Earlier quoted context omitted.
I don't think pursuing a career in academia was what he had in mind when he said 'do mathematics'. In fact he said specifically: >Teaching mathematics shouldn’t be about sending everybody to a Ph.D. program. That’s a very narrow view of what it means to do mathematics.
Doing serious research level mathematics (or any other science for that matter) is almost impossible outside academia. Doing serious work is hard and mentally draining. Mere mortals can't motivate themselves to do it unless the incentives align just right (i.e. being in academia). Also it takes all your time so forget about it if you have another job. I know this from personal experience. After I joined industry I th…
In physics these laymen are generally called 'crackpots' for a good reason because the context one needs to understand to generate new results is so huge.
Mathematics is different - all one needs is pen, paper, a mind that's attuned to generating logically whole statements and a problem that just won't let go of ones mind. The results are likely in geometry or some other approachable field and not in some more obscure subject.