4 December 2022

Concrete AfterMath

A few weeks ago, we were studying what Gauss had done in kindergarten.

Now we're studying stuff beyond his Ph.D. thesis.

– Concrete Mathematics, page 212, "graffiti" in the margin.

… everything can be made rigorous, but our goal right now is to expand our consciousness beyond conventional algebraic formulas.

– Concrete Mathematics, page 323.

Force Majeure

My plans to write the notes while reading the book had failed. Not because I didn't want to or was too lazy. But because a bunch of guys in the kremlin – pumped up by the fascist and imperialistic ideals, called in short rashism – had decided that a neighboring country didn't deserve to be an independent entity and its citizens deserved to die at their will. By a chance, I was staying that winter in Odesa, one of the cities that were bombed early in the morning on the 24th of February, so all the following events – known quite well to virtually everyone thanks to the beloved Internet – pushed me on edge.

Obviously, the survival mode was not productive for comprehending and writing about delicate matters, mathematics included. Still, I continued reading the book when the situation had stabilized a tiny bit; though strictly sans knocking my head against each derivation, and without taking the notes. Enough stress was enough.

And, oh surprise, surprise! The "take it easy" style of reading yielded curious results. Many derivations were way too often easier to understand looking at them with less of a focus. As a painter steps back to observe a picture as a whole from a distance, I now step back – mentally – to grasp a general idea behind the words and symbols, and then get closer again to see a more precise picture based on the just received general information about it. Such "focus shifting" had helped me a lot during Concrete Mathematics comprehension.


The foundational background I had in the fields of discrete mathematics and mathematical analysis was not sufficient for reading Concrete Mathematics with ease. But it was enough for getting through the book with significant effort.

Accordingly, I think that the intended audience of the book is the readers with a stronger background in mathematics than just foundational. It is compliant with the fact that the book

… is based on a course of the same name that has been taught annually at Stanford University since 1970. About fifty students have taken it each year – junior and seniors, but mostly graduate students

… the time seems ripe to present the material to a wider audience (including sophomores).

[emphases are mine]

– Concrete Mathematics, Preface.

Despite all this, Concrete Mathematics is recommendable for reading by any curious non-mathematician person. The flicker of genius can be seen in the book's contents even without full comprehension of each derivation (which is still desirable, of course). This is due to the significant part of the text's value laying in a field of problem-solving methodologies in mathematics.

Besides, Concrete Mathematics charms by following a specific rhythm. Each chapter of the book starts with the more elementary sections to then cover more and more complex (and more generalized) aspects of a topic. Thus, a reader shouldn't despair of a text becoming hard to understand. One should simply "swim" through it. There will be an opportunity to take a breath: reading the opening sections of the next chapter.

Moreover, this book is not just a collection of separate topics, but rather a crime-story with many intertwining lines, observable from a different angle with each new piece of evidence found.

Hence, an attentive reader will definitely feel the exquisiteness of this book. I wholeheartedly agree that

… the book has turned out to be a tale of mathematical beauty and surprise …

And its contents confirms that

… mathematics is fun, and we aren't ashamed to admit the fact.

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