Mr Babbage’s Secret

Transcribed from the printed issue; not yet reviewed. Read it in the PDF of the issue, page 61

When Charles Babbage died in 1871, the ‘Times’ remarked in his obituary that he had lived to be almost 80 “in spite of organ-grinding persecutions”. A prolonged opponent of street musicians and other noise pollution was how society remembered him — the genius who declared, looking back upon his life: “The great object of all my inquiries has ever been to endeavour to ascertain those laws of thought by which man makes discoveries”.

Today, Babbage is recognized as the great pioneer of the digital computer. Modern electronics has made his dream come true. To appreciate the significance of this evolution, let me draw a historical analogy with the introduction of logarithms, one of the first practical means for facilitating more advanced computations.

In 1627, after years in the treadmill of arithmetic, Kepler finally published the ‘Rudolphine Tables’. Based on the life-long labour of Tycho Brahe, these tables also contained logarithms, for the first time put to astronomical uses. Kepler became acquainted with Napier’s logarithms some ten years earlier, and he wrote enthusiastically: “A Scottish baron has appeared on the scene (his name I have forgotten) who has done an excellent thing by transforming all multiplication and division into addition and subtraction…”. This enthusiasm did not please Maestlin, Kepler’s old professor, who rebuked him: “It is not fitting for a professor of mathematics to manifest childish joy just because reckoning is made easier.

Since Napier did not as first explain the underlying principle, Maestlin’s remark was not without merit. The potential of technological advances is in the novel principles they uncover. Babbage’s perhaps most successful book: ‘On the Economy of Machinery and Manufactures’, support this thesis. Published in 1832, in the midst of the industrial revolution, it is hailed today as a scientific landmark in economics as well as in manufacturing engineering. In it, Babbage carried the concepts and abstract models of description from the world of natural phenomena into the realm of man-made systems. He said in its preface:

“Having been induced during the last ten years, to visit a considerable number of workshops and factories, both in England and on the Continent, for the purpose of endeavouring to make myself acquainted with the various resources of mechanical art, I was insensibly led to apply to them those principles of generalization to which my other pursuits had naturally given rise”.

We are also in the midst of an industrial revolution, brought about by the modern computer. Within a few decades, we have witnessed a rapid change from electromechanical relays to vacuum-tubes, transistors and complex chips. Yet Ohm’s and Kirchhoff’s laws have not changed. Independent of the technological stage, they have ruled the electrical behaviour of the hardware. The means and the emphasis have changed, but the basic principles still hold.

What a contrast to the development on the software side! From the very beginning, we have identified the aim of our teaching with programming — the dull drilling of current technology. Over the years, the adherents of each individual programming language have fought, in the words of Oehlenschläger, the nineteenth century Danish poet, for each their own “little Hassan with the crooked legs”.¹ Notwithstanding that education is simpler than re-education, it is re-education which has carried the day. For we have taken no heed of Babbage’s advice to apply “those principles of generalization to which my other pursuits had naturally given rise”.

These other pursuits suggest that we are facing a two-fold problem in our teaching for the future. We need to distinguish between:

(1)

A theory, giving the principles or laws of data and data-operations.

(2)

An art, establishing the empirical guide-lines and preferred standards for systems design.

To recognize this dichotomy between laws of data and guidelines of systems design is of vital importance; for laws of data and data-operations are independent of the changing technology, whereas the art of systems design is not. Both problem areas present formidable challenges which must be met here and now, since each in its way demands adjustments, changes, or even reorientations in our curricula. Note, though, that I am not talking about the development or invention of entirely new ideas or principles. Because, as I see it, the inventions are made, tested and documented in the traditional sciences. What we need to do is to collect them, adapt them and bring them down from the level of the highly-trained specialists.

Piet Hein,² my fellow countryman who pins down truth by humour in his “Grooks”, once said that humour cannot be defined: — for those who were likely to define it, had no sense of humour. Data is another familiar concept which escapes a rigoristic definition, at least in the scientific sense. The mathematician Gödel proved that in any logically consistent system, there had to be some undefined terms basic to the system. The notion of data is one such basic term common to the various sciences. However, since Galileo the important thing has been, not definition nor explanation, but measurement. For something to classify as a measurement, it must be verifiable, and hence repeatable. In other words, we must be able to identify certain invariant properties or relationships.

The notion of data is closely related to the scientific concept of measurement, since observation and the recording of observations are the only link between the real world and a corresponding theoretical model. Yet, the concept of measurement is strangely absent from our discussion of data and their handling by computers. The fact remains, though, that some of the most important properties of data derive from the invariant scale-form underlying the associated measurements. Common scale-forms are defined across the disciplines, from psychology and sociology at one end to physics and engineering at the other. It lies in the scale-form that it is meaningful to subtract calendar years, but not telephone numbers, and that it is meaningless to take the ratio of calendar years. In fact, scale-forms help us to classify data-operations.

But data are more than mere recordings of individual measurements. I am thinking here of the relationships between data, the so-called data-structures, which always depict certain invariant properties of the system under consideration. The computer is merely a tool whose justification lies in its application for solving problems. The data-structure “measures” the interconnections or interrelations in the applications. Lists, tables, boxes and nested arrays are geometrical representations of these systems properties. Geometry is the fundamental model of description in any scientific discipline, so it would be natural to expect that there should also exist a geometry of data. Indeed, there does; and APL illustrates part of it.

The geometry of data differs from the classical geometries on two accounts. Its spaces are finite, and its binding property is not distance but well-ordering. That is, every non-empty subset has a first element. Certain properties of these subsets, that I might perhaps call data-spaces appear strange at first, because common-sense notions of the conventional metric space break down, such as the usual conception of dimensionality. However, since the geometry of data satisfies Felix Klein’s definition of a geometry, put forward in 1872 in his famous Erlanger Program, this formulation paves the way for the enunciation of laws of data, as we use the APL notation to push this geometry into an associated algebra.

The laws of data are concerned with preservation of properties under various operations. They do not compare with, and they are essentially different from the well-known natural laws of Newton, Hooke, Ohm, Kirchhoff, and all the other classical physicists. Yet, analogies to the data laws, classified mathematically as algebraic invariants, are abundant in modern phsyics and chemistry. The power of adopting this approach is that we are now able to introduce laws in various areas of application: production engineering, management, administrative systems, and modern control theory, for which it has not been possible to formulate laws in the axiomatic sense of classical physics. To illustrate, the bill of materials, the cost aggregation of an assembly, the scheduling of processes, and the state-equations of a control system, may now be explained as similar, so-called isomorphic, data-structures.

Thus, as I have tried to indicate here, a new perspective has opened up towards a geometrical theory about laws of data and data-operations. This theory is in the mainstream of all the other sciences and their areas of application. It is teachable and testable in the conventional scientific sense, and it is essentially independent of the stage of current technology.

Let me now turn to the other problem area we have to face in our education, namely the art of systems design. The building-up of systems is encountered in many disciplines. Still, it has been cultivated in its purest form in engineering, for the essence of this topic area is design. Consequently, many of the basic ideas and approaches to be used in connection with the computer may be introduced by emulation of the time-honoured engineering approach.

However, for our universities and high-schools to adopt the spirit and attitude of design is perhaps the most difficult challenge in our time. The training of our teachers has been geared towards analysis of existing systems emphasizing problems with a unique solution. Now, the computer will force us to complement this type of education with the experimental or cut-and-try approach so characteristic of the design process. We have to emphasize the scaffolding, how to solve multiple-answer problems, and to establish criteria for evaluating the feasible solutions.

Of course, what is important here, is not the actual design, but rather the development of an attitude in the mind of the student: an intellectual courage and certain rather formal working habits. What makes this such a perplexing task, however, is the fact that the student does not command any discipline well enough to actually solve a meaningful design problem. Faced with a series of often trivial and usually uncorrelated problems, the attention is focused on all kinds of side issues, so that when the first excitement of using a computer is over, the desire to learn has become a desire to leave.

What is needed, therefore, is an area of application requiring no or very little prerequisite knowledge, yet complex enough to admit of meaningful or even exciting problems. Also, it should be possible gradually to introduce a rich and inspiring variety of mathematical solution techniques emphasizing the geometrical view on data and data-operations.

This problem as well as the other challenges I have mentioned in relation to the computer education in our universities and high-schools, have much occupied my colleagues in FinnAPL. The excellent ‘FinnAPL Idiom Library’ which they published in 1981, is a tangible result of their thoroughness in this endeavour. So, naturally, during a visit to Helsinki a few years ago, I found myself in the midst of a heated discussion on this vitally important educational issue. My Finnish colleagues were so enthusiastic that, in the spirit of the moment, I somewhat rashly accepted their challenge, later politely formulated as an invitation, to write a book showing how Babbage’s cryptographical hobby could be brought to bear on the teaching of computers. This is the book ‘Mr. Babbage’s Secret’ which, thanks to the gracious support of IBM in the Nordic countries, has been published on the occasion of this conference.

Newton said that if he had seen further than others, it was because he had stood on the shoulders of giants. He immersed himself in the past for the very same reason that the engineering constructor digs into the ground — to secure the future. The more powerful the tool, the greater its potential possibilities, the more we need to have perspective and inspiration. Consideration of the great scientists of the past provides this longer view. From their work on the kind of problems confronting us now, we can learn, to quote Alfred Whitehead, “to see what is general in what is particular and what is permanent in what is transitory”. It is this aspect that makes history fascinating. It is here we may ascertain, as Babbage expressed it, “those laws of thought by which man makes discoveries”.

This historical perspective brings us round a full circle to the starting point of Charles Babbage’s early work on computers. What he and his contemporaries wanted, and what the modern end-users want, is to use the computer as a problem-solving tool. Their interest and effort are in the application. They have no desire to become computer specialists in order to transform their problem formulation into a form acceptable to the computer. Instead, they want the communication with the computer to take place in a form which is natural relative to the problem at hand.

For two reasons in particular, APL represents a breakthrough in this direction. One feature is the reintroduction into computer programming of the fundamental fact that, basically, people perceive data as geometrical objects. The other feature is APL’s elegant and simple mode of interactive communication, offering the user the opportunity of gradually improving his design by experiments. It is noteworthy how well these two features meet the two basic educational needs — in the area of a geometrical theory of data, and in the art of systems design.

I do not intend here to go into the actual contents of my book, except to say that it is centered around the notion of laws of data and the use of the computer as a workbench. APL is applied as a mathematical notation as well as a programming tool to make the approach operational. In fact, some thirty user-defined APL functions are documented in the book and, with a little patience, everything I have done, can be done in a 64 K workspace. Half of the figures are in APL, so together with the terminal sessions APL occupies slightly more than one quarter of the total number of pages. Yet it is no textbook nor an introduction to APL. It is a tale based entirely on historical facts, many of which are published here for the first time.

When Hans Christian Andersen had written his first tales, he turned to the physicist Hans Christian Oersted for advice, whether or not he should publish. Oersted was enthusiastic, and thus became directly responsible for launching Andersen unto his life’s work. What a strange combination: that the discoverer of electromagnetism should counsel the poet on tales for children. Perhaps it was not so strange after all; for Andersen’s tales have that special quality, often lost in translation, that they are ambivalent. Submerged in each tale for children there is also one for adults. Undoubtedly, this was what fascinated Oersted.

Babbage’s preoccupation with secret writing exhibits a similar trait: an exciting hobby and a fascinating scientific work. This similarity made any other form than the tale inconceivable. In Babbage’s endeavours there would be something to inspire teacher and student alike. For he had fun.

In my introduction, I touched upon Babbage’s battle against street-musicians. Let me therefore conclude by recounting a true episode from one of the times Babbage had hauled an offender before a magistrate. The latter asked Babbage whether a man’s brain would be injured by listening to a hand organ, to which Babbage retorted: “Certainly not, for the obvious reason that no man having a brain ever listened to street-musicians”.

Are we listening, or are we just playing it by ear?