We argue that the logico-mathematical reasoning as an ideal to be aimed at by all forms of reasoning, guaranteeing absolute certainty, only holds in the "protected" environment of mathematics. More importantly, this ideal of reducibility of all forms of reasoning to logico-mathematical proofs is not reached in mathematics, and even if this ideal of reducibility would be possible, we consider it not desirable. The "ecological" approach we are proposing here runs counter to this ideal of reducibility. It is a metaphor with some interesting properties, as we will argue at the end of this presentation.
Towards an ecology of proofs and arguments
COMIJN, Hans; FRANCOIS, Karen; VAN BENDEGEM, Jean Paul
In this presentation, we want to stress that philosophers, logicians and mathematicians have, starting with Aristotle, via Descartes, up to Kant and so many others wrongly emphasized the logico-mathematical reasoning as an ideal to be aimed at by all forms of reasoning, guaranteeing absolute certainty. We claim this only holds in the "protected" environment of mathematics. However, this ideal of reducibility of all forms of reasoning to logico-mathematical proofs is not reached in mathematics. Moreover, even if this ideal of reducibility would be possible, we consider it not desirable. The "ecological" approach we are proposing here runs counter to this ideal of reducibility. It is a metaphor with some interesting properties, as we will argue at the end of this presentation.
Ideal of reducibility of all reasonings to logico-mathematical proofs
The reducibility of all reasonings to logico-mathematical proofs is still a common view among philosophers, logicians and mathematicians. Moreover this reducibility is firmly believed to be the ideal to be strived for.
To see how firmly rooted this belief is, one merely needs to consult some mathematics manuals. Most of those manuals still propose the formal-axiomatic method as the one and only acceptable method and try to convince the reader that this method can always be applied, at least in principle. If not yet convinced, one can follow some of the professional interactions mathematicians and logicians have among each other. One glance at, e.g., the discussion list FOM (Foundations Of Mathematics) http://www.cs.nyu.edu/mailman/listinfo/fom/ will convince one completely that the ideal of reducibility is very firmly rooted among mathematicians, logicians and philosophers.
It is important at this point to indicate what an ideal can do. As long as something can last as an ideal, it can function as a point of reference for the desirability to reach this ideal - no matter how unreachable the ideal may be in practice. In other words, if an ideal (or ideals) exists, and/or persists, it produces a way of looking at the world: one always sees how something should be ’? travers’ what is given. Without this ideal, one is confronted with, or one sees things as they are "in front of us", that is, one has a better chance of paying attention to the things, as they are, in reality. At least that is what we hope.
We think that the ideal of reducibility, the belief that all forms of reasoning are in principle reducible to logico-mathematical proofs and should ideally be reduced to them, shortcuts the possibility of a fruitful public proof. In the next section, we will locate some of the historical roots of this ideal. After that, we will argue that the ideal of reducibility is not reached within mathematics. Finally, we will counter this ideal with an ecological approach to reasoning.
Historical roots - Descartes
We owe at least some of the cultivation of the ideal of reducibility 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 (Descartes takes this faculty by which one is capable of grasping truth in an immediate way for granted) 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. We introduce here the first three rules to explain the concept of logical truth and how this idea cultivated the ideal of reducibility of all reasonings to logico-mathematical proof.
In his first rule, Descartes gives a description of his program:
Rule 1 The purpose of intellectual inquiry should be to reach solid and true judgements about everything that occurs. (our translation)
In the second and third rule, Descartes gives us the epistemological constraints to obtain certain knowledge. It is in rule 2 that the core of the logical truth comes into the picture and in rule 3 the place and status of mathematics within the sciences.
Rule 2 We should attend only to those objects of which our minds seem capable of having certain and indubitable cognition. (our translation)
Descartes reduces the question of what is interesting to know to the question of what is logical truth. Only the things of which we can have certain and indubitable cognition are interesting to know, according to Descartes. In Descartes’ politics of representation, mathematics has a crucial role, because it seems to him the only method that can achieve certain knowledge by the method of deduction.
Rule 3 Concerning things proposed, one ought to seek not what others have thought, nor what we conjecture, but what we can clearly and evidently intuit or deduce with certainty; for in no other way is knowledge acquired. (our translation)
This 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. The idea of Cartesian logical truth cultivated the ideal of reducibility.
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.)
Counterargument - the ideal of reducibility is actually not reached even in mathematics
Just how strong the Cartesian truth procedure has been in Western knowledge production, has become clear in, among other areas, the studies of mathematical practises. Even today, if one asks the question: "What is it that mathematicians do all day long?", the answer will be: "Looking for proofs." What a proof is then, is clear to all (a connected series of statements, at least one being the statement to be proved and every step in the proof to be justified either because it is an axiom or the result of the application of one of the logical rules).
In addition, the idea that we have a clear idea of what proofs are and what they are about is supported by the fact that it is possible to have formal versions of the notion of proof. Let Proof(X,A) stand for the relation that expresses that X is a mathematical proof of A. Then all kinds of formal statements can be written down that reflect properties that ’good’ proofs have and should have, such as:
If Proof(X,A), then A
Or If Proof(X,A) and Proof(Y, A implicates B) then Proof(Z,B)
(where Z is the result of joining together X and Y)
The study of mathematical practices has long been neglected, not in the least because the ideal of reducibility persisted (compare with religious matters where the ideal of mind over matter denies completely the aspect of the body, while where this ideal is absent, these bodily aspects will be considered as aspects "just" like any other aspect).
In recent years however, the study of mathematical practices has taught us a lot. For one thing, it is clear now that most, if not nearly all interesting mathematical proofs do not satisfy the formal standards. It is sufficient to take any textbook on any mathematical topic and turn to the middle of the book; it is then immediately obvious that the ’proofs’ presented there are not proofs in the formal sense.
Of course, one might argue that these ’proofs’ can always be rewritten in the formal style. Apart from the fact that the practical feasibility may be seriously doubted - the formal counterpart of Andrew Wiles’s proof of Fermat’s Last Theorem would be very, very long indeed - for the purpose of this presentation it is sufficient to note that mathematicians themselves do not do it. Mathematicians do not invest part of their time in rewriting existing ’proofs’ as formal, correct ones.
Moreover, mathematicians spend quite some time on activities that seem strange from the formal proof perspective. They spend a lot of time on:
a) informal proofs - proofs that are to be distinguished from real proofs where one believes that it is possible to rewrite the proof in all formal detail. Mathematicians do spend time on proofs of which they know that they are not formally correct, yet lead to a correct result. Mathematicians do this because it gives him or her an ideal of what the result could be.
b) career induction - the idea that, if you have to prove a universal statement of the form ("n)A(n), then it is worthwhile investigating A(1), A(2), up to some finite number k. Formally speaking there could only be one case where such an approach is interesting, namely, if it turns out that one of the special cases does not hold. Thus, by searching for proofs for special cases, the mathematician gains some insight into the kind of proof elements and proof concepts that will be needed if a proof of the universal statement is ever to be found
c) mathematical ’experiments’ including visualizations and computer graphics. Such experiments cannot be considered formal proofs because the translation of a mathematical problem involving infinite domains (such as the real numbers) to the computerscreen consisting of a finite set of pixels must involve approximations, but an image can of course reveal certain aspects of a mathematical object.
d) mathematical arguments that involve probabilistic considerations
e) ’proofs’ involving the use of computers - the second part of the proof of the four-color theorem exists of a computer listing, presenting the details of a computer program that has actually checked a finite set of maps and said: "yes, I have checked them all and it’s OK" . This is not a formal proof, but obviously a mathematical argument.
f) mathematical arguments involving non-mathematical elements - for example the situation whereby a statement is believed to be false because it has implications that, although strictly mathematically speaking are correct, nevertheless are considered to be paradoxical on philosophical and/or non-mathematical grounds.
Although this list is not complete, we believe it shows that the ideal of reducibility of all reasonings to logico-mathematical proofs is not reached in mathematics. Mathematical arguments are different from logico-mathematical proofs, and such arguments are abundant. Mathematicians spend more time on mathematical arguments than on logico-mathematical proofs. Such arguments do allow mathematicians to convince themselves of the truth, falsity, provability or refutability of particular mathematical statements.
Conclusion
First, logico-mathematical proofs and not ’proofs’ or arguments are, all things considered, rather atypical in actual occurrence.
Secondly, we prefer therefore to introduce the concept of mathematical arguments. This concept retains the connection with the logico-mathematical proof, which now appears as an borderline case, one end of a continuum, but there being no longer any reason to expect certainty of mathematical statements, since an argument supports a statement but does not (necessarily) prove it. In short, it presents mathematical activity as a fallible activity, thereby reducing the philosophical importance attached to such questions as what the possible source of mathematical certainty can be. Along more modest lines, this approach at least helps to bridge the unfortunately still existing gap between formal-mathematical reasonings on the one hand and informal-argumentative reasoning on the other.
Finally in a constructive mood, we want to propose an "ecological" approach. This metaphor helps to explain why it is possible that a mathematical (or scientific, for that matter) proof is not necessarily a proof for the non-mathematician or non-scientist,, e.g., when it becomes a public proof. In addition the eco-metaphor readily "explains" the following phenomena: If a proof is made, constructed, thought up, invented, whatever, in a scientific setting or "niche" and it then moves into the public realm (e.g., in the court room), the proof itself does not really change (e.g., as a text or a report, it remains just that). The crucial feature, however, is that the proof ends up in a new environment (hence the "eco"-approach). Therefore, a proof that survives excellently in a particular environment need not survive in another. If, e.g., a particular way of statistical reasoning is not accepted in a courtroom, then that kind of proof is thereby "doomed" (even though it is mathematically valid). A proof in a new environment can meet other "species" that are (in some sense) very close to it. More concretely, proofs will "compete" with arguments, rhetorical constructions, and the like. It thereby offers a new route to integrate, rather than oppose, formal proofs, logically justified and arguments as studied in argumentation theory (Cha?m Perelman, Stephen Toulmin, Henry Johnstone, Douglas Walton, to name but a few, although some of these authors strongly defended the opposition).
Afterthought: the search for absolute truth within a democratic constitutional state has to be given up in favor of a truth-process that brings all interested and informed parties together and has them negotiating, not about truth, but about interest. Fragmented logical truths, logico-mathematical proofs can have a supportive role in this process, just as much as other truths and proofs (political, emotional, material, esthetical, ...) do.f
REFERENCES
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Comijn Hans, Fran?ois Karen and Van Bendegem Jean Paul are members of the
Center for Logics and Philosophy of Science
University of Brussels
Pleinlaan 2, B-1050 Brussel
Belgium
e-mail: hcomijn@vub.ac.be, karen.francois@vub.ac.be, jpvbende@vub.ac.be
http://www.vub.ac.be/CLWF/