Mathematics as well as philosophy have a very long tradition which results in several problems hindering the communication between each other. Both have developed many subfields some of which require long periods of study in order to enable a person to both understand some of the main problems in both fields as well as to acquire the tools with which to approach these problems.
This is important in the sense that the respective scientific sub-community bases its values on the fluency in its respective language including the familiarity with its sophisticated objects and tools.
This leads to three problems:
Firstly if a philosopher starts to deal with mathematics, she is most often not taken seriously by mathematicians because of the lack of fluency in the mathematicians’ field or subfield. “You don’t understand what I am doing so you cannot tell me anything of relevance for my work.” We should not neglect this argument but take it seriously, independently of the quality of the philosophers work!
The same holds the other way around: if a mathematician starts to do philosophy (this holds for history as well and much less in education which will be treated later on) the general impression of philosophers is one of dilettantism which also has to be taken seriously.
This should remind us that neither philosophy nor mathematics is something easy to grasp and learn and that it takes time to be enculturated into the respective communities and hence to be taken seriously by its members – to make them listen and adhere to ones arguments.
Secondly both philosophy and mathematics are quite segregated in themselves. Both have a big variety of sub-disciplines and a simple combinatorial question arises: Can we do philosophy of mathematics in general or do we have to do analytical philosophy of the axiomatic method or phenomenology of geometry or hermeneutics of topology or any other combination of sub-disciplines in both mathematics and philosophy?
This raises the immediate question of the unity of both disciplines. What is philosophy about (or better: what should it be about) and what is mathematics about? Only if we can find those unifying items we can attempt to bring them together. Of course this makes only sense if we have solved the first problem. Or maybe it could even help us to solve the first problem!?
Thirdly these ‘communities of practice’, however scattered they might be, have developed historically a set of questions, problems, methods, theories and programs (and a system of values!). These are constitutive for the respective community and hence of enormous value – hence everything outside of these enterprises needs a big political effort in order to be introduced and considered as a valuable question, problem, method, theory or program.
The approach to this last problem, though difficult in itself, seems clearer than the resolution of the other two. This problem is also not as symmetrical as the first two ones. It deals more with the question how to enter (as a philosopher) into the mathematical community or into a certain mathematical sub-community.
There is another issue which is more intrinsic to both domains. There are difficulties in accepting each other due to the different nature of approaching problems. Mathematicians search for solutions. Philosophers very seldom provide solutions to the problems they pose. If they happen to provide solutions, they are not of the type mathematicians are used to. Another and maybe less important aspect is that mathematicians are used to formalisms and the level of formalization is also a value in itself. Very little of philosophy is formalized and it is by no means a goal there.
Mathematicians are used to use diagrams and sketches and to make further and further abstractions in order to generalize their statements more and more. Maybe an attempt to use a more diverse set of representational tools than only ordinary plain prose could be a chance to make philosophical texts more appealing to mathematicians.
This difference in the approach to problems is on one side a big problem, on the other side it leads sometimes even to animosity – at least from the side of mathematicians. Statements by prominent mathematicians are quite strong in this respect. Here are two examples:
The following remark is attributed to Karl Friedrich Gauss: “When a philosopher says something that is true then it is trivial. When he says something that is not trivial then it is false.”
Contemporary mathematician and Fields medalist Jean Bourgain used during a lecture (‘Visions in Mathematics’, Tel Aviv 1999) the phrase “this is a philosophical remark” for everything which was not directly related to the mathematical demonstration. The strange thing is that he could have avoided (could he really?) all these unnecessary commentaries but he didn’t do so. Nevertheless the impression was one of a very dismissive attitude towards those “philosophical remarks”.
The good news is that the existence of animosity (and not just ignorance) indicates that there might be more points of contact than initially supposed by both sides.
- A certain number of questions arise out of the above thoughts:
- Are there aspects in mathematics which need a philosophical treatment? Where the issue is not (only) to find a solution?
- Are mathematicians aware of these problems?
- How can the situation be changed so that these problems start to be perceived as relevant?
- How can these problems become important?
Obviously a political effort is needed. An effort has to be undertaken aiming at a change of values or of the hierarchy of values of the mathematical community (or of a mathematical sub-community). But for this task we should know the current values of the mathematical community, their hierarchy and their changes in the past.
Can philosophy also profit from this? Who is the ‘queen of sciences’? In this respect philosophy has been successfully contested by mathematics! The consequences from this are well presented in the two-cultures-debate started by C. P. Snow.
There is another important problem. Philosophy is not anymore the sole science/domain which might be useful to reflect upon mathematics. It has split – and this has various extrinsic reasons – into Philosophy, Social Sciences like Sociology or Anthropology, Psychology, Cultural Studies, History, Semiotics, etc. It is really a mess! And this mess is also reflected in what has been published as reflections upon mathematics. There are sociological studies (Bettina Heintz), psychological attempts (Hadamard, Borovik), attempts from the standpoint of semiotics (Rotman, Marcus, myself), history of mathematics develops quite well in recent years (M. Epple, L. Corry and many others), etc. All this scattering seems to be a problem of philosophy as well.
Does the publishing of more and more texts mean that we are on a good track? The pity is that mathematicians still don’t care.
In order to approach these questions I first must clarify what I mean by philosophy and what I mean by mathematics.
Philosophy has initially been an enterprise to reflect upon the world we live in. It has developed a big number of tools which provide a language but also techniques to describe, evaluate, make comparisons, etc. One big advance in philosophy during the 20th century was that issues of communication and of language have been given a much more prominent role than before (Semiotics, The New Rhetoric and Wittgenstein’s work) and that aspects of training and habituation have become an issue in epistemology (Polanyi’s work). It is interesting that both aspects are linked to time which seems to be in conflict with the claim of universal and hence time-independent insight.
The quest for universal insight is a main driving force in both domains – philosophy and mathematics. But an insight which is detached from anybody who has eyes to look and see does not make much sense. We could imagine a trivial thought experiment: does a library with all the universal insights but without any readers make sense? There will be nobody around to judge or even to approve, that there are really universal insights available in this library.
From the very moment when a person, a reader, a partner in communication enters the scene all issues of communication become indispensable.
It turns out that it would be more useful to consider all these universal insights as tools. A tool is something that becomes relevant (as a tool) from the moment somebody begins to use it. Before this moment it might be something that could be observed but nothing more.
In order to start to use the tool one needs to know how to use it. It needs some kind of instruction. This means that somebody has to give (write) an instruction and somebody else has to learn from this instruction and to practice. The latter includes of course all aspects of learning and training.
We may conclude that universal insights in the domains of philosophy and mathematics seem to be universal only for the learned and trained ones.
It is important to know that I do not want to enter a debate concerning the objectivity or subjectivity of results in both domains, especially in mathematics. This is the reason why I use the term “insight” – a notion which is clearly human-related. By the way, it would be interesting to ask the question, whether a computer or a computer program might have something like an insight and what this might mean.
The discussion of insights gives me the opportunity to introduce a tool which comes into use in some of the items discussed in this encyclopedia. The tool I mean is the use of self-referencing as a means for reflection.
In our case the issue is to gain insight about insights. It may sound strange at first but it works. This is a type of self-reference which does not lead to an infinite regress. If we have gained insight about insights we shall call this a ‘second order insight’. An insight about a second order insight is itself again a second order insight. The point is that there is a difference to the kind of self-reference which leads to infinite regress: a second order insight leaves the potentially closed system related to the first order insight. The tools used in order to gain a second order insight can be used again for gaining an insight about this second order insight. The issue is that when a second order insight is investigated it is itself treated as a first order insight.
To describe the above situation in plain words:
- Step 1: we have understood a mathematical relation.
- Step 2: Next we want to understand what it means to understand a mathematical relation – we want to understand what understanding means.
- Step 3: After having achieved this, the question of understanding what the understanding of understanding means does not bring any new understanding different from the one in step 2.
The method of self-reference shows in general, that a closed system is not possible. If self-reference leads to paradox, it means that a system has been artificially considered to be a closed one.
There is a fine and profound example from biology: A caricature shows a scientist who rushes into the room to tell his colleagues that he has discovered the gene which determines the belief that everything is determined by genes.