Another Word For It Patrick Durusau on Topic Maps and Semantic Diversity

MathB.in

From About MathB.in:

MathB.in is a website meant for sharing snippets of mathematical text with others on the web. This is a pastebin for mathematics. This website was born out of a one night hack on Sunday 25, 2012.

Posting and sharing

A new post can be composed by visiting the home page and writing or pasting code in the box on the left hand pane of the page. Once a post is composed and submitted, the page is saved and it becomes accessible with a new unique URL. The new page looks similar to this page and it has a unique URL of its own. The URL can be shared with anyone on the web and he or she will be able to visit your post.

Code

The post can be composed in a mixture of plain text, LaTeX, Markdown and HTML. HTML tags commonly used for formatting text elements are supported. For a demonstration on how LaTeX is rendered, see the demo page. To quickly get started with posting math, see the tutorial.

Bug reports and suggestions

If you come across any bugs, or if you have any suggestions, please email Susam Pal at susam@susam.in or report an issue at https://github.com/susam/mathb/issues.

Your mileage will vary but drawing on Principia Mathematica without citation will leave any government agents tracking your posts in the wilds of 20th century logicism. Unlikely they will damage anything.

If your Principia notation skills are weak, consider The Notation in Principia Mathematica to translate proofs into late 20th century logic notation.

Wild Maths – explore, imagine, experiment, create!

From the webpage:

Mathematics is a creative subject. It involves spotting patterns, making connections, and finding new ways of looking at things. Creative mathematicians play with ideas, draw pictures, have the courage to experiment and ask good questions.

Wild Maths is a collection of mathematical games, activities and stories, encouraging you to think creatively. We’ve picked out some of our favourites below – have a go at anything that catches your eye. If you want to explore games, challenges and investigations linked by some shared mathematical areas, click on the Pathways link in the top menu.
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The line:

It involves spotting patterns, making connections, and finding new ways of looking at things.

is true of data science as well.

I’m going to print out Can you traverse it?, to keep myself honest, if nothing else. 😉

Enjoy!

Book of Proof by Richard Hammack.

From the webpage:

This book is an introduction to the standard methods of proving mathematical theorems. It has been approved by the American Institute of Mathematics’ Open Textbook Initiative. Also see the Mathematical Association of America Math DL review (of the 1st edition), and the Amazon reviews.

The second edition is identical to the first edition, except some mistakes have been corrected, new exercises have been added, and Chapter 13 has been extended. (The Cantor-Bernstein-Schröeder theorem has been added.) The two editions can be used interchangeably, except for the last few pages of Chapter 13. (But you can download them here.)

Order a copy from Amazon or Barnes & Noble for $13.75 or download a pdf for free here. Click here for a pdf copy of the entire book, or get the chapters individually below.

From the Introduction:

This book will initiate you into an esoteric world. You will learn and apply the methods of thought that mathematicians use to verify theorems, explore mathematical truth and create new mathematical theories. This will prepare you for advanced mathematics courses, for you will be better able to understand proofs, write your own proofs and think critically and inquisitively about mathematics.

For a 300+ page book, almost a steal at Amazon for $13.75. A stocking stuffer for Math/CS types on your holiday list. For yourself, grab the pdf version. 😉

Big data projects are raising the bar for being able to think critically about data and the mathematics that underlie its processing.

Big data is by definition too large for human inspection. So you had better be able to think critically about the nature of the data en masse and the methods to be used to process it.

Or to put it another way, if you don’t understand the impact of the data on processing, or assumptions built into the processing methods, how are you going to evaluate big data results?

Just accept them as ground level truth? Ignore them if they contradict your “gut?” Use a Magic 8-Ball?, a Ouija Board?

I would recommend none of the above and working on your data and math critical evaluation skills.

You?

I first saw this in a tweet by David Higginbotham.

“How Not to Be Wrong”: What the literary world can learn from math by Laura Miller.

From the post:

Jordan Ellenberg’s “How Not to Be Wrong: The Power of Mathematical Thinking” is a miscellaneous romp through the world of quantitative reasoning. You can tell just how modular the book is by the way bits of it have been popping up all over the Web of late, promising to explain such mysteries as why so many handsome men are jerks or why an athlete’s performance always seems to suffer a drop-off after he signs a big contract. This pull-apart quality may sound like a bug, but in fact it’s a feature. It makes “How Not to Be Wrong” a rewarding popular math book for just about anyone.

Ellenberg is a professor of mathematics at the University of Wisconsin-Madison who spent “some part of my early 20s thinking I might want to be a Serious Literary Novelist.” (He even published a novel, “The Grasshopper King.”) He has a popular math column at Slate. So Ellenberg can write, and furthermore he brings a set of references to the subject that will make many a numbers-shy humanities major feel right at home. He explains why B.F. Skinner’s “proof” that Shakespeare was not particularly inclined toward poetic alliteration was incorrect, and he is as likely to refer to Robert Frost and Thomas Pynchon as to such mathematical titans as R.A. Fisher and Francis Galton. He takes the playful, gentle, humorous tone of a writer used to cajoling his readers into believing that they can understand what he’s talking about.

At the same time, those who like to pull out a pencil and a piece of paper and work through a few equations for the fun of it should not be utterly put off by the aforementioned cajoling. Interested in a brush-up on plane geometry, the implications of the Prime Numbers Theorem and Yitang Zhang’s recently announced proof of the “bounded gaps” conjecture about the distribution of primes? Pull up a chair. On the other hand, if any sentence containing the term “log N” makes you go cross-eyed and befuddled, you need only turn a few pages ahead to another chapter, where you can read about how two separate math cartels gamed the Massachusetts state lottery.
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Laura’s review will leave you convinced that “Have you read ‘How Not to be Wrong’ by Jordan Ellenberg?” should be on your technical interview checklist.

Or at least you should pull examples from it to use in your technical interviews.

Despite all the hype about “big data,” “web scale,” etc., the nature of coherent thinking has not changed. The sooner you winnow out candidates that thing otherwise the better.

Rings — A Second Primer by Jeremy Kun.

From the post:

Last time we defined and gave some examples of rings. Recapping, a ring is a special kind of group with an additional multiplication operation that “plays nicely” with addition. The important thing to remember is that a ring is intended to remind us arithmetic with integers (though not too much: multiplication in a ring need not be commutative). We proved some basic properties, like zero being unique and negation being well-behaved. We gave a quick definition of an integral domain as a place where the only way to multiply two things to get zero is when one of the multiplicands was already zero, and of a Euclidean domain where we can perform nice algorithms like the one for computing the greatest common divisor. Finally, we saw a very important example of the ring of polynomials.

In this post we’ll take a small step upward from looking at low-level features of rings, and start considering how general rings relate to each other. The reader familiar with this blog will notice many similarities between this post and our post on group theory. Indeed, the definitions here will be “motivated” by an attempt to replicate the same kinds of structures we found helpful in group theory (subgroups, homomorphisms, quotients, and kernels). And because rings are also abelian groups, we will play fast and loose with a number of the proofs here by relying on facts we proved in our two-post series on group theory. The ideas assume a decidedly distinct flavor here (mostly in ideals), and in future posts we will see how this affects the computational aspects in more detail.

I have a feeling that Jeremy’s posts are eventually going to lead to a very good tome with a title like: Mathematical Foundations of Programming.

To be used when you want to analyze and/or invent algorithms, not simply use them.

Methods of Proof — Contradiction by Jeremy Kun.

From the post:

In this post we’ll expand our toolbox of proof techniques by adding the proof by contradiction. We’ll also expand on our knowledge of functions on sets, and tackle our first nontrivial theorem: that there is more than one kind of infinity.

Impossibility and an Example Proof by Contradiction

Many of the most impressive results in all of mathematics are proofs of impossibility. We see these in lots of different fields. In number theory, plenty of numbers cannot be expressed as fractions. In geometry, certain geometric constructions are impossible with a straight-edge and compass. In computing theory, certain programs cannot be written. And in logic even certain mathematical statements can’t be proven or disproven.

In some sense proofs of impossibility are hardest proofs, because it’s unclear to the layman how anyone could prove it’s not possible to do something. Perhaps this is part of human nature, that nothing is too impossible to escape the realm of possibility. But perhaps it’s more surprising that the main line of attack to prove something is impossible is to assume it’s possible, and see what follows as a result. This is precisely the method of proof by contradiction:

Assume the claim you want to prove is false, and deduce that something obviously impossible must happen.

There is a simple and very elegant example that I use to explain this concept to high school students in my guest lectures.

I hope you are following this series of posts but if not, at least read the example Jeremy has for proof by contradiction.

It’s a real treat.

Methods of Proof — Contrapositive by Jeremy Kun.

From the post:

In this post we’ll cover the second of the “basic four” methods of proof: the contrapositive implication. We will build off our material from last time and start by defining functions on sets.

Functions as Sets

So far we have become comfortable with the definition of a set, but the most common way to use sets is to construct functions between them. As programmers we readily understand the nature of a function, but how can we define one mathematically? It turns out we can do it in terms of sets, but let us recall the desired properties of a function:

• Every input must have an output.
• Every input can only correspond to one output (the functions must be deterministic).

Jeremy continues his series on proof techniques.

Required knowledge for reading formal CS papers.

Methods of Proof — Direct Implication by Jeremy Kun.

From the post:

I recently posted an exploratory piece on why programmers who are genuinely interested in improving their mathematical skills can quickly lose stamina or be deterred. My argument was essentially that they don’t focus enough on mastering the basic methods of proof before attempting to read research papers that assume such knowledge. Also, there are a number of confusing (but in the end helpful) idiosyncrasies in mathematical culture that are often unexplained. Together these can cause enough confusion to stymie even the most dedicated reader. I have certainly experienced it enough to call the feeling familiar.

Now I’m certainly not trying to assert that all programmers need to learn mathematics to improve their craft, nor that learning mathematics will be helpful to any given programmer. All I claim is that someone who wants to understand why theorems are true, or how to tweak mathematical work to suit their own needs, cannot succeed without a thorough understanding of how these results are developed in the first place. Function definitions and variable declarations may form the scaffolding of a C program while the heart of the program may only be contained in a few critical lines of code. In the same way, the heart of a proof is usually quite small and the rest is scaffolding. One surely cannot understand or tweak a program without understanding the scaffolding, and the same goes for mathematical proofs.

And so we begin this series focusing on methods of proof, and we begin in this post with the simplest such methods. I call them the “basic four,” and they are:

• Proof by direct implication
• Proof by contradiction
• Proof by contrapositive, and
• Proof by induction.

This post will focus on the first one, while introducing some basic notation we will use in the future posts. Mastering these proof techniques does take some practice, and it helps to have some basic mathematical content with which to practice on. We will choose the content of set theory because it’s the easiest field in terms of definitions, and its syntax is the most widely used in all but the most pure areas of mathematics. Part of the point of this primer is to spend time demystifying notation as well, so we will cover the material at a leisurely (for an experienced mathematician: aggravatingly slow) pace.

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Parallel processing, multi-core memory architectures, graphs and the like are a long way from the cookbook stage of programming.

If you want to be on the leading edge, some mathematics are going to be required.

This series can bring you one step closer to mathematical literacy.

I say “can” because whether it will or not, depends upon you.