We talked at the end of last class about the uncertainty of math. In the schools that I have grown up in, there persists a distinct contrast between the accuracy of math and science versus the humanities. In the former, there was "one right answer," while the latter is left up for interpretation. However as made evident by "The Calculus" reading, the fundamental concepts that "allow" calculus to work are not clear cut logic.
In both Newton's and Liebnitz's calculus, there is debate on the validity of finite limits expanding to infinity and infinitesimals (respectively). If we question a math that is so persistent in describing the world around us and in predicting systems, is there actually a black and white outcome? In other words, just because the math works, does that mean that the answer given is solely right or wrong? Or does the justification between right and wrong come from what the answer means to us, an interpretation? For me, I think (but am not sure) the vindication of the answer arises from the ability to predict what is going to happen next. This requires a faith in the answer you found, sort of like religion. Nevertheless, I think that math to us has no meaning without the interpretations we give it, requiring some subjectiveness.
One note: after further reading multiple sources seem to point to nonstandard analysis as an answer to the problem of infinitesimals, but debate in the validity of certain fundamental principles in math and science is not just present in calculus.
As a future math teacher I find those questions you pose to be particularly interesting because it does effect how we teach in the classroom. We have these equations we use, and we know that if we use them it will give us an answer, but never really pose to our students, nor do they ask, if these equations are correct or if the answers we get are correct. This article then raises some good questions in that, we shouldn't take things on faith, we need to look at everything under a magnifying glass.
ReplyDeleteI guess this article did a great job pointing out, that even though Newton is considered one of the greatest thinkers of history, people still asked questions about his work. It wasn't enough to go on his word alone, we had to pick it apart bit by bit, and that was when people started seeing some of the logical problems in Newton's work. It is this attitude of making sure someone is actually right versus someone just trying to sound smart. It is up to future educators to make sure we instill inspiration into students so that they can make sure we keep seeing growth in the various fields.
To quote: "In other words, just because the math works, does that mean that the answer given is solely right or wrong?"
ReplyDeleteWow, what a question. As a passionate and often frustrated student of of the schools of physics and mathematics I find this question creeping up on me often. When I look at physical problems and put a mathematical framework onto them, there seems to be some disconnect (for me anyway) as to whether or not the world truly works like the model or if the model just happens to match the behavior I observe within my ability to understand it. Quantum mechanics seems to really draw these types of questions due to its uniquely nondeterministic approach to solving problems (at least in the formulation I've been taught and read up on). One cannot help but wonder if the electron's position/momenta is fundamentally stochastic (random) or if it obeys some underlaying deterministic, but drastically over complicated process that we as humans are simply not privy to? If that is the case, does our model really give us a "right" answer to our questions of how the universe works?
It is my personal belief, and I do think it is a belief or leap of faith, that the world does its own thing and our models merely try to mimmic it in a manner that is comprehensible to a thinking thing. This goes beyond questions pertaining to physics, however, as we can similarly try to place mathematical models onto numerous other subjects that are purely human such as language, politics, social structure, sport and more. While a given model may give an extremely detailed account of what is observed it is certainly not the only way to understand the thing at hand. Some understanding seems to come from intellectual pursuit, but others come from action and cannot be translated easily into mathematical jargon. For example, I like to slack line (a variant of tightrope walking). While I can project a logical, mathematical model upon the mechanics underlaying how I am able to walk on the rope, such effort would never give me the intuition or grace that comes with merely practicing the action. Command over the limbs to do minor adjustments and a sense of the rope are truly lost in such an understanding.
Anyway it is late and I need to contemplate the back of my eye sockets for a while.
Jason, your point about mathematics being like a religion really struck me, and got me thinking. Are not religons, mathematics, the sciences, etc all really sides of the same coin? In the sciences, for instance, one must have faith that the theories that have been made throughout time are actually true. I mean, thats why they are called theories in the first place, because we haven't been able to completly prove them. So, when someone argues that science has conquered religion, really they are only saying that they think their belief system is better than someone elses belief system.Reason becomes merely another method by which humans attempt to attain truthful information about their world. It does not discount revelation, for example - it is merely another belief system. "Hard facts" are really just observations, then, made by imperfect human beings that "seem" to be true. Mathematical "facts" are just the same. Just because the language of mathematics (which, as we discussed, was a creation of humans as well) seems to reveal something about the way the universe works, how do we know that the very language we created actually does an accurate job of describing what we perceive to be natural phenomena? I think this is interesting to think about, and perhaps shows that priests and scientists are all really doing the same thing.
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