Poets and Peasants
David Corfield’s book Towards a Philosophy of Real Mathematics spends some time (effectively the whole book) addressing the question of what makes a particular mathematical idea important. I could sit in my chair this evening and invent a dozen new mathematical constructions, none of which would stand a chance of entering mainstream mathematics. In his Mathematician’s Apology, Hardy makes an informal stab at identifying importance in mathematics but, like any great artist, he knows good work when he sees it but has difficulty in saying how he knows.
Corfield builds on Imre Lakatos and others and, on page 205 of his book, enumerates five principles that can be used to assess the importance of a mathematical idea (concept). Two of these five, the first and last, can be considered “applied” (rather than “pure”) and the last is:
(5) When a development reasonably directly leads to successful applications outside of mathematics.
Now, where are these applications going to fall? Unless we’re in for another bout of metaphysical poetry, my guess would be not within the arts. I can’t see Corfield’s first example after the list of criteria, the Atiyah-Singer Index Theorem (see here for a popular account of this theorem), making a major impact on most painters or even poets. I suspect that Corfield’s criterion expects “successful applications” in science and, in particular, physical science.
A couple of blogs ago, I wrote of the bridge over the river at Wakefield and the way the water displays the answers to complex and coupled partial differential equations with ease. I see the physical world which the physical scientists study as an instance of a class defined by those differential equations. If this is so, then the equations are not a description of the real world, they are the real world and our sense perceptions detect the instance because they cannot detect the underlying class. In this case Corfield’s fifth criterion is inverted: the mathematics is useful if it aligns with the definition of the class, an instance of which we perceive.
Talking, as I was, about poets and scientists I’m sure that my reader knows of Babbage’s remark to Tennyson but it always bears repeating:
“In your otherwise beautiful poem, one verse reads,
Every moment dies a man,
Every moment one is born.
… If this were true the population of the world would be at a standstill. In truth, the rate of birth is slightly in excess of that of death. I would suggest [that the next edition of your poem should read]:”
Every moment dies a man,
Every moment 1 1/16 is born.
Strictly speaking, the actual figure is so long I cannot get it into a line, but I believe the figure 1 1/16 will be sufficiently accurate for poetry.”
I have always had an affection for this, particularly after reading O’Shaughnessy’s Ode (set by Elgar as The Music Makers) which ends
For each age is a dream that is dying.
Or one that is coming to birth.
Does anyone else worry about that “or” at the beginning of the last line? I always use “and” when I recite it to avoid the possibility of leaving us without a dream for a while (or, worse still, having two simultaneous dreams if one comes to brith before the other dies).