By Randall Chu, Monash University
Whenever we pose questions with the word `why’, we are seeking an explanation or line of reasoning to justify a particular assertion being made. Indeed, we can continue to ask `why’ over and over again, after each answer is given, until the most basic and fundamental reasons are reached. My intention in this piece is to draw attention to what mathematics is about, and share why I believe mathematics is interesting, and worth studying. To do this, I’d like to motivate our discussion with this question: “Why does one plus one equal two?”
How would you respond to this question? This is actually a little more tricky to answer than it might first appear, because our long-standing familiarity with the rules for counting allows us to take them for granted, without ever needing to worry about why they in fact work, at all. We `know’ that two comes after one, and we `know’ that when we take one orange and put it next to another orange, we then have two oranges. It is likely that you would relay the manner of counting to a toddler in a similar way.
Let us take a deeper look at the statement `one plus one equals two’ from a mathematician’s perspective. ‘One’ and ‘two’ are objects (with the familiar label number being a name for the type or class of object). `Plus’ (addition) is an operation that is done on two objects and produces another object (the type of object in view here of course, is the number). `Equals’ (equality) is a relation between two objects (in this case, numbers), which we know to hold when the two objects are the same. The statement is saying that the object resulting from `one plus one’ is the same as the object ‘two’.
Our next step then, is a matter of definition: to define what a number is, to define addition for numbers and to define the notion of equality. This is not a trivial task (to get an idea of the technicalities involved, read the Wikipedia article on ‘Peano axioms’ , which shows one way of doing this). Although rather academic in nature, the definitions of numbers themselves, provide a more fundamental answer to the original question posed, than the rules of counting, which depend on those definitions.
I like mathematics for many reasons. Briefly, in no particular order, they are:
Rigour. When a mathematician makes an argument (proof), it is tight and logical, with careful attention to definitions and underlying assumptions. A mathematical proof explains `why’ something holds on a logical basis, and is much more convincing than conjecture or opinion. Once a proof has been verified, it is adopted by the mathematical community and added to a growing body of knowledge.
Usefulness. Mathematics finds tremendous application in developing technology and decision-making. To name a few instances, results from mathematics are used to keep planes in the air, construct a working GPS system, predict the weather, model the stock market, and help keep your electronic transactions secure.
Beauty. Not all arguments or proofs are gory in detail. Some are very simple and elegant, yet reveal something profound. For example, look up a proof of Euclid’s Theorem , which states that there are infinitely many prime numbers.
Links: https://en.wikipedia.org/wiki/Peano_axioms  https://en.wikipedia.org/wiki/Euclid’s_theorem
Randall Chu was one of the recipients of a 2016/17 AMSI Vacation Research Scholarship.