Carlo Cellucci

Sapienza University of Rome | la sapienza · Department of Philosophy

The question of whether mathematics depends on experience, including experience of the external world, is problematic because, while it is clear that natural sciences depend on experience, it is not clear that mathematics depends on experience. Indeed, several mathematicians and philosophers think that mathematics does not depend on experience, and...

In this section we introduce second-order theories, an extension of first-order theories to second-order logic.

In this section we introduce a basic kind of formal languages, first-order languages.

In this section we introduce some primitive recursive relations and functions which are meant to handle codes of finite sequences.

In this section we give a proof of Gödel’s First Incompleteness Theorem, based on the Traditional Fixed-Point Theorem.

In this section we consider a noteworthy subclass of computable functions, the class of primitive recursive functions.

In this section we introduce languages that allow quantification not only on individual variables, but also on predicate variables.

In this section we introduce first-order theories, an elementary counterpart of mathematical axiomatic theories.

In this section we consider the interpretation of a first-order language, which consists in fixing a domain and a function that assigns a meaning of the appropriate sort for each individual constant, function constant and predicate constant.

In this section we consider Tarski’s two Undefinability Theorems, which establish in two different ways that the property of being a sentence true in N is not expressible in L PRA.

According to heuristic philosophy of mathematics, one of the tasks of the philosophy of mathematics is to give an answer to the question: What is the nature of mathematical objects? This question has received several answers. The chapter discusses them and argues that they are inadequate. Then, it offers an alternative answer: mathematical objects...

An essential aspect of the contrast between mainstream philosophy of mathematics and heuristic philosophy of mathematics concerns the method of mathematics. According to the former, the method of mathematics is the axiomatic method, according to the latter, the method of mathematics is the analytic method. So the question of method is a central ele...

An alternative to mainstream philosophy of mathematics is heuristic philosophy of mathematics, according to which the philosophy of mathematics is concerned with the making of mathematics, in particular discovery, and the method of mathematics is the analytic method. Heuristic philosophy of mathematics is not to be confused with other views of math...

According to heuristic philosophy of mathematics, one of the tasks of the philosophy of mathematics is to give an answer to the question: What is the method of mathematics? And its answer is that the method of mathematics is the analytic method. The chapter examines the analytic method, its mathematical origin and medical origin, its original formu...

According to heuristic philosophy of mathematics, one of the tasks of the philosophy of mathematics is to give an answer to the question: What is the nature of mathematical notations? The prevailing answer to this question is the precision-conciseness view of mathematical notations. The chapter discusses this answer and argues that it is inadequate...

From antiquity, a number of methods have been put forward as an alternative to the analytic method. The chapter examines Aristotle’s analytic-synthetic method and its relation to the analytic method. Then, it examines Pappus’s analytic-synthetic method and its relation to reductio ad absurdum. Finally, it examines the material axiomatic method, a b...

According to heuristic philosophy of mathematics, one of the tasks of the philosophy of mathematics is to give an answer to the question: Why is mathematics applicable to the world? The question has received several answers. The chapter discusses them and argues that they are inadequate. Then it offers an alternative answer, but also underlines the...

According to heuristic philosophy of mathematics, one of the tasks of the philosophy of mathematics is to give an answer to the question: What is the nature of mathematical demonstrations? This question has received several answers. The chapter discusses them and argues that they are inadequate. Then, it offers an alternative answer: mathematical d...

According to heuristic philosophy of mathematics, one of the tasks of the philosophy of mathematics is to give an answer to the question: What is the nature of mathematical definitions? In the past century, the prevailing answer to this question has been the stipulative view of mathematical definition. The latter has been so absolutely prevailing a...

According to heuristic philosophy of mathematics, one of the tasks of the philosophy of mathematics is to give an answer to the question: What is the nature of mathematical explanations? Two different kinds of mathematical explanations can be distinguished: mathematical explanations of mathematical facts, and mathematical explanations of empirical...

According to heuristic philosophy of mathematics, one of the tasks of the philosophy of mathematics is to give an answer to the question: What is the nature of mathematical beauty? The chapter argues that a piece of mathematics is beautiful if it provides understanding, namely recognition of the fitness of the parts to each other. Beauty is relevan...

The chapter describes characters, origin, and goal of mainstream philosophy of mathematics, namely the philosophy of mathematics that has prevailed for the past century. According to it, the philosophy of mathematics cannot concern itself with the making of mathematics but only with finished mathematics, namely mathematics presented in finished for...

According to heuristic philosophy of mathematics, one of the tasks of the philosophy of mathematics is to give an answer to the question: What is the nature of mathematical diagrams? In the last few decades, the prevailing answer to this question has been the axiomatic view of mathematical diagrams. The chapter discusses this answer and argues that...

Mainstream philosophy of mathematics and heuristic philosophy of mathematics yield two different views of theories, the axiomatic view, and the analytic view. The chapter describes them and argues that the axiomatic view is inadequate, only the analytic view is adequate. Then, it discusses some questions raised by the analytic view of theories, nam...

According to heuristic philosophy of mathematics, one of the tasks of the philosophy of mathematics is to give an answer to the question: In what sense is mathematics knowledge? The chapter argues that mathematics is knowledge in the same sense as all other knowledge, namely it is a function of life that is essential to the survival of individual o...

In the analytic method, hypotheses are obtained by non-deductive rules. Obtaining them is not a sufficient condition for discovery, because discovery requires showing that the hypotheses are plausible. Nevertheless, obtaining hypotheses is a necessary condition for discovery and, in this sense, non-deductive rules can be said to be rules of discove...

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....

The terms of a mathematical problem become precise and concise if they are expressed in an appropriate notation, therefore notations are useful to mathematics. But are notations only useful, or also essential? According to prevailing view, they are not essential. Contrary to this view, this paper argues that notations are essential to mathematics,...

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 statemen...

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 defini...

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...

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 bring unanimous agreement by all qualified experts; mathematicians' proof is...

This monograph addresses the question of the increasing irrelevance of philosophy, which has seen scientists as well as philosophers concluding that philosophy is dead and has dissolved into the sciences. It seeks to answer the question of whether or not philosophy can still be fruitful and what kind of philosophy can be such. The author argues th...

This chapter examines the methods to acquire knowledge. The methods considered are the analytic method, the analytic-synthetic method, the material axiomatic method, and the formal axiomatic method. The chapter also examines the original formulation of these methods, and the relations between them. Since deductive and non-deductive rules play a cru...

One of the main questions about knowledge is the relation of knowledge to reality. This question is particularly important for the heuristic view of philosophy, according to which philosophy aims at knowledge and methods to acquire knowledge. In the course of the history of philosophy, the question of the relation of knowledge to reality has receiv...

Since antiquity, several philosophers and scientists have claimed that the aim of science is truth. This raises the question: What is truth? A popular answer is that a proposition is true if it corresponds to the facts to which it refers. This is the concept of truth as correspondence. However, the concept of truth as correspondence does not provid...

How is knowledge acquired? In the past century it has been generally believed that this question is vacuous, because there is no method for acquiring knowledge, the basic laws of science can be reached only by intuition, then, from them, conclusions are deduced and compared with experience. This belief has had a very negative impact on logic, the m...

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. In...

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...

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...

The heuristic view of philosophy is opposed to the foundationalist view, according to which philosophy does not aim at knowledge and methods to acquire knowledge, but only at justifying already obtained knowledge, by providing a foundation for it. The foundationalist view assumes, first, that there is immediately justified knowledge and all other k...

The main motivation of the foundationalist view is to save knowledge from sceptical doubt. This chapter maintains that such motivation is unjustified. Indeed, according to a widespread opinion, absolute scepticism is irrefutable, since no logical argument can be advanced against it. On the contrary, the chapter argues that logical arguments can be...

According to an influential tradition, perception is a passive process, determined entirely by the features of the external world. On the contrary, this chapter maintains that perception is problem solving by the analytic method, hence it is an active process. Since, in the analytic method, hypotheses are obtained by non-deductive inferences, this...

Several mathematicians have maintained that mathematical beauty plays an important role in mathematical research. This raises the problem: What is mathematical beauty? This chapter supports the view that a piece of mathematics, demonstration or theorem, is beautiful when it provides understanding, meaning by this the recognition of the fitness of t...

Not only knowledge plays a central role in human life at all levels, from survival to improving the quality of life, but, since antiquity, several people have claimed that knowledge is the purpose and meaning of human life. This chapter maintains that this claim is unjustified, and that all attempts to show that human life has a purpose and meaning...

In the past century, the prevailing view has been that the main problem of epistemology, or theory of knowledge, is to give a definition of knowledge, and the prevailing definition of knowledge has been that knowledge is justified true belief. This contrasts with the fact that, since antiquity, the definition of knowledge as justified true belief h...

The conclusion summarizes some of the main theses of the book. In the past century, the view that philosophy aims at knowledge and methods to acquire knowledge has been abandoned, and this has contributed to the increasing irrelevance of the subject. The book attempts to revive this view. This requires a rethinking of knowledge. According to it, kn...

The view that mathematics is problem solving has been challenged by the claim that, in the twentieth century, mathematics has been reduced to theorem proving. This raises the question: Is mathematics theorem proving, or problem solving? The purpose of the present chapter is to answer this question, which is a philosophical question about the nature...

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 problem...

As a response to the increasing irrelevance of philosophy, this chapter lists the characteristics philosophy should have in order to be fruitful. This results in a view of philosophy that may be called the heuristic view, according to which philosophy aims at knowledge and methods to acquire knowledge. In listing the characteristics philosophy shou...

The distinction between axiomatic demonstration and analytic demonstration provides a basis for dealing with the question of mathematical explanations. One may speak of mathematical explanations in two different senses: mathematical explanations of mathematical facts and mathematical explanations of empirical facts. This chapter mainly deals with m...

The aim of this chapter is to give answer to the question: What is mathematics about? The chapter maintains that, being problem solving by the analytic method, mathematics is about objects which are hypotheses human beings make to solve mathematical problems. Therefore, mathematical objects exist only in the minds of the mathematicians who hypothes...

According to the heuristic view, philosophy is not essentially different from the sciences. Some people oppose this view, claiming that philosophy is a humanistic discipline which aims to make sense of ourselves and of our activities, contrary to the sciences, which have nothing to say about this. In this chapter it is argued that these claims are...

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,...

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...

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: t...

Contrary to the belief of several mathematicians that philosophy is irrelevant to mathematics, this chapter argues that philosophy is relevant to it, because it may expose the inadequacy of some basic mathematical concepts, it may provide an analysis of some basic mathematical concepts, and may help to formulate new rules of discovery. What is rele...

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 m...

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 th...

This monograph addresses the question of the increasing irrelevance of philosophy, which has seen scientists as well as philosophers concluding that philosophy is dead and has dissolved into the sciences. It seeks to answer the question of whether or not philosophy can still be fruitful and what kind of philosophy can be such. The author argues tha...

The nature of the scientific method has been a main concern of philosophy from Plato to Mill. In that period logic has been considered to be a part of the methodology of science. Since Mill, however, the situation has completely changed. Logic has ceased to be a part of the methodology of science, and no Discourse on method has been written. Both l...

This is a review of the book by P. Garavaso and N.Vassallo, Frege on Thinking and Its Epistemic Significance. In it I point out that the book does not discuss Frege’s logicist project or his philosophy of language but rather Frege’s views on thinking, independently of an analysis of the whole of his logic or his views on sense and reference. The Au...

This article describes and compares four models of science: the analytic-synthetic model, the hypothetico-deductive model, the semantic model, and the analytic model. It also briefly discusses to what extent each of these models is capable of accounting for models in science.

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’evoluzio...

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 ha...

The view that the subject matter of epistemology is the concept of knowledge is faced with the problem that all attempts so far to define that concept are subject to counterexamples. As an alternative, this article argues that the subject matter of epistemology is knowledge itself rather than the concept of knowledge. Moreover, knowledge is not mer...

Three decades ago Laudan posed the challenge: Why should the logic of discovery be revived? This paper tries to answer this question arguing that the logic of discovery should be revived, on the one hand, because, by Gödel’s second incompleteness theorem, mathematical logic fails to be the logic of justification, and only reviving the logic of disc...

From antiquity several philosophers have claimed that the goal of natural science is truth. In particular, this is a basic tenet of contemporary scientific realism. However, all concepts of truth that have been put forward are inadequate to modern science because they do not provide a criterion of truth. This means that we will generally be unable...

In a very influential paper Rota stresses the relevance of mathematical beauty to mathematical research, and claims that a piece of mathematics is beautiful when it is enlightening. He stops short, however, of explaining what he means by ‘enlightening’. This paper proposes an alternative approach, according to which a mathematical demonstration or...

KreiselGeorg and TakeutiGaisi. Formally self-referential propositions for cut free analysis and related systems. Dissertationes mathematicae (Rozprawy matematyczne), no. 118, Polska Akademia Nauk, Instytut Matematyczny, Warsaw1974, 50 pp. PäppinghausPeter. A version of the Σ1-reflection principle for CFA provable in PRA. Archiv für mathematische Lo...

This paper discusses the approaches of Frege, Nagel, Hanna and Cooper to reason, logic and their relationship, it points out their limitations and outlines an alternative approach hopefully not subject to those limitations.

Can philosophy still be fruitful, and what kind of philosophy can be such? In particular, what kind of philosophy can be legitimized in the face of sciences? The aim of this paper is to answer these questions, listing the characteristics philosophy should have to be fruitful and legitimized in the face of sciences. Since the characteristics in ques...

The paper distinguishes between two kinds of mathematics, natural mathematics which is a result of biological evolution and artificial mathematics which is a result of cultural evolution. On this basis, it outlines an approach to the philosophy of mathematics which involves a new treatment of the method of mathematics, the notion of demonstration,...

This chapter examines the relation of logic to method and knowledge. Let us first consider that of logic to method. Method is a means of solving problems, thus a means of discovery. There are various kinds of methods, but an important distinction is that between algorithmic and heuristic methods. While algorithmic methods guarantee to solve problem...

Despite the limitations of Frege’s and Gentzen’s analysis of deduction, throughout the twentieth century mathematical logic has been extolled, and the importance of Aristotle’s logic downplayed. For example, Russell states that mathematical logic gives thought “wings. It has, in my opinion, introduced the same kind of advance into philosophy as Gal...

Frege’s view, that there cannot be a logic of discovery but only a logic of justification based on deduction, and that the goal of logic is the study of deduction, has had a deep impact on the relation of logic to method. Such relation was a very strict one at the origin of logic and from the sixteenth to the first half of the nineteenth century. B...

Logic is generally considered to be the study of the principles of valid inference, thus a subject, like papyrology or botany. But logic is not only that, it is also that problem solving capacity that virtually all organisms have as a result of biological evolution, and without which they could not survive. Thus logic is not only a subject, but als...

As it has been mentioned in Chapter 4, Aristotle does not completely dismiss the analytic method but replaces it with the analytic-synthetic method. The reason Aristotle gives for this replacement is that the analytic method “does not allow one to know anything in an absolute way, but only on the basis of a hypothesis.” The latter is not necessaril...

When dealing with logic, the following three questions naturally arise:

In Chapter 10 it has been argued that Frege’s analysis of deduction does not achieve his ideal of atomizing deduction. A better approximation to this ideal is provided by Gentzen’s analysis of deduction. In order to describe it, we need to fix some terminology and notation about first-order languages.

On the basis of what has been said in the previous two chapters, this chapter examines the relation of logic to evolution, language and reason, Chapter 17 the relation of logic to method and knowledge, and Chapter 19 the relation of philosophy to knowledge.

As we have seen in Chapter 2, Aristotle claims to be the originator of logic as a subject. But what is Aristotle’s logic? There are two alternative views, the deductivist view and the heuristic view. This chapter deals with the former, Chapter 7 with the latter.

This chapter examines the relation of philosophy to knowledge. Such relation has been a fairly critical one since the Scientific Revolution of the seventeenth century, because science has invaded several areas which were traditionally part of philosophy, thus making the latter problematic and in need of legitimation. In fact, a great deal of philos...

While, throughout the seventeenth and eighteenth century, the quest for a logic of discovery is a live question, the situation essentially changes with Frege. Although strongly influenced by Kant, Frege excludes induction and analogy from the domain of logic. For him, there cannot be a logic of discovery but only a logic of justification based on d...

In Chapter 2 it has been stated that Aristotle’s choice of the title ta analutika was intended to suggest a connection of Aristotle’s Analytics with the analytic method of Hippocrates of Chios and Hippocrates of Cos. In fact the analytic method, as formulated by Plato, and the Greek practice of discussion in the agora, in courts or in political deb...

In Chapters 4 and 17, deductive and non-deductive rules were distinguished in terms of the fact that non-deductive rules are ampliative while deductive rules are non-ampliative.

It is widely held that the core of the Scientific Revolution of the seventeenth century was a revolutionary change in the scientific method which involved a break with Aristotle. Thus Cohen states that Galileo earns the title of “founder of the scientific method of inquiry.” This is unjustified, because Galileo’s method is really Aristotle’s analyt...

In this final part of the book several rules of discovery are considered, that is, non-deductive rules for finding hypotheses to solve problems. Of course, finding hypotheses is not a sufficient condition for discovery. The latter requires hypotheses to be plausible, and the plausibility test procedure of Chapter 4 involves operations beyond simple...

In Chapter 20 two of the oldest and better known rules of discovery have been considered: induction and analogy.

While not involving a break with Aristotle as regards the scientific method, the Scientific Revolution of the seventeenth century gave rise to a considerable debate over the nature of logic. In particular, it led to a quest for a logic of discovery. The most important thinkers of the age took part in such a debate. Some of them considered a logic o...

After considering the nature of reason and knowledge, this chapter examines the relation of reason and knowledge to emotion.

In Chapter 2 we have seen that, for Parmenides, Plato and Aristotle, logic is discursive thinking but, in order to acquire knowledge of the universe, the human mind also needs intuitive thinking, which is the highest faculty.

The limitations of mathematical logic and the divorce of logic from method suggest that a different approach to logic is necessary, based on an alternative logic paradigm.

As it has been said in Chapter 3, Plato’s conception of science is based on the view that the analytic method is the scientific method.

This volume examines the limitations of mathematical logic and proposes a new approach to logic intended to overcome them. To this end, the book compares mathematical logic with earlier views of logic, both in the ancient and in the modern age, including those of Plato, Aristotle, Bacon, Descartes, Leibniz, and Kant. From the comparison it is appar...