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On the other hand, it has turned out that to some extent it is possible to bring mathematical methods to bear on philosophical questions concerning mathematics.

The setting in which this has been done is that of mathematical logic when it is broadly conceived as comprising proof theory, model theory, set theory, and computability theory as subfields. Thus the twentieth century has witnessed the mathematical investigation of the consequences of what are at bottom philosophical theories concerning the nature of mathematics.

When professional mathematicians are concerned with the foundations of their subject, they are said to be engaged in foundational research. When professional philosophers investigate philosophical questions concerning mathematics, they are said to contribute to the philosophy of mathematics. Of course the distinction between the philosophy of mathematics and the foundations of mathematics is vague, and the more interaction there is between philosophers and mathematicians working on questions pertaining to the nature of mathematics, the better.

The general philosophical and scientific outlook in the nineteenth century tended toward the empirical: platonistic aspects of rationalistic theories of mathematics were rapidly losing support. Especially the once highly praised faculty of rational intuition of ideas was regarded with suspicion. Thus it became a challenge to formulate a philosophical theory of mathematics that was free of platonistic elements.

In the first decades of the twentieth century, three non-platonistic accounts of mathematics were developed: logicism, formalism, and intuitionism. There emerged in the beginning of the twentieth century also a fourth program: predicativism. Due to contingent historical circumstances, its true potential was not brought out until the s. However it deserves a place beside the three traditional schools that are discussed in most standard contemporary introductions to philosophy of mathematics, such as Shapiro and Linnebo The logicist project consists in attempting to reduce mathematics to logic.

Since logic is supposed to be neutral about matters ontological, this project seemed to harmonize with the anti-platonistic atmosphere of the time.

The idea that mathematics is logic in disguise goes back to Leibniz. But an earnest attempt to carry out the logicist program in detail could be made only when in the nineteenth century the basic principles of central mathematical theories were articulated by Dedekind and Peano and the principles of logic were uncovered by Frege. Frege devoted much of his career to trying to show how mathematics can be reduced to logic Frege He managed to derive the principles of second-order Peano arithmetic from the basic laws of a system of second-order logic.

His derivation was flawless. However, he relied on one principle which turned out not to be a logical principle after all. Even worse, it is untenable.

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Russell himself then tried to reduce mathematics to logic in another way. So Russell postulated that only properties of mathematical objects that have already been shown to exist, determine classes. Predicates that implicitly refer to the class that they were to determine if such a class existed, do not determine a class. Thus a typed structure of properties is obtained: properties of ground objects, properties of ground objects and classes of ground objects, and so on. This typed structure of properties determines a layered universe of mathematical objects, starting from ground objects, proceeding to classes of ground objects, then to classes of ground objects and classes of ground objects, and so on.

Unfortunately, Russell found that the principles of his typed logic did not suffice for deducing even the basic laws of arithmetic. He needed, among other things, to lay down as a basic principle that there exists an infinite collection of ground objects. This could hardly be regarded as a logical principle. Thus the second attempt to reduce mathematics to logic also faltered. And there matters stood for more than fifty years. In it, Wright breathes new life into the logicist project.

This relation of one-to-one correspondence can be expressed in second-order logic. In particular, Basic Law V is not needed in the second part of the derivation. If that is so, then at least second-order Peano arithmetic is reducible to logic alone.

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These principles are called abstraction principles. At present, philosophers of mathematics attempt to construct general theories of abstraction principles that explain which abstraction principles are acceptable and which are not, and why Weir ; Fine But these weak background theories only allow very weak arithmetical theories to be derived from Basic Law V Burgess Intuitionism originates in the work of the mathematician L. Brouwer van Atten , and it is inspired by Kantian views of what objects are Parsons , chapter 1. According to intuitionism, mathematics is essentially an activity of construction.

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The natural numbers are mental constructions, the real numbers are mental constructions, proofs and theorems are mental constructions, mathematical meaning is a mental construction… Mathematical constructions are produced by the ideal mathematician, i. But even the ideal mathematician remains a finite being.

She can never complete an infinite construction, even though she can complete arbitrarily large finite initial parts of it. This entails that intuitionism resolutely rejects the existence of the actual or completed infinite; only potentially infinite collections are given in the activity of construction. A basic example is the successive construction in time of the individual natural numbers. From these general considerations about the nature of mathematics, based on the condition of the human mind Moore , intuitionists infer to a revisionist stance in logic and mathematics.

They find non-constructive existence proofs unacceptable. Non-constructive existence proofs are proofs that purport to demonstrate the existence of a mathematical entity having a certain property without even implicitly containing a method for generating an example of such an entity. The characteristic feature of non-constructive existence proofs is that they make essential use of the principle of excluded third. In classical logic, these principles are valid. The logic of intuitionistic mathematics is obtained by removing the principle of excluded third and its equivalents from classical logic.

This of course leads to a revision of mathematical knowledge. For instance, the classical theory of elementary arithmetic, Peano Arithmetic , can no longer be accepted. Instead, an intuitionistic theory of arithmetic called Heyting Arithmetic is proposed which does not contain the principle of excluded third. Although intuitionistic elementary arithmetic is weaker than classical elementary arithmetic, the difference is not all that great.

There exists a simple syntactical translation which translates all classical theorems of arithmetic into theorems which are intuitionistically provable.

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In the first decades of the twentieth century, parts of the mathematical community were sympathetic to the intuitionistic critique of classical mathematics and to the alternative that it proposed. This situation changed when it became clear that in higher mathematics, the intuitionistic alternative differs rather drastically from the classical theory. For instance, intuitionistic mathematical analysis is a fairly complicated theory, and it is very different from classical mathematical analysis. This dampened the enthusiasm of the mathematical community for the intuitionistic project.

David Hilbert agreed with the intuitionists that there is a sense in which the natural numbers are basic in mathematics.

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But unlike the intuitionists, Hilbert did not take the natural numbers to be mental constructions. Instead, he argued that the natural numbers can be taken to be symbols. Symbols are strictly speaking abstract objects. Nonetheless, it is essential to symbols that they can be embodied by concrete objects, so we may call them quasi-concrete objects Parsons , chapter 1.

Perhaps physical entities could play the role of the natural numbers. For instance, we may take a concrete ink trace of the form to be the number 0, a concretely realized ink trace to be the number 1, and so on. Hilbert thought it doubtful at best that higher mathematics could be directly interpreted in a similarly straightforward and perhaps even concrete manner.

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Unlike the intuitionists, Hilbert was not prepared to take a revisionist stance toward the existing body of mathematical knowledge. Instead, he adopted an instrumentalist stance with respect to higher mathematics. He thought that higher mathematics is no more than a formal game. The statements of higher-order mathematics are uninterpreted strings of symbols. Proving such statements is no more than a game in which symbols are manipulated according to fixed rules. Hilbert thought that there can be no reasonable doubt about the soundness of classical Peano Arithmetic — or at least about the soundness of a subsystem of it that is called Primitive Recursive Arithmetic Tait And he thought that every arithmetical statement that can be proved by making a detour through higher mathematics, can also be proved directly in Peano Arithmetic.

In fact, he strongly suspected that every problem of elementary arithmetic can be decided from the axioms of Peano Arithmetic. Of course solving arithmetical problems in arithmetic is in some cases practically impossible. Hilbert realized, albeit somewhat dimly, that some of his convictions can actually be considered to be mathematical conjectures.

For a proof in a formal system of higher mathematics or of elementary arithmetic is a finite combinatorial object which can, modulo coding, be considered to be a natural number. But in the s the details of coding proofs as natural numbers were not yet completely understood. On the formalist view, a minimal requirement of formal systems of higher mathematics is that they are at least consistent.

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Otherwise every statement of elementary arithmetic can be proved in them. Hilbert also saw again, dimly that the consistency of a system of higher mathematics entails that this system is at least partially arithmetically sound. So Hilbert and his students set out to prove statements such as the consistency of the standard postulates of mathematical analysis. Otherwise the proof does not increase our conviction in the consistency of mathematical analysis.

And, fortunately, it seemed possible in principle to do this, for in the final analysis consistency statements are, again modulo coding, arithmetical statements. So, to be precise, Hilbert and his students set out to prove the consistency of, e. It turned out to be more difficult than they had expected. In fact, they did not even succeed in proving the consistency of the axioms of Peano Arithmetic in Peano Arithmetic.