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Dr. Roger Roybal
Math 331
03/28/2017
Proof Based Mathematics
Proofs are one of the most important element of mathematics. In mathematics, proofs are a
sequence of logical, mathematical statements which have been previously proven, which lead to
a sought conclusion, called the claim. Proofs are used to irrefutably show that a claim is true. In a
proof based mathematical world, any claim which is not entirely true in every possible case is
false. It is important that one understands that in these modern times all of mathematics is proof
based. Without the certainty of proof based mathematics all theorems and claims are subject to a
certain level of doubt, as they may stand on incorrect assumptions or false conclusions.
In ancient times, there were many cultures which used non-proof based mathematical
systems. For example, the mathematicians of ancient India had no knowledge of nor interest in
the Greek concept of proof. They instead focused on computation, developing many methods for
arithmetic calculation that we still use today. While the lack of the restraints that proof based
mathematics places upon a mathematical system or method may leave more room for creativity,
it often results in developments which are incorrect or incomplete. For example, the Indian
mathematician Āryabhata gave the volume of a pyramid as half the product of the base and the
altitude[Eves], when it is in fact one third the product of the base and the altitude.
The ancient mathematicians of Classical Greece are well known as the creators of proof
based mathematics, starting with Thales of Miletus, and culminating with Euclid’s Elements.
Ancient Greek mathematics utilized proofs to push their knowledge of geometry to new heights
unattained by their contemporaries the world over. The Greeks operated almost entirely through
geometry, and as such lacked any major developments in algebra.
An example of a geometric proof in the style of the ancient Greeks is as follows:
The sum the first n cubes is equal to a square grid of unit cubes with sides equal to the sum of
the first n numbers. One can see that an n cube is composed of n many square grids of unit cubes
with sides equaling n. One can re-arrange the cubes to form the square grid of unit cubes as
claimed. [Proofs]
(Image by Wikimedia hobbyist illustrator Cmglee, Creative Commons) [Cmglee]
With the knowledge that this claim has been proven we can easily simplify related
problems with this knowledge. For example, say one wanted to weigh four cubes of solid marble,
each with sides longer than the last by the length of the side of the smallest cube. One could
simply weigh the smallest cube and multiply by the square of the sum of the number of cubes, in
this case the weight of the smallest cube multiplied by sixteen. If one tried to solve the same
problem intuitively one might guess any number that seems reasonable. This may result in a
quicker answer, but if the intuitive solution is wrong it may cause more harm than help.
There are a great many advantages to utilizing proof based mathematics over non-proof
based mathematics. Above all, proof based mathematics can produce results that are certainly
true. If the solution obtained is incorrect then a mistake was undoubtedly made. The only
advantage of solving a problem by intuition, other than speed, is the ability to get a solution,
though possibly an incorrect one, when current mathematical knowledge is not enough to lead
one on a path to the solution. If one must solve a problem without the knowledge needed to
produce a correct result, then intuition may lead you to a solution, which may be better than
leaving the question ‘unsolved’.
One should always aim to solve problems with proofs rather than by intuition, especially
students attempting to solve problems like the one discussed previously. Solutions by proof
result in knowledge, and the acquisition of knowledge is the purpose of the student.
Works Cited
Cmglee. "User:Cmglee." Wikipedia. Web. <https://en.wikipedia.org/wiki/User:Cmglee>.
Eves, Jamie H. "Chapter 7 - Chinese, Hindu, and Arabian Mathematics." An Introduction to the
History of Mathematics. 6th ed.: Saunders College. 228. Print.
"Proofs Without Words." Art of Problem Solving. Web.
<https://www.artofproblemsolving.com/wiki/index.php?title=Proofs_without_words>.