Objects

Our everyday experience relates to things like jam jars, pencils, and traffic lights, whose meaning for us arises from their uses, which in turn are related to their construction.

For example, a pencil is the result of a special manufacturing process involving certain ingredients: wood for the shaft, a graphite-clay mixture for the lead, and lacquer for the grip. Its uses (including drawing and writing as well as back scratching) are determined by the properties of the ingredients and how they are combined during production.

In the conceptual world of mathematics, mathematical objects are the counterparts to things in the real world. They, too, are made from other objects, with their properties resulting from the manufacturing process and the ingredients used, each of which is described symbolically (i.e., with symbols).

For example, in the conceptual framework of the Peano axioms, the number 4 is constructed as the successor of the number 3, which is symbolically represented by a function evaluation of the successor function, i.e., 4:= successor(3). The function evaluation combines the two ingredients 3 and successor, with 3 (the so-called argument) written in parentheses after the function name. The symbol 4 is then declared with the definition symbol := as an abbreviation for the symbolic combination.

If you examine the ingredients of a pencil more closely, you’ll find that they, in turn, are made up of other ingredients (down to molecules, atoms, and elementary particles). The same applies to mathematical objects: 3 = successor(2) and 2 = successor(1). Object 1, like the successor function, is an initial object in the theory of natural numbers (according to Peano), which simply exists and is not formed from other objects.