Mathematics with Typewriters
What you're suggesting is certainly doable, and was frequently done in it's day, but it isn't the sort of thing you want to subject yourself to while you're doing your Ph.D. (and probably not even if you're doing it as your stress-releiving hobby on the side.)
I several decades of heavy math and engineering experience and really love typewriters. I even have a couple with Greek letters and other basic math glyphs available, but I wouldn't ever bother with typing out any sort of mathematical paper using a typewriter these days.
Unless you're in a VERY specific area that doesn't require more than about 10 symbols, you're highly unlikely to be pleased with the result and it's going to require a huge amount of hand drawn symbols and be a pain to add in the graphs and illustrations. Even if you had a 60's+ Smith-Corona with a full set of math fonts using their Chageable Type functionality, you'd spend far more time trying to typeset your finished product than it would be worth.
You can still find some typewritten textbooks from the 30s and 40s in math and even some typed lecture notes collections into the 1980s and they are all a miserable experience to read. As an example, there's a downloadable copy of Claude Shannon's master's thesis at MIT from 1940, arguably one of the most influential and consequential masters theses ever written, that only uses basic Boolean Algebra and it's just dreadful to read this way: https://dspace.mit.edu/handle/1721.1/11173 (Incidentally, a reasonable high schooler should be able to read and appreciate this thesis today, which shows you just how far things have come since the 1940s.)
If you're heavily enough into math to be doing a Ph.D. you not only should be using TeX/LaTeX, but you'll be much, much, much happier with the output in the long run. It's also a professional skill any mathematician should have.
As a professional aside, while typewriten mathematical texts may seem like a fun and quirky thing to do, there probably isn't an awful lot of audience that would appreciate them. Worse, most professional mathematicians would automatically take a typescript verison as the product of a quack and dismiss it out of hand.
tl;dr in terms of The Godfather: Buy the typewriter, leave the thesis in LaTeX.
a reply to u/Quaternion253 RE: Typing a maths PhD thesis using a typewriter at https://reddit.com/r/typewriters/comments/1js3cs5/typing_a_maths_phd_thesis_using_a_typewriter/
