The deeper hidden message

in Maxwell’s equations

Why have Maxwell’s Equations survivedfor SO long? by Ivor Cat

t

{DECEMBER 1985

In November’s article 1 I in-vestigated Maxwell’s Equa-tions, generally regarded as thegreatest mathematical achieve-ment in science.*In the 1830s, Faraday

discovered electromagnetic in-duction, thus closing the loopbetween electricity andmagnetism. This discoverypaved the way toward the rapidgrowth of electricity-based in-dustrialization and the hightechnology which shapes to-day’s world.

By making the key discover-ies of their era, uneducatedtechnicians like Michael Fara-day and James Watt threatenedthe scholastic myth, that allprogress, including scientific

progress, needs must use therigour and discipline controlledand celebrated by academics in

*And if mathematics is the highestflowering of science, then Maxwell’s

Equations become the greatest achieve-ment in alt science.

places like Cambridge Univer-sity. The ultimate in scientificrigour (rigor mortis?) was heldto be mathematics. Biographyand History of Science writingsspawned in academia presentthe thesis that, lackingmathematics, Faraday couldnot and did not really effect hiediscovery of electromagnetic

induction. Rather, he stumbledinto it, but it could only be pro-perly exploited decades later,after Professor Maxwell had

I placed a mathematical struc-ture upon Faraday’s fumbling,unscholarly ideas. Thus, accor-ding to the Platonic interpreta-

tion of history, Professor Max-well, not Faraday the techni-

j

cian, paved the way for massive

|

exploitation of electromagnet-

!ism in transformers, motors

and generators. The deeperhidden message in Maxwell’sEquations is that, do what they

i will, the local yokels will not

| replace mathematical academia

as the fount of knowledge and

|

progress.

190 Ci • i k. • i i •

In a previous article 1 I posedtwo questions:

Do Maxwell’s Equationscontain any informationabout the nature of electro-magnetism?Why do academics and

practitioners generally thinkthat Maxwell’s Equationsare useful?

I am sure you will have foundmy answers unsatisfactory.The reason is that they were

j

based on certain assumptions,and failed to dig deeply enough

;

into the underlying motivation,1 psychoses and myopia withincontemporary science.The underlying battle for the

;

soul of science is between thei practical engineer on the onehand and the Platonic puremathematician on the other.

-

i

For his part, the mathematiciansees this battle as more impor-tant than search after truth or

! technology-fuelled search afternew sources of wealth. For him,

I

the important thing is Form;the purity and beauty of hisworld, and his ability to controland manipulate it intellectually.(The profane aspect of this ideais the desire to impose a struc-ture onto any ‘discipline’ suchthat it is easy to teach and,

1 more importantly, easy to set1 exam questions on). One FRS1 told me that physical realityI was composed of sine waves,land this encapsulates the

mathematician’s attitude to our

world.

A good example of anacademic with the mathemati-cian’s attitude is Sir JamesJeans. He was highly regardedin the 1930s both as a Cam-

i

bridge academic and as apopulist, much like Sir FredHoyle in the 1950s. In his book“The Mysterious Universe’’, 3

Jeans gives a clear view of the

attitude of the Platonic mathe-matician discussed in the last

paragraph.

“Bv ‘pure mathematics’ ismeant those departments ofmathematics which are crea-tions of pure thought, of reasonoperating solely within her ownsphere, as contrasted with ‘ap-

plied mathematics’ whichreasons about the externalworld, after first taking some

i

supposed property of the exter-nal world as its raw material’’

On the next page, Jeans goeson to write,

“. . . the universe appears to

have been designed by a puremathematician.”

The important thing is not toponder over the possible con-tradiction between these twostatements, but to grasp thementality underlying them,

f

This mentality, usually in a bet-ter camou..flaged and lessgrotesque form, is what made

!

possible the survival of mathe-

|

matical absurdities like Max-

C« • i i. • i'i • 191

well’s Equations for such a longtime.

Jeans then goes on helpfullyto point out the flaw in his

argument:

“This [last] statement can hard-ly hope to escape challenge onthe ground that we are merelymoulding nature to our pre-conceived ideas. The musician,it will be said, may be soengrossed in music that he

would contrive to interpretevery piece of mechanism as amusical instrument; the habit ofthinking of all intervals asmusical intervals may be so in-grained in him that if he felldownstairs and bumped onstairs numbered 1, 5, 8 and 13he would see music in his fall. Inthe same way, a cubist paintercan see nothing but cubes in theindescribable richness of nature- and the unreality of his pic-tures shews how far he is fromunderstanding nature; his cubistspectacles are mere blinkerswhich prevent his seeing morethan a minute fraction of thegreat world around him. So, itmay be suggested, themathematician only sees naturethrough the mathematicalblinkers he has fashioned forhimself. We may be remindedthat Kant, discussing thevarious modes of perception bywhich the human mind ap-prehends nature, concluded thatit is specially prone to see naturethrough mathematical spec-tacles. Just as a man wearingblue spectacles would see only ablue world, so Kant thought

that, with our mental bias, wetend to see only a mathematicalworld. Does our argument mere-

I lv exemplify this old pitfall, if' such it is?

“A moment's reflection willshew that this can hardly be thewhole story. The newmathematical interpretation ofnature cannot all be in our spec-tacles - in our subjective way ofregarding the external world -since ij it were we should haveseen it long ago [my italics). Thehuman mind was the same inquality and mode of action a cen-tury ago as now; the recentgreat change in scientificoutlook has resulted from a vastadvance in scientific knowledgeand ot from any change in thehuman mind; we have foundsomething new and hithertounknown in the objectiveuniverse outside ourselves. Ourremote ancestors tried to inter-pret nature in terms of an-thropomorphic concepts of their

‘ own creation and failed. The ef-forts of our nearer ancestors tointerpret nature on engineeringlines proved equally inadequate.Nature refused to accommodateherself to either of these man-made moulds. On the other

[

hand, our efforts to interpretnature in terms of the conceptsof pure mathematics have, sofar, proved brilliantly suc-cessful. It would now seembeyond dispute that in someway nature is more closely alliedto the concepts of puremathematics than to those ofbiology or of engineering, andeven if the mathematical inter-pretation is only a third man-made mould, it at least fits ob-jective nature incomparably bet-ter than the two previouslytried."

c..192

Professor Einstein argued

similarly in 1949‘:

“. . . the approach to a more

profound knowledge of the basicprinciples of physics is tied up

with the most intricate of

mathematical methods.”*

I have put the weak point inJeans’ argument above initalics. The mathematicizationof science developed with a

vengeance as a result of the

professionalization of educa-

tion. Dr Ivor Grattan-Guinness

once pointed out to me that thedecline, or ossification, of

science into ‘maturity’ was anecessary result of the in-

troduction of universal educa-

tion in the mid-19th century,

because it caused the growth of

a powerful group with a vestedinterest in knowledge, the pro-

fessional teachers.

Basil Bernstein'' says that a

body of knowledge is property,with its own market value andtrading arrangements, to be

protected by the social group

which administers that body of

knowledge.

If only those who lived off abody of knowledge could makethat knwledge more secure,their careers and pensionswould be protected. Twostrategems were open tothem*' 78 :

*In passing, it is worth noting from page

02 of the same book, where Kinsteinwrites: "The special theory of relativity

owes its origin to Maxwell’s equations of

the electromagnetic field." In the

literature we repeatedly come acrossassertions that Maxwell’s Equations play

a pivotal role in science.

- to freeze the knowledgebase so that it would not be aprey to the ebbs and flows ofthe real world, and- to develop the thesis thatany change in. or extensionof. the knowledge base couldonly be properly effected bythe professional ‘knowledgemagicians', ‘knowledge doc-

tors’ or 'knowledgebrokers’, with their special,

skilled, occult ways ofpushing forward the boun-daries of knowledge.’

It would of course be less ef-

fective for the professional

group of knowledge brokers

merely to bless or condemn in-fluxes of new knowledge. (Ad-mittedly they do do that. All myattempts to publish work on

electromagnetic theory and on

computer architecture (USpatents 3913072 and 4333161)

were blocked for more than ten

years by learned journal

referees, who are by definitionknowledge brokers). Theknowledge brokers’ powerwould be greater if they re-

quired that new knowledgearise in their own prescribedstyle, preferably devised by one

of their members, a knowledge

professional. An early exampleof this in my own publications isthat under threat of. firing by

my boss, who was also a Fellowof the IEEE, 1 was compelled"

to include a ghastly, recondite,

mathematical last section, writ-

ten by someone else, in my1967 IEEE paper 1 ".

C* • iv • 1 1 . 193

We have reached the follow-ing point in the argument.

Under cover of claiming tomaintain standards of scholar-ship, or to maintain rigour,

knowledge brokers (1) blockthe ingress of new knowledge,parti cuarly revolutionaryknowledge in the Kuhniansense 11

, and also (2) they makea last-ditch, bitter defence of

old, discredited knowledge, likeMaxwell’s Equations.

"Unfortunately, however, whenthe body of knowledge is biggerand the rate of inflow of newknowledge is smaller, more andmore of the activity within theknowledge |basej becomes‘celebration*, more and moreceremonial rather than exercise

in depth. As a result, a differentcalibre of person is attracted to

that large knowledge, lackingthe ability to understand and de-fend a body of knowledge withmany levels of meaning. Theyare maintenance men’ ratherthan ‘builders’. The centralbody of knowledge ossifies,becomes brittle and then disin-tegrates.'’^

We need to realise that thecardinals who suppressedGalileo did not need to be com-petent theologians or scientists;

they only needed a much nar-rower competence, the abilityto distinguish between the or-thodox and the heretical, inboth content and in styled Asto style, it is worth pointing outthat possibily the ability topublish radically new, revolu-tionary 11 knowledge in the oldaccepted style would prove thatafter all the new knowledgewas not truly revolutionary. Soarguments about style, whichare regularly lodged against mywriting, including my last arti-cle, 1 create a beautiful Catch 22situation where no newknowledge can be published.

In this penultimate paragraphI mention in passing TheLateral Arabesque, ‘Arabes-que’ having the meaning ascrib-ed to it by Dr Peter 11 ratherthan its dictionary meaning. Inthe engineering sense, the sup-posed situation where academiacontrolling a discipline - elec-tromagnetic theory for example- maps onto the real subject, is

194 C.A.M

unstable. If at any moment theprofessors administering adiscipline happen to be weak inone branch of it, they will tendnot to examine their students init, and so will tend to select out.those up and coming studentswho have that sub-discipline astheir strength. Positive feed-back down the generations ofstudents will further the retreatfrom that particular sub-discipline. (Sir James Jeans andEinstein could be said to be tell-ing us very wordily thatacademia have selected outbudding scientists who showeda grasp of the physics, ratherthan the maths, of their sub-ject.) Similarly, the whole ofacademia will move deeper anddeeper into any misconceptionor abeiration, and there is nocorrective force. In my view,The Lateral Arabesque’ makes

it possible for an academic sub-ject s content to end up with nooverlap at all onto the real sub-ject from whence that branch ofacademia sprang. I have justcompleted four years as Prin-cipal Lecturer in a College ofFurther Education, where I wasstruck by the lack of any signifi-cant link between the HigherTEC syllabuses that I taught

and the real subject, electronic-design, in which I had been ear-ning my living in industry forthe previous 20 years. As aminor example, academiaevolved the myth thatdissatisfaction among logicdesigners with the indeter-minate state of an R-S bistableif driven on both its inputs atthe same time led to thedevelopment of the J-Kbistable; then that the instabili-ty of the J-K led to the develop-ment of the Master-Slave J-K,regarded in academia as theRolls-Royce of bistables. A niceidea, but with no historicalfoundation.

A larger example would beacademia’s fixation on Quine-McCluskey, something noteven heard of, let alone used,by engineers in the real worldof logic design. Although I wasin the best position possible tointroduce or alter syllabuses,being on the County Commit-tee, during my four years as aP.L. I failed to change one wordof one syllabus. I struggledvery hard to do so.

1 o sum up. Professionaliza-tion of knowledge leads to avested interest in knowledge,

C.A.M 195

which leads to the disintegra-

tion of competence among the

knowledge professionals as

well as the prevention of the

ingress of new knowledge.Something like this syndrome

is needed to explain the

survival of Maxwell’sEquations for so long.

Moving graphics help to il-

lustrate the subtleties of elec-

tromagnetic theory. For informa-

tion on the availability of

videotapes, please write to Ivor

Catt, IS King Harry Lane, StAlbans AL3 4AS.

of Living Philosophers, 1949. p.17

5. Bernstein, B., Gass, Codes and

Control, vol. 1. R.K.P., 1971.

6. Catt, I., The rise and fall ofbodies of knowledge. The Informa-

tion Scientist

,

vol. 12, no. 4. Dec.

1978. p.137.

7. Catt. L, The scientific receptionsystem as a servomechanism, Jour-

nal of Information Science vol. 2,

1980. p.307.

8. MacRoberts. M.H. and B.R.,

The scientific reception system,Speculations in Science andTechnology , vol. 3, no. 5. 1980.

p. 573.

9. Catt, l., Computer Worship. Pit-

man, 1973. p.64.10. Catt. 1.. Crosstalk (noise) in

References digital systems. IEEE Trans. Elec-

1. Catt, I., The hidden message in twn. Computers, vol, EC-16 1967.

Maxwell’s Equations. Electronic & p.743.Wireless World, Nov. 1985, p.35. 11. Kuhn, T.S., The Structure of

2. Catt, 1., The new bureaucracy. Scientific Revolutions. University

Wireless World, Dec. 1982, p.47. of Chicago Presf , 1962.

3. Jeans, J., The Mysterious 12. ref. 9, footer e, ?.14.Universe, CUP 1931, p.113. 13. Peter, L.J. and HuU R., The4. ed., Schitpp, P.A., Albert Eins- Peter Principle. Souvenir, 1969.

fcein, Philosopher-Scientist, Library p. 38.

196 (j • 1 i. • 1

A mathematicalrake’s progressIvor Catt looks back on how he nearlybecame a maths addict

JANUARY 1986

In my article of lastNovember, I showed thatMaxwell’s Equations, so

long thought to contain theheart and essence of electromagnetism, told us virtually no-thing about the subject. Then,in my December article, Idiscussed the academic mafia’svested interest in knowledge (seepanel). Here I try to discover whothis group of charlatans*, themaths pushers, are. How does ayoung student grow up tobecome part of the social groupwho live by mathematicalnonsense like Maxwell’s Equa-tions, and who conspire to pre-vent the development of a scien-tific subject in a proper, physical

way?

’The Shorter Oxford English Dictionaryentry for this word is particularly apt:

Charlatan 1. A mountebankwho descants volubly in the street;esp. an itinerant vendor of drugs,etc. . . .

2. An empiric who pretends towonderful knowledge or secrets.... a quack.

However, if we then look up the entry forEmpiric, the whole picture backfires on us.

Concern about this questionled me to look back on my owneducation. What pressures wereexerted on me to become amathematical rake?

My experience indicates thatthe slide is similar to that of thedrug addict — a number of small,apparently innocuous, slipsdownward, culminating in totalseparation from reality. As weprogress through school and col-lege, we are fed a series of po-tions, each more heady than thelast.

The process started with thecalculus. My introduction to it, atthe age of 15, was worrying anddisorienting. It was part of thegreat disaster which I thoughthad overtaken me in my first fewmonths in the sixth form.Whereas I had always been good

at maths, I found the first fewmonths in the sixth form confus-ing. Even though Sam Richard-son was a very good teacher, andI had help from my mother, abrilliant mathematician, athome, I couldn’t understand thebasis of what we were learning inmathematics, particularly thecalculus.

G-.A.M 197

This was a new experience forme. Previously, I had always

found maths easy, and scored

high marks. Now, suddenly, itwas different. This was seriousbecause if I tried to retreat from

maths into some other field, allnearby subjects were based on

maths anyway. There seemed tobe no escape from my new-foundinadequacy in mathematics. Asthe first half-year exams ap-proached I became more andmore worried, because still Icouldn’t grasp the basis of what Iwas being taught.The flaw in the calculus

package is what I now recogniseas the reductionist fallacy a

misconception which underlies

and undermines westernphilosophy.* The error is tothink that ‘the whole is the sumof the parts’, no more; that lots of

bits of string are quite as useful

as (and the same thing as) a long

piece of string. Putting it another

way, the problem of discon-

tinuities was ignored. I was rightto worry.

A whole array of misleading,damaging concepts slipped in

with i, or j as we electricalengineers call it. “Two for theprice of one”; if a+jb * c + jd,

then a = c and b = d; so we can dotwo jobs at once. Pretty, but a

delusion, similar to the illusion

that we can drive better afterdrinking, and for the samereason — our vision is blurred.

"Titus, H.H., Living Issues in Philosophy.

American Book Company, 1964, pp.148,

527, 540 etc.

Hot on the tail of j came thatawful array of cons under the ap-

propriate descriptor ‘sine’. I shall

not develop this theme fully, but

only repeat that one FRS went sofar as to say that “Physical reali-

ty is composed of sine waves”. In

fact, the sinusoidal wave, which

is a camouflaged circle, is

Ptolemy’s pure, circularepicycles fighting back againstKepler’s less pure, more real,ellipse. Kepler, who himself lov-ed the idea of the ‘harmony of thespheres', saw a more pure ‘equalareas in equal time’ rather than a

distinctly un-heavenly, earthy,

(we would say ‘real’,) ellipse.

The Wireless World July 1981editorial, ‘The decline of the

philosophical spirit’, contrasts

the nineteenth century, whenscientists were interested in and

capable of distinguishing bet-

ween the physically real and themere mathematical construct,

and today, when scientists nolonger know or care about thedifference, and have even

developed a philosophy of

science which confuses them.*

An example of the destructiveeffect of sine is the way in whichit suddenly appears, unannounc-

ed and without justification, on

the second page of a text book

discussion of the t.em. wave.

In the event, my first half-yearexams in the sixth form didn’t

seem too hard, and I felt that I

must have scored over 50%,

•Popper, K. Conjectures and Refuta

tion*. R.K.P., 1963, p.100

198 C.A.M

which would give me a breathingspace in which to re-plan myfuture. To my astonishment, Ilearned that I had scored 99%and 92%.However much I might think I

didn’t understand what was go-

ing on in maths, the marks I

scored ‘proved’ otherwise. Myhigh scores told me that I was stillgood at maths, as I had always

been. However, a nagging suspi-

cion remained with me thatsomething was amiss. I doubtedwhether I could really have mis-

judged the situation so badly.

Today, I believe that I was cor-rectly judging the situation, and

it was my exam marks that werewrong. I was being brainwashed

into the belief that under-

standing was unnecessary, even

impossible; that success meant

the ability to manipulate the

symbolism of the subject, not to

understand it. I was beingencouraged, the initial carrot be-

ing high exam marks, to turn thehandle of the mathematical bar-

rel organ, and not to ask too

many awkward questions.I seemed to learn my lesson,

and later on, when taking A-levels, I gained a State Scholar-

ship in maths although only 17

years old. This was a remarkable

achievement, and should have

secured my loyalty to the admini-strators of the mathematical

myth. However, I was already

questioning the usefulness of

some of this maths, particularly

the interminable geometry (since

dropped) in the Cambridge Open

exam, and so at Cambridge I

decided to leave my strong sub-ject, maths, and read engineer-

ing.* My background must havemade me particularly sceptical.My mother had scooped the lot,gaining the top ‘first’ in maths in

London University, but the

payoff to her in benefits in later

years proved minimal.

The next piece of blatantbrainwashing occurred during

my engineering course in Cam-bridge. We had a lot of ther-modynamics, which was very

mathematical. One day I asked

my tutor, Professor Binnie, whatpractical interpretation I could

place upon an equation contain-

ing a college of terms involving

the three e’s — energy, enthalpyand entropy. His answer was

that I should not bother to look

for a physical interpretation, but

should merely regard it as a piece

of algebra to be manipulated ac-

cording to the rules of algebra. I

was shocked by this, and I re-

main shocked today. Had I left

maths and taken up engineering

for nothing?

Whereas drawing, ordraughting, was strong in the

Cambridge Engineering Faculty

and seemed to occupy a large

part of our time, being the only

subject you were not allowed to

fail, electricity was weak, rating

only one lecture a week, or at

*1 love the Heaviside remark; “Whether

good mathematicians, when they die, go to

Cambridge, I do not know.” — Heaviside,0., Electromagnetic Theory, Vol. 3,

Dover, 1950. (First published 1903.)

C.A.M. 199

most two. One suspects that con-

servative Cambridge of the

1950s hoped that this new-

fangled electricity thing would

prove to be a flash in the pan, and

go away soon. (Gaslight, I have

been told, was very pleasant;

much softer on the eye than elec-

tric light.) I suspect that my later

success in elecromagnetic theory

resulted from the lack of

teaching in it that I had sustained

while at college.

We did not cover the LaplaceTransform, and this set me apart

from upstart graduates from red-

brick universities, who enjoyed

discovering how backward Cam-

bridge was. I was lucky in this

omission, because I now feel that

transforming is one of the

destructive mathematical techni-

ques in engineering that increa-

ses the divorce from reality, and

which is the legacy to engineers

Mhemtlcal nnfla

The twisting of historical fact in

the bands of the academic mafia is

beautifully illustrated by the case

of the discovery of the electro-

magentic theory of tight. Obvious-

ly, a mathematician would tike us

to believe that the proposal that

tight was electromagnetic in

nature resulted from i subtle

manipulations of his electro-

mageotic equations by Professor

Maxwell the mathematician. In

fact, Whittaker1 says that the pro-

posal that light is electromagnetic

came from Faraday in 1851, when

Maxwell was 20- Now it might beasserted that the vague sugges-

tion by Faraday was confirmed

and strengthened by Maxwell’s

mathematics. However,Chalmers2 says that there is an er-

ror in Maxwell's calculations.

from mathematicians. Whereas

to me it is obvious from first prin-ciples that to get constant cur-

rent through a capacitor* you

need a continually increasing

voltage, I recently found that for

a student of Laplace this is the

conclusion of a lengthy piece of

complex calculation.Thus was the stage set for

Maxwell’s Equations, that

phoney apology for elec-tromagnetic theory, which held

sway for a century and so befog-

ged the subject.

There is a similarity between

the maths pushers and drug

pushers. Both entice the victim

with promises of Elysium. Both

gradually increase the dose. In

both cases, there is nothing at the

end of the rainbow.

* using theory N

which “led Pierre Duhem to ac- Icuse Maxwell of adjusting his

calculation so that he could arrive

at a theory of tight which he {or

should we say Faraday?) alreadyhad in mind.”

The truth appears to be that the

idea preceded the maths; the

maths was force-fitted onto the

idea, like the ugly sister’s shoe;

and then the mafia claimed the

maths generated the idea. The

prince was not hoodwinked, and

neither should we be. This racket,

of forcing mathematical liturgy

onto a reluctant discipline, con-

stantly recurs in science, perhaps

reaching its most grotesque in so-

called 'computerscience’ courses.

1. E.T. Whittaker, A History of theTheories of Aether and Electricity.

Nelson, 1961, p.194.

2. Chalmers, A.F., Maxwell and the

displacement current, Pkysics Educa-

tion, vol. 10, 1975, p.45.

0 .. • is200

C * K* 201

ELECTRONICS & WIRELESS WORLD JANUARY 1986

ENERGYTRANSFER

In a realistic physics all interaction

should be treated as a discrete

cause-effect process, which is

intuitively self-evident from the

principles of physical reasoning but

it is not realised in present day

theorising. Such a discrete process

generates a power series as the

interaction propagator passes to-

and-fro and standard analytical

techniques can be used to

determine whether the series will

converge to a finite limit or not.

(The information accumulated in

the series coefficients corresponds

to the field exchange particles of

high energy physics.)

If it tends to an asymptototic

equilibrium the power series may

be replaced by a more compact

function from which the power

series can be regenerated. The

reversible correspondence between

power series and generating

function representing it is only

valid within the radius of

convergence for the iterative

process of the cause-effect

interaction.

This is the crucial problem for

advance in theoretical physics: to

develop a non-equilibrium

behaviour with mathematically

self-consistent formulae using

continuous variables. Self

consistency means that we assume

that an equilibrium can exist

between the variables (such as in a

circuit) then proceed to calculate

their would-be values.

For instance; Newtons laws ofmechanics and gravitation andEinstein’s relativistic modification

of them, and de Broglie’s ‘wave’

description for electrons in atoms,

the formulae of electromagnetic

theory and electronic circuitdesign. These all correspond toasymptotic equilibrium of adiscrete interaction: the conditionsof interaction under which theseself consistent limiting descriptionsare valid (as a calculation short-cutto a solution that exists is nevereven thought about.The example I can quote is of

feedback in a control system whichI derived longhand in WW Dec.1983. A finite time delay wassupposed in a feedback controlsystem and its response to the

simplest possible stimulus of a unit

step was described using discreteelement analysis.

The ‘operational amplifier’formula was rederived includingvalid limits for convergence, which

is essential to understand the

negative-feedback instability (so

intuitively obvious) and why it isnot predicted by the self-consistent

op-amp formula. The selfconsistently derived op-ampformula is beautifully simple but

wrong in principle, since the

solution cannot exist (meaning well

defined finitely stable behaviour)

outside the region of convergence.

(As a matter of interest the

Nyquist formula is just a special

instance of the Curie-Weisssusceptibility law for a loose —coupled system and fits into a

wider scheme of ‘phase transition’' formulae which includes therelativistic formulae for the

limiting speed of light to material

motion.)

202 C • it • Js

i •

Upon similar reasoningconcerning what mathematicalrepresentations mean, the formulaeof relativity and general relativityare calculation short-cuts. In all

probability Einstein got the right

answers for the wrong reasons.The de Broglie wave formula forequilibrium behaviour of electrons

in atoms is a piece of brilliantinsight but falls into the sameclassification of self consistency. Amore fundamental mechanics isneeded if we are to unify physics.

The Catt anomaly is based upona misconception about the

mathematical formulation ofelectromagentism; classical

electromagnetism and its quantum,relativistic modifications can onlybe valid as a description forasymptotic equilibrium.

The response of any system(transmission line of interacting

particles) to the transient of a step

stimulus is a non-equilibrium

phenomenon, hence the ‘anomaly’vanishes as meaningless when it isrecognised as an attempt to applyan equilibrium theory to a non-equilibrium situation. What isneeded is a more general non-equilibrium theory from which toderive the equilibrium one as aspecial case.

E and M displacement currentare not confused, as Mr Cattsuggests (Letters, April). Theyrefer to two separate aspects of aninteraction, the asymptotic regimeand the initial transient effectsrespectively where at the ultimatediscrete scale of reality the

mathematical sophistication ofFourier analysis is beautiful but

useless.

Most worrying of all is thethought that the mathematicalformulation of our physical laws(using continuous variables andcompacted into generally

applicable differential equations)implicity precludes certainphenomena from our reasoningwhich anyone with an ounce ofphysical insight would see asintuitively obvious though perhapsno so easy to quantify.Algebra is just a noise-free

information processing system (ormedium) and logical manipulationsdo not add anything that isessentially ‘new’ beyond thatphysical insight which wasencoded into the originalformulation of the theory. As withany computing language if you putrubbish in you will get rubbish out;however, one needs to overcomethe psychological barrier of thebrazenly sophisticated formulae sospuriously created if we are tochallenge the conclusions of thetheory with physical reasoning toattempt unification.

One needs to discussmathematics in everyday languageto truly understand what a formula(usually in a commutative orasymptotic cause-effect logic)represents and not to float adrift ina sea of assumedly understood.P. J. Ratcliffe

Stevenage

Hertfordshire

C.A.M 203

ENERGYTRANSFERThe reversed polarity of the Halleffect, referred to by R. Petzeratt

in the November issue, is notconfirmed to p-type

semiconductors. The samephenomenon is found in certainmetals, such as zinc and cadmium.The mater was a major anomalyuntil the late 1920s, when it wasaccounted for in quantummechnics.

As regards Catt’s viewsconcerning electrical conductivity,

he seems to have confused myreference (in the February issue) to

electrons entering and leaving the

conduction band (from and to thevalence band) with the metal strips

or “bands” ins triplines. Thissuggeests lack of appreciation of

the physical realities, except for

the out-dated “cannonball”

electron from classicalelectromagnetism, that participate

in the physical processeds in

conduction, such as photons and

psi waves. My main point was, andremains, that the velocity of

electrons does not form a strong

enough foundation from which tomake such pronouncements as“the death of electric current" and

“electric charge does not exist”

(December 1980).G BerzinsCamberley,

Surrey.

WHAT'S IN ATHEORY?CKlr Ivor Catt always makes

cheerful light rfeadihg, especially

when he joints out the obviousnessof some physical statements whichare usually mentioned only withbated breath. Typically, in E&WW for November 1985 hereminded us that only uniquelyelectromagnetic information

contained in the two Maxwellequations relating E and H invacuum lies in the theoreticalartefacts no, €o; from them one canextrapolate to derive “the

impedance of free space" as2d * Mol£o and “the universalvelocity of light” as C = H\f^o£o-The phenomehon is by no means

confined to electromagnetic theory.S6Yne of your readers might care totry the following unconventional

derivation in the wave theory ofmatter:—

A completely generalexpression for 3-dimensional

Waves in any linear, continuousmedium is provided by

v 2v + (t)2

- 4' = ° ‘M)where (psi) is “whatever it isthat oscillates” in the wavesand u,k are the wave velocityand frequency respectively.The last afe classed as“unobservables” by the theory,so we will simply eliminatethem by means of the waveaxiom (self-evident truth) u =nv, obtaining

v 2v + (7)^-0 -Next, on the apparent evidenceof experiment we assume theserhatter-wives obey themomentum precept p = mv =h/X, so that

* 24> + = 0. ...(3)

Now the momentum p = mv is

204 C.A.M

directly related to the particle’s

kinetic energy K = 'hntif* bythe form p? = 2mK;substitution for pr leads

inexorably to

V 2V + ^-K-v = ft ...(4)

Finally, since the total dynamicenergy of any particle is defined

as W = (K + U), evidently K= (W - V) and therefore

V 2V + ^P-(W- U).y = a/r

...(5)

Here we have derivedSchrodinger’s Wave Equation, thatfamous, basic statement of the

Wave Mechanics which studentsmay approach only with extremestreverence. Given two furtherassumptions it leads to

Schrodinger’s mathematical model

of the atom — represented as aninfinite set of spherical harmonics— which is said to “explain” theatomic line spectra.

The derivation given above maybe unfamiliar, but it is both simple

and watertight. Step by numberedstep it consists of the following

physical assumptions:—

(1) let us suppose that linear

matter-waves exist;

(2) Then these matter-wavesmust obey the wave axiom;

(3) The matter-wavelength of aparticle is related in this

particular way to its mechanicalmomentum (de Broglie);

(4) The taws of Newtonianmechanics are to apply; and,

(5) A particle’s energy is partkinetic, part potential.

Given at (1) the idea of matter-waves and at (2) the axiom whichmust follow from that, the onlyreference made to any property ofthe waves, namely p = h/X, occursat (3). The derivation is in factcomplete at this point, the othertwo steps being added to facilitateits practical application. We notethat the derivation, and therefore

the wave equation itself, isinvariant with respect to anydefinition of u, so that “any old

waves will do” for the wave theoryof matter. That is why a WaveMechanician will give you an oddlook if you ask him about thefrequency of his matter-waves. Noris any textbook willing to tell uswhether these matter-waves aretransverse or longitudinal. Thetheory doesn’t know about suchthings, and it gets away withoutcaring about them either!As Mr Catt has pointed out in

the case of the electro-magnetic

theory, so here in the case of the

wave-mechanics, and so also as iswell known in the case of specialrelativity, the key theoreticalformulations are found to rest upontwo, and only two, particular andradical physical assumptions. Therest is algebra.

Wr A Scott MurrayKippford

Galloway

,• i i. • l 1 205

ELECTRONICS & WIRELESS WORLD FEBRUARY 1986

ENERGYTRANSFERP. L. Taylor ( Wireless World, p. 15October 1985) likes the choicebetween e.m. wave energy transferthrough space, as required by thePoynting vector, or through wires,as required by the Slepian vector.There is. in fact, a third choiceadvocated by Cambridge ProfessorG. H. Livens. Writing 'On the fluxof energy in radiation fields' at p.313 of his 1926 book ‘The Theoryof Electricity’, published byCambridge University Press, heargued in favour of an alternativeto Poynting’s theory. Wraves do notneed to carry energy at their speedof propagation. Their generationmerely adds energy to a commonpool of energy in the field mediumat the locality of the transmitterand their absorption draws on thatpool in the locality of the receiver.I like this third alternative, becauseit is easier for me to picture

creation a a big splash in anexisting smooth pool of energythan as a big bang appearing fromnowhere in a complete void.J.N. KidmanSouthampton

MAXWELLThe consistence of Ivor Catt’smisrepresentation of Maxwell’s

laws is remarkable. Whateverdeficiency they do contain, if any,

it is certainly not at the elementary

level claimed in the issue of

November 1985. (‘The HiddenMessage in Maxwell’s Equations’).The basic problem remains

apparent ignorance of vectors and

the role they play in Maxwellian

theory. Ivor Catt seems to thinkthat Maxwell's laws are some kindof elaborate hoax supported by anestablishment conspiracy tosuppress ‘alternative’ theories. Healso believes that equations (9) and(10) in his latest diatribe representthe views of the conspirators. Ifthey did, he would indeed have apoint. But, sad to say, thewindmills that this exuberantknight errant is tilting at aresignificantly different from thereality of Maxwell!The mistake he makes is

fundamental and disastrous. It isentirely necessary to modern emtheory that the E field vector isperpendicular to the H vector.Why, then, do equations (9) and(10) not show this? Without thedirection property of a vector, emtheory would fail to account forsuch simple phenomena asreflection. Ivor Catt attaches somemystical importance to Z0 ; anyonewho was properly conversant withEM theory would not. Z0 is derivedfrom the magnitude of the E and Hvectors; their directional propertyis eliminated and most that isuseful in the theory with it. Z0 isnot a ‘primitive’: it lacks

directional information.

Ivor Catt’s difficulties with theexpression of physical concepts inmathematical form do not seem tobe confined to electrical matters.True enough, if he walks along theplank far enough in the direction(hooray for direction!) V, ‘h’ doesindeed decrease but so does ‘x’.Sorry, Ivor old son, but you arewrong again, as you walk along theplank it is because it is goingbackwards underneath you that itleads to that sinking feeling.Dermod O’ReillyAntwerpBelgium

206 C.A.M

MARCH HiHfi

RAKES' PROGRESSI fed that Mr Catt (January 1986)has missed the purpose ofmathematics in science.Mathematics provides a formaldescription of a theoretical modelfor empi rically d iscovered results

.

Such a descrip [ ion is necessary forpredicting other results from thosealready d iscovered

,

Since the mathematicaldescription is of a model, it cannotbe expected to fit the world, wartsand all. The model will deal with anidealisation

; in particular, since themodel is not a description of all andeverything, it will entail

concentrating on some effects andIgnoring others. Hence theproblems with definingfe.g.)amptres experimentally. Further,formulisations based on “real"numbers entail using limits whichare impossible experimentally..Overcoming these problems meansinvent! ng a bet ter model

,one which

includes more forces, or can bedescribed by quantisedmathematics.

The universe ls quite complex,,and it is often useful to deal i nidealisations which ignore"irrelevant" forces. Presumably,the next round of "complete" orunified theories will bring togetherastrophysics r electromagnetism andsubatomic physics, however, Esuspect that pred ict ing simpleeffects from such a theory will bei ery complicated,

It is for precisely the abovereasons that it is wrong to attack theuse of mathematics in computingscience. Here, one is dealing with asituation which is wholly

understood, since it has beentreated Synlhet ical ty

. One ca nproduce complete mathematicaldescriptions of (eg.) the languagesin use. It is the very artificiality olthe computing science world thatallows this accurate formalisation,in contrast to the situat ion gene rallyin physics.

ENERGY

TRANSFERRecently, since Ivor Catt firstelaborated his iconoclastic views,several Wireless World roadershave surfaced with their ownproblems relating to electriccurrent theory. To these I wouldlike to add my own.At school, f was taught that the

resistance of a conducting wirewas inversely proportional lo thecross-sectional area of the wire.RalMol/r2 (where r = radius of

the wire) This seemed reasonableSince, just as with water runningdown a pipe, the thicker thewire/pipe, the more space therewould be for (he charge/water topass through.

Later, I learned that electriccharge only flowed along thesurface of a conducting wire.(down to a certain small "'skindepth’ 'J, This seemed to squarewith the idea that the residualcharge on a conductor is to befound on the surface. However,this would lead one to imply thatthe resistance would be inverselyproportional to the cross-sect ioiialperimeter of the wine,Ralfr

G* A.M 207

Furthermore, does a hollow

conducting wire (with wall

thickness greater than skin depth)have the same resistance as a solicconducting wire of the sameradius?

Ivan L. E. Fant

Stoke NewingtonLondon N6 _