# Mathematics  ## Metadata - Author: [[Timothy Gowers]] - Full Title: Mathematics - Category: #books ## Highlights - It turns out that even very simple models of systems of interacting particles behave in a fascinating way and give rise to extremely difficult, indeed mostly unsolved, mathematical problems. ([Location 341](https://readwise.io/to_kindle?action=open&asin=B000SEP2T2&location=341)) - An array of gates linked by edges is called a circuit, and what I have described is the circuit model of computation. The reason ‘computation’ is an appropriate word is that a circuit can be thought of as taking one sequence of 0s and 1s and transforming it into another, according to some predetermined rules which may, if the circuit is large, be very complicated. This is also what computers do, although they translate these sequences out of and into formats that we can understand, such as high-level programming languages, windows, icons, and so on. ([Location 365](https://readwise.io/to_kindle?action=open&asin=B000SEP2T2&location=365)) - A collection of points, some of which are joined by lines, is a mathematical structure known as a graph. Figure 5 gives a simple example. It is customary to call the points in a graph vertices, and the lines edges. ([Location 393](https://readwise.io/to_kindle?action=open&asin=B000SEP2T2&location=393)) - When devising a model, one tries to ignore as much as possible about the phenomenon under consideration, abstracting from it only those features that are essential to understanding its behaviour. ([Location 400](https://readwise.io/to_kindle?action=open&asin=B000SEP2T2&location=400)) - These are two senses in which mathematics is an abstract subject: it abstracts the important features from a problem and it deals with objects that are not concrete and tangible. ([Location 404](https://readwise.io/to_kindle?action=open&asin=B000SEP2T2&location=404)) - What matters about the black king is not its existence, or its intrinsic nature, but the role that it plays in the game. The abstract method in mathematics, as it is sometimes called, is what results when one takes a similar attitude to mathematical objects. This attitude can be encapsulated in the following slogan: a mathematical object is what it does. ([Location 427](https://readwise.io/to_kindle?action=open&asin=B000SEP2T2&location=427)) - the theory of quantum mechanics, for example, depends heavily on complex numbers. ([Location 618](https://readwise.io/to_kindle?action=open&asin=B000SEP2T2&location=618)) - shall discuss many concepts, later in the book, of a similar nature to these. They are puzzling if you try to understand them concretely, but they lose their mystery when you relax, stop worrying about what they are, and use the abstract method. ([Location 666](https://readwise.io/to_kindle?action=open&asin=B000SEP2T2&location=666)) ## New highlights added August 1, 2023 at 7:02 AM - As the last few paragraphs illustrate, the steps of a mathematical argument can be broken down into smaller and therefore more clearly valid substeps. These steps can then be broken down into subsubsteps, and so on. A fact of fundamental importance to mathematics is that this process eventually comes to an end. In principle, if you go on and on splitting steps into smaller ones, you will end up with a very long argument starts with axioms that are universally accepted and proceeds to the desired conclusion by means of only the most elementary logical rules (such as ‘if A is true and A implies B then B is true’). What I have just said in the last paragraph is far from obvious: in fact it was one of the great discoveries of the early 20th century, largely due to Frege, Russell, and Whitehead (see Further reading). This discovery has had a profound impact on mathematics, because it means that any dispute about the validity of a mathematical proof can always be resolved. ([Location 744](https://readwise.io/to_kindle?action=open&asin=B000SEP2T2&location=744)) ## New highlights added August 2, 2023 at 7:53 AM - If we want to understand the statement completely, this phrase forces us to ask what sort of object the square root of 2 is. And that is where infinity comes in: the square root of 2 is an infinite decimal. ([Location 944](https://readwise.io/to_kindle?action=open&asin=B000SEP2T2&location=944)) - The neat infinite statement is ‘x is an infinite decimal that squares to 2’. The translation is something like, ‘There is a rule that, for any n, unambiguously determines the nth digit of x. This allows us to form arbitrarily long finite decimals, and their squares can be made as close as we like to 2 simply by choosing them long enough.’ ([Location 1014](https://readwise.io/to_kindle?action=open&asin=B000SEP2T2&location=1014)) - The way to understand instantaneous speed is to exploit the fact that the car does not have time to accelerate very much if t is very small – say a hundredth of a second. Suppose for a moment that we do not try to calculate the speed exactly, but settle instead for a good estimate. Then, if our measuring devices are accurate, we can see how far the car goes in a hundredth of a second, and multiply that distance by the number of hundredths of a second in an hour, or 360,000. The answer will not be quite right, but since the car cannot accelerate much in a hundredth of a second it will give us a close approximation. ([Location 1038](https://readwise.io/to_kindle?action=open&asin=B000SEP2T2&location=1038)) - This is another example where the abstract approach is very helpful. Let us concentrate not on what area is, but on what it does. ([Location 1068](https://readwise.io/to_kindle?action=open&asin=B000SEP2T2&location=1068)) - We find (and this is not quite as obvious as it seems) that as we take finer and finer grids, with smaller and smaller squares, the results of our calculations are closer and closer to some number, just as the results of squaring better and better approximations to become closer and closer to 2, and we define this number to be the area of the shape. ([Location 1089](https://readwise.io/to_kindle?action=open&asin=B000SEP2T2&location=1089)) - Another way of putting it, which is perhaps clearer, is this. If a curved shape has an area of exactly 12 square centimetres, and I am required to demonstrate this using a grid of squares, then my task is impossible – I would need infinitely many of them. If, however, you give me any number other than 12, such as 11.9, say, then I can use a grid of squares to prove conclusively that the area of the shape is not that number: all I have to do is choose a grid fine enough for the area left out to be less than 0.1 square centimetres. In other words, I can do without infinity if, instead of proving that the area is 12, I settle for proving that it isn’t anything else. The area of the shape is the one number I cannot disprove. ([Location 1098](https://readwise.io/to_kindle?action=open&asin=B000SEP2T2&location=1098)) - High-dimensional geometry is yet another example of a concept that is best understood from an abstract point of view. Rather than worrying about the existence, or otherwise, of twenty-six-dimensional space, let us think about its properties. ([Location 1130](https://readwise.io/to_kindle?action=open&asin=B000SEP2T2&location=1130)) - Multi-dimensional geometry is also very important in economics. If, for example, you are wondering whether it is wise to buy shares in a company, then much of the information that will help you make your decision comes in the form of numbers – the size of the workforce, the values of various assets, the costs of raw materials, the rate of interest, and so on. These numbers, taken as a sequence, can be thought of as a point in some high-dimensional space. What you would like to do, probably by analysing many other similar companies, is identify a region of that space, the region where it is a good idea to buy the shares. ([Location 1274](https://readwise.io/to_kindle?action=open&asin=B000SEP2T2&location=1274)) ## New highlights added August 3, 2023 at 7:49 AM - The people who write these letters have no conception of how difficult mathematical research is, of the years of effort needed to develop enough knowledge and expertise to do significant original work, or of the extent to which mathematics is a collective activity. By this last point I do not mean that mathematicians work in large groups, though many research papers have two or three authors. Rather, I mean that, as mathematics develops, new techniques are invented that become indispensable for answering certain kinds of questions. As a result, each generation of mathematicians stands on the shoulders of previous ones, solving problems that would once have been regarded as out of reach. If you try to work in isolation from the mathematical mainstream, then you will have to work out these techniques for yourself, and that puts you at a crippling disadvantage. ([Location 1978](https://readwise.io/to_kindle?action=open&asin=B000SEP2T2&location=1978))