The nature of something is described by its structure, the relationships and interactions between its structural components, and the relationships and interactions between it and external environments. By the toolness of mathematics, I mean the tool nature of mathematics.
There are two basic concepts associated with tools: their forms and their functions. For example, ancient humans had specific functions and forms in mind when they made tools out of stones. Modern tool-making has not veered from this form and functionality relationship. Usually, the intent of a tool inventor is to create the best form for his or her tool so that it can efficiently perform the function for which it was invented. Some tools take many years and many inventors to perfect. Mathematics is such a tool. It is yet to be perfected although its development has been very impressive. Nonetheless, the toolness of mathematics consists of a very well defined and sophisticated form and function. This form is constituted by pattern thoughts. Pattern thoughts are thoughts with specific arrangements and about specific objectives. The specific arrangements of mathematical thoughts are axiomatic and theorematic in essence and the specific objectives these thoughts address are the problems posed by natural and man-made phenomena. The function of the toolness is problem-solving.
The toolness of mathematics is summarily represented by the word MATH which is an acronym for Mathematical Modeling, Analysis, Techniques and Hand-shaking.
Each letter in MATH represent a structural component described as follows:
Mathematical modeling: the formulation of a mathematical model to describe the phenomena of interest.
Analysis: pricise definition and analysis of a mathematical problem .
Techniques: the use of appropriate mathematical methods and procedures to obtain a solution for the problem defined.
Handshaking: the reconciliation of the result obtained from the mathematical model with experimental data.
The pairing of the axiomatic and theorematic form of mathematics with its problem-solving function is a great realization of the form-function duality (Toolness) of Mathematics.