Wintry perversion

The great-grandmother of airports, Heathrow was closed yesterday. There will be tremendous chaos, but not of the type that characterizes war, or a natural disaster such as a large earthquake. Somewhat disturbingly, some part of me is actually enjoying seeing how the part of the world that I detest grinds to a halt.

I am of course speaking about the make-believe world where well-heeled executives jet around busily, shiny briefcase in hand, converging at precise locations and timings to discuss how they're going to screw the rest of us over again. I enjoy seeing how a few inches of snow manages to stop our great, big capitalist machine in its tracks, and how we will all actually survive this catastrophe. Even better is the irony that the man stranded at the airport might, just a few days ago, have been trying to convince everyone of his superior abilities to foresee the future.

Meanwhile, business is as usual in the rest of the universe, proving that 1000 sweatshop workers and 100 paper-pushers didn't really have to pay for their CEO's Christmas vacation to London. Of course, when the snow melts, we shall all scurry back to our designated positions, work doubly hard to clear the backlog, just in time to ensure that all the cogs in the machine will be turning nicely for my flight from Gatwick on New Year's Day.

Hey mum, erm... I'm studying to become a physicist, not a physician

Apart from the usual incredible memories I have from my trip to Iceland, which you can find pictures of on some other people's blog/Facebook page, I have also brought back two other things. The first is URTI, which I think is medical jargon for "the unknown infection that's stuffing up your nose". A few panadols later, I am up-and-running, ready to serve out some Icelandic humour (that's the second thing I brought back).

Unfortunately for my mum, I am not training to become a physician. I trust that she has figured that out over the years. As I have myself made a host of English language gaffes, I am not entitled to berate her for the unintentional mix up with "physicist". However, and this is no by-product of mathematical snobbery, the following accidental rediscovery of the trapezium rule cannot be forgiven:

A mathematical model for the determination of total area under glucose tolerance and other metabolic curves.

1. M M Tai

1. Obesity Research Center, St. Luke's-Roosevelt Hospital Center, New York, New York.

Abstract

OBJECTIVE--To develop a mathematical model for the determination of total areas under curves from various metabolic studies. RESEARCH DESIGN AND METHODS--In Tai's Model, the total area under a curve is computed by dividing the area under the curve between two designated values on the X-axis (abscissas) into small segments (rectangles and triangles) whose areas can be accurately calculated from their respective geometrical formulas. The total sum of these individual areas thus represents the total area under the curve. Validity of the model is established by comparing total areas obtained from this model to these same areas obtained from graphic method (less than +/- 0.4%). Other formulas widely applied by researchers under- or overestimated total area under a metabolic curve by a great margin. RESULTS--Tai's model proves to be able to 1) determine total area under a curve with precision; 2) calculate area with varied shapes that may or may not intercept on one or both X/Y axes; 3) estimate total area under a curve plotted against varied time intervals (abscissas), whereas other formulas only allow the same time interval; and 4) compare total areas of metabolic curves produced by different studies. CONCLUSIONS--The Tai model allows flexibility in experimental conditions, which means, in the case of the glucose-response curve, samples can be taken with differing time intervals and total area under the curve can still be determined with precision.

For the record, Dr Tai's mathematical discovery was published in a top medical journal, peer-reviewed by a number of world-leading experts, and, gasp..., cited over 100 times by other researchers. If you have bothered to sift through the abstract, you might have suspected that the confidently-named "Tai's Model" is really the "trapezium rule", which is something we have all learnt in secondary school when the teachers were preparing us for calculus.

I actually found this via some other blog, which had scathing comments about plagarizing integration from Newton. Actually, the blogger was probably being kind. Newton's invention of the calculus certainly required quite a profound insight. Independent discovery of integration, even if it took place centuries after Newton's, might arguably require some cognition. The definite integral roughly refers to the area under a curve, and through the Fundamental Theorem of Calculus, one can relate this geometrical quantity to the slope of the curve, i.e, differentiation That is at least Newton's insight. As for the trapezium rule (or in modern terminology, Tai's Model), the insight is that the area under a curve lies between the following two quantities: (1) the area of the trapeziums that you can squeeze under the curve, but almost couldn't, and (2) the area of the trapeziums that you can't but almost could. Most people will probably have played with one of those toys with holes of different shapes and corresponding pegs for those holes. A pleasant lesson from this toy is that you cannot squeeze a square peg into a round hole, and then you develop the commonsense intuition of area (which, incidently (or incidentally?), may not be quite right since it leads to the Banach-Tarski paradox). Equipped with this, you might discover, as Dr. Tai did, that you can approximate an area by breaking it up into little rectangles and triangles, and then summing up their individual areas. It might be advisable to check if your surgeon is familiar with Tai's Model before your operation.

At the other end of the spectrum, some people in the medical profession do have a wonderful sense of humour:

Inappropriateness of the concept of a price

Over a coffee today, I made a bold statement about economics, or more specifically, the fatally-flawed concept of pricing.

I claimed that using numbers (i.e. dollars and cents) to "measure" the value of goods is fundamentally wrong. I have always believed this statement to be obvious, although I have had a hard time convincing people. Often those with the economics education that I lack will introduce complicated words like "utility", "fluidity", "non-linearity", etc. But let's not get bogged down by these distractions. I have a simple situation: In front of me is a cup of coffee, a cup of tea, and a cake. I prefer the coffee to the tea, the tea to the cake, and the cake to the coffee. Well, then, there is no consistent way of pricing these three objects with numbers! Apart, of course, from giving all of them the same price.

A common way out of this conundrum is to say that the notion of value is subjective. Fine. But I think there is an objective way to value things. Just not with numbers. Here's what I think is wrong with using numbers in pricing objects. Suppose you claim that the cup of coffee is worth 2 dollars. Do you know what you have just done? Based on the premise that there is a meaningful numerical price to any valuable object, you have just simultaneously compared the value of the cup of coffee to everything of value in the universe! You would have made, implicitly, the claim that the cup of coffee is more valuable than anything, anywhere in the world, that you will possibly place a $1 price tag on. Now, is that cup of coffee still worth $2?

The problem, I feel, is not that numbers are inadequate for use in pricing, but that numbers are overly adequate for that purpose. For numbers are not as innocent as they seem. They are extremely well-behaved. They have the property that if Number A < Number B, and Number B < Number C, then necessarily Number A < Number C. For all choices of numbers. Now, at some point in our lives, we must all have faced the paradoxical situation in which you prefer product A to B, B to C, and C to A. For instance, suppose I have a choice of going out on a date with Amy, Betty, and Cindy. I prefer Betty's smile to Amy's. But I like Cindy's legs even more. Yet Amy has this mysterious aura about her that draws my attention. How can I  attach numbers/price tags on the value of these three dates? Well, the point is, you simply can't! Unless you have a preference that goes like A<B<C, there is just no way to use numbers consistently in a A<B<C<A situation.

Furthermore, it is completely natural not to have a "most-preferred date". But this is precisely what you have to commit to when you put price tags on, for among any three numbers, there is surely a biggest number! In this sense, the ordering that humans attribute to valuable objects, simply does not have the requisite properties of numbers! Why then, insist on using dollars and cents to label objects? Incidently, in the more primitive system of barter trade, such inconsistencies never arise, because one only ever compared the relative value of goods, and never made the leap of faith of operating on a dollar-and-cents system.

Now, the economists-armed-with-calculus (I'm trademarking this phrase) will come at my throat with all sorts of jargon to explain away this inconsistency. In retaliation, I could present some nasty algebra to expose the inherent divergence in the properties of valuation, and the properties of numbers. I find it a burden to try to look for the correct "price" for everything, and have it all work out consistently. Why not just concede that numbers are not the appropriate mathematical objects to use?

This is not to say that pricing is not useful. It certainly saves us some trouble, especially when we never had any preferences to begin with. It also allows us to feel a sense of satisfaction when we have more of these dollars than our neighbours. But to take the bold move of developing an entire theory based on dollars-and-sense might just be misguided. 

*At least two possibilities can arise from my little rant. Quite likely, I am naively commenting on a field that I have little expertise in. Or, maybe, I have made a truly profound observation, for which I stake a proprietary claim to. In any case, this is just my 2 cents worth.