Roux’s Forms of Mathematization

Sophie Roux. Forms of Mathematization (14th-17th Centuries). Early Science and Medicine, Brill Academic Publishers, 2010, 15, pp.319-337

As a mathematician I am intrigued by notions of ‘mathematization’. This fascicle presents a point of view that seems to me similar to that of many social scientists.

[Mathematization] was founded on the metaphysical conviction that the world was created pondere, numero et mensura, or that the ultimate components of natural things are triangles, circles, and other geometrical objects. This metaphysical conviction had two immediate consequences: that all the phenomena of nature can be in principle submitted to mathematics and that mathematical language is transparent; it is the language of nature itself and has simply to be picked up at the surface of phenomena.

The assumption that reality is measurable or consists of geometrical objects is no longer tenable, yet the consequences seem to still hold: ‘mathematization in British. or mathematisation (ˌmæθɪmətaɪˈzeɪʃən)’ is variously described as being:

  • Reduction to mathematical form (Mirriam-Webster)
  • The act of interpreting or expressing mathematically, or the state of being considered or explained mathematically. (Collins)
  • The characterization of objects and events in terms of mathematical relationships. (Various papers on sciencedirect.)

Also:

Language as an empirical phenomenon (just like many other empirical phenomena) is described in mathematical terms to obtain a ‘model’, which is investigated using mathematical means and the results are then projected back on the phenomenon. (We can also understand this mathematization as a matter of extracting the structure of the phenomenon in the form of a mathematical object.)  [Linguistics and Philosophy Jaroslav Peregrin, in Philosophy of Linguistics, 2012]

That is, mathematization seems to presuppose that the real phenomena of interest has a structure that is capable of being represented mathematically. This contrasts with that of a blogger:

the mathematization of a problem or area of study consists of applying mathematical ideas to that problem or field so as to think more precisely or clearly about things. …. Whether any human can follow the argument and understand the subject matter is another question

The main point of difference being to what extent the representation of a problem or field assists in thinking more precisely or clearly. The fascicle opines:

Grand narratives such as this [the effectiveness of mathematization] are perhaps simply fictions doomed to ruin as soon as they are clearly expressed.

Perhaps. But really?

The fascicle makes a non-controversial observation:

In general, the term “mathematization” refers to the application of concepts, procedures and methods developed in mathematics to the objects of other disciplines or at least of other fields of knowledge. A definition of this kind seems to assume that there is an agreement, first, on what is mathematics, second, on the profits that various disciplines can make out of its application and, third, on the relevance of the very notion of application. But there are many good reasons to think that such an agreement might be difficult to achieve.

It goes on:

There was never a working definition of mathematics in general; even at the time when the traditional definition of mathematics as the “science of quantities” or “magnitudes in general” emerged and was commonly accepted, there were different conceptions of quantities, and consequently different ways of conceiving of the unity of mathematics. Now, if the second-order question of how to define mathematics was ever raised, it is because it is a fundamentally complex field, that included various domains from its very beginning and that kept developing new domains throughout history.

Wikipedia notes that the term ‘mathematics’ still has no ‘generally accepted definition’. The fascicle characterises mathematics thus:

[With] the emergence of calculus and of infinite series, new domains began to be explored. In these circumstances, [to social scientists] it seems inevitable to admit that we should neither look for a definition of mathematics in general, nor think of mathematics as a unified field of knowledge, but, rather, submit to an historically situated and empirical determination of mathematics: what should be called “mathematics” is the activities of those who called themselves or were called by others “mathematicians”. As tautological and circular as it may appear, such a determination is not without consequence on how we should conceive of mathematization.

No reason for this baleful conclusion is offered, other than its seeming inevitability, so any other view is at least as credible. (I shall offer one, later.)

Arithmetic, in as far as it is the practice of numbers, generates a first form of mathematization: what we call “quantification” consists in capturing in numerical form certain aspects of material things. Such a capture requires indeed measurements, concrete apparatus and a growing concern for precise and standardized data, but also graphical techniques to present numerical results and intellectual techniques of approximation and averaging. Of course, quantification may be only peripherically related to the disinterested search of laws in natural philosophy: the alledged benefits of quantification are sometimes practical.

Even today, mathematization is often conceived of as mere quantification, and hence mathematization is often thought to imply various beliefs (such as measurability) that are, in fact, not mathematically credible. Such concepts are easy to refute. The current fascicle needs to be taken more seriously. For example:

Practices may refer to the non-verbal commitments shared by mathematicians that help them defining a scientific style and constituting an intellectual community; practices in this sense are opposed to explicit beliefs and assumed to be invisible to the mathematicians themselves.

The fascicle regards all kinds of mathematicians as mathematicians, so that most mathematicians are part of some broader community of scientists, technologists, engineers and ‘practical people’ rather than forming some relatively closed community of mathematicians. Hence their ‘non-verbal commitments’ are not just or even mainly mathematical: they are at least heavily influenced by the culture of the community in which they work. Contrary to what the above paragraph seems to suppose, there are often (as with other professions) some very clear issues that arise due to the ‘opposition’ (or at least differences and seeming contradictions) between the commitments of mathematicians as mathematicians and their commitments to their wider communities, for example between their professional integrity and their role as employees. These issues are far from ‘invisible’: even in academia they are a frequent topic of discussion.

[Mathematical] practices can be identified with practical mathematics, as contrasted with pure mathematics and which refers to the real world, with its economic interests, practical concerns, material instrumentation, local settings and complex social networks.

Quite so.

Comments

It seems to me that the social sciences generally have some very important things to say about ‘mathematical practices’. In particular, however highly one regards mathematics ‘as such’, mathematicians who are a part of a community (e.g., a work-force) are subject to much the same influences as other professionals, and one should not simply assume that they are as open and honest about their mathematics and its implications as one might wish. If ‘mathematization’ is what mathematicians do, and if we can’t trust mathematicians then clearly we can’t trust mathematization. In this case, mathematization might even seem to be a bad thing, in so far as creates an artefact that can only be understood and used by a relative few, and largely excludes people who otherwise seem to have essential insights into real problems.

If, as a mathematician, I try to adopt the viewpoint of social scientists, then it seems ‘inevitable’ that mathematization of ‘wicked problems‘ is going to make things worse.

But could one devise or develop communities in which mathematicians could be more open and honest, and what might the implications for ‘mathematization’ be? Could it not be a ‘force for good’?

It seems to me that there is a key point to be made:

Working mathematicians often seek to mathematize a problem or an area of interest, whereas – as mathematicians – all they can properly do is to mathematize a conception of a problem or a theory about an area. It seems to me that the difference matters.

For example, the role of mathematicians in economics and finance has been criticised, not unreasonably. But mathematizing a wrong theory isn’t going to correct it. It may make it clearer, less ambiguous. It may enable more efficient methods, or even totally new instruments. But if the theory is wrong, more efficient or more effective misguided methods may simply lead one to realise the inadequacies of the theory sooner.

This leads to another key point:

  1. Mathematizing a theory considered as if it were a straightforward description of some ‘real system’ is essentially using mathematics as a tool for developing the theory. It may accidentally reveal inconsistencies in a theory, but not other mistakes.
  2. An alternative is to consider a theory together with a description of how the theory is justified and tested. Mathematizing the theory in this broader context (i.e., ‘as a theory’ not just ‘as a description’) can then reveal the extent to which the theory is justified and has been tested.

My own experience is that whenever I try to mathematize an existing theory (which I do in the second sense) I have always found it wanting, even when the theory has already been mathematized in the first sense. Practitioners often initially disagree about whether this is ‘a difference that makes a difference’, and even those who think that it could make a difference (including me) recognize that there are other issues that could prevent a properly mathematized theory making a difference. But it seems to me that in considering a whole range of areas that have been mathematized in the first sense, there could be substantial payoffs in at least some of them in mathematizing in the second sense. But such mathematization alone is not enough: all it really does is to reveal ambiguities, imprecisions and potential or actual mistakes before brute experience does: one still needs appropriate subject-matter ‘experts’ to resolve the areas for investigation that have been revealed.

My experience is that for sufficiently complex problems one never ends up with a ‘reduction to mathematical form’, and of course such a reduction is not even possible in principle for ‘wicked problems’. But one can use insights from mathematization (in the second sense) to reduce the wickedness, which often seems to me to be a ‘good thing’.

Dave Marsay

P.S.

I see that I offered to give a characterisation of mathematics. I think the core concept is ‘logical proof’. There are a few different schools of thought within mathematics as to what types of proof are appropropriate for which subjects. If we pick one, say x-proof, we can define an x-claim as one which has an x-proof. An x-method is one which x-provably can only produce x-claims. An x-logic is one which is used in x-proofs. An x-conjecture is one which for which no x-proof of falsehood is known and which is amenable to x-logic, and which we have some reason to think may be x-provable. An x-heuristic is a method which we x-conjecture often produces reasonable x-conjectures. An x-mathematics is the collection of the above x-types. I have never found the particular notion of x-proof to be that important, as long as it is genuinely ‘logical’ and not simply an appeal to some supposed authority.

P.P.S. What applies to mathematics seems to have some truth when applied to other disciplines. All disciplines seems to interpret situations in their own terms and make recommendations. To me, the the key question is always: have they cosnidered whether there discipline is adequate, or the extent to which its recommendations might usefully be complemented by those from other disciplines? How should the recommendations be caveated? More generally, a common practice is to treat different disciplines as ‘viewpoints’ and to meld their conclusions togther to reach some sort of ‘common sense’ resolution. But when is this adequate? Could logic play a special role?

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