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.
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 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 Cannabis.
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
3.
the INVERT GATE or NOT 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.
E.G. 1, 3-VARIABLE KARNAUGH MAP.
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 cocaine, 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 and enjoy the sight of hot
steamy beefcake 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 B&Q 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 a 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.
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.