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History of Math #279
Project 1
Concept of Proof: A proof is a set of steps that attempts to clarify a concept in a certain manner
that can be verified and repeated. A person that reads a proof should be able to go through it,
hopefully understanding the steps, and then be able to demonstrate what was proven in a similar
manner. Similar to a theory, a proof needs to be shown that it is true in a way that the outcome
can reproduced over and over again.
Proof based mathematics vs. non-proof based mathematics: One major difference between proof
based mathematics and non-proof based mathematics is the idea about what is good enough to
complete the given equation or operation. In non-proof based mathematics, a sufficient “proof”
that an equation works is that the equation gives the desired results time after time. An example
of this can be seen, for example, when needed to pour cement to create the outer lining of a pool,
a constructor might use integration to find the area that the pool will be and not understand the
proof of integration. In those types of cases, math is being used to do some function, and the
correctness of the concept is “proven” by the results. As we know, it is very unlikely that a
constructor would use an unproven method of calculation, but the way one might use a formula
without understanding how it works shows an example of non-proof based mathematics. One
example we can see of non-proof based mathematics is from the Babylonians.
In the case of the Babylonians, scribes set out tables of figures representing all of the different
possible values for an equation. Though the tables were helpful when making calculations, they
were still pretty inefficient when compared to proof-based mathematics carried out by the Greeks
(Eves). While the Babylonians were able to look at their tablets to determine values, a good
proof would have made them able to understand what would have worked for a general case.
Proof based mathematics allowed for more efficient calculations in addition to providing a more
concrete foundation for the idea.
How Greek mathematics was helped by using proofs and how it was hindered: An obvious way
of how Greek mathematics was improved by using proofs is seen by the amount of mathematical
content the Greeks understood. Past ideas that were useful became concrete concepts that
allowed for further and further development of more advanced properties. Looking at Euclid’s
Elements, we are able to see a logical “flow” of information that leads to a better understanding
of more advanced material. Looking directly at Book IV of Elements, we can see that many
proofs in this book rely on propositions from the previous books. For example, looking at
proposition 10, we can see the Euclid had used propositions from Book III, specifically
propositions 32 and 37, when showing that a given triangle is tangent to a circle (Joyce), as
shown in the following diagrams:
Euclid used his previous propositions to reinforce the further understanding of geometric
properties in this way.
One wonders though, how did proofs hinder the Greeks mathematics? As we can see, Greeks
relied mostly on geometry. This insistence on geometry prevented Greek mathematics from
advancing towards different paths of understanding. Because the Greeks possessed only a small
understanding of irrational numbers, they overemphasized the use of geometry in their
mathematics and associated proofs. As Morris Cline explains in “Mathematical Thought from
Ancient Times”, “Had the Greeks faced the irrational number, they might have furthered the
development of arithmetic and algebra.” The Greeks, relying on geometry remind you, “were
induced to think that only geometry offered a secure foundation for the treatment of any
magnitude whose values might include irrationals”, Cline expands.
An example of a proof: Parting from Euclid’s Elements, let’s take a look at the summation
question:
Prove that: 13 + 23 + … + n3 = (1 + 2 + … + n)2
If we look at each portion individually, we can attempt to understand the whole problem. Let’s
look at the left side of the equation (for 1-4):
As we can see, each “cube” is split into its layers, and then subdivided for the next step of
placing then into a perfect square. Before we get to that, lets do some calculations:
13 = 1
23 = 8
33 = 27
43 = 64
All of them added together, we achieve the following: 1 + 8 + 27 + 64 = 100. Now lets look at
the right side of the equation graphically (from 1-4 again):
As we can see, every rectangle from the left side of the equation is accounted for, thus proving
the formula. For safe measure, let’s do the calculation to make sure it works:
(1 + 2 + 3 + 4)2 = (10)2 = 100
Sure enough, we get 100 again, which matches the left side of the equation. The benefit of
solving this problem by proof is that we now have a clear explanation of why the sum of cubic
areas adds up to the sum of all sides added together and then squared. It sets up a good base for
proving similar problems by using the geometric images to expand their problem. The
disadvantage to solving a problem by proof is that it can take longer and the need for rigor can
mean that even though you can “see” a solution to a problem, that if you do not understand the
formal relationships of its components, there is not much luck in defining an adequate proof.
Better or worse to solve by proof for student: Though solving by proof would lead to a greater
understanding of how an equation works, sometimes it’s very difficult for some students to
visualize how something works. Each student is different and some may be better at proofs than
others, which is why I believe it is better for students to solve problems intuitively. It would be
great if students were asked to solve intuitively and given the option to also solve by proof for
their own benefit. Students who are truly out to better their understanding will take on the
challenge and attempt to solve by proof.
Works Cited
Eves, Howard. An Introduction to the History of Mathematics. Holt, Rinehart and Winston, Inc,
1990.
Joyce, DE. Euclid's Elements. 1996.
Kline, Morris. Mathematical Thought from Ancient to Modern Times, Vol.1. New York: Oxford
University Press, 1990.