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^{2}= ma, the canceling of the small m is actually a bit nuanced because you have to assume the gravitational mass is equal to the inertial mass (though it's supported by experiments). I'm so used to seeing mass as just mass so I'm having a bit of trouble understanding the need for separating the two.

He also said mass can macroscopically be defined by the acceleration caused by an applied force. Is this where the separation came from, the fact that one mass is defined by the gravitational force, and one by the inertial force?

If they weren't equivalent, what implications would there be? Assuming F = ma is universal, would the gravitational force be a "subforce"? Or perhaps the gravitational mass is a "submass"? He also brought up people who were looking for the 5th fundamental force, but I didn't understand the context so I don't know what it was really about.