David Hilbert: Biography, Facts & Inventions

Instructor: Betsy Chesnutt

Betsy teaches college physics, biology, and engineering and has a Ph.D. in Biomedical Engineering

David Hilbert was one of the foremost mathematicians of the twentieth century. He revolutionized many areas of mathematics and inspired an entire generation of mathematicians. In this lesson, learn about his life and his contributions!

David Hilbert and the Twenty Three Problems

A new century was just beginning as the 1900 International Congress of Mathematicians convened in Paris, France. Among the mathematicians in attendance was a young German named David Hilbert. At the conference, Hilbert made a revolutionary presentation. He identified twenty three of the most important unsolved problems in mathematics and challenged the world's mathematicians to find their solutions during the next century.

These problems became known as Hilbert's problems, and they inspired an entire generation of mathematicians. Today, some of Hilbert's original twenty three problems have been solved, but there are others that mathematicians are still working on over one hundred years later!

David Hilbert was one of the most influential mathematicians of the early twentieth century
David Hilbert

David Hilbert did a lot more than just present problems for other mathematicians to solve. He was one of the most important and influential mathematicians of the early twentieth century, and he made contributions to mathematics that are still important today.

Early Life and Education

Hilbert was born in 1862 in the Prussian city of Konigsberg, which is now part of western Russian known as Kaliningrad. From an early age, he was interested in mathematics. He eventually enrolled in Konigsberg University in 1880 and graduated with a Ph.D. in mathematics just five years later. Even at this early stage of his career, he was making a big impact on the mathematical world.

Konigsberg during the time period David Hilbert lived there
Konigsberg during the time period David Hilbert lived there

While still a student at Konigsberg, Hilbert began collaborating with fellow mathematicians Adolf Hurwitz and Hermann Minkowski, and the three continued to work together after Hilbert graduated. He remained at the University of Konigsberg and worked as a Senior lecturer of mathematics until 1895. Most of his work during this time dealt with number theory and abstract algebra. However, he also famously developed an algorithm to generate fractals that would fill a shape like a rectangle or cube by drawing a single line that never intersected itself!

Contributions to Mathematics

In 1895, Hilbert relocated to the University of Gottingen, where he joined some of the most prominent mathematicians of the time as a professor of mathematics. At Gottingen, he turned his attention to geometry and published a book titled The Foundations of Geometry in 1899. In it, he identified errors in Euclidean geometry, which mathematicians studied for thousands of years. He clearly laid out new axioms that removed these errors, and this was a significant breakthrough that cemented his position as one of the world's most important mathematicians.

Hilbert also did groundbreaking work with equations. His finiteness theorem proved that it was possible to separate all equations into a finite number of equation types. These types could then be combined to generate an infinite number of equations.

Although he successfully proved that equation types existed, he was not able to identify or produce them. This led many mathematicians to disregard his work, but it was actually quite revolutionary. Hilbert's finiteness theorem laid the foundation of a whole new way of thinking about abstract algebra and had a profound influence on the course of mathematics in the twentieth century.

One of Hilbert's most important contributions was the development of what is now known as Hilbert space, in which he developed methods to extend the techniques of vector algebra and calculus to spaces with any number of dimensions. Hilbert space remains very important in modern mathematics and was used by physicists to develop the mathematical basis of quantum mechanics!

Hilbert's Program

As his career progressed, Hilbert became interested in making mathematics more consistent and logical and less reliant on intuition. He believed that all mathematical problems could ultimately be solved and that the foundations of mathematics must be based in logic, so he set out to determine exactly what these logical foundations really were.

His early work with developing new foundational axioms for geometry showed him that the best way to approach any scientific subject, including mathematics, was to use an axiomatic approach. To Hilbert, this made it easier to analyze mathematical concepts. This axiomatic approach to mathematics was known as Hilbert's program.

Hilbert also believed that for axioms to be valid, they must be proven to be consistent and independent of each other. In the early 1900s, he developed a new way to prove that axioms were consistent. These new proofs were called consistency proofs, and they are an important part of mathematics today.

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