Definitions, Definitions, Do We Need Them?
The role of definitions in mathematics and theory
Alfonso Bedoya was not a theoretician, but an actor. He is perhaps most famous for uttering the lines:
“Badges? We ain’t got no badges. We don’t need no badges. I don’t have to show you any stinking badges!”
He played Gold Hat, in the The Treasure of the Sierra Madre.
Today I want to talk about definitions.
We all think mostly about theorems and proofs. We try to understand proofs of others, we try to prove new theorems, we try to advance the field. Yet definitions are not the main stars. I cannot imagine a paper getting into the main conferences that had only a new definition, but perhaps that should happen. Our attitude is a little like this: To paraphrase Bedoya:
Definitions, definitions, we don’t need no definitions.
Yet definitions are critical, but strange. Unlike a proof, a definition cannot be “wrong.” Yet a definition can be “silly.” A great definition can be more important than a new proof, and that is why I want to talk about them.
A major part of mathematics, of all kinds, is played by definitions. Every area of math has many definitions that must be learned if one is to be able to understand that area. Here are some key points, in my opinion, about definitions in general:
Definitions can be eliminated. A definition can always be eliminated from a proof. Essentially, a definition is a shorthand; a definition allows us to make arguments clearer and shorter. For example, the statement that there are an infinite number of primes
could be written without using the definition “prime.” But, the statement would be longer and less readable:
This is a important role that definitions play: they help reduce the size of proofs. There is some nice work in proof theory on the role of definitions in reducing proof size. I will discuss this role in the future.
Definitions can be “silly”. Any definition is allowed; however, some are much more useful and productive than others. Consider the definition: is a special prime if and only if , , and are all prime. Is this a good definition? No. Because is special if and only if . We will see other, less trivial, examples of “silly” definitions in a moment.
Definitions can be “critical”. Definitions shape a field; they affect the way we think, and what theorems we even try to prove. A good definition can suggest new research, can open up new directions, and can have a profound effect on progress.
Definitions need justification. While any definition is “valid,” some are more useful than others. Often we read something into a definition that is not justified. So a good question to ask about any definition is does it actually capture the property that we are trying to “define?”
I will continue the discussion of the last point by giving a series of examples.
What is a function? This is one of the hardest definitions to get “right.” Early mathematicians argued over what was the right definition: some insisted that any “rule” was fine, others that a function had to have an analytic form, others that it had to be given by a series of some kind.
What is a set? Georg Cantor gave a beautiful definition:
A set is a Many that allows itself to be thought of as a One.
The trouble with his definition is that it is not precise enough. Unfortunately, there are many paradoxes possible with a naive definition of set. The most famous perhaps is due to Bertrand Russell: consider the set
Is a member of ? Either answer yields a contradiction.
What is a “nice” function? Define a continuous function on the interval to be -Lipschitz provided,
Similarly, continuous functions which satisfy the below property are said to be Lipschitz of order , or Hölder continuous functions,
These conditions are named after Rudolf Lipschitz and Otto Hölder.
There is a story: a student is giving his thesis defense at Princeton University. He has studied the class of Lipschitz functions where —why not study these too—he reasons. His thesis has many neat results, and he has proved that these functions are closed under many interesting operations. The story continues that a visiting professor raises his hand, during the talk, and asks:
Can you give me an example of any function in this class that is not constant?
The sad answer is no. End of talk, end of thesis. Hopefully, only an urban legend.
What is a curve? An obvious definition of a curve is a mapping from to define by,
where and are continuous functions. This seems to be well defined and reasonable. The problem is that such “curves” have a non-intuitive property: a curve can fill the entire square. Giuseppe Peano discovered that a space-filling curve was possible.
This discovery showed that the definition was not “correct.” In order to avoid space filling curves one needs to add more to the definition. For example, if the maps and are smooth, then space filling curves are impossible.
Let’s consider two definitions from computer science: sorting and security.
What is a sorting algorithm? Suppose that someone sells you a sorting algorithm: the algorithms takes groups of records and sorts them according to any key that you like. What does it mean to be a correct sorting algorithm? I once asked someone this question—their answer was:
- The algorithm’s output must be a permutation of the input records. The algorithm cannot change the records in any manner at all.
- The algorithm’s output must have the records in the correct order based on the given key. If the keys of the output records are
Is this the right definition of a sorting algorithm? Would you buy this algorithm from me? Or is this definition “wrong”?
The definition is not precise enough; if you used this algorithm to do many tasks that arise, there would be a problem. The issue is that the definition does not insist that the sorting algorithm be stable. A sorting algorithm is stable if it retains the order of the records that have the same key.
For example, quicksort is not stable, but merge is stable.
Consider three records
Suppose we sort the records on the second key: the last names.
Then, we sort on first names. This could be the outcome:
This is not what we wanted: the records are not sorted according to last names and then first names. A stable sort would have yielded the correct order:
What is a secure protocol? Modern cryptography has stated many definitions that attempted to capture some notion of security. Then, later on the notion was discovered to be deficient: usually the definition sounded right, but lacked the exact property that was wanted. So a new definition was created and the process continues
I will say more about this in the future.
Justification of Definitions
If definitions can be “wrong,” how do we know that they are “right?” The best methods that I know are based on two kinds of evidence.
Equivalence Approach: One method of getting some confidence that a definition is right is to show that it has many different equivalent definitions. This is done throughout mathematics. For example, there are many very different—but equivalent—definitions of a continuous function. Some use and ‘s, others use topology, and others use In computing there are many equivalent definitions of the complexity class NP, for example.
The fact that very different definitions yield the same concept is good evidence that the definition is the right one.
Consequence Approach: Another method for increasing confidence that a definition is correct is to look “consequences.” For example, the fact that the original definition of a curve could fill a square was not expected. One could argue that the “right” definition of a curve would not have this property.
Another example is the definition of continuous functions. The fact that they are closed under certain basic operations gives one a good feeling that the definition is right. If our definition of some property is not closed under reasonable operations—operations we feel it should be closed under—then the definition probably is not capturing our intuition.
Any definition is valid. Some definitions are interesting, some are silly, some definitions are useful. But, getting the right definition is not easy. Often it is quite hard to get the definition that captures the property that you are after.
The open question you might think about is: are the definitions you work with the right ones? How do you know that they make sense? I would argue that more thought should go into definition justification. Often—especially in computing—we are given a definition without any supporting arguments.