Andrzej Lubomirski
On the philosophy of mathematics

W: Władysław Krajewski (ed.), Polish Essays in the Philosophy of the Natural Sciences, D. Reidel Publishing Company, Dordrecht, Holland, 1982, s. 239-248.
It is well known that any philosophical reflection, irrespective of the specific problems it refers to, differs from scientific activity by two important (even if not constitutive) features.

First, any philosophy, as opposed to any scientific discipline, involves a certain idea of its own nature. The mathematician, of course, can also ask himself the question, what is mathematics (and, consequently, what is the right way of cultivating it)1;
1In fact, history shows these questions as always accompanying the purely 'technical' activity of the greatest mathematicians; the relevant ideas are developed, for example, in the papers of scholars such as Gauss, Galois, Riemann, Poincaré, Hilbert, Cantor and others.

however, neither the question itself nor considerations leading to possible answers to it, belong to the area of mathematics. They accompany this discipline but themselves remain situated within the scope of reflection of a different kind, one that, generally speaking, is usually considered philosophical. As far as philosophy is concernded, it is not so, for the the question about the essence of philosophy belongs to philosophy itself and the reflection on its nature forms an indispensable component of philosophical reflection.

That is where the second difference springs from, which is particularly important from the methodological point of view.

In fact it is evident that both question about the nature of philosophy and the question about the nature of, for instance, mathematics can be answered differently and, indeed a variety of answers have been offered. In the case of mathematics or any other scientific discipline, however, this fact does not seriously affect the way of cultivating it, insofar as differences in ideas offered by two mathematicians on the nature of mathematics (such as, for example, philosophical controversies between Poincaré and Hilbert, or between Cantor and Kronecker) do not genereally lead to any difficulties in identifying their actual research practice as belonging to the same area of knowledge. It is not so as far as philosophy is concerned, as views on its nature presented (implicitly or explicitly) in philosophical texts differ so significantly from one another that only a serious intellectual effort can make it possible to reveal a principle on the basis of which these texts can be classified as an intellectual unit distinguishable from the whole of the human knowledge under the name of philosophy.

It is, therefore, understandable that any attempt to state what philosophy is, or what a specific branch of philosophy is, usually leads to serious difficulties and can never succeed in a short, clear and unambiguous answer to the question posed.

The above remarks are valid, in particular, for the branch of philosophy which is usually called the philosophy of mathematics and which is the subject of the present paper: our intention is to answer the question, what the philosophy of mathematics is or rather what it should be.

The phrase 'it should be' is used here because the present article is meant to be a proposal rather than a report on the state of facts and as such is, to a certain extent, of a normative and arbitrary character (although, of course, we consider it important to bring the proposed sense of the term as close as possible to the meanings that are functioning in the literature of the subject).

This circumstance also suggests a definite conception as far as the method of presentation is concerned: it seems convenient to proceed in a synthetic and negative way rather than in an analytical and positive one. Therefore we shall start from the discussion of difficulties which are encountered when - on the basis of texts which for various reasons have been classified as belonging to the philosophy of mathematics - an attemopt is made to construct a general image of the philosophy of mathematics as an autonomous area of knowledge.

These difficulties consists, generally speaking, in the fact that there exists a considerable variety of views on what is the subject-matter and the method of the philosophy of mathematics, as well as its role and its relation to other domains of human thought. It will not be necessary to present the whole variety of views here2;
2See, for example, Beth, E. W., The Foundations of Mathematics, Amsterdam: North-Holland 1959, especially part IX, and Mathematical Thought, Dordrecht: Reidel 1965, especially Chapters VIII and IX.

the discussion will therefore limit itself to outlining three fundamental oppositions, the elements of which should be considered extremes of a whole range of standpoints pertaining to the understanding of the term 'philosophy of mathematics' and orderd according to a certain criterion (different, obviously, for each of the oppositions).

Thus, first, there is on the one hand something that can, generally speaking, be called a philosophical reflection about mathematics, arrived at by the mathematicians themselves on the margin (considered by them more or less important) of their purely mathematical research activity. The philosophy of mathematics thus understood is open: it has no set limits and may pertain to any problem that appears as accompanying mathematical research activity, that a mathematician finds worth considering and that he (intuitively) classifies as philosophical3.
3Therefore, such a philosophy of mathematics is not systematic and does not provide complete conceptions; it is more a chaotic set of ideas, remarks, and suggestions. Note, however, that, as it is cultivated by those who continue to be in constant close contact with the true reality of mathematics, it proves, it seems, particularly important.

There appears, however, on the other hand, an opposite view according to which the philosophy of mathematics should be a discipline with a relative autonomy, professionally and systematically developed by specialists; it should have its (clearly defined) specific subject, its specific point of view, and its specific method; briefly, it should not be cultivated in the form of sponaneous reflection, but as a programmed and organized branch of knowledge4.
4This idea is involved, for example, in Piaget's 'genetic epistemology' (see, for example, Piaget, J., Introduction à l'épistémologie généttique, vol. 1 (La pénsée mathématique), Paris: Presses Universitaires de France 1950, and Piaget, J. (ed.), Logique et connaissance scientifique, Paris: Gallimard 1967). On the other hand, it also springs from the conception of philosophy as defended by Carnap and his successors (see Carnap, R., The Logical Syntax of Language, New York: Harcourt Brace 1937 and Foundations of Logic and Mathematics, Chicago: University of Chicago Press 1939).

The thesis of the second of the three oppositions mentioned above is a consequence of a more general view on the nature of philosophy, one that claims the specific character of philosophical cognition as incapable of reduction by any means to the scientific one (see, for example, Husserl, E., Philosophie als strenge Wissenschaft, "Logos" 1 (1910-1911), s. 289-341). It is not important that philosophical cognition thus understood is depreciated or, contrarily, valued higher tham scientific knowledge; what proves to be important here is only the fact that, according to this view, philosophy in general and the philosophy of mathematics in particular is an attempt to answer questions which can, at the present state of science, be neither posed nor solved within the field of any scientific discipline (see, for example, Russell, B., Introduction to Mathematical Philosophy, London: Allen and Unwin, 1919). There also exists, however, an oppostie view which centers around the tendency to dismiss, so to speak, the philosophy of mathematics as philosophy.Such a dismissal should be carried out so that the philosophy of mathematics, still as an autonomous discipline, wuld be cultivated either according to the rigors usually imposed on scientific knowledge or so that problems traditionally considered philosophical problems of mathematics would, after a proper reformulation, be coped with within the field of the respective scientific disciplines (such as foundations of mathematics, logic, psychology, etc.)5.
5The conception of the epistemology of mathematics as a branch of scientific knowledge (and not of philosophy in the traditional sense of the word) is developed, for example, by Piaget (cf. the references cited above in note 4). We find the same idea (although realized in another way) in contemporary positivism.

Finally, the third of the oppositions lies in a distinction between the philosophy of mathematics, which aims at nothing other than a description of various aspects of mathematics (such as, for instance, the nature of mathematical entities, the structure of mathematics as a whole, the actual origin of mathematical notions, methods applied in research practice, mechanisms of development of mathematics, etc.), and a normative philosophy of mathematics the intention of which is to set guidelines, norms, suggestions and restrictions directed at mathematics itself (such as those pertaining to the correctness of various procedures in defining mathematical concepts, to justifiability of methods of proof, etc.).

There is no place here either for a broader analysis of these conceptions of the philosophy of mathematics or for a detailed discussion of the problem as to which of the positions is supported by more important arguments. For the time being the only purpose of our considerations has been to point out that these arguments, as well as positions that have been based on them, can and have been differentiated.

Now we will point out that the conception of the philosophy of mathematics as a set of do's and dont's adressed at mathematics and mathematicians - would hardly be acceptable if these norms were to be based on philosophical assumptions, independent of mathematical knowkedge. History provides numerous examples of situations where extra-mathematical ideas exerted a significant influence on the development of mathematics. Even if it were so, however, these ideas have always originated from purely mathematical problems or attempts at their solution; a philosophy of mathematics that would want to neglect it and impose norms that have not beeon rooted in the actual research practice of mathematicians would be condemned to inevitable sterility.

It also does not seem reasonable to suggest a conception of the philosophy of mathematics as a branch of human knowledge that is preprogrammed once and for all as far as its subject and methods are concerned, if anything more was aimed at than merely a most general outline of the type of problems that could be dealt with. Such a conception would have to neglect totally the fundamental fact of the historical changeability of the scope of the philosophy of mathematics, the changeability consisting not only in expanding it s scope (natural for every field of knowkedge) but also in a specific phenomenon of partial 'dephilosophization' of some problems (the problem of infinity, for example), springing from the earlier periods of the history of the philosophy of mathematics amd originally included in its scope. There is no reason to suppose that any of the two opposite tendencies will disappear. It seems, therefore, that any attempt to program the philosophy of mathematics will be dismissed by history as a complete failure.

First of all, however, we believe that philosophy in general and the philosophy of mathematics in particular can and should remain just philosophy, i.e., cognition specific and irreducible to scientific knowledge in any sense of the word. We do not deny that philosophy should take into account results of scientific research.6
6It is, for instance, evident that every serious philosophy of mathematics has to take into consideration results obtained in the area of the foundations of mathematics. Cf. Beth's work cited in note 2; see also Beth, E.W., Piaget, J., Épistémologie mathématique et psychologie, Paris: Presses Universitaires de France, 1961 and Wang, H., From Mathematics to Philosophy, London: Routledge and Kegan Paul, 1974.

We reject philosophy if taken to be purely specuative knowledge. We think, however, that it ought to have its own problems and its own methods, which are not, cannot and should not be scientific. In short, we want to cultivate philosophy as philosophy and are not distressed that we are not going to obtain results as certain and as precise as those obtained in scientific knowledge (cf. Wang, H., From Mathematics to Philosophy, London: Routledge and Kegan Paul, 1974).

If, therefore, we were to answer the question, what philosophy in general and philosophy of mathematics in particular has to deal with, we would say that it ought to deal with just what it has been dealing with since antiquity. We do no see a better definition here (although we are aware of reservations that have been and can be formulated in this respect) than the one referring to the traditional division of philosophy into three fundamental domains: ontology, i.d., the gereal theory of being; epistemology, i.e., the theory of knowedge; and axiology, i.e., the general theory of values, that is, of everything which is considered desirable and which, thereby, determines the objectives of human activity.

Thus, we shall distinguish, first, the ontology of mathematics, the intention of which is to work out a general view of the nature of mathematical entities and the mode of their existence. Common sense suggests that they belong to a sphere of being different from that to which we classify the reality we contact through our senses. The point, however, lies in whether this is so and, if it is so, in what sense can we speak of the existence of mathematical reality.

The fundamental opposition of views on this problem has always been between the standpoint, for historic reasons called, Platonism and the one usually named constructivism.7
7What is meant here is constructivism considered as a philosophical position and not as a 'technical' direction of the development of mathematics.

Both of them take various, more or less radical forms; we, however, will concentrate here only on the basic difference.8
8For more details, see, for example, Benacerraf, P. and Putnam, H. (eds.), Philosophy of Mathematics, Englewwod Cliffs, N.J.: Prentice-Hall, 1964, and Heyting, A. (ed.), Constructivity in Mathematics, Proceedings of the International Colloquium 'Constructivity in Mathematics', Amsterdam, Amsterdam: North-Holland, 1957. See also related analyses in the works of Beth and Piaget cited above.

According to Platonism, mathematical entities form a reality which exists, or even preexists, independent of any human consciousness; they exist in a specific way, characteristic merely of mathematical entities; they are as they are per se, irrespective of whether anybody wants to contact them in any possible way. According to constructivism, on the other hand, mathematical entities are constructs, products of the human mind; they do not exist apart from the knowing subject; their nature and their mode of existence cannot be explained without reference to appropriate acts of cognition, i.e., to processes these entities result from.

It is clear that the choice of an answer to the question about the mode of existence of mathematical entities leads directly to a definite interpretation of the central problem of the second big branch of philosophy of mathematics, i.e., of its epistemology: the question about the nature of the relation between the subject and the object of cognition in mathematics.

This problem should be considered in two aspects. On the one hand, the point is in what way is objective knowledge possible in mathematics, objective here being understood as referring somehow to a mathematical reality of any kind; on the other hand, in what way is contact between the subject and the object possible at all. In the case od constructivism (according to whose views, as we have already mentioned, the object cannot be separated from the act of cognition with which it is connected and through which it exists) the problem of contact seems to have an obvious solution. However, the question, in what sense can we speak of the objectivity of knowledge remains open and extremely difficult. Platonism, in turn, seems to solve the objectivity problem (in the sense that if knowledge is possible at all, it is objective). It is, however, extremely difficult to answer the question about how contact is possible between the subject and a mathematical reality preexisting independently of any consciousness and any knowledge. In other words, the central problem of Platonism is the question, how any knowledge is poossible in mathematics, while the central problem of constructivism is how any ignorance is possible in it.

Therefore, each of the two positions explains or at least makes it possible to exlain something that requires explanation and with which the opposite position cannot successfully cope; one the other hand, each of them leads to insoluble difficulties. Thus, in a sense, the aim of epistemology can be perceived in constructiong a synthesis of these two conceptions which would preserve what can be seen as an achievement in each of them and remove their basic shortcomings. In fact, all the great philosophies of mathematics (Leibniz and logicism, Kant and the doctrine of a priori synthetic judgments, positivism, etc.) can be interpreted as attempts at such a synthesis; none of them, however, so it seems, managed to solve the problem completely.

We do not intend to criticize these great theories here; a suspicion, however, arises that their failures have been connected with their synchronic character.9
9In fact, the possibility of the historical philosophy of mathematics is involved in Kant's theory; it was realized, however, only in the neo-Kantian, German tradition (particularly by Cassirer) and in French idealism (by scholars such as Brunschvicg, Boutroux, Gonseth, Piaget and others). On the other hand, a historical approach is, in general, one of the basic ideas of Marxist philosophy. As far as the philosophy of mathematics is concerned, however, this idea - with the exception of Marx's Mathematical Manuscripts (published in Russian in "Pod znamenem marksizma" 1933, no. 1) - has not so far been developed in a sufficiently complete and systematic way.

If mathematical knowledge is understood as a static relation between a static, petrified mathematical reality and an unchanging subject, constant as far as structure and functioning are concerned, then the opposition of Platonism and constructivism becomes an insoluble contradiction. We believe, however, that a chance to solve the problem will appear when cognition is understood as a process of evolution of the relation between subject and object, where both elements of the relations change in time, both as far as their structure and as far as the way of their interaction are concerned.

In such a diachronic or historical perspective (and only in it) does the third fundamental domain of philosophy of mathematics, i.e., its axiology, obtain it full significance. Here axiology is understood as a theory of all values (such as generality, rigor, evidence, effectiveness, intelligibility, etc.), regulating the cognition process in mathematics, that is to say, as an analysis of phenomena connected with the historical changeability of value systems determining, in various periods of the history of mathematics, the main directions of mathematical thinking. The axiology of mathematics thus understood would be an indispensable complement of the epistemology of mathematics. For, on the one hand, it is useless to hope that a reasonable description of the functionning of the subject of mathematical cognition will be possible without pointing out respective values that determine it (as it is not enough to say that such and such a cognitive procedure has been carried out, it must be stated why it has been carried out and why it has been done in that particular way). On the other hand, it is not enough to say what values have been functioning; it must also be demonstrated in what ways they could be and actually have been realized in a given historical situation.

Therefore, the philosophy of mathematics, as it is understood here, is based on two fundamental assumptions. First, the categories of subject and object of cognition not only cannot be analyzed but cannot even be defined independently of each other. The very concept of the object of cognition incudes the idea that it is an object for a certain subject, just as the concept of the subject of cognition includes the idea that it is a subject confronted by a certain object. Whatever is a subject, is the subject inasmuch as it is oriented toward a certain reality and inasmuch as it enters contact with it. If, on the other hand, we speak of the existence of this reality (and especially of the existence of mathematical entities), then we have to mean not an absolute existence but an existence from the point of view of the knowing mind. In other words, the central category of the philosophy of mathematics is a relation between the subject of mathematics and the mathematical reality and not these elements considered separate from each other.

Second, this relation is always to be considered historically relative; the subject and the object of cognition in mathematics are constantly mutually constructing and reconstructing each other. Then, from the point of view of the diachronic philosophy of mathematics, what proves to be the central problem is an analysis of mechanisms responsible for the evolution of the fundamental subject-object relation.

Speaking of the subject and object of cognition in mathematics we therefore have to consider these categories as relative in a twofold way, i.e., to one another and to history. What is important is no pre-existing mathematical reality but the one to which the subject of mathematics is intentionally oriented in a given historical situation, just as we are not interested in any subject in general but in what, in a given situation, is the subject confronted by a certain mathematical reality.

We cannot develop these ideas more broadly here; we must, however, at the end of the present paper, at least touch on the two fundamental., difficult problems to which aproval of the above suggested philosophy of mathematics might lead.

The first of the two is connected with the very idea of the subject of cognition in mathematics. It is clear that from the philosophical point of view no psychologically or sociologically understood subject (i.e., the mind of a particular mathematician or any kind of group consciousness) can be meant here but only something that would be an analogue of what Jean Piaget in his genetic epistemology calls "sujet épistémique" and what is defined through reference to what he calls "operational structures" in which - if they could be determined for mathematics in general - the whole system of connections between the subject and the object of mathematics would be coded.10
10See Piaget, J., Introduction à l'épistémologie généttique, 3 vols, Paris: Presses Univeritaires de France, 1950, and Piaget's papers in Piaget, J. (ed.), Logique et connaissance scientifique, Paris: Gallimard 1967; see also Beth, E.W., Piaget, J., Épistémologie mathématique et psychologie, Paris: Presses Universitaires de France, 1961, chapters VII-XII.

Therefore, one of the basic tasks of the philosophy of mathematics would be to extrapolate properly the ideas of Piaget; we have to say, however, that, although in general these ideas look stimulating, the task, even if it can be done, is definitely going to be very difficult.

At least equally difficult is another problem, one that is a key to the philosophy of mathematics understood as it is understood here. The point is whether, and in what way, the process of the evolution of the subject-object relation can be handled as a historical one.

The last term can and has been understood in various ways, often very broadly and vaguely. For us, however, such an understanding of historicism is important, according to which it should be accepted that a truly historical approach to an object that is changing in time should lead, generally speaking, to the construction of an intelligible image of this object's evolution as a process constituing, first, a historical unit to which a certain sense i s ascribed (a unit which cannot be divided into fragments along the time axis without the loss of that intelligibility) and, second, as a process that is necessary (i.e., such that that sense is the only one possible).

The above postulate implies two other conditions, both of them necessary, if not sufficient; first, the object of a historical process thus understood should necessarily be conceived as an autonomous system (as only then can the mechanisms of history be deduced from its own course); secondly, every process that we would like to call history should be distinguished as such, whose structure cannot be deduced from any distinguished state (any frame of reference) to which past and future are referred and through which past and future acquire their sense, but merely from the process itself as a whole. However, it is evident how difficult an attempt to fulfil either one of the two conditions will be.

Moreover, both of the two problems mentioned are not the only difficult questions involved in the conception of the philosophy of mathematics that has been suggested here. But, as long as there is the slightest chance to answer them, we will tend to believe that the perspective accepted in the present paper shoulsd not be too easily rejected, as each theory - which wants not to be a pure speculation nor a mere fragment of scientific knowledge but which is to become a philosophical reflection on the true reality of mathematical knowledge - has to accept the simple fact that mathematics is o product of man, man with everything he has ever produced is a product of history.