The Language of Mathematics

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 1 March 2010

The grammar of mathematics

• Definitions are the foundation of mathematics.
• Theorems are the landmarks of mathematics.
• Proofs are the explanation of mathematics.

Learning to read, write and speak mathematics is a skill that anyone can learn. Like all languages, it requires lots of practice to use it fluently.

Like all languages, the grammar of quality mathematical communication is very rigid. The grammar of a definition is:

1. A noun is a       such that
   1. (a) If       then      , and
   2. (b) If       then      , and
   3. (c) If       then      , and ...

An adjective is most conveniently defined by putting it in the form of a noun:

1. An adjective noun is a       such that
   1. (a) If       then      , and
   2. (b) If       then      , and
   3. (c) If       then      , and ...

Sometimes definitions of adjectives take the form:

1. Let S be a noun.
2. The noun S is adjective if S satisfies
   1. (a) If       then      , and
   2. (b) If       then      , and
   3. (c) If       then      , and ...

The words “let” and “assume” are synonyms for “if”. The grammar of a lemma, proposition or theorem (or any other statement) is:

1. If       then      .

Two special constructions in mathematical language are:

1. There exists       such that      .

and

1. There exists a unique       such that      .

It is impossible to prove a statement without being able to write down the definitions of all the terms in the statement.

How to do proofs

There is a certain “formula” or method to doing proofs. Some of the guidelines are given below. The most important factor in learning to do proofs is practice, just as when one is learning a new language.

1. There are very few words needed in the structure of a proof. Organized in rows by synonyms they are:

 

To show

Assume, Let, Suppose, Define, If

Since, Because, By

Then, Thus, So

There exists, There is

Recall, We know, But

 

2. The overall structure of a proof is a block structure like an outline. For example:

 

To show: If   A   then   B   and   C  .

Assume:   A  .

To show:

(a)   B  .

(b)   C  .

(a) To show:   B  .

               .

               .

               .

     Thus   B  .

(b) To show:   C  .

               .

               .

               .

     Thus   C  .

So   B   and   C  .

So, if   A   then   B   and   C  .

 

3. Any proof or section of proof begins with one of the following:

    

   (a) To show: If   A   then   B  .

   (b) To show: There exists   C   such that   D  .

   (c) To show:   E  .

    

4. Immediately following this, the next step is

    

   Case (a) Assume the ifs and `to show' the thens. The next lines usually are

   • Assume   A  .
   • To show:   B  .

    

   Case (b) To show an object exists you must find it. The next lines usually are

   • Define   xxx  .
   • To show:   xxx   satisfies   D  .

    

   Case (c) Rewrite the statement in   E   by using a definition. The next line is usually

   • To show   E'  .

A useful guideline is, “Don't think too much.” Following the “method” usually produces a proof without thinking. Most of doing proofs is simply rewriting what has come just before in a different form by plugging in a definition.

 

There are some kinds of proofs which have a special structure.

Proofs of equality.

To show:   A=B   .

Left Hand side:   A = ⋯   

                    = ⋯   

                    = ⋯   

                    = ⋯   

                    = expression   .

Right Hand side:   B = ⋯   

                    = ⋯   

                    = ⋯   

                    = ⋯   

                    = the same expression   .

Proofs by contradiction.

(*) Assume the opposite of what you want to show.

               .

               .

               .

End up showing the opposite of some assumption (not necessarily the (*) assumption).

Contradiction.

Thus Assumption (*) is wrong and what you want to show is true.

Counterexamples.

To show that a statement, “If        then       ”, is false you must give an example.

To show: There exists a        such that

(a) it satisfies the ifs of the statement that you are showing is false.

(b) it satisfies the opposite of some assertion in the thens of the statement that you are showing is false.

Proofs of uniqueness.

To show that an object is unique you must show that if there are two of them then they are really the same.

To show: A THING is unique.

Assume X₁ and X₂ are both THINGs.

To show: X₁ = X₂.

Proofs by induction.

A statement to be proved by induction must have the form

If n is a positive integer then   A  .

The proof by induction should have the form

Proof by induction.

Base case:

To show: If n = 1 then   A  .

               .

               .

               .

Thus if n = 1 then   A  .

Induction step:

Let N be a posititive integer and assume that if n is a positive integer and n < N then   A  .

To show:   A  .

This last to show line contains exactly the same statement A as in the original statement except with n replaced by N.

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