[jvvw]

I suspect there is some confusion here by the use of 'first principles' in the title. As a former mathematician when I see 'first principles' I think of axiomatic approaches to mathematics and the study of subjects like analysis, algebra and geometry from those first principles. I suspect however that what you want is good ground in the foundations of mathematics necessary to understand common applications of mathematics, which is quite different.

It is also a difficult question to answer without some context of your current mathematical understanding. Do you know any calculus? Any linear algebra? If you don't, those would be good places to start as they underpin many areas with applications of mathematics. Bear in mind too that Mathematics is a huge subject in that even if you take to it naturally, you're not going to acquire a breadth and level of understanding without a fair amount of study. Looking back I probably put a lot of hours in my youth into really understanding linear algebra fully for example to the level that I could teach it at a high-ranking university, and that was with the help of people whom I could pester with my questions and the incentive of exams to take.

The other approach is to look at the areas you want to understand and then work out what topics you need to study to fully understand them. Cryptography is worlds away from electromagnetism for example. Looking at cryptography, are you interested in public key cryptography or symmetric key cryptography? If the former, then you need to start learning number theory and if the latter, knowing some statistics is probably more relevant.