Empirical Proof Schemes
Human concept formation is based on observing examples. So it is natural to use examples as a means of getting a "feel" for a mathematical statement. But there is an incorrect use of examples. That is to use them as the basis for justification of a statement. Such a use produces an empirical proof. Sowder places empirical proof schemes under two types: The perceptual proof scheme and the example-based proof scheme.
The perceptual proof scheme
George Polya says: "If you have to prove a theorem, do not rush. First of all, understand fully
what the theorem says, try to see clearly what it means. Then check the
theorem; it could be false. Examine the consequences, verify as many
particular instances as are needed to convince yourself of the truth.
When you have satisfied yourself that the theorem is true, you can start
Yes, this is the use diagrams must be put to, unless they are needed in a proof as an illustration. Beginners in mathematics sometimes go a step further. They use the geometric figure or illustration to justify the theorem or statement.
This is the perceptual proof scheme - the tendency to reason from geometric and other diagrams. A valid proof must begin from given assumptions and use axioms, and statements that are already proved.
A diagram or illustration is only an aid. The old dictum is to the point: "Geometry is the ability to reason well from false diagrams." Perhaps, the old dictum is a bad translation of a good Greek wording. Let us say: The discipline of geometry entails the ability to reason well from false diagrams.
Examples-based Proof scheme
Examples are the greatest aides to reasoning. One observes the patterns and relationships, makes conjectures, and proceeds to construct arguments. An example is usually a particular case of a problem. The constructed proof is what establishes it for the general case.
Claiming that the illustration provided by an example suffices as proof is an Example-based Proof scheme. For instance, the polynomial n2-79n+1601 = (n-40)2 + (n-40) + 1 produces primes only for 0≤n≤79. So if you checked only values of n among 0 to 79, and stated that the expression yields only primes for all n would not be a valid proof. Incidentally, to check that m is prime, one only needs to check that m is not divisible by primes up to √m (why?). It is part of the mathematics folklore that no non-constant polynomial f(n) with integer coefficients can be prime for all n, or for all sufficiently large n.
"For all sufficiently large n" means that we can prove the existence (we don't have to find it!) of an integer N so that for all n > N, the polynomial f(n) will not be always prime.
Let us a take a more dramatic example! .
Is 1 + 1141n2 a perfect square, for some positive integer n?
If you tried illustration as convincing proof, you will be out of luck: the expression is not a perfect square for integers 1 through 30693385322765657197397207, but it is a perfect square for the next integer! (I hope I have typed the digits right; but you can check by finding the square root of the integer above the one given.)
Trusting a pattern should only be used as an encouragement to work out a proof. So what is a proof? That is our discussion below.
A valid mathematical proof is of this type. Sowder classifies these under two types: transformational proof schemes and axiomatic proof schemes. I think mathematicans, in general, would not disagree with this classification. I say this because I can not prove the contrary - namely that there are other types of proofs.
Transformational proof scheme
This type of proof is an argument appropriate to the problem but set, often, in a more general context. As an example, let us show that f(n)= n2-79n+1601 is not always a prime for all sufficiently large n.
Let N be an integer greater than 79 and x > N.
f(x) = x2 - 79x + 1601 = y, say.
Note that y > 1 (why?)
f(ry+x) = (ry+x)2 - 79(ry+x) + 1601 is divisible by y for every integer r (why?).
Therefore, since f(ry+x) tends to infinity with r, there are infinitely many composite values of f(n).
This beautiful proof is not this writer's. It is adapted from the proof in Hardy of Theorem 21 in his book - it is the theorem quoted earlier of non-constant polynomial with integer cofficients not producing primes for all n. The problems on a Mathpath Quiz would not entail a situation where only a proof of this beauty or level of technique would provide the solution.
What we need to note here is that the proof did not call in any axioms or theorems except a fact that can be used without explicit mention, namely, that a product of integers is indeed divisible by the factors of the multiplicands.
Many of the proofs required in MathPath Quiz problems are transformational proofs.
Axiomatic proof scheme
This is a procedure by which one proves - irrefutably establishes - a result by calling in present assumptions, prior results, axioms, and definitions, as needed.
It is perhaps the most distinguishing aspect behind a mathematical theorem or proposition. Without it, the statement of the theorem would remain a conjecture.
Want an example of an axiomatic proof? Consider the most celebrated mathematics book of all time - The Elements by Euclid. It has 465 propositions. And each proof in the Elements is an axiomatic proof! (A few of those proofs are invalid due to not having enough axioms to start with.)
The ordering and wording are carefully made in the axiomatic proof. The two-column proof in school geometry is an example of this. The teacher asks for the two-column proof so that the statement in the left column is implied by the one in the right column. Once the student reaches university, or if the teacher knows that the student already writes proofs correctly, then the two-column format is allowed to be merged as a one-column format filling the width of the page.
With this we are not kicking out transformational proofs. They are fine where they suffice! In fact, where they suffice, they would be the shorter and, perhaps, the simplest proof.
P. J. Davis, Are there Coincidences in mathematics?, American Mathematical Monthly 88(1981)311-20
G. H. Hardy and E. M. Wright, An Introduction to
the Theory of Numbers, Fifth Edition, Oxford University Press, 1978
D. L. Medlin, Concepts and Conceptual Structure, American Psychologist 44(1989)1469-81
G. Polya, How to solve It 2nd ed. Princeton University Press, 1957
L. Sowder, Types of Students' Justifications, Mathematics Teacher, November 1998
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Last updated - December 24, 2004