Obtain Plausible Statement B from Plausible Statement C by

setting MD 0.

At this point the reader may scratch her head. Have we not already proved Plausible Statement C in Section2.4? It she looks back, she will see that we did indeed prove Plausible Statement C, but we did so by assuming Plausible Statement A!

16 _{The reader will recognise elements of Sections2.3and2.4but, as I warned her there, these} elements will be combined in a very different way.

Fortunately, it is possible to prove Plausible Statement C directly, though, as the reader may by now expect, the proof depends on exploiting the Fundamental Axiom.

Theorem 11.5.3. [The mean value inequality] Ifjg0(t)j  M for all t, then jg(b)  g(a)j  Mjb  aj for all a and b.

For reasons of tradition as much as anything else, the mean value inequality is usually deduced from a slightly more powerful result known as the mean value theorem.

I strongly advise the reader to make a note of every time the mean value inequality (or some similar result) is used, since otherwise it is easy to lose sight of its importance. I also advise the reader to exercise great care whenever she proves the theorem, since it is fatally easy to produce circular arguments.

11.6 What next?

It is generally agreed that the intermediate value theorem, the theorem on the existence of maxima and the mean value theorem represent the centre of any first course in analysis. Such a course will also discuss and justify rules for dealing with infinite sums. It will also give a definition of an integral which does not depend on any ‘intuition’ concerning the properties of ‘area’. It turns out that the Fundamental Axiom is also required to show that_abf(t) dt exists whenever f is continuous,17 _{but that the rest of elementary ‘integral calculus’}

is easily justified. The fundamental theorem of the calculus is now proved in two stages. The proof that ‘differentiation reverses integration’ is, essentially, that given in this book and the proof that ‘integration reverses differentiation’ uses the mean value inequality as outlined in the previous section.18

There are many books that set out the contents of a first course in analysis. Burkill’s A First Course in Mathematical Analysis [1] is elegant and to the point. If you have a good background in calculus, this may well be the book for you. If you are less well prepared or you prefer a more discursive style, then Spivak’s Calculus [6] is excellent. Both books have good exercises.19

17 _{We might expect this, since the intermediate value theorem shows that}x

1(1/t) dt with x > 0 takes every real value as x varies.

18 _{Some careful writers underline the point by referring to the results as the first and second} fundamental theorems of the calculus, but this useful distinction has not caught on. 19 _{These books have my strong recommendation, but it is best to discuss further reading with}

someone who knows you personally and choose a book which fits in with whatever your present or intended future educational institution does. If you want to widen your mathematical background, Courant and Robbins’ What is Mathematics? [2] remains unsurpassed.

Once the first course in analysis is out of the way, we are free to advance towards the broad sunlit uplands of modern analysis and the rest of advanced mathematics.

11.7 The second turtle

Rigorous calculus based on the Fundamental Axiom was completely successful in banishing the paradoxes which threatened to block further progress. How- ever, mathematicians being what they are, they set about looking for a second turtle on which to stand their first turtle. Recall that I said that rigorous calculus depended on the Fundamental Axiom (plus the usual rules of arithmetic and the usual rules of inference). If you look at the this statement sufficiently suspi- ciously, you may begin to wonder what ‘the usual rules of inference’ actually are.

Of course, this is a question about mathematics in general and not just analy- sis. The second turtle has to support not only the turtle of analysis, but the turtle of geometry, the turtle of arithmetic and any other turtles that mathematicians wish to study. It turns out that, in trying to make clear the rules of mathematical inference, we also have to make clear the nature of a mathematical object and draw up rules about how to construct one mathematical object from another. The study of mathematical inference is called mathematical logic and the study of how mathematical objects are constructed is called set theory. Analysts tend to find that set theory impinges rather more on their working practices than mathematical logic. Standard set theory itself rests on a number of axioms which are nicely set out in the classic text Naive Set Theory [3] by Halmos.

Starting from these axioms we can construct the positive integers, the ratio- nal numbers and the real numbers in such a way that the usual rules of arithmetic and the Fundamental Axiom hold. By constructing the real numbers rather than postulating their existence we seem to evade Russell’s reproach20 _{that ‘The}

method of “postulating” what we want has many advantages; they are the same as the advantages of theft over honest toil.’ However, in order to perform the constructions, we have to accept the axioms of standard set theory. The major- ity of mathematicians accept these axioms, not because they are obvious, but because no one can see any way to deduce them from something which is more obvious. A minority (though a respectable minority) seek a different second turtle but, at least for the moment, it seems unprofitable to seek a third turtle.