On Vulgar Numbers

[Note:  The title is a pun.  Cf. vulgar fractions, i.e. common fractions (as opposed to continued fractions, etc.)]

Among Harvard math-majors, back in my day (before your time -- don’t ask), the commitment to abstraction was intense and unquestioned.   We had barely begun to shave -- and many of us had never been laid -- but we took to abstraction like -- like a duck to water, like a kitteh to cheezeburger.   It was to some extent an end in itself:  not the pitch of wisdom by any means, but temporarily no more harmful than any other young-man’s idealistic infatuation, the blaue Blume or what-have-you.  No sooner did we learn about a thing, than we wanted to generalize it.  Actual numbers had been replaced by x and y in junior-high algebra;  now we rushed to embrace the abstract structures in which these variables lived:  groups, rings, fields … The brighter among us (I was not of their number) before graduation  came to embrace  what its own practitioners called “abstract nonsense” (category theory, “diagram-chasing”), the night or perhaps twilight  in which all structures are grey.  As for the everyday numbers of science -- ungainly things with decimal points in them -- they were as attractive as a turd on a sidewalk.

Accordingly, we had no truck at all with what we called “apple-math” (punning on Appl. Mathematics, the course-catelogue designator, plus the notion of counting up fruit).  Such a department did exist, so we were told, probably somewhere out behind the barn;  but our steps would never take us there.   Oddly, the College itself shared our prejudice, it would seem, since Appl.-Math majors were denied the B.A. shared by math majors and literature majors alike:  they got a B.S., along with (presumably) majors in Sports Medicine  or Veterinary Science.   It told the select world:  “These fellows know how to ply a slide-rule, but they have not received the education of a gentleman.”

[For a glimpse of freshman calculus at Harvard, back when the world was young,
click here:
http://worldofdrjustice.blogspot.com/2011/12/adventures-in-algebraic-geometry.html

And for sophomore year -- the legendary "Math 55", fabled in song and story:
http://worldofdrjustice.blogspot.com/2013/03/andrew-gleason-in-memoriam.html ]

Note:  Our Harvard freshman prejudice  has been shared by others.  A great Chinese mathematician reminisces:

Arriving in California at the age of twenty… I had no idea of what direction to pursue.  I was initially inclined toward operator algebra, one of the more abstract areas of algebra, owing to my vauge sense that  the more abstract a theory was, the better.

-- Shing-Tung Yau, The Shape of Inner Space (2010), p. 35

Appendix for physicists:

The most practical person must realise that abstract arguments (by which we really mean  arguments with a tremendously wide range of applicability  are a necessity … now that science has grown so vast.  If the engineer is willing to overcome his nostalgia for the practical, and embark on the study of Lagrange’s equations in a apirit of abstraction, he will be rewarded by having at his disposal a powerful tool for the study of electrical networks, which are not ‘dynamical systems’ in the ordinary mechanical sense, but nonetheless behave as if they were.

-- John Synge & Byron Griffith,  Principles of Mechanics (1942, 1959), p. 411

Graduate students in theoretical physics … are very often impressed with “formalism” -- the formal apparatus of their subject. … I suffered … from an infatuation with beautiful formalism.  Working with Viki Weisskopf was a most effective remedy against the excesses of such an infatuation.  He never ceased to harp on the importance of … understanding, by means of simple arguments, the physical meaning of a theory …

-- Murray Gell-Mann, “The Garden of Live Flowers”, in Selected Papers (2010), p. 27

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Relief for beleaguered Nook lovers!

http://murphybros.blogspot.com/p/read-murphy-on-your-nook.html

We now return you to your regularly scheduled essay.

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Here, now, is a very funny reflection of a variety of attitudes about our number-friends:

http://xkcd.com/899/

Explanatory footnotes here:

http://www.explainxkcd.com/2011/05/16/number-line/

Re the locus “2.9299372”, the explainer acutely comments:

2.9299372 is a President's Day reference because it is the average of e and pi just as the American President's Day is always observed on a random day between George Washington and Abraham Lincoln's birthdays.

There is, however, another layer to this.   The notion of “observed value” is strictly from experimental science, especially physics and chemistry.  Thus, the “observed mass of the proton”.   In that sense, you could speak of an “observed” value of pi, meaning the length of a tape-measure wrapped around a circle of diameter one.  So here an image is conjured up, of e and pi being within experimental error of each other.   If you just consider them as regular old numbers, like the mass of the proton and the mass of the neutron, that even makes sense -- presumably there was a time when those two masses were not neatly differentiated.    But to a pure mathematician, the notion is riotous -- risible:  e and pi each thrones separately and uniquely in a starry empire of which they are sovereign.  Their structural status is everything;  their particular numerical value -- nothing.

There is, indeed, a big difference in ontological status between “observed quantities” and the pure numbers of mathematics.   As, in the case of mass, we can’t judge well visually ourselves, so we use a scales (or a cloud-chamber) and let that “observe” it for us.  In the case of color, sometimes the scientific intstrument is just our eyes.  So, “3.8 kg.” and “vermilion” have similar ontological status.  Whereas a number (in particular an integer, or pi), in pure mathematics, has no observational status at all.  Indeed, if anything, pi observes you -- it’s liable to pop out at you without warning.  You’ll be sitting there innocently summing up an infinite series (say of the reciprocal of squared integers), and it turns out to be pi-squared over six.

Commenting on “if you encounter a number higher than this, you’re not doing real math”, the explainer gets things backwards, I think.  It is not that discrete-math practitioners don’t think that the rest is not real math;  it’s that mathematicians in general, of the abstractophilic stripe, have little to do with big numbers in general, because these do not (like pi, or 2) constantly pop up when you’re not looking for them.   True, number theorists consider indefinitely large integers:  but collectively, with little interest in this one or that one in particular.   Thus, the Goldbach conjecture applies to all integers at once.  Should someone ever find a smallest counterexample, then that integer will be enshrined, for a time; if there are infinitely many counterexamples, its significance will shrivel.  Intense interest in a menagerie of specific large integers tends to be best represented, I suspect, among idiot-savants.

There are, however, a couple of comparatively recent developments  that really do care about -- or at least come across -- specific large integers:  namely, the (successful) classification of all finite simple groups, and  investigations of the exceptional simple Lie groups.

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Finally, a rich index to a plethora of math-related posts:
http://worldofdrjustice.blogspot.com/2014/08/miscellanea-mathematica-ter-renovata.html