Information & Computational Sciences

Numbers

By on 04/11/2015 in ICS

I grew up with a love of numbers. The simple process of counting things leads, in a few short steps, to a fascinating world of complexity. Define a few operators, and numbers come alive, forming patterns, and then exceptions to the patterns that lead to new operators and new numbers. From the natural numbers (1, 2, 3, …), subtraction leads to negative numbers, multiplication the primes, division the fractions. Numbers and groups of numbers are described using a range of terms. I have already mentioned ‘negative’, ‘prime’ and ‘natural’, all of which have a world of meaning wrapped up in them through their associations with other contexts in which these words are used, but to these may be added words such as ‘rational’, ‘golden’, ‘square’, ‘complex’, ‘real’ and ‘irrational’. (The horror of the Greeks at the discovery of these last numbers, when trying to find the ratio of the circumference of a circle to its diameter, is legendary.) David Wells’s Curious and Interesting Numbers devotes a 231-page book to the wonder of numbers.

It is often said that everything in a computer is a number. This isn’t quite true. Rather, everything in a computer is the presence or absence of something, be it charge, magnetism, or light, depending on the medium of storage. It just so happens that sequences of presences and absences can be interpreted as numbers using the binary counting system, using, say, 1 for presence, and 0 for absence. Even then, the numbers in the computer’s memory may have multiple meanings. The same sequence of bits (binary digits), depending on context, could be an instruction to the computer’s central processing unit to do something, a string of alphanumeric characters to display to the user, a colour of a pixel in an image, or a number in the sense we normally mean by the term: an integer, or a floating-point number (a subset of rational numbers that is the closest a digital computer can get to representing real numbers). This flexibility in interpretation makes computers post-modern machines: there is no absolute perspective from which to interpret the numbers – they can only be understood in relation to the currently running program.

Scientists tend to like numbers because they measure things, allowing them to be compared with each other, modelled using equations, and used to make predictions that are quantified. There is even theory about the proper use of numbers when measuring things – which comparison and arithmetic operators can be meaningfully applied – depending on what the number represents. For example, numbers can be used as labels, such as a customer number. With customer numbers, it only makes sense to say that one customer number is equal to another, or is different from it. Adding customer numbers together is meaningless. On questionnaires, we are often asked to indicate the strength of our agreement or disagreement using a number from 1 to 7, say. All that matters there is the order. If 1 means strongly disagree, and 7 means strongly agree, we can meaningfully say that if someone answers 2 to one statement, and 3 to another, then they disagree with the former statement more strongly than the latter, but it would be wrong to argue that the strength of agreement with the latter question is one and a half times more than that of the former. It would also be meaningless to say that the mean agreement of the two statements is 2.5: we could just as easily have used a scale from -3 to 3 (where the same meaningless mean would have been -1.5) instead of 1 to 7, because all that matters is their order.

Scientists have been able to measure numbers that are fundamental to the workings of the universe, including the speed of light, the gravitational constant, and even Planck’s constant, which ironically tells us something fundamental about the limits to our ability to measure things. Perhaps, then, it is inevitable that scientists have tried to come up with ways of using numbers to measure themselves. This is the field of scientometrics. Two popularly-used metrics include the ‘impact factor’, which is supposed to measure of how significant a journal is, and the ‘h-index’, which was intended to be used to rank individual scientists. Both, in one way or another, depend on citations – authors making reference to your work in their own – which at least shows that someone else has read what you have written and considers it worth mentioning the fact.

However, both have been the subject of controversy (the h-index is arguably indirectly discriminatory on the basis of gender, as is the impact factor due to differences in impact factors of journals and gender balance in different disciplines) and game-playing (where editors of journals deploy a number of strategies to increase the impact factor, and authors self-cite to increase their h-index). Still, their use persists, including in recruitment and promotion of scientists, awarding of grant funding, and in assessment of departments, universities and institutes. Indeed, such is the extent of the belief in the validity of these numbers, that some argue that attempts to manipulate them constitute scientific fraud. Though certainly poor academic practice, spurious citation is hardly in the same league as fabricating experimental results or plagiarism.

Science is as much a social process as it is an endeavour to increase human knowledge. Like numbers in computers, numbers in social systems need to be understood in context, not only the context of their measurement, which constrains what we can meaningfully do with them mathematically, but also the wider social and political context, which constrains how we interpret and use them. In fact, sometimes the context tells a far more interesting story than the number, which is why some comparisons are better and more rationally made using qualitative judgements rather than on the basis of contested metrics that lend a false sense of precision. To someone who loves numbers, seeing them misused is a horror in its own right, quite apart from any injustice that misuse may create.

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