The Forms of Thought

25 07 2010

In defining a methodology for Applied XQ, I keep testing out and questioning David’s approach against existing or past approaches.

Descartes’ interest in Geometry, for instance … his formation of a moral stance made up of four maxims to guide his thinking:

  • To obey the laws and customs of his country
  • To be the most constant and resolute in his actions that he could
  • To always try to conquer himself rather than his fortune; to change his desires, rather then the order of the world
  • To try to make the best choice of work, without prejudice to other men’s

(Paraphrased from Rene Descartes, A Discourse OF A METHOD For the well guiding of REASON, And the Discovery of Truth In the SCIENCES, LONDON, Printed by Thomas Newcombe MDCXLIX. (1649)

(Beginning of Part III))

Nowhere in mathematics or logic, philosophical mathematics, or philosophical logic that I’ve been able to find to date, is there a precise correlation for his premise that maths processes mirror our mental processes.

There is one interesting observation, though … and that comes from a chance internal gyroscopic intuitive leap of faith – towards the work of D’Arcy Thompson.

Speaking of form, he writes:

“The form, then, of any portion of matter, whether it be living or dead, and the changes of form which are apparent in its movements and in its growth, may in all cases alike be described as due to the action of force. In short, the form of an object is a ‘diagram of forces’, in this sense at least, that from it we can judge of or deduce the forces that are acting or have acted upon it: in this strict and particular sense, it is a diagram – in the case of a solid, of the forces which have been impressed upon it when its conformation was produced, together with those which enable it to retain its conformation; in the case of a liquid (or of a gas) of the forces which are for the moment acting on it to restrain or balance its own inherent mobility. In an organism, great or small, it is not merely the nature of the motions of the living substance which we must interpret in terms of force (according to kinetics), but also the conformation of the organism itself, whose permanence or equilibrium is explained by the interaction, or balance of forces, as explained in statics.”

Quoted from D’Arcy Wentworth Thompson, Abridged / Edited by John Tyler Bonner, ‘On Growth and Form’, CUP, 1961, reprinted 2004, page 11. Snippet view online at Google Books here: 

Here’s a thought …

Our thought processes are reactions, not actions.

(forgive the exposition of it, but I’m keen to show you along the path I’ve discovered)

If maths processes mirror our thought processes,

If addition and multiplication equate to agreement and expansion / subtraction and division equate to negation and disagreement,

If these mental positions can be seen as ‘forms’ of thought,

If these ‘forms’ of thought can be seen as equivalent to D’Arcy Thompson’s views of ‘forms’

Then we can see that these ‘forms’ of thought could also be ‘diagrams of forces’.

And if the thought processes depend on an agonist and an antagonist, internal and external inputs,

Then the forms of thought reflect diagrams of forces – internal and external – to which  our thought processes conform.

ie our thought processes are reactions, not actions, which raises the question as to whether the subject of our attention should be our thought processes, or the character of the forces which enable their conformation.



checking your sums

20 07 2010

At the algebraic level, you can check your answer by substituting in your numerical answer for the unknown, and then finding out whether the left and the right side are equal, as initially proposed.

There are other, simpler examples of this process. Times tables. 4 x 8 is 32. How can you tell? Well:

1. it sounds right :)

2. 4 x 10 is 40, so two fours less than this is 32

3. count up in eights: 8, 16, 24, 32… the fourth item is 32

4. count up in fours: 4, 8, 12, 16, 20, 24, 28, 32… the eighth item is 32

Two different ways of getting the same answer suggests you’ve got the right answer. Like the general rule of thumb that if you find a fact in two “independent” sources, they greatly increase the chance of them being right.

This should be provable simply with Bayes’ Theorem.

One instance, me thinking there is another side to maths, simply makes it peculiar or unique. Two instances, and we have some validity, and so on. The jump from one to two is huge. This goes for a lot things. 2020worldpeace springs to mind…

two ways of thinking about division

20 07 2010

I may have covered this, but while tutoring it became crystal clear. Consider:

8 / 2 = 4

Actually, when I look at it like this, it immediately smacks of fraction. However, if we were using the more normal divide sign, there are two ways of interpreting it. At least, perhaps.

1. “how many two’s are in eight?”

2. “what is eight divided by two?”

The difference becomes particularly graphic when we consider fractions:

2 and 1/2 divided by 1/2… is 5

This is a bit of a jump, mentally, for most people. Which of the above interpretations fits? ie

1. “how many halves are there in two and a half?”

2. “what is two and a half divided by half?”

In my mind, the first makes sense. It’s to do with shapes, or words even.

“how many half-slices of pizza are in two full pizzas and one half-slice?”

“how many pairs in eight?”     :  :  :  :

This is very different from thinking about division as cutting, ie

“what is eight divided by two?” or ” what is eight divided into two groups?”   : :      : :

Now consider a bunch of kids in a room, and at one time they may be thinking of division one way, and then there’s another way being explained to do something. The way of thinking must be contextualised, rather than there being a rule that is true for all time and for all things.

Of course, this is simply the difference between 4×2 versus 2×4. Wow. I thought these were the same. But they aren’t. At least, not in terms of division. In the first way, we have four groups of pairs, and in the second way we have two groups of four. In terms of XQ, and the two different ways of thinking about addition (the action being on the plus, the action being on the equal sign), the action here is in terms of the mind. The divisor determines the number within each group, or, the number of groups. Again, counting in terms of things, and counting more in terms of a higher order, cuts, divisions.

OMG. Can we teach primary teachers to have this sensitivity? Why not? It’s pretty simple really. It’s a matter of listening to what a kid is doing. Noticing, that’s all. And the best way to show that you understand is by giving them more that they can do well, which gives them confidence, and then giving them things that they can’t seem to get with the same thinking methodology. Or perhaps getting two kids who happen to be doing the two different ways confidently to engage and see what they make of it.


A perfect example of Applied XQ (perhaps)

14 07 2010

I wrote an email to David which I subsequently posted up on his 2020worldwalk blog at his request. In doing so, I noticed the appearance of an amazingly powerful rhetorical device – in which language, structure and meaning came together so beautifully, it took my breath away when I realised what was happening …

>> there’s a difference between giving up on what you instinctively know is not working, and giving up on a certain position and redistancing to reframe, realign, redefine, re-whatever – you fill in the suffix/term. Re … what????

What could work in terms of collectivist structure will emerge spontaneously as you well know – no child was ever born able to walk, but children do have self-righting systems and a will to walk.<<

I pointed this out to Giles – he didn’t GET it … at first. Hmmmm …

From flocking to fucking hell

14 07 2010

Today started well – I was flocking, or at least playing with the idea. Creativity was flowing, I was in the groove, relaxed. By the end of the day, I’d been swept up in the eddies and currents of an organisation and system at odds with me. I ended up in a dualistic, antagonistic state with a client, close to discord. The situation is redeemable and in the long run a learning point, not a catastrophe. A similar thing happened yesterday when two stances collided – mine, flexible, yet seething with a sense of injustice versus my adversary’s which was unmoving, with a parallel sense of righteousness. The only option was to walk away.

This is neither here nor there – what is more interesting is my awareness of it, contrasted with an ‘other’ state it is juxtaposed against as another option.

My thoughts are as follows:

In both situations I felt I was in a state of disharmony internally and the external relationship was one of discord. I felt acutely aware of an internal self-righting mechanism unable to balance. The image of the gyroscope arose spontaneously – just like a gyroscope juddering, falls out of balance, it needs to be set in motion again from a point of stillness. In terms of consciousness, this could equate to the still small point of dissolution, of stasis, or absorbtion, perfect balance, from which perfect spin can be unleashed.

So here’s a question for David – if XQ is about maths processes being a mirror of the processes within our minds, what is the mathematical process which is equivalent to the above?

What is the mathematical equivalent of balance? For that matter, of love, trust, terror, grief, purity, of pain?

I had a sense I could not get to =

But a maths problem cannot solve itself.

If the process of thought // maths processes

and processes are operators, eg +, -, =

with the goal being to solve, to find an answer, a resolution, the main operative being =

and we instigate the process mathematically, what is the instigator of the process mentally?

What is the hidden spinner of the gyroscope of consciousness?

Reason? Rationality? Intuition? Other (eg sixth sense, higher wisdom, ???)

Applying the thought experiment of the 5 dots and the sum allows different viewpoints to emerge, a shift in consciousness, but a clearly defined framework, // maths is lacking to allow balance in consciousness to be achieved. It exists, I’m sure – will defining it be possible? desirable?


13 07 2010

I promised David I’d post up the zen parable of ‘the vessel and the grain’ (although it’s not about them at all):

Skill creates the vessel to be filled.

But until the vessel is filled, its purpose is not fulfilled.

And until the vessel is created it has no purpose to fulfil.

If the grain for the vessel grows before the vessel,

It has no containment,

It spills and disperses.

A vessel without grain is barren;

Grain without its vessel is scattered,

And feeds neither mind nor body.

As the grain grows mysteriously,

The vessel is prepared.

The vessel-maker’s skill sings the song of the growing grain.

The grain, filled full of earth and sky,

Thrusts up to its fulfilment.

When the grain is ready,

The vessel must be ready.

The vessel-maker must not lag behind.

Thoughts on Reading Alex’s Adventures in Numberland

10 07 2010

Picked up a copy of Alex Bellos’ book, Alex’s Adventures in Numberland. Readable, observant, well-written.

I learned some new terms …

If I focus on a group of 5 carrots, I’d be thinking in Cardinal terms. If I were counting from 1 to 20 , I’d be thinking in Ordinal terms. (p23)

pug – xep xep – ebapug – ebadipdip – pug pogbi

The Munduruku count ‘one’, ‘two’, ‘threeish’, ‘fourish’, ‘fiveish’. (p34)
Their words for the numbers one to five have as many syllables as the number up to four, then drop off – does 5 become a handful? A bunch? A pinch?

Logarithmic thinking is the ability to compare ratios rather than quantities, figure out best odds in a survival situation, and linear thinking – the kind of thinking we use when we contemplate numbers in sequence. We think differently when counting and comparing. (Bellos, pp 19 FN ref to p 190) Wow!

It reminds me of 38 Parrots, a classic Russian cartoon featuring a group of jungle friends – a snake, a monkey, a parrot and an elephant . In one episode, the elephant asks, ‘how many nuts do I need to pick to make a huge pile?’ … after some ordinal / cardinal to-ing and fro-ing, the monkey provides the definitive answer .. ‘lots’. It’s not so much the words, but the complete conviction with which they are uttered that makes the statement so compelling.

We put so much emphasis on training cardinal and ordinal perceptions of number yet we instinctively use logarithmic thinking to get through our daily lives – time passes slowly when we’re bored, it rushes when we’re having fun. It ceases to exist when we’re in the zone / groove. Perspective changes the size of objects, although the distance between them is the same. We can process this without a problem. I can tell without counting the floors in a building whether I want to take the stairs or the lift and estimate how long it’ll probably take.

I think this explains the success behind many advertising and marketing strategies – they target the logarithmic parts of our brain, bypassing the cardinal / ordinal functions. Wow!

Would an awareness of and increased familiarity with these thought modes make us less gullible? I believe it might – just have to devise a few exercises to address this.

Bellos points to a recent study (Only referenced as a 2008, John Hopkins University and Kennedy Krieger Institute collaborative study and not listed in the bibliography. It really annoys me that editors are paid good money to edit books and get away with substandard work in so many books that are published nowadays) that shows ‘a strong correlation between a talent at reckoning [logarithmic thinking – guessing whether there are more blue or yellow dots in different groupings] and success in formal maths. The better one’s approximate number sense, it seems, the higher one’s chance of getting good grades. This might have serious consequences for education. If a flair for estimation fosters mathematical aptitude, maybe maths classes should be less about times tables and more about honing skills at comparing sets of dots.’ (p 33)

Not so much of a wow – what would the use of that be if no connection is being made at a functional, internal level?

Can you give the answer to 7 x 7 or 8 x 7 as fast as you can 5 x 10 or 6 x 4?
How do you work out the sums?
What’s going on inside your mind?

In Munduruku mode, we find it easy to think of 1, 2, 3 – even 4. This is borne out by Lawrence Potter (Mathematics Minus Fear, Marion Boyars publishing, London 2006, pp 34-40) in terms of his observation of relative difficulty of multiplication / times table retention and recall. 1, 2, 3, 4, 5 are fine. 6, 8, 10 are easy – double the relevant proportional sums in the lower groups. 9 works back from 10, but 7 seems the hardest to recall. It certainly checks out in my own mind.

Bellos mentions Dyscalculia – it seems that Dyscalculics can be very good at logic and geometry that ‘prioritize [sic] (I hate the cultural infiltration of American spellings in the UK edition of a book that is published by a UK-based publisher with offices also in the States) deductive reasoning or spatial awareness rather than dexterity with numbers or equations.’ (p 39)

A final  observation – I’m observing a selective function operating here – filtering out things which don’t interest me – traffic cones in the introduction (pp 8-9); experiments with animals (pp 21-28); experiments with kids; perceptions of mathematical processes (pp 28-30). Logarithmic processes working as I read … and comparison and recall as well. On p 30, Bellos outlines similarities between visual depictions of quantity in Roman, Chinese Indian number systems. He doesn’t mention Arabic numbers (ie written in Arabic script). There was a better expose’ of the correlation between script and number which went up to 10 displayed as part of the 1,001 Inventions exhibition of Islamic culture at the Science Museum earlier this year. Mental note to contact them to get the information. When I do I’ll add it to this post.

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