By our definition, mathematics began in Greece around 500BCE. Before Greece, primitive mathematics was recorded in Babylonia and Egypt. After Greece, mathematics was practiced mostly informally, without much deduction, until the beginning of the formal period. Formal mathematics began in the late 19th century.
What Came Before Mathematics?
People around the Eastern Mediterranean had been working with problems involving quantities since nearly 3000BCE, when the earliest discovered writing examples were produced. Before writing was known, counting boards with pebbles (later the abacus), pots with pebbles, and marks on stone may have been the computers of choice, likely going back to the dawn of the Neolithic in some form. Quantities have always been important to us, whether it be for commerce, astronomy, logistics, taxation, recipes, or making sure all the cows (or loaned monies) come home.
The Greek mathematicians did not start with a blank sheet of paper. Much of what we attribute to the classical Greeks undoubtedly had its roots in Babylonia over a millennium before. The Babylonians devised algebraic methods for solving simultaneous linear equations and quadratic equations.
The Greeks added rigor, proof, and further generality and abstraction, the essential ingredients of what we now consider mathematics. They also admitted incommensurables to their science.
What Is Naive Mathematics?
The axiomatic basis of mathematics is fundamental to the validity of the mathematical science. But most mathematicians proceed intuitively. They practice naive or pragmatic mathematics locally in their specialties, with no explicit reference to axioms and primitive notions. The formal axiomatic superstructure is too cumbersome to drag along on a daily basis. For example, once we understand the axioms that establish the integers as instances of sets, we can forget this and use our intuitive notion of counting numbers and their arithmetic.
Mathematics shows us its most beautiful aspects when we approach it with intuition and naïveté (idealism). Two centuries ago, the practice of mathematics was not nearly as rigorous as it has now become. Yet much beautiful, profound, and lasting mathematics was constructed in this period. The intuition behind it served us well. Now we have the rigor and formalisms that place our results on a sound footing, but we still do much of our creative work naively (informally and intuitively).