Posted 9 April 2008
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 many books and articles on the web about creating proofs.
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Contents
Presentation of proofs. This is a short general article about how proofs are written in math texts.
Mathematical reasoning. This is about the logical structure of proofs, and for each type of structure, how it is presented in a narrative proof.
Forms of proof. There are several basic logical
forms of proof (modus ponens, contradiction, contrapositive and so on). This section describes the forms and also the
conventions used in writing proofs in the various forms.
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.
Each proof step must follow from
¨ previous proof steps,
¨ definitions, or
¨ theorems that have already been proved.
The proof step must follow using a correct logical method
of deduction, covered in detail in Mathematical Reasoning.
Presentation
Most proofs at the college level or higher are written in narrative
form, resembling an essay. To read
such proofs you need 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.
Status
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.
