This is a final wind-up chapter for this course, on frequently asked questions in mathematics. Recommended reading, with lots of scope for opinions!
Is a mathematician past it by the age of 30?
G. H. Hardy, writing in 1940 in his A Mathematician's Apology, section 4, said "No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game," and "I do not know of an instance of a major mathematical advance initiated by a man past fifty". Ignoring the now politically incorrect use of "man" here, this is in no doubt in large part true. Younger people may have more energy and perhaps a new vision, and one of the features of good mathematics is that vision is required to be able to guess what might be possible. Also, as Gowers says, society expects different kinds of work (in universities: teaching, administration, management) from older members of staff.
However Hardy's remarks, and A Mathematician's Apology in particular, are so well known I can't help thinking that his stated opinion on this matter has contributed widely to people's belief in it and the political and managerial decision on what the roles of an older mathematician should be. In other words, it is a bit of a self-fulfilling prophesy.
A mathematician working over a period of, say, 20 years will certainly gain a lot of knowledge and experience, and that is also highly useful. My view is that, if you can bring your advantages of wider knowledge and greater experience to bear, your mathematical contributions can be very great even well after the age of 30 or 40, or even 50.
Why are there few women in mathematics?
I have little to add to what Gowers' says, except to comment that the situation has changed slightly but noticeably in the right direction in the last 10 or 20 years or so (though there is still a long way to go).
I see no reason for the low numbers of women, except of course reasons in society as a whole, and these are obviously issues that we should all continue to work on.
Do mathematics and music go together?
There are two types of mathematicians: those that are good at music and those that aren't. There are two types of musicians: those that are good at maths and those that aren't. The connection is not necessary, and many good mathematicians are hopeless at music (but many are surprisingly good).
There is a direction to this "connection": it is not uncommon to find a very good professional mathematician who is also a strong amateur musician. It seems to be much rarer to find a very good professional musician who has anything more than a very basic school level ability in maths. But this is just an (unjustified) observation of my own based on the people I have met over a number of years. Maybe you can prove me right? (or wrong?)
As a mathematician with strong interests in music myself, the answer I would give to the question as stated is a very strong YES: they do go together, but having said this there is absolutely no reason why you cannot be a good mathematician if you are poor at music (or vice versa).
A mathematical mind does have a certain advantage in learning music initially: the notation for music is quite abstract and some of the "music theory" that you must learn is mathematical. This is the only connection that I personally see, and it only gives the budding musician with a mathematical mind the very briefest of head-starts in music. I see nowhere obvious where knowledge of music actually helps mathematics.
But, extending on the idea of mathematics in music (the mathematics of musical theory), many composers (in the 20th century particularly) had mathematical "theories" for their work. In most cases these are simply combinatorial, and would not count as "good mathematics" (e.g. in G.H.Hardy's sense of this phrase). Personally, I did work quite seriously on some mathematics connected with music quite recently: I was using mathematical knowledge concerning group theory to suggest new methods of composition in music. I think the work was even quite successful as music, and the mathematics was slightly above the normal level for "maths in music" (being approximately 3rd year UG level). In this regard I suppose I am highly unusual in even having even attempted something like this. But the success was not just down to mathematical skills, but also to some skills in music obtained independently.
Why do so many people positively dislike mathematics?
I do not know!
Gowers' comments here are very interesting, and worth reading.
Interestingly, this seems to be a modern thing. G. H. Hardy, writing in 1940 in section 10 of his A Mathematician's Apology, said "The fact is that there are few more 'popular' subjects than mathematics". If it is true that mathematics has been getting more and more unpopular over the last 100 years then that is very sad indeed.
In the popular media, films, etc., mathematicians or mathematically gifted people are very frequently portrayed as mentally subnormal or significantly "different" to "normal" people in terms of mental health issues. (E.g. "The Rain Man", "A Beautiful Mind", or many others.) This is also quite sad, and not at all representative of most mathematicians.
Do mathematicians use computers in their work?
I don't disagree with what Gowers says, including supporting him in the opinion that computers will become increasingly important in future.
Computers are certainly used as a medium for communication. This includes typesetting. Mathematical typesetting is rather complicated, so this is a major use of computers in mathematics. (Unfortunately, for a large number of different reasons, most of the "big" computer software companies, such as Microsoft, Apple, Google,..., have very noticeably and deliberately not been supporting mathematics in their software. Of all the main web browsers, only Firefox contains maths support. This situation occurs despite the fact that Microsoft, Apple, Google, etc., all negotiated in the last round of web standards for maths and agreed the final document. But after this they chose not to implement it.)
Obviously some computer systems do calculations on numbers in a similar way to (but more complicated than) calculators. This can be helpful, and numerical methods in mathematics are very much used in some areas of applied mathematics, but the majority of pure mathematicians do not use these systems at all.
Computer algebra systems are increasingly common. These systems do algebraic manipulations on mathematical expressions rather than numbers, and can be useful. But again, only a small minority of mathematicians find them useful for their research.
A few specialist areas of mathematics have specialist tools. This includes finite group theory, and (for example) group theorists often use specialist computer programs (such as "GAP") for their calculations in groups.
Another growing area is computer proofs, which has already been mentioned.
Despite all this, most pure mathematicians do not use computers except as a communication device and typesetting tool.
Is mathematics beautiful? or pretty?
Some mathematics is very beautiful, and some arguments are extremely pretty! I hope that either Gowers or I will have been able to persuade you of this!
Which mathematicians have won the Nobel Prize?
This is not on Gowers' list of Frequently Asked Questions, but I think it should have been put there. Maybe it is not such a frequently asked question, but if it is not then I suspect it is not for the following reason: very few people are sufficiently interested in mathematics to ask a question when they expect to get a long long long boring boring list of mathematicians and intricate details of their highly complicated work in proving various far-too-difficult-to-understand questions.
To back this up, I would add that you can regularly hear casual and dismissive remarks about such-and-such a person "winning the Nobel Prize for mathematics" in the popular media, films, TV etc.
The correct answer is that there is no Nobel Prize for mathematics and therefore no mathematician has won a Nobel prize for mathematics.
That's not to say no mathematicians have won a Nobel Prize. http://planetmath.org/listofmathematicianswhohavewonthenobelprize gives a list, but arguably it is incomplete or inaccurate: some of these people don't really count as "mainstream mathematicians". It includes Bertrand Russell, for example, who won the Literature Prize, and did start off in mathematics when young (though most people think of him as a philosopher). http://www-groups.dcs.st-and.ac.uk/history/Societies/Nobel.html gives another list which includes some physicists as well as mathematicians. Sometimes it is difficult to find the boundary between Physics and Maths, but mathematicians of this kind who have won the prize did so for their contribution to Physics.
Perhaps the best known mathematician to win a Nobel Prize is John Nash (the subject of the film "A Beautiful Mind"), who was a mathematician but won the Prize for Economics. The film contains some incorrect references, such as (as I recall) Nash's character being made to say that he should have been given the Nobel Prize for his work in number theory. This is impossible, and is an example of poor film scripts colouring the modern view of mathematics. In the film script, Nash's film character appears not to have understood how Nobel Prizes work; but the real Nash would certainly have known that work in number theory would not ever attract the Nobel Prize committee unless it had applications in one of the Nobel Prize disciplines.
If you are a mathematician and you want to win the Nobel Prize, it seems that your best bet is to learn some economics. Nash won other prizes in mathematics, but it is probably true that few if any of the other Nobel Economics Prize winning mathematicians would be rated so highly in the the Mathematics community.
The closest there is to a "Nobel Prize for Mathematics" is probably something called the Fields Medal. It's not quite the same, as there are different conditions attached (including the age of the recipient). One well-known recipient of the Fields Medal is a certain Timothy Gowers.