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Stephen Tashi

Science Advisor

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- TL;DR Summary
- Many topics in mathematics can be studied from the viewpoint of performing a sequence of steps on sequences of symbols. What are the limitations, if any, of only using two types of sequential order? (-order in the symbols and order in the steps).

Formal logic can be studied from the viewpoint of rules for deriving strings of symbols from other strings of symbols. Computability can be studied from the viewpoint of a machine operating on a 1 dimensional tape. It seems that, in math, we can handle computation and deduction in two dimensions. A string of symbols has a order, so we can think of it as being written from left to right. A process of deduction proceeds in some order, so we can visualize its steps as being written in lines, going from the top of the page to the bottom. Is there any mathematics that requires another ordering (a "third dimension") to formulate?

Things like matrices, tensors, and graphs can, for convenience, be represented as multi-dimensional configurations of symbols. However, I think that they could also be formulated (from the viewpoint of defining them in a computer language, or describing a formal logical system for manipulating them) as strings of symbols in a sequential order that can be changed into other strings by performing steps in a sequential order.

Is there math where multi-dimensional arrays of symbols or steps are absolutely necessary?

Things like matrices, tensors, and graphs can, for convenience, be represented as multi-dimensional configurations of symbols. However, I think that they could also be formulated (from the viewpoint of defining them in a computer language, or describing a formal logical system for manipulating them) as strings of symbols in a sequential order that can be changed into other strings by performing steps in a sequential order.

Is there math where multi-dimensional arrays of symbols or steps are absolutely necessary?