What are formalisms used for?

 

Robert Marty, Honorary Professor, University of Perpignan (France),   robert.marty@bbox.fr

This text provides an answer to Jon Awbrey’s question about the role of axiom systems in the search for scientific truth.

Jon Awbrey writes[1]:

« We do not live in axiom systems. We do not live encased in languages, formal or natural. There is no reason to think we will ever have exact and exhaustive theories of what’s out there, and the truth, as we know, is “out there”. Peirce understood there are more truths in mathematics than are dreamt of in logic and Gödel’s realism should have put the last nail in the coffin of logicism, but some ways of thinking just never get a clue.

That brings us to the question:

What are formalisms and all their embodiments in brains and computers good for? »

1. Peirce asked himself this question...

Here are two texts among the 76 definitions of the sign[2]:

1.1 Text n°30 « So then anything (generally in a mathematical sense) is a priman (not a priman element generally) and we might define a sign as follows:

A « sign » is anything, A, which,

(1) in addition to other characters of its own,

(2) stands in a dyadic relation r[3] , to a purely active correlate, B,

(3) and is also in a triadic relation to B for a purely passive correlate, C, this triadic relation being such as to determine C to be in a dyadic relation, s[4], to B, the relation s corresponding in a recognized way to the relation r.

In the which statement the sense in which the words active and passive are used is that in a given relationship considering the various characters of all or some of the correlates with the exclusion of those only which involve all the correlates and are immediately implied in the statement of the relationship, none of those which involve only non-passive correlates will by immediately essential necessity vary with any variation of those involving only passive correlates; while no variation of characters involving only non-active elements will by immediately essential necessity involve a variation of any character involving only active elements. And it may be added that by active-passive is meant active and passive if the entire collection of correlates excluding the correlates under consideration be divided into two parts and one part and the other be alternately excluded from consideration; while purely active or passive means active or passive without being active-passive. « 

1.2 Text n°31 « This definition avoids the niceties for the sake of emphasizing the principal factors of a sign. Nevertheless, some explanations may be desirable. But first for the terminology. I use « sign » in the widest sense of the definition. It is a wonderful case of an almost popular use of a very broad word in almost the exact sense of the scientific definition. [...]

I formerly preferred the word representamen. But there was no need of this horrid long word. [...]

My notion in preferring « representamen » was that it would seem more natural to apply it to representatives in legislatures, to deputies of various kinds, etc… I admit still that it aids the comprehension of the definition to compare it carefully with such cases. But they certainly depart from the definition, in that this requires that the action of the sign as such shall not affect the object represented. A legislative representative is, on the contrary, expected in his functions to improve the condition of this constituents; and any kind of attorney, even if he has no discretion, is expected to affect the condition of his principal. The truth is I went wrong from not having a formal definition all drawn up. This sort of thing is inevitable in the early stages of a strong logical study; for if a formal definition is attempted too soon, it will only shackle thought. [...] (highlighted by me)

I thought of a representamen as taking the place of the thing; but a sign is not a substitute. Ernst Mach has also fallen into that snare. » (Semiotics and Significs, pp. 192-193 – Letter to Lady Welby (Draft) presumably July 1905).

It should therefore be noted that Peirce had refrained from formalizing before 1905 for fear of « blocking the path of research »…literally…

This formalized definition is also included in the text n°66 (MS 793, On signs, undated)

And in another form in text n°74:

« A mental representation is something which puts the mind into relation to an object. A representation generally (I am here defining my use of the term) is something which brings one thing into relation with another. The conception of third is here involved, and therefore, also, the conceptions of second or other and of first or an. A representation is in fact nothing but a something which has a third through another. We may therefore consider an object:
          1. as a something, with inward determinations;
          2. as related to another;
          3. as bringing a second into relation to a third. » (MS 810, On the formal Principles of Deductive Logic, undated)

 Obviously, Peirce’s reflections on the desirability of producing formal definitions must be linked to this truth that is « out there », at the end of an undefined path. For all this is perfectly in line with the pragmatist perspective: to establish a formal definition is effectively to stop the movement of thought and run the risk of it becoming encysted in this form if it is not sufficiently confronted with the existing and the facts it claims to inform. But are we not faced with a necessity that imposes precisely in any scientific process formal moments to ensure that the path is the right one so that the scientific community concerned – the only guarantor of the collective march towards universality – can find a consensus after debate? In other words, to answer Jon clearly, I will say that formalisms are obligatory passages, discontinuities, levels from which it is not excluded, for some of them (I think of mathematical formalisms) that they can be an ultimate level…

 2. A famous example from anthropological research

 The mathematician André Weil comments below on a mathematical work he did at the request of the ethnologist Claude Lévi-Strauss:

 « From what sociologists working « in the field » have observed, the marriage laws of Australia’s indigenous tribes contain a mixture of exogamic and endogamic rules whose description and study pose sometimes complicated combinatorial problems. Most often the sociologist gets away with listing all the possible cases within a given system. But the Murngin tribe, at the northern tip of Australia, had developed a system so ingenious that Lévi-Strauss could no longer unravel its consequences. In desperation he submitted his problem to me.

The most difficult thing for the mathematician, when it comes to applied mathematics, is often to understand what it is and to translate the data of the question into his own language. Not without difficulty, I ended up seeing that it all boils down to studying two permutations and the group they generate. Then an unforeseen circumstance appeared.

The marriage laws of the Murngin tribe, and many others, include the principle that « Any man can marry his mother’s brother’s daughter » or, of course, the equivalent of her in the tribe’s marriage classification. Miraculously, this principle means that the two permutations in question are exchangeable, so that the group they generate is abelian. A system that at first glance threatened to be an inextricable complication thus becomes fairly easy to describe once a suitable rating is introduced. I dare not say that this principle was adopted to please mathematicians, but I must admit that I still have a certain tenderness for the Murngins » (Paul Lavoie, Claude Lévi-Strauss et les mathématiques)[5]

In short, concludes Lévy-Strauss, we show that marriage laws of a certain type

  »can be subjected to algebraic computation, and how algebra and substitution group theory can facilitate its study and classification. » (Lévi-Strauss,1981[1949] Les structures élémentaires de la parenté. Paris, Mouton, p. 257)

Hence Paul Lavoie’s observation:

« The mathematical solution of an anthropological problem and, consequently, its contribution to Lévi-Strauss’ structuralist approach, highlight the applicability of new mathematics to the human sciences – mathematics that is different from the classical tools of calculation and statistics » (Paul Lavoie, Claude Lévi-Strauss and mathematics)

With this example from anthropology we have a completed case of mathematization of a formless doctrine:

« an formless doctrine is, since it is a doctrine, i. e. a given opinion to know how to communicate, a more or less coherent system of concepts, relating to a sector or field of human experience. If there is a system, even in the inactive state, it is clear that there is a form » (Canguilhem, Georges. 1972. La mathématisation des doctrines informes, p.8, Paris: Herman)

From this point on, we can give an answer, at least partial, to the question asked: formalisms serve to give form to the formless and mathematical formalisms, in particular axiomatizations, go much further, because they bring from the outside the universality that is necessarily lacking in doctrines marked by their location in restricted fields of human experience.

Returning to Peirce and his somewhat complicated definitions of the sign mentioned above and taking into account the developments of later mathematics, allow me to recall that the primitive concepts of homological algebra ( categories, functors) allowed me first to match the classes of Peirce’s signs using certain functors in a one-to-one correspondence with them, and following, by using natural transformations to obtain « naturally » universal classifications[6] for the classes of signs. Moreover, this axiomatization extends to all kinds of complications of semiotic theory, whether hexadic signs or decadic signs and more generally n-adic signs. Obviously, we can continue to import other algebraic notions into semiotics, such as the sums and products of diagrams in a algebraical category that allow us to give shape to other problems encountered in the analysis of meanings, their production and their communication.

3. Conclusion

The axioms systems are necessary passages to go « out there », and care must be taken, when submitting them to criticism, not to « throw the child out with the bathwater », because they are essential tools that make it possible to make our ideas clearer in order to think more accurately about the objects of knowledge