A nice workout typically jolts you out of bad day and makes you alive. For me,a good book does the same when I have no energy to step out of home. Last few days I have not been keeping well and this book has kept my mind active despite the dullness around.
In this delightful little book, Timothy Gowers delves in to the philosophical differences between people who are happy with the notions of infinity, curved spaces, square-root of –1, N dimensional spaces, etc, from those who find them disturbingly paradoxical. There are many amongst us who question the very purpose for thinking about quantities that do not exist in their experience. Why bother about to think about infinity? It’s just a mathematician’s symbol for something that is not finite. Why bother about anything beyond the 3 dimensional experience objects? Can you show me a 4 dimensional object? If not, why should I spend my time understanding a four dimensional geometry ?, goes the usual stance. They say, god created integers and rest of the math is basically human mind’s construct. So why bother developing a mathematical approach to our daily life situations, if it’s just a theoretical construct , a ghost of our imagination ? The book’s underlying theme is that math is a happy ghost which will only help you in understanding stuff better.
This book does not talk about history of math/ math disciplines, nor does it talk about various theorems or proofs of a specific subject. Its central purpose is to urge the reader to think mathematically, in other words, think abstractly. What is abstraction in mathematics?
One type of abstraction occurs in model building. When devising a model, one tried to ignore as much as possible about the phenomenon under consideration, focusing only those features that are essential for understanding the behaviour. In examples relating to Physics, let’s say a projectile motion of a stone, one abstracts away all the forces except the initial velocity, angle of projection and effect of gravity. While studying behaviour of gases, one abstracts away all the interactions of particles and considers individual particles moving in a chamber with no interactions. Despite removing most of the effects present in the real life scenario, the variables and the model used could just be the right kind of abstraction necessary to study it appropriately.
There is another type of abstraction which one finds in mathematics, which is the subject of this book. This abstraction is much deeper. One can easily relate this type of abstraction to let’s say a a chess piece, for example black king. If one were to ask whether black king exists, the question might make sense in the existential sense but goes no further. Yes, you see a chess piece which represents a black king.It begets an immediate question , “ What does a black king do in chess? ”. This is a far more interesting question as it talks about the role of black king than some platonic existential stuff. Mathematics is similar to the situation above where mathematical objects by themselves might not mean anything from a existential point of view. A mathematical object is what it does.
Take for example N, the set of natural numbers. For some numbers like 1, 2, 3, till some finite number you can probably visualize the number. Additions of these small numbers might make sense but once you get to slightly bigger numbers , a simple addition like 243 + 786 does not make any existential sense. Well, for any set of natural numbers, once you decide on the associate, commutative and distributive laws, then the result of 243 + 786 follows from these rules. There is nothing more than that. Numbers need not be very large before we stop thinking of them as isolated objects and start to understand them, through their properties, through how they relate to other numbers in a number system. So, after learning the basic number system at a school level, we are supposed to look at number “system” in the sense of what they do.
Similar is the case of negative numbers, fractions, irrationals, complex numbers. If you want to understand square-root of 2 , it is better to understand to what it does. Square root of 2 does not mean anything. It is not something we can see somewhere in reality. So, what should be the attitude towards such mathematical objects ? Well, it basically solves the equation X*X = 2. That’s it .Introduction of irrationals and defining properties is purely an abstract exercise but which gives us ways to solve equations. Same is the case with complex numbers. “i”, the symbol by itself does not carry any meaning, but in the context of solving an equation, X*X+ 1 = 0, it makes tremendous sense. i, a completely abstract thing , is used very heavily in the theory of quantum mechanics. It provides one of the best illustrations of a general principle: if an abstract mathematical construction is sufficiently natural, then it will almost certainly find a use as a model. There are umpteen such objects in math like infinity, logarithms, exponentials, fractional powers which need to be viewed from the perspective of, “What they can do? “.
Mathematics as a subject is built axiomatically. There are a set of axioms to begin with and mathematicians / scientists create statement called proofs that are built from these axioms. So in one sense, math is one subject where the disputes in principle, “can end”. If you and I debate about a theorem, we can dig deeper and deeper till all the axioms are laid on the table. As long as the logical structure is intact, we can say that theorem is proved. At the last stage where axioms are seen, mathematicians stop the argument. Now one can ask, “why not debate about axioms in mathematics?”. The most important thing that matters to mathematicians is less the truth aspect of axioms and more their consistence and usefulness. In that sense, it is very much different from let’s say economics. Two economists can debate about Monetarism and Neo-Keynesianism till their last breath and still not reach a conclusion.
Let me end my take on this book by giving a nice example which shows the usefulness of abstraction.
Visualize a 4 dimensional cube. How many edges does it contain?
To answer the above question, you got to understand the term “visualize”. In our everyday parlance, visualize is something we do with the help of our mind to “see” stuff which is not yet in our consciousness. For a mathematician, visualization has a different meaning. An object or concept that cannot be visualized by a mathematician, means it is something for which he needs to stop and calculate. I will try to make it clear in the context of this example. One can “just see” that for a three dimensional cube, there are 4 edges round the top, 4 edges round the bottom, 4 going from top to bottom,making it 12 all. Now a mathematician “can see” a 4 dimensional cube in this way. He would say :
“ I can think of a four-dimensional cube facing each other , with corresponding vertices joined by vertices (in the fourth dimension) just as a three dimensional cube consists of two squares facing each other with the corresponding vertices joined. Although I do not have a completely clear picture of 4 dimensional cube, I can still “see” that there are 12 edges for each of three dimensional cube, and eight edges linking the vertices together . This gives a total of 12+12+ 8 = 32. “
Thus an answer to the above question can be obtained even if such an object is beyond our practical experience.In the above case, visualization enabled an answer to the question. Not always. Mathematicians spend considerable time developing theorems/tools/lemmas to deal with higher and higher level of abstraction. These help in bridging the gap between “What mathematicians can see” Vs “What mathematicians cannot see ?”
So, next time you see a mathematical object described via properties and rules, this book will serve as a gentle reminder that that those rules and properties constitute the abstraction that is needed to work on them. Existential questions such as “what is i?”, “What is α?“, “What is logarithm?” are of no relevance.
If you want to understand from a mathematician’s point of view, “What is abstraction ? ” & “Why is it important to develop a certain sense of abstract thinking? ” , this book is spot on.