looked exactly right amid the shimmering, and then had to read off separation and position angle from graduated scales on or near the micrometer barrel.
Do this often enough for one single, adequately fast-moving, binary system - say, twenty times over fifteen years, if the orbital period is itself ten or twenty or thirty years - and you can after a lot of desk mathematics determine the ratio of the masses of the two stars.
If you can additionally get a distance measurement for your two-star system, and if you have from some reliable worker's Cavendish torsion balance a reliable measurement of the Universal Gravitational Constant, you can get also the individual masses of each of the two stars (in, for instance, kilograms).
The distance measurement you can have from semiannual parallax comparisons - in which Struve was also among the pioneers - coupled with a knowledge of the Sun-earth distance.
It is at this point, with kilogram determinations for each of the two stars in hand, fair to say that your desk has sailed over the boundary from mere Astronomy - silly, fluffy, purely phenomenological "So what is this angle on the celestial sphere?" Astronomy - to the solid domain of Astrophysics.
Struve taught at Tartu University, or in its Czarist appellation the Universitas Dorpatiensis, from 1813 until his 1839 move out of Estonia to the nascent Pulkovo observatory. I believe Struve's principal contribution in stellar binaries to be a catalogue entitled Stellarum duplicium et multiplicium mensurae micrometricae, comprising micrometric measurements of 2174 systems over the period from 1824 to 1837.
I gather that the "filar micrometer", now superseded by electronic methods, was tedious: as you fought sleep and cold, possibly perched on a ladder, you had to transfix the infuriating, boiling, puddles of light which are stars in a poor atmosphere under high magnification with thin, sinister lines of spider silk. You had to rotate one or two things until everything looked exactly right amid the shimmering, and then had to read off separation and position angle from graduated scales on or near the micrometer barrel.
Do this often enough for one single, adequately fast-moving, binary system - say, twenty times over fifteen years, if the orbital period is itself ten or twenty or thirty years - and you can after a lot of desk mathematics determine the ratio of the masses of the two stars.
If you can additionally get a distance measurement for your two-star system, and if you have from some reliable worker's Cavendish torsion balance a reliable measurement of the Universal Gravitational Constant, you can get also the individual masses of each of the two stars (in, for instance, kilograms).
The distance measurement you can have from semiannual parallax comparisons - in which Struve was also among the pioneers - coupled with a knowledge of the Sun-earth distance.
It is at this point, with kilogram determinations for each of the two stars in hand, fair to say that your desk has sailed over the boundary from mere Astronomy - silly, fluffy, purely phenomenological "So what is this angle on the celestial sphere?" Astronomy - to the solid domain of Astrophysics.
To the left of the postcard, my drawing pins anchor a couple of papers from Cambridge University. There is a 20-year old sheet, headed "The Institute: Past and Present" (a visitor's introduction to the Institute of Astronomy, said to be on the Madingley Road: this has, then, to be some modest distance away from the Mathematical Bridge, Trinity Great Quad, and the other celebrated Cambridge sights). Adjoining it is a sheet of similar age and provenance, which has long been admonishing me, as a reminder of what can be achieved by those who work hard:
UNIVERSITY OF CAMBRIDGE
PREPARATION
FOR RESEARCH IN
MATHEMATICS
For over 60 years , the Faculty
of Mathematics has run a one-year
advanced programme called
Part III of the Mathematical Tripos.
This provides graduates with the
further training
and knowledge needed for original research. /.../
Over 100 students are admitted
each year. To join them, request
further information from /.../
At the left edge of the notice board, affixed to a clean letter-sized white sheet, is the heavily yellowed obituary of my 1970 Dalhousie University teacher Prof. Max Edelstein, by his daughter, British Columbia mathematician Leah Keshet ("LIVES LIVED: Michael Edelstein", from the Toronto Globe and Mail of 2003-02-07).
Prof. Keshet has for her part, in an act of extraordinary kindness, sent me two books from her distinguished father's personal library. Those books, however, live not by the desk in my big writing-and-sleeping room, but some metres away, in my compact book-lined "library room".
Prof. Keshet has for her part, in an act of extraordinary kindness, sent me two books from her distinguished father's personal library. Those books, however, live not by the desk in my big writing-and-sleeping room, but some metres away, in my compact book-lined "library room".
Additionally - and this is what I want at present specially to stress - since some point in 2016, there is a sheet with, among other material, a quotation from Albert Einstein. In a letter of 1897, Einstein, then aged about eighteen, wrote as follows:
Die anstrengte geistige Arbeit
und das Anschauen Gottes Natur
sind die Engel, welche mich
versöhnend, stärkend und doch unerbitterlch streng
durch die Wirren dieses Lebens führen werden.
Under this I have also placed - I conveyed this same material to a child for whose intellectual welfare I am in some measure responsible, and to whom I will return in this essay - my translations, into Estonian (which he does not read) and English (which he does read):
Pingeline vaimne töö ja Jumala looduse vaatlemine
on need inglid kes, trööstitavad, julgustandvad,
kuid tagasiastumatult karmid, saavad olema
minu teejuhtideks läbi selle elu kära.
Strenuous intellectual labour
and the contemplation of God's Nature
are the angels which, consoling,
strengthening, and yet implacably severe,
are going to lead me through the tumult of this life.
****
If the life of Mohandas Gandhi is a witness to Justice, then the life of Prof. Albert Einstein is a witness to Truth.
While barely a university graduate, and at that early point by no means a prof, Einstein deduced Special Relativity from conceptual puzzles regarding moving electrically charged bodies.
It is astonishing that none of the profs before him had worked the (glaring) conceptual puzzles thoroughly, but that's how it is. One can see that there must be something deep, somewhere, and I am surprised that high academic authorities before Einstein's day were relaxed and mellow: since a charge moving through the rest frame of the laboratory is embedded in a magnetic field, and since that field is absent in the rest frame of the charge itself - the laboratory whizzes past, and the sole field is electrostatic- the very question whether a magnetic field is present must be relative to choice of frame. In the end, it turns out - Einstein got this clear, along with the length-contraction and time-dilation results for which Special Relativity is more loudly celebrated - that magnetic "forces" are not a fundamental kind of force.
And I rather think, though I have not worked through it or looked it up, that the celebrated null result of the Michelson-Morley experiment, viz., that the speed of light, in contrast with the speeds of bullets and arrows, is independent of source motions and target motions, was worked out a priori by Einstein at this same epoch. Incredibly (so, I think the maths goes, though I think it subject to correction): when you ponder, you find that the speed of light has to be constant if other very basic accepted facts, mere truisms, are to remain true.
If I am right on this, then Michelson-Morley, hailed on television programmes as a key experiment, takes on merely the aspect of a lab confirmation for something proved with pencil and paper on a foundation of lab truisms. (It is still, of course, still necessary to do the Michelson-Morley experiment. But I reiterate that the experiment becomes now a simple precaution against some oversight in what was key, the pencil-and-paper part.)
If I am right, then the Michelson-Morley history must be a vindication of the overriding importance of mathematics - a theme which I will be developing further, later in this essay.
It is astonishing that none of the profs before him had worked the (glaring) conceptual puzzles thoroughly, but that's how it is. One can see that there must be something deep, somewhere, and I am surprised that high academic authorities before Einstein's day were relaxed and mellow: since a charge moving through the rest frame of the laboratory is embedded in a magnetic field, and since that field is absent in the rest frame of the charge itself - the laboratory whizzes past, and the sole field is electrostatic- the very question whether a magnetic field is present must be relative to choice of frame. In the end, it turns out - Einstein got this clear, along with the length-contraction and time-dilation results for which Special Relativity is more loudly celebrated - that magnetic "forces" are not a fundamental kind of force.
And I rather think, though I have not worked through it or looked it up, that the celebrated null result of the Michelson-Morley experiment, viz., that the speed of light, in contrast with the speeds of bullets and arrows, is independent of source motions and target motions, was worked out a priori by Einstein at this same epoch. Incredibly (so, I think the maths goes, though I think it subject to correction): when you ponder, you find that the speed of light has to be constant if other very basic accepted facts, mere truisms, are to remain true.
If I am right on this, then Michelson-Morley, hailed on television programmes as a key experiment, takes on merely the aspect of a lab confirmation for something proved with pencil and paper on a foundation of lab truisms. (It is still, of course, still necessary to do the Michelson-Morley experiment. But I reiterate that the experiment becomes now a simple precaution against some oversight in what was key, the pencil-and-paper part.)
If I am right, then the Michelson-Morley history must be a vindication of the overriding importance of mathematics - a theme which I will be developing further, later in this essay.
In middle age, Einstein extended Special Relativity to General Relativity, explaining the force of gravity (in stark contrast, it must be added, with the other three fundamental physical forces) as a mere deformation in local spacetime geometry.
And still later, he expended years, ultimately decades, in a heroically failed quest for a Unified Field Theory.
2. Cultural despair:
(a) gales of unreason; (b) the "Peak Science" fear
Einstein's witness provides a beacon by which to steer as our cultural darkness deepens. I have remarked in my "Life's Business" essay at http://www.metascientia.com/DNNN____business/JBNN___mission.html, but will here remark again (adapting two key paragraphs) that even in this relatively early stage in the oppression of science, the "sinister gales of Unreason" blow strong:
(a) Of the tiny handful of young people that David Dunlap Observatory stellar spectroscopist Prof. Robert Garrison and I used to be mentoring together, fully two suggested to me that NASA could have faked the moon landings. And I have heard the same from at least one, perhaps two, members of the general public. Much can be forgiven the general public, addled as it is by a diet of television infotainment. Students from the science departments, however, are a more serious matter, since much is given them by their universities, and correspondingly much is expected. If a third science-anchored student some day comes up with the suggestion of a NASA hoax, I should first remark, politely, that Prof. Garrison is himself an author of a paper on NASA Apollo rock samples, and then politely ask by what concrete mechanisms the student believes so respected a prof got duped.
(b) I keep reading that there are people who believe the universe to be a mere six thousand years old. Such allegations have even swirled recently, I think without plausible denial from the relevant quarters, around a former Canadian federal party leader, somewhat antedating Mr Stephen Harper's circle. If I were to meet such a person, I would ask, politely, what we are seeing when we examine M31, the Andromeda Galaxy, an easy object for binoculars, with progressively better telescopes. With a sharply imaging ground-based telescope rejoicing in an aperture of 3 or 5 or 8 metres, M31 looks like a pancake of occasionally resolved stars rather than a simple gaseous nebula. It indeed looks like a pancake of stars similar to our own Milky Way galaxy. However, a pancake of Milky Way dimensions, and yet of the observed M31 angular extent, has to be so far away that it takes light not six thousand but hundreds of thousands or thousands of thousands of years to reach our eyes. That fuzzy distance conjecture is sharpened to a value of 2.7 million light years (Gieren et al., 2013) by apparent-brightness measurements of Cepheid variable stars in M31, whose intrinsic brightnesses the professionals believe they know well from studies of more local Cepheids.
Of course there are also compelling reasons for thinking we see in light, and "hear" in radio, objects two thousand or four thousand times farther away than M31.
Of course there are also compelling reasons for thinking we see in light, and "hear" in radio, objects two thousand or four thousand times farther away than M31.
And other examples abound. Leaving today's David Dunlap Observatory heritage-conservation fiasco temporarily aside, I here merely select, rather randomly, a pair of further examples, out of a wide field. The first involves a Good Guy (speaking for scientific truth, in the teeth of corporate power), the second a guy not so good:
- Dr David Healy - as a whistleblower whose stance on Prozac cannot have been comfortable to manufacturer Eli Lily, and whose University of Toronto appointment got revoked in 2000, under controverted circumstances (one of the University's medical arms was a recipient of Eli Lily funding);
- Mr Donald Trump - as a commenter on climate science (this politician, in fact would-be world leader, is on Twitter record as saying "The concept of global warming was created by and for the Chinese in order to make U.S. manufacturing non-competitive")
****
Even apart from the gales of Unreason, we have to note a kind of cultural despair - a kind of pervasive suggestion that science, having risen so high, now has nowhere to go but down.
How, it might be argued (indeed a friend argued this for me over tea less than a month ago) is continued substantive progress in astrophysics possible? Surely we are at, or are approaching, the age of Peak Observatories, in which it becomes harder and harder to fund the first-rank telescopes - the behemoths with five or more times the now-modest aperture of the 1935 David Dunlap Observatory 1.88-metre reflector? Surely we cannot, in this age of growing economic turmoil, with NASA more and more under the axe, hope for further "Great Observatories", significantly extending the capabilities of the Hubble, Compton, Chandra, and Spitzer missions?
The argument is put forcibly by humanist, neo-pagan theologian, and social critic John Michael Greer, in his 2014-11-26 blog posting "Dark Age America: The Suicide of Science", at http://thearchdruidreport.blogspot.ca/2014/11/dark-age-america-suicide-of-science.html:
/.../ the grand designs of intellectuals in a mature society normally presuppose access to the kind and scale of resources that such a society supplies to its more privileged inmates. When the resource needs of an intellectual project can no longer be met, it doesn’t matter how useful it would be if it could be pursued further, much less how closely aligned it might happen to be to somebody’s notion of the meaning and purpose of human existence.
Furthermore, as a society begins its one-way trip down the steep and slippery chute labeled “Decline and Fall,” and its ability to find and distribute resources starts to falter, its priorities necessarily shift. Triage becomes the order of the day, and projects that might ordinarily get funding end up out of luck so that more immediate needs can get as much of the available resource base as possible. A society’s core intellectual projects tend to face this fate a good deal sooner than other, more pragmatic concerns; when the barbarians are at the gates, one might say, funds that might otherwise be used to pay for schools of philosophy tend to get spent hiring soldiers instead.
Modern science, the core intellectual project of the contemporary industrial world, and technological complexification, its core cultural project, are as subject to these same two vulnerabilities as were the corresponding projects of other civilizations. Yes, I’m aware that this is a controversial claim, but I’d argue that it follows necessarily from the nature of both projects. Scientific research, like most things in life, is subject to the law of diminishing returns; what this means in practice is that the more research has been done in any field, the greater an investment is needed on average to make the next round of discoveries. Consider the difference between the absurdly cheap hardware that was used in the late 19th century to detect the electron and the fantastically expensive facility that had to be built to detect the Higgs boson; that’s the sort of shift in the cost-benefit ratio of research that I have in mind.
A civilization with ample resources and a thriving economy can afford to ignore the rising cost of research, and gamble that new discoveries will be valuable enough to cover the costs. A civilization facing resource shortages and economic contraction can’t. If the cost of new discoveries in particle physics continues to rise along the same curve that gave us the Higgs boson’s multibillion-Euro price tag, for example, the next round of experiments, or the one after that, could easily rise to the point that in an era of resource depletion, economic turmoil, and environmental payback, no consortium of nations on the planet will be able to spare the resources for the project. Even if the resources could theoretically be spared, furthermore, there will be many other projects begging for them, and it’s far from certain that another round of research into particle physics would be the best available option.
3. The irrationality of proceeding
from "Peak Science" fears to general pessimism
The efficacy, in the world of physical science, of some kind of Law of Diminishing Returns, in some form or other, is incontestable. But I will now argue the illogicality of proceeding from this to a comprehensive pessimism.
Consider, for a moment, not science but literature. Consider the standpoint of some hypothetical humantiies scholar taking stock of Greek and Latin authors, from the hot and dilapidated streets of Rome, in the summer of 430. (This was the summer in which Augustine of Hippo died.) Have we not, asks the hypothetical scholar, now attained Peak Literature? Have we not, in a millennium of intense effort, tried everything worth trying, in fully two languages?
Epic? Been there, done that. First, in the magically archaic diction of that unknown poet, or succession of poets, that we in our ignorance simply call "Homer". Then, more than a half millennium later, in a startling adaptation of that diction to the sophisticated political milieu of our own initial Augustus, in the Aeneid.
Stage tragedy? Been there, done that: Aeschylus, Euripides, Sophocles, in variation upon variation. And later, Seneca.
Stage comedy? Yes, through Aristophanes, Plautus, and successors.
Lyric poetry? Yes, in both languages.
Prose? Yes: and even in radical forms, as with the Hippo bishop's searing first-person "Confessions".
In future, says this hypothetical scholar, we can hope only for derivative and imitative works - say for a second Statius, as a second pale reflection of Virgilius.
Is this hypothetical A.D. 430 pundit right?
What we have here, we moderns have to reply, is a failure of imagination. The possibilities even of epic are not exhausted by the Greek and Roman models. We see very different possible approaches upon reading, for instance, Beowulf. While Keats is for us moderns a writer not utterly unlike Catullus, nobody brought up on the hypothetical theorist's circa-430 restricted diet of Graeco-Latin lyric would find it easy to imagine Gerard Manley Hopkins.
What we have here, we moderns have to reply, is a failure of imagination. The possibilities even of epic are not exhausted by the Greek and Roman models. We see very different possible approaches upon reading, for instance, Beowulf. While Keats is for us moderns a writer not utterly unlike Catullus, nobody brought up on the hypothetical theorist's circa-430 restricted diet of Graeco-Latin lyric would find it easy to imagine Gerard Manley Hopkins.
Analogously, then, I say, for science, including even the various branches of physics. Emerging limitations in technology mean only that some directions of scientific movement, out of an indeterminately vast ensemble of possible directions, are closed off. At some point, whether in the next few years (this I personally find too pessimistic) or in the next few decades (as I personally believe), technology will stagnate, and even regress. It will at this point become impractical to fund telescopes with bigger apertures, or supercomputing clusters with more nodes, or particle accelerators with more beam power. Depending on how severe our political and social crises become, in an era of deepening fuel shortages and rising seas, we may indeed face either a moderate or a severe technological contraction. (For what it is worth, my personal hunch is that we are one or two or three decades into an overall stagnation - masked, however, by some continuing, decelerating, advances in cyber technology - and that a severe contraction impends some decades from now, in a setting of increasing sociopolitical turmoil.)
Let us, then consider what happens when technology stagnates. To keep things simple, we consider a technological Steady State. But my argument can be modified, admittedly at the expense of clarity and vividness, to fit also the case of a technological decline, even (once my argument is developed in the imaginative spirit required by Humility in the face of the Unknown) a severe one.
With telescope, computer, and particle-accelerator power frozen, what remains possible?
The fundamental driver of physical science is not technology (important though technology is), but mathematical insight. The history of mathematics shows how things can change foundationally, even in offices equipped with nothing beyond paper and pencil.
The record shows that already within traditional Euclidean three-dimensional geometry, startling things can emerge, for the creative.
Consider, for example, a sphere, sitting in some immovable hemispherical cradle. We are free to move the sphere around in any way we please, twisting it this way and that - first rotating it, for instance, in the right-hand-rotation sense, through an angle of 33 degrees, around a ray pointing from the centre of the sphere to the tip of the Washington Monument., and then rotating it in the left-hand-rotation sense through an angle of 271 degrees around a ray pointing from the centre of the sphere to the centre of the Moon. We perform some large number, say four thousand, of such rotations, right-handed and left-handed, through angles great and small, around four thousand very disparate rays.
Now, we ask: Is there some single rotation which could have taken the sphere from its initial to its final configuration, saving us all the trouble of four thousand separate twistings? Equivalently: Is there some diameter D of the sphere such that after all the four thousand manipulations are complete, D remains unmoved?
Although we might suspect the answer to be "Yes", the answer is not obvious. Even the imaginative Greeks were not, so far as is now known, imaginative enough to have posed the question.
The question was posed, with a proof for "Yes", by Leonhard Euler, in surprisingly modern times - so recently as 1776. (Apparently it goes, in his formulation - my programme of studies indicates that I should in the next month work through his short, Wikipedia-recapitulated, proof - Quomodocunque sphaera circa centrum suum convertatur, semper assignari potest diameter, cuius directio in situ translato conveniat cum situ initiali. So he is using my second, unmoved-diameter-D, formulation.)
If we exit the confines of Euclidean geometry, the history of mathematics becomes more startling still.
The concept of a set is readily introduced to an eight-year-old. (I ascertained this through practical work with a great-nephew, in the Christmas of 2014, in the Ottawa area - with the same child as got the Einstein quotes from me, framed under glass as a gift, with an accompanying photo of Einstein's Princeton day-of-death 1955-04-18 desk.) The set of Canadian provincial capitals is a set with exactly 10 elements. The set of unicorns presently residing in Canada is a set with exactly 0 elements. And this latter set, conveniently called the "empty set", is the very same as the set of dragons presently residing in Belgium.
As a next point in the child's guided investigation, we introduce the concept of a subset: A is a subset of B if, and only if, there exists no element of A which fails to be an element of B. The set of Canadian provincial capitals is thus a subset of the set of Canadian cities. Further, the empty set is a subset of the set of Canadian provincial capitals (and indeed the empty set is a subset of every set). Further, the set of Canadian provincial capitals is a subset of the set of Canadian provincial capitals (and indeed any set is a subset of itself).
We now ask a question which in Christmas of 2014 I did not press in full generality with my great-nephew. It was appropriate not to lean on him, but merely to wait for another of his oft-repeated spontaneous demands, "Cousin Toomas, could we do some more math?" For any number n in the set {1, 2, 3, ... } (the set of positive integers), we ask: If an arbitrary fixed set S has exactly n elements, then how many elements are in the set of subsets of S? The set {Toronto, Winnipeg} has as its subsets the empty set, the single-element set {Toronto}, the single-element set {Winnipeg}, and the entire set {Toronto, Winnipeg}. So, in general, a set of exactly two elements has exactly four subsets.
Similar examples illustrate the truths that a set of exactly three elements has exactly eight subsets, that a set of exactly four elements has exactly sixteen subsets, that a set of exactly five elements has exactly thirty-two subsets, and so on.
In general (this is the point which my great-nephew and I left uncovered, with Christmas playing some distracting role), for any n in the set {1, 2, 3, ...}, and for any arbitrary fixed set S, if S has exactly n elements, then the set of subsets of S has exactly 2-to-the-power-n elements.
Finally, we ask a question which, elementary though it is, was perhaps not posed by anyone before Georg Cantor (1845-1918). What happens if we consider the set of subsets of an infinite set? What happens if, for instance, we consider the set of subsets of {1, 2, 3, ...} itself? The odd positive integers are one subset of this set. The even positive integers are another. The empty set is a third; the six-element set {5, 7, 11, 12, 13, 14} is a fourth; and so on.
Say that sets T and U, whether finite or infinite, are "equipollent" if and only if the elements of T can be paired off one-to-one with the elements of U - in others, can be paired off with, so to speak, no polygamy, no polyandry, no bachelors, and no spinsters. Cantor's question then is the following: Is the set of subsets of {1, 2, 3, ...} equipollent with that simpler thing, the mere set {1, 2, 3, ...}?
A simple reductio-ad-absurdum proof shows the answer to be in the negative.
I will belabour the proof a little here, for the possible benefit of some children or their parents. The proof should in any case be accessible to Grade Three or so, or at any rate to Grade Eight or so. It is what the professionals call a Diagonalization Argument, in a child-in-livingroom setting best done with infinite rows of bits - "1" for "Yes, this individual in the infinite progression 1, 2, 3, ... is a member of this particular subset", "0" for "No, this individual in the infinite progression 1, 2, 3, ... is not a member of this particular subset." The four just-mentioned subsets of {1, 2, 3, ...} are then coded with, respectively, the rows 101010101010101010101..., 010101010101010101010101010..., 0000000000000..., and 00001010001111000000... . Suppose, per absurdum, that the sets of positive integers can be successfully paired one-to-one with those simpler things, the positive integers. Then write down at the top of your paper some initial segment in that infinite row of bits that is a representing code for the particular set-of-positive-integers successfully - so we are imagining, per absurdum - paired with that simpler thing which is 1. Next, write down some initial segment in that infinite row of bits that is a representing code for the set-of-positive-integers successfully (as we are imagining) paired with that simpler thing which is 2. Next, write down some initial segment in that infinite row of bits that is a representing code for the set-of-positive-integers successfully (as we are imagining) paired with that simpler thing which is 3. And so on. Now consider the "Perverse Infinite Row of Bits", or PIROB, obtained by reversing the first bit in the first row, the second bit in the second row, the third bit in the third row, and so on - "reversing" here meaning "altering 1 to 0, and 0 to 1". (If the four just-mentioned subsets are paired with, respectively, the positive integers 1, 2, 3, and 4, then the PIROB has, as its first four bits, 0011 - for it is cunningly designed to disagree with the first row in its first place, to disagree with the second row in its second place, to disagree with the third row in its third place, and to disagree with the fourth row in its fourth place.) The PIROB itself certainly represents some, perhaps finite and perhaps infinite, set of positive integers, the "PIROB Set". (In the example here being developed, the PIROB Set lacks 1, and lacks 2, and contains 3, and contains 4, ... ) Can the PIROB Set possibly be wedded to 1? If not, then can the PIROB Set possibly be wedded to 2? If not, then can the PIROB Set possibly be wedded to 3? to 4? to 5? to 6? to 7?)
A simple reductio-ad-absurdum proof shows the answer to be in the negative.
I will belabour the proof a little here, for the possible benefit of some children or their parents. The proof should in any case be accessible to Grade Three or so, or at any rate to Grade Eight or so. It is what the professionals call a Diagonalization Argument, in a child-in-livingroom setting best done with infinite rows of bits - "1" for "Yes, this individual in the infinite progression 1, 2, 3, ... is a member of this particular subset", "0" for "No, this individual in the infinite progression 1, 2, 3, ... is not a member of this particular subset." The four just-mentioned subsets of {1, 2, 3, ...} are then coded with, respectively, the rows 101010101010101010101..., 010101010101010101010101010..., 0000000000000..., and 00001010001111000000... . Suppose, per absurdum, that the sets of positive integers can be successfully paired one-to-one with those simpler things, the positive integers. Then write down at the top of your paper some initial segment in that infinite row of bits that is a representing code for the particular set-of-positive-integers successfully - so we are imagining, per absurdum - paired with that simpler thing which is 1. Next, write down some initial segment in that infinite row of bits that is a representing code for the set-of-positive-integers successfully (as we are imagining) paired with that simpler thing which is 2. Next, write down some initial segment in that infinite row of bits that is a representing code for the set-of-positive-integers successfully (as we are imagining) paired with that simpler thing which is 3. And so on. Now consider the "Perverse Infinite Row of Bits", or PIROB, obtained by reversing the first bit in the first row, the second bit in the second row, the third bit in the third row, and so on - "reversing" here meaning "altering 1 to 0, and 0 to 1". (If the four just-mentioned subsets are paired with, respectively, the positive integers 1, 2, 3, and 4, then the PIROB has, as its first four bits, 0011 - for it is cunningly designed to disagree with the first row in its first place, to disagree with the second row in its second place, to disagree with the third row in its third place, and to disagree with the fourth row in its fourth place.) The PIROB itself certainly represents some, perhaps finite and perhaps infinite, set of positive integers, the "PIROB Set". (In the example here being developed, the PIROB Set lacks 1, and lacks 2, and contains 3, and contains 4, ... ) Can the PIROB Set possibly be wedded to 1? If not, then can the PIROB Set possibly be wedded to 2? If not, then can the PIROB Set possibly be wedded to 3? to 4? to 5? to 6? to 7?)
With that tricky, negative, answer firmly in hand, via an admittedly rather tricky Diagonalization Argument, a dizzying prospect opens up. The positive integers are one infinite set; the set of sets of positive integers is found through our tricky reductio ad absurdum proof to be a set not equipollent with it (but, so to speak, "more numerous", "superpollent", "larger"); the set of sets of sets of positive integers is found by a somewhat similar style of reductio ad absurdum to be bigger still - we do best now to resort not to rows of bits, but instead to the perhaps-beyond-elementary-school language of mappings, saying "suppose, per absurdum, this set can be mapped one-to-one onto that set" - and so on. Infinity itself, then, is found to come in infinitely many different sizes.
[To be continued in the upload of UTC=20160607T0001Z/20160607T0401Z, with more points on the overriding importance to physical science of that technology-independent discipline which is mathematics; and with points regarding the miseries and joys - the concrete, three-colours-of-pencil, practicalities - of private mathematics study; and with a moral on joy and suffering drawn from Catholic-praxis author Dorothy Day.]
