Saturday, January 29, 2011

Real as Rocks

Behold a stone.
A parable:
Practical Pete knows only that you can use this kind of rock to build sturdy houses.
Chemical Clem knows only that this kind of rock has a high iron content.
Geologist George can tell you it is metamorphic in origin.
Laboratory Larry can tell you its crystalline structure.

Relativists tend to stop there, and “celebrate” the tribal knowledge of whatever faction, each equally valid and independent of the rest -- even overt contradictions make no difference to them.   And we are with them part-way, in that we don’t particularly privilege the science-sounding assessments of Clem et ilk, over that of Practical Pete.  The thesis of Realism about Rocks is simply that, should Clem learn a bit about the building trades, he will come to agree with Pete on the suitability of this rock for dwellings.  And if Pete takes a lab course, he’ll wind up agreeing with Larry.  And so forth.  For the qualities lie, not in us, but in the rock itself.  But as for the politics of the thing (which we don’t stress, since the relativists are not intellectually interesting opponents): whoever had an integrated synthetic overview of all those properties (with perhaps a few more needed to explain the interconnections among the properties, e.g. the crystaline structure related to the lack of friability and hence suitability for sturdy building), has an unambiguously superior understanding of that rock than does someone who (like our friend the hedgehog) knows only One Big Thing.
            We would further assert -- and this is not definatory of Realism, but is more the result of observation and experience -- that none of these summations of views can exhaust all there is to know and say about that rock, precisely because it is so real -- discovered, not invented.  (You might almost say:  Begotten, not made…)

Abel is a complete empiricist.  Various practical considerations (perhaps, estimating how much fencing material shall be needed for each new circular field) have led him to make ever-more-precise measurements on the ratio of the circumference to radius of a number of circles of different sizes.  (By good fortune, these occur naturally on his planet, in great abundance.)  All the measurements tally exactly to within experimental error.

Baker lives motionless in a lightless world, and knows nothing of shape or size.  But since (by way of compensation) his lifespan is a billion years, he has plenty of time to explore the wonders of infinite series.  And he was astonished to learn that four times the alternating sum-and-difference of the reciprocals of the odd natural numbers (one minues a third plus a fifth, etc.) turns out to be six times the sum of the natural numbers squared; and furthermore, equal to the product of  (well, consult your favorite book of math tables, for many more surprising examples.)

Charlie is a wave-function, living in a world of waves.  He quickly hits upon e to the i-pi-theta as key to his world.

Now Doug (just to make things interesting) has no particular intellectual pretentions or attainments at all:  but one day God takes him aside and reveals to him the infinite decimal expansion of pi.  Puzzled, Doug accepts the revelation.

Realism about pi is exactly parallel to realism about rocks.   The longer and more accurately Abel pursues his experiments, the more his measurements will converge on Doug’s value.   The farther out Baker calculates his infinites series, the more they will agree with what was revealed.  Deeper mathematical insights will connect the geometric and trigonometric and wave-theoretical and number-theoretical properties of pi.

            William James (The Principles of Psychology (1890), vol. II, p. 334) makes the same point about the nearly endless ways of describing a thing, taking as his example a piece of paper -- “a combustible, a writing surface, a thin thing, a hydrocarbonaceous thing…” -- concluding “the reality overflows these purposes at every pore”.  
All true; though the example, being an artefact rather than a natural object, proves less than it might, since the object in question was made for certain purposes -- our purposes -- by us;  and thus made (by no means begotten), its other qualities flow only so far.               
Having invented the coffee-cup to hold our coffee, or tweezers to tweeze, we do not subsequently discover that said cup is the perfect containment structure for magnetic monopoles, nor that tweezers are the precise shape required to receive meaning-of-life narrowcasts from the Orion nebula.   Whereas a rock might prove to have any number of further, unexpected properties: suitably ground, it might be medicinal; suitably sliced, it might reveal ancient fossiles; further analysis might show it to have reached our solar system from far away, preserving evidence of different, earlier  settings of the physical constants elsewhere in the cosmos.  And the wonderful properties of pi, likewise, may never be exhausted: because we found it, we didn’t make it;  He did.

(Such considerations illuminate the celebrated "unreasonable effectiveness of mathematics" -- not just in the physical sciences, but in distant-lying parts of mathematics itself.  It's all from the same workshop.)

So, our motto:
Transcendental Numbers -- As Real as Rocks. ©

1 comment:

  1. Didn't you just love the part in Sagan's "Contact" where the expansion of pi contains a rasterized encoding of the text and diagrams for a manual to build a wormhole transportation machine? Summarized here by Wikipedia:

    "In a kind of postscript, Ellie, acting upon a suggestion by the senders of the message, works on a program which computes the digits of π to record lengths and in different bases. Very, very far from the decimal point (10^20) and in base 11, it finds that a special pattern does exist when the numbers stop varying randomly and start producing 1s and 0s in a very long string. The string's length is the product of 11 prime numbers. The 1s and 0s when organized as a square of specific dimensions form a rasterized circle."