The inherent 2-dimensional nature of formal math?

[Stephen Tashi]

TL;DR

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?

 

[fresh_42]

A proof is often more a flowchart than a linear sequence. This is a third dimension because of the GOTOs. I assume you did not mean that. You probably don't mean multi valued logic either.

I'm not sure as to how far your questions is simply equivalent to our linear time. I suspect it is: one causes the other.

 

[Stephen Tashi]

A proof is often more a flowchart than a linear sequence. This is a third dimension because of the GOTOs. I assume you did not mean that. You probably don't mean multi valued logic either.

An interesting aspect of mult-valued logic and other exotic logics (quantum logic, fuzzy sets) is that when people write mathematical articles about exotic logics, they reason in terms of ordinary logic. I suppose they must because mathematics deals with two truth values.

Reasoning can be represented as a flowchart that simultaneously offers different sequential versions of a proof. Can we linearly order the possible sequences?

I'm not sure as to how far your questions is simply equivalent to our linear time. I suspect it is: one causes the other.

Its a rather deep question. In logic, the premises must proceed the conclusion. Is this dogma of logic ultimately tied to the requirement that logic must work correctly in describing physical processes? Or pehaps to the phenomena that machinery that makes correct deductions about the world must be implemented by physical processes and so must have an order in time.

 

[pbuk]

Any array with a countable number of symbols in a countable number of dimensions can be mapped to a linear sentence, so I think the question becomes:

Is there any mathematics where an uncountably infinite number of symbols and/or steps is necessary?

The answer, it seems to me, must be no. However we might be able to think further about such an hypothetical system. For instance there would be no analog of Gödel numbering - perhaps we could construct a complete logic? Or perhaps it would not be possible to construct a consistent logic?

 

[Quasimodo]

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).

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?

What makes you think that? The ink on a piece of paper has thickness ( a third dimension ), the 0's and 1's in a computer are multi-dimensional electric currents etc.
1 and 2-dimensions are just mathematical abstractions and I do not know how a 2-dimensional being could exist to do any mathematics there.
Ideas and mathematics are a by-product of a 3D brain, your question is equivalent to asking if a dream has a rectangular or triangular shape, hence meaningless.

 

[Stephen Tashi]

What makes you think that? The ink on a piece of paper has thickness ( a third dimension ), the 0's and 1's in a computer are multi-dimensional electric currents etc.

I'm making a distinction between computation and computers - and between mathematics and mathematicians. Maybe it takes 11 dimensions to implement computers and mathematicians.

 

[Jarvis323]

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?

One possibility is a system where all mathematical objects can exist or not separately in an infinite number of "existential dimensions" or likewise can have separate truth values in infinitely many dimensions (in a sense). For example, if existence or truth is only relative to other mathematical objects, and your axioms had no unconditionally existing objects at all?

Then at least you might be able to have the situation where you cannot guarantee that a finite sequence of statements won't change in meaning depending on if another is added after. For example, each new statement adds new additional meaning to the previous statements.

Or maybe you could also have a system where you can branch off from a line of statements at position x, along another dimension, so that every statement at a position before x becomes modified by every statement in the new dimension; Something like a high dimensional crossword puzzle.

I'm not sure these type of systems make sense or have been studied or not?