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Produced by Charles Wells      Revised 2014-12-27
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CONTENTS

Proofs

The nature of proofs

The role of proofs in math

Presentation of proofs

Forms of proof

Mathematical reasoning

And, or, not

Conditional assertions

Universally true assertions

Existence statements

PROOFS

This is the top page of the chapter on Proofs.

Proofs cause problems to people new to abstract math:

• To verify that a mathematical statement is correct, you must prove it. Evidence does not count in math.
• What is a proof? Many newcomers to abstract math don't know that they don't know what a proof is.
• How can you tell if a proof is correct? Proofs are based on mathematical logic, but they are usually presented in narrative prose and do not mention many of the rules that justify the reasoning.
• How do you create a proof? Abstractmath.org does not go into this very much. There are good books and articles about creating proofs.

The nature of proofs

A proof is an argument intended to persuade other mathematicians of the correctness of a statement. A proof has two main aspects: Its logical structure, and its presentation (the way it is written).

Logical structure

A proof has a precise logical structure:

• A proof is a series of statements that I will call proof steps.
• Each proof step must follow by a correct logical method of deduction from
• previous proof steps,
• definitions, or
• theorems that have already been proved.

The logical methods of deduction are described briefly in Mathematical reasoning. Some of the ways these methods are expressed in written proofs are described in Forms of Proof.

Presentation

Most proofs at the college level or higher are written in narrative form, resembling an essay. To read such proofs you must learn how to extract the logical form of the proof from the narration. (This is called the translation problem.) To do this, you must be familiar with the conventions used to write narrative proofs and with the rules of logical deduction, since most simpler deduction rules are used without mentioning them. The articles Presentation of Proofs and Forms of Proof gives some examples of how narrative proofs work.

The role of proofs in math

At the beginning of this article, I wrote: To verify that a mathematical statement is correct, you must prove it. Evidence does not count in math. This section spells out in more detail what I mean.

The only way to verify that a claim about mathematics is correct is to prove it. Numerical evidence or the fact that the claim is true in some physical situation is suggestive but is not a verification.

See Evidence that fools you for some examples.

Mathematical proofs are public documents that can be challenged and defended using known principles. Any step in the proof can be challenged and a defender of the proof must analyze the step in more detail to show that it is correct - or to discover that in fact the step is incorrect and the proof is wrong!

This description of proofs shows how math is a part of science.

• Every claim in math can be tested: Is there a proof of the claim?
• The idea of proof is so precise that something that claims to be a proof can be checked to see if it really is a proof.

So the proof plays a role that is in some ways analogous to the role of experiments in other sciences. (This analogy should not be pushed to far, though.)

It is an observable fact that when mathematicians argue about whether some proposition is correct, they may argue heatedly for awhile but eventually the argument comes to an abrupt and total end: One party realizes that they have made a mistake, or both parties realize they had different definitions for one of the concepts they were arguing about.

Arguments among specialists in (say) literature or history are not like that - such arguments can and do go on for the lifetime of the participants.

Evidence that fools you

• The Perrin function $P:\mathbb{N}\to\mathbb{N}$ has the property that for the first $271,440$ integers, if $n$ divides $P(n)$, then $n$ is a prime. However, $271,441$ divides $P(271,441)$, but $P(271,441)=521^2$.
• Let $\pi(n)$ be the number of primes less than $n$. The prime number theorem (in one form) says that $\pi(n)$ is approximately \[\text{Li}(n):=\int_2^n\frac{dx}{\ln x}\] In fact, for every value of $n$ that we know of, $\pi(n)\lt\text{Li}(n)$, although it is known that that there is a number $N\lt 10^{371}$ such that $\pi(n)\gt\text{Li}(n)$. The number of atoms in the universe is estimated to be $10^{82}$.

References on how to come up with proofs

• How to Read and Do Proofs, by Daniel Solow. Fourth Edition, John Wiley and Sons, Inc, 2005. Much of the abmath article on Mathematical reasoning is drawn from this book.
• Proofs in mathematics, by Alexander Bogomolny. Many good examples and discussions.
• Tim Gowers' What is deep mathematics? has a detailed discussion of how you come up with two proofs concerning continuity (one easy and one hard).
• Useful behaviors for proofs on this website.
• Velleman, Daniel J. How to Prove It: A Structured Approach. Cambridge University Press, 1994.
• Wikipedia on mathematical proof




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