{∅, {∅}, {∅, {∅}}} : Rage Against the Meat Grinder

Please be advised that this thread may appear unrelated to the more "existential and philosophical" themes of this website and may be skipped by those having no interest in such things.
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On the first page of the Preface to "Concrete Mathematics: A Foundation for Computer Science,"  there is mention of a paper written by a seemingly pretentious Brit, John Hammersley:

It was a dark and stormy decade when Concrete Mathematics was born [1989].  Long-held  values  were  constantly  being  questioned  during  those  turbulent years; college campuses were hotbeds of controversy.  The college curriculum itself was challenged, and mathematics did not escape scrutiny.  John Hammersley had just written a thought-provoking article On the enfeeblement of mathematical skills by `Modern Mathematics' and by similar soft intellectual trash in schools and universities;  other worried mathematicians even asked, Can mathematics be saved?  One of the present authors [Donald Knuth] had embarked on a series of books called The Art of Computer Programming, and in writing the  first volume he  had found that there were mathematical tools missing from his repertoire; the mathematics he needed for a thorough, well-grounded understanding of computer programs was quite different from what he'd learned as a mathematics major in college.  So he introduced a new course, teaching what he wished somebody had taught him.

The course title Concrete Mathematics was originally intended as antidote to Abstract Mathematics, since concrete classical results were rapidly being swept out of the modern mathematical curriculum by a new wave of abstract ideas popularly called the New Math. Abstract mathematics is a wonderful subject, and there's nothing wrong with it:  It's beautiful, general, and useful.  But its adherents had become deluded that the rest of mathematics was inferior and no longer worthy of attention.  The goal of generalization had become so fashionable that a generation of mathematicians had become unable  to  relish  beauty  in  the  particular,  to  enjoy  the  challenge  of  solving quantitative problems, or to appreciate the value of technique. Abstract mathematics was becoming inbred and losing touch with reality; mathematical education needed a concrete counterweight in order to restore a healthy balance.

But what exactly is Concrete Mathematics?  It is a blend of CONtinuous and disCRETE mathematics.  More concretely, it is the controlled manipulation of mathematical formulas, using a collection of techniques for solving problems.  Once you, the reader, have learned the material in this book, all you will need is a cool head, a large sheet of paper, and fairly decent handwriting in order  to  evaluate  horrendous-looking  sums,  to  solve  complex  recurrence relations,  and to discover subtle patterns in data.  You will be so fluent in algebraic techniques that you will often find it easier to obtain exact results than to settle for approximate answers that are valid only in a limiting sense.

The  major  topics  treated  in  this  book  include  sums,  recurrences,  elementary number theory, binomial coeffcients, generating functions, discrete probability, and asymptotic methods.  The emphasis is on manipulative technique rather than on existence theorems or combinatorial reasoning; the goal is for each reader  to  become  as  familiar  with  discrete  operations  (like  the greatest-integer  function and finite summation) as a student of calculus is familiar with continuous operations (like the absolute-value function and infinite integration).

Notice that this list of topics is quite different from what is usually taught nowadays in undergraduate courses entitled Discrete Mathematics.  Therefore the subject needs a distinctive name, and Concrete Mathematics has proved to be as suitable as any other.

The original textbook for Stanford's course on concrete mathematics was the "Mathematical Preliminaries" section in The Art of Computer Programming.  But the presentation in those 110 pages is quite terse, so another author (Oren Patashnik) was inspired to draft a lengthy set of supplementary notes.  The present book is an outgrowth of those notes; it is an expansion of, and a more leisurely introduction to, the material of Mathematical Preliminaries. Some of the more advanced parts have been omitted; on the other hand, several topics not found there have been included here so that the story will be complete.

Note that I had purchase volume one of The Art of Computer Programming (Fundamental Algorithms) back in 1999, but it was way over my head.  I have even returned to Section 1.2, Mathematical Preliminaries, since then, and still managed to only be able to work through a handful of the exercises.  Hence, this Concrete Mathematics is something of a "Holy Grail" to me.

My apologies for the lengthy quote from the preface.   My intention of the initial post of this thread is to draw attention to my aversion to a particular view of mathematics which strongly favors applications (engineering, physical science) over theory.   You can read the entire 1968 article by John Hammersley as I found a pdf version which I linked to at the start of this post.  Again, that link:  On the enfeeblement of mathematical skills by `Modern Mathematics' and by similar soft intellectual trash in schools and universities.

When I got to the sixth page, I felt the need to pause so as to document my research, since I experienced a kind of epiphany that had to do with me not becoming upset by opposing views, but to simply recognize that I may have a much different opinion than the author.  I cringed when Hammersley refers to "the consumers of mathematicians," meaning universities and "employers of graduates".   I will continue reading the rest of the paper eventually, but for the time being, I will let the brute fact sink into my skull that I am more than a little "put off" by Hammersley's views.  He claims that focusing on set-theoretical notation and the like "leads to an enfeebled mathematical skill."   he is referring to, of course, computational skill, or CALCULATING.   This is where I am odds with this view, for I am not fond of his equating "mathematical skill" with "computational skill".  He is looking at mathematics as a tool for engineers and businessmen, not as a discipline in and of itself.  He seems to have a definite contempt and disdain for the theoretical.

I quote him directly here.  Mind you, he states the following with an air of great authority:  "It is more important to find the right solution to a problem than to understand the logic of the method.  This is especially so for people who use mathematics as a tool in some other context.  Similarly, one can well use an electronic computer without being obliged to understand its circuitry or even much about its software.  There is force behind Bertrand Russell's aphorism that a mathematician never knows what he is talking about, nor whether what he is saying is true."

I will continue this thread as I make my way through the article.  I do wish to work through the classic 1994 second edition of Concrete Mathematics, but I feel I will not be prepared for such an engagement for another couple years, at the very least --- God willin' and the crick don't rise, that is ... (as Holden is fond of reminding me).