Hilbert Asks, Is Mathematics Complete, is it Consistent, and is it Decidable? : History of Information (2024)

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What did Hilbert believe about mathematics? ›

Hilbert's program

He wanted mathematics to be formulated on a solid and complete logical foundation. He believed that in principle this could be done by showing that: all of mathematics follows from a correctly chosen finite system of axioms; and.

Means it contains all possible true statements (completeness), it is free of contradiction (consistent) and that it's all deterministic, means there is a way to fully calculate an equation (decidable).

Kurt Gödel was an Austrian mathematician known for successfully refuting the lifework of the German mathematician David Hilbert. In his doctoral dissertation, he proved that it is impossible to use the axiomatic method to construct a mathematical theory that entails all of the truths of mathematics.

Hilbert's problems are a set of (originally) unsolved problems in mathematics proposed by Hilbert. Of the 23 total appearing in the printed address, ten were actually presented at the Second International Congress in Paris on August 8, 1900.

At a conference in Paris in 1900, the German mathematician David Hilbert presented a list of unsolved problems in mathematics. He ultimately put forth 23 problems that to some extent set the research agenda for mathematics in the 20th century.

In 1922, Hilbert introduced the new subject of proof theory for addressing the problem: viewing proofs in formalized theories as objects of investigation, the goal being to establish - using only restricted finitist means - that such proofs cannot lead to a contradiction.

Statement of Hilbert's program

Completeness: a proof that all true mathematical statements can be proved in the formalism. Consistency: a proof that no contradiction can be obtained in the formalism of mathematics.

If a theory is complete and has recursive axioms, then it is decidable. This is because if the axioms are recursive, then the proofs are as well. This gives you your effective procedure. We can also have decidable and complete theories.

A logical system is consistent if no true statement can contradict another true statement, in other words assuming the first statement true will not allow a proof that the other statement is false. A logical system is complete if every true statement can be proved true and every false statement can be proved false.

His mistake was to pose the problem of showing that mathematics, beginning with Peano Arithmetic, is consistent, rather than to ask whether it is consistent.

Which of Hilbert's problems are unsolved? ›

The problems were all unsolved at the time, and several of them were very influential for 20th-century mathematics. Hilbert presented ten of the problems (1, 2, 6, 7, 8, 13, 16, 19, 21, and 22) at the Paris conference of the International Congress of Mathematicians, speaking on August 8 in the Sorbonne.

Born in what was then Austria, on April 28 1906, Gödel died in Princeton, New Jersey on January 14 1978, having developed a paranoia that he was being poisoned and, as a result, starving himself to death (an altogether odd end for one of the greatest logicians the world has ever known).

David Hilbert (1862 – 1943) was one of the most influential mathematicians of the 20th century.

The German mathematician David Hilbert contributed to many branches of mathematics, including invariants, algebraic number fields, functional analysis, integral equations, mathematical physics, and the calculus of variations. He lived most of his life in Germany.

1. Riemann Hypothesis. The Riemann Hypothesis, proposed by Bernhard Riemann in 1859, is a central problem in number theory, and discusses the distribution of prime numbers. The hypothesis focuses on the zeros of the Riemann zeta function.

In physics and cosmology, the mathematical universe hypothesis (MUH), also known as the ultimate ensemble theory, is a speculative "theory of everything" (TOE) proposed by cosmologist Max Tegmark. According to the hypothesis, the universe is a mathematical object in and of itself.

Western philosophies of mathematics go as far back as Pythagoras, who described the theory "everything is mathematics" (mathematicism), Plato, who paraphrased Pythagoras, and studied the ontological status of mathematical objects, and Aristotle, who studied logic and issues related to infinity (actual versus potential) ...

Plato believed geometry and mathematics exist in their own ideal world and that certain shapes (now known as the Platonic solids) were associated with the classical elements from which the world was made: earth, fire, air, water, and the universe.