Essay 1) Discuss Euclidean constructions, the three classical problems, and their role in the history of Greek mathematics.
 

Euclidean constructions are the shapes and figures that can be produced solely by a compass and an unmarked straightedge. Although these tools were indeed simple, their range of abilities seemed unlimited. Not only could they produce a multitude of angles and lengths, but also elegant-looking regular polygons and a wide variety of 2-D shapes with desired area. These basic tools seemed to be able to do or produce anything. When, after countless attempts, they were unable to solve the three classical problems of trisecting an angle, doubling the cube, and squaring the circle, the Greeks were forced to reach out to new and more complicated instruments. It was the inadequacy in these three problems that helped make mathematicians realize that some aspects of mathematics could not be done with real world instruments and that sometimes it must rely on purely symbolic methods. Yet, some still held on to the belief that the traditional compass and straightedge could answer all their questions and were persistent in their efforts to find the means. However, after almost 2000 years of use, the limitations of the compass and unmarked straight-edge were discovered by one of the most abstract and symbolic areas of mathematics: field theory and abstract algebra. These subjects showed, once and for all, exactly what the time-honored tools could and could not do. And amongst the things they could never do were the three classical problems.

Essay 2) Discuss Euclid's Elements.
 

Greek mathematics is thought to have reached one of its highest points in the form of Euclid's Elements. For, even though, it is disputed as to how much of the work contained therein is actually original ideas and proofs by Euclid, he did manage to gather together a wide range of knowledge at his time on such topics as planar and solid geometry and number theory. His scheme of organization was a very concise and straightforward format that allowed the ideas to be seen as a logical progression. Further, it was the aesthetic beauty and surety of this logical progression that paved the road to future mathematicians demanding that everything be proved using well-defined objects and agreed-upon axioms.

Essay 3) Give an outline of the history of Greek mathematics from the time of Thales to the collapse of the University of Alexandria.
 

Starting with Thales in the 6th century BC, Greek mathematics was founded in proof. Soon after Thales in the 5th century BC, the Pythagoreans created a whole society dedicated to mathematics and the logic of numbers. Afterwards, one could say mathematics went into some sort of a cocoon, with it only having minor advances between 500 BC and 300 BC. However, it was after this period that the University of Alexandria was founded and mathematics began to flower starting with Euclid's Elements around 300 BC. Another boost of knowledge came from Archimedes, possibly the greatest mathematician of antiquity, in the 3rd century BC. In addition, the centuries to follow would bring forth other great mathematicians such as Eratoshenes, Appolonius, and Hipparchus. However, the great era of Greek mathematics would come to end somewhat abruptly when in 46 BC Caesar accidentally has the university extremely damaged and badly burned. Moreover, even though the university continued until the Arabs destroyed it in 410 AD, it would only produce commentators of mathematics and ceased to produce new and fruitful ideas.

Essay 4) Briefly give the history of logarithms. How has the use of logarithms changed? Why is it still important for students to be familiar with logarithms?
 

The first evidence of simple logarithms was in the Jain's Dhavala commentary where it suggests that this Indian culture may have had logarithms but never put them to any practical use. The next indicator of logarithms came in Michael Stifel's Arithmetica integra in 1544 where he suggested the use of arithmetic progressions to understand geometric ones. However, nobody truly invented the logarithm until Napier did in the 17th century. Shortly thereafter Napier and Briggs gave the logarithm its familiar base of ten and proceeded to produce large tables of logarithms of natural numbers. Originally, logarithms were invented as a way of working with large numbers and geometric sequences. It wasn't until much later that they were thought to be the inverse of exponentiation. Logarithms are still important today in the most basic analyses of natural sciences such as biology and medicine.

Essay 5) What were some lost mathematical texts? Why was this possible in the 17th century and before?
 

There are plenty of examples of lost texts through the ages. Most of these texts, such as all the Elements prior to Euclid's and a large portion of Archimedes's works, come out of antiquity and we only know of them through some of the commentators of the works in later years. In addition, there were probably a myriad of texts burned at the Library of Alexandria that we don't even know existed. Texts continued to be lost right up until the late 15th century when, for example, some of Robert Recorde's texts were lost. The loss of texts, for the most part, can be blamed on the lack of the printing press. Before this invention, books had to be hand-copied and so texts of a sub-par nature were simply not good enough to have multiple copies made. But after the coming of the printing press, it was just as easy to make 100 copies as it was to make 5 copies and so it became less likely for texts to be lost.

Essay 6) Could Fermat, along with or instead of Newton and Leibniz, be considered to be an inventor of calculus?
 

I believe that Fermat could easily be considered as an inventor of calculus. His ideas on differentiation were new and creative, in contrast to Newton and Leibniz building integration on ideas around since Archimedes. He was the first to use the idea of something that was so close to zero it practically was yet was still okay to divide by. Sure, as rigorous logic goes, this is completely ridiculous, but no one at that time had the key idea of a limit and Fermat's tiny quantities allowed mathematicians to at least build a working model of calculus. Further, he also was to have found integrals of some simple expressions. In a way, I would say that Fermat is an inventor of calculus but not a founder in that he came up with a lot of the key ideas but did not lay them on solid logical footing.

Essay 7) Since so much of calculus was developed by others before Newton and Leibniz, why are they considered to be the inventors of it?
 

I don't think that Newton and Leibniz are deemed the inventors of calculus for their few creative contributions. Instead, I think Newton and Leibniz are similar to Euclid in that their fame comes from their organization and use of the ideas available. Newton used the calculus to derive an assortment of physics equations, while Leibniz bestowed upon calculus its splendid notation. It is for these reasons, in addition to there original contributions to integration that earned them the recognition they receive today.

Essay 8) Trace the history of integration from the Greeks up to 1700.
 

The earliest signs of integral calculus came during antiquity with Euxodus' method of exhaustion and then Archimedes' method of equilibrium. The ideas then lay dormant for centuries before Stevin and Valerio used methods similar to Archimedes' in the 16th century. Soon after, Kepler used crude methods of integration to compute the volumes of a number of solids, as well as in the areas involved with his second law. Early in the 17th century, famous mathematicians such as Fermat, Pascal, and Wallis made small advances in the area of integral calculus, along with Barrow proving the Fundamental Theorem of Calculus. Finally, in the late 17th century, both Newton and Leibniz invented what we know today as integral calculus.

Essay 9) How has challenging the axioms lead to mathematical discovery?
 

Challenging the axioms has lead to some of the major discoveries in mathematics. The oldest challenge was that against Euclid's fifth postulate. This of course gave way to non-Euclidean geometry that was actually later found to be physical reality. Other challenges have led us to ideas of noncommutative algebras and indefinable objects. Now, it seems that challenging the axioms is one of the more popular thing to do. A mathematician will decide to change one key idea and see how long he or she can run with it without reaching a paradox.

Essay 10) Which, if any, of the three philosophies of mathematics listed in the text do you believe?
 

I personally cannot say that I believe whole-heartedly in any of the three theories. Upon first look, intuitionism seems very attractive. I like the idea of mathematics built upon reality, i.e. starting from the fact that we can see that one thing and another thing is two things and so on. It gives mathematics an overall purpose of understanding the world around us. However, it seems vastly limited in its approach. Logicism also seems appealing since it puts mathematics as a subset of logic and gives it the ability to go to any extent it wishes. Yet, it seems like mathematics should encompass logic and not the other way around since what often seems logical can be shown to be mathematically incorrect. Finally, formalism is neat in that it treats mathematics as a game played out on paper with a bunch of nifty symbols and rules of using them. This idea is attractive, but once again, we find a hole in the game in that there are never enough rules to completely describe mathematics. Therefore, in the end, I find none of the philosophies particularly motivating. They all seem to have their plusses and negatives and that some day man will come to the realization that all philosophies are founded on axioms and mindsets that can be chosen to be true or false and the philosophy will still be sound.