Ontology and Semiotics

Professor Juan Godino from Granada University proposes to put mathematical problems at the center of attention for Mathematics Education Research.

I am not familiar with the details of his proposals, but here I write ideas on these two words.

Ontology is part of Philosophy, Semiotics is part of Linguistics. The former is older than the latter. The study of language as a discipline requires more mathematics than old fashion Philosophy. The questions are sharper, and so should the answers. My concern here is with numbers.

Counting numbers were given by God, according to some mathematician, Kronecker. There were debates already about counting and new made up numbers. I do believe that all numbers are made up, but I certainly can see the difference between one, two, three, and zero, infinity and square root of two.

I have been thinking lately about continued fractions, and the representation of irrational numbers as periodic structures, The issue springs out of quadratic versus cubic equations, as Lagrange and Jacobi understood many years ago. I won't go into details here, but do feel compelled to settle in my mind the difference between computer numbers and real numbers.

Writing mathematics for a digital computer, one cannot use infinite processes because no one has an infinite time to wait. The physicality of these instruments forces us to clearly state our meaning, (semantic) ,with clear conventions (semiotics) and separate it from the object (ontology). I hope this states a clear cut practical matter, and not a fuzzy logic problem.

Good symbols, good conventions, will take us a long way in thinking clearly. A computer needs standards to specify a number. Humans get together give a standard, and then the thing is put in hardware. This is definitely more, than trying to tell your mother what square root of two is, unless she happens to be a mathematician.

I believe that there is nothing but the text. Whatever those conventions are, that is what numbers are. The rest is metaphysics.