David Hilbert’s Contributions in Mathematics

David Hilbert

David Hilbert was a German mathematician and physicist, who was born on 23 January 1862 in Konigsberg, Prussia, now Kaliningrad, Russia. He is considered one of the founders of proof theory and mathematical logic. He made great contributions to physics and mathematics but his most significant works are in the field of geometry, after Euclid. Hilbert’s first work was on invariant theory, and in 1888, he proposed the Basis theorem. Before Hilbert, Gordan proved the Basis theorem using a highly computational approach, but finding it difficult, Hilbert adopted an entirely new approach for proving the Basis theorem. He also submitted a paper proving the finite basis theorem to Mathematische Annalen. In 1893, while Hilbert was in Konigsberg, he started a work Zahlbericht, on algebraic number theory. He published Zahlbericht in 1897, which was a great blend of the works of mathematicians Kummer, Kronecker, Dedekind, and Hilbert’s ideas. Even the present-day topics like ‘Class field theory were also contained in his work. John David North had described Hilbert and his work in a letter and he wrote,

It is now nearly thirty years since David Hilbert died, in Gottingen, at the age of eighty-one, the greatest mathematician of his generation. His attitudes and methods were so influential in the world at large that there can scarcely be a mathematician who is not familiar with some aspect of his work. With the exception of two books published alone (Grundlagen der Geometrie and Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen), four more published with the collaboration of others (R Courant on mathematical physics, W Ackermann on logic, S Cohn-Vossen on geometry, and P Bernays on mathematical foundations), and about twenty papers, most of which would have meant duplication, Hilbert’s monumental collected works were first published in three volumes between 1932 and 1935.”

Hilbert’s 23 problems were presented by him at the Second International Congress of Mathematicians in Paris. These problems are still unsolved and were a challenge for all the mathematicians of that era. Hilbert’s problems included the continuum hypothesis, the well-ordering of the reals, Goldbach’s conjecture, the transcendence of powers of algebraic numbers, the Riemann hypothesis, the extension of Dirichlet’s principle and many more. However, many of the problems were solved in his era and every time a problem was solved, that day was celebrated as a major event in mathematics. In 1909, Hilbert researched integral equations which were similar to the 20th-century research on functional analysis (branch of mathematical analysis dealing with functions). His work on integral equations helped him establish the basis for his work on infinite-dimensional space, later called Hilbert space, a concept that is useful in mathematical analysis and quantum mechanics. He also contributed to the field of physics by his reports on kinetic gas theory and the theory of radiations. Many people believed that in 1915, Hilbert discovered the correct field equations for general relativity, five days before Einstien submitted the paper, but never claimed priority, and several other scholars claimed that the proofs given by Hilbert do not contain field equations. Hilbert contributed to many branches of mathematics, including invariants, algebraic number fields, functional analysis, integral equations, mathematical physics, and the calculus of variations. In 1930, Hilbert got retired and in 1934 and 1939 two volumes of “Grundlagen der Mathematik” were published which were intended to lead to a ‘proof theory.’ After giving his last lecture in Göttingen, he never stepped into the institute. In 1942, he fell and broke his hand while walking on the streets of Göttingen. After this accident, his health deteriorated and this incident was a major factor responsible for his death in the year 1943. In a life span of 81 years, he received many awards and honours. He was awarded the Bolyai Prize in 1910 and elected as a fellow of the Royal Society of London in 1928. In 1930, the city of Königsberg made him an honorary citizen of the city. In 1939, he was awarded the Mittag-Leffler prize by the Swedish Academy of Sciences. In 1901, Hilbert was also elected as an honorary member of the London Mathematical Society and for the German Mathematical Society in 1942. His tombstone is marked with his own words, “Wir müssen wissen, Wir werden wissen – We must know, we shall know” that he uttered while announcing his retirement from the Society of German Scientists and Physician.

1. Hilbert’s Basis Theorem

In Algebra, Hilbert’s basis theorem states that a polynomial ring over a Noetherian ring is Noetherian. Algebraically,

If R is a ring, let R[X] denote the ring of polynomials in the indeterminate X over R. Hilbert proved that if R is “not too large”, in the sense that if R is Noetherian, the same must be true for R[X].”

Hilbert’s basis theorem did not give an algorithm to produce basis polynomials for any given ideal, it just showed that such polynomials should exist. He showed that although there are an infinite number of possible equations, they can be split into finite equations to form a set of equations that will help to produce all other equations. However, he didn’t produce any such set but said that it must exist.

2. Hilbert’s Program

In the 1920s, David Hilbert proposed a new approach for the foundation of mathematics. According to him, mathematics should be formalised in axiomatic form along with consistent proof and the proof itself was to be carried out using only what Hilbert called “finitary” methods. He believed that to develop a scientific subject properly, an axiomatic approach is required and it will help in developing the theory of the subject independently. Hilbert’s work on the foundation of mathematics had been developed from his work on the geometry of the 1890s, in his textbook, Foundations of Geometry (1899). Hilbert’s program is called Formalism in the philosophy of mathematics. Hilbert’s program included the formulation of all mathematics according to well-defined rules, completeness, consistency, conservation, and decidability. In 1919, Hilbert wrote,

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.”

3. Hilbert’s Space

Hilbert Space is the generalisation of Euclidean space, which extends vector calculus and algebra to infinite spaces. Hilbert Spaces provided the basis for important contributions in mathematics and physics. A Hilbert space H is a real or complex inner product space that is also a complete metric space with respect to the distance function induced by the inner product. It is used in theories of partial differential equations, quantum mechanics, Fourier analysis, and ergodic theory (a branch of mathematics that deals with trajectories of dynamical systems). Hilbert space is considered to be most significant in the study of functional analysis, specifically for the spectral theory of self-adjoint linear operators.

Hilbert Space

4. Hilbert Curves

In the 1890s, Hilbert developed an algorithm of continuous space-filling curves in multi-dimension. The Hilbert curves were inspired by the space-filling Peano curves, discovered by Giuseppe Peano in 1890. Hilbert curves have several applications as they give mapping between 1D and 2D space. These curves are widely used in the field of computer sciences. Hilbert Curves are also used to determine the range of IP addresses, used by computers and can be mapped into a picture.

File:Hilbert-curve rounded-gradient-animated.gif - Wikimedia Commons

5. Mathematical Physics

Hilbert was a pure mathematician and believed that physical problems can not be solved without applying mathematical concepts. He did lots of research on mathematical physics and most of his research from 1907 to 1912 was based on this topic. After some time, he developed an interest in physics and studied kinetic gas theory and radiation theory. Throughout the study of physics, he tried to apply the mathematical rules to physics with rigour. One of his colleagues, Richard Courant, wrote a book “Methoden der mathematischen Physik” (Methods of Mathematical Physics)  that included some ideas of Hilbert as well and added his name as the author. Hilbert said,

Physics is too hard for physicists, implying that the necessary mathematics was generally beyond them. The Courant-Hilbert book made it easier for them.”

6. Hilbert’s Axioms of Geometry

In 1899. Hilbert proposed a set of axioms of geometry in his book Grundlagen der Geometrie (The Foundations of Geometry). These axioms were introduced to remove flaws in Euclidean geometry. Hilbert gave 20 axioms that are stated below.

1. Incidence

  • For every two points, A and B there exists a line a that contains them both. We write AB = a or BA = a. Instead of “contains”, we may also employ other forms of expression; for example, we may say “A lies upon a”, “A is a point of a”, “a goes through A and through B”, “a joins A to B”, etc. If A lies upon a and at the same time upon another line b, we make use also of the expression: “The lines a and b have the point A in common”, etc.
  • For every two points there exists no more than one line that contains them both; consequently, if AB = a and AC = a, where B ≠ C, then also BC = a.
  • There exist at least two points on a line. There exist at least three points that do not lie on the same line.
  • For every three points A, B, C not situated on the same line there exists a plane α that contains all of them. For every plane, there exists a point that lies on it. We write ABC = α. We employ also the expressions: “A, B, C lie in α”; “A, B, C are points of α”, etc
  • For every three points ABC which do not lie in the same line, there exists no more than one plane that contains them all.
  • If two points A, B of a line ‘a’ lie in a plane α, then every point of ‘a’ lies in α. In this case, we say: “The line a lies in the plane α”, etc.
  • If two planes α, β have a point A in common, then they have at least a second point B in common.
  • There exist at least four points not lying in a plane.

2. Order

  • If a point B lies between points A and C, B is also between C and A, and there exists a line containing the distinct points A, B, C.
  • If A and C are two points, then there exists at least one point B on the line AC such that C lies between A and B.
  • Of any three points situated on a line, there is no more than one which lies between the other two.
  •  Let A, B, C be three points not lying in the same line and let ‘a’ be a line lying in the plane ABC and not passing through any of the points A, B, C. Then, if the line ‘a’ passes through a point of the segment AB, it will also pass through either a point of the segment BC or a point of the segment AC.

3. Congruence

  • If A, B are two points on a line ‘a’, and if A′ is a point upon the same or another line a′, then, upon a given side of A′ on the straight line a′, we can always find a point B′ so that the segment AB is congruent to the segment A′B′. We indicate this relation by writing AB ≅ A′B′. Every segment is congruent to itself, that is, we always have AB ≅ AB.
  • If a segment AB is congruent to the segment A′B′ and also to the segment A″B″, then the segment A′B′ is congruent to the segment A″B″, that is, if AB ≅ A′B′ and AB ≅ A″B″, then A′B′ ≅ A″B″.
  • Let AB and BC be two segments of a line a which have no points in common aside from the point B, and, furthermore, let A′B′ and B′C′ be two segments of the same or of another line a′ having, likewise, no point other than B′ in common. Then, if AB ≅ A′B′ and BC ≅ B′C′, we have AC ≅ A′C′.
  • Let an angle ∠(h,k) be given in the plane α and let a line a′ be given in a plane α′. Suppose also that, in the plane α′, a definite side of the straight line a′ be assigned. Denote by h′ a ray of the straight line a′ emanating from a point O′ of this line. Then in the plane α′, there is one and only one ray k′ such that the angle ∠(h, k), or ∠(k, h), is congruent to the angle ∠(h′, k′) and at the same time all interior points of the angle ∠(h′, k′) lie upon the given side of a′. We express this relation by means of the notation ∠(h, k) ≅ ∠(h′, k′).
  • If the angle ∠(h, k) is congruent to the angle ∠(h′, k′) and to the angle ∠(h″, k″), then the angle ∠(h′, k′) is congruent to the angle ∠(h″, k″), that is to say, if ∠(h, k) ≅ ∠(h′, k′) and ∠(h, k) ≅ ∠(h″, k″), then ∠(h′, k′) ≅  ∠(h″, k″).
  • If, in the two triangles ABC and A′B′C′ the congruences AB ≅ A′B′AC ≅ A′C′∠BAC ≅ ∠B′A′C′ hold, then the congruence ∠ABC ≅ ∠A′B′C′ holds (and, by a change of notation, it follows that ∠ACB ≅ ∠A′C′B′ also holds).

4. Parallels

  • Let ‘a’ be any line and A a point not on it. Then there is at most one line in the plane, determined by ‘a’ and A, that passes through A and does not intersect ‘a’.

5. Continuity

  •  If AB and CD are any segments then there exists a number n such that n segments CD constructed contiguously from A, along the ray from A through B, will pass beyond the point B.
  • An extension of a set of points on a line with its order and congruence relations that would preserve the relations existing among the original elements as well as the fundamental properties of line order and congruence (Axioms 1-3 and 5-1) is impossible.

7. Hilbert’s 23 Problems

Hilbert problems are the set of 23 problems in mathematics, some of which are unsolved to date. He presented these problems at the Paris conference of the International Congress of Mathematicians. A few less than half of the problems were presented at the Second International Congress of Mathematics, and later on, the rest of the problems were added to the list. Some of the problems were solved soon after he introduced them, some of the problems are partially solved and some of them have been discussed throughout the 20th century. In 2000, the 24th problem was discovered in Hilbert’s original notes by a German historian, Rudiger Thiele. His unsolved problems are often known as The Riemann Hypothesis, The Kronecker-Weber theorem extension, The problem of the topology of algebraic curves and surfaces

8. Hilbert’s Theorems

Theorem on Differential Geometry

Hilbert gave this theorem in 1901. In differential geometry, the theorem states that there exists no complete regular surface S of constant negative Gaussian curvature K immersed in R³

Theorem on Cyclic Extensions

This theorem is also known as Hilbert’s Theorem 90 because this was the 90th theorem in Hilbert’s book, Zahlbericht. The theorem states that if L/K is an extension of fields with cyclic Galois group G = Gal(L/K) generated by an element σ, and if α is an element of L of relative norm 1, that is

N (α) = α. σ(α ). σ²(α )…….{σ}^{n-1}(α ) = 1, then there exists b in L, such that

α = σ(b)/b

Irreducibility Theorem

Hilbert gave this theorem in 1892. In number theory, this theorem states that every finite set of irreducible polynomials in a finite number of variables and having rational number coefficients admit a common specialization of a proper subset of the variables to rational numbers such that all the polynomials remain irreducible.

Hilbert’s Nullstellensatz

This theorem talks about the zeroes of a polynomial. It states that if p is some polynomial in k[{X}_{1},………, {X}_{n}] that vanishes on the algebraic set V(I), i.e. p(x) = 0 for all x in V(I), then there exists a natural number r such that pr is in I.

Theorem of Invariants

Hilbert's theorem of invariants states that the algebra of all polynomials on the complex vector space of forms of degree d in r variables which are invariant with respect to the action of the general linear group GL (r, C), defined by linear substitutions of these variables, is finitely generated.

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