## Background

You have undoubtedly noticed that in college we don’t use the two column proof method you (may) have seen in high school geometry. Here is a particularly egregious example of it.

 To prove $1 = 0$ Proof $0 \times 1 = 0 \times 0$ Multiply both sides by the same thing $0 = 0$ Zero times anything is zero q.e.d.

To avoid such errors we prefer to treat a proof as an argument designed to convince the reader, not as a ritual exercise for which you fill in the details.

So, there are many possible formats for a proof. But one thing a proof must be, it must be understandable. In particular, it must consist of complete sentences: subject and predicate, just as you learned in English class.

But mathematics consists of notation that abbreviates common speech. In particular, an equation is a complete sentence. It might be true, false, or conditionally true, for some values of any variables it contains. Other such sentences involve inequalities among numbers, or set membership, $x \in S$.

## Propositions

In mathematics we have a particular fondness of sentences that have an additional property, namely that it is possible to decide whether they are true of false. When pressed, we call such sentences propositions. For example $\sqrt{4}=2$ is true. But, regardless how pretty is looks

 $x = \frac{-b \pm \sqrt{b^2 -4ac}}{2a}$ (1)

this equation is neither true nor false, it all depends on the variables. For instance

 $ax^2 + bx + c = 0 \implies x = \frac{-b \pm \sqrt{b^2 -4ac}}{2a}$ (2)

is almost a proposition. Note that $x$ occurs on both sides of the $\implies$ sign. So it’s a dummy variable. But the parameters $a,b,c$ are still "floating". And even adding something reasonable like like $(\forall a,b,c \in \mathbb{R})$ isn’t enough unless we allow complex numbers.

## Quantifiers

In the previous paragraph we reminded you of quantifiers you learned about in a college course on elemntary logic, like MA347 at Illinois. In a formally quantifies proposition we prevent ambiguities by using the $\forall \exists$ statements in parentheses, at the beginning of a proposition, also in parentheses.

In practice, we’re usually less fussy, and append some conditions at the end of a sentence, or even leave it to the context of the discussion. So, you might what to do.

Here is a principle which you can apply as you evaluate your own argument:

## Mathematical Notation

You will be expected to use correct mathematical notation. For this we have some rudimentary text authoring tools. However, even though we read certain symbols using common phrases, the symbols should never be used instead of these phrases, outside of a mathematical proposition.

Thus, do not use $=, \exists, \implies$ in an English sentence instead of the verb "is", or when you mean "for some", or "implies".

In the same breath, please do not use an arrow when you don’t know anything better. Even arrows have specific meanings within their contexts, as for limits in calculus.

Don't replace common English phrases with a mathematical symbol that happens to be "pronounced" the same way.

## Logical argument

Of course your arguments seem logical to you. But perhaps not to the reader. So we build our arguments up in a logical order. You don’t necessarily have to obey the King’s injunction to Alice:

Start at the beginning, and go till you come to the end; then stop.

But to jump around the argument too much is bad too. Sometimes we work from the hypotheses a while, and then work backward from the conclusion a bit, until we meet in the middle, like building the transcontinental railway. But if you do this, you must say so. Such prefaces as

-To prove ...
-This follows if ...

establish the clues for where that line fits into your argument.

## Gaps in the arguments

It is never possible to write out every possible reason and step in an argument. Some, or even much of it must be "left to the reader" to fill in. Sometimes, this ends up being impossible. Then the proof is said to have developed a gap. Even very famous proofs begin life with gaps that other mathematicians become famous for in filling the gaps.

However, you are just beginning to be mathematicians. So the gaps in your proof will be of a different kind. They will be unwarranted inferences for which you do not in fact know the reasons. You’re "jumping to conclusions". How do I know that? Well, experience. Is it always true? No, I’m not mind reader. So, here’s the trickiest principle of them all.

## What Exactly Are You Doing?

The reader is never in a good position to guess what you’re doing. Even if the problem is clearly stated in the book, or the notes, you need to restate it, preferably in your own words. Only then can the reader tell what you are doing. Thus problem solutions like these are not acceptable

• The answer to problem 1.23.b is yes, because we did this in class.

You say what the problem asks you to prove, and then you prove it.

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