Math Number: 280
Project 1: Greek Proof-Based-Mathematics
In high school, students are introduced to the concept of a mathematical proof. Many
students do not know why they are proving geometry theorems, but do it nonetheless. From then
on, the idea of proving something in math lingers over their heads, but they still do not exactly
know what that means. This paper aims to explain what proofs are and why proof-based
mathematics are important by using examples from ancient history and also present more
modern proofs.
A proof employs basic mathematical truths in conjunction with logic to show specific
mathematical relations and identities to be true in all circumstances. Different types of proofs
rely on different logical structures. However, all proofs stem from the desire to know why
mathematical relations and identities are true.
Proof-based mathematics differs from non-proof -based mathematics primarily in its
central focus. As mentioned above, all proofs stem from a desire to know why certain
mathematical truths are true instead of just employing these truths. This facet of proof-based
mathematics can especially be seen in the advent of Greek mathematics. Howard Eves notes that
this characteristic stems from the fundamental Greek mindset itself when he says, “The empirical
processes of the ancient Orient, quite sufficient for the question of how no longer sufficed to
answer these more scientific inquiries of why” (Eves 72). In other words, Eves here notes that
the primary way the world did math could not satisfy the curiosity of Greek mathematicians.
Their desire to know “why” rather than just “how” impelled the Greeks to develop a proof-based
mathematics that would naturally be set apart from any type of empirical mathematics.
As such, Greek mathematics benefitted from using proofs because they provided for a
systematic way to investigate, record, and propagate mathematical truths. These benefits of
proof-based mathematics are easily seen in the lasting fame and applicability of Euclid’s
Elements of Geometry. Many high schools still use theorems found in Euclid to teach elementary
geometry (Eves 145). The fact that the proofs in Euclid’s elements have withstood the testing of
thousands of years and thousands of mathematicians indicates that the way Greek
mathematicians approached proofs benefited them by allowing them to arrive at true conclusions
in a systematic way repeatable by other mathematicians.
The benefits of proof-based-mathematics also hindered the Greeks by proving previously
held philosophical worldviews false. For instance, when the Pythagorean school proved that
irrational numbers exist, they immediately knew that their entire definition for a whole-numberbased proportion was flawed because of how strong the proof was for irrational numbers. In fact,
the Pythagoreans knew that the evidence for their incorrect reasoning was so strong that they hid
the evidence for years because of the logical scandal it produced (Eves 84). In this way, proofbased mathematics hindered Greek mathematics because showing that a conclusion is logically
necessary based upon given premises does not mean the premises are entirely correct. In other
words, the Greeks had to be extremely careful that their proofs relied upon absolute truth and not
speculation. As seen in the Pythagorean example above, this task was complicated by not
knowing when to question foundational beliefs.
To better grasp what a proof would look like, this paper will now demonstrate how
someone could go about proving mathematical identities involving π. For example,
π/4=arctan(1/5)-arctan(1/239). To prove this equation, which is known as Machin’s formula, one
must also know the series expansion of arctan(x). Plugging in Machin’s formula to the series
expansion of the arctan will quickly show that this is a function that is alternating and
decreasing. Thus, subtracting the sum of the first n terms from pi will be less than the value of
the next term. This works because the n+1 term will always be greater than the difference of pi
and the first n terms. Since each individual term will get progressively smaller, this effectively
means that the difference between them will be how many digits out they will differ, allowing
the mathematician to know the digits of pi out to whatever digit the n+1 term reaches.
(https://www.math.brown.edu/~res/M10/machin.pdf)
Another proof example is of Viéte’s formula. Viéte’s formula is an infinite product that is
represented as (2/pi)=(sqrt(2)/2)(sqrt(2+sqrt(2))/2)(sqrt(2+sqrt(2+sqrt(2)))/2)… The proof for
this relies on finding the ratio between the length of a 2^{n}-gon inscribed within the unit circle
and the length of 2^{n+1}-gon inscribed within the unit circle. The 2^{0}-gon is the diameter
counted twice. Taking this ratio and multiplying together for successive values of n allows one to
find this ratio because pi is a ratio of a circle’s circumference to its diameter. In essence this
process allows for the computing of the ratio of a 2^{0}-gon, which is twice the diameter to a
2^{\infinity}-gon, which is basically the circumference of a circle. The tricky part of this proof is
making sure that the square roots are placed appropriately, because this is an equation for 2/pi
instead of just pi. (Rummler 858).
There are two main advantages to solving these problems with proofs instead of by
intuition. The first is that the calculation of pi is a task that remains extremely difficult to do
without using some ratio of the circumference to the diameter of the circle. Although this is an
intuitive idea, finding the correct decimal values remains challenging. Thus, having a proof to
determine the value of pi allows for a certainty in value even if pi itself is hard to calculate. It is
also difficult to understand why certain portions of the proofs are true. For instance, to derive
Machin’s formula one must be extremely familiar with complex numbers, which is a difficult
concept to understand intuitively. Therefore, creating a proof for pi allows for a more rigorous
and accurate method of calculating pi.
The main disadvantage of solving the value of pi using proofs is that it is somewhat
inaccessible to the man on the street. While pi remains a concept worthy of attention and
scholarship, many have trouble grasping the implications of what it is and why it is important. In
this way, the proof method is disadvantageous to promulgating mathematical truth.
As such, the typical, non-mathematical student would benefit more from intuitively
solving a problem because it would allow for easier application outside of an intellectual
community. It seems that many people do not like math and as such want to use as little math as
possible in their daily lives. Since constructing mathematical proofs not only involves using
basic mathematical truths but also extremely creative thinking within rigorous rules, most
students would not care enough about the math to put in the time to fully learn and understand
what they are doing. However, all students should have to take some more intuitive mathematics
because it trains the mind to think in a way useful for general life. In short, although the purpose
of an education is ever increasingly to produce workers who know what they are doing, it still
retains its old purpose of enhancing the life of the mind and equipping students with tools that
will help them in their life.
In conclusion, proof-based mathematics is important because it allows for the knowledge
of the absolute truths in mathematics in a way that is easy to repeat and distribute to other
mathematicians.
Works Cited
Eves, Howard. An Introduction to the History of Mathematics. Thomson, 1990. Print.
https://www.math.brown.edu/~res/M10/machin.pdf
Rummler, Hansklaus. “Squaring the Circle with Holes” The American Mathematical Monthly,
vol. 100, No. 9, 1993