What is mathematics?

A very general preliminary answer follows from a simple observation: we are all doing mathematics since early childhood.

What is mathematics?

Mathematics is a human activity.

But what distinguishes mathematics from other activities, or more specifically, what is the purpose of doing mathematics?

To answer this question, let us think of an exemplary situation where we have a certain amount \(m:=32\) of money and we want to buy the maximal, affordable number \(n\) of items each having price \(p:=7\).

Here, doing mathematics means the following: we first translate the situation into the mathematical statement $$(n\cdot p\leq m)\wedge((n+1)\cdot p>m)$$ which means that the money suffices to buy \(n\) items and that one additional item exceeds our budget. Next, we derive the useful equivalent statement \(n=4\) based on valid mathematical arguments. It allows us to ensure the proper choice by simply counting the items in our shopping cart.

Summarizing similar experiences in an abstract form without introducing too many details at once, we can clarify the new question like this:

What is the purpose of doing mathematics?

The purpose is to find useful mathematical answers.

Our example already demonstrates that mathematical answers come in the form of statements which are shown to be true using mathematical arguments. Truth alone, however, does not make the answer useful. For example, the alternative statement \(n=\min\{x-1\,|\, x\in\mathbb N:x\cdot p> m\}\) can also be derived but does not help us to check that we have the correct amount in our cart. We summarize these observations as follows:

What is a mathematical answer and what makes it useful?

A mathematical answer is a statement which is “true” according to established mathematical inference rules. Its usefulness is not governed by rules but depends on the intent of the questioner.

By not explaining the notions “statement” and “inference rule” we can keep our answer short but naturally provoke follow-up questions.

To explain the first notion, we look back at our example where mathematical statements are formed using relation symbols like \(=, \leq, >\) between numbers or the conjunction symbol \(\wedge\) between statements.

What is a mathematical statement?

It is a sequence of symbols which is formed using established language rules based on descriptions of mathematical objects and/or mathematical statements.

The meaning of a mathematical statement follows from the structure of the inference rules which control its truth state and which are influenced by its state. For example, according to one inference rule \(A\wedge B\) is true if both statements \(A\) and \(B\) are true. Another rule concludes the separate truth of \(A\) and \(B\), if \(A\wedge B\) happens to be true. Due to these rules, the conjunction \(\wedge\) plays the role of the word “and” in mathematical statements.

What is a mathematical inference rule?

It is a rule which assigns the value “true” to certain mathematical statements provided other related statements are already known to be true. In this way, inference rules generate the meaning of mathematical statements.

Finally, it remains to explain “object descriptions” which are introduced as building blocks of mathematical statements. In our example, the statement \(n\cdot p\leq m\) connects two such descriptions \(n\cdot p\) and \(m\) with the order symbol \(\leq\). Both descriptions act as references to numbers – the mathematical objects of the statement.

Some of these descriptions like \(n\cdot p\) are themselves composed out of separate object descriptions \(n\) and \(p\) with the product symbol \(\cdot\) refering to a specific recipe applied to the participating objects.

What is a mathematical object?

The result of applying an established recipe for object formation to suitable mathematical objects as ingredients. To start object formation, initial objects are not constructed but described with reserved symbols and charaterized by statements which are true without justification (axioms).

At the end of this quite general top-down analysis we can summarize the key components which characterize the human activity called Mathematics:

• Mathematics generates answers using a limited set of inference rules.
• The answers are formulated in a specific language about mental objects.
• Statements of the language are generated using a limited set of grammar rules.
• The objects are generated using a limited set of object formation rules.

In particular, doing mathematics requires some skills which have to be practiced:

• Problem descriptions have to be translated from everyday language to mathematical statements.
• Mathematical answers have to be translated back to everyday language.
• All inference-, grammar- and object formation rules have to be remembered and mastered.

The language \(\small\mathbb M\)ATh comprises a complete set of rules for object formation, grammar and inference. The concrete syntax is close to the conventions used in beginner courses of university mathematics. Details will be worked out in subsequent posts.