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 relevant points to reasonable length.
Revision history:
- UTC=20160615T1624Z/version 1.1.0: Kmo added several paragraphs, trying to do better justice to logician-philosopher Saul Kripke's "Naming and Necessity", and to bring out more clearly the idea of physical science as an investigation of "substantial essences" (and he reserved the right to upload, over the coming few days, without formal documentation in this revision history, nonsubstantive revisions, as version 1.1.1, 1.1.2, ... ).
- UTC=20160614T0047Z/version 1.0.0: Kmo finished an interrupted uploaded of 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, ... .
4. The Coming Dark-Ages Triage of Physical Science
The prospects for physical science, in our ongoing civilizational decline, resemble the prospects for a republic on the brink of destruction by Stalin.
What may we have to face?
Over the next four decades or so, at any rate assuming a continuation of trends and conditions from the last four decades (and, in particular, assuming no widescale thermonuclear war), we face no apocalypse. We may expect the standards of high schools - relevant to science as a source of supply for university admissions offices - to sink still lower. We may expect the current scandals, notably around inadequately refereed journals, to continue. This is a development which will make the already rather cynical public appraisal of scientists increasingly like the current cynical public appraisal of those social outcasts who are the lawyers, the politicians, and the property developers. With this ongoing decline in public esteem for science, eloquently analyzed by John Michael Greet on 2014-11-26 at http://thearchdruidreport.blogspot.ca/2014/11/dark-age-america-suicide-of-science.html, we may expect a concomitant contraction in public funding.
And I for one would expect public superstition to rise correspondingly, as with the gullible people already making it commercially realistic for "psychics" to advertise their services here in Richmond Hill, Ontario.
But I am interested in the present essay in the long term, over the next two or three or four centuries. Our best understanding of that long term suggests a contraction in the available supply of fossil fuels, and yet also (because we are at present abundantly burning the abundant supplies we have, rendering them temporary) a change in climate. What can the social environment of science be like in a world in which even the Monument and Bank Tube stations, perhaps even some or all floors of the present Bank of England, are submerged at high tide?
We may well imagine that at this stage governments and literacy, as we now know them, will have faded.
In place of Number Ten Downing Street, the Oval Office, and the like, we may reasonably predict warlords. In place of the British Library, we may reasonably predict what - total illiteracy? I would envisage, rather, a world in which the powerful brandish their literacy as a badge of authority, keeping finely bound classics on their manor-house shelves, and turning their nasty little surviving universities (their Yales, their Oxfords) into socially exclusive social-networking tools, into which only they and their special allies penetrate easily. The serfs would in this scenario be supplied with only as much education as might be needed for them to meet manorial needs.
If, in that impending Dark Age, serfs can be distracted with forms of pseudo-education, in which they take degrees in "Media Studies" and the like (rather than learning such potentially empowering things as Roman law, Church history, modern languages, and differential equations), then all the better. But in that warm, humid, impoverished, ill-lit world, pseudo-educating the serfs will perhaps prove beyond the economic means of most jurisdictions.
In an era in which education is radically de-democratized, there will be a demand less for science than for engineering. Here the more able of the serfs will get something. Any warlord of administrative ability will appreciate, for example, the value of aerial reconnaissance - by drones with cameras, if sufficient engineering skills have survived, but otherwise through balloon techniques familiar to Generals Grant and Lee.
And any warlord worth his salt will appreciate the value of communications. We may well imagine things stabilizing at a level appreciably higher than the merely Victorian. Transistors - so dependent on pure materials and clean-room technique - may perhaps be gone. But what is likely to remain, as an empowering technology in a violent world, is the suitcase-sized Morse transmitter, its little "peanut valves" glowing gently on the strength of chemical batteries. Such a set would be skilfully operated, even skilfully maintained, in the field by some private or lieutenant, born in some trim little cottage on the manorial estates, and trained in the relevant engineering at his lord's expense.
So our question pertains to physical science, not to engineering. And our question is not whether physical science is doomed in the next four decades. Rather, the question is whether physical science is doomed over the long term - say, over the next four centuries.
****
Republics on the brink of destruction by Stalin come in different varieties. At one extreme in the autumn of 1939 is Estonia. At a different extreme is Finland.
In that autumn, Estonia found itself incapable of conserving anything. With Moscow's ultimatum in hand, the careful defensive preparations of the entire 1920-1939 period were rendered irrelevant. (A deal had been made with the UK, I think early in 1939, to upgrade a part of the air force to state-of-the-art Spitfires; bridges had long been built with cavities in their piers, to facilitate quick destruction by Estonian land forces at the instant of retreat; military radio is likely to have been solidly engineered, since the civilian telephone service, equipped with shortwave, was itself offering duly wealthy civilian subscribers the operator placing of long-distance calls to Japan and Malaya; and there was a volunteer "Defence League" over and above the formal armed services. All this, and much else, had to be set aside, instantly.) It was clear to the government of the day, after deliberations in Tallinn lasting an hour or so, at any rate not for even as long as two hours, that just one single outcome was possible in the event of hostilities with Moscow.
Finland, to whom Stalin handed a similar ultimatum that autumn, likewise made a calculation correctly fitting its circumstances. Its circumstances, however, were different and less bleak. In Finland's case, Stalin's ultimatum was rejected, with resistance over the next few months successfully mounted - and this even though, in the terrifying event, Karelia was lost, and (contrary to the rational expectations of non-Finnish analysts) the Mannerheim Line was breached.
The Republic of Physical Science does not, over the long term, confront annihilation in the manner of 1939 Estonia. Neither, on the other hand, does it rejoice in the prospect of survival in the robust manner of post-1939 Finland. Our task here is to locate the truth which lies somewhere between these two extremes. Much will have to be lost - not just (to pursue the Finnish comparison) Viipuri and its Karelian hinterland, but perhaps Turku and Helsinki. And yet something - some austere little Oulu, some harsh little Rovaniemi, so far north that the December sun scarcely rises - can be kept going.
Where must our priorities lie?
Where must our priorities lie?
****
Physical science is in part an investigation of matters that are prima facie contingent. What, for example, is the value of the Hubble Constant, which yields the recessional speed, say in our galaxy-centre rest frame, of an external galaxy at some given distance? (On looking this up today in you-know-what, I found a surprisingly large recent recalibration: apparently, although the ESA Planck Mission published 67.80±0.77 km/s per megaparsec on 2013-03-21, the Hubble Space Telescope people published 73.00±1.75 km/s per megaparsec on 2016-05-17: https://en.wikipedia.org/wiki/Hubble's_law#Observed_values .)
I presume that as far as any of the profs know, there is no reason for calling the modal status of the Hubble Constant value anything but "contingent". In some state of affairs logically consistent with all the known fundamental physical laws, the Constant, I presume, has one value. In some other states of affairs logically consistent with all the known fundamental physical laws, the Constant, I presume, has a different value. And I presume it is even in some sense possible that the "Constant" assumes different values at different stages in the history of the cosmos.
A similar, more humble, example of a radically contingent matter is the present pressure (say, in newtons per square metre) at the core of our own Sun.
A more important part of physical science, however, looks not into contingent particulars, such as the pair just cited, but into universal principles.
Before developing my argument regarding conservation priorities in a new scientific Dark Age, I want to highlight the special status of such principles. I wish to do so by exploring a question too little discussed - the question of the modal status of laws. Are they, like particular facts, contingent? Or do they possess, at least in some of their aspects, some relevant element of necessity?
Consider the principle that the acceleration experienced by a body in an inertial frame is directly proportional to the force on the body times the inertial mass of the body. If forces, masses, and accelerations are measured in the usual "SI" system of units, or again in the alternative "cgs" system that has been prevalent in astrophysics, this becomes the familiar F= ma, i.e., F = kma with constant k becoming the purely numeric quantity 1.
I had the good luck to have, as a tutor in first year, the University of Toronto low-temperature experimentalist Cindy Krysak. She was not satisfied with us, her tutorial group, until we were willing to chant, loudly, in the manner of a football crowd, when she asked "So what's the first thing to remember on your midterm?", "F EQUALS M A! F EQUALS M A!"
What is here contingent, and what is here a matter of definition? It is no contingent matter that all triangles have three sides. A four-sided polygon would by definition not be a triangle. Analogously, we may well feel, something which was not related to force and acceleration as a factor-in-a-proportionality, as obediently chanted by Cindy's mob of frosh, would no longer qualify as an "inertial mass".
The importance of laws in the edifice of physical science becomes all the more striking when we realize, upon further reflection, that the concept of mass conceals conceptual depths, ultimately linked to General Relativity or cognate disciplines. An elementary Newtonian study of astrophysics, as in orbital mechanics, reveals the need to introduce a concept correlative with inertial mass, "gravitational mass". But then arises a question pressed by the General Relativity gurus: why do the two quantities, introduced to physics in such observationally disparate ways, seem always equal?
Or consider, again, Coulomb's electrostatic law, according to which the attractive or repulsive force between two charges (each of negligible spatial extent) is inversely proportional to the square of their separation. What is here contingent, and what is here necessary? Without having pondered this beyond a second-year level, I would suspect on the one hand that the constant of proportionality (once we fix on, say, SI units) is contingent, and that on the other hand some deep necessity, perhaps of a geometrical origin, attaches to the "2" exponent. Perhaps the "2" is at some deep level related to the fact that the area of a sphere increases as the square of its radius. For this fact about areas has the consequence that the illumination-per-square-metre of a sphere enclosing a lamp, or analogously the electrical "Illumination"-per-square-metre of a sphere enclosing a charge, is inversely proportional to the square of the radius. Perhaps to imagine that the exponent is not 2 but, say, 2.000001 is to imagine geometry as being something other than what it in physical space inevitably is.
Admittedly, the suspicion of a geometrical necessity does not release us from the obligation to do experiments, as a check on the correctness of whatever a priori reasoning we may construct. On today consulting my copy of that thoroughly laboratory-anchored exposition, at second-year level, which is W.J. Duffin's Electricity and Magnetism (McGraw-Hill, fourth edition, 1990), I find references to several workers, from Cavendish in 1772 (in an investigation not published until 1879) up to Williams-Faller-Hill (said by Duffin to be in Phys. Rev. Letters 26 (1971)).
If physical science, on its observatory-dome side, finds reason for thinking that "laws" change from one part of the cosmos to another, then this will only mean that the supposed laws are not true laws - that either (a) there are no laws at all (beyond, that is, the minimal, negative law that There Are No Non-Minimal Laws) or (b) the local "laws" are locally varying instances of some underlying universal principle which itself qualifies as a law.
It seems to me that up to 1990, when I left professional academic philosophy, there had been insufficient discussion of the modal status of physical laws. A promising beginning had, indeed, been made by Saul Kripke (1940-) at Princeton, who had pointed out the noncontingent character of such propositions as "Gold has atomic number 79." Any element with a different number of nucleus protons, Kripke had remarked in his well-titled "Naming and Necsssity", could not be gold. If there is a possible world in which an element of atomic number 82 has the mass, ductility, conductivity, and colour and sheen, and so forth, of gold, then in this possible world it would be lead, not gold, which possessed the so-to-speak Golden Phenomenology.
Although it is in a sense true that "Gold has atomic number 79" is a definitional necessity, one hesitates to say here (as one unblushingly says for "Every triangle has three sides") that it is a definitional necessity of a banal kind - that it is a, so to speak, "mere" definitional necessity. I wrote incautiously in the 1.0.0, 1.0.1, 1.0.2, ... series of versions of this essay, calling Kripke's proposition "perhaps straightforwardly definitional". But the ancients and mediaevals, ignorant of the atomic nucleus, nevertheless succeeded in attaching a duly definite meaning to the term "gold". In the admittedly none-too-clear language of their Aristotelian tradition, we might say that they correctly meant the term "gold" to signify some "substantial essence" unknown to them, and explored only in modern times.
It might be further suggested that physical science is quite generally the study of "substantial essences", and that much remains to be done. What, for example, is a "substantial essence" of a Linnean species of tree, such as the red maple, Acer rubrum? Something will have to be said here about DNA.
The underlying conceptual problems here assume a certain - to me unfathomable - complexity when it is noted that the DNA can remain intact even in a dead specimen of Acer rubrum. Perhaps the full story of the "substantial essence" will involve some duly mathematical considerations of something like entropy flows. (I did at one point read a book with the promising title Entropy for Biologists, but cannot claim to have retained from it anything beyond television-and-magazine-level trivialities.)
A short remark which I published in a philosophical journal (Analysis 37 (1977), pp. 147-148, under the title "Disturbances") might - or might not - be helpful. I remarked that a living thing resembles a knot, in that just as a knot can migrate along a rope, so a living thing migrates through consignments of chemicals, progressively ingesting and excreting stuffs. As a knot is in a rope, so, I suggested, is a living thing "in" its constituent matter. I quoted in this context, as potentially useful, Aristotle's Metaphysics 1034a6 dictum that that the "form" of a human being is "in" his flesh and bones"; Aristotle's comparison of living tissues with flowing water at de Gen. et Corr. 321b24-25; and Aristotle's suggestion at Topics 127a3-8 that a wind is better defined as a movement in air than as air in movement. Perhaps substantial essences of living things involve "forms" - in some as-yet-unknown sense to be rendered precise through mathematical insights both into DNA and into entropy - migrating "through" stuffs?
One imagines that even the DNA story (leaving the mathematics of entropy aside for the moment) will bristle with subtleties. We have in DNA a strange approach of biology to formal computer science. But how mature, as a branch of mathematics (as opposed to the mere engineering of hardware, in the now-hoary 1940s-onward von Neumann/Harvard machine-architecture tradition) is today's computer science? And how well understood are the mathematical formalities of DNA? One does, as a layman, get the impression that while those parts of the double helix which encode proteins are at some level understood, unclarities abound in the remainder, and that some of the professionals are nowadays uneasy with the dismissive term "junk DNA".
****
Consider the principle that the acceleration experienced by a body in an inertial frame is directly proportional to the force on the body times the inertial mass of the body. If forces, masses, and accelerations are measured in the usual "SI" system of units, or again in the alternative "cgs" system that has been prevalent in astrophysics, this becomes the familiar F= ma, i.e., F = kma with constant k becoming the purely numeric quantity 1.
I had the good luck to have, as a tutor in first year, the University of Toronto low-temperature experimentalist Cindy Krysak. She was not satisfied with us, her tutorial group, until we were willing to chant, loudly, in the manner of a football crowd, when she asked "So what's the first thing to remember on your midterm?", "F EQUALS M A! F EQUALS M A!"
What is here contingent, and what is here a matter of definition? It is no contingent matter that all triangles have three sides. A four-sided polygon would by definition not be a triangle. Analogously, we may well feel, something which was not related to force and acceleration as a factor-in-a-proportionality, as obediently chanted by Cindy's mob of frosh, would no longer qualify as an "inertial mass".
The importance of laws in the edifice of physical science becomes all the more striking when we realize, upon further reflection, that the concept of mass conceals conceptual depths, ultimately linked to General Relativity or cognate disciplines. An elementary Newtonian study of astrophysics, as in orbital mechanics, reveals the need to introduce a concept correlative with inertial mass, "gravitational mass". But then arises a question pressed by the General Relativity gurus: why do the two quantities, introduced to physics in such observationally disparate ways, seem always equal?
Or consider, again, Coulomb's electrostatic law, according to which the attractive or repulsive force between two charges (each of negligible spatial extent) is inversely proportional to the square of their separation. What is here contingent, and what is here necessary? Without having pondered this beyond a second-year level, I would suspect on the one hand that the constant of proportionality (once we fix on, say, SI units) is contingent, and that on the other hand some deep necessity, perhaps of a geometrical origin, attaches to the "2" exponent. Perhaps the "2" is at some deep level related to the fact that the area of a sphere increases as the square of its radius. For this fact about areas has the consequence that the illumination-per-square-metre of a sphere enclosing a lamp, or analogously the electrical "Illumination"-per-square-metre of a sphere enclosing a charge, is inversely proportional to the square of the radius. Perhaps to imagine that the exponent is not 2 but, say, 2.000001 is to imagine geometry as being something other than what it in physical space inevitably is.
Admittedly, the suspicion of a geometrical necessity does not release us from the obligation to do experiments, as a check on the correctness of whatever a priori reasoning we may construct. On today consulting my copy of that thoroughly laboratory-anchored exposition, at second-year level, which is W.J. Duffin's Electricity and Magnetism (McGraw-Hill, fourth edition, 1990), I find references to several workers, from Cavendish in 1772 (in an investigation not published until 1879) up to Williams-Faller-Hill (said by Duffin to be in Phys. Rev. Letters 26 (1971)).
If physical science, on its observatory-dome side, finds reason for thinking that "laws" change from one part of the cosmos to another, then this will only mean that the supposed laws are not true laws - that either (a) there are no laws at all (beyond, that is, the minimal, negative law that There Are No Non-Minimal Laws) or (b) the local "laws" are locally varying instances of some underlying universal principle which itself qualifies as a law.
It seems to me that up to 1990, when I left professional academic philosophy, there had been insufficient discussion of the modal status of physical laws. A promising beginning had, indeed, been made by Saul Kripke (1940-) at Princeton, who had pointed out the noncontingent character of such propositions as "Gold has atomic number 79." Any element with a different number of nucleus protons, Kripke had remarked in his well-titled "Naming and Necsssity", could not be gold. If there is a possible world in which an element of atomic number 82 has the mass, ductility, conductivity, and colour and sheen, and so forth, of gold, then in this possible world it would be lead, not gold, which possessed the so-to-speak Golden Phenomenology.
Although it is in a sense true that "Gold has atomic number 79" is a definitional necessity, one hesitates to say here (as one unblushingly says for "Every triangle has three sides") that it is a definitional necessity of a banal kind - that it is a, so to speak, "mere" definitional necessity. I wrote incautiously in the 1.0.0, 1.0.1, 1.0.2, ... series of versions of this essay, calling Kripke's proposition "perhaps straightforwardly definitional". But the ancients and mediaevals, ignorant of the atomic nucleus, nevertheless succeeded in attaching a duly definite meaning to the term "gold". In the admittedly none-too-clear language of their Aristotelian tradition, we might say that they correctly meant the term "gold" to signify some "substantial essence" unknown to them, and explored only in modern times.
It might be further suggested that physical science is quite generally the study of "substantial essences", and that much remains to be done. What, for example, is a "substantial essence" of a Linnean species of tree, such as the red maple, Acer rubrum? Something will have to be said here about DNA.
The underlying conceptual problems here assume a certain - to me unfathomable - complexity when it is noted that the DNA can remain intact even in a dead specimen of Acer rubrum. Perhaps the full story of the "substantial essence" will involve some duly mathematical considerations of something like entropy flows. (I did at one point read a book with the promising title Entropy for Biologists, but cannot claim to have retained from it anything beyond television-and-magazine-level trivialities.)
A short remark which I published in a philosophical journal (Analysis 37 (1977), pp. 147-148, under the title "Disturbances") might - or might not - be helpful. I remarked that a living thing resembles a knot, in that just as a knot can migrate along a rope, so a living thing migrates through consignments of chemicals, progressively ingesting and excreting stuffs. As a knot is in a rope, so, I suggested, is a living thing "in" its constituent matter. I quoted in this context, as potentially useful, Aristotle's Metaphysics 1034a6 dictum that that the "form" of a human being is "in" his flesh and bones"; Aristotle's comparison of living tissues with flowing water at de Gen. et Corr. 321b24-25; and Aristotle's suggestion at Topics 127a3-8 that a wind is better defined as a movement in air than as air in movement. Perhaps substantial essences of living things involve "forms" - in some as-yet-unknown sense to be rendered precise through mathematical insights both into DNA and into entropy - migrating "through" stuffs?
One imagines that even the DNA story (leaving the mathematics of entropy aside for the moment) will bristle with subtleties. We have in DNA a strange approach of biology to formal computer science. But how mature, as a branch of mathematics (as opposed to the mere engineering of hardware, in the now-hoary 1940s-onward von Neumann/Harvard machine-architecture tradition) is today's computer science? And how well understood are the mathematical formalities of DNA? One does, as a layman, get the impression that while those parts of the double helix which encode proteins are at some level understood, unclarities abound in the remainder, and that some of the professionals are nowadays uneasy with the dismissive term "junk DNA".
The full story of the "substantial essence" of Acer rubrum might at any rate involve a vindication of taxonomists in the school of Linnaeus. Although ignorant of DNA and entropy flows, they insisted on some deep generic similarity, and on some concomitant deep specific difference, between, for instance, Acer rubrum and its cousin Acer saccharum (the sugar maple). Their procedure here recalls the still earlier study of chemical elements: the mediaevals insisted on some deep difference between gold and its look-alike pyrites, even while being unable to say what the difference consisted in.
****
In my admittedly ill-informed view, some suggestion that not all necessities in physical science are straightforwardly definitional emerges not only from Kripkean necessities anchored in "substantial essences", but additionally from the failure of properly strenuous efforts made, from Frege onward, to exhibit mathematics as a collection of merely definitional truths.
The programme of Frege, soon taken up also by the young Bertrand Russell (1872-1970), was to exhibit the full sweep of physics-pertinent mathematics as an edifice of tautologies, each having the metaphysically reassuring emptiness of the only-trivially-necessary "All triangles have three sides." It is perhaps fair to say that the Frege-Russell programme failed, and that the metaphysically troubling spectre of necessities-which-are-not-merely-definitional survived.
Frege's offered edifice was found by Russell to be internally inconsistent. Russell, writing to a length of three fat volumes with Alfred North Whitehead (1861-1947), claimed to restore consistency through a revision of Frege's theory, dividing the possible properties of individuals into a hierarchy of logical "Types".
I gather that to recover, as the pinnacle of the imposing edifice, the corpus of physics-pertinent mathematics, is to proceed from purely logical constructions right out to Peano's axioms for the arithmetical structure of the set {0, 1, 2, ... }. From this point onward, I gather, the well established results of other workers kick in, as when one defines the negative numbers; and then the rationals; and then the reals; and then the derivatives and integrals of the more suitably housebroken univariate real functions; and so on, onward and upward, through Div and Grad and Curl in multivariate real calculus; eventually also with physics-pertinent constructions, including calculus, in the complex-number plane.
To recover the corpus of physics-pertinent mathematics, Russell found it necessary to introduce a new postulate, the "Axiom of Reducibility". His Axiom, in my admittedly ill-informed view, is not plausibly called a principle of logic, but a piece of substantive mathematics.
If, now, physics-pertinent mathematics stubbornly retains (in light of Frege's and Russell-Whitehead's adverse experiences) a necessity deeper than the merely definitional, might the same not be true (I ask) for physical laws themselves, or at least for some aspects of those laws - if not, for instance, for the value of Coulomb's proportionality constant in some selected system of physical units, then at least for his so-tidy value "2" for that denominator exponent?
I suggested that Coulomb's exponent might be anchored in necessities of geometry. It will be objected that geometry was itself found by Riemann and Lobachevsky to be infected with contingency, in researches ultimately calling into question even the inevitability of classically accepted formulae for areas of surfaces. Indeed (it will be objected) the very question whether physical space is Euclidean or non-Euclidean nowadays gets one answer near a massive body and a different answer when massive bodies are removed.
To this objection, I reply that if I am wrong in my suggestion about Coloumb, nevertheless - whatever may be the case with the contingency, in General Relativity, of the local curvature of space - possibilities for deeper necessities remain. For example, it as far as I have read necessary that when that closed non-self-intersecting single-sided single-edged loop which is a Moebius strip is cut with scissors along its median, it does not fall apart into two closed non-self-intersecting loops. It instead opens up, and I imagine cannot be prevented even by God from opening up, into a single, rather vigorously twisted, closed non-self-intersecting loop.
And the mere fact that space (as opposed to Minkowski spacetime) has three dimensions might be a necessity in its own right. At any one instant - we keep time out of this, staying purely spatial - at any point P in space, there can be at-least-and-at-most three vanishingly thin mutually perpendicular metre sticks with their zero ends touching at P. Although we can describe, in the framework of linear algebra, spaces of 4, 5, 6, ... dimensions, our own physical space (however locally varying its Euclid-defying curvature) seems resolutely 3-dimensional. This limitation does not appear a mere contingent matter.
Or is even this wrong? Has String Theory, or some similar physics formalism, shown a way to lift the limitation to three spatial dimensions? All I really dare say here is that the questions are weighty.
Whatever the precise mixture of contingency and necessity in the world of physical laws, it is at any rate evident that the laws constitute the core of physical science, with such prima facie contingencies as the value of the Hubble constant tending to lie more on the periphery.
****
We can now see where conservation priorities must lie as we decide, so to speak, how far behind the breached Mannerheim Line our forces shall retreat.
[To be continued, and perhaps concluded, in the upload of UTC=20160621T0001Z/20160621T0401Z. As I have outlined the writing, I am to finish this essay with some remarks on the organization of science research and the organization of science education. My envisaged additional remarks, on the practicalities of my own private programme of work in mathematics or physics - the resort to three colours of pencil; the MI-6-worthy filing system; the principle of reading more than one author on any single given topic; and above all the precept "Close the book, work it out alone, then open the book" - is best deferred to some subsequent essay, at some either near or distant point in 2016.]
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