Proceedings of the XXIII World Congress of Philosophy

Brouwer introduced a principle called the bar induction to develop his intuitionism including the theory of choice sequences. In order to justify the bar induction, Brouwer supposed a fundamental assumption on the range of canonical proofs. The assumption, however, has been controversial and discussed in papers. Especially, we have to explain the reason why Brouwer introduced the fundamental assumption. In this paper, we point out that Brouwer’s argument is very close to the method of the Ω-rule due to Buchholz, which has been a main tool in infinitary proof theory since 1970’s. Based on this observation, we explain why Brouwer needed the fundamental assumption.

Philosophy of mathematics demands a strong sense of necessity for mathematical truths. But this demand seems to collide with the existence of different but true and consistent mathematical beliefs. Following the works of Benacerraf, Field, Balaguer and Putnam, I argue that philosophers of mathematics can either accept the plurality of interpretations of mathematics, but deny that every interpretation corresponds to an object, or, they can claim that every theory isolates a unique and only object. Facing this quandary, the philosopher of mathematics submit their theories to several difficulties and are led to adopt either a coherence or a correspondence theory of truth. A series of questions are entailed and need an answer to provide an impressive and complete theory of mathematical objects.

The subject of my address is a presentation of the crucial threads of Plato’s theory of Ideas. Accordingly, this proposal conjoins topics of epistemology and ontology, but foremost allows us to consider the issue of interconnection between mathematics and ontology. I will present Plato’s conception of measure-Idea, which has crucial significance for understanding the status of being of Ideas. The face of Plato’s teachings about Ideas that will also be analyzed is linked with his “unwritten doctrine” (agrafa dogmata). This theory refers mainly to teaching about numbers and ideal figures in addition to tutelage about principles of being, the One and the Infinite Dyad. It will let us depict in a precise way the very essence of Plato’s understanding of mathematics and explain the status of being which he attributed to mathematical objects. It will relate to critique of the interpretative stand declaring that in Plato’s philosophy we deal with so-called “two worlds” theory. Epistemological topics will also be given much attention. I consider my address as an invitation to discussion about the contemporary understanding of Plato’s philosophy, including the part of it that is accepted in many current versions of mathematical Platonism.

The aim of the paper is to show the complexity of problems concerning mathematical proof, and, in particular, the extramathematical aspects of the process of proving theorems in mathematics. The proof can be treated as a tool enabling the justification of a given theorem. In this sense the proof can be perceived as a tool persuading others to accept this proof as an element of mathematical knowledge. The justifying aspect of mathematical proof is most important for mathematics. It is a ‘certificate’ of the truthfulness of the considered theorem. However, apart from the persuasive aspect, the role that proofs play in mathematics is much more complex. Proofs can serve as a way of ‘making accessible’ a ‘fragment’ of mathematical reality to a mathematics adept person. By following attentively the proof progression, a beginning mathematician performs a ‘reconstruction of mathematical discovery’, thus participating – though ‘secondarily’– in discovering mathematical truth. In this way he can reach the essence of how mathematics functions as science. This participative character of mathematical proof – understood as a pass to participate in a mathematical experience – is definitely essential for the process of mathematical education. The third dimension of the functioning of mathematical proof in mathematical practice is its ritualistic character. Proving – once again – the already proven theorems, repeating once more the steps of the proving process accepted earlier, not only do we again get stronger in our conviction that the given theorem is true, but we also deepen our understanding of the performed reasoning. Such a conduct can be a component of ‘mathematical way of life’ of mathematicians. The process of carrying out a proof can be perceived as some celebration, ritual, ceremonial, present in life of mathematicians allowing them to appreciate beauty and elegance of conducted proofs, alongside allowing them to keep their brains ‘in aptitude’.

In his recent book Peirce and the Threat of Nominalism, Paul Forster (2011) presented how Peirce understood the nominalist scruple to individualise concepts for collections at the cost of denying properties of true continua. In that process Peirce showed some vibrant problems, as for example, the classic one of universals. Nonetheless that work is still incomplete; as long as that should be adequately related with what Peirce called his ‘scholastic realism’. Continuity is started by the theory of multitude and frees his analysis from any constraints of the nominalist theories of reality as integrated by incognizable things-in-themselves. His theory of multitude, instead, can be derived with mathematics: By drawing in the work of the ways of abstraction in diagrammatic reasoning made by Sun Jo Shin (2010) and in continuum theories by Cathy Legg (2010) I will show the device of diagrammatic reasoning as a plausible pragmatic tool to represent those continua and make sense of his scholastic realism. The analysis of continuity is a perfect example of how the method of diagrammatic reasoning helps unblock the road of philosophical inquiry and also helps to clarify other problems as, for example, the applicability of Mathematics. General concepts define continua, and, while the properties of true continua are not reducible to properties of the individuals they comprise, they are still intelligible and necessary to ground any science of inquiry.

The paper deals with the ontological status of number. The authors are convinced that it is useful to discuss the concept of number within the framework of the Aristotelian division of being into substance (ens in se) and accident (ens in alio). Number can thus be taken as ens in se or ens in alio. Aristotle and Thomas Aquinas believed that number is an accident and their concept is explained in the first part of the paper. In the second part it is shown that the Aristotelian concept is not correct. However, if number is not an accident then it seems that it must be identified with a Platonic entity (ens in se). In the third part the authors reject this Platonic conclusion that Frege seems to have defended. In the final part, a possible solution is shown: From the logical point of view number is an object but from the ontological point of view it is an entity that depends on linguistic structure (ens in alio).

The paper distinguishes two types of Platonist approach, namely the Traditional one and the Robust one. In relation to this distinction I am going to argue that if the ontology of mathematics is intended to be defended plausibly in a Platonist way then this cannot be done according to the Traditional version. This will draw our attention to the plausibility of the Robust version. The plausibility of the two versions of Platonism will be examined in relation to one of the central problems of the philosophy of mathematics, namely the truth-proof problem. The surveying of the truth-proof problem will bring to the surface the prima facie plausibility of the Platonist approach, as well as the apparent accessibility problem of it. Focusing on the accessibility problem will help us to identify two conditions that have to be met by any particular access theory of Platonism. These will be the reducibility condition, and the matching one. The Traditional version will appear an insufficient philosophical theory in relation to the two former conditions. The insufficiency will be demonstrated in the area of the incompatible mathematical theories, namely in the area of Euclidean and hyperbolic geometries. It will turn out that Robust Platonism can escape the squeeze of these conditions, so can it save the original prima facie plausibility of the Platonist approach.

The relation between mathematical intuition and formal representation of mathematical knowledge is considered in the framework of D. Hilbert’s program in the foundations of mathematics. The notion of sign is connected with both categories: on the one hand, it is an object of intuitive comprehension, thereby represented immediately to mind, on the other hand – the part of formal structure. A double nature of sign plays the special place in D. Hilbert’s finitism. It is shown that D. Hilbert relies heavily on some epistemological characteristics of sign in order to make elementary mathematical structures as the base of mathematical knowledge. Thus Hilbert’s views of intuitive contents of proofs by means of mathematical induction assume that concrete examples of mathematical induction are captured by intuition provided that predicates entering into the proof are in a sense elementary, namely, primitive recursive. According to W. Tait, Hilbert’s finitism, which consists in an assumption of intuitive comprehension of basic mathematical operations and objects, is reduced to acceptance of primitively recursive reasoning. Hilbert relied on the existence of some valid philosophical consensus in what can be considered as correctness of mathematical reasoning. Hilbert had admitted that there is some philosophical ‘minimum’ which is acceptable by all who are ready to accept mathematics.

Применение математики в механике, астрономии, физике, биологии, социологии, психологии и в других областях научного знания, способствовало проникновению в научный аппарат указанных областей знания таких понятий, как число, функция, производная, дифференциал, интеграл, структура, система и т.д.. Математизация процесса научного знания становится определяющим фактором того, что теория той или иной сферы научной сферы может называться научной. В процессе математизации научного знания должны соблюдаться необходимые условия, как в содержательной теории, так и в выбранных математических методах. Они отражают реальность и тем придавать высокую точность предсказанию и описанию процессов.

Theories which born during interior development of mathematics later obtain empirical interpretation and become a part of applied science. The reasons for this are still not clear, although many mathematicians and philosophers (E. Wigner, M. Steiner, R. Hersh and others) put forward their hypotheses. We believe that solution the problem should involve investigation of mathematics as a sort of evolving system. Two systems may be subordinated; that is, if the first one is a primary and fundamental one, then the other one is secondary and adjusted to the first. We propose that substantial sciences are primary and formal sciences are secondary. There are reasons to think that mathematics in its interior development has intention to physics. Secondary system may have changes of two sorts: those which are requested by the primary system, and those which are free of the requests of the primary system. Analyzing biological systems we see that interior changes of the system, which are not caused by its current needs, are determined by its further purposes. Each living system carries a “model of future” in itself, and it tends to this future by its free changes. We think that the living systems development logic may be transferred to conceptual systems, also. If we consider mathematics as a conceptual system which is secondary in relation to physics, then we receive a natural explanation of the possibility of mathematical anticipation.

The features of development of the subject-matter of mathematics can be viewed from the standpoint of the study of mathematics as objectification in Marx’s sense – as a relatively independently self-reproduced sphere of activity. For an explication of this development it is necessary to define concepts of “world”, “object”, “subject-matter”, “subject-actor”, “reality”, “structure”, “model”, “objectification”, “ontology”. In the process of objectification of the domain of objects of mathematics takes place a transformation of forms of existence and creation of new ontologies, with respect to new types of realities. The development of mathematics as a sphere of human activity necessarily contains subjectification (reproduction of a type of subject-actor as necessary substructure of every social objectification). The results of mathematical activity are fixed as objects and reproduced in the intersubjective, objectivized domain of mathematics, because the subject-actor itself is practically incorporated into the structure of the world as an object. Mathematics as practically realizable objectification of knowledge is modelling of ontologies. The immediate subject-matter of mathematics is the (realized in representation) manifold of abstract structures of subject-actor, whereas the mediated object of mathematics is the manifold of object structures of the world.