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In the last few decades there has been a revival of interest in diagrams in mathematics. But the revival, at least at its origin, has been motivated by adherence to the view that the method of mathematics is the axiomatic method, and specifically by the attempt to fit diagrams into the axiomatic method, translating particular diagrams into statements and inference rules of a formal system. This approach does not deal with diagrams qua diagrams, and is incapable of accounting for the role diagrams play as means of discovery and understanding. Alternatively, this paper purports to show that the view that the method of mathematics is the analytic method is capable of dealing with diagrams qua diagrams, and of accounting for such role.

In the past century the received view of definition in mathematics has been the stipulative conception, according to which a definition merely stipulates the meaning of a term in other terms which are supposed to be already well known. The stipulative conception has been so absolutely dominant and accepted as unproblematic that the nature of definition has not been much discussed, yet it is inadequate. This paper examines its shortcomings and proposes an alternative, the heuristic conception.

According to a view going back to Plato, the aim of philosophy is to acquire knowledge and there is a method to acquire knowledge, namely a method of discovery. In the last century, however, this view has been completely abandoned, the attempt to give a rational account of discovery has been given up, and logic has been disconnected from discovery. This paper outlines a way of reconnecting logic with discovery.

Given the large literature on Frege, one might believe that it would be impossible to say anything essentially new on the subject. This book contradicts this belief, calling attention to Frege's in...

The question that is the subject of this article is not intended to be a sociological or statistical question about the practice of today’s mathematicians, but a philosophical question about the nature of mathematics, and specifically the method of mathematics. Since antiquity, saying that mathematics is problem solving has been an expression of the view that the method of mathematics is the analytic method, while saying that mathematics is theorem proving has been an expression of the view that the method of mathematics is the axiomatic method. In this article it is argued that these two views of the mathematical method are really opposed. In order to answer the question whether mathematics is problem solving or theorem proving, the article retraces the Greek origins of the question and Hilbert’s answer. Then it argues that, by Godel’s incompleteness results and other reasons, only the view that mathematics is problem solving is tenable.

After giving arguments against the claim that the so-called Big Data revolution has made theory building obsolete, the paper discusses the shortcomings of two views according to which there is no rational approach to theory building: the hypothetico-deductive view and the semantic view of theories. As an alternative, the paper proposes the analytic view of theories, illustrating it with some examples of theory building by Kepler, Newton, Darwin, and Bohr. Finally, the paper examines some aspects of the view of theory building as problem solving.

Per trattare la questione dell’applicabilita della matematica questo articolo distingue tra matematica naturale, cioe matematica innata, e matematica artificiale, cioe matematica come disciplina. Esso sostiene che la matematica naturale e applicabile al mondo perche i sistemi di conoscenze di base su cui si fonda, essendo un risultato dell’evoluzione biologica, sono adatti a certe proprieta matematiche del mondo. Dall’altro lato, la base dell’applicabilita della matematica artificiale al mondo sta nella rivoluzione filosofica di Galileo, la decisione di limitare la fisica allo studio di alcune proprieta del mondo di carattere matematico. Ma, come l’applicabilita della matematica naturale, anche l’applicabilita della matematica artificiale dipende dalla nostra costituzione biologica, e percio in ultima analisi dall’evoluzione biologica

According to Bernard Williams, philosophy is a humanistic discipline essentially different from the sciences. While the sciences describe the world as it is in itself, independent of perspective, philosophy tries to make sense of ourselves and of our activities. Only the humanistic disciplines, in particular philosophy, can do this, the sciences have nothing to say about it. In this note I point out some limitations of Williams’ view and outline an alternative view.

The various methods to acquire knowledge are the basis of alternative models of science. With regard to science, one may speak of models in different senses, but the two main ones are models of science and models in science. Models of science are representations of how scientists build their theories. The chapter considers four models of science: the analytic-synthetic model, the hypothetico-deductive model, the semantic model, and the analytic model. Models in science are representations of empirical objects, phenomena, or processes of some area of science. The chapter discusses to what extent each of the above four models of science is capable of accounting for models in science.

Contrary to a philosophical tradition which maintains that sense perception is the starting point of knowledge, this chapter maintains that problems are the starting points of knowledge. This raises the questions: What are problems? How are problems posed? How are problems solved? The chapter gives an answer to these questions, arguing that problems are questions to be investigated in order to solve them; problems are posed by analysing the terms of the problem, namely the conditions that must be met to solve it; problems are solved discovering their solutions by the analytic method, therefore, knowledge is problem solving by the analytic method. The chapter also argues that the view that knowledge is problem solving by the analytic method permits to give an answer to the question: Is there any a priori knowledge?

Reuben Hersh is a champion of maverick philosophy of mathematics. He maintains that mathematics is a human activity, intelligible only in a social context; it is the subject where statements are capable in principle of being proved or disproved and where proof or disproof brings unanimous agreement by all qualified experts; mathematicians’ proof is deduction from established mathematics; mathematical objects exist only in the shared consciousness of human beings. In this paper I describe my several points of agreement and few points of disagreement with Hersh’s views.

This chapter considers the relations of knowledge to objectivity, certainty, intuition, deduction, and rigour. It argues that knowledge cannot be objective in the sense of being totally independent of any subject, but only in the sense of being as independent as possible of any particular human subject; that knowledge cannot be absolutely certain, since it can only be plausible; that knowledge is not obtained by intuition, not even fallible intuition; that knowledge cannot be obtained merely by deduction, but requires non-deductive reasoning; and that knowledge cannot be obtained by sticking to an abstract ideal of rigour, since what is important is not rigour but fruitfulness, hence the concept of rigour is better replaced with that of fruitfulness.

In the analytic method, knowledge is obtained by use of deductive and non-deductive rules. Since such rules are not plausibility preserving, they may give rise to errors. On this basis, contrary to a tradition according to which error is heterogeneous to knowledge, this chapter maintains that error is homogeneous to knowledge and inherent to it. Indeed, error is even homogeneous and inherent to logical and mathematical knowledge. The chapter also maintains that, though error may be a serious problem for logic, mathematics and science, it may also be fruitful. For, an analysis of the causes of error may provide useful indications for forming new hypotheses, and may also help to formulate new concepts.

As an alternative to the view that the aim of science is truth, this chapter maintains that the aim of science is plausibility, and specifically to make plausible hypotheses about the world, namely hypotheses such that the arguments for them are stronger than those against them, on the basis of the existing knowledge. The chapter argues that this meets all the difficulties of the view that the aim of science is truth. It also argues that plausibility is different from truth, probability and warranted assertibility, but is to a certain extent related to Aristotle’s endoxa. Moreover, the chapter discusses the relation between scientific knowledge and common sense knowledge, arguing that, rather than being opposed to common sense knowledge, scientific knowledge gives an explanation of our common sense knowledge, providing an interpretation of it.

This chapter discusses the relation of knowledge to mind. It distinguishes between two views of knowledge, the view of disembodied knowledge and the view of embodied knowledge. According to the view of disembodied knowledge, the mind is separate and independent of the body, and knowledge belongs to the mind alone, it is entirely based on ideas or representations of the mind, and its object are ideas or representations of the mind. Conversely, according to the view of embodied knowledge, the mind consists of certain capacities of the body, and knowledge is a natural process based on these capacities, which are implemented through processes external to the mind. The chapter maintains that the view of disembodied knowledge is untenable and only the view of embodied knowledge is defensible.

The view that mathematics is theorem proving, hence the method of mathematics is the axiomatic method, and the view that mathematics is problem solving, hence the method of mathematics is the analytic method, lead to two different concepts of demonstration: axiomatic demonstration and analytic demonstration . This chapter highlights the limitations of axiomatic demonstration. It maintains that analytic demonstration is the basic concept of demonstration and that, in terms of this concept, it is possible to deal with the question of the depth of demonstrations and theorems. On this basis, the chapter characterizes a view of mathematics, called the heuristic view, in terms of the following features: mathematics is problem solving by the analytic method; mathematical objects are hypotheses human beings make to solve mathematical problems by the analytic method; and analytic demonstration is the basic concept of demonstration.

This chapter discusses the relation of mathematics to the world considering two questions: What is the relation of mathematical objects to the world? Why is mathematics applicable to the world? As to the first question, the chapter maintains that mathematical objects are not obtained by abstraction from sensible things, or by idealization from our operations of collecting objects. They are hypotheses we make to solve mathematical problems by the analytic method, several of which have an extra-mathematical origin. As to the second question, the chapter maintains, on the one hand, that the applicability of natural mathematics to the world is due to the fact that natural mathematics fits in certain mathematical properties of the world. On the other hand, the applicability of artificial mathematics to the world is due to several factors, starting with the decision of modern science to confine itself to dealing only with some phenomenal properties of the world, mathematical in kind.