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conceptual
A proof is conceptual if it is an argument that makes use of one's mental representation or geometric
insight. It is opposed to a proof by symbol manipulation.
Example
Let m and n be positive integers, and let r be m mod n. One can prove that
GCD (m,n)= GCD (m,r)
by showing that the set of common divisors of m and n is the same as the set of common divisors of n
and r (easy). The result follows because the GCD of two numbers is the greatest common divisor, that is,
the maximum of the set of common divisors of the two numbers, and a set of numbers has only one maximum.
I have shown my students this proof many times, but they almost never reproduce it on an examination.
Example
Now I will provide three proofs of a certain assertion, adapted from [].
The statement to prove is that for all x, y and z,
(
(x>z)⇒((x>y)∨(y>z))
1)
Conceptual proof
We may visualize x and z on the real line as in this picture:
There are three different regions into which we can place y. In the left two, x>y and in the right two,
y>z. End of proof.
This proof is written in English, not in symbolic notation, and it refers to a particular mental representation of the structure in question (the usual ordering of the real numbers).
Symbolic Proof
The following proof is due to David Gries (private communication) and is in the format advocated in
[]. The proof is based on these principles:
P
(Contrapositive) The equivalence of P⇒Q and ¬Q⇒ ¬P.
(DeMorgan) The equivalence of ¬(P∨Q) and , ¬P¬Q.
The equivalence in any totally ordered set of ¬(x>y) and x≤y.
In this proof, " ¬" denotes negation.
Proof: ,,
(x>z)⇒((x>y)∨(y>z)) ≡
by P.*
¬((x>y)∨(y> z))⇒¬(x>z) ≡
y)¬(y>z))⇒¬(x>z) ≡
by P.* three times
((x≤y)(y≤z))⇒(x≤z)
by P.*
(¬(x>
which is true by the transitive law.
This proof involves symbol manipulation using logical rules and has the advantage that it is easy to
check mechanically. It also shows that the proof works in a wider context (any totally ordered set).
Another conceptual proof
The conceptual proof given above provides a geometric visualization of the situation required by the
hypothesis of the theorem, and this visualization makes the truth of the theorem obvious. But there is a
sense of "conceptual", related to the idea of conceptual definition given under elementary, that does not
have a geometric component. This is the idea that a proof is conceptual if it appeals to concepts and theorems at a high level of abstraction.
To a person familiar with the elementary rules of first order logic, the symbolic proof just given becomes a
conceptual proof (this happened to me): "Why, in a totally ordered set that statement is nothing but the
contrapositive of transitivity!" Although this statement is merely a summary of the symbolic proof, it is
enough to enable anyone conversant with simple logic to generate the symbolic proof. Furthermore, in my
case at least, it provides an aha experience.