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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difference over time

3 02 2012

The late great Gregory Bateson drew attention to the simple equivalence:

What he meant was, if you close your eyes, and run your finger over a crack, you can sense the crack only because your finger is moving. If it is still, whether on the crack or not, there is no differential, and as such, there is no sense — you can’t tell whether your finger is on the crack or not.

It has made my mind go in a few explorative directions, some of which may test the reader’s ability to conceive as it may transgress certain conceptual boundaries.

Consider:


That is, value is an estimation of the difference between state of mind at time t and at t+1. This subjectively derived enumeration is a measure of experience. In a crude way, it could be used to determine the learning experience, how much I have learned over a period of time.


That is, the experiences of person A and B are different, given the same time. There is something multiplicative about this. Two different experiences, or indeed as many experiences are there are people, of the same time.

Combine this with the squaring observation above, and consider a group of seven people experiencing the same “meeting”, and the reflective evaluations of each other’s experience. Things get out of hand rather quickly, and not simply in terms of the mathematical number of handshakes (7! is it?), but in terms of the psycho-dynamics. There are orders of complexity here.

Which leads to a rather intuitive leap to map it against buddhist “piles” or “heaps”, which I have considered to be emergent levels of complex organisation:

form inorganic matter .
sense ∆ form awareness of form
perception ∆ sense awareness of change of form, and yet maintenance of “form identity”
volition ∆ perception awareness of change of sense, “making sense”
consciousness ∆ volition awareness of change of perception, continuity of “perceptual identities”
“social”? ∆ consciousness awareness of change of consciousness, true acknowledgement of equivalent “other”

I’ve no idea how useful this is. And it is too linear for my liking. Bees and ants, for example, seem to suggest to me that their “perception” may be embodied at a collective level rather than at the individual. It is too easy to think of hierarchy here, of progress. The buddhists give clear warning to the illusion of our projective mind, the thought center, and seem to seat consciousness at a lower level, heart-mind. Having the simplicity of an animal, a self-aware animal, rather than an animal that runs after mental objects and thinks of themselves as something different.

This kind of thinking might enter into wise^0 territory. Just wanted to include this in order to exercise the possibility that any mathematician will tend to import math that has been applied to the world of objects, whereas there may be a different form of maths, or application of it, that is better suited for the exploration, and perhaps modelling, of consciousness and social dynamics. With this in mind, the table above suggests rates of change, eg

Which is simply reminiscent of acceleration, as d/t^2. And if you find your mind responding sceptically, simply consider your experience of music. Distinctions in time. And the different layers of complexity. It is in this are, subjective experience, that math needs to evolve a simplicity and subtlety that is lacking in its application to physical systems.

Can we allow ourselves to even think this…?





the maths of uncertainty

3 02 2012

This is definitely peculiar. Absolutely no idea how useful it might be. I used to be a math teacher, and only recently did I realise that an equation is not read like a normal sentence. To keep things simple, consider the following expression, or sum:

Any school kid will tell you, you don’t start out from the left, like it was a sentence, with eg 2×3. You follow the procedure, BODMAS, in order to unpack the expression.

Consider what this means in time. The procedure of BODMAS sets the order by which the equation is unpacked in time. The mental sequence of processes. So, in the above expression, you complete (4+7) and (6-3) simultaneously, then multiply them respectively by 3 and 2. and then you subtract the second from the first, and then multiply the whole lot by 2, which gives 54. Or I guess you can multiply all the brackets and then simply sum them up. Whichever way you do it, the answer is the same, 54.

Now consider the following statement:


If we consider it from left to right, we get (10+4) then multiply by 3, which gives 42. Or, if we follow our usual BODMAS protocol, we do the multiplication first (4×3) and then add 10, the correct answer.

But what would a maths look like that holds both correct, at the same time?


That is, where both answers, 42 and 22, were correct?

And if this doesn’t strike you as being particularly useful, you are in popular company. However the following expression does pique a curious line of inquiry:


Which derives a possible simultaneous answer of 0 and 1. And because the digit in the middle can be a 1 or a 0, it might be more clearly written as:


And if we multiply this expression by itself:

And the answer to this is the truth table for the AND function:

AND (x)

0

1

0

0

0

1

0

1

And if we add it to itself:

We derive the boolean OR function:

OR (+)

0

1

0

0

1

1

1

1

This is not entirely unsurprising, since our original expression simply changes the order of the operations, thus producing the effect of sequenced logic gates.

What is interesting, at least notionally, is that this is contained in a single arithmetical expression. I know that variables can perform this function in maths, and set theory, but to have it in a simple sum, is… peculiar. And I can’t help but relate this to chaos equations, which after a suitable number of iterations, tend to a constant, or two constants, or a set of constants. That, and qubits.








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