A bouncing ball captured with a stroboscopic f...
A bouncing ball captured with a stroboscopic flash at 25 images per second. (Photo credit: Wikipedia)

Mathematical models are supposedly descriptions of a real phenomenon. The descriptive and predictive power of a model depends on how well the model represents the real phenomenon. Extreme precision is not necessary for a good models, so long as it doesn’t vary wildly or deviate from the real phenomenon. If the accuracy of the measurements or observations of the phenomenon are less than the deviation of the model from the real phenomenon, then the model suffices for the purposes.

For instance, a stone thrown upwards or a ballistic round fired from  a cannon roughly follow a parabolic trajectory and the model (in this case a simple algebraic equation) is often accurate enough. However other effects, such a the resistance of the air to the passing of the object and the curve of the earth have to be accounted for in the model if the accuracy of the measurements is such that deviations from the model caused by these effects can noticed.

FN 57 ballistics 100yd
FN 57 ballistics 100yd (Photo credit: Wikipedia)

I’m going to draw a slightly artificial distinction here between ‘mathematical effects’ and ‘physical effects’. By mathematical effects I mean effects like the curvature of the earth (and also, the distance to the centre of the earth), both of which affect the geometry of the model. By physical effects I mean things like air resistance, and the roughness of the missile, which can’t be directly deduced from the physical situation and have to be assessed by experiment. Of course in many cases others have studied the effect of things like air resistance and their results can be plugged into our model to enhance its accuracy.

English: Diagram of simple gravity pendulum, a...
English: Diagram of simple gravity pendulum, an ideal model of a pendulum. It consists of a massive bob suspended by a weightless rod from a frictionless pivot, without air friction. When given an initial impulse, it oscillates at constant amplitude, forever (Photo credit: Wikipedia)

Mathematical effects are ultimately based on physical ones. For instance Newton’s Law of attraction between two masses is a physical effect represented by a mathematical equation – the product of the two masses and the gravitational constant divided by the square of the distance between them gives a measure of the gravitational attraction between them. On the surface of the earth, where the vertical movement of a thrown stone is negligible compared to the distance between the centre of the earth and the stone, this means that we can ignore the variation of the trajectory due to this effect since it is so small and use the mathematical model of a parabola for the projectile’s trajectory.

It turns out that simple parabola is useful as a model only for simple cases where the velocity is low and the distances are small, and the accuracy of measurement is low. For artillery purposes a model based on a simple parabola is not accurate enough. To drop a shell on someone’s head, where you know the distance, you need to factor in not only wind resistance and the curve of the earth, but also such factors as wind direction and strength and even then a sudden gust of wind could put your aim off. The model that artillery men used is contained in a set of tables which were built up over years of experience.

Cannon Model - Part of my military models coll...
Cannon Model – Part of my military models collection (Photo credit: Wikipedia)

It is clear, I think, from the above discussion that models are pragmatic constructs. If a model doesn’t work you merely change it or replace it with one that suits your purposes better. That doesn’t mean that the old model is totally abandoned. After all, the artillery man doesn’t need his complicated tables when all he wants to do is shoot a basketball through a hoop.

Some models are purely descriptive and non-quantitative, such as the economic ‘supply and demand’ model. This is usually depicted by a graph showing one line sloping down from left to right crossing another line sloping up from left to right. The upwards sloping curve is the ‘supply’ curve and the downwards sloping curve is the downwards sloping one. The vertical axis is marked ‘Price’ or similar and the horizontal axis is marked ‘Quantity’ or similar. Rarely are there any tick marks or values on either of the axes.

Supply and demand
Supply and demand (Photo credit: Wikipedia)

The trouble this model is that it is, to my mind, too vague and woefully incomplete to be really useful. Firstly, the lack of any quantitative units means that any usage of the model must be qualitative and prevents it from being useful in any real situation. Secondly, while the trends of the supply and demand curves may be generally in the directions usually shown, this is not generally true, especially if the demand or the price moves far from the current ‘equilibrium’ point. Thirdly price changes are usually discussed in terms of change in demand, whereas the opposite is probably more usually true, and demand is driven by price. Fourthly, the shape of the curves does not stay static and they change with time, often unpredictably. Fifthly, there are many more external influences that are likely to have a bigger effect on price than simply supply and demand. Monopolies and monopsonies have huge effect on prices, and supply and demand can have little or no effect in these situations. The validity, if any, of the model is limited to a very restricted domain of situations.

The biggest criticism of this economic is that it doesn’t lead to quantitative models. It doesn’t direct strategies and few people, I’d suspect, actually use the curves for anything, except economics lecturers.  It is not alone in the economics field, though, as there appear to be no models which are quantitative, valid in more than a small domain, and generally accepted in general use. It’s possible that there never will be.

And Again? Throw the Ball!
And Again? Throw the Ball! (Photo credit: wharman)

I’ll close off by mentioning two other usages of the word ‘model’. There are many more usages, but I’ll leave those for now.

Firstly there is the catwalk model – young ladies and some men who acts clothes horses for fashion and ‘haute couture’. I’ve no problem with that except the usual one, that the models are thin to the point of anorexia, and sometimes the clothes stray to the bizarre side of the street. These young people, should they catch the eye of the fashion industry, may make many millions of dollars. The people who pay them these dollars must feel that they get some benefit from the payment, which brings us back to economics, supply and demand!

Model (Photo credit: vpickering)

Secondly there is the constructional meaning of the word – where people construct sometimes exquisite copies of objects at a much smaller scale and of different materials to the original. Often these models are placed in context in models of the usual surrounding of the original – a model train may run on a complex layout with stations, signals, bridges and so on. Often as much care is lavished on the model’s surroundings as on the model itself. Many of these are true works of art.

"Carlton J. Dearborn, S2c [cements a stri...
“Carlton J. Dearborn, S2c [cements a stringer on the fuselage of balsam model of Stuka Dive Bomber at Camp Smalls, U.S. Naval Training Station, Great Lakes, IL. Dearborn teaches sailors to identify enemy and Allied aircraft].”, 03/13/1943 (Photo credit: The U.S. National Archives)
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