Dear Scott Guthery,
i enjoyed your essay a lot. I agree much on your statement about rational numbers.
Although in the pysical theories are "flaws" too, but they are effect of physicist being conditioned to use wrong mathematics.
To me it looks like we must at least go back to Pythagoras Mathematics, but scientific community is in fear to do so, because physics and mathematics then become connected to political influence and economic system.
Physics about space and time is Philosophy ... not attached to Reality (physical)
"Physical" Physics is different mathematics, as each number (each °1°) must be connected to a "measurement" (empirical data).
irrational mathematics : 1+1=2 ???
rational mathematics :
1+1=1 (1 bread + 1 butter = 1 small meal)
1+1+1+1+1 = 1 (1 bread + 1 butter + 1 egg + 1 orange juice = 1 big meal.
one can find 1 piece of butter and create 1/3 + 1/3 + 1/3 = 3* 1/3 = 1 Piece
same concept is possible for Astronomy:
one find one sphere (sky) and can find 4 (or 4 Million) stars (shining objects) and use them as 4 million "clocks". Using them, there a 4 (million) different times at the same time.
in general, (as said by others too) "math" is our universal language and also "animals" are using maths as language: They use "waves" (sound, EM) to communicate.
Our mathematics (language-puzelling might be completely disconnected from physics at least for 2020 years.
Best wishes
Manfred Pohl