theory and problems
∗José Seoane
Instituto de Filosofía
Facultad de Humanidades y Ciencias de la Educación Universidad de la República
Montevideo-Uruguay
ABSTRACT:
There is a contrast between concepts which may be treated in accordance with the criteria of mathematical rigor and concepts which are not susceptible of such a treatment. We will call theoretical concepts to the former and pre-theoretical to the latter. In mathematical world, sentences which relate theoretical and pre-theoretical concepts (in a determinated way) are denominated thesis; sentences which relate only theoretical concepts (in a determinated way) are denominated theorems. Elucidatory processes in mathematics have thesis as their principal output. I intend to establish in this paper that the introduction of the concept of mathematical elucidation has an important theoretical value to the effects of studying a certain special type of intended conceptual relation and a certain type of justificatory argumentation for it. The analysis of the contrast between thesis and theorems will allow us to construct a context where the interest for those conceptual relations and their supporting justificatory mechanisms arises naturally. Then, I will attempt to offer some structural features of mathematical elucidation qua conceptual relation and their impact on the strategies of elucidatory justification. This is what I grandiloquently call theory in the heading. I will suggest also a classification of the problems which a theoretical reflexion as the one proposed may contribute to clarify and I will make some brief observations on some paradigmatic examples of each one of the categories of the
classification constructed. This work should be considered as a modest definition of a sort of research program.
∗ This essay is a synthesis and reformulation of two conferences presented by the author. The first one in the VII South Cone Philosophy of Formal Sciences Symposium in the Federal University of Santa María, Brazil, in November 2003. I was greatly benefited on that occasion by the valuable comments of O.
Chateaubriand, J. da Silva, A. Lassalle Casanave, J. Legris, W. Sanz, F. Sautter y T. Simpson. The second conference was presented that same year on occasion of the I Forum of the GFLM, Facultad de Humanidades, Universidad de la República, Uruguay. In that opportunity, the students Ignacio Cervieri, Claudia Márquez, Luciano Silva and Ignacio Vilaró commented the exposition. I am very grateful by their worthy comments; undoubtedly, these have helped me improve the final version of this paper. The errors that subsist, of course, are exclusively my responsibility.
1.
Introduction
The concept of philosophical elucidation or conceptual analysis has a venerable tradition. Among the relatively recent authors who have considered this problem we can mention Carnap and Quine. Among the earlier authors who have pondered this question, Kant and Husserl are usually mentioned. The concept I would like to discuss is a close relative of that renowned ancestor. However, it is not my intention to justify its significance by means of lineage1. I intend to establish that the introduction of the concept of mathematical elucidation has great theoretical value to the effects of studying a certain special type of intended conceptual relation and a certain type of justificatory argumentation for it.
There are different ways of introducing our notion; an elegant and precise way of doing this is to appeal to the traditional opposition between theses and theorems. The analysis of the contrast between these two types of accepted mathematical statements will allow us to construct a context where the interest for those conceptual relations and their supporting justificatory mechanisms arises naturally. Then, I will attempt to offer some structural features of mathematical explication qua conceptual relation and their impact on the strategies of elucidatory justification. This is what I grandiloquently call (as I cannot find a better word) theory in the heading. I will suggest also a classification of the problems which such theoretical reflexion as the one proposed may contribute to clarify and I will make some brief observations on some paradigmatic examples of each one of the categories of the classification constructed. This work should be largely considered as a modest definition of a sort of research program.
1 This does not mean to renounce to exploit, according to my needs and capability, the referred philosophical tradition.
2. Theses and theorems: the traditional viewpoint
A reasonable starting point in order to present the traditional opposition between theses and theorems is to start by distinguishing between concepts which may be treated in accordance with the criteria of mathematical rigor and concepts which are not susceptible of such a treatment. Only with the purpose of a clear presentation, we will call theoretical concepts to the former and pre-theoretical to the latter. In fact, the traditional distinction was introduced in more restrictive terms than these, namely formal concepts and pre-formal concepts, where the former were understood as formalized notions and the latter as non-formalized notions. This last classification certainly provides a solid support to the traditional distinction thesis-theorem. However, I believe it is possible to exploit a less rigid conceptual classification, that is, the already mentioned theoretical / pre-theoretical concepts, in order to establish the core of the traditional viewpoint. Then, which is this core?
Let the concepts a,b∈FC (formal concepts), R be a relation between concepts, if the justification of an assertion “aRb” is accomplished by a certain appropriate justification mechanism JM1, such an assertion is a theorem. Graphically
Diagram 2.1
a,b∈FC aRb JM1
theorem
Let the concepts c∈PFC (pre-formal concepts), d∈FC, S be a relation between concepts, if the justification of an assertion “cSd” is accomplished by an appropriate justification mechanism JM2, such an assertion is a thesis. Graphically
Diagram 2.2
c∈PFC, d∈FC cSd JM2
thesis
Following this point of view, JM1 should be a proof while JM2 cannot be a proof. This means
Diagram 2.4
a,b∈FC c∈CPF, d∈FC aRb cSd
JM1 JM2
(proof) (not proof) theorem thesis
This dominant viewpoint is present (perhaps without much articulation) in several authors. For example, Church writes (italics J.S.)2:
2 See Church, A. [1936] – cited in Mendelson, E. [1990].
This definition is thought to be justified by the considerations which follow, so far as positive justification can ever be obtained for the selection of a formal definition to correspond to an intuitive notion.
Firstly, this brief text presents the leading roles of the elucidatory relation: the intuitive, pre-formal notion (in our terminology: pre-theoretical) and the formal, rigorous notion (in our terminology: theoretical). Secondly, it expresses some caution or limitation in regard to the possibilities of justification of an assertion with such protagonists and it identifies the origin of such a limitation as the different nature of the concepts that are at work.
The following passage belonging to Kleene – in reference to Church’s thesis (CT) – is more eloquent (italics J.S.)3:
Since our original notion of effective calculability of a function is a somewhat vague intuitive one, [CT] cannot be proved... While cannot prove [CT] , since its role is to delimit precisely a hitherto vaguely conceived totality, we require evidence cannot conflict with the intuitive notion which it is supposed to complete…
What we pointed us a limitation in the possibility of justification appears explicitly here: Church’s thesis cannot be proved. Again, the reason of such limitation seems to lead us to the contrast formal/pre-formal.
In brief, from the standard viewpoint, the contrast between theorems and theses is justified by the contrast of the different justification mechanisms which support them and the necessary character of such a diversity is founded on the different categorial nature of the concepts related in both cases.
3 See Kleene, S. [1952] – cited in Mendselson, E. [1990].
3. Mathematical Elucidations
The nature, structure and function of JM1 (proofs) has deserved a relatively profuse discussion4; one of the main goals of these pages is to promote the interest for the nature, structure and function of JM2. This mechanism falls under the scope of a more complex concept which I call mathematical elucidation. Such a name, obviously, intends to delimit the general problem of elucidation, limiting the question to a specific type of elucidatory relation characterized by the presence of a pre-theoretical explicandum and a theoretical explicatum – in the sense suggested above5. Therefore we could say that this name intends to limit the degree of generality of the study object – broadly we could say we are restricted to consider explicata belonging to the mathematical field. However, as it has been suggested in section 1, it is not all: our concept can capture a specific type of intended conceptual relation together with its corresponding type of justification.
Certainly no one questions the significance of conceptually clarifying JM1 but someone may question the interest of doing the same for JM2. To put these doubts into questions: which is the interest of the theses and, consequently, of their justificatory mechanisms? Does the phenomenon only happens in a special mathematical area?
E. Mendelson6, in an inspiring article on Church’s thesis, presents an open list of
“possible” theses. I suspect that it is sufficient to mention the following to briefly justify the relatively ubiquity of the phenomenon:
4 A proof of this are, for instance, Detlefsen‘s classic collection of philosophical studies – See Detlefsen, M. [1992a] and Detlefsen, M. [1992b].
5 As I anticipated in section 2, I will use the pre-theoretical/theoretical contrast instead of the traditional formal/pre-formal. The reader may suspect the reasons but the explicit justification of such a preference comes into sight in the next section.
6 See Mendelson, E. [1990].
Peano’s thesis
function function (pre-formal) mathematical elucidation (set theory)
Weierstrass’ thesis
limit limit
(pre-formal) mathematical elucidation (δ-ε definition)
Church’s thesis
computable function partial recursive function (pre-formal) mathematical elucidation (or equivalents)
Tarski’s thesis
logical consequence logical consequence (pre-formal) mathematical explication (model-theoretical)
This list can be extended considerably. For instance, it is reasonable to understand as theses several statements which relate pre-theoretical concepts as “set”, “proof”, “rule of inference”, “natural number” with their acceptable mathematical counterparts7. We frequently find in philosophical literature discussions which sometimes assume explicitly,
7 Other examples can be found in Epstein, R. L. and Carnielli, W. A. [1989].
and often implicitly, such statements. Hence, I believe that the presence of the phenomenon in different areas of mathematics is justified; I will now turn to its interest.
A specially eloquent aspect regarding the interest of the understanding of the justificatory mechanisms of theses is the interaction between theses and theorems.
An example will illustrate this point: the interaction between the completeness theorem of first-order logic and Tarski’s thesis.
In his famous 1930 article, Gödel presents the question which (as he understands) should be answered, that is, the mathematical problem which should be solved (italics J.S.)8:
Whitehead and Russell, as is well known, constructed logic and mathematics by initially taking certain evident propositions as axioms and deriving the theorems of logic and mathematics from these by means of some precisely formulated principles of inference in a purely formal way (that is, without making further use of meaning of the symbols). Of course, when such a procedure is followed the question at once arise whether the initially postulated system of axioms and principles of inference is complete, that is, whether it actually suffices for the derivation of every true logico-mathematical proposition, or whether, perhaps, it is conceivable that there are true propositions (which may even be provable by means of other principles) that cannot be derivable in the system under consideration.
The theorem gives an affirmative answer to such question. Gödel expresses it as follows9
For the formulas of the propositional calculus the question has been settled affirmatively; that is, it has been shown that every true formula of the propositional calculus does indeed follow from the axioms given in Principia mathematica. The same will be done here for a wider realm of formulas, namely, those of the “restricted functional calculus”.
8 See Gödel, K. [1930], page 583.
9 See Gödel, K. [1930], page 583.
It is not my intention to provide a historically exact reading of the original presentation. My only interest is to comment that if the concept of “truth” is interpreted in the paragraph above following a pre-theoretical sense, the theorem’s standard reading is obtained. Furthermore, it is comprehensible why it was called “completeness” and not merely “coincidence” theorem. In other words, the theorem seems to ensure that every consequence (in the intuitive, pre-theoretical sense) of the axioms is actually provable in the system and not merely that two technical notions are co-extensional –i.e model- theoretical and proof-theoretical notions of logical consequence.
Under such interpretation it may be said that the “conceptual task” of the theorem is sustained in the double “action” of specific cases of what here are called thesis and theorem. Thus, it could be summarized – where “|=” and “|−” denote, respectively, the technical model-theoretical and the proof-theoretical notions of logical consequence:
theorem
|=ϕ → |−ϕ
thesis
the |= concept ensures an adequate mathematical elucidation of the concept of logical truth.
The “completeness” result seems to be founded, to use a metaphor, in the sum:
theorem + thesis. And, then, the justification of the result (understanding it in traditional conceptual terms) depends on both justificatory mechanisms; using our terminology:
proof + mathematical elucidation10. This makes evident a specially important interaction between theses and theorems, and therefore, granted the relative ubiquity of the theses, it shows the outstanding role of mathematical elucidation in mathematical
10 For reasons of expositive convenience I will use the expression “mathematical explication”, when the context does not allow confusion, either to refer to the justification mechanisms or to the conceptual relation in question.
practice. I believe these observations may justify the interest of studying the referred phenomenon.
4. Some theory: an autonomous analysis of mathematical elucidations
Let us return to the contrast between the mechanism which allows the justification of the theses (elucidatory justification) and the mechanism which allows the justification of theorems (proof). When we look at diagram 2.4 two attractive possibilities arise immediately: either attempt to emphasize such contrast between one and the other, or else attempt to diminish or lessen it. As a result, the traditional viewpoint may be understood as subscribing to the first possibility; the approach defended by Mendelson can be interpreted as an example of the second option11. The basic support of this author’s standpoint lies in the difficulties of defending the equivalence between mathematically acceptable concepts and formalized concepts, and then, the implausibility of identifying mathematical proof with formalized proof.
I will suggest a third alternative to the conceptualization of the problem, which is not, in a manner of speaking, equidistant from both radical positions; in some way it can be considered to recapture (as we have said before) the core of the traditional viewpoint.
The contrast between theses and theorems is founded, from the traditional viewpoint, in the formal/pre-formal opposition. The problem lies in the fact that this opposition cannot be easily defended in its initial crude, dramatic formulation. However, it is possible to offer – as I said – a less rigid but at the same time equally useful characterization for the purpose of supporting the fundamental contrast. This distinction is what allows to distinguish, in this specific field, between theoretical concepts (that is, defined according to the criteria of mathematical rigor) and pre-theoretical concepts (whose characterization does not respect those criteria).
11 See Mendelson, E. [1990].
The theoretical/pre-theoretical distinction is not identifiable with the opposition formal/pre-formal. Let us consider briefly the main aspects in which the distinction proposed diverges from the traditional one.
Firstly, the existence of rigorously defined mathematical notions may be admitted even if they are not formalized. Then, a concept or a formal theory certainly are theoretical but the converse is not valid. Secondly, the proposed distinction is sensitive to the historical context. That is, we must consider theoretical concepts as definable concepts according to mathematical rigor and pre-theoretical concepts as not definable as such in a given historical context.
Thirdly, it could be admitted that the borderline between both sub-classes is somehow diffuse: perhaps, given certain concepts we are not always in condition of classifying them under one category or the other. Expressly, we could think about a difference of degree, with paradigmatically clear cases but with arguable points. The cases in which we are interested are, precisely, those where that difference is notable and, then, talking about “pre-theoretical” and “theoretical” has a clear meaning.
This approach to the thesis/theorem contrast which relativizes the formal/pre- formal opposition certainly represents a change from the traditional viewpoint. Yet, the most original change I suggest does not refer to this “infrastructural” level of the traditional viewpoint, but to the rejection of an assumption which (in my opinion) is shared by both (traditional and anti-traditional) radical positions: the conceptual heteronomy of the analysis of JM2 in relation to the analysis of JM1. If we recall the diagram I used to describe the traditional position (perfectly endorsed by Church’s and Kleene’s citations) JM2 is characterized, thus, in terms of complement: it is a no-proof.
Nevertheless, Mendelson is not interested in the peculiarities of JM2 either (even if his argumentation deserves a more careful analysis12), inasmuch as he attempts to paraphrase the elicidatory justifications as proofs, he tries to dilute those peculiarities and transform thus the elucidatory justification into a common mathematical proof. In a few words: I
12A very first approach to it appears in Seoane, J. [1993].
suggest a conceptually autonomous analysis of JM2:, that is a conceptually autonomous analysis of what I have called “mathematical elucidation” as justification mechanism.
This analysis will exhibit, as an outstanding virtue, intellectual sensitivity to the characteristics of specific argumental strategies destined to justify theses. Such strategies have, in turn, a strong connection with the conceptual relation working in them.
5. Some theory: mathematical elucidation as relation
How should the conceptual relation referred to above be understood? It is important to note there is not just one way of understanding it. I will sketch one way which could be called – following Coffa13 – tarskian. However, in several aspects, the present proposal differentiates from Coffa´s. I will start by describing it in quite traditional philosophical terms14 :
Pre-theoretical concept 1adequacy conditions 2 theoretical concepts
The pre-theoretical concept works as explicandum15. Sometimes the identification of this concept requires a specification effort; in Logical Foundations of Probability, Carnap talks about “clarification”, placing that stage outside the explicatory scheme. Among the adequacy conditions, those destined to guarantee the intensional and extensional
“fidelity” emerge from the analysis of the explicandum; although these are not the only type of adequacy conditions, it is also possible to include among them the same other clauses destined to capture different requirements (arising from, for instance, the theoretical context). The theoretical concept (explicatum) has to satisfy those conditions:
its “theoretical” character (in the sense this term is used here) guarantees some kind of
13 See Coffa, A. [1985].
14 This description, as other aspects of these developments, is inspired in the legendary debate Coffa- Simpson on the nature of philosophical explication. It was this debate what called my attention on the problem, so I am substantially indebted to that discussion.
15 Perhaps, instead of talking about explicandum and explicatum it would be more convenient to talk, as T.
Simpson suggests, about de elucidandum and elucidatum.
epistemic superiority regarding the explicandum. The satisfaction of the adequacy conditions by the explicatum ensures its material adequacy with regard to the explicandum. Note that this analysis of the elucidatory relation introduce two types of
“connection” or “relational array” among the stages involved – this is the reason for the use of sub-indexes in the diagram.
An elucidatory process, in general, has a more complex structure, but it can be reduced to the articulation of “atoms” or “moments” which can be characterized, basically, by the diagram above. Thus, although the elucidatory processes can be understood as a relation between concepts i.e. between explicandum and explicatum, they are better understood as a series or sequence em1 ... emn of elucidatory moments (with n≥ 2) such that em1 is the explicandum and emn is the explicatum. Each emi (with 1<i≤n) is considered as an elucidation of emk or a reformulation of emk – where k=1 or k is the sub-index of the last element of a finite sequence of reformulations whose first element is em116
.
An elucidation is structured as we have described above; as it may be obvious, we have attempted to capture the eventually complex character (i.e. conformed by several
“stages” or “states”) of the rigorization efforts of an intuitive concept by this series of elucidatory moments. If mi and mi+1 (with 1<i≤n) are both explication (in this specific sense) the difference may be due to the superiority of mi+1 with respect to mi regarding its behavior in connection to the sub-1 relations and/or the sub-2 relations.
A reformulation, on the other hand, consists of a revision or suppression of certain intuitive features (founded by argumentation) of a previous concept – either the original explicandum, or its revised version. The idea of including this type of concepts in the category of elucidatory moments corresponds to the interest to capture the
“elucidatory dynamics”, that is, the interaction between rigorization and intuitive ideas.
Eventually it could be necessary to consider the case of having more than one reformulation of the original explicandum - that justifies the option of a variable sub-
16 Naturally, this definition can be made rigorous but such an effort is not necessary here.
index in the characterization above. Perhaps it will be worth stressing the fact that such reformulations are specially important transformations of the pre-formal concept – consider the case, for instance, in which some of its outstanding properties are abandoned17.
This basic model can be used for analytic or historical purposes. In the first case, it is necessary to justify rationally (in terms of precision, systematization, theoretical interconnection gain, etc.) the passage from emi to emi+1. In the second case, in addition to this, a historical justification of that passage should be supplied – i.e. the historical documentation which proves the existence and order of the sequence should be supplied.
The measure in which an analytic process and a historical process may coincide can be called its factuality degree18.
We can distinguish different elucidatory modes or modalities according to the different interest or emphasis given to sub-1 or sub-2 relations19. The interest in both types of relations, and specially, in the sub-1 relations characterizes the tarskian model.
This model provides certainly a privileged role to the pre-formal concept.
These general observations allow to obtain an idea of how to understand the conceptual relation which falls under the concept of mathematical elucidation. Now I will briefly discuss the justification strategies induced by this relational conception.
17 Perhaps a measure of such significance can be seen in its incidence in the adequacy conditions.
18 This notion appears in Kotsier, T. [1991], page 13.
19 For the distinction between elucidatory modes, see the classic Coffa, A. [1975]. See also Seoane, J.
[2004].
6. Some theory: mathematical elucidation as justification mechanism
If the conceptual relation in question is considered to be expressible by a thesis (as it has been suggested here), the question about these justification strategies can be reformulated as: how are the theses justified?
When we consider the theses from an extensional perspective, the question which arises immediately is which relation they impose to the extensions of each concept. It may be reasonable to require co-extensionality. Carnap claims – regarding the relation he called “explication”, which is notably a more general relation than the “mathematical elucidation” proposed here – that such a requirement is not acceptable20. If I understand Carnap correctly, the reason for not accepting this requirement derives exactly from the contrast in terms of nature between the concepts involved. Given that the pre-theoretical concept is ambiguous, there will be points for which it will be impossible to tell if they fall under the concept, but with regard to them, given the character of the explicatum, it must deliver a verdict, that is, they fall or they do not fall under its extension. If I may introduce a metaphor, in virtue of its epistemological probity, the theoretical concept cannot reproduce the indulgence of the pre-theoretical concept. Where the pre-theoretical concept hesitates, the theoretical concept must answer with certainty 21.
If we insist on considering the explicandum and explicatum relationship as extensional, it is necessary to consider other possibilities. Carnap, for instance, refers to Naess’ conception which seems to require that the extension of the explicatum is strictly included in the extension of the explicandum.
20 See Carnap, R. [1950], chapter I.
21 It is interesting to see that even in the case where the theoretical concept is not considered to be absolutely precise (a notion even difficult to characterize) the above argument could be correct, since, regarding the extensional behavior explicated, the formal concept seems to have to behave univocally.
A type of extensional requirement which in my opinion is compatible with the relational dimension presented above can be described as follows. The theoretical concept must capture every instance of the pre-theoretical concept legitimated by the adequacy conditions. In turn, the theoretical concept cannot have instances which do not satisfy the adequacy conditions, i.e. not legitimated by the adequacy conditions. These can be thought as requirements of the conceptual relation captured by the notion of mathematical elucidation. But I would like to call the attention on the elucidatory justification strategies induced by these requirements. This illustrates (eloquently) the strong connection between the two dimensions of the concept of mathematical elucidation.
In the relevant cases the extensions of the relational concepts can be understood as infinite classes, and therefore, the justification of the existence of extensional relations between explicatum and explicandum as the ones described above, requires an argumentation which allows to ensure analytically, in a manner of speaking, the explicatum’s required good extensional behavior. In these cases, the intensional relation between explicatum and explicandum would work as a sort of guarantee of extensional correction. This relation is essentially mediated by the adequacy conditions. From the point of view of the justification mechanisms, this determines a decisive attention to sub- 1 relations: the (material) adequacy conditions must follow analytically from the pre- formal concept. Yet, although that attention is decisive, it is not exclusive: in addition we should prove that the adequacy conditions are analytically satisfied by the explicatum and this means we should turn our attention to the sub-2 relations. The adverb “analytically”
is used here in a broad sense; from the methodological viewpoint, crucial in this context, it pretends to cover very different forms of justification, from philosophical conceptual analysis (a resource typically used to prove the sub-1 relation) to mathematical proof (an available resource, sometimes, to prove sub-2 relation) 22. The most important aspect to note is that the characterization of the explicatory relation provides, thus, a relatively
22 I prefer talking in singular in this case as sub-1 could be understood as the binary relation “follows analytically from” and sub-2 as “are satisfied by”. The advantage of the plural is that it can contemplate situations that are more general: the description of the adequacy conditions given previously (as capable of capturing certain requirements of the theoretical context) evidences the value of understanding sub-1 as an array of relations rather than a unique relation.
strong, general and plastic justificatory structure. Furthermore, it could be said that this is only one methodological “face”: the “positive” face. That is, the impact of this structure on, strictly, the legitimation of a thesis. However, the other methodological “face” is not less important: it is the “negative” or “critical” face. That is, the impact of this structure on the criticism or the attack on a thesis. Two fundamental critical strategies, induced by the intended extensional relation, are the undergeneration criticisms and the overgeneration criticisms. The fist ones consist of the identification of legitimate instances – i.e. which pass the test imposed by the adequacy conditions – of the pre- theoretical concept which do not fall in the extension of the theoretical concept23. The second ones consist of the identification of instances of the theoretical concept which are not legitimate – i.e. are not adequate on the basis of the control imposed by the adequacy conditions24.
Note that despite the provisional and open character of our theory we have managed to give “sufficient structure” to the concept of mathematical elucidation not only in the relational level, but also in the field of specific justification mechanisms.
7. Problem Categories
I have partially described how, from a certain viewpoint, the structure of mathematical elucidations is conceived. I will comment briefly some problem categories as study object motivated by the attention to this phenomenon and the developments presented.
23 I have tried to show that this is precisely the strategy developed by Tarski in his criticism of the syntactical notion of logical consequence (qua explicatum of the respective pre-theoretical concept) in his famous 1936 article. The “point” which legitimately falls under the pre-theoretical concept and which is not captured by the theoretical concept is, in this case, an argument in which the ω-incompleteness becomes manifest. See Tarski, A. [1936], pages 410-413. The problem is discussed in detail in Seoane, J. [2003].
24 I have tried to show that this is precisely the strategy developed by Tarski in his criticism of the substitutional notion of logical consequence. See Tarski, A. [1936], pages 415-416. The problem is discussed in detail in Seoane, J. [2003].
Firstly, a series of methodological questions, or if preferred, of a general conceptual nature. As far as I can see, these are of utmost significance from the philosophical point of view. I will only mention one of them: the question of the contrast between, to reintroduce our first scheme, the justification mechanisms JM1 y JM2, that is, the finer comprehension of the relations between the concepts of proof and mathematical elucidation as justification mechanism. This study may be oriented towards the detection of characteristic or specific structural features or towards the comprehension of the respective functional peculiarities. To put it more clearly, we can consider the specific characteristics of justification mechanisms or resources at work in each case or we can consider the specific function which play, in the general dynamics of the development of mathematical knowledge, proof-theorem and elucidations-theses. It is evident that a deep reflection could embrace both aspects and attempt to clarify their relations.
Secondly, the development of a certain kind of work, which can be called, field work becomes promising. Field work deals with the historical-conceptual exploration of the structure, foundation and function of particular theses in the history of several branches of mathematics. Just to give a single example, the aforementioned Peano’s thesis, given the long history of the pre-theoretical concept of function, becomes an attractive object of study from this perspective25.
Thirdly, there are important disagreements about the specific elucidatory evaluations. A traditional debate has been the one arising from Church’s thesis; it does not seem unreasonable to understand it as a case of the problematic elucidatory evaluation26. In recent years Tarski’s thesis has been object of the special attention in the literature, mainly, due to the impact of Etchemendy’s critical arguments27. The explicitation and consequent consideration of the elucidatory methodological
25 An effort oriented by this type of methodological questions and which deals with the development of Tarski’s thought on the notion of logical consequence can be found in Seoane [2002] and [2003].
26 A discussion on the historical and conceptual aspects of Church’s thesis can be found in Ertola Biraben, R. C. [1996].
27 The main account of that criticism is found in Etchemendy, J. [1990].
assumptions in the evaluation of the theses is a relevant challenge inspired by the previous developments28.
8. Final observations
It would be very helpful, perhaps, to describe (very briefly) the general framework in which these modest reflections are inscribed and to lay emphasis on a suggestion revealing the extent to which the notion proposed can be considered a general framework.
The basic problem underlying these developments is that of the articulation between the pre-theoretical and theoretical views. When the problem is isolated and idealized, the articulation between the pre-theoretical and theoretical concepts is obtained. Several types of relations between them can certainly be identified but the elucidatory relation is one of special significance. Limiting the problem to the realm of formal sciences, I have dealt with mathematical elucidation. Straightforwardly, the basic conceptual problem consists of how to understand such elucidations. As I have claimed, we should not only consider a certain special type of intended conceptual relation but also the correspondent justification mechanisms.
An aspect which deserves attention is that the purported conceptual gain of this type of inquiry may not reside exclusively in the attainment of an optimal elucidatory model, that is, a single, precise, strict definition of mathematical elucidation. This is a way of developing the current proposal, but not the only one. Since it also may be thought as a construction program of notions of mathematical elucidation, or if other term is preferred, of elucidatory models (with the restriction of the rank of the protagonists i.e. theoretical/pre-theoretical, in the sense these terms have in the present context). Specially, as not all the justificational practices of mathematical elucidations
28 I have tried to make partially explicit the explicatory structure of Etchemendy’s criticism in Seoane, J.
[2004] but a better understanding of the debate arising from a finer methodological reconstruction is something that – to my reckoning – has not been developed yet.
(historically given) may admit to be reconstructed productively in this way. Given that, from my viewpoint, relational description and the induced justification mechanisms characterize a notion of mathematical elucidation, the analysis of justificatory practices may also lead to a certain relational description. Thus, the study of historically given explicatory justifications may open a field of investigation consisting of the comprehension of the specific economy of different models (i.e. relational description and induced justification mechanisms) and their comparative study.
Nevertheless, the promotion of analytical attention to the phenomenon which I have called mathematical explication and the development of some methodological and conceptual distinctions related to it, which I have presented here, may perhaps be a fruitful collaboration to deal with problems as the illustrated above; that at least is my hope.
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