The mind has the capacity to believe that it has personal identity. Are you saying that everything that you believe to be true necessarily is true? Sounds a bit solipsistic to me!
No; I don't think Belz's analogy is quite right.
Calculation is information processing. It doesn't matter what is doing it or how; if the answer is correct, whatever the mechanism is, is a calculator.
The same applies to personal identity, consciousness, self-awareness, ego, whatever you want to call it.
Then demonstrate it! The only "proof" PM could apparently find was that other people believed it! Me too.
There are a number of mental properties that we can determine very simply and directly by asking people.
"Are you awake?" is one such question.
"Can you understand me?" is another.
And then we have, "Do you experience personal identity?"
By the very definition of these experiences, an affirmative answer is objective evidence in itself.
We wouldn't take a single answer as sufficient evidence, any more than 2+2=4 would be enough to confirm that a calculator really works (it might always return 4 when you press the = button). But it's easy to ask further questions to get confirmation. And we do ask, and we do get confirmation.
Personal identity is verifiably, objectively real, and it's a brain process.