How would one go about calculating starting gravity without a hydrometer? I know that there is a way to mathematically determine this, which is what I'm looking for.
The most accurate calculation I have found is the one from the following link, which requires measuring both Brix with a refractometer, and FG with a hydrometer. Using those inputs, this calc will result in % ABV, which based on a dozen or so comparisons I have determined to be accurate within about 0.4% on average.
The actual formula:
ABV = [277.8851 - 277.4(FG) + 0.9956(Brix) + 0.00523(Brix^2) + 0.000013(Brix^3)] x (FG/0.79)
Then you can either stop there if ABV is all you care about; and/or you can back-calculate what the OG must have been based on standard ABV calculators; and/or you can estimate the OG based on the ABV, based on knowing that as a general rule, the ABV is pretty close to the last two digits of the OG. For example, a 6.1% ABV beer usually requires approximately a 1.061 OG wort; or for 4.7% your OG is probably pretty close to 1.047, etc. So when you know your ABV, you basically have an idea of the OG based on it.
In any case, at best, you'll only be accurate within about 0.4-0.5% ABV or 0.004-0.005 gravity points, but it will give you a good ballpark figure.
AND...... if all that isn't enough, or you really really don't want to use a hydrometer at all but only a refractometer, well then just check out Part II of this page:
You will need some other tool to measure the gravity, like a Refractometer.
If you do not use any measuring tool, you can get an estimate based on the ingredients, but it will never be precise (although it should not be too far). Most brewing softwares (like beersmith) will calculate this for you after you enter ingredients.
You can calculate and estimate and be pretty close. But you have to measure it to know what it really is. Its dependent on volume and dissolved sugar. Calculations are pretty close if you are working in extracts. Calculations can be pretty far off if you're working in all-grain and you don't have your process dialed in.