the ferrer diagram of 12 as monochrome trains…

‘Ferrer’ means ‘complete breakdown’ that is all.

This below is a breakdown of 12 showing its prime factors:


Each row contains ‘trains’ of the same colour. You would ask children to ‘make as many trains of the same colour as you can find that will fit underneath twelve’

(or something else of the same meaning…)

you can call 12 a composite number if you wish…

if a number only has trains of white and itself it is called a prime number…here’s a few:

ferrer 2 comp primes

however the main thing here is to notice that

the number twelve has a very rich internal structure

If you are completely and utterly aware of this, many things become easier to see, conversely, or should I say, inversely,

if you are not totally and instantly aware of the inner structure of 12


(particularly with operations involving fractional parts)

let’s hear it for the rods…

the ultimate zen assessment – pseudo object oriented reality…

Warning – the analogy to object oriented programming is FALSE

(if you want to see the point of this post go here)

Everything you and I perceive takes place in the mind. Your senses provide information to your mind. Your mind becomes aware of what you call reality due to information supplied by your senses which is then subsequently processed according to your established interpretative neural networks. These networks are evolved through awareness and analysis of these perceived conditions during your evolution as a perceiving, conceiving being… i.e. whatever you have experienced and then ‘made of’ these experiences.

Clearly, this involves your total historical and psychic environment. It is then clear that ‘this certain something’ that ‘you’ perceived was an interpretation of an interpretation, and was only an image of ‘the real thing’ whatever that was. This certain ‘real thing’ cannot be known absolutely. That is why great masters such as HUANG PO pointed out the ‘error’ of conceptual thought processes:

“There is no  “self”, no “other”. There is no “wrong desire,” no “anger,” no “love,” no “victory,” no “failure.” Only renounce the error of conceptual thought processes and your nature will exhibit its pristine purity-for this alone is the way to enlightenment.” HUANG PO, Wan Ling Record 24, p.86.

This ‘error’, is merely the knowing that what is perceived as ‘the truth, the Absolute Real Reality,’ even in its brute external form, as Searle would describe it, is NOT IT ITSELF. It is only ‘one interpretation of it’, and this is all we CAN KNOW. We cannot know ‘IT ITSELF’, because for us there is no ‘it itself.’ All we can know is what we perceive and then conceive through interpretation. This is relative reality and is different for all beings. This is ‘our world’… This is ‘my world’, this is ‘your world.’ This is why it is said that we ‘create the world.’ This is why there are as many cities you live in as there are perceivers of the city. There is not ‘a city’. There are no unique events.

Clearly there are ‘events’ at some level. There are earthquakes, there are floods. There are divorces. There is love. You will be hurt by the master’s stick. Yet you are the perceiver. You ‘create’ your specific take.  You create ‘your’ world. The external, the unknowable, is the CLASS which is knowable to us only as a fragment or ‘taste’. The specific, our individual  realities, are the INSTANCES.


huatou: I am the world

the flip law 2

it works for multiplication and addition but not subtraction and division.

in multiplication it works for fractions as operators too

play yourself…you must own flip it as second nature

ps the normal name for flip it is the commutative rule

pps dressing is not commutative – you don’t put your socks on over your shoes

the flip law 1

This rule is profound and will change many things, illustrated here with a few rods:


language and rod domain (with a bit of number):

two threes is just as long as three twos, and also they have the same volume

number domain with some signs:

2 x 3 = 3 x 2 = 6

al-jebr domain with signs:

a x b = b x a = c

more and more abstract, more and more general

disambiguation blurb: in the ‘real’ world, two green rods is not the same as three reds. This is why some people object to agreeing that 2 x 3 is the same as 3 x 2. They are correct. However, in the number domain, the answer, which is a pure number, is not affected by the order. The product as a number is invariant to the transformation. If you want another example, it is as invariant as taking a homotopy group functor on the category of topological spaces. You probably don’t need this information. As most people working on calculations are looking for ‘the answer’, one can say that for all intents and purposes, calculationally speaking,

the order of operations in multiplication is irrelevant

further more, if you wish to mention it, and which also makes no difference to the product, the number sentences, transformations or equations, whatever you want to call them, contain the sign ‘x’. This sign is called an operator and it has to be attached to something. It has to be attached either to the first numeral or the second in this case. If it is attached to the first numeral, like this ‘2x’, this ‘whole’ is again called an operator, in this case a ‘doubling’ operator. In language it says ‘two lots’ or ‘two groups’, so this, 2x  3 says

two lots of three or two threes

2x  3

with little children, the easiest and most meaningful form is by using this choice in the attaching of the operator


a) just like in reading, one reads the first numeral first and

b) one doesn’t have to hold the first number in the mind to the same degree as the form below whilst reading the second number. (Saying ‘two threes’ seems somehow less complicated for little learning minds than saying ‘two multiplied by three’)

c) one doesn’t even have to mention the ‘x’ in language, merely recognise it ….two threes

nevertheless the product is still invariant to order

if the operator is attached to the 3, we get 2  x3 which says:

two multiplied by three

2  x3

most teachers call this the ‘correct’ way, but it is just one way

the x3 becomes a ‘trebling’ operator

so, in summary, and for the benefit of little children:


and this 2 x 3 with the operator ‘x’ in the middle, at first means ‘two threes’ to little people

later, with much practice, it looks like ‘two threes and three twos’ at the same time

(ps you can introduce all the other ways of saying it whenever you feel it’s appropriate)


what exists?





Peter, I heard you once say that places don’t really exist. What did you mean by that?

Well, they do and they don’t. I mean, take Moulay Brahim. You’ll find it marked on a decent map just to the south of Marrakesh, just off the S501, and there’s no doubt that if you told a friend that that’s where it is you’d get no weird responses. Everyone would agree. It’s one of Searle’s ‘brute facts’. However, once you start to ask questions, such as “what’s it like in Moulay Brahim?”, or go there and experience it for yourself, you’ll have to agree that because of the particular things that you are bound to experience, including any preliminary information or stories or expectations that you might seek out, the Moulay Brahim that becomes represented in your mind will be unique and different from everybody elses’ representation of the place. Because the representation is not a ‘place’ is it? We can’t BE Moulay Brahim, we can only create images and feelings and memories. Where are these memories and why they are different for us all is indeed very interesting. In fact don’t you think it’s a wonderful idea that whatever you personally construct Moulay Brahim to be is a unique gift entirely and totally yours?

Yeah, I see what you’re saying but it sounds a bit scary too. If you look in a guide book you’ll see things like the shrine and it’ll probably mention places to eat and so on……..

Certainly. Searle called those kinds of facts, non-mental brute physical facts, and really guide books should be considered to be just that. Lists of brute facts. You see once you are introduced into the equation, interpretation begins and this will be unique to you. How can it be any other way? You will also experience things that will be different to the person who wrote the ‘guide’ book. Maybe you’ll meet an interesting helpful person or maybe you’ll be mugged. You might get food poisoning or you might have the most wonderful meal of your trip. The sun might be out or the weather might be uncharacterically cold for the time of year and so on. You might be feeling a bit down or not too lively etc etc. All these factors will colour the ‘representaion’ you will construct about Moulay Brahim. There are as many Moulay Brahim’s as there are people who’ve been there. There’s no escape. In fact everything is always like this. It’s great.

OK, but what was it like for you though when you actually went, if you can put it like that?

Of course you can, but the point is that you mustn’t ‘believe’ my version, go and see for your self. It was OK. My partner and I were travelling with two friends who lived in Marakesh and they knew a few tricks such as eating a ‘safe’ meal straight from a steaming hot tagine with some freshly baked bread. That was very tasty and a welcome interlude. We wandered round for a while looking at horribly nasty bits of dried creatures hanging in stalls. I mean we didn’t talk to people much as we don’t speak hardly any arabic, and I know I felt uncomfortable about taking photos of poverty stricken people, so I hardly did. Some people looked happy, some appeared to tolerate our presence, no more. I really had little idea of what was going off in their heads, what they thought of us, what they did. We were basically aliens, wandering round on an alien planet. We bought a few things, had a coffee. Whilst walking round a few back streets the girl in the picture and a few of her little friends threw stones at us and tried to hit me with a stick. It was a priveledge to be there but we were tired and in a way it was a relief to get back in the car and head off back to Marakesh. It sounds a bit pathetic really.


Well, its different..


cover sets 1 how to sort the gold from the dross

1 cow + 1 cow makes 2 cows.

1 sheep + 1 sheep makes 2 sheep.

What’s one cow + one sheep? Mmmm…you cant do this unless you make a new category, called ‘animals’. Then:

1 cow + 1 sheep makes 2 animals



‘ANIMALS’ is a COVER SET and exists at the n+1 level.

‘COWS’ and ‘SHEEP’ are COVERED by the category above.

In this little diagram there are 2 animals at the nth level.

The DIMENSION of the nth level is 2.

If you added in meerkats and duck-billed platipuses (there appears to be no collective noun for platypus), the DIMENSION would be 4.

This is how mathematicians define DIMENSION.

Cover sets and dimension are going to come in useful when we start to create and criticise CURRICULAE… ahead…

You see you don’t want CATEGORY CONFUSION. That sounds like it could make you ill. However there is a lot of IT about. For example, you probably heard quite a lot of politicians with diametrically opposite views saying the same thing such as;

‘We want the best possible education for the children in our society’

Yes, true, we do, but the problem is they are relying on CATEGORY CONFUSION at the n+1 level to misle you into believing they are doing the RIGHT THING. Everybody of every persuasion agrees at that level but the point is what lies ‘beneath’ at other levels, what dimension they are referring to and what SPECIFICALLY do they mean.

I saw a T shirt once: ‘CAN’T YOU BE A BIT MORE F****** SPECIFIC’, excuse the french.

COVER SETS and DIMENSION are two of the tools that will sort out the idiots from the smart guys.