## The Set of All Mathematics

At the panel discussion on “On Being Human” (see previous post) a student  had an interesting refutation of my assertion that mathematics is a product of our humanness and doesn’t exist outside of us as a platonic ideal or set of ideals.   I managed to have a couple of conversations with him after the meeting and here is my rendition of his idea.

Consider the space of all possible mathematics, that is, all possible sets of definitions and axioms that are consistent.  He called these structures.  He and I and all other human beings are doing math with one of these structures.  By doing math he means proving theorems and even adding new definitions and axioms.  I would argue that the particular definitions and axioms that we actually use are those that can be embodied and understood with our particular neural structure.  The student contends that the space of all possible mathematics exists  outside of the universe and even outside of all possible universes – a sort of meta-universal object.  How cool!

Here is the student’s own rendition:

Authors note: I use category theory because of the work of Alexander Grothendieck; the less mathematically sophisticated reader can use set for category and get 99% of the ideas.

To start we look at our universe as a vector space typically considered R3 with time typically considered a ray.  On this “stage” we have “players”: the category of “players” partitions into the following three subcategories: matter, energy, and information.  The science mathematics is the study of information, specifically subcategories of information, and studies the compatibility of their elements.  The powercategory of information is inherent in our universe, and compatibility (internal consistency) is a statement about the powercategory of information. (a function to {true, false}.)  Therefore, the science mathematics is inherent in our universe.

Now let’s consider the category of all possible universes (M).  So, there exists an Im (the information subcategory for universe m)   for each m in M.  The infinite union of  Im over M is the category (I)  all information in all universes.  Now we can define compatibility function (c).   An element of the powercategory of I is called true (is compatible) if it is an element of the powercategory of In for some n in M . We can now define meta-universe as the union of all m in M and say mathematics is the study of the function c which is inherent to I which in turn is inherent to our meta-universe.

E. Orion Spero