Philosophers in Rags :: We dive deeper, and come up muddier.

 My plan is to tackle a few books at once and just go with the flow.  If I live long enough, I might get back to the rational trigonometry after filling in some gaps, facing some old fears (of writing and reading proofs - is it not like writing or reading computer programs?), and basically getting to the roots, the foundations ... the foundations of abstract reasoning.

In From Mathematics to Generic Programming, on page vii (Author's note), Alexander Stepanov writes:

The separation of computer science from mathematics greatly impoverishes both.  The lectures that this book is based on were my attempt to show how these two activities—an ancient one going back to the very beginnings of our civilization and the most modern one — can be brought together.

While the co-author writes:

While Alex comes from a mathematical background, I do not. I’ve tried to learn from my own struggles to understand some of the material and to use this experience to identify ideas that require additional explanation. If in some cases we describe something in a slightly different way than a mathematician would, or using slightly different terminology, or using more simple steps, the fault is mine.

In the section, What This Book Is About, on page 2 or so, section 1.1:

So where does this generic programming attitude come from, and how do you learn it? It comes from mathematics, and especially from a branch of mathematics called abstract algebra. To help you understand the approach, this book will introduce you to a little bit of abstract algebra, which focuses on how to reason about objects in terms of abstract properties of operations on them. It’s a topic normally studied only by university students majoring in math, but we believe it’s critical in understanding generic programming.  In fact, it turns out that many of the fundamental ideas in programming came from mathematics.
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The abstractions that appear in abstract algebra largely come from concrete results in one of the oldest branches of mathematics, called number theory. For this reason, we will also introduce some key ideas from number theory, which deals with properties of integers, especially divisibility.

This was the direction I had been moving in around 2016 or so, but then I had resurrected a passion for linear algebra, analytic geometry, computer algebra systems, continued fractions - then plunging into the long commitment to revisit "school mathematics" in terms of the "set theoretical presentation" from high school ("New Math" ---> "modern" [abstract] algebra ?), and even a dive into Stroustrup's C++ text, working on the exercises as diligently as a model prisoner/student.   

So, regardless of Dr. Notowidigdo's sense that I have not studied formally a good part of what math majors study, I am confident I can handle the material Stepanov presents.  I was exposed to formal Mathematical Reasoning courses and the like.   Yet I will humbly submit that it would not hurt me to delve deeper into that world of fields, rings, and groups - yes, I get it - concepts underneath "school mathematics" but never presented to students.   Such presentations are reserved for math majors.

Hung-Hsi Wu thinks all those training to become high school (or even elementary school) teachers, ought to have the deeper view (with proofs and understanding) so that they might better communicate to the students.

See What can we do about our dysfunctionalschool mathematics curriculum?, Preface and To the Instructor from Rational Numbers to Linear Equations.

Now, Stepanov's Elements of Programming is a different story, but I am considering pecking away at that book along with From Mathematics to Generic Programming.   Such are the kinds of texts I wish to understand.

Lastly, I am going to commit to the text, Contemorary Abstarct Algebra, if only because the most resources are available for that text.  It's in its tenth edition, so there are free resources available for the 7th and 8th editions, as well as labs and definitely more support than just videos on ZooTube.

I think that a disciplined informal study of Abstarct Algebra will not only help me understand Stepanov's work better, but it would also help me decipher (better) where it is Norman Wildberger might be coming from.   He certainly uses much terminology from abstract algebra.

Why is it I must have a plan?  I'm just chipping away at my ignorance, trying to practice authenticity in a world that rewards phonies and charlatans with a sense of unearned confidence, like the managers in Ligotti's novel of corporate horror, My Work is Not Yet Done.   

The gangsters of the world might mock the likes of me, but I would like to imagine there are a handful of academics who would sympathize with my anti-academic/anti-careerist/anti-professional attitude.

Mike,
I to be honest, was keen to check Wildberg Math especially it is  vetted by you. but I, likewise, am not a fan of videos.

Anyway that was not a lost effort, you left good trails. for example the link to Zeilberger's opinions was an eye opening, I've gone through half of his opinions.

It is interesting that the method he describes on opinion 65 is almost the same way you are doing math.

I applied this principle a few weeks ago when I taught the Buchberger algorithm to my Algorithmic Discrete Math class. First, they had to, during class, compute the Grobner basis of {x+y,x^2+y^2}, all by hand! Then, as homework, also by hand (no cheating please!) they were asked to find the Grobner basis of {x+y+z,x^2+y^2+z^2,x^3+y^3+z^3}. Then, they were asked, using Maple, to perform the same calculations step by step, computing S-polynomials and reducing. Then they were encouraged to try and program their own version, even though it is unlikely to be as efficient as the built-in Maple command. Finally, for ever after, they were given permission to use the built-in command gbasis, and use it as a complete Black Box, without worrying about the details, enabling them to do new research.

I am sure you gone through his site, these opinions might be of interest to you 37, 46.

I find your posts about math illuminating, it sounds like you reached the climax in your hero journey story studying math.

stay well and away from masses

Completing the square as an alternative to plugging values into Quadratic Formular


Source Code for Command Line Program attached as text file.


It can be saved as cts.cpp, compiled and used (./cts).

Maybe my mathematically-driven code has been as close as I will come to having created "works" of "MATHCRAFT."

This begs the question:  What is the ART of Mathcraft ?

I would like to coin the term, mathcraft to mean this vague notion of "Programming as a Mathematical Discipline" ... where years of reflection about fundamental concepts deepen understanding, and that honest kind of genuine understanding is its own reward.

There does not have to be material rewards for technical understanding or intellectual development.  The reward must be the satisfaction of exercising the higher mental faculties, thereby enjoying the cerebral stimulation of engaging in abstract reasoning.   

One can see how this might rescue the mind from some of the bigger philosophical problems, such as, whether life is even worth living. 

From the comments about how mathematics will be an OLD person's "game" as the field becomes an experimental science.

I reflect upon how Wildberger questions the spatialization of geometry, and what a defiant challenge this notion poses to The Established Church of Academic University Mathematics.

Even if you are a mainstreamer, you can change fields. Like Lippman Bers who taught himself Teichmuller theory in his mid and late sixties. Or, like Norman Levinson, who at 64 proved that at least one third of the zeroes of Zeta(s) lie on the critical strip. Better still, you can constantly keep creating NEW fields, making sure to be excused out of the bandwagon once they become `fashionable', like 88-year-old Israel Gelfand, who in collaboration with Vladimir Retakh and Robert Wilson created non-commutative linear algebra a few years ago.

But perhaps most important, since math will soon become an experimental science, where experience, maturity, and patience are just as important as `brilliant flashes', we are guaranteed that math will become more and more an old PERSON's game.

Why must everything be considered a game of egotistic intellectual prick waving?  I would think such a game-mentality would drive some youth to suicide, thinking that it is just the luck of the draw.  Does the author understand that there is a severe elitism in the academic and professional arenas, that there are masses of human creatures on this earth who will not have access to education or nutrition.  They may have children who will not survive adolesence simply from lack of spirit to withstand the pressures which bombard their senses on a daily, weekly basis.

I would like to inject a little humility into the Physical Sciences, just as I think, maybe, Holden would like to rattle the foundations of Mathematics itself.   What points were the authors of "Burn Math Class" or even "Weapons of Math Destruction" driving home?

< Sorry for ranting ... I will have to look into names.  >

I appreciated his practice of seeing a function as a machine.

output = ABSTRACT_MACHINE( input )

As for Math Destruction, that is how the Corporate Elites have trusted the systematic computer-generated analysis of our personal characters and personalities to the point where we will be inhumanely judged by Artificial Intelligence DATA.   This will back-fire or it will demonize those who possess the attributes of those who have challenged their authority in the past.

Just my fifteen cents.