> So far, your definitions of understanding seem to be whatever vibe you are going for in the statement and I would urge you to think about what a sensible mathematical definition of understanding is so that it can sensibly be assessed on neural networks.

Any reasonable definition of understanding is not dependent upon "whatever vibe you are going for", but instead must include at least an English dictionary definition of "understand" such as:

  to grasp the meaning of[0]

And, for further clarification, "grasp" can be defined as:

  to lay hold of with the mind[1]

Which makes an equivalent term-expanded definition of "understand" to be:

  to lay hold of with the mind the meaning of

As such, there is no "sensible mathematical definition of understanding", unless you possess a complete mathematical model of the human mind.

>> Understanding is a state of mind

> This is meaningless, it is a circular definition at best.

See above to as to why there is meaning in what I wrote.

>> It exists entirely within the individual and nowhere else

> Then why are we talking about it? What is the point if it is something that can only be defined per individual?

I like to think analyzing fundamental premises, often implicit, explicitly can help to identify fallacious positions.

> We are still not getting anywhere because you have not prescribed criteria to determine whether [an LLM] understands.

My apologies for being opaque. Let me clarify:

  LLMs do not "understand".  People interpreting LLM output
  are the only entities involved which can "understand",
  because "understanding" exists strictly within each
  person who possesses it.

0 - https://www.merriam-webster.com/dictionary/understand

1 - https://www.merriam-webster.com/dictionary/grasp