In this paper I intend to show how the concept of an axiomatic system and the concept of an experiment may both apply to a philosophy. Since the days of Descartes and Spinoza the notion of an axiomatic philosophy has come to look somewhat naive, and in this century it may appear to have been discredited even in Mathematics, particularly by the work of Goedel. As for an experimental philosophy, this is not be confused with empiricism, which philosophy is as ready as any other to deny the experimental nature of itself. But first let me give an outline of the concepts of an axiomatic system and of an experiment. In order to make it easier for you to see the parallels I am going to draw between them, I will pick out four points in each. An axiomatic system is the ideal of a mathematical system, whereas my characterization of an experiment may be taken as the ideal aimed at in Physics, although I believe it to be applicable to any systematic branch of knowledge, e.g. History.
1) Fundamental Concepts
An axiomatic system has a certain set of fundamental concepts. In general, concepts are endowed with meaning derived from involvement in other contexts than the one immediately under consideration. Such associations prevent a concept from being treated simply as a building block of a particular system. To be a concept in an axiomatic system it must be free from external signification. One must somehow cut the flow of information it expresses.
2) Axioms
Between these fundamental concepts there are interposed a certain set of fundamental relations, the axioms of the system. The axioms implicitly define the fundamental concepts, so that they do not have any meaning other than that supplied by the axioms. By ‘implicitly define’ I mean define in an analogous sense to the way that a set of three algebraic equations with three unknowns defines the unknowns. Since we must ignore any external associations which we make with the concepts, the axioms must be separated from any external verification, their truth simply being something given.
3) Proving theorems
Everything else in the system is then deduced from the axioms solely as a matter of definition and logical necessity. The system is closed under such processes of deduction, i.e. the system simply is the system of theorems and their proofs.
The processes of deduction may be considered as the system unfolding from its own basis (the fundamental concepts and axioms). But these processes do not involve any experience other than the unfolding itself. As far as the system is concerned, the fundamental concepts must not have other properties than those needed to satisfy the given axioms, so that it is true that the axioms implicitly define the fundamental concepts – the proofs simply make the defining explicit. The role of the axioms in the proofs is to establish the relations which, in their totality, will confer meaning on the concepts being used.
An axiomatic system must be consistent. This consistency is demonstrated by the theorems proved, by the fact that no contradictions arise: i.e. that a theorem and its negation cannot both be proved. But for a system to be fully axiomatized requires more than this negative criterion of consistency, of avoiding contradiction: the system must demonstrate its own consistency. In the unfolding of the system it must be shown that there is no statement which cannot, at least in principle, be demonstrated by the system to be true or false. Such an occurrence would mean that the original axioms did not exhaustively define the fundamental concepts. Thus Goedel’s notorious theorem was important in challenging the view that all mathematical systems may be rendered axiomatic, doing so by disproving this for one of the most important examples: arithmetic. For every finite axiomatic basis of arithmetic there are statements which can be made which, while they may seem to be indisputably true, nevertheless are necessarily unproveable from the original axioms – one must add further axioms to cover them. The fact that we may consider the statements as true shows that arithmetic derives part of its meaning for us from something other than any proposed finite axiomatic basis. (The numbers mean something more to us than we could cover by a finite set of axioms.)
4) Change of Basis
Theorem proving can lead to the selection of a set of concepts and a subset of the set of relations proved to hold between them such that these can be taken to be a new basis of the same system, i.e. as the fundamental concepts and axioms which generate an axiomatic system that coincides with the original system.
1) Investigation
The investigation is the discourse where the questions emerge and the object of the experiment is delimited. The object, as something sought, as something yet to be made known, lies beyond the investigation. The investigation does not contain the essence of its own significance. If the investigation has meaning, it comes from the contact made with, and openness towards, its object. Thus the investigation has a sense of direction rather than a method. The investigation involves the gradual acceptance of hypotheses which have been tested by experiment, while continuing to entertain and examine rival hypotheses, and to suspend judgement in respect of many problems.
2) Observation
Thus the object gives the investigation its particular meaning. But how can it be an object for the investigation, and yet remain beyond it? This is made possible by the notion of observation, whereby the object of the investigation is not directly known or specified in the investigation, but indirectly in terms of how it is to be observed. Thus an opposition is developed between hypotheses, statements made in investigation, and data, statements made in observation. Presupposing that observation takes place within the framework specified in investigation, data are elementary statements which do not have a guarantee more reliable than themselves. In this sense data are provisionally unproblematic. Given the context from which data are taken, and allowing for the margin of imprecision involved, there is no sense in casting doubt on the effectiveness of some simple actions, the clarity of some direct statements, and the reliability of some immediate verifications. It makes no sense to imagine that any two people, relating to the object of the investigation in the same way (i.e. in the same situation with regard to observation), could be in disagreement about what they are doing, observing, and verifying.
3) Results
The hypotheses are tested by the data. This means that investigation and observation must be integrated together. So far we have set them up as facing each other in an irreducible duality. But each requires the other for its completion. This integration must be understood in terms of subsuming particular cases (data) under a general law (hypothesis).
The investigation is able to have the notion of its object being independent of itself only through the activity of observation whereby new data is brought into it. Thus the interpretation of the data is the necessary means by which the investigation gropes towards its object. The data are the traces of the object in the investigation, the points of contact.
Conversely, observation does not take place independently of there being an investigation. In investigation, situations are defined and isolated for observation, for extracting data. This definition and isolation follows from the delimiting of the object of investigation, the situations selected being where the object is to be made contact with. In order to be used, these data must be integrated into the investigation.
Data do not take on their meaning spontaneously, independently of the totality of the experiment. Nor must observation be thought of as a self-contained activity, as if all data together formed a little world of its own. For an immediate observation to take on its full meaning it is necessary that it can be interpreted into the framework of an investigation which is not reducible to a mere collection of data. Indeed the hypothesis/data opposition is something set up in investigation, and it is the way it is set up which provides the interpretation of data, although the interpretation is not fully determined in advance of the experiment, but it is rather its result.
Just as the experiment must first be understood at the level where the questions emerge, so also consideration of the data returns us to the investigation with some information to be evaluated. The data are what they are only for answering questions not formulated in observation, and the answers they provide assume their meaning only after being translated into the language of the questions.
The need to interpret data self-consciously can happen at the most elementary level. For instance, since general acceptance of the Theory of General Relativity and of Wave Mechanics, the notion of experimental repeatability can no longer be said to hold strictly even in principle. In the case of Wave Mechanics, the attempts to grapple with this in terms of the concept of probability, talking in terms of a large number of repetitions of the same experimental conditions etc. are well discussed. But the consequences of General Relativity in this matter are just as profound, because it undermines assumptions of homogeneity in space-time. Such homogeneity is implicit in any notion of repeatability. General Relativity still holds on to the notion of space-time being locally homogeneous, but the comparison of experimental conditions must involve more complicated assumptions about the state of the Universe (including its whole history since the Big Bang). The sense in which data can be publicly shared (i.e. be unproblematic) now involves much more explicitly a general framework of understanding, a framework which the concept of repeatability concealed.
4) Development
The results of the experiment are applied, particularly to the development of the experiment through refinements in technology.
The data are given meaning by the context of the investigation, but their form is to some extent fixed independent of the investigation so that it is not immediately altered by every development of the investigation. This fixity is possible because they are "closer" to the object and therefore in some degree represent the continuity of the investigation (which is based on having a constant object).
The meaning of the data is always open to modification in the light of further developments, because their integration into the investigation is never complete, allowing as it does for some imprecision and unreliability.
Indeed, what is tested by the experiment is not just hypotheses, but the correlation of investigation and observation by means of which data is interpreted – the result may be understood entirely in terms of increasing the precision of this correlation. The correlation arises in the setting up of an opposition between investigation and observation, but that does not mean that the correlation is ready-made. It is true that some aspects of the data must be taken as adequate embodiments of theoretical entities, but that does not mean that the correctness of this correlation can be taken completely for granted in advance. However plausible it may otherwise be, the decisive argument for it can only come as a result of the experiment. Even then, the experiment is not conducted once and for all with absolute precision, and so the correlation is still not perfected.
Data arise through the use of organs of perception, the setting up of experimental situations, and further enhanced by the manufacture of measuring devices and other accessory equipment. I shall refer to these collectively by the word ‘instruments’. The instruments embody the relation between investigation and observation, and so bring about and sustain their mutual independence. The instruments have the function of managing the encounter (of being the interface) between the hypotheses proposed and the data observed.
The development of the instruments must be based on the results of other experiments. The use of the instruments presupposes a much wider context (wider than the branch of knowledge in which the experiment is being performed which guarantees its validity. But the results of the experiment also reflect back on the understanding of the instruments used in it.
It may happen, for instance, that a later experiment shows that the readings of a measuring device must be systematically corrected. Since it is a systematic error, the first experiment was not wasted effort. The original data can be taken up again and an amended interpretation made. If the experiment itself has prompted the re-adjustment, then it may happen that the original data develop their informative content for the investigation only across a series of virtually distinct experiments (only "virtually distinct" since there would be no new data).
The precision of the instruments employed in an experiment is necessarily limited. But as the investigation proceeds, the instruments become more precise and the hypotheses approximate more closely to the truth. Even in the case of Physics "exact science" was always a misnomer. Nevertheless, exactitude is not renounced as an infinite goal, i.e. a goal to which, for any degree of precision, the investigation approximates, given sufficient time.
In order to be able to talk of improved precision, there must be some continuity in observation while there is a change in the investigation (i.e. in the hypotheses accepted or rejected), and so mutual independence between them is necessary for progress. The interpretation of data in investigation must be free to be modified in the light of the results of experiment. Such modifications have repercussions on how observation is seen within investigation, but not sufficiently to deprive the results of the experiment of the ability to guide the way in which the investigation should develop.
Let me illustrate some of the points I have been making here about the role of the instruments by using the context of Newtonian Mechanics. Clearly time is of central importance as a measurable entity in Mechanics. To the question ‘How is time to be measured?’ there is the answer: ‘It is simply the time which a watch shows.’ But this answer is inadequate, since the watch need not be a good one, i.e. one which can be relied on to give sufficient precision. The answer therefore should be modified by the following reservation: ‘… provided that it is a good watch.’ But this provokes a new question: ‘How does one know and prove that a watch is good?’ Hence one is lead to the problem of the manufacture of the good watch.
The development of the watchmaking craft may be explained in terms of an ideal, the isochronal oscillatory system, and of attempts to realize this ideal as best we can, with the highest precision possible. The ideal has taken the form of the theoretical balance wheel in which the conditions of a perfectly elastic oscillation are assumed to be satisfied. Now the study (not necessarily theoretical) of the perfectly isochronous balance wheel explicitly presents itself here as a special application of the laws of Newtonian Mechanics (which is taken to be an already established discipline). Thus the manufacture of a good watch turns out to be based on two foundations: the knowledge and application of the laws of Newtonian Mechanics on the one hand, and the experimental research and the testing of the best materials and procedures on the other. It is therefore clear that the watch which is consulted does not give Mechanics a previously determined way of measuring time without the help of Mechanics.
Having given a brief sketch of the concepts of an axiomatic system and of an experiment, let me now suggest a possible application of each in turn to a philosophy. If you consider that what constitutes a philosophy is a much more contentious issue than either of the concepts sketched, you may feel disappointed by my omission of any sketch of it, and you will have to be content, to begin with, with substituting the philosophy of your choice, and considering how much my remarks apply to it. In so far as my remarks fail so to apply, I hope that you will agree that it has fallen short of a philosophic ideal.
1) Theory as Concept
Philosophies may appear to have an endless variety of fundamental concepts, and few would appear to have just one. But we are not, in this matter, confined to considering as the fundamental concept what a philosophy may explicitly declare as such, even if we have no intention of questioning any declaration that philosophy may make about itself. In general we may be able to see the same system as axiomatic in more than one way – there is a choice of bases, and the set of fundamental concepts, and the set of axioms, may fail even to overlap between the different axiomatizations.
I propose here to take the philosophy to have just one fundamental concept, its theory, i.e. the totality of what it is putting forward as true. Any other concept it may put forward must be included in this one, and since a philosophy must make some attempt to demonstrate the truth of its claims, it must have something to say about the nature of its claims, i.e. about its theory. Thus the theory is a concept with the theory, although this does not necessarily mean that it has to be included among the fundamental concepts of an axiomatization. The fact that few philosophies put forward discussion of their own particular theories as central to them should not be allowed to deceive us. Often, for instance, the fundamental role is disguised beneath discussion about the nature of Philosophy in general, and the word ‘Philosophy’ can often be little more than a blanket term for the activity for which the discussion itself is the archetype.
2) Exposition as Axiom
It may seem strange that, since there is just one fundamental concept, there is any possibility of an axiom, for an axiom is a fundamental relation between fundamental concepts and we have chosen to have only one fundamental concept. But this fundamental concept requires to be related to itself. Just as the theory requires to be related to us through a philosophic work or works (i.e. its exposition) in order to be something we can discuss, i.e. to have some content, so also it requires to be related to itself through that same exposition in order to be something it can discuss, i.e. to be a fundamental concept for itself. Thus the exposition of a philosophy may be taken to be its one and only axiom. The theory does not have any content not supplied by its exposition, and its truth must be established independently of anything outside its exposition.
3) Project as Proving Theorems
The entire philosophy is then deduced from the exposition solely as a matter of definition and processes of deduction, i.e. the philosophy simply is the system of such processes launched by the exposition.
This system of processes may be described as the project of expounding the theory, as the theory unfolding itself. But this project does not produce anything other than the exposition, since the exposition is the theory as expounded. As far as the philosophy is concerned, the theory must not be anything more than what is expounded in the exposition. This may seem to be contradictory: the project is described as being generated by the exposition, yet all it can generate in turn is the same exposition. In my general characterization of an axiomatic system this paradox was defused by distinguishing between an implicit and explicit definition of the fundamental concepts. The distinction is not so clear in this case because, in the obvious sense of the distinction between implicit and explicit, the exposition is explicit about determining the content of the theory.
Suppose, therefore, that the exposition is in some sense incomplete. Then there will be room left for the project to do something, viz. to complete the exposition. But let us further suppose that the exposition explicitly acknowledges its incompleteness and is geared towards establishing the project, and so ensuring that it will be completed. In generating the project, the exposition is both incomplete and also, in anticipating (bringing forward) its completion, complete. If the distinction between the exposition and the project is integral to the philosophy, i.e. the paradox just described is a permanent, necessary feature as the theory continues to unfold, then we have an infinite project which allows us to talk meaningfully of an infinitely long, yet in some sense unified, text generated by it, the exposition.
If the philosophy’s arguments are to have any worth the philosophy must be consistent. For an infinitely long exposition we can relax the immediate implications of this in terms of the avoidance of contradictions, for any contradiction can always be defused by the introduction of new distinctions (rather as the distinction between implicit and explicit was introduced). Absolutely rigorous argument and avoidance of all contradiction in the exposition (and indeed also perfect clarity of expression and comprehensiveness in the exposition) are not relinquished as targets to be aimed at, but they may be regarded as infinite goals, goals that can only be said to be attained in so far as the infinitely long exposition is complete.
Further, we require that the philosophy be fully axiomatized, i.e. that it demonstrates its consistency, something most mathematical systems cannot do. As an infinite goal, this means that the exposition must argue that there can be no statements formed in the project whose truth or falsity is not ultimately to be settled by its relation to the exposition. The notion of truth exemplified by axiomatic systems is sometimes described as the coherence theory of truth. In the case of the infinite axiomatic system we are discussing, this coherence has become the organic unity of the project founded on the exposition. (The exposition is the project’s principle of unity.)
4) Regeneration as a Change of Basis
The text of the exposition derives its meaning from the context in which it is written and read. It is a carrier of its cultural background which may prevent the philosophy from unfolding itself further. To become philosophic, the exposition must therefore become free in its contingent associations (i.e. we do not have to eradicate all contingency, but any particular contingent detail must be disposable, i.e. treatable as contingent). The exposition must be self-transcending. Thus separated from any external validity, whether intuitive or experimental, it is from its coherent development according to its own internal principles that a philosophy derives its own legitimacy.
Hence the original text must be rewritten, the new version being regarded as an exposition of the same theory not simply because it is already anticipated as a step on the way to completion of the original text, but because it is an interpretation of the original text. No philosophic work, no matter how well written and thorough, can transcend time and culture without the aid, not just of translation and scholarly notes, but of reinterpretation and reassessment.
1) Project as Investigation
The project of expounding a philosophic theory may be considered a philosophic investigation, part of the general enterprise called Philosophy. The project is a discourse where certain philosophic issues are raised and the theory appears as the resolution to be sought, the object of the investigation. The theory, as something yet to be expounded, lies beyond the project.
2) Exposition as Observation
Thus the theory gives the project its meaning. But the theory cannot be an object of discussion except through reference to its exposition. Even a hypothetical theory must be supposed as having, hypothetically, an exposition. One may think in terms of first having a theory and then later expounding it, but the claim to have a theory only makes sense in terms of anticipating its later exposition. So it follows that the theory cannot lie beyond the project, unless the exposition also does, at least as something complete, and so we must set up an opposition between the project and the exposition.
Presupposing that the exposition is written and read within the context of the project, the statements in it are transparent in meaning. In this sense the text is provisionally problematic. Given the circumstances of the project, and allowing for the margin of imprecision involved, there is no sense in casting doubt on what is meant.
3) Theory as Result
The project and the exposition, both considered as incomplete, find their completion in each other. Their integration may be understood in terms of producing a theory that subsumes under itself. It is able to be treated as a particular through its exposition, hence the latter acts as data in the experiment.
The fact that the project must find itself integrated into the exposition, follows from the fact that the project, as an investigation, regards the theory, its object, as independent of itself. The latter attitude is only possible if the exposition can be taken to be already in existence, albeit in an incomplete form. Indeed, in so far as the project may be said to have an attitude at all, that attitude must be expressed in the exposition, and the notion of the project has no coherence except in being related to the exposition. The exposition is the necessary means by which the project has the theory as its object. The exposition is the trace of the theory in the project.
The converse, that the exposition finds its completion in the project, may seem trivial, since that is how the project was initially characterized – the project of expounding the theory, and hence of completing the exposition. But it should be noted that it is not any more trivial than that no observation takes place outside the context provided by an investigation.
The exposition does not have meaning independently of the context in which it is written and read, which is provided by the project. The project is able to provide the context by delimiting the nature of the theory – how the theory is to be sought. While producing the exposition, the project must also be integrating it into the ever widening context of itself.
Nor must the exposition be thought of as self-contained, as if the exposition formed a little world of its own. Assertions in it do not take on their meaning spontaneously, independently of the totality of the project (i.e. of all contexts and at all times). For the exposition to take on its full meaning it is necessary that it can be interpreted into a framework of a project which is not reducible to a mere text. Indeed the project/exposition opposition is something set up in the project, and it is the way it is set up which provides the interpretation of the exposition, although the interpretation is not fully determined in advance of the completion of the exposition, but that interpretation is rather its result (it is theory).
Consideration of the exposition returns us to the project with some information to be evaluated. The exposition is what it is only for answering questions raised in the project, and the answers it provides assume their meaning only after being re-applied to the questions in the project.
4) Regeneration as Development
The unproblematic character which the exposition has does not imply that its meaning in the project is fixed. On the contrary, because the words always hold only to some degree of precision and reliability, their meaning in the project is always open to modification in the light of further developments.
Part of the text of the exposition may be taken as fixed independent of the project so that it is not immediately altered by every development in the project. It has already been shown that the exposition must be presupposed as already partly written. The fixity of the text arises from the fact that the project can only exist in terms of relating to the exposition and one must suppose a definite text to which things are being related for there to be any project. Nevertheless, the meaning of this text is always open to modification in the light of further developments, because of its dependence on the context within which it is read; it never being completely understood.
What is tested by the philosophy is not just tentative explanations, but the correlation of the project and the exposition: one is concerned with increasing the degree to which the project realizes what is said of it in the exposition, and the degree to which the exposition captures what is going on in the project. Even as the correlation is improvedd and its perfection is attainable as an infinite goal, nevertheless perfection appears to become ever more remote.
The instruments in a philosophic experiment can only be words, collectively a language. It is language that must embody the correlation between the project and the exposition, and so sustain their mutual independence.
The development of language must be based on the result of the experiment. The generation of the exposition by the project, and the generation of the project by the exposition together (which two-way process may be described as self-regeneration) realize a development of the language which they use. Even without a change in the words themselves, the incompleteness of the exposition implies that reference to the exposition must change in its import.
The precision of language is necessarily limited, a fact only naive logicians might dispute. But as the project proceeds, words may be made more precise, and we do not have to renounce absolute precision as an infinite goal of the project. In order to be able to talk of improved precision, there must be continuity in what the exposition is taken to be (its wording must be in some fixed) while there is a change in the project (i.e. in what the theory is understood to be), and so the mutual independence of the project and the exposition is necessary for progress. The way of interpreting the exposition in the project must be free to be modified in the light of the theory. Such modifications have repercussions on how the exposition is seen within the project, but not sufficiently to deprive the theory of the ability to guide the way in which the project should develop.
The expressed intention of this paper was to show how a philosophy may be, and should be, both an axiomatic system and an experiment. I hope I have at least shown that such a joint possibility is plausible. That a philosophy ought to be both I have not spent much effort justifying since that really depends on the actuality of a philosophy being both. If it is not plausible that there could be such a philosophy, it is difficult to see how there ought to be one. And the only real "proof" that such a philosophy is possible must be given by a philosophy which is indeed both. Thus I can only suggest the plausibility of its existence here, and point you to a work entitled ‘Regenerating Philosophy’.