In a previous article (Pie in the Sky), I reviewed a book by George Lakoff and Rafael Núñez called "Where Mathematics Comes From." In that article, I give what I think is a fair treatment of their argument that "There must ultimately be a biologically based account of the mechanism by which [mathematical idea] are created, learned, represented, and used." (pg 347 of the original hard cover version)
The authors have a great deal of respect for mathematicians, but they do not believe in the "Romance of Mathematics." To them, the "Romance of Mathematics" is the idea that mathematics is eternal, non-human, and fundamental to the universe. They are afraid of this Romantic idea because, to them, "The Romance of Mathematics is not a story with a wholly positive effect. It intimidates people. It makes mathematics seem beyond the reach of even excellent students with other primary interests and skills. It leads many students to give up on mathematics as simply beyond them. The Romance serves the purpose of the mathematical community. It helps to maintain an elite and then justify it. It is part of a culture that rewards incomprehensibility, in which it is the norm to write only for an audience of the initiated--to write in symbols rather than clear exposition and in maximally accessible language..." (pg 341)
While I agree that most papers on mathematics are beyond my understanding (even though I studied physics for roughly 20 yrs), this has nothing to do with the fundamental question: is mathematics eternal, non-human, and fundamental to the universe?
So, I think that it's important to review what other great thinkers have believed, and their arguments for those beliefs.
First, we'll discuss Plato.
While it's hard to really know what Plato believed, his character of Socrates seems to believe that mathematical ideas are innate, and that the job of the philosopher (or teacher) is to help the student realize what is already inside of them. In the Meno dialogue, Socrates helps the slave boy (Meno) "remember" what is the correct relationship between the length of the sides of a right triangle. For Plato (or Socrates), this is just an example of a larger story. The real story that Plato is trying to teach us is the following: we have a soul, but it takes a teacher to show us and to remind us of our soul. The eternal is inside of us; it just takes a little bit of guidance to remind us.
For Plato and Socrates, the goal of the study of mathematics is not in its applications. The goal of the study of mathematics is to remind us of the eternal forms. From a study of the eternal forms, we start to understand our own "soul", and we are reminded that the eternal is not something to be afraid of. As Socrates would say, 'Only you can damage your soul.'
Plato would also argue (via Socrates in the Thaetetus dialogue) that there can be no such thing as language unless there were a common underlying, eternal form. For the Plato and other Greeks, this underlying, eternal form was mathematical if the language was music and the underlying, eternal form was moral if the language was politics.
But is there any proof that mathematics is eternal? I know of no proof to show that mathematics is eternal, let-alone non-human or fundamental to physics. It's been a long time since Plato made his arguments for eternal forms, but there has still yet to be a proof of the eternal forms. My own opinion, though, is not to throw the baby out with the bathwater. Plato's ideas have been misused by hundreds of different religious organization, but there is likely an underlying truth: that we have eternal, mathematically based souls.
Second, we'll discuss Pythagoras.
Pythagoras was one of the first great mathematicians, and he formed a colony of people who engaged in an ascetic life of silence, study and vegetarianism. Many of the people in the colony believed in metempsychosis, i.e. reincarnation. They believed that their souls would survive after the death of their bodies (not unlike in "Being John Malkovich.") Pythagoras was also one of the first, if not the first, to figure out the mathematics behind musical harmony. The goal of musical training and mathematical training was to remind people of their underlying, eternal souls.
While there is clearly a relationship between harmony and mathematics, the question is: is that proof that we have eternal souls? I don't think that this is a proof. For me, a proof that we have eternal souls would have to be an actual mathematical proof showing: (a) that we have a soul and (b) that the soul can not be destroyed. We are a far way from proving this, even if it is true.
Until then, I think that it is still an open question, and a good question to debate.
Thirdly, we'll discuss Lakeoff & Núñez's arguments in "Where Mathematics Comes From."
In their book, they are trying to remind us that there is no proof that mathematics is eternal, no proof that mathematics is non-human, no proof that mathematics is not just invented, and no proof that the physical world operates mathematically. As well, they feel that there is a price to be paid for believing in the "Romance of Mathematics", just as their is a price to be paid in their minds for believing in an organized religion. (They are not big fans of large, organized religions, as am I for that matter. But as I stated earlier, I think it's important not to throw the baby out with the bathwater.) They believe that humans have evolved the capability to invent mathematical ideas, and that the ideas can't be proved to be true...they can just be shown to have been useful for further evolution. To them, "Mathematics is a mental creation that evolved to study objects in the world...mathematics is fundamentally a human enterprise arsing from basic human activities." (pg 350-1) To Lakeoff & Núñez, mathematics is culture-dependent and species-dependent. To them, mathematics means different things to different people and different animals.
The problem I see with this theory of mathematics is that it ignores those aspects of the world which are countable and the same, regardless of the subject viewing the system. One example is the entropy of a system. The entropy is the logarithm of the countable symmetries of a system. Subjects moving a different, relative velocities will count the same amount of entropy generation in an irreversible system. To me, group theory and the study of symmetries is eternal, non-human, and fundamental to how the physical universe works. The study of group theory (for me) is a reminder of the eternal soul. It's not the same concept of soul as believed by Plato or Pythagoras because I believe that the purpose of the soul is to grow and replicate. (I don't believe in reincarnation into a different body. I believe that souls can self-replicate and grow...either biologically or non-biologically.)
For me, the problem with Lakeoff & Núñez's arguments is that it ignores the idea that self-replication (i.e. evolution) could be a mathematical concept. They seem to think that evolution is just a random phenomena. Instead, self-replication is (likely) mathematical in nature. Certain sets of differential equations can support self-replication, and certain sets of differential equations can't support self-replication. I understand that, today, we have no proof of an eternal soul, but that does not mean that we could not develop a proof sometime in the future. Until there is proof one way or another, this is still an interesting area of discuss and debate. I welcome your thoughts on the matter.