A characteristic feature of modern technology is the conception of new possibilities for making artefacts of any kind: buildings, bridges, ships, aircrafts, cell phones, etc. One can also think of the formation of, for instance, production processes, management procedures, information processing, schemes for surveillance, etc. For any such construction it is important to provide a blueprint of what to make, and mathematics is essential for this. In all forms of technology, mathematics-based blueprints open new spaces for possibilities. The hectic development taking place in computing and information technology is an example of this.

Naturally, one might consider to what extent mathematics also limits the space of technological possibilities by eliminating what could not be formulated in this very particular language. Also in this sense, mathematics plays a defining role in technological development. The very formation of technological possibilities represents a main example of a mathematics-reality transition.

Often in political discussions, it becomes argued that certain economic decisions need to be taken, in order to maintain economic stability. Most often, outputs from a mathematical model have structured such arguments. Models for providing economic forecasting are applied by governments, political parties, research institutions, banks, and companies.

A model of this kind is loaded with information based on statistics. It is as well designed through a huge number of equations, which represent mathematical insight, economic priorities, political perspectives, as well as particular market interests. Altogether, the components of the model establish an economic universe, which becomes used for providing economic forecasting. Certain values become assigned to certain parameters and hypothetical implications become identified by running the model. A broad experimentation with different possible inputs brings about a reading of economic alternatives including their implications. On this basis, one might claim that some “necessary” economic initiatives have been identified. This “necessity”, however, is a constructed necessity. It is fabricated by means of the model. The model itself is not any economic reality, but it plays the role of reality. It establishes a symbolic reality as a parallel economic universe. The “necessary” actions are identified within this parallel universe, but applied in real life. In this sense we have to deal with a model-based formation of economic rationality.

A similar formation is found in management and engineering. One finds it when new forms of production become implemented, when possibilities for outsourcing are discussed, and when promotion campaigns are launched. There is a mathematical formation of rationalities in medicine as well as in warfare. Such formations, create actions of profound political significance. We have to deal with power exercised through mathematics.

If we look at any industrial produced artefact – shoes, cars, computers, plastic bags, etc. – mathematics is crucial for their design and production.

Let us just consider the industrially produced plastic bag that I use when going to buy fruits and vegetables. It is a colourful bag, weaved of plastic stripes. It is very strong and I can put into it as much as I am able to carry. It is an industrially produced bag and I try to imagine the machinery for producing such bags. I remember the section in the Deutsches Museum, in Munich, that shows the history of weaving. Many tools were used before the first spinning machine was constructed in 1764. In fact, it is surprising to see the great homogeneity of tools applied century after century in this production and also to see that the first weaving machine not only represents a discontinuity in this development, by being a machine, but also a big homogeneity by applying the usual tools. The machines, however, were rapidly developed. Machine power was added and the weaving machines took tremendous dimensions. Later, computational techniques were added and the automatisation became tremendously sophisticated. My plastic bag has been produced by machinery of this kind – machinery which represent a complex mathematics-based development.

However, not only artefacts, but also the very structures of production, become formed through mathematics. Let me refer to one such specific feature of production, namely robotting.8 Some initial steps in the configuration of workers as “robots” were presented by Frederick W. Taylor, in The Principles of Scientific Management, published in 1911. Here, he describes how the worker, Schmidt, became subjected to the principles of scientific management. Schmidt was loading pig iron onto a railway wagon and it was observed what an average worker was loading per day. Taylor and his staff had, however, analysed the working process in all its atomic parts and got to the conclusion that a new way of organising the whole process would more than triple the loading efficiency. A new working algorithm, prescribing what to do and when to do it, became presented to Schmidt. This way he became configured as a robot.

Taking a look into any industrial production today, one finds human robots and proper robots united in complex processes of production. The distinction between what is completed by human beings and what is done by machines is not easy to maintain. In fact, in terms of efficiency it becomes irrelevant to make any such distinction, as it is the combined machinery that is optimised. The whole configuration of production could only take place through an extensive use of mathematical modelling, including mathematics-based cost-benefit analyses. This way, mathematics defines the whole formation of structures of production. This also applies to the production of my colourful bag.

Mathematics-based discursive acts include a reconfiguration of authority. Claims expressed in numbers more easily turn powerful. This is an example of symbolic power exercised by politicians, governments, and institutions with access to mathematical models.

However, authority is not only formed through the power of argument, it is also established through the power of controlling. As a particular example, one can consider the technology of surveillance.9 Surveillance means observing in order to control what is taking place. However, efficient surveillance also means remembering or registering what has been observed.

Surveillance is crucial for making a population governable. One can register names of people; their actions, and in particular their criminal acts. You can register people's income, debt, fortune, all kind of important information for determining and controlling tax payment. One can register people's health conditions, and try to make medical treatment more effective. One can register opinions and political priorities and try to make political campaigns more powerful. You can register students' performances, and on this basis, try to establish efficiency with respect to the distribution of students in streams of further education; and in the end provide a formation of a more productive coming work force. One can register the workers' performances, and try to optimise production processes.

All such forms of registering are mathematics-based. They are crucial for making a population governable. It makes part of the formation of authority.

Through mathematics one can see many things. At the same time, mathematics representsthe systematics of ignoring and overlooking. As an example, one can think of mathematics-based scientific managements. As just referred to, this management imposes a mathematical perspective on the process of production. One see production as a mechanical process which one could try to optimise; and workers become configured as elements in this mechanical process.

This configuration also means that many features of being a worker become ignored. That the worker has a family, has responsibilities, and has personal interests and priorities, do not make part of the mathematic-based representation of the worker as an element in the production process. In general, mathematics does not represent human beings as being human. Instead, mathematics facilitates a mechanical perspective being on nature, economic issues, socio-political affairs, or on human beings. In all such cases, mathematics operates as a means for both seeing and overlooking.

As an example of formation of possibilities, one can think of travelling and navigation. If we consider the period of the so-called big discoveries, one constructed more or less reliable maps; and before any mapping could be done, one tried to identify a route or a direction. One could go West. But what could that mean staying in a boat in the middle of the ocean? Many tools for navigation were developed, not only the compass but also the sextant, besides a range of calculation devices.

We can go further back in time and think of the Vikings who reached Iceland, then Greenland, and finally America. They navigated, although without compass, but certainly they used a range of techniques and tools for reading the stars. Thus, the firmament of stars was a rather stable, although rotating, dome above the all too capricious ocean. We can also consider the crossing of the ocean performed by the Polynesians. They populated the thousands of islands in the Pacific and demonstrated a most remarkable capacity of navigating.

The mathematics involved in navigation provides a range of examples of ethnomathematics, and one can think of navigation as an example of ethnomathematics-based discursive acts.10 Navigation does not include any sharp distinction between mathematics and reality. Quite contrary, navigation represents a complex mix of interpreting and doing something; it represents as well a mathematics-based formation of possibilities. Similar remarks apply as well to any form of navigation today, including the most advanced systems operating in space missions, as well as in GPS used, for instance, when tourists try to find their way around.

Let us consider procedures for dividing. Making a division is a social process. It takes place in families, communities, in larger contexts. It is a process that becomes elaborated into extreme complexity in tax systems and in public welfare systems. Naturally, it is an open question what to consider as a fair division. One can think of the mathematical algorithm for division as a formal suggestion, although very simplistic, for what fairness could mean.

Let me refer to an example that was told to me by a researcher in ethnomathematics: “Members of a group of Indians were presented with the following question. Three men were out fishing together. They caught a total of 36 fishes. How many did each of them get? The Indians looked at each other and asked: What kind of fish was it?”

To mathematics teachers this answer might sound funny. But it can instead be seen as demonstrating the complexity of the notion of “fair division”. This notion is far from always grasped by mathematical division. For instance, the division could concern very small fishes that better had to be prepared all together. Or big and tasty fishes that need to be distributed according to the size of the families. The mathematical algorithm of division provides one interpretation of what a fair division could mean, but just one interpretation that might overlook a range of parameters that also need to be considered.

One can see several ethnomathematical approaches as attempts to provide rationales for making fair divisions. As an example, you can think of techniques for measuring lands, as for instance developed in sugarcane farming.11 One aim of this measuring is to estimate what could be a fair payment in total; another aim could be to make a fair division among the workers. And certainly, at any time one could raise the socio-political question about what a “fair division” could mean.

Many ethnomathematical studies have addressed the construction of artefacts.12 As an example, think of basket weaving. Through a range of studies, Paulus Gerdes has demonstrated that mathematical competences are represented in weaving practices. In his book Otthava, he analyses such practices in the Makhuwa culture, located in the Northeast part of Mozambique.13 Beyond the weaving of baskets and handbags, Gerdes investigates the weaving of hats, fish traps, containers, trays, mats, and several other artefacts as well. He tries to identify the kind of knowledge that becomes acted out through weaving practices, for instance, concerning symmetries, spirals, and cylinders. Thus, Gerdes provides a profound study of the different layers of tacit knowledge that forms such practices, and among such layers he finds mathematics.

Such research approaches, however, have been object for a heated discussion. Do we, in fact, have to deal with any application of mathematics? It has been claimed that we, instead, have to deal with a projection of mathematical ideas into the weaving process. The weaving process includes a range of procedures and competences that form a complex amount of tacit knowledge. But it is only the outsider, the onlooker, the mathematician, who identifies the mathematics of this tacit knowledge. Thus, it has been claimed that the ethnomathematics included in the weaving of baskets is an invention; it is an external reconstruction. The ethnomathematics in such practices has been falsely added.

Let us again consider my industrialised produced plastic bag. Does it include mathematics? Well, in the sense that enormous amounts of mathematics are put in operation along the development of weaving machines. Mathematics is part of the fabrication of the machines, in the organization of the production process, and in the computational automatisation of it all. In this sense, my plastic bag is rich of mathematics.

Who is aware of this mathematics? Well, I do not think of any such thing when I use the bags. This is no surprise. The operators involved in the production, then? Well, they are operating the machines. They are observing that everything works properly. They are checking the computer screens. I would, however, be surprised if the operators find that they are doing mathematics. What, then, about the engineers who have configured the machines for this particular production? They might have used mathematics explicitly. However, basically, the engineering mathematics involved in the industrial production of plastic bags does not emerge in the situation. It stays as a tacit knowledge inscribed in the machinery in operation. This observation relates to processes of demathematisation, which are accompanying the mathematisation of society.14

Naturally, one may claim that this engineering mathematics has been explicit at least at some stages in bag-weaving process, while the basket-weaving mathematics has been imposed on the situation. However, I do not find the difference between basket-weaving mathematics and bag-weaving mathematics to be that crucial. In both cases, we have to deal with a mathematics that forms processes of production.

Authority can be exercised through procedures and regulations. We can think of procedures for taxation. The identification of the amount tax to be paid by a person is far from just a descriptive statement, it is an act, it is a political act, and it represents power and authority.

More generally, authority becomes exercised through claims about how to do things. As an example one can think of the definition of procedures for production. We find examples in modern and industrialised forms of production, but one can also think of many other examples. Let me just refer to one example presented by Aldo Parra Sánchez (2012), who for a period stayed together with groups of Indians in the Andes, in Colombia. In particular, he followed the plantation of corn, which during centuries has been the principal food ingredient in this region.

The plantation and cultivation of corn is emerged in routines and rituals. For instance, corn has to be planted, not in long and parallel rows, but in a huge spiral covering the whole field. The straight lines that we are so used to see in industrialised farming are machine made, while the spiral is hand made. But there might be more into the spiral than just handcraft. This way of organising the plantation can be interpreted as an expression of an understanding of what it means to engage with nature. The spiral is an expression of tradition and culture, combined into certain ethnomathematics.

One can see the organisation of production, as initiated by scientific management, as an expression of particular metaphysics relating to exploitation. One can as well see the organisation of the cultivation of corn as an expression of a metaphysics, although a quite different one. Different as they are, both cases can be read as examples of authority formation. In the case of the corn production, we have to deal with an authority rooted in tradition, culture, and metaphysical ideas. And the ethnomathematical expression of this authority provides a pattern for how to do things. In the case of industrial production, we also find an authority – here expressed through standards and algorithms for production, steered by criteria for maximising profit.

Ethnomathematical studies have concentrated on showing what one should come to see and grasp through ethnomathematical insights and techniques, and maybe first of all: what one should be able to do.

There are not many ethnomathematical studies that concentrate on revealing what an ethnomathematical perspective on a certain phenomenon might ignore and overlook. In fact, now thinking about it, none such studies come to mind. Anyway, I see ethnomathematics, as mathematics in general, as forming not only what one sees, but also what one overlooks.

This issue, however, is in need of being explored much further. And there are many issues that one could consider. House building: Are there any possibilities for construction that tend to be ignored, due to the ethnomathematical outlook? Basket weaving: It has been carefully studied in what sense weaving techniques condense ethnomathematical insight and competence. Are there, however, any such insight that represents limitations and obstructions for the weaving? Health care: Are there, due to the ethnomathematical perspectives, possibilities that have been ignored with respect to the potentials of the natural medicine available? Sugar cane farming: It has been carefully studied how areas have been measured and salary for field work has been calculated. Are there elements of such calculations that may serve the workers' interest, or serve the employers' interests?

I find it important, in order to provide a more complete picture of techniques that are rooted in ethnomathematics, that the possibilities of ignoring and overlooking also become broadly explored.