Historical Context | Beginnings →

## Introduction

**On this page…** (hide)

A collection is like a puzzle. It raises many questions without the need for a single word. This is the opposite. Here are posed quite a few words even before the questions are fully articulated.

Consider this. You are in a new land with which you are not yet very familiar. You stumble across a strange dark place you have never visited. There is a door. You open it and enter.

Inside it is so dark you cannot make out more than the beginning of a path. As you take a few more steps on your left a light soundlessly turns on illuminating a glass case. In it is a pile of sand. You ponder why it is there, but cannot make it out. As you walk on the light extinguishes and a new light illuminates the next case. In it is a small group of stones. In the next, a rope with knots. In the next a clay object with symbols on it. In the next a set of beads on a string. In the next strange symbols scribed on a decaying yellowing material. Soon an assortment of devices with gears, dials, and various displays emerge…

You are puzzled. Retracing your steps in the gloom you find a dimly lit switch. Hesitatingly you decide to flick it to the other position. Now next to each case you see a softly illuminated ageing picture. Each picture shows a human figure doing something with the objects - although what the purpose is is not clear. One or two figures are naked. Most are dressed in clothing of different fabrics, perhaps from different times. In this half light you see branching passage ways with different figures, clothing, and seemingly ethnic features. You are seeing more and yet it is not clear what it really all means.

Getting to what seems to be the end of the display you see in the last case something you recognise. It is a pocket calculator. So, you surmise, perhaps this may seek to illustrate the developments in calculation technology which ended with the abrupt transition to personal electronic mathematical calculators in the early 1970s.

This seems a plausible explanation of the purpose of the display, but why these particular objects, and what does their arrangement signify? Is there a card (or catalogue essay) somewhere to provide more context and explanation? This is one such essay.

The above story is of course a metaphor for the personal collection of objects which is the subject of this website. This essay seeks to provide a context and at least partial explanation of the choice and meaning of them.

### 1. Initial observations.

As you already will know following those early personal calculators came the personal computer. In various converging guises (including even phones) these diffused unparalleled computing power across the planet. But even the most sophisticated modern computers at heart (though not on their ever more functional surface) simply did a few things extremely fast (logic operations such as “if”, “and” and “not” and arithmetic operations of addition, subtraction, multiplication and division ). Of course on top of this are layers of sophisticated programming, memory and input and output. Prior calculating technologies, whilst much more limited in speed, flexibility and adaptability, were nevertheless similarly restricted to the same simple arithmetic operations. For this reason, whilst mathematics encompasses much more than arithmetic it is not necessary to consider all the historical development of its more elaborate analytic structures.

It is also clear that this collection is in some sense a history. It is laid out along a time-line. There is an obvious progression in complexity, sophistication and style from the earliest to the later devices. It is possible to construct histories of technical devices such as calculators as some sort of evolution based on solving technical problems with consequent improvements in design. But this strips away much that may be important in why they were invented and used. The invention and design of technologies depends in major part on what they were to be used for. There are a number of aspects to this and some others will be dealt with elsewhere in this site. But clearly one important factor shaping the need for, role, and design of calculators has been the parallel developments in mathematical reasoning, which in various ways they have been made to assist. In deference to later considerations we might add a qualifier - which they have been made to assist “perhaps”!

So how much of this history of mathematical reasoning is necessary? Thankfully much of the huge corpus of extraordinary development in mathematics need not be considered. For example, important though they are we do not need to talk about the development of set and group theory, nor of the development by Hilbert of the mathematics of infinite dimensional vector space that made the modern formulation of quantum mechanics possible. Nor need we do we need to study the creation of the tensor theory which enabled Einstein in general relativity to write his wonderfully neat field equations for the shape of space time of this universe.^{1} It is sufficient to note that many modern challenges - from the prediction of climate under the stress of global warming, to the simulation of a nuclear reactor accident, to the deconstruction of DNA - could not occur without enormous numbers of calculations of addition and subtraction (multiplication and division) which can only be carried out in workable times with the use of ever faster calculating devices.

The focus need only be on a tiny simple bit of mathematics - and then primarily on the numerical calculation required to carry out practical applications of mathematical analysis. Yet oddly, it seems in doing so we come across many of the curly issues that we would have to think about if we were focussing on the whole evolving field of mathematical thought. The history of mathematics is itself a field of scholastic study which can be developed from many perspectives. These include those from the mainstream of philosophy and history of science^{2} through to the sociology of science.^{3} Even though this discussion here focuses on only a tiny “arithmetic core” to mathematics it will be important to at least take some account of this literature and its insights.

### 2. Did increases in the power of mathematics lead the development of calculators? Was it the other way round?

The question is important, but the answer is a bit more complex. First there is the question of what caused the developments in mathematics! It might be assumed that mathematics developed through a process that was entirely internal to itself. It might, for example, have occurred because people could ask questions which arose within what is known in mathematics, but needed to develop new mathematics to answer them. This is certainly part of the story. Yet the literature on history of mathematics tells us this cannot be all.

The idea of ‘mathematics’, and doing it, are themselves inventions. The decision about the sort of problems mathematical thinking might be applied to depends also on social considerations. Beyond that there is a shifting story about who might legitimately be taught what is known about mathematics and who should address themselves to such questions.

Similarly at different times and in different cultures have been very differing ideas about the value of invention. At some moments the mainstream view has been that the crucial task is to preserve the known truth (for example, as discovered by some earlier civilisation, or as stated in a holy book). But at other times and place much greater value has been placed on inventing new knowledge. But even when invention is in good standing there can be a big question of who is to be permitted to do that. And even if invention is applauded it may be still true that this may only be in certain areas considered appropriate or important. In short, a lot of factors can shape what is seen as “mathematics”, what it is to be used for, and by whom.

As an illustration it is worth remembering that astrology has until relatively recently been considered both a legitimate area of human knowledge and a key impetus for mathematical development. Thus E. G. Taylor writes of the understandings in England in the late sixteenth century: “The dictum that mathematics was evil for long cut at the very roots of the mathematical arts and practices. How were those to be answered for whom mathematics meant astronomy, astronomy meant astrology, astrology meant demonology, and demonology was demonstrably evil?”^{4} Indeed, it was noted that when the first mathematical Chairs were established at Oxford University, parents kept their sons from attending let they be ‘smutted with the Black Art’.^{5} However, despite these negative connotations, practioners of “the dark arts” played a strong role in developing and refining instruments and methodologies for recording and predicting the movement of “star signs” as they moved across the celestial sphere.

One of the key features of the contemporary world is its high level of interconnection. In such a world it is easy to imagine that developments in “mathematics” which happen in one place will be known and built on almost simultaneously in another. Yet that is a very modern concept. In most of history the movement of information across space and time has been slow and very imperfect. So what at what one time has been discovered in one place may well have been forgotten a generation or two later, and unheard of in many other places. So, talk of the evolution of mathematics as if it had a definite timetable, and a single direction is likely to be very misleading.

We can only know where development occurred from where there is any evidence remaining. Even this reveals a patchwork of developments in different directions. No doubt this is but a shadow of the totality constituting a complex pattern of discovery, invention, forgetting, and re-discovery all according to the particular needs and constraints of different cultures, values, political structures, religions, and practices. In short, understanding the evolution of calculating machines is will be seen to be illuminated by investigating it in the context of the evolution of mathematical thinking. But that is no simple picture. The history of developments in calculators and mathematics has been embroidered and shaped by the the social, political and economic circumstances in which they emerged. And at times, mathematical developments have shaped developments in calculators, and times, vice-versa.

### 3. What is a calculator?

The above raises the question of what is to be meant by a “calculator”. “Calculator” could be taken to mean a variety of things. For some, it may conjure up an ‘app.’ on an iphone for doing a range of calculations. For others it may evoke the small digital calculating devices (such as the Hewlett Packard HP-35) which became pervasive in the last three decades of the twentieth century. For others it may include the motorised and before that hand-cranked mechanical devices that preceded the electronic machines. It is difficult to see where the line should be drawn in this regress all the way back to the abstract manipulation of ‘numbers’.

*In this discussion, “calculator” is used as shorthand for “calculating technology”. In particular it is taken to mean any physically embodied methodology, however basic, used to assist the performance of arithmetic operations (including counting).* Thus a set of stones laid out to show what the result is if two are added to three (to give five), or if in three identical rows of five what the outcome is of multiplying five by three (to give fifteen) will be regarded as a simple calculator. So too, will the fingers of the hand, when used for similar purpose, and even the marking of marks on a medium (such as sand, clay or papyrus) to achieve a similar result.

This approach is certainly not that taken in all the literature. Ernest Martin in his widely cited book “The Calculating Machines (Die Rechenmaschinen)” is at pains to argue of the abacus (as well as slide rules, and similar devices), that “it is erroneous to term this instrument a machine because it lacks the characteristics of a machine”.^{6} In deference to this what is referred to here is “calculators” (and sometimes “calculating technologies or “calculating devices”), rather than “calculating machines”.

Thr decision to apparently stretch the concept of calculator so far reflects a well known observation within the History and Philsophy of Science and Technology that in the end, technique and technology, or science and technology, are not completely distinct categories. Technologies embody knowledge, the development of technologies can press forward the boundaries of knowledge, and technological development is central to discovery in science. As Mayr says in one of many essays on the subject, “If we can make out boundaries at all between what we call science and technology, they are usually arbitrary.”^{7} Indeed, as will be described later, the mental image that mathematics is the work of mathematicians (‘thinkers’) whilst calculators are the work of artisans (‘practical working people’) is an attempt at a distinction that falls over historically, sociologically, and philosophically.

### 4. A discussion in two parts.

In keeping with the analysis I have contributed to elsewhere (in a book by Joseph Camilleri and myself), human development, will roughly be divided into a set semi-distinct (but overlapping) epochs, preceded by a “pre-Modern Era” spanning the enormous time period from the birth of the first modern homo-sapiens to the beginning of the “Modern Period”. This beginning is set as beginning (somewhat earlier than is conventional) in the *middle of the sixteenth century*, with the “Early Modern Period” continuing from the *mid-sixteenth to late eighteenth century*, and the “Late Modern Period” stretching forward into the twentieth century, and *terminating around the two world wars*. From thereon, the world is regarded by Joseph Camilleri and myself as entering a *period of transition*^{8} (but there is not much need to focus on that here).

In relation to the collection of objects, for which this discussion forms a context, the content similarly breaks effectively into two major parts. The first part, which looks at the relationship between the evolution of calculating and calculators in the pre-Modern period, forms a backdrop, which important as it may be, does not refer at all to specific objects in the collection. As for the collection, its objects are drawn in their entirety from what, in the above sense, can be considered the Modern Period (the earliest of these objects being from the early seventeenth century).

Historical Context | Beginnings →

^{1} See for example http://mathworld.wolfram.com/EinsteinFieldEquations.html or for more explanation http://physics.gmu.edu/~joe/PHYS428/Topic9.pdf (both viewed 26 Dec 2011) (↑)

^{2} See for example, Eleanor Robson, Jacqueline A. Stedall, *The Oxford handbook of the history of mathematics*, Oxford University Press, UK, 2009 (↑)

^{3} See for example, Sal Restivo, *Mathematics in Society and History: Sociological Inquiries*, Kluwer Academic Publishers, Netherlands, 1992 (↑)

^{4} E.G.R. Taylor, *The Mathematical Practitioners of Tudor & Stuart England 1485–1714*, Cambridge University Press for the Institute of Navigation, 1970, p. 4. (↑)

^{5} John Aubrey quoted in Taylor, ibid, p. 8. (↑)

^{6} Ernest Martin, *The Calculating Machines (Die Rechenmaschinen)*, 1925, Translated and reprinted by Peggy Aldrich Kidwell and Michael R. Williams for the Charles Babbage Institute, Reprint Series for the History of Computing, Vol 16, MIT Press, Cambridge, Mass, 1992, p. 1. (↑)

^{7} Otto Mayr, “The science-technology relationship”, in Barry Barnes and David Edge (eds), Science in Context, The MIT Press, Cambridge USA, 1982, p.157. (↑)

^{8} Joseph Camilleri and Jim Falk, *Worlds in Transition: Evolving Governance Across a Stressed Planet*, Edward Elgar, UK, 2009, pp. 132–45./ (↑)