Fermat’s Only Theorem?

Pierre de Fermat. Image: public domain, via Wikimedia Commons.

Every mathematician, or student of the subject, has heard of Pierre de Fermat’s last theorem, but surely such a talented mathematician has more to show for himself than a proof that he didn’t share with anyone. Fermat’s life and personality is interesting in and of itself, so while this written analysis is primarily a recognition of his other accomplishments, it can only be made more interesting by mixing in background on who Fermat was. While his other contributions to mathematics may not satisfy humanity’s craving for drama, Fermat did contribute more to mathematics than his most famous challenge.

Fermat was heavily influenced by François Viète, a French mathematician and lawyer. Fermat’s methodology in mathematics is described as classical Greek with apparent influence from Viète, and because his methodology consists of sending theorems to his buddies via letter with little to no proof attached, one could only assume the Greeks are responsible for his desire for arrogant competition, and the tender romanticism of the now lost art of letter-writing came from France. Although Fermat claimed that the proof for all his theorems was in the pudding, no one can find the pudding. Other credible mathematicians such as Karl Gauss had their doubts, but there is something particularly entertaining about a man flaunting claims in his peer’s faces and challenging them to prove something that he may or may not have. It’s a little suspicious, did he prove his theorems and simply enjoy rubbing his friend’s inability to in their faces? Well he was a lawyer so it’s possible. Did he trick (or inspire depending on how positively you like to think) other mathematicians into proving things that he couldn’t?

Fermat Invented analytic Geometry and contributed to the development of Calculus inspiring other magnificent minds such as Isaac Newton. In fact Newton admitted that some of his early ideas came from “Fermat’s way of drawing tangents” (Pierre De Fermat, Wikipedia). A manuscript of Fermat’s was published in about 1679 in Varia Opera Mathematica (Pierre De Fermat, Wikipedia. The title of his is manuscript “Ad Locos Planos et Solidos Isagoge” which is Latin, one of many languages that Fermat was conversant in including Greek, Italian and Spanish. If you translate that to English it reads an introduction to plane solid loci. The text can be more specifically interpreted as a classification of curves as: plane, solid or linear. Plane curves being straight lines and circles, solid curves being ellipses, parabolas and hyperbolas and linear curves being described kinematically with some sort of condition (Ad Locos Planos Et Solidos Isagoge).

Moving on to differential calculus, Fermat developed a technique known as adequality that he used for determining maxima, minima and tangents to a curve. Adequality can be defined as approximately equal and is denoted with the ‘~’ symbol (Adequality, Wikipedia). Basically Fermat would compare a function, say f(x), to something approximately the same, say f(x+Ɛ). He’d set them equal to each other which is of course risky and Fermat was likely under oath, so it must be the case that he was being meticulously careful not to be held in contempt of the court when he decided to instead set them adequal. Then by canceling out like terms, dividing by Ɛ and “solving for x” he’d derive his value for a maxima or minima.

The Fundamental Theorem in Calculus is arguably one of the most important concepts in mathematics and Fermat had a hand in inspiring it. By evaluating the integral of general power functions Fermat produced a formula that ended up being useful to Newton when he developed the fundamental theorem of calculus. Fermat evaluated such integrals by reducing them to a sum of geometric series.

In 1654 Fermat teams up with Blaise Pascal. Fermat and Pascal collaborated on a classical probability problem known as the problem of points. The problem is a game of sorts. It has two players. Each player has equal chance to win each round. There is a prize pot and the players agree that a particular number of round wins leads to total victory. The game is interrupted and of course that leaves the question of who gets what amount of the prize. The concept of a “fair” division must be established based on how many rounds each player has won so far and the probability that that player was going to win the overall pot. Fermat figured out how many possibilities were left based on how many rounds were left and charted these possibilities then based his “fair” division in proportion to the probability of these possibilities. Fermat’s solution was inefficient and inaccurate as the number of rounds left gets large and so Pascal made some big improvements on it. Nonetheless this collaboration earned Fermat and Pascal the title of founders of probability theory on the grounds that this game laid the groundwork for probability theory (Pierre De Fermat, Wikipedia).

 

Yes Fermat’s Last Theorem is his most notable and interesting accomplishment, but Fermat contributed much more to the world of mathematics than this. He inspired other great minds with his own work, irritated others with his arrogance and sass, and helped laid the ground work for some of the biggest mathematical concepts by getting his hands dirty with proofs. Even though his pudding will likely never be found, his proofs/challenges (or lack thereof) inspired or tricked other mathematicians into some of the greatest discoveries in mathematics and some less magnificent discoveries as well.

.

 

 

Pierre de Fermat – his last challenge (Fermat Quotes, Rugusavay).

 

 

 

 

 

 

 

“Ad Locos Planos Et Solidos Isagoge | Work by Fermat.” Encyclopedia Britannica Online. Encyclopedia Britannica, n.d. Web. 13 Feb. 2015.

“Pierre De Fermat.” Wikipedia. Wikimedia Foundation, n.d. Web. 13 Feb. 2015.

“Adequality.” Wikipedia. Wikimedia Foundation, n.d. Web. 13 Feb. 2015.

“Pierre De Fermat Quotes.” Rugusavay. N.p., n.d. Web. 13 Feb. 2015.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s