Well, the title of the blog is computational information geometry. It includes the word “computational”. Of course, one aspects is to bring computational geometry vistas to information geometry by fostering algorithmic techniques. Another aspect, is to ponder what can be done with computations in the field of information geometry. The basic elements being numbers, let us realize that computational reals are of measure zero in the set of reals.
A real is said computationable if we can enumerate all its bits one by one (up to infinity). Emire Borel in the 1920’s already felt strange with this notion (that came in time for Turing later): consider all YES/NO questions (say, in French -:) ) and let each bit of the Borel real B answer the corresponding question.
Being able to compute such a real would yield insights to all questions, including all yes/no questions of science, technology in the future, etc. We expect B by contradiction thus to be uncomputable. Otherwise, imagine having this oracle (Is there advanced life in other planets, will human have wings and fly someday, etc.)
Two major dawbacks of course is the notion of infinity in digital expansion and the notion of time in languages. Actually, to construct a non computable real, one uses Cantor diagonal arguments…
Anyway, in Computational science, one has from time to time to think of the basic notion of computable numbers. There are the basic atoms of any computational science.
Frank.
