I enjoyed listening to your group’s discussion on “Why do people hate mathematics”. The first part was about why some people like math while others don’t, and if some are naturally good at it while others aren’t. The second part questioned whether math was an invention or a discovery.
It seems obvious that people who work in the engineering, scientific, and computer fields, for example, must enjoy working with math, otherwise it doesn’t seem likely that they would have chosen those careers. But I don’t necessarily agree that people in those fields, even the very successful ones, were naturally good at mathematics. When I was in grade school I didn’t particularly like math and my grades reflected it, while my sister seemed to have a natural talent for it. It wasn’t until I got interested in science in High School and realized that I needed math to really understand it that I became interested in it. I guess you could say that some people are born with a love for math, while for others , like me, it’s an acquired taste.
As I mentioned the second part of your discussion questioned whether math was an invention or a discovery. If you haven’t already read it there’s an interesting Internet article: “Is Mathematics Invented or Discovered ?”, by Derek Abbot. Basically he list 2 options.
1. Math is innate. The structures of mathematics are intrinsic to nature. Abbot refers to this viewpoint as “Mathematical Platonism”.
2. Math is a human construct. It is a product of the human mind and we make mathematics up as we go along to suit our purposes. Mathematics is not discovered, it is invented.
He says that originally he used to be a Platonist and thought all mathematical forms were real and waiting to be discovered. Then he had a moment of enlightenment and converted to “non-Platonism”. According to this viewpoint all mathematical models are only approximations of reality. These models eventually fail, they go through a process of revision, and we invent new models as needed.
Another interesting Internet Article is “The Nature of Reality”, by Mario Livo. He poses the question “Math: Discovered, Invented, or Both?” His personal belief is that humans “invent” the mathematical concepts (like numbers, shapes, sets, lines, and so on,) by abstracting them from the world around them. They then go on to “discover” the complex connections among the concepts that they had invented; these are the so-called theorems of mathematics.
At the end of Livo’s article there is a link to the PBS Nova program: “The Great Math Mystery”. It gives an interesting and informative account of the subject.
I think that I mentioned recently that I was reading a book “Our Mathematical Universe” by Max Tegmark, that deal with things like the Multiverse (parallel Universes). I guess he could be classified as an “extreme mathematical Platonist”.
He starts with the assumption that there is an external physical reality completely independent of humans. He believes that the majority of physicist favor this long-standing idea, though it’s still debated. For example, Metaphysical solipsist ,who assert that nothing exist external to “this one mind”, and since “this one mind” is the whole of reality, then the “external world” was never anything more than an idea, therefore they reject it flat out. Also supporters of the Copenhagen interpretation of quantum mechanics may reject it on the grounds that there’s no reality without observation. Albert Einstein is reported to have asked Niels Bohr, the main supporter of the Copenhagen interpretation, whether he realistically believed that the moon does not exist if nobody is looking at it! To this Bohr replied that however hard he (Einstein) may try, he would not be able to prove that it does, thus giving the entire riddle the status of a kind of infallible conjecture – one that cannot be either proved or disproved. This is but one of the many friendly battles Einstein had with Bohr. Even though Einstein was one of the founders of quantum mechanics he was convinced that it was only an approximation to a deeper reality, while Bohr believed it was an essential cornerstone of physics. As hard as Einstein tried to trick Bohr and find flaws in the theory it seemed like Bohr was always one step ahead of him.
What’s most controversial about Tegmark’s ideas is that he believes that mathematics can be used to explain all of nature. Assuming that an independent external reality exist, physicists try to describe its behavior with mathematical equations. There most successful theories so far have been Einstein’s general theory of relativity for gravity and Schrodinger's equation for quantum mechanics. But up to now there are branches of science (e.g. geology, biology, psychology, and sociology) whose behavior can’t be directly described using mathematical equations. Some scientist, like Tegmark, hope and believe that someday there will be a purely mathematical theory that will accomplish this task. This so-called “Theory of Everything”, TOE, is the Holy Grail of theoretical physicists.
Tegmark recounts how he has been criticized and ridiculed by many of his fellow scientists for wasting his time with this “hopeless task”. But even if such a theory is never found, others think the endeavor itself is worthwhile. In the search who knows what new scientific knowledge can be gained by new generations of Newtons and Einsteins.