Philosophy of mathematics


Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship to other areas of philosophy, particularly epistemology and metaphysics. Central questions posed include whether or not mathematical objects are purely abstract entities or are in some way concrete, and in what the relationship such objects have with physical reality consists.
Major themes that are dealt with in philosophy of mathematics include:
  • Reality: The question is whether mathematics is a pure product of human mind or whether it has some reality by itself.
  • Logic and rigor
  • Relationship with physical reality
  • Relationship with science
  • Relationship with applications
  • Mathematical truth
  • ''Nature as human activity''

    Major themes

Reality

Logic and rigor

Mathematical reasoning requires rigor. This means that the definitions must be absolutely unambiguous and the proofs must be reducible to a succession of applications of syllogisms or inference rules, without any use of empirical evidence and intuition.
The rules of rigorous reasoning have been established by the ancient Greek philosophers under the name of logic. Logic is not specific to mathematics, but, in mathematics, the standard of rigor is much higher than elsewhere.
For many centuries, logic, although used for mathematical proofs, belonged to philosophy and was not specifically studied by mathematicians. Circa the end of the 19th century, several paradoxes made questionable the logical foundation of mathematics, and consequently the validity of the whole of mathematics. This has been called the foundational crisis of mathematics. Some of these paradoxes consist of results that seem to contradict the common intuition, such as the possibility to construct valid non-Euclidean geometries in which the parallel postulate is wrong, the Weierstrass function that is continuous but nowhere differentiable, and the study by Georg Cantor of infinite sets, which led to consider several sizes of infinity. Even more striking, Russell's paradox shows that the phrase "the set of all sets" is self contradictory.
Several methods have been proposed to solve the problem by changing of logical framework, such as constructive mathematics and intuitionistic logic. Roughly speaking, the first one consists of requiring that every existence theorem must provide an explicit example, and the second one excludes from mathematical reasoning the law of excluded middle and double negation elimination.
These logics have less inference rules than classical logic. On the other hand classical logic was a first-order logic, which means roughly that quantifiers cannot be applied to infinite sets. This means, for example that the sentence "every set of natural numbers has a least element" is nonsensical in any formalization of classical logic. This led to the introduction of higher-order logics, which are presently used commonly in mathematics.
The problems of foundation of mathematics has been eventually resolved with the rise of mathematical logic as a new area of mathematics. In this framework, a mathematical or logical theory consists of a formal language that defines the well-formed assertions, a set of basic assertions called axioms and a set of inference rules that allow producing new assertions from one or several known assertions. A theorem of such a theory is either an axiom or an assertion that can be obtained from previously known theorems by the application of an inference rule. The Zermelo–Fraenkel set theory with the axiom of choice, generally called ZFC, is a higher-order logic in which all mathematics have been restated; it is used implicitly in all mathematics texts that do not specify explicitly on which foundations they are based. Moreover, the other proposed foundations can be modeled and studied inside ZFC.
It results that "rigor" is no more a relevant concept in mathematics, as a proof is either correct or erroneous, and a "rigorous proof" is simply a pleonasm. Where a special concept of rigor comes into play is in the socialized aspects of a proof. In particular, proofs are rarely written in full details, and some steps of a proof are generally considered as trivial, easy, or straightforward, and therefore left to the reader. As most proof errors occur in these skipped steps, a new proof requires to be verified by other specialists of the subject, and can be considered as reliable only after having been accepted by the community of the specialists, which may need several years.
Also, the concept of "rigor" may remain useful for teaching to beginners what is a mathematical proof.

Relationship with sciences

Mathematics is used in most sciences for modeling phenomena, which then allows predictions to be made from experimental laws. The independence of mathematical truth from any experimentation implies that the accuracy of such predictions depends only on the adequacy of the model. Inaccurate predictions, rather than being caused by invalid mathematical concepts, imply the need to change the mathematical model used. For example, the perihelion precession of Mercury could only be explained after the emergence of Einstein's general relativity, which replaced Newton's law of gravitation as a better mathematical model.
There is still a philosophical debate whether mathematics is a science. However, in practice, mathematicians are typically grouped with scientists, and mathematics shares much in common with the physical sciences. Like them, it is falsifiable, which means in mathematics that if a result or a theory is wrong, this can be proved by providing a counterexample. Similarly as in science, theories and results are often obtained from experimentation. In mathematics, the experimentation may consist of computation on selected examples or of the study of figures or other representations of mathematical objects. For example, when asked how he came about his theorems, Gauss once replied "durch planmässiges Tattonieren". However, some authors emphasize that mathematics differs from the modern notion of science by not on empirical evidence.

Unreasonable effectiveness

The unreasonable effectiveness of mathematics is a phenomenon that was named and first made explicit by physicist Eugene Wigner. It is the fact that many mathematical theories have applications outside their initial object. These applications may be completely outside their initial area of mathematics, and may concern physical phenomena that were completely unknown when the mathematical theory was introduced. Examples of unexpected applications of mathematical theories can be found in many areas of mathematics.
A notable example is the prime factorization of natural numbers that was discovered more than 2,000 years before its common use for secure internet communications through the RSA cryptosystem. A second historical example is the theory of ellipses. They were studied by the ancient Greek mathematicians as conic sections. It was almost 2,000 years later that Johannes Kepler discovered that the trajectories of the planets are ellipses.
In the 19th century, the internal development of geometry led to definition and study of non-Euclidean geometries, spaces of dimension higher than three and manifolds. At this time, these concepts seemed totally disconnected from the physical reality, but at the beginning of the 20th century, Albert Einstein developed the theory of relativity that uses fundamentally these concepts. In particular, spacetime of special relativity is a non-Euclidean space of dimension four, and spacetime of general relativity is a manifold of dimension four.
A striking aspect of the interaction between mathematics and physics is when mathematics drives research in physics. This is illustrated by the discoveries of the positron and the baryon In both cases, the equations of the theories had unexplained solutions, which led to conjecture of the existence of an unknown particle and the search for these particles. In both cases, these particles were discovered a few years later by specific experiments.

History

The origin of mathematics is of arguments and disagreements. Whether the birth of mathematics was by chance or induced by necessity during the development of similar subjects, such as physics, remains an area of contention.
A perennial issue in the philosophy of mathematics concerns the relationship between logic and mathematics at their joint foundations. While 20th-century philosophers continued to ask the questions mentioned at the outset of this article, the philosophy of mathematics in the 20th century was characterized by a predominant interest in formal logic, set theory, and foundational issues.
It is a profound puzzle that on the one hand mathematical truths seem to have a compelling inevitability, but on the other hand the source of their "truthfulness" remains elusive. Investigations into this issue are known as the foundations of mathematics program.
At the start of the 20th century, philosophers of mathematics were already beginning to divide into various schools of thought about all these questions, broadly distinguished by their pictures of mathematical epistemology and ontology. Three schools, formalism, intuitionism, and logicism, emerged at this time, partly in response to the increasingly widespread worry that mathematics as it stood, and analysis in particular, did not live up to the standards of certainty and rigor that had been taken for granted. Each school addressed the issues that came to the fore at that time, either attempting to resolve them or claiming that mathematics is not entitled to its status as our most trusted knowledge.
Surprising and counter-intuitive developments in formal logic and set theory early in the 20th century led to new questions concerning what was traditionally called the foundations of mathematics. As the century unfolded, the initial focus of concern expanded to an open exploration of the fundamental axioms of mathematics, the axiomatic approach having been taken for granted since the time of Euclid around 300 BCE as the natural basis for mathematics. Notions of axiom, proposition and proof, as well as the notion of a proposition being true of a mathematical object, were formalized, allowing them to be treated mathematically. The Zermelo–Fraenkel axioms for set theory were formulated which provided a conceptual framework in which much mathematical discourse would be interpreted. In mathematics, as in physics, new and unexpected ideas had arisen and significant changes were coming. With Gödel numbering, propositions could be interpreted as referring to themselves or other propositions, enabling inquiry into the consistency of mathematical theories. This reflective critique in which the theory under review "becomes itself the object of a mathematical study" led Hilbert to call such study metamathematics or proof theory.
At the middle of the century, a new mathematical theory was created by Samuel Eilenberg and Saunders Mac Lane, known as category theory, and it became a new contender for the natural language of mathematical thinking. As the 20th century progressed, however, philosophical opinions diverged as to just how well-founded were the questions about foundations that were raised at the century's beginning. Hilary Putnam summed up one common view of the situation in the last third of the century by saying:

When philosophy discovers something wrong with science, sometimes science has to be changed—Russell's paradox comes to mind, as does Berkeley's attack on the actual infinitesimal—but more often it is philosophy that has to be changed. I do not think that the difficulties that philosophy finds with classical mathematics today are genuine difficulties; and I think that the philosophical interpretations of mathematics that we are being offered on every hand are wrong, and that "philosophical interpretation" is just what mathematics doesn't need.

Philosophy of mathematics today proceeds along several different lines of inquiry, by philosophers of mathematics, logicians, and mathematicians, and there are many schools of thought on the subject. The schools are addressed separately in the next section, and their assumptions explained.