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[ Continued from last week, and I'm feeling considerably better, thank you. ]
In your mathematics paper, between the equations and the striking new translation of The Canterbury Tales, include some definitions. Begin by defining ``normal'', ``unity'', ``orthogonal'', and ``inner product'', because they've been defined in every mathematics paper ever and what will a couple more definitions hurt? Then you can add clarifications, like ``the unified orthogon is the maximal subset of the minimal superset which covers but does not exclude all the subsets composed of non-unity elements of the power set basis subspace excepting but not excluding the span of the orthogonal dual basis'', which sends readers back to the numbered equations' reassurance that H equals the sum of all the components of H, whether serif or gothic. Some definitions should introduce subscripts, ideally ``5'', ``*'', or the pair ``3, 3'', the most mathematical of subscripts.
You might include graphs, to remind you how awful your paper-writing software makes including graphs. The first should be a wavy scribble with a modest peak, proving the existence of grid marks. Figure two should be an isometric view of a tetrahedron with arrows. Figure three should be disconnected open dots that rise sharply, drop to the bottom, then rise back to level. The fourth figure should be scattered dots on a log-log plot and a dashed line of interpolation never threatening to go near any of them. The fifth picture should be some hypnotic repeating pattern in really old artwork. The eighth picture should be a polygon, making the reader search for the sixth and seventh, the mad fools.
Also try a few tables. The left column should be several Greek alphabet levels with superscripts of + and -, and subscripts of up to four characters. Do not repeat any combination of subscripts. Include several numbers beside each figure, but include a hyphen or ``NA'' in at least one of each row's columns. Repeat until this explains how you've proven the specific heat of clay.
Proofs lie at the heart of most papers, surrounded by a protective layer of boldface, the rib cage, a little filled-in box at the lower right corner, and one or more ventricles. The important part is the declaration of something like ``Unified3,3 rotation star matrices are decomposable into linear diagonalizable rotatable non-star matrices diffeomorphic to the quaternion GL5''. Follow this with between two-thirds and three-quarters of a page of algebra, which can be as simple as factoring 15 in a difficult way, and explain the conclusion is obvious now. Any proofs not having enough words should be called Lemmas and itemized as deductions on your income tax.
Next comes explaining the analysis. For example: you do a Laplace transform, modifying the original equation by turning Pierre-Simon, Marquis de Laplace into a penguin. After some manipulations do the inverse Laplace transform, resulting in a different, angrier penguin. If this is unsatisfying try the z-transform, giving you a napping penguin with angry dreams. Use a Hamming filter, surrounding the napping angry penguin with pork products until it is satisfied. Then run away before it remembers how tired it is of the fetishization of bacon in modern comedic culture.
Spend no fewer than four pages reporting surprises and discoveries. For example, perhaps in the middle of your wall calendar you discovered a sticker page for marking important dates like scheduled oil changes, visits to the Museum Of Art You Will Never Visit, or the expiration date to eat your air conditioner by. Your readers will never imagine the murder was actually a suicide, discovered coincidentally by the man the jilted fiancee was going to elope with, and hastily covered up to protect the vicar, who didn't know anything about it before the police were called.
Remember to passive-aggressively thank your granting organization. Then include six citations per page of your paper; this requires thinking of credible-sounding titles. Journals may be identified by sets of three to five units of one to four letters each, such as ``Am J Diff Let'', ``New Brit Cndy Tree'', ``Sing Or Clap'', and ``Turb Hamr''.
Remember to list one of your co-authors as contact person and submit to any reputable journal, if you find any.
Trivia: The International Geophysical Year was conceived in James Van Allen's home as a Third International Polar Year on 5 April 1950; the closing date for analysis and publication of data collected during the Second Polar Year, 1932-1933, was the 31st of December, 1950. Source: This New Ocean: The Story Of The First Space Age, William E Burrows. (And looking it up on Wikipedia reveals that we did have a Third Polar Year after all, 2007-08. I never got a memo about this. Did any of you?)
Currently Reading: The River At The Center Of The World: A Journey Up The Yangze, And Back In Chinese Time, Simon Winchester.
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(no subject)
Date: 2011-01-29 11:32 pm (UTC)Scientific American goes to great trouble with graphics to explain the hypothetical organization of elementary particles predicted by string theory, but doesn't print the equations for SU(2) and E(8). Wait. I took Abstract Algebra in college and I can't recall if group theory even uses equations.
And when the writer asserts that "physicists are searching for a single equation that can explain everything" this is a bit of a stretch if the reader's notational limit is the sigma in the vector form of F=ma, and doesn't even know what the Nabla and Div in Maxwell's Equations really mean. The writer could demonstrate how the General Theory of Relativity is much more elegant when expressed in tensor notation (whatever that is), a sentiment which I think Charles Sheffield expressed (somewhere in his McAndrew and Proteus stories).
Obviously the writer needs to expand the equation, explaining what each part does, and what the implications are. "This bit means that a charged particle in a magnetic field spirals in a certain way; this other phrase relates how the strength of the field relates to the radius of the spiral; and this third clause describes the distribution of chocolate sprinkles on a spherical scoop of ice cream."
(no subject)
Date: 2011-01-29 11:49 pm (UTC)Publishing lore claims that each equation, other than E = mc2, cuts sales in half. That obviously can't be taken literally, but people definitely let their math anxiety scare them away from even simple mathematical work. There are some pop-mathematics books which get into equation work, though those are usually of the mathematical-puzzle-book variety.
The exceptions that come to mind are, from my experience, those that try explaining group theory or game theory to the world. Those probably get some exceptions because so much of group theory lends itself to hugely symmetric pictures, and so much of game theory lends itself to cartoons of game-show contestants at Price is Right podiums, so both look less scary.
I think the ideal of mathematics exposition has to be Richard Feynman's QED, which explains quantum-mechanical phases in a wonderfully clever way (something along the lines of ``don't worry what these arrows mean or why they're rotating but trust me that this works'') so that if you have the mathematical background to do the path integrals over complex integrands required you see exactly what to evaluate, and if you don't have the background you at least have a procedure you can follow if not understand for the work. But Feynman had an expository skill almost nobody could hope to have.
I have seen General Relativity written out in tensor notation and it certainly looks more elegant than the alternatives, but I've never gotten the intuitive feel for reading tensors that I have with vectors, so it ends up looking beautiful but not actually useful to me. At least a part of this is I can't stand the convention of dropping the summation signs and indices; it may be a useful savings of time once one's familiar with the expressions, but it's getting to that point that I want to do.
(no subject)
Date: 2011-01-31 02:06 am (UTC)"Oh no, this book explaining a complex concept attempts to show me exactly how complex it is. I don't want to know anymore!" People are odd, aren't they? How do math authors expect to entice newcomers into professional mathematics without showing them what they'll be working with? OTOH, that's not the concern of the publisher.
How about an appendix? If you isolate the math, can it still frighten the buyer? And in this age of ebooks, how about an entirely separate edition with the math? Or some use of hyperlinks to conceal it. Obviously the audible edition would forego the math, because reading off "the integral from x equals 0 to infinity of the expression one-half, x-squared, dx" would quickly grow tiresome. Matrices would serialize even less well.
I have a Feynman biography, but I've never read his lectures. Why can't other math-writers steal his techniques? Perhaps the two skills (clear writing, and deep understanding of the math-heavy physics) don't overlap often. That certainly seems to be the case at work, where I'm usually tapped to rewrite methodologies for our economic estimates, and I have to repeatedly ask, "What proxy did you use here? How did you go from annual to quarterly?" And a few equations could certainly clarify things -- I've seen them in the methodologies from other economic vendors, so it's not unprecedented.
Tensors, what's a tensor ... entries at Wolfram Mathworld explains the what, but not the when or why ... aha, they're just multidimensional scaling factors. Space-time geometry is esoteric, but describing the stress-strain of a physical object is understandable. Stress an ice cream sandwich in the Z direction, and the strain appears not only in Z (flatter) but also X and Y (wider).
It shouldn't be difficult to explain from scalar scaling factors (e.g., F=kx springs) to vectors, matrices and general tensors. There'd be a diversion in matrix math -- this is a rule for matching the elements, here's a longhand R1C1 notation, and here's progressive amounts of shorthand. And another diversion to symbolic math packages, with a warning that you should really get a feel for working things out by hand before you trust the software.
(no subject)
Date: 2011-02-04 12:09 am (UTC)I don't really know how much pop-mathematics editors do expect people to read the equations alongside the regular work. There are certainly levels in which almost no equations are suitable --- Carl Sagan's Cosmos as an example is full of evocative writing (such as talking about how an Oort cloud comet might be moving as fast as a propeller-driven airplane, so far away the Sun would have no visible disc to the human eye), but if it has one equation per chapter I'd be surprised. And there are books, mostly recreational mathematics or histories of particular problems (Fermat's Last Theorem, the Four-Color Map problem), where a fair number of equations are welcome.
Feynman's Lectures don't make any attempt at being popularizations, and for that matter barely manage as undergraduate textbooks. They're awfully good for graduate students, though, who already know the subject and who need a clearer understanding of them. But I'm thinking particularly of QED, describing Quantum Electrodynamics, as the expository pinnacle.
To some extent other authors can't just swipe Feynman's style because not every problem lends itself to that description. QED is pretty much a calculus-of-variations along paths with a rotating phase factor, and that can be described in terms of arrows and summations without losing too much essential accuracy. But how would you describe, say, the Shannon Sampling Theorem and why it's true in terms of things that aren't just the mathematics recoded?
Plus, good writing is hard, and it's certainly not selected for in mathematics. There's some pressure to write coherently, but `coherent' is a loose condition compared to `narratively interesting'. And they say nobody marginal ever got tenure on the grounds of their prose style, so you're left with either people who have the raw talent or who develop it for their own amusement; most people are going to leave their writing ability at whatever it is without trying.
I know the way you write down tensors and what they're supposed to mean, but I've never had a problem to work with that required them enough to grow comfortable with the things. Particularly bugging me is the clear difference in meaning between superscripts and subscripts in the expressions surrounding them, and I haven't got an intuition about what they mean (although that Noether book I picked up this week is helping). I don't think it coincidental that this is one of those subjects I've never seen popularized deeper than, well, your paragraphs above. Those are fine but they don't prepare you for the path of a free-falling object near a black hole.