Last
edited 12/26/2008 11:02:00 AM
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.
¨ 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? A major
emphasis here at abstractmath.org is on how to recognize a correct proof
when you see it.
¨ How do
you create a proof? Abstractmath.org
does not go into this very much. There
are good books and articles on the
web 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 structure:
¨ A proof is a series of statements that I will call proof steps.
¨ A proof gives a reason that each proof step is correct. (More about this in
Each proof step must follow from
¨ previous proof steps,
¨ definitions, or
¨ theorems that have already been proved.
Methods of deduction
The
proof step must follow using a correct
logical method of deduction. There are several of these: modus ponens,
contradiction, contrapositive and so on.
They are covered in detail 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.
All this is covered (with examples) in Presentation of Proofs.
The role of proofs in math
Here is why proofs are important:
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 an example.
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. In
saying this, I am not dissing literature or history. Well, not very
much.
Mathematical reasoning
Forms of proof
Presentation of proofs
References on how to come up with
proofs
The math reasoning part of this website owes a lot to Daniel Solow’s How to Read and Do Proofs. (Fourth Edition, John Wiley and Sons, Inc, 2005.) This book is an excellent introduction to proofs for those new to abstract math.
¨ Alexander Bogomolny on proofs. 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.
¨ WikiHow on proofs.
¨ Wikipedia on mathematical proof
Appendix: Abraham Lincoln and proofs
In the course of my law reading I constantly came upon the word “demonstrate”. I thought at first that I understood its meaning, but soon became satisfied that I did not. I said to myself, What do I do when I demonstrate more than when I reason or prove? How does demonstration differ from any other proof?
I consulted Webster’s Dictionary. They told of ‘certain proof,’ ‘proof beyond the possibility of doubt’; but I could form no idea of what sort of proof that was. I thought a great many things were proved beyond the possibility of doubt, without recourse to any such extraordinary process of reasoning as I understood demonstration to be. I consulted all the dictionaries and books of reference I could find, but with no better results. You might as well have defined blue to a blind man.
At last I said,- Lincoln, you never can make a lawyer if you do not understand what demonstrate means; and I left my situation in Springfield, went home to my father’s house, and stayed there till I could give any proposition in the six books of Euclid at sight. I then found out what demonstrate means, and went back to my law studies.
(I learned of this passage from John Armstrong’s post on The Unapologetic Mathematician.)