Throughout this post I shall use the British shortening of the word "mathematics".
I struggle with maths. Well, ok, as the proud possessor of a maths 'A'-level and a finance MBA I suppose I don't struggle with it that much, really, compared to many people. But in my head I struggle. I look at a page of equations and my eyes glaze over. I can't look at a graph and imagine the equations that explain it. Nor can I look at an equation and work out what shape of graph it would create. I can visualise the forces in a mechanical system, but converting that into algebra is a bit of a problem as I can never remember what the letters stand for. I have the same problem with econometrics too - even when the terms are clearly documented at the top of the page, by the time I'm halfway down it I've forgotten what they mean and have no idea what the equations are supposed to show.
Many people have the same problem when faced with a musical score. But I don't. I can look at a page of complex musical notation and hear in my head how the music would sound. And as far as I remember I have always been able to do this. Because I can hear the music on the page, I can sight-read anything. It took me a long time to realise that the reason I find it difficult to teach sight-reading is that the majority of people can't look at the score and hear the music as I do, so they have to do it in a completely different way that is foreign to me.
So, you see, I have a problem with mathematics that I don't have with music. But I don't think this problem has anything to do with my ability to "do" maths. I think it is a language problem.
Let me explain. Both written music and maths are systems of notation. They are both ways of describing the workings of real phenomena in an organised and comprehensible manner. But they are not the phenomena themselves. The full score of a Beethoven symphony is no more "that" symphony than a DSGE model "is" the economic phenomenon it portrays. In each case the phenomenon is only partially explained by the notation that is used to explain it. And the further removed the system of notation is from the reality of the phenomenon it is attempting to describe, the less accurate it is. Much contemporary music, for example, is not easily rendered in traditional Western musical notation.
Similarly, languages are systems of notation. They are a way of "making present" thoughts and ideas in a manner that is comprehensible by others. But they are only a partial representation of those thoughts and ideas, and there may be other, better ways of representing them. The use of symbols in maths is itself a derivative of language: but I would argue that maths is actually a language in its own right. Mathematical symbols are, if you like, "words", and the equations that are constructed from them are phrases and sentences: the manipulations and transformations that the equations go through are the mathematical equivalent of grammar.
In languages there seems to be a definite "transition" between having a working knowledge of the language and being fluent. That transition occurs when people stop mentally translating from one language to another in order to understand, and simply understand IN the language. People describe this as "thinking" in the language. A similar transition seems to occur in reading, when people move from reading the words and mentally visualising what each word represents, to reading (or perhaps "living") the story. My ability to read a page of musical notation and hear the music that it represents also appears to be similar: I don't need the notation translated into something else in order to understand it.
Does the same sort of transition happen in maths? I think it does. Unlike me, real mathematicians do not need their symbols translated into English (or whatever their native language is) in order to understand what they mean: they understand directly what those symbols represent. And they understand directly what equations represent, too, and how they can be manipulated and transformed to create different effects. I wish I did. I need the transformation explained in English in order to understand it in maths. I don't "think" in maths as I do in both English and music. I have a "working knowledge" of maths: but in music, I am fluent.
Which causes me a problem when someone dismisses music as "just maths", and insists that maths is the fundamental language that underpins everything and I really should understand it in order to be a proper musician. I find myself rebelling against the notion that a language in which I don't feel comfortable is "better" than a language in which I am fluent. That's like telling me that I should write in Latin, not English, because English is (partly) derived from Latin. If the purpose of language is to communicate thoughts and ideas and describe real phenomena, surely the "best" language to use is the one in which I am most at home? Any other would be less effective, simply because I do not know it as well.
But I have another, and perhaps more fundamental, problem with maths. The person who criticised me for not being fluent in maths as I am in music was making a value judgement about maths as a language. He assumes that mathematical language is ALWAYS the best way of describing and explaining real phenomena - including musical sounds and patterns. I think this is completely wrong and very dangerous.
Earlier in this post, I commented that both maths and music are at best partial explanations of the phenomena they represent. They are approximations. The trouble is, we tend to confuse the paper approximation of the phenomenon with the phenomenon itself - and in so doing we ignore or eliminate those aspects of the phenomenon that don't fit with the system of notation that we have used. Generally in Western music we understand this now, though for a long time we did not: generations of music teachers drummed into their students that they must "play what is on the page", leaving little opportunity for personal expression. But the advent of forms of music such as jazz, that could not be adequately portrayed by Western musical notation, has forced musicians to develop other forms of notation that better represent the newer musical forms - and perhaps more importantly, has forced people to learn music by listening and copying rather than reading. But you will still find jazz standards forced into Western notation in sheet music books produced for use by the general public. They are twisted, tormented mockeries of the real thing.
Historically scientists have shown themselves very willing to invent new systems of mathematics if the old ones don't fit with new discoveries. But at the moment we appear to have some idolisation of mathematics going on. I don't believe that all real phenomena are adequately explainable by our mathematics: I think that other languages - including ones we haven't invented yet - may be equally useful. But if we insist, as my critic did, that everything is "just maths", then we exclude the possibility of expressing real phenomena in other ways, and in that way potentially limit the means by which we can better understand the universe we inhabit. This, to me, would be a serious loss.
And there is another loss, too. In Western music we have experimented with an entirely mathematical approach to composition, and rejected it - because although it resulted in melody and harmony, it lost something fundamental. It lost its humanity. Music is made by people for people: when we reduce it to a series of graphs and a set of mathematical formulae programmed into a computer, we have eliminated its purpose. I suppose musicians of the future could simply be handed a graphical trace of sound patterns and told to re-create them, but this would simply be reheating someone else's creation, rather than creating music themselves. It is the approximate nature of musical notation that makes it possible for performers to express their own personalities through their playing or singing and share with the composer in the creation process. It is this that makes every performance of a Beethoven symphony different, that means that Beethoven symphonies are to a degree still being written today. Music that does not admit the possibility of variation from performance to performance is inhuman.
The dominance of mathematics in other disciplines such as economics tends to exclude people like me, who are not fluent in that language, and in so doing it makes those disciplines elitist. And the problem with elites is that they become divorced from reality: they regard their creations as set in stone and invariable, but the real world varies and changes, and humanity with it. Just as mathematical dominance in music resulted in inhuman musical creations, while advanced mathematics dominates economic thinking, economics is in danger of becoming inhuman.