Today I lectured about the Hydra game by Laurence Kirby and Jeff Paris (Accessible Independence Results for Peano Arithmetic, Kirby and Paris, Bull. London Math. Soc. 1982; 14: 285-293). For the occasion I implemented the game in Java. I am publishing the code for anyone who wants to play, or use it for teaching. (more…)
Paul Taylor has published a revised version of his `lambda`-calculus for real analysis. I recommend it to anyone who is interested in real analysis, be it a computer scientist, numerical analyst, or just a “true” analyst.
The first, second, and third time I talked to Paul I could not understand a word of what he was saying, and that’s not just because he is a native speaker of English English. I only began to “get it” when he visited me in Ljubljana. So I think it’s perhaps worth explaining a bit what this “`lambda`-calculus for real analysis” is about.
The famous theorem by Cantor states that the cardinality of a powerset `P(A)` is larger than the cardinality of `A`. There are several equivalent formulations, and the one I want to consider is
Theorem (Cantor): There is no onto map `A -> P(A)`.
In this post I would like to analyze the usual proof of Cantor’s theorem and present an insightful reformulation of it which has applications outside set theory. (more…)
Recently there has been a discussion (here, here, here, and here) on the Foundations of Mathematics mailing list about completeness of Peano arithmetic (PA) with respect to “small” sentences. Harvey Friedman made several conjectures of the following kind: “All true small sentences of PA are provable.” He proposed measures of smallness, such as counting the number of distinct variables or restricting the depth of terms. Here are some statistics concerning such statements.
Spaces of higher-order functions are fascinating mathematical objects that we do not know enough about. What are they and what is known about them?
Computer algebra systems (CAS), such as Mathematica, are complex systems that have been evolving for a couple of decades. They are advertised as advanced mathematical tools, and users expect them to be such. They are the next-generation calculators. But they also suffer from serious design flaws.
I have created a blog category in which I will stick all my research papers. The idea is to allow people to comment on the papers and have an opportunity for discussion. There are some obvious advantages to this, such as: bug reports, opinions, references to related and relevant topics, a paper and a discussion about it are found in the same place, etc. Right now I do not see any obvious drawbacks, so I hope it will turn out to be a good idea.
This is just to let you know that I moved the blog to a new server, which will be up even when I am away on holidays. The new url is http://math.andrej.com/. If someone knows how to tell all those RSS feeds that seem to be accessing my blog that they should go to the new address, please let me know. Right now I have redirects set up, but that is not an acceptable long-term solution.
I have found a good way to write math in web pages. ASCIIMathML is a piece of javascript that translates simple-minded Latex-like ASCII math to MathML, but only if the browser supports MathML. Since the input syntax is very simple, the expressions are quite readable in the raw form, as well.
For example, if I type
`forall x in RR exists y in CC. (1-x^2 )/sqrt(1+y^4)=1`
it is seen as `forall x in RR exists y in CC. (1-x^2 )/sqrt(1+y^4)=1`. If you are going to post to the blog, you may be interested in the ASCIIMathML syntax reference page.
To enable MathML on your computer, install mathplayer plugin
if you are using Internet Explorer. For Firefox and Mozilla, you have to install math fonts.
I recently stumbled upon a blog on machine learning by a good friend of mine, John Langford. The blog has gathered together a community of people who discuss various topics (not limited strictly to machine learning). Naturally I wanted to have a blog, too.
I devote a lot of my time to thinking about the relationship between mathematics and computation. There are two sides of this, which can be expressed by a the slogan “Computable mathematics and mathematics of computation”. Computable mathematics is about how to do mathematics with computers, while mathematics of computation is about mathematics that describes properties of computation in a mathematical, abstract way.
If this is a subject that interests you, I invite you to join me.
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