The correlated mathematician - a man in disguise. Or how to reach closure withouth mathematical facts?

We try to answer the question what is the matter of concern when we talk about correlated human. I consider the growing traceability mostly a blessing because it makes a Tardean/Latourian sociology possible. The only concern is that a sociology of mathematics has been a lot more difficult because traces have been erased. The second matter of concern deals with the problem of how to reach closure when the majority of mathematical objects appear to be mathematical things, when mathematical facts appear to be matters of concern.

When faced the topic is correlated man, one has to wonder: ’What is the problem? Where is the matter of concern?’ One could warn for a growing traceability, but since Gabriel Tarde and Bruno Latour, I consider the growing traceability a blessing rather than curse. The growing traceability makes Tardean/Latourian sociology possible. However, such a sociology of math has been a lot more difficult because traces have been erased. The correlated mathematician, much more than the correlated scientist, has been a man in disguise, and this doesn’t come to the benefit of mathematics. The second matter of concern I can see is a different one. It deals with the problem of how to reach closure when the majority of mathematical objects appear to be mathematical things, when mathematical facts appear to be matters of concern.

Introduction

Unlike Genetically Modified Organisms, I must admit ’correlated man’ did not ring any instant alarm bells. Whereas I was concerned about GMO because of what it could do when I ate them, correlated man was no matter of concern to me.

It was only after some thinking and talking with other people that correlated man turned into a matter of concern to me. In two respects.
1. The first thing that came to mind was the growing traceablity, but after Tarde and Latour I consider this growing traceability as a blessing, rather than a curse. For those who have nothing to hide, a growing traceability is not a danger, and it allows for a Tardean/Latourian sociology of translation. The only concern I do have in this respect is that a Tardean/Latourian sociology of mathematics is made very difficult because traces in mathematics are faster an more rigorously erased in mathematics than in other sciences. I will explain this
2. The second matter of concern deals with the quest for closure when many mathematical facts or mathematical objects appear to be mathematical things. I will deal with this subject much less, because Karen will talk about this in more detail.

1. Growing traceability

The first possible concern that came to mind was the growing traceability. Because of the web, electronic payments, … many more aspects of human societies have become more traceable. Like many other people, when thinking about this growing traceability, I had the initial territorial reflex that the sharing of so much personal information is something I should worry about very much.

Tarde and traceability

True, the law-men and women have shown the perils of growing traceability, but since Bruno’s talk in Paris before the summer, and reading Social Law - Les Lois Sociales Tarde consequently, I consider a growing traceability much more as an opportunity, as a blessing than as problem, a curse. For people who have nothing to hide, growing traceability is a blessing.

I will not elaborate on this in much detail - Bruno has done so in his presentation and in his text: Tarde and the End of the Social, and his to be published Dialogue sur deux syst?mes de sociologie. Just very briefly:

Tarde (a very interesting sociologist who, after Durkheim, was degraded to a mere precursor, and is now largely forgotten) tries to analyse human societies, and for him, the best case of such an analysis is the history of science, because traceability of science is complete. For all the other aspects of human societies, the paths that leads a monad to its spread (we would say the actor and its network) may be lost or erased through custom and habits. There is one exception however, which makes it the most telling example for social theory, and that is the way scientific practice goes from one tiny brain in an isolated laboratory all the way to become the race’s common sense, is very well documented. The tracability of science is complete :

’’As to the scientific monument, probably the most grandiose of all the human monuments, there is no possible doubt. It has been built in the full light of history and we can follow its development almost from its first inception until today. (…) Everything in it finds its origin in individual action, not only the raw material, but also the overall views, the detailed floor plans as well as the master plans ; every thing, even what is now spread in all the cultivated brains and taught in primary schools, has begun in the secret of a solitary brain.’’ p.125 LS

So Tarde saw that in the case of science, we have continuous traceability. One can see how scientific things were invented, discovered, passed around, shaped and reshaped, translated, …

This continous traceability allows a sociology of translation. To follow those traces is a Tardean sociology that is completely different from a Durkheimean sociology. Such a sociology answers these questions:
- Are the representations accurate?
- Are the representatives legitimate?
- How to make multiplicity coming to one?
- How to solve the problem of composition?
- How to compose the mathematical world to include mathematicians, laws, axioms, students, …
- Where to assemble?
- How to compose the assembly?
- How to make others speak?
- How to choose the representatives?
- How to ascertain public proofs?
- What sort of rhetoric to develop?

Thanks to the web, we have traceability of many more things, and the possibility of a Tardean sociology of many more things. Now that we have the web, many more things have become traceable. Thanks to the web, we can produce traceability, which before was only possible for scientific activities. That is why we are blessed with the web.

Traceability of mathematics

It is true that in the case of science, much more than other aspects of human societies, there is a high traceability, that is one can see how scientific things were invented, discovered, passed around, shaped and reshaped, translated, … but I my guttfeeling tells me the same cannot be said about mathematics. While the correlated scientist may be a well-known person, the correlated mathematician remains mainly a man in disguise, and mathematical traces are erased much more swifly, rigorously and permanently than in other sciences.
Consequently, a Tardean/Latourian sociology of translation isn’t an easy task.

Tempel. Pure math, applied, math education. No problem with priestly knowledge. But we don’t know what is going on in there. What are the rules to be allowed, who gets allowed, who is the priest, what is said there? Not much is known about this. Mathematicians do not talk about it. Pure mathematicians do not understand each other. Any reflexive analysis of mathematics has been another form of mathematics. We don’t even know who build the temple, who were the
- Much less than in other sciences, we don’t know who built the temple
- And we don’t know what is going on inside the temple
Solomon’s Temple was the first temple in Jerusalem which functioned as a religious focal point for worship and the sacrifices known as the Korbanot (sacrifice in Judaism).
- Mathematical knowledge has been very priestly, but it is a problem that also in mathematics education knowledge is transferred in a very priestly way. Think of the way mathematics was taught to you in high school. As if it was nothing else than problem solving, calculations (no mention of different kinds of algebra, … (For example, it would be better to teach the plurality of mathematical knowledge and a sociology of mathematics in a Tardean/Latourian tradition as soon as children learn about proof (different proofs, arguments, …). Maybe not when they are learning to count, but at a certain age one has to be taught that mathematical knowledge is not unified, … Compare with Feyearabend who argues in favor of the separation between school and science.)
- The history of math, if taught at all, is taught as an idee?ngeschichte where mathematicians of flesh and blood have no important role, and the laws, axioms, … play no important role whatsoever. For example, it took until 1975 until J. Fang, founder and initial editor of Philosophia Mathematica, the only professional journal to exclusively focus on philosophy of mathematics, envisaged with it the setting up of "a meeting place, however precarious at the initial stage, for the four groups of specialists with some positive interest in interdisciplinary studies: historians, mathematicians, philosophers, and sociologists" (Fang/Takayama [1975], p.5, our emphasis). Furthermore, prominent contemporary historians of mathematics, such as Bell, Struik, Kline, Grattan-Guinness or Glas, have indeed increasingly proved sensitive to the matter.
- Other names: David Bloor, Harry Collins (Bath School) Eric Livingstone, Restivo, Hersch
- However, focus has been largely on what is going on outside the temple; the influence of math on society and of society on mathematics. Much less has been said on the society of mathematicians themselves, and the closer to the holies of holy, the less is known. One would expect more from a sociology of math than the illumination of mathematical elements in culture, i.e., the influence exerted by mathematics on society and vice versa, as laudable and important an effort as this may be. That is, (s)he would also be interested in mathematics as culture, i.e., in the discipline’s internal (but possibly externally influenced) social dynamics of knowledge creation: who are the priests, who are the high priests, who gets in, who doesn’t, which ideas, axioms, … are tossed around, who is sacrificed (or used as korbanot). So now, we know smth about the temple, how it was constructed, but almost nothing about what is going on inside the temple. A focus on which would be due to help him in properly evaluating the epistemological significance of mathematics as human practice.

Closure withouth mathematical facts

How to reach closure when correlations no longer appear to be mathematical facts, objects, but mathematical things.

Let me start of with a preliminary note. Following Heidegger in Die Frage nach dem Ding (1962), we’d like to define a ’thing’ as a place of encounter, a place where we are assembled around, where we discuss around. Heidegger’s definition of a thing is a very interesting and fruitful definition, because it allows us to use the word in contradiction with the commonly used word ’object’.

We owe at least some of the cultivation of the ideal of mathematization to Descartes. In Regulae Ad Directionem Ingenii (1628), Descartes gave birth to a new methodology that has survived with success until now. He introduced mathematics as the purest of sciences through which one can achieve certain knowledge. Intuition and deduction are, according to Descartes, the two core operations by which reason achieves certain knowledge. This means that all knowledge must be either intuitively obvious or be deducible from other claims which are intuitively obvious. This analytico-synthetic methodology is based on the reduction of the unknown to the known, in the same way that conclusions of mathematical proofs are deducible from the premises.
The Cartesian truth-procedure has had a decisive influence on the western knowledge production and was successfully elaborated, even in areas where absolute truths appear to be unreachable. For example, in statistics, up till this day, a distinction is being made between descriptive statistics (observation, gathering, processing of data of a vast amount of objects) and mathematical or inferential statistics (the formulation of conclusions about populations, based on samples, and with the application of probability calculus to determine the trustworthyness of the conclusions reached.), and many people (scholars and laymen alike) still think there is a direct link between the true, undubitable knowledge of correlations in descriptive statistics, over inferential statistics, to decision making procedures.

However, faced with
- the foundational crisis
- the insight that most, if not nearly all interesting mathematical proofs do not satisfy the formal standards (visualisations, use of computer, … cfr talk in Paris)
- the problems of descriptive statistics and inferential statistics (one can never outrule coincidence, nor is it possible to control all other variables), the ideal of mathematization is reduced to pieces.

The mathematical world proves to be an assembly of things (compare the explosion of the space shuttle with the foundational crisis), not an assembly of objects. Just as nature, the mathematical world is not an outside reality, but an inside collective; it is not unified but multiple; not indisputable but as disputed; not as inanimate but as giving shape. We go from matters of fact to matters of concern, also in the mathematical world.
It is clear that with such a mathematical world there can be no direct, clear link between correlations, via inferential statistics to decision making procedures

The problem at hand then is that we lose the modernist foundations of mathematics to back up our decisions, more, we lose the firm ground that sets us free from having to make decisions. And we still have to make decisions. How can we include correlations in our decision making procedures. That seems to me the problem.