Not unrelated

Submitted by Treatid on Sat, 11/18/2017 - 00:36

1. A and B are unrelated.

2. A and B are related.

3. A and B are Meta-Related.

"A and B are unrelated." has problems as a statement. Somewhat along the lines of "This statement is false."

A and B are, for instance, explicitly related by virtue of being in the same sentence and specifically connected by the phrase 'A and B'.

Let's take this to an extreme. Any two concepts that you can think of must be related to each other by virtue of you. That you thought of both these concepts creates a relationship between these concepts.

Any two concepts that could potentially be expressed are related in so far as they can both be expressed.

This isn't just nit-picky pedanticness.

Modern mathematics relies on the idea of consistency to determine the value of a a theory/theorem. An inconsistent theorem is garbage - it doesn't tell us anything useful (Principle of Explosion).

There are many, many inconsistencies within modern mathematics (E.g. Euclidean and non-Euclidean Geometry are defined to be inconsistent with one another). The general approach is that these inconsistencies are not a problem provided each system is isolated from any other system that it might be inconsistent with.

Being Pedantic

If we take it t0 an extreme, no mathematical theorem is truly, utterly isolated from any other theorem. At the very least, any mathematical theorem is related to any (all) other mathematical theorems as they are all part of mathematics.

As such, claiming that two theorems are unrelated is clearly a non-sequitur. Obviously what is meant is that the theorems are sufficiently unrelated that they can be considered independently. No-one is seriously claiming that there is no relationship of any kind - merely that the relationship is so weak it can be dis-regarded.

Let's define the degree of relationship between A and B

All we have to do is enumerate all the possible relationships between A and B (most of them will be indirect), then measure the degree of all those relationships, sum them up and show that the total is above or below the threshold for "sufficiently unrelated".

Easy.

Oh - we probably also need to define the value of "sufficiently unrelated", and make sure that the quality of each relationship is measurable in such a way that comparing it against "sufficiently unrelated" makes sense.

Straight off the top - enumerating every possible relationship between A and B implies omniscience.

Then we have the issue that in the entire history of humanity, nothing has ever been unambiguously  defined (honest - see Foundational Crises in Mathematics).