Many students starting to study mathematics need some time until they realise the exact role of formal definitions in (esp. pure) mathematics. I love how Lara Alcock puts it: “when mathematicians state a definition, they really mean it.” (How to Study for a Mathematics Degree, OUP 2012, Chapter 3).
What does this mean? Say the definition gives some properties defining a ‘thingamajig’. Then:
- everything which satisfies all the properties given in the definition is a ‘thingamajig’;
- every ‘thingamajig’ satisfies all the properties given;
- everything we can prove using only these properties will be true for any instance of a ‘thingamajig’.
This is not true for “dictionary definitions”. So how can we get a feeling or intuition about a new mathematical definition we might meet? The first thing to do is to look at, or think of, some examples. What kind of things are ‘thingamajigs’? And also to look at edge cases: when is something just still a ‘thingamajig’, but if I change it a bit then it isn’t any more? Looking at counterexamples gives us some good delimiters to our “example space”. So we need to fill our example space and intuition with common examples as well as “weird” and borderline examples. This does not always happen all at once, but students who make an effort to think of, play with and remember such examples will get a good intuition more quickly.
Another important step is to recognise whether we are importing some “intuition” about this mathematical ‘thingamajig’ from the everyday meaning of the word. In her book, Lara Alcock has some excellent examples which illustrate this problem. Does “increasing function” mean “strictly getting bigger” or does it counter-intuitively include “staying the same or getting bigger”? Even for entirely new concepts, it is easy to get hold of some “false intuitions” via the name or non-exact analogues we may think of.
The last important part to mention is how to use the actual formulation of a mathematical definition. This addresses both the “false intuition” problem and also the third bullet point above. When we want to prove something about a generic ‘thingamajig’, i.e. something that is true for any example of a ‘thingamajig’, then we have to use the properties stated in the formal definition. It is ok to use our intuition we have formed about ‘thingamajigs’ to help us come up with what might be true, or to help us explore why a certain statement may be true. But to prove it, we have to start off from the formal definition and use logical deductions and/or algebraic reformulations to reach our claim. Often, definitions are stated in a way which makes it easier to manipulate these concepts. Compare (again taken from Lara Alcock’s book ‘How to Study for a Mathematics Degree’, Section 3.5)
- A number is even if it is divisible by 2.
- A number n is even if and only if there exists an integer k such that n=2k.
The first one is what one might say as a sentence when explaining. It is of course perfectly precise. But the second definition gives us a way to algebraically manipulate and work with even numbers. This will often be the case in how definitions are stated.
Lara Alcock’s summary of her Chapter 3 on definitions includes:
- Mathematical definitions are precise, they do not admit exceptions, and they often provide algebraic notation which we can use to construct proofs.
- To understand a definition, it is useful to think about objects to which it does apply and objects to which it does not.
- Definitions do not necessarily correspond to intuitive meanings of concepts, especially for ‘boundary’ cases. You should be alert to this; if a definition does not match your intuition, your intuition needs to be amended accordingly.