A mathematical scratchpad (even further scratched)

[There simply isn’t time, at least before retirement, to integrate each thought-balloon as it bubbles up -- a proto-insight or pre-idea -- into the appropriate essayistic context in finished form.   Yet to leave these on the desktop equivalent of a desk drawer is to tempt the Reaper.   Therefore I shall place some of them here -- philosophical post-it notes;  mathematical Zettel.]

[Cf. De stultitia]

Our focus  in the essay of that name, is on the plight of those sorry souls (99.9999999 % of us) who fail to grasp what Grothendieck, or Witten, or whom-have-you, saw easily enough.

Distinct from, though related to, this, are questions of which (at the forefront of science) we are permitted a glimpse,  but which nobody understands.  As:

On cosmogenesis:

The whole vast imposing structure  organizes iteself  from absolutely  nothing.

This is not simply  difficult to grasp.   It  is    incomprehensible.

-- David Berlinski,  “Was There a Big Bang?” (1998), collected in :  The Deniable Darwin (2009), p. 229

And:

All this leaves us  where we so often find ourselves.  We are confronted with certain open questions.  We do not know the answers, but what is worse, we have no clear idea -- no idea whatsoever -- of how they might be answered. 

But perhaps that is where we should be left:  in the dark, tortured by confusing hints, … and a sense that, dear God, we really do not yet understand.

-- David Berlinski,  “God, Man, and Physics” collected in :  The Deniable Darwin (2009), p. 270

~

The hardest part of a subject is the beginning.  Once a certain stage is passed, we gain confidence  and feel that, if need be, we could carry on by ourselves.

-- John Synge & Byron Griffith,  Principles of Mechanics (1942, 1959), p. 506

Alas, that has not been my experience at all.

Any technical subject is like a whirligig, which rotates faster and faster until the centrifugal force throws you off.   It’s like the Peter Principle, everyone eventually reaching his own personal level of incompetence;  only, in math and in physics, these levels stack indefinitely towards heaven, so that a few of us can ascend quite a ways, before we are finally out of our element.

[Cf.  Any Ideas? ] 

Recent years have seen striking developments in the conceptual organization of mathematics.  There developments use certain new concepts  such as “module”, “category”, and “morphism”  which are algebraic in character.

-- Saunders MacLane & Garrett Birkhoff, Algebra (1967; 3^rd edn. 1999), p. vii

The reason they speak here of new “concepts” rather than additional structures  is that the notions referred to do not exist merely within algebra, but serve to organize other mathematical fields as well.

~

In an exterior view of the finished product, we see structure mathematics as largely logical or deductive:  P entails Q.

But from the interior standpoint of the practicing mathematician (and here, though we refer to the ‘actio’ sense of mathematicizing as opposed to the actum or product, the interest is not psychological but ideational), a key verb is rather motivate:  P motivates Q.   An illustration of this special vocabulary:  “The desire to extend Fourier L² to L^p spaces  motivates the Riesz interpolation theorem.”

~

More vocabulary from the conceptual domain:  thrust, as in the following passage

The Heisenberg uncertainty principle:  The mathematical thrust of the principle can be formulated in terms of a relation between a function and its Fourier transform.  The basic underlying law, formulated in its vaguest and most general form [i.e., its most intuitive formulation], states that a function and its Fourier transform cannot both be essentially localized.

-- Elias Stein & Rami Shakarchi, Fourier Analysis (2003), p. 158

~   ~   ~

[Cf.  On Depth]

When I made my original discovery of radiation from black holes, it seemed a miracle that a rather messy calculation should lead to emission that was exactly thermal.  However, joint work with Jim Hartle and Gary Gibbons  uncovered the deep reason.

-- Stephen Hawking, in: Stephen Hawking & Roger Penrose, The Nature of Space and Time (1996), p. 44

For the mathematician, contrasting with messy  are simple and elegant -- yet in the following, even these don’t get you to the yonder side, where Depth dwells:

Having derived the equation for the vibrating string, we now explain two methods to solve it:

(1) using traveling waves;

(2) using the superposition of standing waves.

While the first approach is very simple and elegant, it does not give full insight into the problem.

-- Elias Stein & Rami Shakarchi, Fourier Analysis (2003), p.  8

String theory is sometimes described as a theory that was invented backwards … People had pieces of it quite well worked out  without understanding the deep meaning of their results. … Math is funny that way.  Formulas can sometimes be manipulated, checked, and extended  witnout being deeply understood.

--Steven Gubser, The Little Book of String Theory (2010), p. 2

[Cf. Consilence in Mathematics]

Horizontal consilience:

… the structure theorem for finitely generated groups -- a fine illustration of conceptual unification.

-- Saunders MacLane & Garrett Birkhoff, Algebra (1967; 3^rd edn. 1999), p. vi

Mathematics is a coherent, interlocking whole, and advances in one area  often lead to advances elsewhere.

-- Ian Stewart,  How to Cut a Cake (2006), p. 89

*

Commercial Break

A private detective  confronts the uncanny;

an ecclesiastical mystery:

Murphy Calls In a Specialist

*

[Cf. Language and Math]

Expressing himself in the language of fluxions and fluents, Newton managed to conceal his insights in a notation that was miraculously maladroit.  Not so Leibniz.  The language of mathematics and mathematics itself  are mutually sustaining.

-- David Berlinski, Newton’s Gift (2000), p. 57

Note:  The first clause of that observation does not actually relate to the point about notation (as opposed to vocabulary), and is silly in itself.  The concepts were new, so obviously any term for these would either be an out-and-out neologism, or a semantic hijacking of an extant word.   There is nothing lexically more rebarbative about fluent and fluxion than about derivative, differential, infinitessimal. 

*

[Cf. What is Mathematics.]

Betrand Russell, in The Principles of Mathematics (1903):

Mathematics is the class of all propositions of the form ‘p implies q’ …

The appended dribble of dots replace additional uninteresting clauses, which rob the sally of its epigrammatic pithiness, while yet failing to throw any light upon the subject.   It is a definition for people with no interest in the dark loamy richness of actual math as such;  and worthy of the author whose massive Principia Mathematica could as well have been titled Why Math Isn’t Interesting After All.

The characterization becomes even less interesting when you reflect that the expression “p implies q”, in logician’s lingo, is mere ‘material implication’ -- what would be better dubbed immaterial implication, since it involves no notion of causation or even logical entailment (and is thus immaterial to any actual problem), but is neither more nor less than another way of saying “either not-p, or q”.

[cf. Axiomatics in its Element]

Two citations illustrating the insight that axiomatizations, though perhaps logically prior, are pragmatically post-hoc:

The order of nature, and the order of logical dependence, are not the same as the order of our discoveries.

-- Morris Cohen & Ernest Nagel,  An Introduction to Logic and Scientific Method (1934)

Not all axiom systems are formal systems, and formalization need not lead to axiomatization.

The axiomatic method is an orderly way of summarizing experience.

-- Hao Wang, Popular Lectures in Mathematical Logic  (1981), p. 11

[Cf. Oligophrenia mathematica]

I am not a mathematician, but a math groupie;  not even a math wannabe (as I once was, back in Math 55), but a math wannedabe.

And in fact, considered coldly, I did not then  even rise to the level of a wannabe, but only a meta-wannabe, a wannawannabe:  someone who wished that his dearest wish was for mathematics, but who, truth to tell, was more interested in history and literature.

[Cf. Minimalism in Mathematics]

On Ramanujan’s notebooks:

There were thousands of theorems, corollaries, and examples.  For page after page, they stretched on, rarely watered down by proof or explanation, almost aphoristic in their compression, all their mathematical truths  boiled down to a line or two.

-- Robert Kanigel, The Man who Knew Infinity, p. 204

The reasons for this were twofold.  Ramanujan himself was not particularly aphoristic.   But he had never absorbed the modern notion of proof, which would take up so much more space;  and as a poor man in India, he suffered from a shortage of paper.

*

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*

[Sui generis]

On Ramanujan, who grew up in India, and in mathematics  was largely self-taught:

He was like a species that had branched off from the main evolutionary line  and, like an Australian echidna or a Galapagos tortoise, had come to occupy a biological niche all his own.

-- Robert Kanigel, The Man who Knew Infinity (1991), p. 61

The unusual career of Ramanujan  is one of the most celebrated biographies in the history of mathematics.   His achievements in the face of relative intellectual adversity as a child of modest means in the rural subcontinent,  are indeed inspiring, and warm the hearts of those in quest of Diversity -- whence, for those who can decipher the trobar clus of modern peri-academic patois, the book’s subtitle,  “A Life of the Genius Ramanujan”.   (Genius he indisputably was;  but the word these days is mainly used to celebrate anyone other than straight white males -- a “genius at basketball” or whatever.)

Yet the larger lesson is not how divergent Ramanujan was, but how much in the mainstream of things:  He did not found a new field of mathematics, he worked within number theory.   And this fact in turn reminds us of two characterizations of math as a whole, on which we have often dwelt:

            (a)  It is not something we invent out of whole cloth, it is something we discover.   This must channel our discoveries, just as the facts of the actual universe  discipline physics.

            (b)  Mathematics has already, for at least two hundred years, been uniquely rich conceptually  among human endeavors.   In a landscape embracing Cantorian set theory, algebraic geometry, and topos theory, it is next to impossible to come up with something unprecedentedly deep and strange;  in any case, Ramanujan did not.   To return to the metaphor:  We certainly treasure our quirky friend the echidna;  but only to someone whose zoological experience extended no further than a European barnyard, would he seem all that aberrant.   In a world of social insects, benthic hypothermophiles, and communal slime-molds, the echidna seems like just one more furry friend.

~

~  Posthumous Endorsement ~

"If I were alive today, and in the mood for a mystery,

this is what I'd be reading: "

I Don't Do Divorce Cases

Murphy on the Mount.

(My name is Ramanujan, and I approved this message.)

~         ~

~

.

The Realist Vernacular

            What follows is neither proof nor argument, nor philosophy of any sort, but rather an exercise in sociolinguistics.  [And as such, a sort of sociological preparation for the thread announced here.]  The point is simply to exhibit a common way of talking among contemporary mathematicians, as well as some physicists and philosophers -- an easy style of conversation  that you would never imagine exists, if most of your acquaintance with science and its philosophy is mediated by figures like Daniel Dennett and Richard Dawkins.

            To cite evidence of theistic talk from the learned men of history, from antiquity through the nineteenth century, would be pointless, since the mode was well-nigh universal, at all times and in all realms.  Yet in our present day, most of the habitués of faculty clubs and coffee-houses  have managed to satisfy themselves, that all those who ever lived, in history, from Pythagorus  to the Einstein of “Der Herrgott würfelt nicht”, were, without exception,  imbeciles, and simply didn’t know what they were talking about, when they talked that way.  At last mankind has seen the light (or the darkness, rather);  we speak only of what is sensible and sniffable, like Donald Trump.  And who should be so rash as to cite the testimony of a mere Plato, or Galileo, or Cantor, or Gödel, against the brass certitude of so eminent a scholar as Sir Christopher Hitchens, Ph.D?

            Therefore I shall limit myself to recent citations.  A few examples from among many, snatched at random from  recent reading.

            Again, note:   I am not implying anything about the theology, per se, of the people here quoted, let alone suggesting that they never miss Mass.  Indeed a Dennett -- a roaring atheist -- can still indulge in such turns of phrases as "we can take advantage of the God's-eye perspective we have temporarily adopted" (the Devil can quote Scripture to his purpose).   So far, this is just corpus linguistics;  but more anon.

(1) Theistic language

 (a) mathematics

Arthur Koestler, The Act of Creation (1964):

Karl Friedrich Gauss described how he finally proved a theorem on which he had worked unsuccessfully for four years:  “At last, two days ago, I succeeded, not by dint of painful effort, but so to speak  by the grace of God.”

Thomas Tymoczko, ed., New Directions in the Philosophy of Mathematics (1986, rev. 1998), p. 219:

In one of the most cited discussions in his much-quoted book, Kuhn [1962] talks of scientific decision in terms of “conversion experience” and “faith”.

Richard de Millo et al., “Social Processes and Proofs”, in Thomas Tymoczko, ed., New Directions in the Philosophy of Mathematics (1986, rev. 1998), p. 271:

The classical view does not require that an ordinary proof be accompanied by its formal counterpart; on the contrary, there are mathematically sound reasons for allowing the gods to formalize most of our arguments.

Similar language is often used in formulating the contemporary concept of a “hypertask”.  Cf. likewise

Shaughan Lavine, Understanding the Infinite (1994), p. 55:

            For Cantor … “countable” meant countable by God.

and similarly

Michael Potter, Set Theory and its Philosophy (2004), p. 250, re the plausibility of the Axiom of Choice:

… generalizing to the uncountable case  by appeal to the idea than an ideal being could achieve the choices required of him (or perhaps Him).

If we adopt some particular postulate system for abstract set theory, and agree that the criterion for accepting an intuitive set-theoretic argument  is that its analogue can be justified by the postuates, then we are  in efect  agreeing that the set of all intuitive sets  endowed with the membership relation  is a configuration satisfying the postulates.  We can have at best  intuitive reasons for believing this.  It is really an act of faith.

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p. 153

Gregory Chaitin, “Gödel’s Theorem and Information”, in Thomas Tymoczko, ed., New Directions in the Philosophy of Mathematics (1986, rev. 1998), p. 306:

If God tells one how many different programs of size less than N halt, this can be expressed as an N-bit base-two numeral, and from it  one could eventually deduce  which of these programs half  and which do not.  An alternative divine revelation would be knowing that program of size less than N which takes longest to halt.

Robin Wilson, Four Colors Suffice (2002), p. 214, recounting a mathematician’s reaction to the immensely long and repetitive, unsurveyable, computer-aided proof of the Four Color Conjecture:

God wouldn’t let the theorem be proved by a method as terrible as that!

 (b) physics

Hermann Weyl, Symmetry (1952):

Contingency is an essential feature of the world.   Clarke  in his controversy with Leibniz  admitted the latter’s principle of sufficient reason, but added that the sufficient reason often lies in the mere will of God.  I think, here Leibniz the rationalist is definitely wrong, and Clarke [the theist] on the right track.  But it would have been more sincere to deny the principle of sufficient reason altogether, instead of making God responsible for all that is unreason in the world.

Abdus Salam (1988):

God created just two dimensions -- one of space and one of time. … At a later epoch, there was a phase transition to four dimensions, plus six internal ones.

Stephen Hawking, A Brief History of Time (1988; 2^nd edn. 1996) p. 91, alludes to Roger Penrose’s paraphrase of cosmic censorship:  “God abhors a naked singularity.”   The (jocularly) theistic language is especially odd and noteworthy, since the epigram here being echoed -- “Nature abhors a vacuum” (Latin:  horror vacui) does not use it.

Stephen Hawking, A Brief History of Time (1988; 2^nd edn. 1996) p. 126f.:

These laws [of physics] may have originally been decreed by God, but it appears that he has since left the universe to evolve according to them  and does not now intervene…

That, then (for that author) leaves only the explanation of the settings of the parameters for initial conditions:

One possible answer is to say that God chose the initial configuration of the universe for reasons that we cannot hope to understand.  This would certainly have been within the power of an omnipotent being, but if he had started it off in such an incomprehensible way, why did he choose to let it evolve according to laws that we could understand?

Simon Blackburn, Think (1999), p. 69, in a section called “A Scientific Model”:

Classical physics identifies the temperature of a gas with the mean kinetic energies of the molecules that compose it.  So in making hot gases, God has only one thing to fix…

Blackburn is a philosopher, not a physicist; as commonly in that community, he is writing in an “as if” mode;  but still, the choice of words is noteworthy.

John Gribbin & Martin Rees, quoted in Edward Harrison, Cosmology (2^nd edn. 2000), p. 489:  Absent the strong anthropic principle,

If there is a unique ‘theory of everything’, then … we would have to accept it as genuinely coincidental, or even providential, that the constants determined by high-energy physics happen to lie in the narrowly restricted range that allows complexity and consciousness to evolve…

(“Providential”… a lovely word…)

Paul Davies, The Goldilocks Enigma (2006), p. 3:

It appeared to Hoyle as if a superintellect had been ‘monkeying’ with the laws of physics. … Like the porridge in the tale of Goldilocks … the universe seems to be ‘just right’ for life.

Paul Davies, The Goldilocks Enigma (2006), p. 236:

Most theoretical physicists are Platonists in the way they conceptualize the laws of physics as precise mathematical relationships  possessing a real, independent existence…

Shing-Tung Yau, The Shape of Inner Space (2010), p. 101:

As Robert Greene puts it, “you’re trying to find the one metric given you by God.”

Michael Atiyah re string theory, quoted in Shing-Tung Yau, The Shape of Inner Space (2010), p. 292:

They’re onto something, obviously.  Whether that something is what God’s created for the universe  remains to be seen.  But if He didn’t do it for the universe, it must have been for something.

This statement is reminiscent of the Principle of Plenitude; and recalls Einstein’s celebrated quip, anent the possible failure of an experiment to confirm his prediction: "Da täte mir halt der liebe Gott leid; die Theorie stimmt doch."

Less seriously, but using a religious metaphor: J. L. Synge, Relativity:  The General Theory (1960), p.  ix:

It is to support Minkowski’s way of looking at relativity that I find myself pursuing the hard path of a missionary.

More serious is the following.  The author is discussing the vexed question of the Collapse of the Wave-Packet upon observation (related to:  If a tree falls in a forest, and no-one’s around, does it make a sound?  -- here rewritten in Oxford terms, replacing the forest by a quad), and quotes the old limerick:

Dear Sir, Your astonishment’s odd;

I am always about in the Quad.

And that’s why the tree

Will continue to be,

Since observed by Yours faithfully, God.

He comments (J. C. Polkinghorne, The Quantum World (1984), p. 67):

Divine reduction of wavepackets would be an overkill, since it would operate everywhere and always, forcing the electron each time to go through a definite slit.  The point about measurement is that it only occurs spasmodically.

Note the predicted empirical consequences of God in the Quad!   Then, between square brackets, the author adds:

This observation is in accord with the classic theological understanding of creation, which sees God as the ground and support of all that is (in our terms, the guarantor of the Schrödinger equation), but not as an object among objects (no collapser of wavepackets).

This aside is in fact central:  by the time he wrote this, Polkinghorne had quit his endowed Chair in physics to become a village vicar.

(c ) analytical philosophy

Michael Dummett, Truth and other enigmas (1978), p. 15, discussing character as (it may be, untested) hidden propensities:

If B still wishes to maintain the necessity of ‘Either Jones was brave or he was not’, he will have to old  either that there must be some fact of the sort to which we usually appeal … or else that there is some fact of an extraordinary kind, perhaps known only to God.

Dummett’s use of the term, to characterise the viewpoint of a hypothetical philosopher inclined to Realism, is so to speak opaque, not representing his own view, which is rather (id, p. 150) that “only a philosophically quite naïve person would adopt  realist view of statements about character”.  Only such, or Saint Peter.

(2) Realist language

Arthur Koestler, The Act of Creation (1964):

Gauss is reported to have said: “I have had my solutions for a long time, but I do not yet know how I am to arrive at them.”

Klaus Jänich,  Topology (1980; Eng. trans. 1984), p. 35:

When topological groups are found in nature [emphasis added], they  are generally not given abstractly as a set G with a composition law and a topology, but concretely, as a group of transformations…

This is not really anymore outrageously realist than saying “When a set of six objects is found in nature…”   (say, the familiar six-pack; although the one at my side is already down to five, and it’s not even noon).

And again, p. 157:

Covering spaces very often “occur in nature”:  that is, one comes across them spontaneously, while studying entirely different problems.

Shaughan Lavine, Understanding the Infinite (1994), p. 160: 

We seem to have nontrivial intuitions concerning the infinite, going far beyond simple things like Extensionality, Pairing, or even Power Set.

Shing-Tung Yau, The Shape of Inner Space (2010), p. ix:

The strength of this discipline [i.e., mathematics] lies not simply in its ability to explain physical reality…, because to a mathematician, mathematics is reality.

If all you know of math is elementary arithmetic, this statement may lack punch.   But in the upper reaches of set theory, topology and so on, it embraces a world of miracles and of monsters.

~ ~ ~

It is important to note, that this is the way Realists talk en famille.  The talk is casual, often not literal, yet is meant in some serious sense.  It is not to be compared with the tawdry pseudo-theological, pseudo-mystical sort of gibberish that popular authors (and even some serious ones, yielding no doubt to the Satanic promptings of the marketing department) use to gin up their pap for the masses -- like “God particle” for the freaking Higgs boson.

It may be objected (it will be objected;  it has been objected) that such expressions, in a modern mouth, are a mere façon de parler.  To which we reply (with Whorf), that a façon de parler tends to cohere with a façon de penser. 

Nor is it a refutation of the point here made, to adduce agnostic pseudepigrapha from any of the gentlemen here quoted.   We are each a walking contradiction;  we contain multitudes.   C.S. Lewis himself  confessed that he tended to be a cranky agnostic at dawn, but a theist later, when he’d had tea and was more himself.

Once again:   The point here is not to assert that so-and-so among our near contemporaries  is or is not a believer.  Indeed it will strengthen my eventual case, if many of them are not in fact believers, yet find themselves attracted, or guided, or driven, to Realist or Theistic language, whether for convenience, or (in the case of Cantor, Gödel, and Einstein) something deeper.

So, a very modest initial move, a sort of pawn to king’s four.  (Only later -- much later -- shall we see if we can capture Satan’s Queen.)   So far we hold simply, that occasional use of Realist or Theist language, in serious discourse, is not  in and of itself  diagnostic for Trisomy 21.

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.

~ ~ ~

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

E8: a Riposte (continued)

[A continuation of this.]

Synge waits until well into his second volume (J. L. Synge, Relativity:  The General Theory (1960), p.  104) to really let rip against Realism; and since he was himself very much a mathematical physicist, rather than an empirical experimenter, his testimony must be respected as coming from within the tent.  He distinguishes “Natural Observations” (NO) from “Mathematical Observations”, and opines:

Between NO and MO  there is a sharp and decisive break.  Only the simplest MO (counting) can be regarded as being NO also … Generally MO involve infinity (irrational numbers, differential calculus, and so on) and so lie outside physics and outside nature.

This is exactly the position of Kronecker (“Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk”).

He then delivers himself of a curious passage, which Irrationalists would seize on with glee (fortunately none of them are reading this):

The peculiar fascination of theoretical physics  lies in the art of forcing meaningful truth out of the meaningless equation NO = MO, which is a symbolic form of the assertion that natural phenomena obey exact mathematical laws.  The true inequality NO =/= MO should not be spoken above a whisper, because it is extremely dangerous.   If believed, it would sever mathematics from physics, and reduce both to sterility through lack of mutual fecundation.  It is whispered here only as an apology to those readers who expect to see the mathematics of relativity [which he presents in great detail] tied to the physics of relativity  by a strong chain of clear thought.  It cannot be done.

These ring like the Night Thoughts of a relativistic physicist, on the eve of taking his own life.

            Despite his conspiritorial tone in that passage, Synge was by no means alone in his reservations. Here is another anti-Realist view from the world of physics:

A. D’Abro, The Rise of the New Physics (1939), vol. II, p. 728:

A hyperspace is obviously a mathematical fiction; and waves that can be represented only in a fictitious space  must themselves be unreal.

Now here indeed is a statement that has been overtaken by events.  In the view of string theory, this hyperspace, far from a mathematical fiction, is a physical fact, the one we live in;  indeed, we must beware lest those compactified but very real extra dimensions someday unfurl in our faces.  --  The point here being, not to make any point whatsoever about cosmic geometry, let alone to proclaim the truth of string theory:  but simply to counsel against that “obviously”, when dismissing the Realist picture.

[concluded here]

E8: a riposte

[a continuation of this]

Writes J. L. Synge,  in Relativity:  The Special Theory (2^nd edn. 1965), p.  163:

According to this hypothesis [viz. that of a unique mathematical structure for nature], the mathematical formulae of physics are discovered  not invented,  the Lorentz transformation, for example, being as much a part of physical reality as a table or a chair.

Hear, hear! The Realist raises his glass with a sigh, basking in the warm glow of these purling words.  -- But suddenly, the author surprises us with a basin of ice-water in the face:

But this hypothesis of a unique mathematical structure for nature  is actually very naïve.  It is the product of the eighteenth century, a period when mathematics was understood much less than it is today, and it is unacceptable to any physicist who has thought about mathematics, or any mathematician who has thought about physics. [Oh, snap!] When understood properly (i.e. as mathematicians understand them) these concepts exist in the human mind and not in nature;  it is a meaningless waste of time to debate whether the ratio of two measured lengths is rational or irrational, or whether matter is continuous or discontinuous, because the concepts of irrationality and continuity belong to a world of the intellect, a world of mathematics, and not to the real world in which phenomena occur and are measured by pieces of apparatus.

(… slow burn…)

And again, p. 207:

Is matter really discrete or continuous? … That question must be regarded as quite meaningless.  For ‘continuous’ is a mathematical word, not a physical word, and has only a very vague bearing on nature;  we must not try to attach physical meanings to mathematical concepts which involve infinite processes.

[…]  The above remarks merely underline the philosophical attitude [quoted above].  It is a theme that bears repetitition.

( Somehow in these last bits, I detect the tones of Dolores Umbridge…)

And p. 308:

We use  now this mathematical representation, now that,  seeking those representations which are convenient to work with  and which yield at least some correct physical predictions.

            Now, we quite agree that it doesn’t make sense to say whether a numerical physical measurement is rational or irrational.   And certainly, in physics, infinities are tricky.   The continuum may indeed not be what the doctor ordered for the texture of spacetime.  And we agree that it may be convenient to employ this or that formalism for this or that particular problem.  (We shall develop this point further in our discussion of the wave picture vs. the matrix methods in quantum mechanics:  but shall draw a very different moral from that of Professor Synge.)   And yet we hold to our Realist position:  which has, moreover, practical consequences, and is not simply a matter of private preference in ontology.

            The Realist claim is that mathematical objects are as real as physical objects.  That is not to claim that any given mathematical object -- be it the continuum, E8, or Klein bottles -- is actually instantiated in this particular physical cosmos, let alone that it applies everywhere and across the board.   Thus, take the continuum.  Our own spacetime may well be granular, not continuous;  the cosmos might be finite, both in diameter and duration.  It might even be downright cellular, as in Stephen Wolfram’s view.

http://en.wikipedia.org/wiki/A_New_Kind_of_Science

Furthermore, the continuum itself is among the most mysterious of mathematical objects;  it is, after all, the eponym of the spectacularly counterintuitive status of the Continuum Hypothesis.  It sticks in the craw even of some mathematicians (though not the ones we prefer to share a beer with), let alone physicists or shopkeepers.

            However.  Considered purely as an object of group theory, E8 has calculable values along certain dimensions of assessment  (algebraic properties): Let us call the dimensions of assessment  "alpha, beta, gamma" … , and the values we calculate  "alpha-0, beta-0", etc.  Thus, for E8 (reading the values out of Wiki), alpha might be “What is its dimension?”, and alpha-0 turns out to be 248;  beta might be “What is its rank?”, and the answer “8”;  “What is its center?” -- “Trivial”; “What is the order of its Weyl group?” -- “696729600” (but you already knew that) …

            Now:  Suppose that spacetime does turn out to have six extra compactified dimensions, and that heterotic string theory is indeed the gospel truth.  Now, those in a position to know, inform us that  in that case  there are only two possible choices for the gauge group of our world.    Recall that earlier gauge groups we’ve met were things like the rotation of a wheel, things whose reality is familiar even prior to their employment as gauge groups:  but now we must choose between  SO(32) and E8 X E8.   The choice must be made on the basis of certain physical predictions.   Assessment-dimension lambda, for instance, might turn out to be physically measurable (specifying, say, the spin of a graviton, or the mass of a magnetic monopole, or what have you);  lo and behold, lambda-0 in E8 X E8 turns out to correspond what we have measured, whereas the different value that falls out of SO(32) does not;  and so forth for some other assessments.

            All right then:  The Realist thesis in this case says simply that adoption of E8 -- which is, in this view, pre-existent, and independent of physics just as it is independent of politics -- is a package deal.   Not every fact and feature about E8 will be physically interpretable, let alone measurable;  but for any that is -- say, omega -- the value as measured experimentally must turn out to be omega-0.   If it doesn’t, we have a problem.  E8 being pre-existent, and not invented by ourselves for our own convenience, it cannot be toyed and tinkered with, selecting some features and rejecting others in a Procrustean attempt to match experiments.   And Synge’s picture of using Newton or Einstein, merely as the mood moves us, as though one were an Allen wrench and the other a pair of pliers, won’t do at all.   If the mathematical structures we seem dimly to glimpse behind the veil of the world  were analogous, not to mountains, but to man-made tools, then we could always kludge one up for the occasion.   Then physics would be the theoretical equivalent of toenail clippers and pinking shears.

[continued here]