Analysis of math and its elements

Question

Can anything in math ultimately be analysed into symbols, equations, formulas, with as exeption perhaps the Euclidean geometry we know with triangles,straight lines etc? Can also proofs, definitions and theorems be analysed in this way? for example, when we say function, continuous, matrix, tensor, metric tensor, continous, symmetric, integrable, bounded all these words can be translated into symbols and formulas.

Can this be done with anything in math except Euclidean geometry and use anything in terms of symbols and formulas when reading and proving and understanding in math? Do mathematicians do this when reading , proving and understanding math?

Should i try to translate any words in math into symbols and formulas? Does this help? Thank you.

Answer 1

This perspective is known as Formalism https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathematics). It is sometimes the case that working out the details of a symbolic argument is tedious and unenlightening, in which case it is easier to simply convince someone that you could in theory write out the details in a purely formal way. It is my perspective that most mathematical arguments are this: attempts to convince the audience that a completely formal symbolic proof exists.

If you're going to make an exception for Euclidean geometry, you might also need to make an exception for graph theory or category theory, however I don't think you need to: geometric figures can be encoded in a linear symbolic language (just think about how you would encode these diagrams in a computer programming language).

Translating math into symbols is excellent practice, both as practice working with symbolic mathematics, and with the topic at hand (they say that a good way to learn is to explain something in different words!).

Just be aware that there's a reason mathematicians don't communicate exclusively in symbols: the key ideas and intuitions behind an argument are usually best communicated in words.