Gemmological Footnote

In 1955, the noted gemmologist and contrarian Sir Nelson Goodman, in a brief note published in the proceedings of the Devonshire Horticultural Society, questioned the received wisdom concerning the colour of emeralds.  “Everyone says that they are by nature green,” he noted, “simply because they have always been green.”  (Note that, in Sir Nelson’s quaint dialect, “been” and “green” actually rhyme.)  “But our observations to that effect, however numerous and of however long standing, may equally be marshaled to support the thesis that they are in fact grue:  which is to say, green prior to 18 November 2018, and blue thereafter.”

In the half century or more that has elapsed since that time, all sorts of philosophical fuss about “projectible predicates” and whatnot (green but not grue  enjoying this rather circular distinction), but in all of that, thinkers lost sight of the basic facts of the matter: are emeralds in fact green, or grue, or what?

We need wonder no longer, for the great day -- 18 November 2018 -- has arrived at last.

And it turns out that  last night, all over the planet, and into the deepest recesses of intersteller space, quite at the stroke of midnight, every last emerald  did in fact turn blue.  Or rather, “grue”.

So now we know.

The Perfection of Penguins: an inductive proof

In mathematical induction, you begin with the base case a.k.a. Induktionsanfang, traditionally symbolized as P(0) (where P stands for “Penguin”, and  0 depicts the egg).   In the present instance, P(0) is Baby Fuzzy.

Kleiner Kuschel-Induktionsanfang

Now, P(1) = Fuzzy’s Mommy;  and they rejoice in a Perfect Love.  Thus, the Induction Hypothesis is established.

Now consider any arbitrary P(k).  In this case, P(k)  is known as Fluffy;  and P(k +1) (the k-plus-first Penguin) is Fluffy’s perfectly adoring mate.   So once again, the induction goes through.

Thus we have established the Perfection of Penguins.

~

Note:  Using the more powerful technique of Transfinite Induction, we can establish the existence of (the wonderful, the unutterable) Zorn’s Penguin.   This, however, requires the Axiom of Choice, in its strengthened form, “Choose Penguinity”.

Ideas: the case of Induction

[We continue with our survey of leading Ideas, begun here.]

(3) Induction

            There are logical problems with practical induction, notorious since Hume.  That we nonetheless freely (and even over-freely) constantly make use of it, might suggest that it reflects merely an innate disposition, reinforced because often successful (the world being, contingently, the way it is), thus logically on a par with the inveterate conviction of salmon that they need to swim upstream and spawn – which also usually works.  So let me creep up on it genetically, via something at least similar to the notion of induction, which I did not always take for granted (hence its eventual attainment may count, albeit marginally, as an Idea).

            For Parents Night, our third-grade teacher put samples on the blackboard  of multiplication problems  mastered by her class.  Everyone in the class could multiply a one-digit number by a one-digit number, she explained; eighty percent could multiply a two-digit number by a two-digit number, and one pupil (unnamed, but in fact your correspondant) could multiply a three-digit number by a three-digit number.  Not unnaturally proud of this accomplishment – which already was my first glimmering of an idea  not widespread in the lowest-common-denominator society of primary school, viz. that individual abilities might differ – I jauntily asked my father, How many digits can *you* multiply?

            The response I expected was something like “five” (he being an adult), or even “seven” (he being a scientist), or maybe even “ten” (what a special Dad!).  Instead he looked puzzled, even embarrassed.  It was not, he said, a well-defined question.  Multiplying numbers of *any* length was just a small homogeneous step up from multiplying slightly shorter numbers.  In practice you might become confused, and have to carefully write things down and check your work (indeed there *is* an individually varying limit as to how formidable a pair of numbers one can multiply in one’s *head*, but that was not the ability under discussion); yet in principle there was no limit to it, just as there wasn’t any highest number one could count to:  if you make it as far as n, you can get to n + 1.

            Now, he didn’t explain it quite as clearly as that, I imagine, but whatever it was he did say  triggered a flash of insight in my small head.  From that moment on  I had a completely different attitude to arithmetic as taught, to school as conducted, even a different attitude towards knowledge and education in general.  The teacher, from being an oracle, shrank to a small frail figure indeed.  And the problems on the blackboard were no longer confined to their chalky two dimensions, but bored outward, into the void.

            So far, there’s been no demonstration of an Idea beyond what we were born with, merely the application of same to phenomena initially conceived to be beyond its reach.  That is, the child began by conceiving the multiplication of two-digit numbers and the multiplication of four-digit numbers  to be as distinct as bipeds and quadrupeds, and we do not conceive six-legged insects and eight-legged spiders and octopuses to be some sort of inductive continuation of these.  But by the time we have arrived at mathematical induction, there does seem to be something new under the sun.  Given (by demonstration or observation) the truth of F(0), and proving that F(k) implies F(k+1), we suddenly, by these two little finite exercises, have access to an infinitity of truths, down to uncharted reaches of the sample space.

And lest you consider that elementary anecdote  a mere trifle of childhood, the smooth and indefinite extensibility of multiplication being obvious to any mature mind,

consider this, from John von Neumann:

In an analogy machine [what we now call an analogical computer, when we call it anything at all, since these have largely fallen by the wayside -- ed.],  a precision of 1 in 10^3 is easy to achieve; 1 in 10^4 somewhat difficult; ... 1 in 10^6 impossible …

In a digital machine, the above precisions mean merely that one builds the machine to 3,4,5, 6 decimal places … The transition .. gets actually  easier …

--- quoted in  James R. Newman, ed. World of Mathematics (1956), p. 2077

Compare further this parable:

http://worldofdrjustice.blogspot.com/2010/12/constructivist-angelology.html

[Continued here.]

Any Ideas ?

  

Those solutions make the greatest appeal to the mind which depend upon ideas  rather than upon calculations. It is sometimes amazing to see what can be produced from a simple idea, and what great consequences may flow from it.  It seems almost incredible  that so much can come  from so little.

-- Louis J. Mordell, Reflections of a Mathematician (1959), p. 40

An essay in this morning’s NYTimes, on the dwindling public status of Big Ideas, prompted me to fret upon the fact that I’ve been sitting on an exploration of the subject for several years, adding a paragraph every few months or so, until (in the fullness of infinite time) it might be ripe (“Pauca, sed matura” -- the motto of Gauss):  thus virtually daring the Reaper not to take me just yet, ere the work be finished.  But the Reaper might have other plans.   And so, as earlier with the subject of Causality, I here publish some preliminary notes.

The essay in question, by the author of a study on such wide-ranging deep thinkers as John von Neumann and Arthur Koestler -- oh no wait, actually his magnum opus is about Walt Disney, nevermind -- is the lead article on the front page of the section I always go to first:  what used to be known as “The Week in Review”, but which, in a bold rebranding, aimed no doubt at recuperating those twentysomethings and panda-handlers and whatever other demographic has been abandoning the print media, has been re-baptized… wait for it…. [But! We! Can’t! Wait for it!  We’re the InstantGratificationGeneration !!!] ….  ‘Sunday Review”.  But on the Times website, it is deeply buried, so we link to it here:

http://community.nytimes.com/comments/www.nytimes.com/2011/08/14/opinion/sunday/the-elusive-big-idea.html

It, itself, dives to no particular depths, seeming more like the musings of someone who is bored with what has recently been on TV (hey man, it’s summer, hold out till the fall schedule).   But it is very, very refreshing, even to see a sort of placeholder of an article on the subject of ideas, rather than the latest vaporings of Michelle Bachmann, or a conspectus of Celebrity Tattoos.

And even at that, in the timid middlebrow venue of the New York Times, as a primary representative of the "Elusive Big Idea", we are offered (along with the inevitable Einstein) ... George Washington Carver.

(True, true, I’m being snarky again.  The Times article in turn used its lead paragraph to be snarky about the Atlantic -- apparently an eyebrow’s-breadth lower on the nation's Middle Brow.  -- Meanwhile somewhere there’s a blog that is snarky about me.  Unfortunately, it is written in Klingon.)

And so, I herewith offer, the beginning of a selection of observations, to be continued if I am spared -- as yet but halfway baked, but which else would have been mere scraps in the Nachlass, while I lay cold upon the slab.

~ ~ ~

ANY IDEAS?  (incipit)

            Basic ideas, like idealized geometric figures, are few.

            --S.J. Gould

Locke, we learn, innovated, in his use of the term idea:

The term ‘idea’ as coined by Locke  and used by Berkeley  does not have its normal sense – a fact much remarked upon by Locke’s contemporaries. … Locke.. defined an idea as ‘whatever is the object of the understanding..’

[H. Robinson, intro to Berkeley, p. xii]

We too shall do so, though not with philosophical intent, but only expository convenience. We shall use the term roughly in the sense of a ‘leading idea’ (French: idée-force), one which really does lead your thought and behavior, as opposed to, say, ‘bright idea’, a balloon that goes poof when it bumps up against the cactus of reality; or idea in the sense of ‘general impression’ (“Any idea how…?”); or ‘proposal’ (“I have an idea!  Let’s go out for tacos.”) .  Idea, in our shorthand sense, denotes battle-scarred propositions that have repeatedly won laurels on the open field.  We shall crown them with a capital:  Ideas.

Our concern is not at all with the (perhaps hopelessly) abstract, and properly philosophical, question of what ideas ARE, simpliciter.  Rather, given an initial intuitive notion of leading ideas, what are some examples?  The answers will differ from epoch to epoch, and nation to nation, and person to person – and to that extent, the question is of scant general interest.  (You might want to stop reading right here.)  I am simply attempting an inventory of the various struts and strengths of my own ideation, and more broadly that of the scientific/intellectual community with which I am conversant.

First, a sketch of what we shall not be considering under this head.  We mean only such ideas as we come to learn, and may explicitly reflect upon, and cite evidence for: thus excluding capabilities or notions that we are more or less born with.  Otherwise many or most lower species might be said to possess Ideas.  In some sense they may, but it is not our sense here.

Nor do we consider Life’s Lessons Learned.  As a result, the roster below may seem skewed towards the brainy and the bloodless, but there is a reason for the exclusion.  I have the sense (and this itself might almost qualify as an Idea) that such life-lessons tend to be, and perhaps always are, merely relative to a time and a culture, and even to be largely conditioned by one’s individual psychology.  The Ideas we consider must be of general validity, so far as we know (things like the Existence of Other Minds, or Induction; rather than, say: “Women – you can’t live with them, and you can’t live without them” – true enough for most of us, but a dictum which singularly fails to obtain on the Isle of Hnork).

So, to our roster, in no particular order.

(1) The Isomorphism of Other Minds

Consider any member of a species of born solipsists.  As time goes on, he would notice a large number of regularities;  and,  if of a logical cast of mind, would soon hit upon the Idea that his conspecifics are mentally quite comparable to himself – that solipsism is in fact false: in a sense, radically so.  Human beings, apart perhaps from autists, are not born solipsists, so this insight does not arrive as an Idea: we innately assume the existence of like minds; it does not strike us as a revelation.  We have to be taught in philosophy class  even to entertain the solipsistic hypothesis as logically possible.

            [In like fashion:  After reading a great many attacks on the notion of free will, both empirical and philosophical, I have nevertheless concluded that some core thereof is epistemologically as irreducible as the Cogito.  But this does not qualify as an Idea, since it merely preserves the naïve assumption.]

            So:  Thus far we have an organizing principle, a basic assumption, but it is not an Idea, because we are born with it; we share it equally with Mike the Milkman.

            There may, however, be a further appreciation of Other Minds  which does qualify as an Idea in our sense.  Namely, that successful communication often depends upon a wealth of shared tendencies and attributes among communicants; failing these, attemps at communication, even elaborate attempts, seemingly inexplicably go awry.  This (quite disturbing) result  contravenes the straightforward assumption that communication succeeds by virtue of its content and presentation.  The model of human communication becomes  not so much the Dialogue concerning Two New Sciences, as rather the antennae-canoodling of ants -- very effective, but whose content is nowise propositional, but lies wholly in chemistry..

            Much could be said on this score, but it is not our concern here to prove or even motivate this.  It is a conclusion I have gradually come to, and it qualifies as an Idea. It may of course just possibly be false, and would indeed be so if even the weaker hypothesis of the Existence of Other Minds (connected to our own) were false – the old brain-in-a-vat scenario.  For that matter, all of mathematics may somehow be a mistake, the physical world an illusion, etc. etc.  We won’t worry about that.  In light of our best intuitions, tutored by reason and guided by experience, it is well-founded, and is an Idea.

(2) There is a Universe

           

            Which is to say: There are more things in heaven and earth, than are dreamt of in the nursery.

            Here the genetic development is straightforward, and well-known.  Initially there is just the Breast, and (in its tragic absence) one’s Blankie.  Then there is the House to be explored, and the Neighborhood, and School; and foreign lands, and even planets; et cetera, et amazingly cetera.

It grows to an Idea, in that it eventually reflects back on items  initially not conceived as so situated.   The cup of coffee on one’s desk is no longer just an object, surmounted by a curl of steam, but the proverbial Universal Cup of Coffee, embedded in a nexus of economic-social-technical relations.   Eventually, every single thing you meet  you may assess in terms of its being an instantiation of a universal.

[Continued here:
http://worldofdrjustice.blogspot.com/2013/06/ideas-case-of-induction.html ]