Foundations of mathematics: Difference between revisions

For the book by Hilbert and Bernays, see Grundlagen der Mathematik.

Foundations of mathematics is the study of the basic mathematical concepts (number, geometrical figure, set, function...) and how they form hierarchies of more complex structures and concepts, especially the fundamentally important structures that form the language of mathematics (formulas, theories and their models giving a « meaning » to formulas, definitions, proofs, algorithms...) also called metamathematical concepts, with an eye to the philosophical aspects and the unity of mathematics. The search for foundations of mathematics is a central question of the philosophy of mathematics; the abstract nature of mathematical objects presents special philosophical challenges.

But the foundations of mathematics as a whole does not aim to contain the foundations of every mathematical topic. Generally, the foundations of a field of study, refers to a more-or-less systematic analysis of its most basic or fundamental concepts, its conceptual unity and its natural ordering or hierarchy of concepts, which may help to connect it with the rest of human knowledge. But the development, emergence and clarification of the foundations can come late in the history of a field, and may not be its most interesting part.

Mathematics always played a special role in scientific thought, serving since ancient times as a model of truth and rigor for rational inquiry, and giving tools or even a foundation for other sciences (especially physics). But the many developments of mathematics towards higher abstractions in the 19th century, brought new challenges and paradoxes, urging for a deeper and more systematic examination of the nature and criteria of mathematical truth, as well as a unification of the diverse branches of mathematics into a coherent whole.

The systematic search for the foundations of mathematics started at the end of the 19th century, and formed a new mathematical discipline called mathematical logic, with strong links to theoretical computer science. It went through a series of crisis with paradoxical results, until the discoveries stabilized during the 20th century as a large and coherent body of mathematical knowledge with several aspects or components (set theory, model theory, proof theory...), whose detailed properties and possible variants are still an active research field. Its high level of technical sophistication inspired many philosophers to conjecture that it can serve as a model or pattern for the foundations of other sciences.

Historical context

See also : History of logic and History of mathematics.

Ancient Greek mathematics

While the practice of mathematics previously developed in other civilizations, the special interests for its theoretical and foundational aspects really started with Ancient Greeks. Early Greek philosophers disputed as to which is more basic, arithmetic or geometry. Zeno of Elea (490 BC – ca. 430 BC) produced four paradoxes that seem to show that change is impossible.

The Pythagorean school of mathematics originally insisted that only natural and rational numbers exist. The discovery of the [[irrational number|irrationality] of √2, the ratio of the diagonal of a square to its side (around 5th century BC), was a shock to them which they only reluctantly accepted. The discrepancy between rationals and reals was finally resolved by Eudoxus of Cnidus, a student of Plato, who reduced the comparison of irrational ratios to comparisons of multiples (rational ratios), thus anticipating Richard Dedekind's definition of real numbers.

In the Posterior Analytics, Aristotle (384 BC – 322 BC) laid down the axiomatic method, to organize a field of knowledge logically by means of primitive concepts, axioms, postulates, definitions, and theorems, taking a majority of his examples from arithmetic and geometry. This method reached its high point with Euclid's Elements (300 BC), monumental treatise on geometry structured with very high standards of rigor : each proposition is justified by a demonstration in the form of chains of syllogisms (though they do not always conform strictly to Aristotelean templates). Aristotle's syllogistic logic, together with the Axiomatic Method exemplified by Euclid's Elements, are universally recognized as towering scientific achievements of ancient Greece.

Platonism as a traditional philosophy of mathematics

The objects of mathematics are abstract and remote from everyday perceptual experience : geometrical figures are conceived as idealities to be distinguished from effective drawings and shapes of objects, and numbers are not confused with the counting of concrete objects. Their existence and nature present special philosophical challenges: How do mathematical objects differ from their concrete representation ? Are they located in their representation, or in our minds, or somewhere else ? How can we know them ?

The ancient Greek philosophers took such questions very seriously. Indeed, many of their general philosophical discussions were carried on with extensive reference to geometry and arithmetic. Plato (424/423 BC – 348/347 BC) insisted that mathematical objects, like other platonic Ideas (forms or essences), must be perfectly abstract and have a separate, non-material kind of existence, in a world of mathematical objects independent of humans. He believed that the truths about these objects also exists independently of the human mind, but is discovered by humans. In the Meno Plato’s teacher Socrates asserts that it is possible to come to know this truth by a process akin to memory retrieval.

Above the gateway to Plato's academy appeared a famous inscription:

Let no one who is ignorant of geometry enter here.

In this way Plato indicated his high opinion of geometry. He regarded geometry as ``the first essential in the training of philosophers", because of its abstract character.

This philosophy of Platonist mathematical realism, is shared by many mathematicians:

"Platonists, such as Kurt Gödel (1906–1978), hold that numbers are abstract, necessarily existing objects, independent of the human mind"

It can be argued that Platonism somehow comes as a necessary assumption underlying any mathematical work.

In this view, the laws of nature and the laws of mathematics have a similar status, and the effectiveness ceases to be unreasonable. Not our axioms, but the very real world of mathematical objects forms the foundation.

Aristotle dissected and rejected this view in his Metaphysics. These questions provide much food for philosophical analysis and debate.

Middle Ages and Renaissance

For over 2,000 years, Euclid’s Elements stood as a perfectly solid foundation for mathematics, as its methodology of rational exploration guided mathematicians, philosophers, and scientists well into the 19th century.

The Middle Ages saw a dispute over the ontological status of the universals (platonic Ideas): Realism asserted their existence independently of perception; conceptualism asserted their existence within the mind only; nominalism, denied either, only seeing universals as names of collections of individual objects (following older speculations that they are words, "logos").

René Descartes published La Géométrie (1637) aimed to reduce geometry to algebra by means of coordinate systems, giving algebra a more foundational role (while the Greeks embedded arithmetic into geometry by identifying whole numbers with evenly spaced points on a line). It became famous after 1649 and pathed the way to infinitesimal calculus.

Isaac Newton (1642 – 1727) in England and Leibniz (1646 – 1716) in Germany independently developed the infinitesimal calculus based on heuristic methods greatly efficient, but direly lacking rigorous justifications. Leibniz even went on to explicitly describe infinitesimals as actual infinitely small numbers (close to zero). Leibniz also worked on formal logic but most of his writings on it remained unpublished until 1903.

The Christian philosopher George Berkeley (1685–1753), in his campaign against the religious implications of Newtonian mechanics, wrote a pamphlet on the lack of rational justifications of infinitesimal calculus : “They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?”

Then mathematics developed very rapidly and successfully in physical applications, but with little attention to logical foundations.

19th century

In the 19th century, mathematics became increasingly abstract. Concerns about logical gaps and inconsistencies in different fields led to the development of axiomatic systems.

Real Analysis

Cauchy (1789 – 1857) started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner, rejecting the heuristic principle of the generality of algebra exploited by earlier authors. In 1821 (Cours d'Analyse) he defines infinitely small quantities as decreasing sequences that converge to 0, which he then used to define continuity. But he did not formalize his notion of convergence.

The modern (ε, δ)-definition of limit and continuous functions was first developed by Bolzano in 1817, but remained relatively unknown. It gives a rigorous foundation of infinitesimal calculus based on the set of real numbers, clearly resolving the Zeno paradoxes and Berkeley's arguments

Mathematicians such as Karl Weierstrass (1815 – 1897) discovered pathological functions such as continuous, nowhere-differentiable functions. Previous conceptions of a function as a rule for computation, or a smooth graph, were no longer adequate. Weierstrass began to advocate the arithmetization of analysis, to axiomatize analysis using properties of the natural numbers. In 1858, Dedekind proposed a definition of the real numbers as cuts of rational numbers. This reduction of real numbers and continuous functions in terms of rational numbers and thus of natural numbers, was later integrated by Cantor in his set theory, and axiomatized in terms of second order arithmetic by Hilbert and Bernays.

Group theory

For the first time, the limits of mathematics were explored. Niels Henrik Abel (1802 – 1829), a Norwegian, and Évariste Galois, (1811 – 1832) a Frenchman, investigated into the solutions of various polynomial equations, and proved that there is no general algebraic solution to equations of degree greater than four (Abel–Ruffini theorem). With these concepts, Pierre Wantzel (1837) proved that straightedge and compass alone cannot trisect an arbitrary angle nor double a cube, nor to construct a square equal in area to a given circle. Mathematicians had vainly attempted to solve all of these problems since the time of the ancient Greeks.

Abel and Galois's works opened the way for the developments of group theory (which will be used to study symmetry in physics and other fields), and abstract algebra. Concepts of vector spaces emerged from the conception of barycentric coordinates by Möbius in 1827, to the modern definition of vector spaces and linear maps by Peano in 1888. Geometry was no more limited to 3 dimensions. These concepts do not generalize numbers but combine notions of functions and sets which are not yet formalized, breaking away from familiar mathematical objects.

Non-Euclidean Geometries

After many failed attempts to derive the parallel postulate from other axioms, the study of the still hypothetical hyperbolic geometry by Johann Heinrich Lambert (1728 – 1777) led him to introduce the hyperbolic functions and compute the area of a hyperbolic triangle (where the sum of angles is less than 180°). Then the Russian mathematician Nikolai Lobachevsky (1792–1856) established in 1826 (and published in 1829) the coherence of this geometry (thus the independence of the parallel postulate), in parallel with the Hungary mathematician János Bolyai (1802–60) in 1832, and with Gauss. Later in the 19th century, the German mathematician Bernhard Riemann developed Elliptic geometry, another non-Euclidean geometry where no parallel can be found and the sum of angles in a triangle is more than 180°. It was proved consistent by defining point to mean a pair of antipodal points on a fixed sphere and line to mean a great circle on the sphere. At that time, the main method for proving the consistency of a set of axioms was to provide a model for it.

Projective geometry

One of the traps in a deductive system is circular reasoning, a problem that seemed to befall projective geometry until it was resolved by Karl von Staudt. As explained by Laptev & Rosenfeld (1996):

In the mid-nineteenth century there was an acrimonious controversy between the proponents of synthetic and analytic methods in projective geometry, the two sides accusing each other of mixing projective and metric concepts. Indeed the basic concept that is applied in the synthetic presentation of projective geometry, the cross-ratio of four points of a line, was introduced through consideration of the lengths of intervals.

The purely geometric approach of von Staudt was based on the complete quadrilateral to express the relation of projective harmonic conjugates. Then he created a means of expressing the familiar numeric properties with his Algebra of Throws. English language versions of this process of deducing the properties of a field can be found in either the book by Oswald Veblen and John Young, Projective Geometry (1938), or more recently in John Stillwell's Four Pillars of Geometry (2005). Stillwell writes on page 120

...projective geometry is simpler than algebra in a certain sense, because we use only five geometric axioms to derive the nine field axioms.

The algebra of throws is commonly seen as a feature of cross-ratios since students ordinarily rely upon numbers without worry about their basis. However, cross-ratio calculations use metric features of geometry, features not admitted by purists. For instance, in 1961 Coxeter wrote Introduction to Geometry without mention of cross-ratio.

Foundational crisis

The foundational crisis of mathematics (in German: Grundlagenkrise der Mathematik) was the early 20th century's term for the search for proper foundations of mathematics.

After several schools of the philosophy of mathematics ran into difficulties one after the other in the 20th century, the assumption that mathematics had any foundation that could be stated within mathematics itself began to be heavily challenged.

One attempt after another to provide unassailable foundations for mathematics was found to suffer from various paradoxes (such as Russell's paradox) and to be inconsistent: an undesirable situation in which every mathematical statement that can be formulated in a proposed system (such as 2 + 2 = 5) can also be proved in the system.

Various schools of thought on the right approach to the foundations of mathematics were fiercely opposing each other. The leading school was that of the formalist approach, of which David Hilbert was the foremost proponent, culminating in what is known as Hilbert's program, which thought to ground mathematics on a small basis of a logical system proved sound by metamathematical finitistic means. The main opponent was the intuitionist school, led by L. E. J. Brouwer, which resolutely discarded formalism as a meaningless game with symbols (van Dalen, 2008). The fight was acrimonious. In 1920 Hilbert succeeded in having Brouwer, whom he considered a threat to mathematics, removed from the editorial board of Mathematische Annalen, the leading mathematical journal of the time.

Philosophical views

Main article: Philosophy of mathematics

Set-theoretical Platonism

Many researchers in axiomatic set theory have subscribed to what is known as set-theoretical Platonism, exemplified by mathematician Kurt Gödel.

Formalism

Main article: Formalism (mathematics)

It has been claimed that "Formalists, such as David Hilbert (1862–1943), hold that mathematics is no more or less than mathematical language. It is simply a series of games...". Indeed he used the words "formula game" in his 1927 response to L. E. J. Brouwer's criticisms:

"And to what has the formula game thus made possible been successful? This formula game enables us to express the entire thought-content of the science of mathematics in a uniform manner and develop it in such a way that, at the same time, the interconnections between the individual propositions and facts become clear . . . The formula game that Brouwer so deprecates has, besides its mathematical value, an important general philosophical significance. For this formula game is carried out according to certain definite rules, in which the technique of our thinking is expressed. These rules form a closed system that can be discovered and definitively stated.".

Thus Hilbert is insisting that mathematics is not an arbitrary game with arbitrary rules; rather it must agree with how our thinking, and then our speaking and writing, proceeds:".

"We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise".

The foundational philosophy of formalism, as exemplified by David Hilbert, is a response to the paradoxes of set theory, and is based on formal logic. Virtually all mathematical theorems today can be formulated as theorems of set theory. The truth of a mathematical statement, in this view, is represented by the fact that the statement can be derived from the axioms of set theory using the rules of formal logic.

Merely the use of formalism alone does not explain several issues: why we should use the axioms we do and not some others, why we should employ the logical rules we do and not some others, why do "true" mathematical statements (e.g., the laws of arithmetic) appear to be true, and so on. Hermann Weyl would ask these very questions of Hilbert:

"What "truth" or objectivity can be ascribed to this theoretic construction of the world, which presses far beyond the given, is a profound philosophical problem. It is closely connected with the further question: what impels us to take as a basis precisely the particular axiom system developed by Hilbert? Consistency is indeed a necessary but not a sufficient condition. For the time being we probably cannot answer this question . . .."

In some cases these questions may be sufficiently answered through the study of formal theories, in disciplines such as reverse mathematics and computational complexity theory. As noted by Weyl, Formal logical systems also run the risk of inconsistency; in Peano arithmetic, this arguably has already been settled with several proofs of consistency, but there is debate over whether or not they are sufficiently finitary to be meaningful. Gödel's second incompleteness theorem establishes that logical systems of arithmetic can never contain a valid proof of their own consistency. What Hilbert wanted to do was prove a logical system S was consistent, based on principles P that only made up a small part of S. But Gödel proved that the principles P could not even prove P to be consistent, let alone S!

Intuitionism

Main article: Intuitionism

"Intuitionists, such as L. E. J. Brouwer (1882–1966), hold that mathematics is a creation of the human mind. Numbers, like fairy tale characters, are merely mental entities, which would not exist if there were never any human minds to think about them."

The foundational philosophy of intuitionism or constructivism, as exemplified in the extreme by Brouwer and more coherently by Stephen Kleene, requires proofs to be "constructive" in nature – the existence of an object must be demonstrated rather than inferred from a demonstration of the impossibility of its non-existence. For example, as a consequence of this the form of proof known as reductio ad absurdum is suspect.

Some modern theories in the philosophy of mathematics deny the existence of foundations in the original sense. Some theories tend to focus on mathematical practice, and aim to describe and analyze the actual working of mathematicians as a social group. Others try to create a cognitive science of mathematics, focusing on human cognition as the origin of the reliability of mathematics when applied to the real world. These theories would propose to find foundations only in human thought, not in any objective outside construct. The matter remains controversial.

Logicism

Main article: Logicism

Logicism is one of the schools of thought in the philosophy of mathematics, putting forth the theory that mathematics is an extension of logic and therefore some or all mathematics is reducible to logic. Bertrand Russell and Alfred North Whitehead championed this theory fathered by Gottlob Frege.

More paradoxes

Gödel's incompleteness theorems, proved in 1931, showed that essential aspects of Hilbert's program could not be attained. In Gödel's first result he showed how to construct, for any sufficiently powerful and consistent recursively axiomatizable system – such as necessary to axiomatize the elementary theory of arithmetic on the (infinite) set of natural numbers – a statement that can be shown to be true, but is not provable by the system. It thus became clear that the notion of mathematical truth can not be reduced to a purely formal system as envisaged in Hilbert's program. In a next result Gödel showed that such a system was not powerful enough for proving its own consistency, let alone that a simpler system could do the job. This dealt a final blow to the heart of Hilbert's program, the hope that consistency could be established by finitistic means (it was never made clear exactly what axioms were the "finitistic" ones, but whatever axiomatic system was being referred to, it was a 'weaker' system than the system whose consistency it was supposed to prove). Meanwhile, the intuitionistic school had not attracted many adherents among working mathematicians, due to difficulties of constructive mathematics.

Partial resolution of the crisis

Most mathematicians either do not work from axiomatic systems, or if they do, do not doubt the consistency of ZFC, generally their preferred axiomatic system. In most of mathematics as it is practiced, the incompleteness of the underlying formal theories never played a role anyway, and in those branches in which they do (such as logic and category theory), they may be treated carefully. Toward the middle of the 20th century it turned out that set theory (ZFC or otherwise) was inadequate as a foundation for some of the emerging new fields, such as homological algebra, and category theory was proposed as an alternative foundation by Samuel Eilenberg and others.

See also

• Mathematical Logic
• Brouwer–Hilbert controversy
• Controversy over Cantor's theory
• Epistemology
• Euclid's Elements
• Liar paradox
• New Foundations
• Philosophy of mathematics
• Principia Mathematica
• Quasi-empiricism in mathematics
• Mathematical thought of Charles Peirce

References

• Anglin, W. S. (1996) [1994]. Mathematics. A Concise History and Philosophy (Corrected 2nd printing ed.). New York: Springer Verlag. ISBN 0-387-94280-7. Chapter 39 Foundations contains concise descriptions, for the 20th century, of Platonism (with respect to Gödel), Formalism (with respect to Hilbert), and Intuitionism (with respect to Brouwer).
• Avigad, Jeremy (2003) Number theory and elementary arithmetic, Philosophia Mathematica Vol. 11, pp. 257–284
• Eves, Howard (1990), Foundations and Fundamental Concepts of Mathematics Third Edition, Dover Publications, INC, Mineola NY, ISBN 0-486-69609-X (pbk.) cf §9.5 Philosophies of Mathematics pp. 266–271. Eves lists the three with short descriptions prefaced by a brief introduction.
• Goodman, N.D. (1979), "Mathematics as an Objective Science", in Tymoczko (ed., 1986).
• Hart, W.D. (ed., 1996), The Philosophy of Mathematics, Oxford University Press, Oxford, UK.
• Hersh, R. (1979), "Some Proposals for Reviving the Philosophy of Mathematics", in (Tymoczko 1986).
• Hilbert, D. (1922), "Neubegründung der Mathematik. Erste Mitteilung", Hamburger Mathematische Seminarabhandlungen 1, 157–177. Translated, "The New Grounding of Mathematics. First Report", in (Mancosu 1998).
• Katz, Robert (1964), Axiomatic Analysis, D. C. Heath and Company.
• Kleene, Stephen C. (1991) [1952]. Introduction to Meta-Mathematics (Tenth impression 1991 ed.). Amsterdam NY: North-Holland Pub. Co. ISBN 0-7204-2103-9.

In Chapter III A Critique of Mathematic Reasoning, §11. The paradoxes, Kleene discusses Intuitionism and Formalism in depth. Throughout the rest of the book he treats, and compares, both Formalist (classical) and Intuitionist logics with an emphasis on the former. Extraordinary writing by an extraordinary mathematician.

• Laptev, B.L. & B.A. Rozenfel'd (1996) Mathematics of the 19th Century: Geometry, page 40, Birkhäuser ISBN 3-7643-5048-2 .
• Mancosu, P. (ed., 1998), From Hilbert to Brouwer. The Debate on the Foundations of Mathematics in the 1920s, Oxford University Press, Oxford, UK.
• Putnam, Hilary (1967), "Mathematics Without Foundations", Journal of Philosophy 64/1, 5–22. Reprinted, pp. 168–184 in W.D. Hart (ed., 1996).
• Putnam, Hilary (1975), "What is Mathematical Truth?", in Tymoczko (ed., 1986).
• Sudac, Olivier (Apr 2001). "The prime number theorem is PRA-provable". Theoretical Computer Science. 257 (1–2): 185–239. doi:10.1016/S0304-3975(00)00116-X.{{cite journal}}: CS1 maint: year (link)
• Troelstra, A. S. (no date but later than 1990), "A History of Constructivism in the 20th Century", http://staff.science.uva.nl/~anne/hhhist.pdf, A detailed survey for specialists: §1 Introduction, §2 Finitism & §2.2 Actualism, §3 Predicativism and Semi-Intuitionism, §4 Brouwerian Intuitionism, §5 Intuitionistic Logic and Arithmetic, §6 Intuitionistic Analysis and Stronger Theories, §7 Constructive Recursive Mathematics, §8 Bishop's Constructivism, §9 Concluding Remarks. Approximately 80 references.
• Tymoczko, T. (1986), "Challenging Foundations", in Tymoczko (ed., 1986).
• Tymoczko, T. (ed., 1986), New Directions in the Philosophy of Mathematics, 1986. Revised edition, 1998.
• van Dalen D. (2008), "Brouwer, Luitzen Egbertus Jan (1881–1966)", in Biografisch Woordenboek van Nederland. URL:http://www.inghist.nl/Onderzoek/Projecten/BWN/lemmata/bwn2/brouwerle [13-03-2008]
• Weyl, H. (1921), "Über die neue Grundlagenkrise der Mathematik", Mathematische Zeitschrift 10, 39–79. Translated, "On the New Foundational Crisis of Mathematics", in (Mancosu 1998).
• Wilder, Raymond L. (1952), Introduction to the Foundations of Mathematics, John Wiley and Sons, New York, NY.

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