We are on a mini-weekend-vacation, having a great time, break bread at a great locally owned sushi restaurant, head out and stop at a half-priced-books, only to find a book that I already had on my list, brand new, great price… awesome! ðŸ™‚

# Tag: mathematics

## Rules-Engines Recommended Wikipedia Reading

You know that advice “trust but verify”? It couldn’t be more true when it comes to the realm of applied or narrow artificial intelligence, specifically with respect to rules-engines. There is more dis-information out there on rules-engines than on any other tech topic that I’ve ever researched. It is really, really shocking the amount of inaccurate, confused, and just plain wrong information out there.

The problem with it is that it makes it harder for people who want to learn more about the topic by wasting their time and presenting the topic as very, very shallow and simplistic. It makes rules-engines very, very easy to blow off as sort of a joke and nothing more than pricey business rules engines. That is sad, too, because rules-engines provide a very nuanced and special programming style all their own that every programmer should at least learn.

In order to avoid this, I tried to diligently record all of the materials that I have found valuable, including Wikipedia pages. My hope in sharing them is that they provide some baseline for getting started, that worked for at least one person. My approach was to read and bookmark internally to the site, and also do research out of the site. It wasn’t a race, I took time to take it in, reading as many times as necessary for it to make sense, including re-reading it over as many days or weeks as I saw fit. The topics are grouped into sections that are somewhat logically related, and definitely do relate to one another, and try to help make it easier to access the topics in the following group.

Rules-engines comes from a rich computational heritage, but strangely get a fraction of a percentage of coverage even in comparison to seemingly fringe topics like Lisp, so I hope that you have as much fun reading as I did, it was really enlightening!

Continue reading “Rules-Engines Recommended Wikipedia Reading”

## Post's Tenth Rule of Computing

## On Rules

Over the last few weeks nights and weekends I’ve played around with Jess studying up on rules engines and expert systems. The approach of focusing on the limited reality defined by the facts and rules in the universe of the engine along with letting go of control of sequencing are surprisingly freeing. Memories of event loops, OOAD, OOP, multimethods, state machines, horribly hand-written code, and event handlers all come to mind. It is fascinating to see how well some things can be simplified. It makes me wonder why we ever choose one approach over another. Supposedly we always use the best tool for the job; but in practice most of us learn one halfway decent approach (“tool”) once and end up using it forever. I suppose that is a human trait and not unique to problem solving with programming languages.

## Some Thoughts on Mathematics

R. L. E. Schwarzenberger, The Language of Geometry, in A Mathematical Spectrum Miscellany, Applied Probability Trust, 2000, p. 112: My own attitude, which I share with many of my colleagues, is simply that mathematics is a language. Like English, or Latin, or Chinese, there are certain concepts for which mathematics is particularly well suited: it would be as foolish to attempt to write a love poem in the language of mathematics as to prove the Fundamental Theorem of Algebra using the English language.

Yu. Manin, Mathematics as profession and vocation, in Mathematics: Frontiers and Perspectives, (V. Arnold et al, ed), AMS, 200, p. 154: The basis of all human culture is language, and mathematics is a special kind of linguistic activity.

A. Adler, Mathematics and Creativity, in The World Treasury of Physics, Astronomy and Mathematics, (T. Ferris, ed), Little, Brown and Co, 1991, p. 435: Mathematics is pure language – the language of science. It is unique among languages in its ability to provide precise expression for every thought or concept that can be formulated in its terms. (In a spoken language, there exist words, like “happiness”, that defy definition.) It is also an art – the most intellectual and classical of the arts.

J.-P. Changeux, A. Connes, Conversations on Mind, Matter and Mathematics, Princeton University Press, 1995, p. 10:

Changeux: Mathematical language is plainly an authentic language. But is it therefore the only authentic language?

Connes: It is unquestionably the only universal language.

Bertrand Russell (1872-1970), Autobiography, George Allen and Unwin Ltd, 1967, v1, p158

It seems to me now that mathematics is capable of an artistic excellence as great as that of any music, perhaps greater; not because the pleasure it gives (although very pure) is comparable, either in intensity or in the number of people who feel it, to that of music, but because it gives in absolute perfection that combination, characteristic of great art, of godlike freedom, with the sense of inevitable destiny; because, in fact, it constructs an ideal world where everything is perfect but true.

Bertrand Russell (1872-1970), The Study of Mathematics

Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.

Aristotle (384 B.C.-322 B.C.), Poetics

Beauty depends on size as well as symmetry.

J.H.Poincare (1854-1912), (cited in H.E.Huntley, The Divine Proportion, Dover, 1970)

The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful.

J.Bronowski, Science and Human Values, Pelican, 1964.

Mathematics in this sense is a form of poetry, which has the same relation to the prose of practical mathematics as poetry has to prose in any other language. The element of poetry, the delight of exploring the medium for its own sake, is an essential ingredient in the creative process.

J.W.N.Sullivan (1886-1937), Aspects of Science, 1925.

Mathematics, as much as music or any other art, is one of the means by which we rise to a complete self-consciousness. The significance of Mathematics resides precisely in the fact that it is an art; by informing us of the nature of our own minds it informs us of much that depends on our minds.

G. H. Hardy (1877 – 1947), A Mathematician’s Apology, Cambridge University Press, 1994.

The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas, like the colors or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics.

Lawrence University catalog, Cited in Essays in Humanistic Mathematics, Alvin White, ed, MAA, 1993

Born of man’s primitive urge to seek order in his world, mathematics is an ever-evolving language for the study of structure and pattern. Grounded in and renewed by physical reality, mathematics rises through sheer intellectual curiosity to levels of abstraction and generality where unexpected, beautiful, and often extremely useful connections and patterns emerge. Mathematics is the natural home of both abstract thought and the laws of nature. It is at once pure logic and creative art.

I.Newton, Letter to H.Oldenburg, the Secretary of the Royal Society, October 24, 1676, in A Source Book in Mathematics, D. J. Struik, ed, Princeton University Press, 1990

I can hardly tell with what pleasure I have read the letters of those very distinguished men Leibniz and Tschirnhaus. Leibniz’s method for obtaining convergent series is certainly very elegant…

Jane Muir, Of Men & Numbers, Dover, 1996.

Gauss: You have no idea how much poetry there is in the calculation of a table of logarithms!

F.Dyson, in Nature, March 10, 1956

Characteristic of Weyl was an aesthetic sense which dominated his thinking on all subjects. He once said to me, half-joking, “My work always tried to unite the true with the beautiful; but when I had to choose one or the other, I usually chose the beautiful.” (Herman Weyl (1885-1955))

O. Spengler, in J. Newman, The World of Mathematics, Simon & Schuster, 1956

To Goethe again we owe the profound saying: “the mathematician is only complete in so far as he feels within himself the beauty of the true.”

O. Spengler, in J. Newman, The World of Mathematics, Simon & Schuster, 1956

“A mathematician,” said old Weierstrass, “who is not at the same time a bit of a poet will never be a full mathematician.”

Jakob Bernoulli, Tractatus de Seriebus Infinitis, 1689 (quoted in From Five Fingers to Infinity, F.J.Swetz (ed), Open Court, 1996)

So the soul of immensity dwells in minutia.

And in narrowest limits no limits inhere.

What joy to discern the minute in infinity!

The vast to perceive in the small, what divinity!

S.Lang, The Beauty of Doing Mathematics, Springer-Verlag, 1985

Last time, I asked: “What does mathematics mean to you?” And some people answered: “The manipulation of numbers, the manipulation of structures.” And if I had asked what music means to you, would you have answered: “The manipulation of notes?”

(via someone’s comments here)

## Understanding the Y-Combinator with Racket

Understanding the Y-Combinator seems to be one of the functional programming right-of-passage things, and for good reason, it is fun. Not all of the articles out there spoke to me, so I took notes while coding it up in a way that I understood it.

My notes are attached in the form of a Scribble here How the Y Works, viewable here, and the source code version here y.

It is Racket, but it is really close to Scheme and if you wrote a couple of macros it would probably run on Clojure pretty easily.

The goal is for there to be no gaps in the code, it should be pretty obvious how each step was taken. Hand-coding it made it clear what was happening. This is what worked for me :).

## Spring 2011 Wrapup

During my Fall 2011 semester I took one class, Applied Linear Algebra. It was a total blast. You just can’t believe some of the wild stuff that you can do with matrices. It was a lot of fun and the hardest that I’ve ever worked, and my grade reflected that.

## When is your math always wrong?

Last year in my Applied Linear Algebra class we were assigned a homework that introduced Backward Substituation, Forward Substitution, Horner’s Method. Something really interesting happened though as a side effect of the assigned work.

Continue reading “When is your math always wrong?”

## One Reason Why Mathematics is Fun

Imagine that you’ve learned a new language that allows you to *see* things in a way that you’ve never before been able to conceive. Imagine the new kinds of questions that you can now ask which you could not before! It is an amazing feeling.

Mathematics is invigorating, inspiring, and fun.

## Fall 2010 Wrapup

During my Fall 2010 semester I took one class, a survey class, Applied Mathetical Analysis. It was a total blast. Sure, every topic covered had entire classes for and books written about them… but that just made it more interesting!

This semester is in my top-three favorite semesters of all time. I worked extremely hard, learned a lot, and will only have fond memories of it.