Matematika kolotočov a roller-coasterov

Tento týždeň som bola na konferencii vo Švédsku o tom, ako učiť inžinierov matematiku. Zaujímavé, miestami inšpiratívne, takže cesta stála za to, aj keď som nezostala úplne do konca (aby som sa stihla vrátiť na Kikin posledný prvácky školský deň).

Jednu z prednášok mala Ann-Marie Pendrill. Učí prvákov fyziku a na začiatku semestra ich berie do miestneho zábavného parku Lisberg. Rozdelia sa do skupiniek, v ktorých potom riešia rôzne úlohy a keď treba (alebo aj keď netreba) sa popritom vozia. My sme tiež dostali takúto sadu úloh, ale z mojej skupiny som bola jediná, kto bol výsledky aj experimentálne overovať…

Kolotoče boli výborné, ale roller-coaster mi stačil jeden:

Vďaka neďalekému koncertu a nie práve ideálnemu počasiu naň neboli žiadne rady. Vedľa mňa sedela Ann-Marie, na ruke náramok, ktorý má ako celoročnú vstupenku, a keď sme vystúpili, zahlásila, že to bola jej 47. jazda… na večeru sme išli až potom.

A Mathematician’s Apology

By G. H. Hardy, 1940, 153 pages, partial text, amazon

I propose to put forward an apology for mathematics; and I may be told that it needs none, since there are now few studies more generally recognized, for good reason or bad, as profitable or priceworthy… A mathematician does need not now consider himself on the defensive… The public does not need to be convinced that there is something in mathematics. All this is in its way very comforting to mathematicians, but it is hardly possible to be content with it. Any genuine mathematician must feel that it is not on … crude achievements that the real case for mathematics rests, that the popular reputation of mathematics is based largely on ignorance and confusion, and that there is room for more rational defence. At any rate, I am disposed to try to make one.

And that is what he gives in this tiny book. More than a third of it is a foreword by C. P. Snow, which introduces Hardy to the reader, introduces him as a mathematician proud of what he’s doing, person with very delicate and unorthodox thinking who places high expectations on everyone and highest on himself.

Then you start reading the essay itself and whether you are a mathematician or not, you’ll be absorbed. For non-mathematicians this is an amazing opportunity to peek into mathematician’s way of thinking, reasoning, motivation and notion of beauty. For mathematician (in addition to all of the above) this is a unique verbalization (sparse, but very much to the point) of what they understand on an intuitive level, but would probably have a hard time expressing so clearly.

Even though you might not agree with all of Hardy’s ideas (I don’t, especially with the those concerning applied mathematics… which is probably due to the fact that applied mathematics is today something very different from what Hardy considered it to be), you cannot help yoursel but admire the way he presents his points.

And what is he actually apologizing for? You need to read that for yourself. Highly recommended!

PS On a similar note (but not as beautifully written), from much younger (but at least as talented) mathematician Terry Tao comes an article What is good mathematics?

Does God Play Dice?

By Ian Stewart, 1989, 317 pages, partial text, amazon

This is a book about chaos. Not the chaos you have in your closet after three years of not cleaning it up but the mathematical chaos. And it is written with almost no equations at all, which makes it highly readable for non-mathematicians, but leaves the mathematicians somewhat hungry by not going deep enough. That was probably the intention, though – to provide interesting enough introduction that the hungry go on reading further on their own.

Also the book is some 20 years old and much have been done in the theory of chaos since – another reason to look for further reading – but it provides a very nice background of how people started to realize that in the nature there actually exist something “between deterministic and random” – something that can be deterministic and random (in a sense) at the same time. And it shows how we encounter this kind of stuff almost everywhere – problems with prediction of weather, movement of asteroinds, fluid dynamics, etc and how it took people from different areas of expertise to find out what it is.

What I also liked about this book was that since it is a popular math book, it takes various side trips, talkes about mathematicians, their lives, about non-mathematicians and their approach to math. It gives hope to an aspiring applied mathematician:

To criticize mathematics for its abstraction is to miss the point entirely. Abstraction is what makes mathematics work. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools: analogy, generality and simplicity. Mathematics is the ultimate in technology transfer…

However, technology transfer needs more than just the technology. Someone has to transfer it. So, while mathematicians should be encouraged to continue doing whatever it is that mathematicians do – whether or not the outside world can understand a word of it – it will remain just an art-form unless enough people are willing to make the effort to apply it to problems outside mathematics. The story of chaos is full of such people. They come from all subjects – physics, biology, engineering, chemistry, physiology, astronomy as well as mathematics. They are the true ‘applied mathematicians’ and they do what that phrase ought to mean. They take mathematics and they apply it.

PS. If God played dice, He’d win.