Today’s article will look at a similar problem to Russell’s Paradox. This paradox, at its heart, is to do with frames of reference. This is a very good sanity test for an Artificial General Intelligence and, at least to me, the resolution of which marks the boundary between AGI and Super AGI (SAGI).

Time to play with DeepThought’s Paradox.

Take a look at the picture below of two squares denoted A. In algebraic form, we would state that the scene represented 2A. However, we can also state that A = A. That is, both squares are the same.

This is where our problems begin.

In the above statement, we are describing the scene from two different frames of reference. The first being an objective view of the quantity and type, the second being the type with quantity information preserved across the comparison.

Both are valid descriptions, however, what are the implications of A being equal to A? Well, that this statement can be reduced to simply A. That is, from this particular frame of reference, there is now only one square instead of two.

We can attempt to resolve this issue by stating that A is not equal A because they are spatially separated. That is, by declaring that there are 2 of them, or 2A. However, we now have the problem of A no longer being just A, but rather a complex mathematical state of both A and Space, or A-Space.

That is, we lose discreteness.

In addition to this, space becomes complex in that we must now derive a mechanism from which A can become 2A. Further, that we have a problem of defining which is the true A, or if that even makes sense.

If we provide space with a mechanism to duplicate, or offer a different view of, A, then this space must in turn be created by an, as yet, undescribed underlying mechanism. This ultimately leads to a form of infinite regress as one underlying mechanism must naturally be stacked upon another ad infinitum.

Even worse, as A can no longer be described independently of space, it now just becomes space.

When reviewing this, it will be tempting to think that there is a false statement, or confusion or equivocation between different levels of abstraction, but that is not the case. The frame of reference is valid, as is the chain of reasoning.

In some sense, this is similar to quantum teleportation and vacuum fields. It also draws to mind the problems in resolving the continuous nature of Einstein’s work with the discreteness of Quantum Mechanics.

In an AGI which arranges facts and arguments in complex dependency chains, this type of reasoning would lead to invalidation of integers, causality, all of mathematics in general and bring into question it’s reasoning.

As such, axioms must be employed as a type of fire-break throughout dependency chains to stop the inconsistencies cascading and marking entire lines of functional reasoning as invalid.

In terms of a Super AGI, I would expect it to provide a unique approach to dealing with the infinite regress. Otherwise, at some point, someone will need to become an expert in banana-squirrel space.