equations themselves are nonlinear

15 02 2012

When faced with an expression, the mind is instructed to perform a sequence of calculations. Consider a simple arithmetical expression:

1+4(4+7)-3(9-3)+2^4-1

The convention is to perform the calculations in a proscribed order: Brackets, Order, Division, Multiplication, Addition, Subtraction. So, the above calculation simplifies to:

1+44-18+16-1

Giving the answer 42. Of course.

Compare this to how you are unpackaging this sentence. In english, we read left to right, arabic right to left, chinese top to bottom. There is a linearity to it. With maths, we stretch into the equation and work from the inside out, as it were. Equations are nonlinear, in terms of the sequence of processes our mind performs.

My brother pointed out over the weekend, that we package sentences into phrases. An interesting comparison, and I got all excited because it provided a little more detail to the general, first order intuition, that algebra is related to language, when I conducted my first deep dive into XQ a few years ago. The manipulation of unknowns, akin to the semantic manipulation of unknowns when listening to wording.

It is important, even at this juncture, to note the different modality of script and vocalised wording. Mathematics, at least algebra, does take scripted form, and I am not sure what the equivalent is in wording. Can we speak mathematics as easily as we speak a language?

Of course, things get way more complex when we deal with generalised arithmetic, and when we examine the complexities of higher order mathematics. Looking at einstein’s field equation, and we can see the expression requires much unpackaging.

Most of the terms need to be expanded, such as lambda being the cosmological constant, einstein’s greatest blunder or so he thought at the time.

Mathematics is more like the creation of a chinese character than it is to writing a sentence. Composing a picture, more than writing a book. There is mental depth to it. In this way, we can take a prosaic interpretation, that the glyphs sequence the processes that need to be performed for the equation to “work”. Or we can take a more poetic interpretation, that the mind must hold various mental functions simultaneously in order for the equation to make sense.

The difference might be greatly overlooked. Scientists perform in the first way, like accountants manipulating spreadsheets, or factory workers simply performing tasks that work. To actually conceive of what is meant, to navigate the concepts, to be creative in such a conceptual space, requires composition, requires sensitivity, requires an experiential appreciation.

And rather than start off with a complex symphony like einstien’s field equations, or indeed any higher level mathematical constructs, might it be wise for us to retrace the conceptual developments that have been made historically? Or alternatively, to grapple with simple arithmetic as does every child, but this time from the vantage point of appreciating what the mind must perform for it to “make sense”.

Specifically, and returning to our first example, there is a sequence in time.

1+4(4+7)-3(9-3)+2^4-1

We perform the (4+7) and the (9-3) and the 2^4 first. And if we can learn to appreciate what function our minds are performing in time as we conduct arithmetic, and perhaps simple algebra, might this suggest a direction of mathematics that is less to do with application to the external world of objects, nor the pure field of formalist play in the flat world of script, but in the reflection of our internal world of thoughts?

3(x+2)-7=11

So when we look at equation, and we observe how our minds perform the various processes, we may observe the sequence in our minds. Perhaps we add seven to ll, then divide by three then subtract two. Perhaps. We get rid of things furthest away and gradually approach the revealing of the unknown. This temporal sequence, this algorithm of subjective processing, so easily objectified in our schools and implemented in computers, is often overlooked. But it did take humanity many centuries of exploration to discover and hone and apply. Mostly for physics, the application to physical objects out there in the “real world”. Perhaps it is time to consider the subjective implications, the effect on the mind that performs these functions, in time?

Undoubtedly, performing specific forms of mathematics exercises our minds. Can we take this a step further? If mathematics consists of formula for mental processing, can we derive formulations that exercise our minds to perform specific mental processes? That is, can we construct mathematical equations have therapeutic value? Equations that heal sick minds?

I have already detailed processes like multiplying by -1, a trick to convert any “negative” experience into a “positive” one, far more useful than the additive strategy. What is the mental equivalent to multiplying by -1? Or by zero, for that matter? To nullify an experience or a thought. And of course, something which maintains it fascination, the root of negative one. Can these have therapeutic value?

Once we feel comfortable with experiencing the effect of arithmetic and simple algebra, we might be able to appreciate the subjective appreciation of euler’s identity, for example, an equation which remains a beautiful and enigmatic formula.

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