Guest Author: Andrea McNally
Anyone involved in the discipline of math can most likely recall one, if not multiple, instances of being questioned on the usefulness of math. Eduardo Saenz de Cabezon addresses this question in his TED talk “Math is forever” (which can be found here). He claims there are three types of responses. First, the attacking one, which states math has a meanings all its own without the need for application. Next is the defensive one, which replies math is behind everything from bridge building to credit card numbers. The third response is where Eduardo claims math’s utility stems from its ability to control intuition, thus making it eternal.
Is math forever? Eduardo seems to think so stating diamonds aren’t forever, a theorem is. Mathematicians spend their lives generating conjectures and searching for ways to prove them. Once a conjecture is proven true though, it becomes a theorem, which is a truth that will remain so forever. Therefore, concepts such as the Pythagorean Theorem and the Honeycomb Theorem will forever be true, regardless of whether or not we are here to acknowledge it. This idea is rooted from Platonism, which is the philosophical view that there are abstract math objects that exist independently from our thoughts. Thus, all math truths are waiting to be discovered and not invented.
There are two main contributors in the world of mathematical philosophy. The first is German mathematician David Hilbert (pictured to the left), creator of Hilbert’s Program. He claimed that all math is formulized in axiomatic form with a proof to accompany it; it is done so by using finitary methods only which gives proper justification for classical mathematic problems. Hilbert believed theories could be developed without the need for intuition and would generate a set of rules and axioms that are consistent so one cannot prove an assertion as well as its opposite. Hilbert, like Eduardo, believed the capabilities of math were limitless.
Hilbert’s work, in turn, inspired the work of Kurt Gӧdel (pictured right) and his Incompleteness Theorems. Gӧdel proved that Hilbert’s concept of a decision procedure that generates axioms cannot be possible; there will always be conjectures that need a proof that may not actually exist. Gӧdel’s first incompleteness theorem proved that math knowledge cannot be specifically summed up and identified. Even the soundest basic rules will have statements about numbers that can’t be verified. It is important to note however, that Gӧdel never had the intention of disproving Hilbert’s program but rather to offer a new view.
So this leaves the math community open to explore if math is created or exists regardless of human recognition. If a tree falls in the woods when no one is around, does it make a sound? If no one has been able to prove a conjecture, does that theorem still exist? Like many schools of thought, there is ambiguity and uncertainty. As an individual in the math community, we all are responsible for looking into the information and opinions and coming to our own conclusions. Yet one thing remains certain, intuition and creativity
are absolutely essential in mathematics.
de Cabezon, Eduardo Saenz. “Math is Forever.” TED. TED, Oct. 2014. Web. 05 Apr. 2016.
Elwes, Richard. “Ultimate Logic. (Cover Story).” New Scientist. 211.2823 (2011): 30-33. Academic Search Complete. Web. 4 Apr. 2016
Linnebo, Oystein. “Platonism in the Philosophy of Mathematics.” Stanford University. Stanford University, 18 July 2009. Web. 05 Apr. 2016.
Peterson, Ivars. “The Limits of Mathematics.” Science News. Society for Science & the Public, 2 Mar. 2006. Web. 5 Apr. 2016.
Zach, Richard. “Hilbert’s Program.” Stanford University. Stanford University, 31 July 2003. Web. 05 Apr. 2016.
Image 1 retrieved from: https://www.bing.com/images/search?q=Incompleteness+Theorems
Image 2 retrieved from: https://www.bing.com/images/search?q=david+hilbert
Image 3 retrieved from: https://www.bing.com/images/search?q=Incompleteness+Theorems