Has it always been there?

2) Can we know if mathematics is invented or discovered?

The question above is a quintessential concept surrounding mankind and its relation to the study of mathematics. Nobody ever honestly asks if math was discovered or invented. We may question theories or aspects of absolutely any other area of knowledge doubt the judgements or assumptions they make, but have discussions and doubt the validity of math? That is an unfathomable concept. Or is it?


Humans have developed throughout history on a basis of mathematical discoveries and revelations, using these to their advantage in advances in all other fields. Mathematics has served as a foundation for this progress. However, a question arises of whether mathematics has always been present on earth, waiting to be discovered or if we in fact as a species have created it and all its derivatives. The amount of theorems, conjectures, proofs, numbers and topics surrounding math are countless, however a vital point comes up. Mathematicians on a daily base struggle and search to find new connections and proofs to problems in their area of math. As we watched in the film "The Proof," Andrew Wiles, math professor at Princeton, spent several years in the search for the answer to Fermat's last theorem. After several obstacles and even a thought-to-be solution, problems continued to arise until he successfully worked them out and found a working proof to the theorem formulated centuries ago. The issue here is whether there was truly only one proof that could work, or if he, using his previous knowledge of certain areas math, led himself to an answer. Andrew followed certain paths and options to arrive at his destination. Could it be though, that using this knowledge, he created a connection between those conjectures and a solution to the theorem? We must consider that it is possible that there were hundreds maybe thousands of solutions to that one problem that could be reached through different approaches or paths. Furthermore, it is believed that Fermat himself actually had a proof for his theorem however it was too big to fit in the margin of his book. Andrew's proof was an extensive 200 pages long. Once again, assuming Fermat was being truthful, this proves that Fermat had a substantially simpler and smaller proof using mathematics and resources in his time, while Andrew's was based completely on 21st century discoveries, therefore they could not have been the same. However, there is still the possibility that there were few theories, these two being included. Fermat could also have made some grave error in his calculations, hence the fact that maybe the proof was always there and it was only Andrew that was finally able to decipher it. 

The reason this question is so puzzling and mind-racking is that society as a whole always takes math as the complete truth. In school it is the one class where there is ALWAYS one definite answer, and everything else is practically: wrong. As opposed to sciences or humanities which are much more subjective and prone to discussion, math is in a sense always exact and precise. There are many pieces of evidence, such as the one above, to prove otherwise. We take many basic math concepts, things we learn very young, to be completely correct and build all our other learnings and understandings of the subject around it. By the presentation given by Mr. Fertig during one of our class sessions, some basic principles or statements that we all believed to be correct and unalienable, were very quickly disproved and invalidated. We experienced the simple statement 1 = 0.9999... be proven in less than 5 steps. Using the basic principles of variables and manipulations of equations, and assuming no advanced math trickery was in place, the simple algebra we were taught throughout elementary school was used to prove to us something we knew for a fact could not be true. This led to an astounding discovery that either the statement was false, which we knew could not be true because it was proven, which forced us to believe that in fact there was a flaw in the algebra that we along with the rest of mankind had and was currently using was skewed. This lead me to believe even more that mathematics was actually created, that we find patterns and suit them and adapt them with rules to fit our needs our desired ends, and not that they are waiting to be discovered. 

There is no doubt that they are essential to mankind, but the constant news of mathematical discoveries could in truth be more of mathematical connections, and one more person using their knowledge etc. to connect or create these connections and proofs.