Chapter 1 of Gowers' book, "Mathematics: a very short introduction" introduces us to the idea of a mathematical model. Mathematical models are not like model ships, model aeroplanes, etc., though in some respects there are similarities.
For example, if we were to want to test out some new idea that might make aeroplanes fly better we might make a model to test it. Whether the new idea works equally well on the model as it would at full size is an interesting question. For example the fantastic new idea we have just thought of may actually be something that doesn't scale to 100th life size for various reasons. (Changing the scale of things, especially in a significant way, often changes the relative properties of fluids such as air or water significantly.)
In a mathematical model, the stuff we make our model out of is "pure mathematics". To what extent the stuff of mathematics is real or not is an interesting and very difficult question. (Some of the discussion in this module will be hinting about what this "abstract mathematical stuff" really is, however the full question is too difficult for a precise answer here.) Using abstract mathematics in this way has major advantages: in principle we know everything about our model (because we defined it perfectly rigorously) and can calculate all its properties using mathematics. So mathematical models are "perfect" in a way that model ships and aeroplanes cannot be.
Mathematical models can be used to obtain predictions about the real world, and can be used to obtain insight into the workings of the real world. Here "real world" just means whatever it is we are modelling. Gowers has a number of very different examples.
One of the problems is how to determine the relevance of the model. Is it a good model? What do we mean by good? In which situations does the model make sensible predictions and in which situations not? There are no simple answers to any of these questions.
E.g., on page 1, Gowers talks about "the best compromise". Determining this best compromise is often very difficult. There are a number of ways you might try to determine if you have a reasonable compromise, and Gowers mentions many of them in his chapter.
The main point is that complicated models take complicated mathematics to analyse, but possibly have a better chance of being "more accurate". A clever view is to decide how much accuracy you need in advance and build this into the model so the model predicts the answer and tells you how accurate it expects that prediction to be. Unfortunately such models with estimates of accuracy are also much more complicated to work with.
Here is a link to a section of a lecture given by the Nobel prize winning physicist Richard Feynmann on modelling and the relationship between maths and physics. (I don't agree with absolutely everything he says, by the way.) Feynman: Mathematicians versus Physicists (accessed 2016-07-27).
Probability models are interesting too, because the predictions they make seem nothing like predictions from deterministic models. For example quantum mechanics is a probabilistic model of the physical universe. What does it mean to make predictions from such a model?
Most answers to this question refer to repeatable experiments. Such as repeatedly tossing a die under identical circumstances. Of course the circumstances in the real world are never identical. So what is going on? For example (writing this in 2016 well before the US presidential election) I came across the statement that "Hilary Clinton has a 75% probability of winning the presidency". What on earth was meant in this statement?
Gowers says some things on these matters, and I shall leave you to think some more for yourselves and explore other sources.
On the behaviour of gases and talking about the molecules of the gas, Gowers says (page 8) "Temperature, for example, corresponds to their average speed." Be careful of this, as it is potentially misleading. The problem is that he doesn't say what he means by "corresponds to". In other words his statement is incomplete and potentially incorrect, since many people would understand by "corresponds to" the phrase "is proportional to". It is essential in academic scientific work to be precise about terms and if there could be any doubt, to define them carefully.
In fact a better statement would be that "Temperature, for example, is determined by the molecules' average speed, and in fact temperature (in degrees Kelvin) is proportional to the square of this speed." Of course this statement is just telling you how to obtain from a model of a gas a prediction about its temperature. It is not saying this model is perfect or its predictions are always correct.
Gowers also discusses models of computers, the brain, and of "real world" problems such as timetabling, and colouring maps. These are all particularly interesting and very close in spirit to much university mathematics.
The idea of abstraction arises for the first time here, because (for example, in the map colouring problem) we choose to represent concrete real-world objects such as countries and colours with abstract objects such as names or letters or numbers. Thus abstraction is some kind of correspondence between real world objects and mathematical objects.
Gowers goes on to say that mathematics is an abstract subject because:
There will be much more on abstraction later.
title: Is a more accurate model necessarily better than a less accurate model?
Definition of terms:
(This section could take up quite a large part of the essay, maybe a half or even more.)
Give your opinion. Several options seem possible.
Give reasons and examples supporting your opinion.
Summarise your position and what you have learned in this discussion.