Essay titles for week 6, 2019-20 version

This web page provides the list of essay titles that for submission at the end of Autumn week 6. Note also that essays will be subject to anti-plagiarism checks.

Think about the titles, read the book and any other relevant materials you can find, e.g. on web pages. Make notes of anything you find that might be interesting or relevant.
There should be plenty of scope for incorporating something you are particularly interested in, but be careful not to be too ambitious, and remember that an essay with limited scope and lots of explanations is going to be much better than one which is missing many of the necessary arguments.
Make a draft "essay plan" on one piece of paper. Include a short introduction on what you will be trying to say, and a conclusion section. Use this plan to inform yourself of what other reading/research you might want to do. Do further reading as necessary. If you find it difficult to think about or discuss any of these points in full generality, try to think of illustrative examples.
It is a good idea to start off by planning two or three different essays. You might find that the first one you try is too difficult and one that starts off looking less promising is actually much better.
Start writing a draft essay. I suggest you start with putting down titles for sections from your plan, and then start to fill in details. You don't have to start writing at the very beginning in section 1 (but if definitions are going to be needed it obviously makes sense to start with these). In fact filling in details for the middle section(s) is usually the best way to start. The opening section and final conclusions section are usually best written at the end.
By Friday of week 4, submit your draft to Canvas.
From this point onwards you will be able to see some other students' draft essays. Make constructive and helpful comments on them. You should expect to get constructive and helpful comments back, including ones from teachers on the module.
By Friday of week 6, submit your final essay for marking to Canvas.

With particular reference to mathematical modelling, say what the difference between Science and Mathematics is.: Gowers Chapter 1, my section on Models including the Feynman youtube clip may be relevant. Use plenty of examples as illustration (you don't need to go into details for any of them). Students interested in Computer Science may like to think about whether Computer Science is science or not!
A deliberately inaccurate model is useless. Discuss.: This essay will require you to be able to think of and describe a number of good examples of mathematical models. These will be best if they are very simple, and you will probably need several. There are many examples in lectures and in the book but you can find more by reading around on the internet. It is essential that you define the terms in your title. Obviously this includes "model" but also "accurate". How might you know that a model is or is not accurate? what part of the modelling process most closely concerns accuracy?
Random mathematical models and probability: how can a probability be a prediction?: Gowers discusses probability as one case of mathematical models. It is interesting to note that such models do not predict a single answer but a range of answers. How does this work? How can experimental evidence support such a model? What can the model tell you about the "real world"? Can you find any bogus or imprecise or incorrect uses of the language of probability in every day usage, and why are these wrong? Note also that Probability can say more than just the average outcome of shaking a dice. It can also say what range that average is likely to be in after (say) 100 throws. Can you explain this?
Focusing particularly on the modelling process itself, write an essay on the shape of a piece of A4 paper.: One of the lectures discussed these considerations. This essay will require you to explore sources outside Gowers' book and these web pages. Don't forget to mention the experimental part of the subject, which might (or might not) support the model, and what the model can tell you about the "real world". Note that there may be more than one model to consider.
Number systems described using axioms.: Explain the axiomatic method for understanding numbers and number systems. Compare a range of different number systems using this method by explaining which axioms do and do not hold. Be sure to distinguish between the axioms and the number system that does or does not satisfy the axioms. What does this method tell you (or not tell you) about numbers? Gowers' Chapter 2 is of course highly relevant, but see my additional pages highlighting the fact that that there are additional axioms Gowers does not give, which may have to be discarded when one passes to a bigger number system.
The game of abstract mathematics.: To what extent is abstract mathematics like a game? Illustrate with one or more games such as Noughts and Crosses, Nim (https://en.wikipedia.org/wiki/Nim), Chess, or some other game(s) of your choice. Try to find analogies or disimilarities with mathematics. You might like to explain which games are more like mathematics and which ones are not at all like mathematics. Ideas of games to think about: Solitaire games for one person e.g. to find a way to remove all the pegs from a board except one; abstract games like Chess with finitely many positions in which draws are possible; abstract games like Hex or Go on an unspecified size of board; physical games like Tennis, Cricket, Football; games of chance like Backgammon, Poker, Monopoly, Bridge etc.