Illogical Logic and football management



Numbers and Agreements is an exposition of the shortcomings of Bertrand Russell’s Theory of Classes and the errant Logic of George Boole upon which our hopes for Civilised entry into Interstellar Commune are currently based.
It simplifies and reduces the Rules currently in industrial use rather than adding to them.
The backbone of the illustrative ideas are driven by periods of scholastic research mainly conducted throughout Glasgow Rangers FC Nine in a Row league title winning run and also the subsequent revolution they inspired and funded within Scottish Football when they introduced the Souness and David Murray partnership.
Although Murray International Metals has as yet been unable to build a working Starship, Glasgow Rangers is still today one of the few premiere teams in the U.K.
(That last statement shows this document was written in 2003 before we found out that Rangers had been having some illogical cash flow practises)

For the purposes of making good sensible examples, however, certain fictional constructs about personnel in the Rangers’ Teams were invented to illustrate why David Murray has been unable, as yet, to fund a successful Starship project by using the current industrial rules of electricity.
Any resemblance to the behaviour of anyone, Living or Dead is entirely incidental, and profuse and humble apologies are ‘a priori’ forthcoming in the event of any offence being taken whatsoever.
We are all realistic, however, and we must all want our sons and daughters to be Astronauts so that we can make a richer world for ourselves and our grandchildren to stay in.
One recent night in the Leith Oyster Bar, having watched Rangers get done over again on some muddy farm track in Transylvania in a futile pre- pre qualifier for the European Cup qualifying rounds – a game refereed by a guy in black doublet and hose, I  began to disassociate, having acquired some of those injuries myself on bad football pitches (that were almost as bad as that one) when my attention was grabbed by a loud, loud discussion about Logic.
This was the Leith Oyster bar and we come here to drown our sorrows on Staropramen an imported beer from somewhere very near Count Vlad’s old castle.
I was shocked at this perceptibly illogical intrusion on the subject of Logic and what is more, I felt like disagreeing with it.
Firstly let me get real here, I’m an agreeable sort of guy. If I agree with anyone and I remember what I said and what he said, and maybe if he had a friend that agreed with him, and I later remember that I agreed with his friend but didn’t say anything at the time, I can have further grounds for discussion later with his friend if I see him in my local bar.
That’s called society amongst people who agree on common grounds.
The grounds for discussion here, are though, that it’s now his round.
Some may write it down in their diaries, or the barkeep may keep a tab, or some may send an e-mail, or leave a message on their answerphone, but in the society of heavy mathematical drinkers who organise themselves around a table in our local winebar – we are a set of people with obvious financial commitments and agreements to our own sense of party.
We all agree on one thing though – that booze is a good thing to drink no matter who hasn’t yet bought us a round.
I’m not only a realist, but a logicist too and I often find it amusing to listen to the erudite conversations of the rich-kids who did not and could not make the money for the lifestyles that they espouse and who had to be getting supplied with free money.

I’m there having a vodka chaser and I’m listening to a discussion between Horace and Oliver  giving themselves and indirectly me, in a manner as loud as their social packaging can endow, the third degree of  ‘technical mince’ oft heard in the Professorial Ward at the Royal Edinburgh Mental Hospital.

I can still agree with myself to internally associate in my own relatively pain-free way with that discussion – and by no means am I feeling too hurt to rethink it for you as I can still tolerate passive intellectual pollution as much as I can passive smoking.
They called the stuff that they were discussing the Principles of the Calculus of Classes and the Rules of Boolean Logic.
Horace and Oliver, who had been at it all night, were upsetting the regulars and they were not buying. In fact they were a waste of a couple of good barstools.
At first I thought that they were going on about commuters and that a new law had been passed by the local council that said that if folks were in a bus queue and the bus turns up – then it’s a free for all and whoever gets on first is ok with the police. They obviously hadn’t ever been waiting for a 12 at the foot of Leith Walk in the rush hour. They said that was the commutative law. That whatever commuter gets on the bus from that queue in whatever order was now ok in law.
I was shocked to learn that they were talking about something called Logic.
For me, who got my early education in Logic from Dr Spock in Star Trek that just didn’t seem right.
Seeing how they might get beaten up outside the pub by several tradesmen for suggesting these things, I advised them to keep their voices down and tried to make it as easy as A,B, C for them so that they would understand the meaning of life  and reality on this planet.
I sagely advised them to come over to the fruit machine as I may have a payout for them.
They followed me over to the games machine where the brightly-lit display showed three fruits: Apple, Banana, Citrus.
Looking around to see if the regulars had again returned to their drinks, I began to explain.
You see these fruits, lets call them A for apple, B for banana and C for citrus orange – ok – do you agree?
Smirking like they were on drugs or something they nodded.
You were saying that A + B = B + A  right, agree ? They agreed.

Well, a lorry loading with two small containers labelled A and B from Holland at the local Ocean Terminal still has the same consignment - unless the order of despatch and drop-offs were under-equipped it wouldn't technically matter which of the containers came off first - as long as the standard hoist and or fork-lifts could be used at the delivery depot. It's all the same if the contents are of equal strength, size, weight, politically correct in trading provenance, and quality, but it might not be.
Technically, therefore - and as far as Mr Capone is concerned, two crates of dope are two crates of dope.  As long as they are on the lorry and arrive on time and are distributed appropriately - business isn't fussed whether they come off as A + B or B + A.

A + B = B + A though isn't a Universal Law - it's Irish bullshit, and I’ll tell you why.
See this fruit machine - well let's talk about fruit.
I’m talking about fruit, only about fruit and nothing but fruit here – so with a fruit example in mind – do you agree to continue? They said yes.
Well suppose I live on a diet of fruit and that I had large quantities of bananas, apples and citrus orange to eat.
The three fruits have differing effects on my digestive system at different times of the day, and, my digestive system at my age can sometimes give me problems.
If I dine on Apples first thing in the morning, the apples can be hard, variously difficult to eat, even if the same variety and the same consignment, and not often sweet. It can be a sour and painful start to the day especially for a guy my age.
If I dine on bananas first thing, they are sweeter, and although they have several flavours during their ripening cycle, and have the same organic tendency to inconsistency no matter how branded, they are more consistent in that they don’t provide any shocks to the palette or the gut. They also have a higher sugar content, and are easy to eat.
If I dine on citrus orange first thing, I will give myself acidic indigestion, and as a result, an excess of reaction in my oesophagus may produce other undesirable effects.
Orange is always more acidic for me than the other fruits.
For me then, B is the best start to the day not A – agreed?  Smirking less they nodded.
Well for me then A+B is not the same as B+A no way.

For others, however, the matter is arbitrary; e.g. if two equal containers at Govan shipyards equally full of pine disinfectant have to be loaded onto one HGV lorry for transportation one labelled A, the other labelled B, then it really doesn't matter (usually) which one gets heaved on first unless unloading conditions at the destination points are very restrictive.  
For George Boole though, in 1874, there was only one way to get the Protestants.
Now Rangers, that's Glasgow Rangers, not Queen’s Park Rangers, had three midfielders that I can associate with an era of Ranger’s supremacy – before they got hacked to death on a muddy farmyard in northern Europe in a totally vindictive pre-qualifier to the qualifying round of the qualifiers for the European Champions League. A fate reserved for all Scottish sides.

These three midfielders in no particular order of preference had attacking qualities and defensive qualities, but (Konterman), Conterman was more of a static mixer that would tend to slug it out in a tight spot. Good for the Scottish style at home, or for a defensive Euro tie, but in Europe, liable to get skinned by tight control and also not prone to supporting lightening raids but could get the odd vital goal.
However, A for Albertz and B for Bronkenhorst were the two that could tear a big rich Euro outfit apart.
So for the purpose of my Associative Law of Rangers Midfielders for dangerous Euro Occasions that needed fire power up-front to maybe make up a lost goal from the first leg, I would start with A and B, then if we got one back, then bring on C as a late substitute to stand and hold it.

Boole, therefore did not create a Law, but a strategy for plenty of footballers for Glasgow Celtic and the Rangers squad for which, today, we are all extremely grateful - God Rest his Universally Irish soul.
Plenty of Boole gives you Order, but also, plenty of Boole gives you disorder too.
The Black Magician from the Order of Thelema, Mr Alastair Crowleigh suggested that 'Do what thou wilt is all of the Law.' which is a bit of a distortion on Civilised Logic.

Well you described this commuter thing that you got wrong then went on to talk about things like Mothers and Children don’t matter and that if they got separated in the rush hour by other commuters helping themselves in a free-for-all at the bus stop then that’s fine.
You upset those folks over there I said, quietly indicating a couple of folks who had recently lost a marriage in a roof leak in their cheap un-insured council flat.
The Actuary mathematics of the insurance companies use this kind of Irish sentience too – concocted by George Boole in 1847. When the insurance company man came to evaluate their claim, he just spouted Boole-shit.

I don’t know who told you that it was a Law of Association, but frankly it seems a bit insensitive to me.
To return to football contracts and not speak of other social contracts, at  Ibrox, no manager that needed attacking firepower with pace would fairly expect Conterman’s style to consistently deliver against AC Milan. So C would not feature in a starting line-up for that kind of match.
It therefore does matter what order the pairings proceed in a European match.
The law of team selection is the same as not breaking up a good pairing at a Leith bus stop. That is – when you said (A+B)+C = A+(B+C) is a rational law – I think you had better watch your back before you get into your BMW out there tonight.
If you had asked anyone at the Leith Walk bus-stop about whether or not that team selection was a good idea or a bad one, they would have said that they didn’t care – either because they had no interest in football due to other social disasters, or that they supported Hibernian and any selection was going to do at Ibrox as long as it lost on the night.
Some folks farther away than Glasgow e.g. in Leith, may not actually bother about what team Rangers will put out – especially if they want them to lose.
Whoever taught you that that Associative Law was a law, was a loser.
When you said that you were doing Sum Rules, it sounded like you meant that the sum of the whole was greater than its parts. This is indeed true, but when I heard you say things like A+1=A after you said the more sensible thing like A+0=A, then I began to wonder.
It sounded like you were going to give Jorg Albertz a ‘One’,  to me – but One What?
If it were foodstuffs good or bad, alive or dead, he would still be Jorg but having digested one big meal he would have put on weight therefore would not technically be the same A for Albertz he was prior to eating the one banquet.

He may take it ill out if he hears you say that he should ‘go forth and multiply’ back to Germany.  Two clones from the same DNA will not produce two identical Albertz's, nor do they add up to identical Albertz's when you say A + A = A. Even if he was cloned at Rosslyn Research Lab, his attributes as a player would be different both biologically, phenotypically and psychologically.
He is though in the set of German football players, but there is only one Jorg Albertz and that’s the specific name of the ‘A’ that we are addressing.
There may indeed be an infinity of look-alike A’s and other objects that are not ‘A for Albertz’.
Albertz found that out trying to get a game for Germany under the manager called Goedel who didn’t think he was good enough because he liked playing in Scotland.
Goedel would eternally number every other player in the national squad before he ever got to Albertz.
Goedel called him on the phone one night and said that if A + not A were to come about then there would be only one player of choice for the German midfield and that was not Albertz, but Oliver Kahn, the flappy and geriatric German world cup veteran.
Jorg, much miffed by such derogatory treatment went off back to Germany.
Where you two guys seem to think that A and Not A equals 1.  

I frankly disagree with that selection for the German midfield, Which One that is,  .. for even Jorg’s clone couldn’t fill that vacancy and we would have to agree to disagree on which hard hitting midfielder could in my opinion.

Albertz’s long association with professional football both in Scotland and in the German top division means that he had obviously developed some sophisticated product rules that he agreed with his agent as the best way of making money and futures.
Your Logic teacher obviously wasn’t a good businessman because he just wouldn’t have got a good executive job anywhere by the sound of it.
In fact where did he get his degrees ???

I agree with your Logic teacher when he says that Albertz times zero salary equals no Albertz, i.e. A.0=0, but I then disagreed that Albertz, if he were forced to marry his male clone replacement that he would be happy to stay at home and be a househusband i.e. A.1=1. Your Logic teacher is suggesting to me that there can only ever be one clone, or one other idea that Jorg doesn’t matter. I don’t agree with that.
Whereas, your teacher of Logic can assert that it may be a matter of personal taste for Jorg, however, if Jorg leaves his clone at Home to do the  unskilled labour and sustain in an indirect way his career on the pitch i.e. A.A=A, I do not agree with that either.

In fact – if you pass me that old newspaper from under the Star Trek ashtray – I’ll illustrate it a bit more for you – in case you have to again visit the Purple Haze Café to replenish your stocks of natural hemp.
I’ll use some jargon in case you show this stuff to your logic teacher, but you can skip to the basic illustrations that I paraphrase using ordinary streetwise ideas that the average bod like myself needs when going shopping.
Only half of George Boole’s stuff is Boole-shit though – but if we cut the crap – we may be able to build computers and NASA shuttles that work, or buy an insurance policy that is safe and not so expensive, or maybe even lots of insurance policies that hardly cost anything.
If I had gone to University and had done Laplace and Fourier Transformations, calculated electromagnetic flux and tolerances in my new Warp Drive design and had gotten a research grant from the CIA and NASA, tooled up with lots of nice flashing lights and various mixing desks, and had been telling my CIA minder girlfriend that one day she would be running the Company, despite her tendency to be inattentive at times even in the Bahamas – and then compiled my years of mathematical expertise using the widely available and socially essential rules of Boole – I might have found myself surplus to requirements, as no doubt would half of Nevada.

Complex mathematical electrical engineering ideas can be Logically modeled using conventional ‘Logic gates’ in the design of circuit boards and are a part of the Octal arithmetic used by Boole in the traditional Boolean circuit designs.

However, In current industrial designs of electrical technology, that are unable to build the Starship Enterprise there are additional levels of sophistication added into the decision making process using such rules that can infer no rational outcome.
Beyond the basic logic switch circuits called;

  1. the AND GATE
  2. the OR GATE
  4. the NAND GATE
  5. the NOR GATE

are the compilation stages of every electrical design process.
The compilation stage uses 3 rules of ‘Boolean Logic’ that are falsifiable and can be contradicted in many different contexts and applications.
These rules do not satisfy the criteria for being Logically sound for the purposes of building the Starship Enterprise or its Warp Drive Reactor. I assert that empirical research on current components and designs may prove them more wrong than that.
These Warp Drive experiments may run out of more than Dilithium Crystals, which is perhaps a good social reason for the Purple Haze café in Leith where more legal social sedatives such as cannabis can be heaved in off the banana boat from Rotterdam instead.
Federation Scientists predict, however, that problems arise when large numbers of Logic gates are being compiled in highly complex electrical engineering projects.  The IEEE dictionary edn. 4.  lists many types of electrical distortion but defines distortion as an undesired change in waveform in terms of;
a.  A non-linear relation between input and output at a given frequency.
b.  A non-uniform transmission across different frequencies.
c.  A phase shift not proportional to the frequencies etc.

Without recourse to speculation on the empirical aspects of current electrical engineering performance, it is possible to illustrate big problems in using some Boolean calculus of classes to compile logic circuits.
It then becomes possible to identify that certain of the Boolean laws are currently in use by reference to the Electronics Engineers Handbook edn. 3. Fink & Christiansen eds. 1989 and, the main material of the faculty of Actuaries in their studies on insurance and viability of social infrastructure.

However, although, the Boolean calculus of classes is used and referred to, only the following laws are deemed adequate to construct future associations within the logic modeling – they being in line with a picture of rational equivalence within the symbolism in use herein.
The Boolean laws deemed true for purposes of building the Starship Enterprise are:

1. Commutative Laws   A + B = B + A,
                                       A.B = B.A

2. Associative Laws     (A+B)+C = A + (B+C)
                                      (A.B).C  =  A.(B.C)

3. Distributive Laws      A.(B+C) = A.B + A.C
                                       A + (B.C) = (A+B).(A+C)

These ideas depicted by the algebra above demonstrate non-arbitrary; additions, associations and shifting of  permutations of classes that do not impose abstract processes onto real world objects. Nor do these processes, unlike the other Boolean rules of; sum, product and absorption, suffer from generalization problems.
The three good rules could be successfully used and employed at a supermarket it being a matter of store policy or agreement what objects could go on what shelves.
The sum rule of Boole, however, that A + A = A would not pass muster with the checkout operator unless A was on special offer.
Taken to the level of circuit board design though, it could be said that although logic gates are logical, as are supermarket checkouts, problems such as attempting to use those Boolean tactics at the checkouts could cause a lot of loading on the Staff at busy times.

The process of A to B through C when compiled by Boolean Rules generates various arbitrary consequences, and they can be described by a series of ‘three variable Karnaugh Maps’ or topically, ‘three variable maps’. These have many different outcomes designated as true or valid.
Within current Electrical Engineering, the rules of Boolean algebra identified below as irrational in the [T] strategies, are still currently in use to contribute to large scale electrical engineering projects and processes.
This strategy as defined by; Fink, D and Christiansen, D,  ‘Electronics Engineers Handbook’ edn. 3, pub McGraw-Hill, 1989, NY, St Louis,
ISBN 0-07-020982-0. As a ‘Minimum-Complexity Combinational Network’, (page 72) .. ‘ The important design aim of reducing network complexity usually leads to lower cost and greater ease of construction (3-51)
‘Minimum complexity may have several meanings, some of which are in opposition. A minimally complex network may be defined as having:
1.  A minimum number of gate elements.
2.  Some of the set of gate elements in a set of.
3.  Fewest number of inter-connections.
4.  Wired with fewest numbers of cross overs on the circuit board.
5.  Minimum total compilation cost.
6.  Easy to maintain and repair.
7.  Highest speed of operation.
8.  Highest reliability.

In section 3-52 of this book the logic circuits in a project are minimised by Karnaugh Mapping using two basic laws of Boolean algebra,
a.  X + notXY = X + Y, which I here deem false.
b.  XY + notXY = Y, which I here deem false.

Both of these assertions predicate on the idea that whilst Y is both visible and identifiable, some other process not identifiable within the assumption that is not X, a third aspect called not X is different from being either X or Y. Cutting down to basics, the two equations read therefore ..

a.  X + ZY = X + Y
b.  XY + ZY = Y

Both of these ideas are used to ‘minimise complexity’ when cutting corners on complex electrical engineering design.
This causes problems that create both real and tenuous relationships amongst the circuits. These states are denoted, 1, 0 or D states and add up in the electronics industry as large amounts of tenuous blocks of 1’s and the occasional and singular D which denotes the only logically real path for the circuit. E.g. (3-53)
These ‘sums of product’, ‘standard sums’ or ‘minterm canonical forms’ help simplify ‘quite unwieldy large networks’ (3-51).
‘When a Karnaugh map is used to find a minimal representation, one tries to combine adjacent 1-squares into larger groups. Each group that can be made which is not properly contained in a larger group is a graphical example of a prime implicant – and compiled on the Karnaugh truth tables. (Page 74, 3-51)

An example of a Karnaugh ‘truth table’ where lots of 1’s should really be adding up to lots of zeros. In this process, the one logically valid process is labeled as per (3-53) as ‘d’ in the compilation results column labeled ‘F’.
In the following table from Fink and Christiansen, 1989, (3-53) various blocks of ones would be deemed to add up to viability.


Gate    A B      C       D F
01. 0 0 0 0 1
02. 0 0 0 1 1
03. 0 0 1 0 0
04. 0 0 1 1 0
05. 0 1 0 0 0
06. 0 1 0 1 0
07. 0 1 1 0 1
08. 0 1 1 1 0
09. 1 0 0 0 1
10. 1 0 0 1 1
11. 1 0 1 0 1
12. 1 0 1 1 1
13. 1 1 0 0 1
14. 1 1 0 1 1
15. 1 1 1 0 1
16. 1 1 1 1 d

In this table from Fink and Christiansen, 1989, (3-53) various blocks of ones would be deemed to add up to viability.
Whereas row 1 could be deemed a logic gate called a ‘nor gate or invert gate’ in circuit design, it would be the ‘d’ in row 16 that defined a true relationship within that complex. If row 1 and row 16 were used concurrently, though, 14 out of the 16 selections for that electrical engineering project would be considered logically ‘unreal’ by Federation Scientists for the purposes of building a safe Warp Drive.

In line with this mathematical modeling, truth tables such as a three-variable Boolean table called the ‘three variable [T] Map’ can be used to describe the progress of connectivity between any two atomic points in a crystal or indeed, anywhere.
i.e. to get the Starship Enterprise to go to Praxis the Klingon Industrial moon to pick up some Dilithium crystals – one naturally assumes that the journey takes the continuous form of A to B through some common C which is perhaps Warp space, or in the case of Deep Space Nine, a Federation trading conduit, a wormhole.
A to B through some common C is as true for starships as it is for; electrons in dilithium rocks, food in Klingon digestive systems or indeed everything else in our Cosmos.

In this case however, it only has one logically true consequence about connectivity within and between the relativity of two points through a common medium. It is one unbroken line of logically real integrity in the relationship – independent of the consequences of arbitrary labels and assignments in an ‘a posteriori’ world of assumptions.
Much of the Boole that George Boole wrote in 1847 C.E. assumes far too much about the world. No doubt Irish whiskey or a severe lack of potatoes was responsible for that.
If we ditch the bad half of it, that is the ideas that talk blarney and keep the good stuff, then, we can use the symbolism he put together without getting the information distorted.
To be sure, then, we should only keep the rules of:
Commutation, Association and Distribution

That the electrovalent dilithium crystal conducts and  does output is in fact ‘a priori’ true. It is not made uncertain by the assignment of labels that have been previously made within sum rules, product rules and absorption rules within a three variable Karnaugh Map.

The three variable Karnaugh map, however, currently at the heart of Boolean logic and circuit design on planet Earth, much to the concern of Federation Scientists is only tenuously correct and approximate and introduces massive amounts of inefficiency into any circuit through the use of the rules of sum, product, and absorption.
These rules: (1. Sum,  2. Product, 3. Absorption) are re-described as follows in terms of a common example from the world of social and empirical objects as produced by the world of electricity and circuit diagrams were no doubt introduced by an evil robot species to thwart a First Contact scenario.
The electrical industry of Earth may wish in their reconsideration of what parts of their current software to keep using and which parts of their current design software to disable, to think of one litre cans of black and white paint and their presence or absence of full strength colours.
The sort of domestic surface colourant one may acquire in an average Homebase DIY Store.
A litre can of paint is an analogy for a Standard Industrial Unit of some energy packet.  Mixing paint e.g. a one litre can of black plus a one litre can of white will result in two one-litre cans of grey paint. In terms of full strength colour being representative of the presence or absence of voltage, and using the Federation Unapproved Boolean laws of; Sum, Product and Absorption,
It can be discerned that the results with very few exceptions are not black and white.
Such rules create problems that if re-iterated over a massive electrical project could lead to distortion effects within the electrical hardware from component overloading and failure with time.

The meta-language that I supply paraphrases the use of these rules in the Karnaugh Maps at the heart of electrical engineering and circuit design. The descriptions are supplied below as analogies of the statements being made. The rules of;

1. Sum,   2.  Product,   3.  Absorption,
    Are deemed insufficiently realistic for use in Federation projects

1.  Sum Rules

Sa. A + 0 = A         [which I deem correct]
Sb. A + 1 = 1         [which I deem false]
Sc. A + A = A        [which I deem false]
Sd. A + notA = 1   [which I deem false]

Sum Rules paraphrased examples:

Sa.      White paint can plus nothing is a can of white paint.  
(a rule that I  deem true)
Sb.     White paint in a can added to absolutely anything else in the   Universe, is always necessary for everything else in the universe that we know or can think of to continue operating.
(a rule that I deem false)
Sc.      White paint can plus another identical can now  equals one can of  paint.
(a rule that I deem false)
Sd.     White paint can missing enables the job of any and every painting  with a can of white paint to be done.
(a rule that I deem false and contradictory)

2.  Product Rules

Pa. A.0 = 0          [which I deem arbitrary]
Pb. A.1 = A         [which I deem true]
Pc. A.A = A         [which I deem false]
Pd.     A.notA = 1        [which I deem arbitrary]

Product Rules paraphrased  examples:

Pa.  White paint can goes  nowhere and can never leave anywhere for any  reason or agreement.
(a rule that I deem arbitrary and circumstantial)
Pb. White paint consignment, can A, applied to one job is a job painted white.
(a rule that I deem true.)

Pc.  White paint poured into a full can of white paint, whether a millilitre or metric tonne fits into the same tin.
(a rule that I deem false.)Pd. White paint, when mixed with absolutely anything in the universe that is not white paint is useful for a paint job.
(a rule that I deem arbitrary)

3.  Absorption Rules
Aa. A + A.B = A               [which I deem false]Ab. A.(A + B) = A            [which I deem false]Ac.          A + notA.B = A +B   [which I deem false]

Absorption Rules paraphrased examples:

Aa.  Absolutely White paint plus white and black paint is equal to absolutely white paint.
(a rule that I deem false.)

Ab. Absolutely White paint plus white paint plus black paint is equal to absolutely white paint.  
(a rule that I deem false.)

Ac. White paint plus (yellow paint (or absolutely anything else in the universe)) plus black paint is equal to the sum of black and white paint.
(a rule that I deem refuted as the minimum error in this example results in an outcome that is coloured green. Green is obtained from a mixture of black, yellow and white. In this example the logic dictates that a grey result is obtained. Green occupies markedly different positions within the optical spectrum than ‘grey’ for example. )

These three rule sets become dependent on assumptions based on observations and associations made after-the-fact i.e. ‘a posteriori’ classifications and produce contradictions and combinations of classifications that assume false universal properties and proportions about the ‘a posteriori’  after the fact world of objects.
The various statements of absolute equivalence where A = A after some operation of addition, augmentation or detraction are False.

The Federation may not play Scottish football as such, but quite obviously they could run a good Scottish Football side that could win tournaments and enable Star Players like Jorg Albertz,  Amoruso, and Tugai to play for their countries and keep the ‘Gers on top.
If some geezer decided that it was time for Jorg to move on, on the basis that he should really half his salary because he was too expensive by making the allegation of Industrial Logic that they could afford two great midfielders like Jorg for the price that he was asking. Jorg would no doubt like to see the clause in Industrial Logic that was causing contractual problems. If the management then show him by  the Boole Sum Rules of;
A + A = A, or even A + 1 = A – he would rightfully sue them via a mobile phone call to his Agent and the FIFA regulatory body.
The Federation would probably back him up.

At that, the two London yuppies who were getting rather discontent at the obvious weakness in their education and grasp on reality interjected.
Ah, but suppose one was something else that Jorg liked.
Seeing that those jokers were trying to subvert the argument by talking krud and attempting to take their flaws onto some other ground like attempting to solicit a sale of some recreational legal high, I realised that their attention span was starting to wane when they said that – well he may get a Mercedes like ours in a sponsorship deal.
I realised that I had to take this dialectic philosophically.
Yes, but although Jorg can be described as precision German footballing engineering, if he buys a car or is given a car – he simply does not become a car either on the pitch or off it ok ??  agreed ??

There is only one way people can get together and talk from different backgrounds – it’s to agree to meet and to communicate in language and terminology that describes the processes that we all have in common.
Here we’re using the life and career of  Rangers’ PowerHouse Teuton Jorg Albertz.
I agree that not everyone can be enthusiastic about Scottish Football.
Do you support an English Football team?
No came back the reply, we like to both go and watch Rugger on a cold Cambridge Sunday afternoon.

Well then, I enthused, do you agree that  league and cup winning Harlequin’s First 15 Rugby Union side minus half its first team and comprising of half of the B side should go into a cup final as an unproven and untried squad against a Wasp’s First 15 that has been on a devastating run through the English league ?? because your Logic teacher told you that A+A.B=A or that A.(A+B)=A.  You see that what your Logic teacher told you suggests that if you field the ball boys and the ‘mobile refreshment provender’ as well you could still bet on Harlequins with a straight face.

I did agree with your Logic teacher on one thing though, and that is if players get dropped from the first team then they are no longer first team players i.e.  A+ (not A.B) = A + B.
I see that your  Logic teacher acknowledges that there is an A and B team but that he cannot distinguish the difference of style, stamina, training, skill, experience, age, injury, speed, size, weight, power, co-ordination and of course, your Logic teacher has missed the point that The Harlequins first team plays better as a congruent unit because it is familiar with all its plus’s and minus’s.

If your Logic teacher was told that he would have to hand over half his salary because a supply lecturer was being brought in to supplement his input to the University department – would he have agreed with his own ‘Rules of Absorption’ ?? I doubt it.

Now, your Logic teacher at .. er where ???  Cambridge University will have come across a theory that all mathematicians like himself cannot solve.
It is that when people say equals – they really mean equal as in it’s the same as.  What he taught you about the calculus of classes is not what it’s about in today’s modern world.
If it was, the Greeks would have been building spaceships, and making better use of natural detergents to wash their togas in.
The main problem with the fires smouldering within our Universe, as you can see from the guy burning that cheese and onion crisp in the candle flame over there is that all is in chaos. If he gets too near the fire there, he will set fire to his beard and go crashing onto the leaflet display stand.
I agree that his friends have chosen to keep the supply of various flavours coming over to see what bright colours may appear, but this may be thought of as drunken and irresponsible behaviour and they may get barred.
Now I know that we as individuals are entitled to agree or disagree, but we have to obey the licensing laws set down by the brewery and the local authority in case the publican and landlord are not allowed to renew their late license and thereby lose out on a lot of money.
Potentially what they are doing in that Startrek ashtray over there where the multicoloured Hogmany fireworks are being rehearsed with the phosphorescent flavours of sparkling burning crisps is not equal to the kind of behaviour that in our experience is likely to get them into the Friday night late nite lock-in drinking and adult video session.

At that, both Henry and Oliver smiled, thinking that I had lost the plot, well then I said, in this place if the management says that we are all equal when we are equal, then they mean that.
If they think we are a load of cobblers they will not ask us to socialise with them.
Here it’s all for one and one for all, but mess up the business by harassing the high-paying clientele and discouraging big spenders by stupid behaviour then you’re outta here.
In this pub, equals means equals and fool means fool.
What your Logic teacher said about Jorg Albertz and the things he allegedly equalled was totally False except where I said otherwise.
Your Logic teacher uses the idea of ‘equals’ in a very loose and bad way.

There is an idea that a guy called Fermat was working on though, and it did rather look like your Logic teacher would have liked Fermat, because although thankfully it was Fermat’s last problem, what Fermat said and did is to use the same loose meaning of equals in the Calculus of Classes when he says that Xn + Yn = Zn has no positive integer solution x,y,z if n is an integer greater than 2. Fermat although now dead left mathematics in the same state with that idea.
If X and Y equals Z then that is a fixed state – agreed ?
Because I think that that idea is far clearer than your Logic teacher said it when he said as e.g1. X+Y=X in the sum rule X+1=X where 1 = Y for instance, or  e.g. 2.  X + not X = 1 where Y = 1.
Fermat said X + Y = Z which to me is clearer, because the end product is specifically different from the two components.
E.g. Black and White equals Grey.

Now supposing we all agree that in this world we like the colour Grey and that Fermat didn’t.  And that Fermat made a big deal about adding tons of black or tons of white into the mixture such that it couldn’t be called Grey anymore. Deep, deep black, isn’t the same as what we call grey. And muted white isn’t what we would call grey either.
You could find these colours down at BQ or Woolworth’s in the pantone colour index where you get your paint mixed by the mixing machine, or as hexadecimal references in your net browser, but if you were communicating either with a shop assistant with an O’ Grade, or a Graphic Designer with a Degree from the Art College at Lauriston place and you wanted these folks to find either extreme version of Fermat’s mixtures you would in both instances not initially ask for Grey paint, but would qualify your request as ‘very, very dark grey, or very, very light grey, but you would never say that you had an average grey no matter which of Fermat’s mixtures that you were asking for.
Generically as a society, when we say Grey we usually mean a neutral Grey which cannot be described  as non neutral  unless we add in the qualifiers of  bright or dark.
Black and White make Grey, Blue and Orange make Grey, Red and Green make Grey, Purple and Yellow make Grey, and as long as they are of equivalent hue and lustre and intensity and frequency and wavelength and observed under the same optical conditions for the human retina and visual cortex then these colours, called complementary colours when mixed together from two original monochromatic sources will produce the same tonality of grey in all instances as far as the human retina can make any distinction. That is a very standardised and unique form of  Grey.
Fermat however, decided to create a last and final problem before he left us to grace the stage of some other backwater planet, by selling every mathematician on the planet for 100 years on the idea that because you can load any of these colours to lose their complementary aspect by piling in magnitude upon magnitude of pigment on maybe the blue, or the green, that when the blue next mixes with the orange and the green next mixes with the red that if we don’t get the usual neutrality of grey we were expecting that he has performed a God-like magical trick.
Neither the shop assistant in B&Q nor the Graphic Designer in the Media Factory would have got away with that in downtown Leith.

I went to school here at Leith Academy where my maths Teacher ‘Johnny B’ used to tell us all about numbers. In fact he counted the class to catch truants and used to whack them senseless with a leather tong hardened and steeped in whiskey.
I used to think that digits were integers because I could count to ten on my own natural abacus and innate decimal numbering system. Not so said Johnny B, disagreeing profusely and going a dark shade of volcanic red in the process. There are positive integers and negative integers. Staring dangerously at me he would say, well there are the ten fingers that you can see and the ten fingers from God that you cannot. Those are positive and negative integers. You see, for every ten things you add up, ten invisible integers will come along with bad little fingers and cancel them out.
That is to what I am referring when I say that there are positive and negative integers. You may call your activities natural work by hand and fingers, but then the work of Satan is to come along when you cannot see him and tear down what you do with 10 demonic fingers that you cannot see -– up close and personal like.
That cannot be natural I enthused … Ah said Mr B, what is natural anyway eh dear boy ????

I always pondered on the nature of natural until I met Mr Fermat.
Pantone Blue number four added to Pantone Orange number 4 will give if mixed in equal proportions, Pantone Grey number 4.
Although all of these colours and processes going into the tins at B&Q are natural substances, neither the blue or the orange used could be simultaneously called natural by Mr B’s definition or indeed neither could the end product be called natural by Mr Fermat either.
We get the refutation of Fermats Last Theorem from this context sensitive analogical model. Even if n is greater than 2 of the K, black weighting  in the CyanMagentaYellow [cmyk] colour system - we still get a knowable and positive solution for any n despite what Fermat says.
The real numbers to equivalent decimal places on optometric equipment that calibrate the frequency of the light emissions in the spectra of the blue and orange paint would have been thought impossible by Mr Fermat.
Although some people could use this frequency index to define objects into agreed classes of colours and calculate precise mathematical outcomes observable by  photometric equipment, and then pass those benefits onto society at large who would then dutifully specialise in other important tasks such that if we phone B&Q and ask for a generic middle grey for a garage door– everybody knows that near black isn’t going to get delivered instead.
Not so for Mr Fermat though –  many decades have passed since he made out that we were insufficient Mathematicians, it has to be said, and I will, that they all pretty much were for falling for that one. ….

      Additional Bibliography.

COHEN M, NAGEL E.   ‘An Introduction to Logic’ Pub. Routledge & Kegan Paul Ltd, 1966.
DAVIS P, & HERSH R. ‘The Mathematical Experience’ pub. Pelican 1981, ISBN   0-1402-2456-4

Russell B and Whitehead AN, 'Principia Mathematica', pub. 1910, Cambridge University Press.


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