An account of some of the less rigorous utterances of applied mathematicians

Howison, Sam

English

Oxford University Research Archive (ORA)

If I remember rightly, $\cos\frac{\pi}{2} =1$

Mathematical Institute Eprints Archive

“If I remember rightly, cos pi2 = 1”
Sam Howison
Mathematical Institute
24-29 St Giles’
Oxford OX1 3LB
UK
July 1, 2004
Introduction
Some time ago, I returned from a year abroad to find a scrap of paper pinned on
the otherwise empty noticeboard in my new office. It said “C is bounded with
an unbounded bound”. The words had been spoken by the previous occupant
during a spirited argument about something or other, now long forgotten, and
had fortunately been recorded. Absurdities like this are more common than one
might suppose, and having started out listening for them, the attuned ear soon
amasses a fair collection (a similar position is seen with Spoonerisms [1]). In
Britain they are called Colemanballs, a reference to sports commentator David
Coleman whose remarks, along the lines of “and for those of you watching who
don’t have television sets, there is live commentary on Radio 2”, gave rise to a
regular column of ‘Colemanballs’ collected by readers of the satirical magazine
Private Eye [2]. In the US I have heard them referred to as ‘jokes’, but this
seems to me to miss the point. A classic Colemanball is above all unintentional;
if the absurdity is compounded by hubris, as in the example of the title, which
I recall was delivered in a particularly superior tone of voice, then so much the
better. Protestation by the perpetrator that “you know perfectly well what I
meant” (as is indeed sometimes the case) may properly be ignored.
Applied mathematics, and in particular the informal and mildly competitive
discussions of workshop or Study Group activities such as those held at Oxford
or RPI, has been a fruitful source of Colemanballs. Perhaps this is because ap-
plied mathematicians are not overly concerned with rigour. Perhaps also many
Colemanballs originate at places like infinity which have no real-world coun-
terpart, and hence tax our skills of translating the abstract into the concrete.
Perhaps we just don’t think before we speak. . . . At any rate a good Colemanball
has an unmistakable ring that is not exactly truth, more artistic verisimilitude.
1
What follows is a selection from the Blue Book, kept in the Common Room
of the Oxford Centre for Industrial and Applied Mathematics. In it, we record
the Colemanballs collected by our spies (who are everywhere); they are set
down verbatim - improvements are forbidden. Readers interested in a similar
collection, this time of comments made by lecturers at Cambridge University,
should see [3]; it includes the classic “Damn, I’m running out of integers”.
Zero and Infinity
Applied mathematicians clearly have trouble with the concept of infinity (pure
mathematicians probably do too, although they will not admit to it; see [4] pp
137-8). Indeed, I suspect that most simply think of it as being rather a long way
off the right-hand side of the page. Surely this was in the mind of the colleague
who decided that “you can always make infinity smaller by multiplying by h”, or
that “the stagnation point might go half-way to infinity and then stop”. Further
evidence for the finiteness of infinity is “the velocity [which] is much bigger than
infinity”, although “in this case you can take infinity to be quite small”. Zero
is generally better understood, being nice and central in the real line, but spare
a thought for the topology behind “it’s infinite there, even though it’s zero”.
Large and small
Even finite numbers can be confusing: “What size is the Peclet number?”; “It’s
106”; “Now remind me, is that big or small?”. For some reason, small numbers
seem to hold more terrors than large ones, especially when encountered as orders
of magnitude, as in “in that frame of reference, O(1) is transcendentally small”,
or “it’s not fair to look at a small point”. Sometimes the two are confused:
the colleague who said “instead of considering 10 as large, let’s consider 10 as
small” might well envy the certitude of judgment of “there are some big ones
and some very infinitesimal ones”, or of “the exponential term is large – in fact
it’s exponentially large”.
Geometry
There are some mighty strange shapes in the Colemanballs geometry. The
“long thin spherical region” is quite common (in two dimensions, “Chapter 3
deals with the thin circular geometry”), as is “the sunfish [which] is shaped
like a nearly spherical disc” (maybe also “like a triangular disc”, which has the
property that “one side of a triangle is always shorter than the sum of the other
three sides”). Curves and lines, too, are unusual: “a hyperbola has a straight
bit”, while “the straight line has by its nature changed its curvature at each
end”. The latter is quite unrelated to “a parabola with fixed curvature”. Even
rigid bodies can move in unexpected ways, like the “rigid pulsating cylinder”,
2
which doubtless has a “rigid flexible boundary”. At least “a circle’s isotropic –
it looks the same from every direction”.
Count-ability
‘There are three kinds of mathematician, those who can count and those who
can’t.’ This was a joke, but deadly serious was the statement that “the only
time you get interesting behaviour is when a is O(1), or very small, or very
large, or one of the two”, and so was “there are 4 quantities: 3 are the same and
one of the others is different”. Also in the latter class were the colleagues who
averred “there’s only one word for that: ‘free boundary problem’”, and those
who observed “that’s interesting: the bubbles rise in pairs of three - no, four”,
and “I think the coefficients always add up to 2, so 1, 2 and 1 will do it”.
Logic
Logic is not the forte of Colemanballs mathematicians, whose seminar titles are
along the lines of “Why are rarely-occurring phenomena not uncommon? A
logician’s viewpoint”. At least “you can’t have s = 0 if it’s not equal to zero,
as it were” was correct, but what could “if 10 corresponds to 23, then 23 is
half of 45” have been intended to mean ? The question “how unique is that
solution?” can be answered positively by “there is only one unique solution”,
but perhaps it is best in the end to admit that “I have only reached inconclusive
conclusions”.
Physics
Colemanballs physics is a topsy-turvy subject. “The solid mass of injected liq-
uid” may have a “stress tensor [which] is completely isotropic; the only nonzero
element is σ23”, although “gas doesn’t have a pressure in its own right”. “Heat
has a perfect thermally insulating role”, or, even weirder, “you’ve got to have
some cold heat”, but even with cold heat, “the temperature at this point never
gets above the maximum temperature”. The system of units in which “I’m
thinking of a time t1/2 and a distance t downstream”, “which gives a length of
10 seconds”, and, conversely, “the thermal timescale . . . is about 25 metres”, is
also that in which “the mass of this thing is about 1 kilometre”. Although “no-
body’s interested in measuring things that can’t be measured”, “they measure
time chronologically”, perhaps to solve a “problem [which] is unsteady in time”.
“When you lean forward, gravity changes”, so that “there is a horizontal gravity
component”; if you do this often enough, “gravity acts in several directions”.
3
Nonlinearity
We all prefer to avoid nonlinear problems if possible1 (“dy/dx = ex . . . looks
pretty nonlinear to me”, in a suspicious tone of voice), so although “when the
nonlinear terms in an equation occur at the same order as the linear terms, and
we find the lowest order term in an asymptotic expansion, the resulting equation
is nonlinear”, the usual result is that “it’s linear except for the nonlinear terms”.
The resulting problem may be “a nonlinear Burgers equation” and may have “a
spectrum that is almost continuous except that it’s discrete”, with “eigenvalues
crossing the imaginary axis at smaller and smaller values of i”.
Singularities
The best way to deal with these is probably to “regularize the problem by remov-
ing the term causing the singularity”; this would be sensible if “the singularity
is latent in the branch point” or if you “don’t like non-analytic singularities”.
A rigorous result
Theorem: Bearing in mind the fact that “it is not unknown for things not
to tend to their limit”, it is possible to “approximate an irrational point by a
sequence of increasingly irrational rationals”.
Proof: “I have an empirical proof of this theorem”.
Mathematical technicalities
Nothing simpler than a periodic function? Well, try the equation with “periodic
solutions within one period” (during which “the phase changes by 180◦ and the
amplitude changes sign”) or “the function g(x) which is slightly periodic”. Still
on a technical note (“x > 0: these sorts of issues really need to be put in
mathematical terms”), we see that “of course, in regions where the yield stress
is not exceeded, the stress is indeterminate (and perhaps not a particularly
useful concept), but it may be assumed differentiable with respect to x”; or at
least it may be “slightly differentiable”.
Constants
Colemanballs constants are not without interest (“if you’ve taken them arbi-
trary, you’ve taken them to be something”). Leaving aside the “small O(1)
constant”, and the “constants [which] are either zero or not zero”, we have “ar-
bitrary constants which can be functions of space and time” (except “in the
present situation [where] the steady–state solution is truly time–independent”)
1We except the valiant but misguided “you can homogenize it but you don’t get a nice
nonlinear thing”.
4
and the nonconstant constant, as in “the viscosity is constant but varying”.
These shifting sands perhaps provoked the assertion “it’s constant for all time”
and the immediate riposte “what, the same constant?”. Maybe the variable
constant is really the mythical c(t) occurring in “we can solve the problem and
find c(t) even though c(t) isn’t in the problem”.
Conclusion
Some of the assertions presented above may be hard to believe (“there are critical
criticisms of this speculation”), and some of the methodology suspect (“there
is a word for that beginning with s: heuristic”). We conclude by reassuring the
reader of the verities that “classical slender body theory has in mind the slender
body”, and that “at the end of the day, the best we can hope for is 1 = 1”.
Acknowledgments
The author, who is not himself guiltless, is grateful for helpful conversations
with, and/or written communications from, Marc Audet, Bruce van Brunt,
Chris Budd, John Byatt-Smith, Jonathan Chapman, David Crighton, Jeff De-
wynne, Gail Duursma, Charlie Elliott, Antonio Fasano, Alistair Fitt, Andrew
Fowler, Ian Frigaard, Paul Glendinning, John Hinch, John Hogan, Malcolm
Hood, Rein van der Hout, Chris Huntingford, Andy Johnson, Guy Kember, John
King, Greg Kriegsman, Andrew Lacey, Sir James Lighthill, Ken Louie, Graham
McCauley, Katherine O’Malley, Tom Mullins, Gideon Ngwa, Hilary and John
Ockendon, Michael Peake, Colin Please (of course), Joel Rogers, Tryfan Rogers,
Immo Rupf, Philip Saffman, Warren Smith, Gil Strang, Louise Terrill, John
Titchener, Don Turcotte, Jonathan Willis-Richards, G. Willmer, Paul Wilmott
and Steve Wilson.
References “in alphabetical order”
[1] Potter, J.M., ‘Dr Spooner and his dysgraphia’, P. Roy. S. Med. 69, 639-648,
1976.
[2] Fantoni, B. (ed.) ‘Private Eye’s Colemanballs’, Pressdram, London, 1982.
[3] Anon. ‘Not the Schedules for the Mathematical Tripos, 1988-9’, Eureka 49,
60-61, 1989.
[4] Davis, P.J. ‘Thomas Gray, Philosopher Cat’, Grafton, London, 1991.
5


If I remember rightly, $\cos\frac{\pi}{2} =1$

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If I remember rightly, cos⁡π2=1\cos\frac{\pi}{2} =1cos2π​=1

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If I remember rightly, $\cos\frac{\pi}{2} =1$