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Mathematics are Base 10. Brian Greene stated that there are 20 numbers that are special to physics. Could there be a system based on those?

I was just watching Brian Greene's talk on string theory, and he was saying that there are about 20 numbers that are special to physics (eg. mass of a proton, electron, neutron, etc), but that there was this question as to why these numbers, out of every number, and why so fine-tuned and precise of a number. Now obviously our current mathematics have been working for us, and that these numbers have worked for us so far. Do you think there will ever be a point where our current Base 10 system doesn't work any more (we have had other base systems in the past)? Any thoughts on if you think an entirely new mathematics could be made based on these 'special' numbers from physics? I have no background in this, and no idea, so just thought I'd throw it out there.