'simply using numbers to designate pitches to illuminate the theory ... ' 'examining each of our 'pitch positions over a two octave span ... ' 
"What we play is life." 
wiki ~ Louis Armstrong

In a nutshell: Thinking the pitch letter names of the diatonic and chromatic scales are already known to the reader, a supereasy way into our music theory is to simply begin the process of matching pitch letter names with a corresponding number. 'C' is now also #1 etc. That's the easy part :) 
The tricky part is that each of our essential components have a bit of a different counting system but the good news is that once a letter name is chosen to be '1'. It stays one. For example, in the key of 'A' is always '1' :) Scales are thought of as 123 not A B C, arpeggios are 135 not A C E, triads are also 135, rhythms are 1234 2234 3234 4234 etc. Then there's the ratios for our intervals; the 2:1 octave basis of all, 3:2 of the perfect fifth to get the wheel to create new pitches a turnin. 'The '440' frequency pitch of today's tuning note, 'A' 440, and its detune to 432 hz. and beyond even, making new musics with a new pitch center of earth tuning. 
So not quite on and on depending (unless you be a jazz leaning artist) but for those so curious for sure, that any of these last ideas can probably become a thesis in academia is cool to know. The knowledge base is deep all is related to one another, forming the loops within our Americana musics. 
Jazz leaning artists. Just to note that this numbers approach can get applied to the shedding method too, ensuring that all of the resources are fairly available over the first 12 frets or so. Scale shapes can be numbered into a loop of 5, there's working with a metronome (really for everybody) there's numbers for the chord voicings; 1 5 b7 3 5 8 barre chord, interval studies; major scale in 3rd's, chord progressions so now, on and on might be true, at least for the foreseeable :) 
Why do? Numerical theory allows us to project any idea onto any pitches of any music, into any key scheme in any style of the past, present and forward. So, anything from anywhere? Yes but we have to flip around the AFA telescope view to get the right focus here. For here are view is all a numerical center :) 
First number. Eggs in a dozen ? 12. Numbers on the face of a clock ? 12 Number of pitches in the chromatic scale ? 12 
One through #15. This goofy looking chart stage left is the numerical linking to the original intent of this 'by the numbers' discussion. Based on an idea sparked to life during a lesson I took with longtime Alaska jazz guitar ally Mark Manners in its writing, each of our numerical positions within a two octave chromatic scale are examined. The half dozen or so letter / number swaps listed above became the narrative basis for discussion. 
"I have to be my own teacher, curious to know the reason for things." 
wiki ~ Laurindo Almeida 
References. References for this page's information comes from school, books and the bandstand and made way easier by the folks along the way. 
References academia Alaska. And when you need university level answers to your questions and musings, and especially if you are considering a career in music and looking to continue your formal studies, begin to ereach out to the Alaska University Music Campus communities and begin a dialogue with some of Alaska's own and finest resident maestros ! 