domingo, 28 de octubre de 2012

are mathematics invented or discovered?

discovery or invention?

In the LinkedIn group about mathematics "Math, Math Education, Math Culture" Opher Liba asked an old and recurrent question in mathematics: Invention or Discovery? This was the title. Many people contributed to this interesting topic. I am not going to make a summary of all the relevant opinions, some of which were quite elaborated. Just let me highlight the new term proposed by Jonathan Visona: innovery (a mixture of invention and discovery). I will limit myself to reproduce my entry. 

I am one of those who think that mathematics is nothing more (and nothing less) than a language: http://www.cut-the-knot.org/language/MathIsLanguage.shtml

Is the language (theormems, propositions, corollaries) already in the grammar (axioms) that specifies it? Well, in some sense it is, but on the other hand you do not care for every possible sentece that can be expressed in the language (leaving aside incompleteness results), but for those sentences or sets of sentences (theories) that are of interest for some practical or theoretical reason.

My opinion is that mathematics are discovered, but the way in which we put everything together and make it "understable" is invented.

Today's quote naturally belongs to J.W.Gibbs: Mathematics is a language.

lunes, 15 de octubre de 2012

main problems III

Other interesting problems

There are other interesting problems that can be studied. I introduce in this post some of them, which I hopefully will try to address in future contributions.

One that I think is very interesting and that I call redundancy can be stated as follows. For a given MGG (matrix graph grammar) decide whether there are redundant productions. A redundant production is one that can be written as a sequence of some of the other productions in the grammar. This is inspired by the notion of base in vector spaces. In essence we are asking to find minimal grammars to express some given language. 

Liveness is a notion from Petri nets, and asks for the (potential) applicability of some production. In fact it can be used as a halting condition for matrix graph grammars, in particular when they are extended using affine productions (a post to come on this). Other concpets can be of interest, also taken from Petri nets, such as boundedness. Enough for today.

Today's quote's from Billy Connely: I have been made redundant before and it is a terrible blow; redundant is a rotten word because it makes you think you are useless.