Why do we not have to prove definitions?

Question

I am a beginning level math student and I read recently (in a book written by a Ph. D in Mathematical Education) that mathematical definitions do not get "proven." As in they can't be proven. Why not? It seems like some definitions should have a foundation based on proof. How simple (or intuitive) does something have to be to become a definition? I mean to ask this and get a clear answer. Hopefully this is not an opinion-based question, and if it is will someone please provide the answer: "opinion based question."

Answer 1

I'd like to take a somewhat broader view, because I suspect your question is based on a very common problem among people who are starting to do "rigorous" or "theorem-proof" mathematics. The problem is that they often fail to fully recognize that, when a mathematical term is defined, its meaning is given exclusively by the definition. Any meaning the word has in ordinary English is totally irrelevant. For example, if I were to define "A number is called teensy if and only if it is greater than a million", this would conflict what English-speakers and dictionaries think "teensy" means, but, as long as I'm doing mathematics on the basis of my definition, the opinions of all English-speakers and dictionaries are irrelevant. "Teensy" means exactly what the definition says.

If the word "teensy" already had a mathematical meaning (for example, if you had already given a different definition), then there would be a question whether my definition agrees with yours. That would be something susceptible to proof or disproof. (And, while the question is being discussed, we should use different words instead of using "teensy" with two possibly different meanings; mathematicians would often use "Zduff-teensy" and "Blass-teensy" in such a situation.)

But if, as is usually the case, a word has only one mathematical definition, then, there is nothing that could be mathematically proved or disproved about the definition. If my definition of "teensy" is the only mathematical one (which I suspect is the case), and if someone asked "Does 'teensy' really mean 'greater than a million'?" then the only possible answer would be "Yes, by definition." A long discussion of the essence of teensiness would add no mathematically relevant information. (It might show that the discussants harbor some meaning of "teensy" other than the definition. If so, they should get rid of that idea.)

(I should add that mathematicians don't usually give definitions that conflict so violently with the ordinary meanings of words. I used a particularly bad-looking example to emphasize the complete irrelevance of the ordinary meanings.)

Answer 2

The other answers did not explain the background of logic that is the key to understanding this issue. In any formal system where we write proofs, we have to use some formal language that specifies the valid syntax of sentences, and we must follow some formal rules that specify which sentences we can write down in which contexts. In mathematics we usually use classical first-order logic, which consists of both the language of first-order logic and classical inference rules. This language is sufficient but extremely cumbersome if we were not allowed to make any definitions.

For example, if we are working in Peano Arithmetic where the only objects are natural numbers, then if we want to prove that an odd number multiplied by an odd number is odd, we effectively have to prove: $\def\imp{\Rightarrow}$

$\forall m \forall n ( \exists a ( m = 2a+1 ) \land \exists b ( n = 2b+1 ) \imp \exists c ( mn = 2c+1 ) )$.

Now certainly we can do this and completely avoid defining "odd", but as the theorems grow in complexity (and this example is an incredibly trivial theorem) it would become simply impossible to refrain from definitions.

What is a definition, then? In first-order logic it can be understood to be simply a shortform for some expression.

Continuing the above example, if for any expression $E$ we define "$odd(E)$" to mean "$\exists x ( E = 2x+1 )$" where "$x$" is a variable not used in "$E$", then we can rewrite the theorem as:

$\forall m \forall n ( odd(m) \land odd(n) \imp odd(mn) )$.

See? Much shorter and clearer.

Answer 3

In a definition, there is nothing to prove because the general form of a definition is:

An object $X$ is called [name] provided [conditions hold].

The reason that there is nothing to prove is that before the definition [name] is undefined (so it has no content). The [conditions] are like a checklist of properties. If all the properties of the [conditions] are true, then $X$ is whatever [name] is.

The reason that a definition can't be proven is that it isn't a mathematical statement. There's no if-then statements in a definition, a definition is merely a list of conditions; if all the conditions are true then $X$ is [name]. Since [name] had no meaning before the definition, you can't even check that [name] means the same as the conditions.

Answer 4

Think about English definitions. They just assign meanings to symbols. It's really the same thing here. If I told you to prove that $1 + 1 = 2$, you would probably object that $2$ is defined as being $1+1$. What more is there to say?

Answer 5

If you want to prove a statement, you need to first tell me what the objects involved in the statement are. For example, if you want to prove that the product of two even numbers is even, you first need to tell me what an even number is, what a product is, ... even what a "number" is. Mathematics is about finding out what relationships/results hold after starting with certain objects that you define. Yes in some sense, the definitions seem to come from nowhere (why do we need imaginary numbers? they're just "made-up"?), but are usually well-motivated (we want roots of negative numbers, solutions to quadratics, etc.).

Answer 6

Mathematics is a kind of exploration of consistent systems. It needs a language to do this exploring. In order to communicate with each other about these systems, one needs common reference points, or things we all agree upon. In daily life we all agree that the word "chair" has a set number of meanings, the most common of which is something to sit on. Often, when we translate from one language to another, we find a problem because one language doesn't have a word for something so meanings become fuzzy or intuitive. This can't work in mathematics, so we have to agree on the meanings of certain things. We define things and agree upon those definitions so that we can move forward and see the ramifications of statements about those definitions.

Euclid, in his Elements started with definitions. For example, "I say a point is that which has location but no dimension." or "I say, a line is that which has length but not width or height." The rest of the Elements are then statements about those definitions.. Back in his time, someone could challenge him and say "I don't see that. Look, I put my finger in the sand and it has size." Euclid might answer, "I can see that. But I think if you follow what I'm saying and see where my statements lead, you'll find some interesting things that are true not just for lines but also for apples and oranges or building things." Euclid's definitions are useful and produce results.

Often for learning mathematics, one needs a book and has to start from the "beginning," however, in mathematical exploration the people doing the innovating didn't start that way. They made discoveries and then had to work backwards or develop a system, and to teach that, they had to start with definitions or a common language, so the student can follow along.

Getting back to what I was saying about the chair, imagine a world in which nobody agrees on the definition of a chair. There would be confusion and a complete lack of communication. Imagine if I say "A chair is something that you sit in." and someone counters, "Really? Prove it to me." or "I need a new office chair," but they defined chair as a device for adding and subtracting numbers, and you defined chair as a place to park your car.

Definitions aren't wrong or right and they don't require proof. They don't say something and they don't arise from a logical progression of ideas. I don't feel that they are intuitive.

You might want to check out Euclid's Elements and see how things are worded there. I think this will help you to get your head around this, and give you a feeling for the roots of mathematics and how things started from ground zero.

Answer 7

Frequently, a definition is given, and then an example or proof follows to show that whatever has been defined actually exists. Some authors will also attempt to motivate a definition before they give it: for example, by studying the symmetries of triangles and squares and how those symmetries are related to each other before going on to define a general group.

A definition is distinguished from a theorem or proposition or lemma in that a definition does not declare some fact to be true, it merely assigns meaning to some group of words or symbols. The statement of a theorem says that "such-and-such" thing is true, and then must back up the claim with a proof.

Answer 8

Definitions assign meanings, not truths. They describe how you are going to talk about stuff. Definitions are basically arbitrary. It does not make sense to try proving them. Axioms describe what you are going to talk about, the identity of some system.

Given a mathematical axiom system, you cannot prove one axiom from the others, but unlike with definitions, that is a matter of proof: basically you show that there is at least one possibility of meeting all the axioms' conditions, and then you show that you can also find one possibility of meeting all the resulting axioms' conditions when replacing one axiom with something incompatible with it, so no axiom is a necessary consequence from the others.

Theorems are necessary consequences from a set of axioms. They trivially include the axioms themselves. They constitute knowledge about the properties of a mathematical system defined by its axioms, described in terms of basic definitions.

Answer 9

You cant prove a definition, because the act of defining is to give a meaning to a particular concept.

For example, the normal English definition of an even number is an integer divisible by 2. That's just what an even number is. We can later prove that if we add even numbers together, we will always get an even number.

---

If we alternatively decide to define an "even number" as a positive integer with all its decimal digits the same (not recommended because it goes against the normal English definition) the very act of defining means that in our language system, this relationship is true.

We can then proceed to prove some facts about these "even numbers."

For example, they can all be factorised into a single-digit integer and an integer with all its digits 1 (I will leave the proof of this to the reader!)

Furthermore, the integers with all digits $1$ are of the form

$\dfrac{10^n - 1}{9}$ where $n>0$.

So we can say that according to our new definition of "even numbers", "even numbers" are of the form $m\left(\dfrac{10^n - 1}{9}\right)$ , where $n>0$ and $0 < m < 10$.

How do you like my new definition of "even numbers"?

Answer 10

Of course definitions, to become accepted as standard useful concepts, undergo some kind of testing process, by examining their consequences to see if they correctly express what was meant to be captured by the definition. This is a more subjective process than proof in the sense of "formally proving theorems", but it does correspond to the original English language meaning of prove as in test, check, verify, attempt to refute.

Unlike theorems, definitions do not come with an objective (or anywhere near objective) notion of correctness. The only judgement to be made about a definition is whether to use it or not. Mapping out the consequences of the definition is way of testing if the definition is effective. Experience in using competing definitions can sort out which ones to keep and which ones to avoid in the long term.

Answer 11

How simple (or intuitive) does something have to be to become a definition?

I don't think the difference between a definition and a theorem is a measure of how true that statement is (or is expected to be). Yes, definitions require motivation; they had to occur to someone. But to me it has more to do with how much you can do with a set of definitions; how far you can extend them; and how much beautiful results one can draw out of them. The crux of mathematics to me is in this process. I believe that is what we study in this discipline. The art(or science) of drawing meaningful and useful conclusions from a set of definitions (or axioms, I choose not to distinguish).

I am starting to learn Topology and these two statements intrigue me.

• A subset $A$ of a metric space $M$ is closed if for every convergent sequence $(a_n)$ of points in $A$ the limit $a$ of $a_n$ lies in $A$
• A subset $A$ of a metric space $M$ is closed if it contains its boundary.

Some treatments give the first statement as a definition and prove the second as a theorem using the first. Some treatments introduce the latter as the definition and prove the former using it.

According to me, what is interesting is purely the process of arriving at one using the other.

The investigation of this process to me is what Mathematics is all about.

"Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true" - Bertrand Russell

So it's not about proving everything you suspect to be true. It's about how much one can do by assuming a few things to be true.

---

Disclaimer: This is very much an answer from an amateur. No heed is given to formal matters of philosophy, logic and set theory.

Answer 12

I may be over-reacting to one sentence in your question ("It seems like some definitions should have a foundation based on proof."), but perhaps you are troubled that a definition might be inherently contradictory.

A rational person will not define something that obviously leads to a contradiction. However, if a set is defined as a collection of objects, you can get all of Cantor's original results before you run into a contradiction that is traceable back to the original definition.

Another way a definition "should have a foundation based on proof" occurs when, for example, you were to consider all functions which satisfy some particular constraint, and then define the yungen-value of such a function to be the global maximum of the function. For the definition to make sense you would need to prove that the constraint actually forces the function to have a global maximum.