Footnotes
Keywords: PatRec

...system.
  There is a school of thought which believes that this shows beyond all doubt that scientists and engineers are crass. Subscribers to this belief are seldom capable of mastering, say, calculus. Who is the crasser?

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...likelihoods
  This use of the term `likelihood' for the value of the gaussian function evaluated at a point is frowned on by statisticians and probabilists, who insist that the likelihood is a function of the parameters of the gaussians (or whatever distributions they wish to use), and not a function of the data. It is obvious that the number obtained is actually a function of two sets of variables, one comprising the data and the other the parameters of the distribution. Statisticians, and some Probabilists, seem to have trouble with this idea and a sentimental attachment to a rather clumsy language. Engineers commonly use the terminology I have employed, and although their jargon is sometimes just as bad, in this case they have mathematics on their side, so I follow their usage.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...analysis.
  Incidentally, when I say `trajectory', I do not mean `path'. The former is specified by where you are at different times, the latter is just the set of points where you were. If your path intersects that of a tiger in the jungle, you might be alright, much will depend on how far away the tiger is when your paths cross, while if your trajectory intersects that of a tiger your prospects are bad.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...language
  And the extensions of the language, used currently for describing objects studied in theroretical physics, such as vector bundles and tensor fields.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...language
  At least this is one interpretation of what he thought he was doing. He may have thought that there is some vagueness in the universe.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...hypothesis''.
  Lagrange is supposed to have piously remarked on hearing this story ``Ah, but it is such a beautiful hypothesis, it explains so much.'' Actually, it explains everything. That's what's wrong with it.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...space
  Although I slithered around the question on every reader's lips:` yes, but what exactly is a cluster?' It's a good question.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...them
  The best bet at the time of writing, is to get yourself a nice Pentium machine and install LINUX, thus getting the best of both worlds.This text is being written on such a machine, and will be compiled by LATEX. The LATEX is explained in the Scientific Writing course, and the installation of LINUX should be done by consultation with your tutor.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...levels
  And for the lucky among us, colour.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...book.
  I promised that this book would be user friendly and am trying to keep my word. Maybe my next book will be academically respectable and properly inscrutable.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...us.
  It is rumored that Probabilists are good people with whom to play poker, because they focus on the probabilities and not the personalities. I am unable to confirm this from personal experience as none of the Probabilists I know will play poker with me. I can't imagine why.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...board,
  It is usual to model dart throwing as selecting a point at random inside a disk.

  This might shock genuine darts players. Of course, real dartboards are made of atoms, and space might come in little lumps for all we know, so this model is only one of several.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...the
  The main objection to starting with a pdf is that not all measures can be represented by pdfs. We can assign a uniform measure to the real line, and also the measure which concentrates every possibility at one point. These are perfectly respectable measures and might well be of use in probability theory, but they cannot be described by pdfs. One way out of this is to allow ourselves to use generalised functions to represent measures, where a generalised function is a suitable sequence of ordinary functions. For instance, we can take a gaussian on the origin and make it take on progressively larger and larger values for the standard deviation, to get a sequence of functions defining the uniform measure on the real line. Or we could make tend to zero so as to get the measure concentrated at the origin. To be more precise, what I should have said is that it would be simpler to start with a measure on the sample space of possible outcomes. This is often done. I shall assume from now on that we are free to use generalised functions to represent measures, or that we can restrict ourselves to the case where the measure can be represented by a (usually continuous) function.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...pdf.
  There is a school of thought which regards this as a very wicked choice of terminology. This is because probabilists have persuaded themselves that the right way to use the term `likelihood' is to mentally drive a nail through the data and then regarding the numbers obtained as a function of the parameters used for specifying a pdf. This is the likelihood. They are, however, making rather a pointless distinction, since they are talking about a function of two variables, data, and parameters of a family of pdfs. What I describe as pointless, they would describe as subtle. My objection to this particular subtlety is that it seems to be concerned with the semantics rather than the syntax: they use the term so as to tell you what they plan to do with the number, not how they compute it. Conversely, when offered my definition above, probabilists have assured me disdainfully that there aren't any parameters, so my definition makes no sense. But there is a perfectly good parameter space, it just happens to consist of a single point. I have discussed this matter with distinguished probabilists: usually, each of us winds up thinking the other rather crass. I may return to this point later if my conscience gets to me.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...randomness
  This leads to the following argument: Of all the sequences that could occur in a game of poker, some are random and others are not. Without getting into details, all hearts first, in order, then clubs, then diamonds, then spades, is one of the non-random sequences. Let us therefore remove these non-random sequences from consideration. Our cards have been shuffled, and if the sequence turned out to be non-random, a complaint that the shuffler had failed to do his job properly would carry. Hence we may assume that our sequence of cards is one of the random ones.

  The arguments advanced in the beginning of this chapter on the probability of filling a flush or a straight at poker are now invalid. They depended on considering all sequences of cards as equally likely, but we have deliberately excluded the non-random ones. How much difference this would make in practice is a little hard to say absent a definition of random.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...stated,
  But see Thom's Structural Stability and Morphogenesis, and Abraham's Foundations of Mechanics.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...Penttila,
  In the common room of the Mathematics Department at the University of Western Australia. The coffee is awful, but the conversation sometimes makes up for it.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...Model.
  This terminology is more used by engineers than probabilists, for whom the term `model' is more often used to denote what I have called a family of models. Thus a probabilist would commonly say that what I have called a family of models indexed by the numbers between 0 and 1 is just the binomial model. Since it is clearly a good idea to have a distinction between the space of things and the individual things in the space, I shall follow what seems to be a de facto tradition among engineers and such roughnecks, that is less confusing than the tradition followed by statisticians and probabilists. Beware of this point when reading the works of the latter.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...model
  Known to followers of E.T Jaynes as the `SureThing' model.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...past.
  What `roughly similar' means is, of course, intimately tied up with transfer of learning and the crucial business of what precisely constitutes a replication of an experiment? These issues are slithered over in the foundations of probabilistic statistics because they are too hard.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...English.
  It isn't that easy, but this is another subject.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...Inequality.
  Proofs done using water are frowned upon in some quarters, usually by people to whom proofs are part of life's mystery, and who want to keep life as mysterious as possible. You may have read some of their books, or tried to.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...Platonists.
  Of whom there are a fair number around. No doubt a cure will be found in due course.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...sort)
  The Intermediate Value Theorem tells you that if you have a continuous function defined on an interval which takes values which are positive at one location and negative at another location, then it must be defined at some other location where it takes the value zero. This codifies an intuitive picture of the real line as resembling a ruler made out of chewing gum and a continuous function as stretching it but not tearing it, and it means that when we want to find the zero of a function by some iterative process, we have usually got grounds for thinking it is there to be found. The price paid for this simplicity and adherence to naive intuitions includes the uncountability of the reals and non-constructive existence proofs which give confidence only to the more naive Applied Mathematicians.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...for
  Well, it's what I want models for. If you have some alternative use for them, good luck to you.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...rapidly
  Well, fairly rapidly. Doing this with axiomatic precision requires careful avoidance of pathologies which trouble the sleep of Pure Mathematicians but which most people wouldn't think of. I shan't ask you to think of them either.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...task.
  Robots are programmed to do similar sorts of things, e.g. shoot down enemy aircraft, and they do it tolerably well.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...model.
  There is altogether too much of people feeling things in their bones in this business.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...cheese?
  Do not try to use it to prove Kraft's inequality.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...chance
  Actually, probability 1.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...meanest
  If your intellect isn't very mean, and if these things are not immediately obvious, you need some remedial linear algebra.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...small.
  Of course, I can compress this information better if I know anything about it, which is the point of modelling the data. Likewise I can compress the parameters better if I know anything about how they are distributed, which gives us the Bayesian prior. And smaller fleas...

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...utility.
  It is true that no computer can add any two numbers. The point is that there are always numbers too big to be represented in the computer, no matter how big the computer is. Since we can make computers which can add numbers in the range we usually care about, this is not quite so devastating as it sounds to the innocent and uncritical.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...military
  That is, he hadn't killed anyone yet.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...down,
  Or a particularly half baked one either, alas. Astrology has been going much longer than Science, which may yet turn out to be a passing fad.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...smooth
  `Smooth' is Mathematician's talk for differentiable as many times as looks like being a good idea

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...class.
  S0 is the zero dimensional sphere for those who don't know; the n-dimensional sphere is the equator of the n+1-dimensional sphere and the 1-dimensional sphere is a circle. The equator of a circle consists, of course, of the two points . Now you know.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...classified.
  As described, it will keep going indefinitely of course, but at least it will do so in a simple way which can be checked on. Always right, it will sit smugly there for ever, so all we need to do is to monitor its torpor.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...hyperplane.
  This is intuitively hard to doubt and if you wish to doubt it, do so, but be careful, you may turn into a Pure Mathematician.

  The price of freedom to doubt intuitively appealing assertions like this is that you get asked to prove them. Go ahead, make my day.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...time.
  This is a legitimate trade off, of course. The extent to which people opt for random searches through huge state spaces, rather than think about what they are doing, tells you how high a price they need to be paid in order to think. For some people, there isn't that much money.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...support
  compact support means that the function is zero except in some bounded region of .

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...easy
  Well, fairly easy. See Simmons in the references for a perspicuous proof.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...Rock
  For those readers unfortunate enough to live somewhere other than Western Australia, it should be explained that Wave Rock is a tourist attraction. It is a rock that looks quite a lot like a wave on the point of breaking.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...Guinea.
  Or, for that matter, the wilder parts of New York.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...Recognisers.
  Some of us are humbler than others, of course. And some of us have a self-esteem that would survive impact by a medium size asteroid.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...polite
  And, let's face it, civilised.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...
  That's all from Chris. You're back with me again from now on.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...alphabet
  If this looks like a concession to the Artificial Intelligentsia, and a descent into rather crude types of representational machinery, this is an error, and I shall put you straight in later chapters, but it has to be done, if only because continuous trajectories are reduced to discrete trajectories a lot of the time. There are other and better reasons, as the persistent reader will discover.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...frequency
  The same effect can often be obtained by using a cheap microphone, but this approach is thought unprofessional by most engineers. They prefer to get an expensive microphone and then put a filter in to make it behave like a cheap one.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...range
  By comparison, the human ear has a cochlea, a rubbery tube filled with fluid, and a membrane down the middle of it, which vibrates when sound strikes the ear, and which because of varying properties of the membrane, responds in different places to different frequencies. The vibration of the basilar membrane is monitored by around 30,000 hair cells. It is thought that this performs something like a fourier transform on the speech signal, but it is known that much more happens than this. The difference between 16 and 30,000 is a little disturbing.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...function
  If you can't tell the difference, don't cross any roads until you have thought about it.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...way
  So much so that one of my more ebullient colleagues has been found threatening to vector quantise people who argue with her.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...algebra:
  Watch out for the next footnote.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...procedure
  Described in English rather than algebra. Algebra has the merit of precision and the drawback that it can take a long time to work out what the underlying ideas are, sometimes rather more than a lifetime. It is generally rather easier to turn an idea from English into algebra or a suitable programming language than to go the other way round, but you may memorise algebra and get the illusion that you understand it when you have merely learnt it. Engineers sometimes use algebra as Lawyers and Medical men use Latin: to impress the peasantry and justify their salaries. Also, it conceals the fact that they can't spell and have awful handwriting.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...hope.
  I suppose that a methodology which allows morons to publish research papers is a fine and egalitarian thing, but it does clog up the journals somewhat. Again, I incline to blame the Mathematicians for a disgracefully inadequate job of teaching Mathematics in Universities particularly to Engineers and Physicists. Of course, the Mathematicians blame the Engineers and Physicists for demanding technical skills without any idea of why the methods work. This leads to Engineering journals keeping the standards high by ensuring that there is a significant mathematical content to each paper. They decide this by counting the number of integral signs, partial derivatives and half the summation signs and divide by the number of pages. This leads to the most banal of ideas, which could easily be explained in English in half the space with twice the clarity, being expressed in algebra. The other part of the problem is an insane ambition which leads folk of limited intellect to imagine that they can do Science and Mathematics. Let them do the so called Social Sciences. That's what they were invented for. Real Science and Engineering and Mathematics are difficult. Ah, I feel lots better now, thank you. You might ask why diatribes like this are found in a simple, academic book. Well, if nobody bitches about incompetence, dogmatism and stupidity, we shall get a lot more of it, and there's already plenty. It's a dirty job, but someone has to do it; it clearly takes a highly developed contempt for the whole concept of political correctness, making the author rather well qualified. Actually, I'm joking. Academic life is really just one big happy family.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...them.
  What this does to the French language shouldn't be done to bushflies.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...trying
  There are two activities traditionally practised by grammarians. The first is to regulate how other people ought to talk and write. Grammars promulgated for this reason are called normative. The second is to try to describe what is actually said or written by some group of people. Such grammars are called descriptive. We are only concerned with the second sort.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...extensively.
  It is important to understand that we don't judge a model by its plausibility in science, we judge it by whether it works or not. Thus we replace woolly arguments between proponents and detractors by careful investigation of the world. This saves a lot of time, and sometimes blood.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...things.
  Or perhaps not.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...about.
  Definitely desperation city. When Einstein remarked that the most amazing thing about the universe was that we could understand it, he was looking at a biased sample. Mostly we can't. It's rather less amazing that there are bits of the universe simple enough to be understood even by thickos like us, and quite typical of human beings to congratulate themselves excessively when they find one.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...limited.
  This is the good news.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...equations.
  The power of Linear Algebra is shown here: No less than four blokes got their names attached to a geometrically trivial observation just because the dimension was higher than three.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...oafs
  Well, the ones who can put up with me tend to be. No doubt it's a biased sample. Of course, they say they are merely healthy and have good digestions and clear consciences as a result of doing honest work.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...reader.
  And if it burns out the brain of the unreflective reader, serve him right.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...see.
  You were warned about function fitting. Don't say you weren't.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...`seminal'
  Others have been even ruder.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...either.
  Linguists have argued that English is not really finite, since you can self embed parenthetic clauses as deep as you like. Thus we have such constructs as the old lady who swallowed a cow in order to catch the long list of other things she swallowed in order to catch the spider which she swallowed to catch the fly she started off with. Similarly for houses that Jack built, and those sentences of German which end with a barrage of verbs. I have found while talking, something I do a lot, to students, and others, that the deepest (and I mean deepest!) level of recursion to which it is practicable (in the sense of keeping the attention of the student, or whoever) to go, is about three. I also find sentences of more than three million words tiresome. Sentences of more than thirty billion words would find no human being living long enough to read them or remember what the hell the beginning was about. It seems safe to conclude that English is actually finite, although the simplification of making it infinite might be worthwhile. But to do so in order to accommodate nursery rhymes seems excessive.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...chapter.
  Of course there are people who will want every term defined short of circularity, but algorithms for most of us are like elephants. We can't give satisfactory definitions of them , but we can recognise them when we see one.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...streets.
  This is important. The damage those guys could do if let loose in business or politics doesn't bear thinking about.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...bibliography.
  It may be remarked that many books on languages and automata are written by mathematical illiterates for mathematical illiterates, which makes them hard to read, and Eilenberg is a notable exception. His book is hard to read for quite different reasons: he assumes you are intelligent.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...on.
  After doing it my way, I discovered that Eilenberg had got there before me, and he explains how somebody else had got there before him. Such is Life.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...them.
  Alternatively, you can go into seedy bars with a clear conscience, knowing that you no longer have to worry about meeting people who will expect payment for slipping you a suitable initialisation.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...indeed.
  If you are used to reading chinese you may feel differently about this, but then I expect you are having a lot of trouble reading the present work.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...long.
  The reason for allowing infinitely long sequences is not that anybody expects to meet one and be able to say with satisfaction, `wow, that was infinitely long', it is more of a reluctance to specify the longest case one expects to get. In other words, it is done to simplify things; the infinitude of the natural numbers is a case in point. There may be some grounds for reasonable doubt as to whether this in fact works: sometimes one merely defers the difficulties to a later stage.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...end.
  Grammarians used to believe that high levels of inflection showed a highly evolved language, probably because Latin was regarded as morally superior to English, German or French. Then it was discovered that Chinese used to be inflected a few thousand years ago but the chinese very sensibly gave it up.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...level.
  Past a certain point it is called `plagiarism'.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...is.
  As Herod said to the three wise men.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...methods
  Or whatever is the pet method favoured by the audience.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...world.
  To a geometer, it looks as though the algebraists have gotten into the cookie-jar when everyone knows they are there to tidy things up and nit-pick about details, not go around being creative. Or, even worse than algebraists, gasp, logicians. When logicians go around being creative, universes totter and crumble.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...calculation
  mere singularity is unimportant; to have one eigenvalue collapsing to zero may be regarded as a misfortune, to have both collapse seems like carelessness

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...world.
  As when one's progeny want one to switch off the Bach so they can concentrate on the Heavy Metal, or vice versa.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...group.
  A Topological Group is a collection of invertible transformations such that the composite (do one then another) of any two is a third, and also having the property that the multiplication operation and the inversion operation are continuous. A Lie Group (pronounced `Lee') is a topological group where the operations are also differentiable and the set forms a manifold, which is a higher dimensional generalisation of a curve or surface. This definition is sloppy and intended to convey a vague and intuitive idea which is adequate for present purposes.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...group
  The term is from differential topology, and may be found in any of the standard texts. It is not necessary to expand upon its precise meaning here, and an intuitive sense may be extracted from the context.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...book.
  But watch out for the sequel.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...cognition.
  Blakemore conjectured it in the paper cited, but it probably struck a lot of people that this was a variant of a Hebbian learning rule.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .

...preference.
  There is a belief in some quarters that God is a mathematician, or why did He make Physics so mathematical? And in particular, He must be a geometer, because most of Physics uses geometry. This is like wanting the Sun to shine at night when we need it and not in the daytime when it's light anyway. The fact is that the most powerful languages allow you to say more interesting and important things. That is why I use geometry.

      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .
      .