Tuesday, December 11, 2018
--- Language and mathematics ---
This year my dual focus will be language and mathematics, but are those really two different things?
There is an ongoing philosophical debate as to whether numbers have an independent existence "out there" in nature. Is there a property of nature called "one"? Is there really a such a thing as a fraction?
Part of my graduate training was in research methodology. I've done a little research myself, mostly as parts of student teams, but I've mostly been involved with helping others develop their studies. Two of the biggest problems I've seen in studies are reification and reductionism.
Reductionism, in this instance, is the tendency of specialists to see their world from the narrow viewpoint of their own area of expertise. A medical issue will always have psychological, social, and environmental elements so a physiologist looking at diabetes might focus on the blood and pancreas and forget all about these other elements, and that can be useful as long as it is kept up front that his results are only part of the story. But if you have to deal with the reality of diabetes, you'd better not forget the other things.
Reification is much more insidious and difficult to guard against - often, it's just ignored.
Science doesn't give truths. It provides models that allow us to understand things that happen in the world and make predictions, but no model is perfect. All models are approximations of reality. A good model preserves as many of the important features of reality as possible so that it's outcomes can be said to be accurate to within certain specified limits. The error can be specified. But there is always error.
We keep models in our heads about how we think the world works. And, hopefully, our models are pretty close to reality. But philosophers and research advisers are there to warn us that the word is not the thing and that the model is not the reality.
And I think that is why some people mistakenly believe that numbers are "real". You can point at a number on a page but that's just ink that's been allowed to soak into paper and dry. To be grammatically correct, "1" is not 1. Fractions are even more problematic. If you break a stick in nature, you don't have two fractional sticks. You just have two sticks, and the "two" only exists in people's heads.
Mathematics is a language, just like English or Spanish or AMSLAN. It has been developed to help us come up with technically correct descriptions about how the world around us works.
In a way, numbers do have a kind of existence, as information, but that existence isn't independent. If there were no minds around to appreciate a zero, there would be no zero. But zero revolutionized our world by allowing us to make very precise "words" to describe very large and very small quantities.
Some machines have parts that must be accurate in size to ,say, 0.01 millimeters, or else the tiny space between the parts would allow enough motion to shake the machine apart. Try expressing 0.01 in Roman numerals (which have no zero).
Our technical understanding of the world, and therefore, our exquisite technology, relies on the language of mathematics, but language it still is.
We have many languages - literary language that allows us to communicate complex messages across both space and time, nonspoken body languages that allow us to communicate exquisitely our emotional intentions, aesthetic languages that let is communicate beauty (and sometimes ugliness)to multitudes. The sciences use the languages of logic and mathematics to communicate ideas with great precision.
Now, as I begin to explore the hard sciences, my first stop will be the "hard language" of mathematics and the softer human spoken languages.
Does mathematics exist "out there"? Well, in fact, it does. As long as there are thinking people "out there", there will be mathematics and I will be carrying it out into the field more and more to explore the intricacies of the world around me.