Let's be clear. Nobody knows who Satoshi Nakamoto is, or if it is even a single person.  There is one thing that anyone claiming to be Satoshi has to prove however, and that is that he knew and communicated with Hal Finney *BEFORE* Bitcoin. And Craig Wright never met Hal, therefore he is NOT Satoshi. Q.E.D.

Now, why do we know this?  The key is not what is in the initial exchanges between Satoshi and Hal Finney. The key is what is *NOT* in the initial exchanges. Any engineer, even one Aspergers, asks personal questions when they come across another talented engineer. Maybe not the first time if they just answer a question, but in a project as big as Bitcoin it will come up.  Just general background things so they can size up who they are talking to and how seriously they should take this person. E.g. am I talking with a PhD student or a smart 16 year old kid? None of this normal personal banter is ever exchanged between Satoshi and Hal. And they had many chances to do it during the course of their email exchanges. There is only 1 plausible explanation for this, and that is whoever Satoshi was, Hal Finney already knew him.  Read the emails for yourself.

Hal knew Satoshi. And he elected to keep his identity a secret to his death. So unless Craig Wright can prove he had some kind of a relationship with Hal, he is not Satoshi. However, the fact that someone named Satoshi Nakamoto lived only a short distance away from Hal Finney leads me to guess that Hal was actually the individual who suggested the Satoshi Nakamoto pen name. Think about how that could have gone. This is entirely conjecture, but I would bet on it being close to the truth:

Hal is sitting in a group with the other like minded people who are strategizing about how to keep Bitcoin anonymous in case the FBI comes looking. One of them says, "we need a pseudonym, something that sounds really foreign."  "How about "Satoshi Nakamoto?", Hal pipes up.  Where did Hal get that name? It could have been intentional because he knew the guy, or it could have just been subconscious. Maybe he accidentally received Satoshi's mail one day. Or perhaps he stood behind him at the grocery store and caught the name on his credit card. Or maybe he just saw it on a voter registration list. There could be dozens of ways he came across his name.  Of course, it is also possible that Hal was not the guy who suggested the name, but the fact that he lived so close to someone with that name suggests that there might be a connection.

In any case, that is secondary. Hal was involved in the development of Bitcoin. Hal both knew and respected whoever Satoshi was. Craig Wright simply does not fit the profile.