Sunday, March 24, 2013

A new kind of square root

The postmodernism generator has been translated to mathematics. Now there is a program called Mathgen that outputs nonsensical papers on the advances of mathematics, complete with theorems and references. It has certain idiosynchrasies that makes it easy to recognize its papers. Authors are often drawn from among the most famous names of mathematics, but usually getting the first initial wrong. Theorems and conjectures are generously attributed to pairs of collegues across history, often using centuries old personalities as authors of brand new theories. Who has ever heard of the Conway-d'Alembert conjecture? Well, now we have.

The tone is exactly as condescending as one might fear: 'Clearly' such and such result follows; 'as every student knows …', and what follows is invariably clear as mud. Proofs are safely omitted because they are 'obvious'.

All this remarkable research, those 'little known results', are published safely beyond accessibility in Transactions of the Kenyan Mathematical Society, South Korean Journal of Integral Category TheoryIranian Journal of Homological PDE, and the like. Surely most of these publications cannot be found at your local library anytime soon.

It is not hard to generate plain gibberish by TeX. Begin with listing a few elementary symbols and operators:

const char *alpha = "\\alpha";
const char *beta = "\\beta";
const char *r_arrow = "\\rightarrow";
const char *sqrt = "\\sqrt";
const char *sup = "^ ";
const char *sub = "_ ";

Put all the symbols in an array, so they can be easily accessed and picked at random. Concatenate several of the symbols into a string and print it. With some luck, the symbol sequence will not break the TeX syntax. This doesn't happen by itself, so next one might like to do something more structured. Elementary functions (program routines, that is) that generate small expressions like x ∈ ℂ2 or f : ℝ → ℝ are not hard to write.

This is a sample of the babbling that results from the mere concatenation of a few symbols and numbers without regard for syntactical rules:

Difficulties arise when operators are used, because these expect arguments. An expression should not end with, say, an empty square root with no argument, as the above formula seems to do. In fact it ends with \sqrt{%0^{\sum}}, but this is apparently beyond the wits of TeX.

The notorious mathgen paper Independent, Negative, Canonically Turing Arrows of Equations and Problems in Applied Formal PDE by M. Rathke contains a larger sortiment of abstruse mathematical symbols in hilarious combinations. Or what about the frequent use of various powers of zero? Already the first formula contains expressions such as 0-4, 05, and other meaningless entities such as tan(∞-1). A judicious use of elaborate idempotent expressions may even accidentally result in a true statement despite the funny appearance.

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