Learning about mathematics. Language of Mathematics by Catalin Barboianu is a wonderful and eye-opening book for those curious to learn an introductory answer to the question: What's math?

Math, simply put, is logic.

Barboianu sets the foundations by first speaking of symbols, axioms, and systems. This is crucially important because having a conversation of logic requires understanding what we use before using logic.

Similar to a cooking recipe, the ingredients are listed first before the recipe. In this case, Barboianu speaks of the book even before speaking of the ingredients or recipe. Here are a few quotes from the book that I found helpful in understanding math.

In logic, language is symbolic in the sense that any logical proposition can be expressed only through symbols without content, even though it can be read with words. This is why the language of logic is characterized as symbolic and formal. The compatibility of the language of mathematics with that of logic often makes us characterize the former as a formal language, however, mathematical formalism has a certain hermeneutics, especially when we talk about the structure of mathematics and the application of mathematics. The reduction of axioms and propositions to symbolic formulas is not a sufficient criterion for reducing mathematics to logic, because mathematical concepts have a more complex nature than their logical formalism, and mathematical language contributes to the knowledge about these concepts.

In mathematics, we cannot operate with judgment about undefined or vaguely defined concepts for which there is a danger of confusion through interpretation or of logical inconsistency. This principle does not apply to other disciplines such as art, history, or even psychology, which make full use of natural language in description and argument.

Any consistent formal system T that describes/models arithmetic is incomplete. That is, there exist propositions in the language of T that can neither be proved as true or denied (called unprovable). Or in other words, 'Any consistent formal system of arithmetic is either inconsistent or incomplete.' I will not detail here Gödel's proof, which is not very complex, but employs the notion of recursive and primitively recursive functions and their properties, notions whose explanation does not have its place here. The proof is based on several substitutions of some variables with certain Gödel numbers, both in the formula that expresses the fact that an arbitrary proof - as a finite string of formulas - has a certain Gödel number, and also in certain formulas resulting from the construction made. At some point, a certain formula with Gödel number j is expressed in the formal system of arithmetic T derived by means of the system (hence provable in T), and under the assumption that T is consistent, it is deduced that the negation of the formula whose Gödel number is j is also provable in T, which contradicts the assumption that T is consistent.

By the time Gödel published these results, many mathematicians believed that a consistent and complete formal system can be built not not only for a number theory, but also for mathematics as a whole (the trend intensified with Hilbert's program of axiomatization of mathematics).

As a result, the incompleteness theorem was a "shock" in the mathematical community and beyond. This established that the price of the consistency of formal language systems is its incompleteness, or in other words, while gaining in safety and rigor, we also lose in scope, diminishing the range of formalism. In addition, Hilbert pointed out that in a formal system of arithmetic, truth is a broader notion than proof; we can't formally prove everything is true (about arithmetic).

'A consistent formal system of arithmetic cannot prove its own consistency' or, in other words, 'Any formal system can prove its own consistency if and only if it is inconsistent.'