Mathematical Statements

When we communicate, we write phrases in sentences. The same is true in mathematics, though we typically write these phrases in mathematical notation. In mathematics, we are primarily considered with the validity of these sentences.

Formal logic is the branch of mathematics that deals with the principles of valid reasoning and argumentation. It provides a systematic framework for analyzing statements and their relationships allowing us to determine the validity of an argument based on its structure. In formal logic, we are primarily concerned with the form and structure of an argument rather than the content itself.

Is an apple a fruit?

If you said yes, that's great but unfortunately, you would technically be incorrect.

Botanically, a fruit is strictly defined "as the seed-bearing structure that develops exclusively from the mature, fertilized ovary of a flowering plant". When put to this anatomical test, an apple fails to satisfy the requirement because the part we eat actually grows from the flower’s swollen receptacle, while only the core represents the ovary wall.

Under this precise definition, an apple is actually not a fruit; only the core is.

This leaves us with a dilemma: where do we actually start building a system of logical truth? Our conclusions can shift depending on which definition someone chooses to work with. How can we ever prove anything with certainty?

To address this issue, mathematics relies on axioms.