Read Through: 'Die Architektur der Mathematik: Denken in Strukturen' by Pierre Basieux
A review of Pierre Basieux's book 'Die Architektur der Mathematik: Denken in Strukturen'
After having read through Gowers's book Mathematics: A Very Short Introduction and gaining some momentum , I decided to read the next book on the subject.
A month or two ago a friend of mine recommended me Basieux's book Die Architektur der Mathematik: Denken in Strukturen, saying it helped him grasping some mathematical concepts we both spent time learning in the past few months better, and so I decided to start reading this book right after having finished reading through Gowers's exemplar almost three weeks ago.
Get the Big Picture
Basieux's book is perfect if you want to learn more about the foundation of math, and more importantly if you want to get a very good overview of it. Just as Gower's book, Basieux's Die Architektur der Mathematik: Denken in Strukturen is a rather slim, but still a quite fascinating exemplar. Since the book's goal is to introduce the reader to the foundation of math and give an overview, it does not go into great detail about the mathematical structures presented, but rather explains the various subjects with just enough details and information, and still manages to teach you a ton of interesting things.
In my humble opinion, sometimes you just really need such an overview that only scratches the surface to fully comprehend a specific subject.
When I bought this book I already had background knowledge about some of the subjects presented, but reading through this book definitely helped me a lot.
I spent the last four months delving into various areas of math at quite a pace, and sadly I missed the one or other connection, because most of the times I did not have a clear picture of the whole thing in front of me.
Just to name you a very, very simple example, a few weeks ago if you would have asked me what the relationship between a metric space and a normed vector space is, I just would have glanced at you with a confused look. In the end it's rather simple, but still, back then I was enough confused with all the other mathematical terms that were thrown at me, I even couldn't understand the rather simple connection between this two terms.
Really, you should have seen my face when I first heard that the norm function induces a metric; it was quite a what the hell does that even mean moment for me.
Now I can tell you why exactly the norm function ||·|| of a normed vector space (V, ||·||) induces a metric, therefore making (V, ||·||) a metric space, too.
To put it simple, let X be a set and let d be a function d: X x X → R (R is the set of real numbers), then the pair (X,d) is called a metric space if d satisfies specific properties, thus making d a metric on X. The norm function ||·|| of a normed vector space (V, ||·||) is a function ||·||: V → R that simply assigns a size to each object of V, and also has to hold some properties. In contrast to the metric d, the norm function ||·|| works on single elements of the set V. Now a function DV: V x V → R can be defined as DV(x,y) = ||x -y||, which holds all the necessary properties of a metric thanks to the fact that ||x-y|| = ||x + (-y)||, and thus making DV a metric on V.
There already exist quite a number of established norms, e.g. the Euclidean or Manhattan norm, but regardless of which norm you are using, be it your own or some established ones, once you define such a norm (implying that it satisfies all the required properties), you can always use it to define a metric, just as shown above.
And exactly this is the reason why a norm function ||·|| of a normed vector space (V, ||·||) always induces a metric, and more importantly, why a normed vector space will always be a metric space, too (the reverse statement, namely that a metric space will always be a normed vector space, too does not always hold).
As I told you before, this is just a simple example, because it was more of a definition issue for me than something else. But still, a lot of things became quite clear for me once I unraveled this knot.
I cannot mention it enough: if you want to get a really good overview of the whole picture, read this book.
Personally, for me reading through this book was a very good and quite eye-opening experience.
But then again, I am just a novice exploring the exciting world of mathematics.