# Why do we not have to prove definitions?

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."

• What does it mean to "prove" a definition? A proof is a demonstration of the truth of a certain claim about something. Definitions are not claims; just like cats are not Dolphins. – user230734 Aug 16 '15 at 23:54
• "Proving" a definition makes no sense, since a definition is a decision to introduce and use a particular concept. But mathematicians have not yet advanced to the point where a motivation of each definition, conforming to the rules of the logic of motivation, follows definitions in the way in which proofs follow theorems. ${}\qquad{}$ – Michael Hardy Aug 17 '15 at 0:18
• Definitions are motivated, I think, not proved. – Akiva Weinberger Aug 17 '15 at 0:31
• @Zduff that's correct. Definitions are like the entries of the mathematical dictionary. They are simply stated facts and don't have any deeper meaning in and of themselves, but when we start putting them to use, we can make some really nice things out of them. – Cameron Williams Aug 17 '15 at 1:48
• Because otherwise it would be turtles all the way down! Also, see the third option in this trilemma – James Webster Aug 17 '15 at 7:33

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.)

• +1 Adding to the confusion is the fact that students are very likely to encounter different definitions for common objects in different textbooks: in one class real numbers are complete "by definition," in the next they satisfy the last upper bound property "by definition," and so on. The student is left with the impression that there is a platonic concept of a "real number" with a hodgepodge of properties, some of which require proof and some of which don't, and no obvious difference between the two. – user7530 Aug 17 '15 at 0:38
• If I ever write a text book, I shall include the term "teensy" as you define it. – PyRulez Aug 17 '15 at 2:44
• 'I don't know what you mean by "glory",' Alice said. Humpty Dumpty smiled contemptuously. 'Of course you don't — till I tell you. I meant "there's a nice knock-down argument for you!"' 'But "glory" doesn't mean "a nice knock-down argument",' Alice objected. 'When I use a word,' Humpty Dumpty said, in rather a scornful tone, 'it means just what I choose it to mean — neither more nor less.' 'The question is,' said Alice, 'whether you can make words mean so many different things.' 'The question is,' said Humpty Dumpty, 'which is to be master — that's all.' -- Lewis Carroll,ThroughtheLookingGlass – Jeffrey Bosboom Aug 17 '15 at 22:43
• @JeffreyBosboom: Humpty Dumpty is actually not justified in saying that, because any language is something that is a common consensus between different people to use certain lexical units to denote certain grammatical or semantic concepts. The Egg therefore cannot claim to be free to choose whatever meaning he likes for his words, otherwise communication would utterly break down. What if his "mean[t]" actually means "do[es]/did not mean"? The Egg is just being a proud character who falls. – user21820 Aug 18 '15 at 4:30
• Except that Humpty Dumpty redefined some words, such as portmanteau, and his definition has since stuck. So he could do what he wanted with his definitions, and often did so successfully – Henry Aug 18 '15 at 12:51

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.

• +1, gets to the heart of the issue, deservesmore upvotes. – 6005 Aug 17 '15 at 15:22
• I want to add that there is a sense in which some "proof" is required in association with a definition, usually in order to make sure that the definition "makes sense". For example, we can define the degree of a polynomial as the index of the largest nonzero term, and this certainly defines an expression, but to show that it has any meaning at all, say to prove that this yields a nonnegative integer, we will need to prove that there is a largest nonzero term (and in the course of that proof you will have to assume that the polynomial is nonzero), which clarifies the "domain" of the definition. – Mario Carneiro Aug 17 '15 at 23:33
• @MarioCarneiro: Yes, but it depends on the exact rules of the formal system. If you always prove unique existential quantification before you define something to be that, then you can indeed define something that uses an instantiation of that. Of course we should intuitively devise definitions backward in the way you describe, but that does not mean that a formal proof must be written backwards. They are two separate things. Since there are a lot of such finer details in any formal system, I didn't mention any of them in my answer. Certainly one has to be very careful when actually doing it. – user21820 Aug 18 '15 at 4:25
• +1 for this wonderful answer. The need for definitions has perhaps never been expressed in so concise and clear manner. – Paramanand Singh Nov 7 '17 at 14:23

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.

• I wouldn't go so far as to say it isn't a mathematical statement. It can even be thought of as an axiom, though it is an axiom involving a new symbol, so the axiom is logically independent from all previous axioms. Many definitions are stated as axioms--e.g., whether you include $=$ in the axioms of set theory or whether you define later two sets to be equal if they contain the same elements. – 6005 Aug 17 '15 at 15:20
• @6005 It isn't a mathematical statement because a statement is a sentence that must be either true or false. – Michael Burr Aug 17 '15 at 16:29
• Certainly it can be regarded as true or false--just involving a previously-undefined symbol, so its truth value is independent of previous assertions. I'm not just arguing here--this is literally the way definitions can be formalized in, say, model theory. How could a mathematical definition not be a mathematical statement anyway? And again see my set theory example. – 6005 Aug 17 '15 at 16:31
• Here's a definition of $1$: $1 := S(0)$. What it is is an axiom. I am asserting that $1$ means $S(0)$, and from this point on will take the statement $1 = S(0)$ to be true. The reason we don't prove definitions is that we take them to be true by default, not because they aren't syntactically true-or-false. – 6005 Aug 17 '15 at 16:33

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?

• 1 + 1 = 2 may not be as simple as you think. Whitehead and Russell spent hundreds of pages on it blog.plover.com/math/PM.html – Brice M. Dempsey Aug 17 '15 at 7:33
• @JamesT.Huggett But that depends on which definition you choose to have for $2$. – JiK Aug 17 '15 at 12:04
• @Jik Which I think is precisely Huggetts point. 2 is very seldom defined as the result of 1+1. The closest you get is that 2 is defined as the sucessor of 1. – Taemyr Aug 19 '15 at 11:16
• @JamesT.Huggett It's inaccurate to say that Principia Mathematica spends hundred of pages proving that 1+1=2. They spendt most of those pages setting up the language in which they are able to prove that 1+1=2. – Taemyr Aug 19 '15 at 11:19

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.

• But how is a definition different from an axiom? – user117644 Aug 16 '15 at 23:50
• An axiom, as I understand it, is something which is considered to be uncontroversially true, whereas a definition is merely an assignment of meaning to a collection of symbols or words, with no assumption of truth. – Ben Sheller Aug 16 '15 at 23:54
• I would be careful with saying "uncontroversially true", but perhaps it's better to say that it is an accepted truth. – Cameron Williams Aug 17 '15 at 1:54
• @mistermarko: See my answer where I explain what Ben means by assigning meaning to an expression. – user21820 Aug 17 '15 at 2:18
• Also, I would not even say what Cameron says. An axiom in a formal system is merely a given statement that can be used as a true statement. In other formal systems that axiom may not be given, or even its negation may be given instead. Of course, we try to choose a formal system with axioms that are generally accepted (especially if they seem to accurately describe the real world), but there is nothing wrong with considering other formal systems with different axioms (and possibly different rules as well). – user21820 Aug 17 '15 at 2:21

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.

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.).

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.

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"?

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.

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.

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.

Definitions are not within math, they within linguistics, on the edge of math.

Saying "A is true of B by definition" simply states that you are assuming (an axiom) that the subject B has a property A, and that you will not be trying to prove it. If the listener does not agree that B has a property A, then the listener needs to stop right there and say "this proof may be valid, but its underlying assumptions do not match mine, so I cannot apply the conclusion to my understanding of what B is without digging further into that assumption."

• That's an odd statement to make. – Asaf Karagila Aug 17 '15 at 15:36
• @AsafKaragila What's odd about it? – Cort Ammon Aug 17 '15 at 18:09
• If you say "A is true of B by definition", you are not assuming anything. The statement A is either true or false depending on what the definition of B says. Of course we often write "A is true of B by definition" when we really mean "We can prove that A is true from the definition of B, but the proof is straightforward and uninteresting, and we are not going to give it in full here because the reader should be able to prove it easily for him- or herself." – alephzero Aug 17 '15 at 18:43
• @alephzero Ahh, I'm used to the latter case being "an exercise for the reader." I'm used to saying "And because the sum of the sequence of numbers X grows without bound, its sum approach infinity, by definition." That definition implies that the listener agrees with your definition of natural numbers. A transfinitist may find your definition invalid, unless they are aware you are using a different set of words than they are. As a rule, I find making a claim by definition rather than by proof quite distasteful, because that's just one more place an error can creep into the math. – Cort Ammon Aug 17 '15 at 19:13
• I think the only thing wrong with saying "And because the sum of the sequence of numbers X grows without bound, its sum approach infinity, by definition." is the two words "by definition". I still remember some good advice on exam technique at university: never write "it follows from X that Y". If you just write "Y", and Y is true, the grader in a hurry will probably give you the credit, even if Y doesn't follow from anything else in your proof ;) – alephzero Aug 17 '15 at 20:26

## protected by Asaf Karagila♦Aug 19 '15 at 16:55

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