A question on language and mathematics.

I've been thinking a bit about big words lately, and why they seem to be so common in theology, philosophy, and physics books. And at this point all my thinking has led to a question. I haven't found a good answer yet, so this blog post is going to have a very open ending. But it's an open ending that should make you think. If that bothers you, don't waste your time here and go read something else. If you have any comments or ideas then definitely post them.

When you read academic books, they often have to spend a significant fraction of their time just defining their terms. It's as if the dictionary definitions of common words have to be redefined depending on their context so that they precisely convey the meaning that the author wants. But even after the word is defined, the author will generally give some exceptions to the definition, or say something about how the definitions still don't completely fit the ideas.

The Cross of Christ by John Stott is one of my all time favorite theological works, mostly because it's the most life changing book that I've read that is also very academic. In this book, he devotes an entire chapter to what we mean by "self-substitution" in theology. There is another chapter devoted to defining the word "self-satisfaction." And in each of these chapters, Stott uses a massive diversity of quotations and words of his own in order to narrow down what he means. But each time he repeats himself and each time he quotes another author, he uses slightly different words that refine his ideas, and look at it from an ever-so-slightly different angle.

The reason why Stott repeats himself from different angles is quite obvious. The most important area to have a very accurate understanding is the cross and the work of Christ. Pascal claims that Christianity is the key that fits the lock of the human heart. God designed our hearts to fall in love with and depend on Him. If that is true, then the only way for us to be completely satisfied is by knowing God and the work of Christ the way it truly is. We need an accurate understanding of the shape of this key, because the better we know Christ the more we will naturally fall in love with Him.

However, language itself is limited. Diderot was right to a certain extent. There is a limitation to the precision that our words can describe concepts, and after reading through Stott's book, it does appear as though he is losing words to describe the wonder of the Cross. This shouldn't be surprising. The gospel is reality, and reality is nearly always very complex. In particular, the cross is a complex idea that should captivate us with wonder for eternity. But how are we to push past our barrier of language?

The curious difference between the way physicists and mathematicians communicate and the way philosophers and theologians communicate is their use of words and notation. Physicists and mathematicians use a very powerfully formulated system of shorthand notation (called variables) and well defined rules of logic that use the shorthand notation (called mathematics) to describe their concepts. On the other hand, theologians and philosophers stick with the dictionary definitions of words to represent basic ideas. They also use a well defined system of logic, but the system of logic is based on words rather than variables. So, physicists and mathematicians use a much more concise system called mathematics to represent their ideas and derive their theorems, while philosophersand theologians have historically used words and theorems that are expressed using language. Are there any other differences between language/reason and mathematics?

There is a distinct advantage of using mathematics over language, and the advantage is simply that mathematics is so much more concise. In a few dozen symbols, one can express all of Maxwell's equations in mathematics. Maxwell's equations completely determine all of electromagenetic theory and the nature of light, all in a few dozen symbols. I can't think of any equivalent statement in philosophy that can be expressed in such a shorthand notation, but at the same time completely determines an entire field of study.

So, why don't philosophers and theologians develop a type of mathematics for their fields? Would that be very beneficial? It definitely seems so at a first glance to me. It definitely seems that there would be fewer fallacies and more efficient work done in the humanities if they simply had their own type of mathematics.

On the other hand, I don't think it would offer any more precision to their ideas. And here's why. Any variable in mathematics or physics is always defined in terms of words. Therefore, it's only as good as the words behind it that specify what it means. The shape and mechanics of an electric field can be described completely by mathematics, but each of those variables in the mathematical expression (such as charge, force, distance, etc) must be defined in terms of dictionary words. Even if they weren't described in terms of words, the metaphysical meaning of the electric field may (or may not) extend beyond simply the movement of charges. If the electric field is a thing in itself, then we can only describe and define the nature of it in terms of words.

Okay, I have just two more comments. The first is more of a question. Why is it that we always have to go back to words? Why do we always need words to define variables in mathematics? Concepts in our minds are not words. We know that because you can have a concept in your mind and call it by whatever word you like. Our brains are not tied down to any specific words. So is there a way to get around that?

Finally, I hope that this blog post will help the physicists and philosophers have a greater appreciation of each other's work. I know so many people who become physicists just because they thrive in mathematics as a mode of communication, and so many people who become philosophers just because they thrive in language as a mode of communication, even though reason and logic are the fundamental roots that connect them both. But the physicist cannot ultimately do without words, and the philosopher may benefit from using a concise and systematic mode of communication like mathematics.