Rome by Different Roads

[In our basic essay on Consilience in Mathematics [which is under reconstruction, following a blogspot glitch; temporarily redirected to this post]  one topic was the way in which a given mathematical topic or object can be approached from quite different angles, or defined in superficially quite different ways, so that it is finally surprising that (after a lot of work and lucubration) you wind up at the same mathematical metropolis.   And just as with the phenomenon of parallax, or triangulation (which we learned in Boy Scouts), the result is a more multidimensional appreciation than we began with.]

We have discussed the notions of abstraction and of generalization as regards the overall ‘topology’ (or geography) of the  mathematical landscape.   Not the same as either of these are cases where two intuitively quite distinct notions turn out to imply each other -- to be provably logically equivalent.  As, the (growing) menagerie of propositions that are equivalent or weakly equivalent to the Axiom of Choice.

The closest notion to this that we have previously touched on is that of  “Rome by Different Roads”, which in the simplest case means merely that the same proposition, say of number theory, can be reached (proven) with distinct sets of tools, e.g. an heavy-machinery analytic proof versus an ‘unplugged’ proof (generally using quite different ideas) which ascetically restricts itself to using only elementary methods.   The proposition you arrive at remains unchanged, the variety of proofs  something of a curiosity.  (As, the many different ways of proving the Pythagorean Theorem, most elegantly involving no more than a single diagram -- a small square tiltedly inscribed within a larger.)   It’s like -- tiens!  this road leads to Rome as well.

Whereas in this deeper phenomenon, the different equivalences -- the different perspective views of the same central Thing -- actually wind up changing or rather sublating what one’s concept of Rome is.

It’s a bit like spooky action-at-a-distance in quantum physics, in which you can’t consider the particles to be quite independently existing, even though they are distinct.   There seems to be looming some supernatent notion, some eka-conception, of which each individual proposition from the pool of those interderivable, are like the sundry avatars of Vishnu, and not Vishnu himself.

The study of computability came to be known as recursion theory, because early formalizations by Gödel and Kleene relied on recursive definitions of functions.When these definitions were shown equivalent to Turing's formalization involving Turing machines, it became clear that a new concept – the computable function – had been discovered, and that this definition was robust enough to admit numerous independent characterizations.

-- Wiki, “Mathematical Logic”

Specifically, equivalent formulations of the intuitive notions of computability, independently arrived-at by different thinkers, include:  Church’s lambda-calculus; Kleene’s general recursive functions; Post’s automata; and Markov algorithms.

For the notion of “independent characterizations”, compare the Heisenberg formalism vs. the Schroedinger formalism  in quantum mechanics.   That one is a cause célèbre, and with few comparable situations in the history of physics;  whereas mathematics is chock-full of people discovering the same mountain via different slopes.

The Wikipedia article on “Representation theory” highlights “the diversity of approaches to representation theory.  The same objects can be studied using methods from algebraic geometry, module theory, analytic number theory, differential geometry, operator theory, algebraic combinatoriecs  and topology.”

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For an application of this theme to the topic of uniform spaces,  consult this essay.

 

Uniform Spaces

[The following does not rise even to the level of an essay-in-progress;  more like a thought-in-progress, or even (saving your presence) a difficult bowel-movement.   But the hordes of typist-elves in the cavernous warehouses of WDJ  have yet to present anything brought to perfection this morning, and I wished not to disappoint the milling crowds that swarm this site each weekend, bringing the whole family, Sister Sue and Fido too, gawking at the glittering thoughtfronts -- the polemics, the poems, the darling little monostichs (these we can all afford) -- while shaking their heads sadly at the Trinitarian Minimalism and Cantorian Realism (out of our price-range) -- all  save one diminutive child towards the back of the bunch, eyes riveted on the prize, instinct with penetrating understanding…]

We saw here the dialectic of mathematical invention (not trying to be too Hegelian here -- think of it as an ensouled pendulum) whereby, beginning with the everyday world we live in -- I almost wrote ‘space’, but that would be to get ahead of our tale -- we abstract from the clutter of minute-to-minute experience, and conceive of it all happening within a space.   We then formalize that space with the Euclidean axioms.   We then familiarize ourselves with this new mind-environment, solving tricky problems and whatnot for a couple of thousand years, then -- since we have long effectively been working in the World of the Unseen -- very lightly generalize to Euclidean spaces of any finite dimension  -- a bit of a stretch biologically, but where, mathematically, everything works pretty much as before.

Meanwhile independently, mathematical analysis had proceeded apace, not necessarily concerned with the geometrical substrate as such, but piling up its own increasingly intricate problematics.   Then by an ideational leap which is of the essence of mathematics, and into which simply listening to lectures and slogging through the problem-sets at the end of the chapters, gives you no insight at all (executive summary:  Mathematicians are like gods), a clutch of bold spirits, bearing in mind certain delicate problems such as infinite sequences of functions and their convergence, generalized the stage on which such pageants play out, from the Euclidean to the general topological.   (The history has here been brutally telescoped.)  Something of the sort was in any case needed to save the Euclidean picture itself, since infinite-dimensional spaces were now required (even by physics),  and the finite-dimensional structures would not generalize in any straightforward way.

General topological spaces being a wildly assorted bag, various restrictions are put on them, for one purpose or another, to allow deduction and calculation.  One of these is metrizability, which we examined in the essay on Urysohn.   That has the advantage of preserving much of our hard-won familiarity with the Euclidean metric, while allowing a vast array of new metrics for particular purposes. (For example:  the by-now-familiar Lorentz metric of Einsteinian spacetime.  Once mind-boggling, yet now -- in this vaster context -- almost cuddly.)  These in turn can be slightly re-generalized, by considering pseudometrics; or further regimented, with the concept of a norm, which in turn may be relaxed into a seminorm;  and so it goes.

~

A quite different and likewise fruitful generalization of metric spaces  is the notion of a Uniform Space, introduced by algebraic geometer André Weil, in “Sur les espaces à structure uniforme et sur la topologie générale” (reprinted in volume I of his Collected Papers as [1937]).   He broaches it with a bang:

La notion de distance  est utilisée dans de nombreux travaux de topologie, [mais] l’on s’explique mal qu’elle soit venue à jouer un pareil rôle  dans une branche des mathématiques  où elle n’est, à proprement parler, qu’une intruse…

On voit apparaître ici  cette hypothèse du dénombrable (dite aussi, on ne sait pourquoi, de séparabilité),  malfaisant parasite qui infeste tant de livres … dont il affaiblit la portée  tout en nuisant à une claire compréhension des phénomènes.  … La conscience d’un mathématicien, s’il en possède [!], doit répugner à faire intervenir une hypothèse superflue …

Strong words !   The notion of metric, he claims, is not simply too restrictive, but is the wrong sort of notion for topology -- a cuckoo’s-egg in the nest.   And indeed, minus the polemics, James Dugundji makes the same point (Topology, p. 200):

A metric … can be regarded a providing a measure of nearness that is applicable throughout the space  … This notion of uniform smallness is not a topological concept :  equivalent metrics specify different sets as being equally small.

… Notice that, even in metric spaces, a continuous map may be uniformly continuous if one pair of metrics is used, but not uniformly continuous when another pair of equivalent metrics is used;  uniform continuity is therefore  not a topological concept.

(“Equivalent” metrics in the sense that they generate the same roster of open sets, which define the topology.)

Contrast a different -- and very fruitful -- restriction on general topological spaces, that of being compact Hausdorff.  This notion is strictly topological in spirit.

Footnote:   For another instance of Gallic arithmophobia, cf. the remarks of Weil’s countryman  Jean Dieudonné, in Foundations of Modern Analysis (1960), p. 141:

The fundamental idea of Calculus [is] the “local” approximation of functions by linear functions.  In the classical teaching of Calculus, this idea is immediately obscured  by the accidental fact that, on a one-dimensional vector space, there is a one-to-one correspondence between linear forms and numbers, and therefore the derivative at a point is defined [horresco referens !] as  number instead of a linear form.

In defense of Sir Isaac Newton, it must be observed, that our worthy ancestor was  quite understandably  interested in how fast something was going, at each moment:  to answer which question, he needed to invent the differential calculus.  Dieudonné, from the vantage point of centuries of progress, is looking ahead to function-spaces and dense subsets of special functions and like that.

~

The passages immediately above  evoke, unbidden, an untoward echo  characteristic of their times (the Thirties; the Sixties):  “unAmerican” and (failure to adhere to) “Chairman Mao’s Correct Line”.   But “topological” is not an all-or-nothing concept;  and we return to sanity  with jolly John Kelly (General Topology), in the chapter titled “Uniform Spaces”:

We deduce from a topological premise (that the space is compact) a non-topological conclusion (that a function is uniformly continuous).  This chapter is devoted to a study of quasi-topological results of this sort.

Even more telling is the remark by George Simmons, author of the superbly pedagogical Introduction to Topology and Modern Analysis (1963):

Some writers deal with the theory of metric  spaces as if it were merely a fragment of the general theory of topological spaces.  This practice is no doubt logically correct, but it seems to me to violate the natural relations between these topics, in which metric spaces motivate the more general theory.

Thus, it is scarcely fair, or psychologically realistic, to denounce the notion of metric as an “intruder” in topology, as Weil does.  Similarly:  you shouldn’t start off with categories and functors  before learning about  ordinary numbers and sets, even if categories prove ultimately more foundational.

That said, there does come a point where actual everyday examples impel one to consider such things as convergence and compactness  in a setting more general than a metric space.  As: pointwise convergence, which is a perfectly familiar non-exotic sort of convergence, but which cannot be seen as convergence with respect to a metric.

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We have thus seen uniform space as a gentle generalization of metric spaces.  Since the point of the latter is often concerned largely with matters of limits and convergence, all we really need to know is what it means to get “closer and closer”;  we don’t need to put a number on how close, each step of the way.   This aspect was highlighted by André Weil, when he debuted the idea of uniform spaces, as a kind of intellectual hygiene.   But in practice,  quite as important to the introducer of uniform spaces is their natural application to topological groups, which come ready-made with a structure amenable to notions of nearness.

But there is more.   John Kelley, in his General Topology (1955), who devotes an entire chapter to uniform spaces, writes:

It should be emphasized that this is by no means the only framework in which uniformity can be studied.  It is possible to study a set X  together with a distinguished family of pseudo-metrics for X, or to distinguish a collection of covers of X where are to be uniform covers (roughly in the sense of the Lebesgue covering lemma).  One may also consider “metrics” with values in a structure less restricted than that of the real numbers.  All of these notions are essentially equivalent.

Such a situation illustrates a recurring intellectual theme of this series of essays (with both philosophical and mathematical applications), tagged as “Rome by different roads”.   There is a section on this notion in our essay Consilience in mathematics (indeed, in one sense, the entire notion of consilience in general  is related to this idea).

Credo (continued)

[continues this]

I see no reason why we should have less confidence in mathematical intuition, than in sense-perception.

          -- Kurt Gödel

            To grant necessary ontological status  only to the non-negative integers, as does Kronecker and his thought-mates, resembles the attitude which prevailed among geometers from Euclid down to Gauss and Lobachevsky, which considered geometry as described by Euclid  to be the only possible geometry.  It was a word with no plural, like “universe”; that both these terms are now pluralizable (the latter in a conjectural, the former in a now very precise sense) represents a triumph of the human spirit, which now knocks tankards with the Invisible, in a toast to Him who made us all, the math along with the meat.  The old geometers believed this because they fetishized the visible: the flat, drab world (well, pied with beauty, true, but drab compared with the full panoply of Riemann surfaces and Finsler space) whose mensuration is approximated so closely by Euclidean geometry  -- at least, that stretch of turf that lies visibly close to hand – that the actual deviations that do exist cannot be detected by ordinary, plain-man means; and those who scoff at the invisible infinite, are very plain men indeed.  And it was this servitude to the locally visible – which is  in particular  to say, to the contingent – which caused the finest minds to fritter fruitlessly after a derivation of the Parallel Postulate from the rest of the Euclidean axioms, a fiasco that lasted literally for over two thousand years, from antiquity down to the nineteenth century. Indeed, it was not until we became familiar with the “invisible” worlds  revealed to us by Lobachevsky, Riemann, Klein and Poincaré  that we became fully clear on the status of Euclid’s axioms, and the distinction between axiomatics and model theory.

            So, to argue concretely:  If the number one is real, then so is a half, for I give you half this pie. And if “one” is real then so must be the square root of two, as being the measure of the hypotenuse of the isosceles right triangle, by the inexorable evidence of the Pythagorean Theorem. (Likewise the square root of 5, 13, 17, etc., and thus their products.) And the square root of two turns out not to be a ratio of natural numbers. Now, Pythagorus himself shrank from this conclusion, and stigmatized the postulant entity as irrational; yet now we take them for granted.   And if third roots or electricity are real, then so are the imaginary numbers required for their description; and if matter is real, and with it atoms, then so is quantum mechanics: which mean that compact operators on Hilbert Space are real:  you cannot see them or touch them, but you can almost hear them, buzzing all around us…  It is a slippery slope (we might almost say, a declivity whose slope is infinite) when we admit the reality of the visible world: it quickly (“quickly”, considered sub specie aeternitatis) drags in all the invisibles wíth it.

            It is true, the atmosphere around those higher turrets is rather rarefied.  Sometimes we become light-headed, and wonder if we are not after all just making some of this stuff up.  Yet no sooner do we begin to doubt our senses – or rather, to put too much trust in our senses, and too little in our carefully nurtured sense of the unseen – than Nature coughs up some concrete correspondence with our most arcane designs.  That same Riemann hypothesis now seems to be in some strange harmony with the energy-levels of atoms.  And as for connections on fibre bundles – meet the gauge fields of particle physics, your twin, separated at birth!

*

            The question remains, whether mathematical entities are, so to speak, real in general, or only real within a particular reality: much as a planet might pursue its course, in our universe but not another.  Now, the examples of mathematical reality adduced thus far, have all been given a clean bill of health by our actual, particular universe.  Hilbert space is as much a part of our daily reality as are porpoises – quite as vibrant, almost as much fun, and much less likely to go extinct.  But the Continuum Hypothesis… ahh.  That’s another matter.  One would really like to have a better handle on that one.  There are models for set theory in which it is true, and models in which it is false.  This tends to make us acutely uncomfortable, since, unlike the logically equivalent but intuitively more ethereal Axiom of Choice, it’s the sort of thing where you feel there ought to be a plain fact of the matter.  Nevertheless, its status may be ultimately no worse than that of the parallel postulate, which is quite placidly and understandably true in some geometries, false in others, and indeed true in our own universe at appropriately small scales (here I mean, of course, not what happens unobserved at infinity, but such local effects as the sum of the angles of a triangle), while false at others.

[concluded here]

E8: a Riposte (concluded)

Let us examine a bit more closely  Synge’s picture of physics as bricolage,  where theories have the intellectual status of just-so stories, and are really little more than pragmatic techniques, or tools -- Newtonian mechanics and relativistic mechanics each useful in its own sphere, like screwdrivers and spoons, but of little interest in their own right.   Now, this is not to knock the status of a toolkit -- my respect for competent carpenters and electricians borders on reverence -- but fundamental physics is not like that.

            Synge presents the Newtonian view as having not been replaced or refuted by relativity;  it rules as before in its own realm.  Newton’s good for some things, Einstein for others, and Wiccan no doubt for others still.   But this view assumes a confusion.  For it is not the case that Newtonism and relativity are independently valid in their own way but incompatible;  rather, Newtonism is the limiting case of relativity, in a way very familiar in mathematics;  its continued use in everyday life is simply a calculational convenience, a shortcut.   To continue the tool metaphore:  Einstein and Newton are not like screwdriver and pliers, but like a hammer, and an old shoe used as a hammer, good enough for the task at hand.

            Furthermore, it is a good thing, not a bad thing, when initially separate paths converge.  If you only know one way to climb a thing,  perhaps it is only a Potemkin mountain -- a paper-maché façade, hollow behind the north slope.   It is quite a relief -- and an ontological ratification -- to meet another mountaineering party that has scaled up the other side.

            The reader may be familiar with the story of how Schrödinger and Heisenberg separately found Rome by different roads.  Let George Gamow tell it, in Thirty Years that Shook Physics (1966), p. 3:

The simultaneous appearance of Schrödinger’s and Heisenberg’s papers  in two different German magazines … astonished the world of theoretical physics.  These two papers looked as different as they could be, but led to exactly the same results concerning atomic structure and spectra.

We are, in hindsight, not overly surprised by this, since by now we most of us accept that there is something there at the quantum level, something real, something other than subjective, to be described.   It is describable by two quite different mathematical approaches, much as our peak may be scaled by walking up the north face  or rappelling up the southern cliffs.    Nor is such ‘duplication of effort’ a waste of time, for  in this instance, not only the factual success, but even the approaches themselves retained their usefulness -- for determining energy levels, Schrödinger’s wave mechanics was calculationally more convenient; and Heisenberg’s matrix methods had the edge when it came to calculated the intensities of the radiated frequencies.   Or, alternately, P.A.M. Dirac, The Principles of Quantum Mechanics (4^th edn. 1958), p. viii:

Quantum mechanics … is known under one or other of the two names ‘Wave Mechanics’ and ‘Matrix Mechanics’, according to which physical things receive the emphasis in the treatment, the states of a system or its dynamical variables.

And (p. 115):

The Schrödinger form is the more useful one for practical problems, as it provides the simpler equations. … Heisenberg’s form for the equations of motion  is of value in providing an immediate analogy with classical mechanics.

Or again (R. F. Streater & A. S. Wightman, PCT, Spin & Statistics, and All That (1964), p. 4):

Throughout this book, states will be described in the Heisenberg picture of quantum mechanics.  The Schrödinger picture is much less convenient for the description of a relativistic theory, because it treats the time coordinate on a very different footing from the space coordinates.

And:

P.A.M. Dirac, The Principles of Quantum Mechanics (4^th edn. 1958), p. 311:

The Schrödinger picture is unsuited for dealing with quantum electrodynamics, because the vacuum fluctuations play such a dominant role in it. … They get bypassed when one uses the Heisenberg picture, and one is then able to concentrate on qualities that are of physical importance.

Dr. Matrix

Dr. Wave

Approaching an abstract but genuine reality from two different theoretical complexes  has its counterpart in different experiments, or different means of calculation, strengthen each other when they arrive at the same result.   Thus Einstein, in his annus mirabilis of 1905, when not inventing Relativity, found it worth his while  to “develop theoretically  three independent methods for finding Avogadro’s number.” (Abraham Pais, Subtle is the Lord (1982), p. 55.)   It was worth his while because, independently of our endeavors, this number is indeed there.

Summarizing:  For epistemology, the fact that two or more radically different approaches each manages to describe the phenomenon of interest, reassures us that we really do have our arms around this thing.   The lesson goes over, I would submit, in cases where what is being described is nothing so tangible as an atom (which Rutherford reportedly saw in front of his face as plainly as a spoon), but rather a four-manifold, or a simple Lie group.

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Afterword.

I recently happened across the following curious passage:

The algebras G_2, […] E_8  are called exceptional.  In 1945, Chevalley remarked  that the existence of these algebras  is a brutal act of Providence  which we must accept blindly.  Perhaps this should be revised today  to assert that the source of these algebras  is the wisdom of the Deity  in allowing the Cayley numbers to exist.

-- Irving Kaplansky, “Lie Algebras”; in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 126

~ ~ ~

There’s one further type of brane in M-theory  that is really surprising.  This brane is the edge of spacetime. … The photons at the edge of spacetime participate in supersymmetric E₈ gauge theory.

-- Steven Gubser, The Little Book of String Theory (2010), p. 95