Unstructured Footnotes
Sunday, August 22, 2004
Hilbert's Basis Theorem Footnote
Adapted from "Hilbert", by Constance Reid, Springer-Verlag, 1996.
Hilbert's Basis Theorem was his solution to Gordan's Problem, a major unsolved problem in invariant theory at the time. It would take too long to describe exactly what everything means, so allow me to just gloss over most of the technical issues. 'Invariants' are certain mathematical objects, each with a finite number of variables, and they were well understood at the time; it was well known that there were an infinite number of invariants. The question, however, was whether one could find a finite collection of invariants from which all the other invariants with the same number or fewer variables could be "constructed" through easily understood processes; this is called a "finite basis" for the invariants.
In 1868 Gordan had managed to show that this was the case for the simplest type of invariants, the binary invariants (two variables), by explicitly constructing such a finite collection. For twenty years, people had struggled with the next step, the ternary invariants (three variables), attempting to produce an explicit finite basis.
Rather than attack the problem directly, Hilbert rephrased the question and asked a seemingly harder question: If an infinite system of forms is given, expressed in a finite number of variables, under what conditions will there be a finite basis for them?
What Hilbert managed to prove was that there would always be a finite basis... but he did not do so by explicitly exhibiting such a basis. Rather, the proof proceded by induction on the number of variables, and by contradiction: assuming that no such finite basis existed, a contradiction was obtained. From this contradiction, one deduces that a finite basis must exist, even though we still do not know what that basis is.
This came as a surprise to most, who of course had been thinking along the lines of a constructive proof. Cayley, who 50 years earlier had laid the foundation of the theory, at first was politely dismissive, saying that the argument had a very interesting idea, but he was not sure Hilbert had actually managed to solve the problem; two weeks later, however, he acknowledged in a letter to Hilbert that his difficulty was an a priori one rather than one with the argument, and now he was convinced that Hilbert had solved the problem. Klein and Minkowski quickly hailed the new method.
Others were not so polite or forgiving: Lindemann called the method "unheimlich" (unconfortable, sinister, weird). Gordan himself dismissed the entire argument in a famous aphorism: "Das ist nicht Mathematik. Das ist Theologie."
Eventually, Hilbert returned to the subject again; based on his argument that a basis did indeed exist, he was able to produce an algorithm that would, in principle at least, construct such a basis. The proof that the algorithm worked, however, depended heavily on his nonconstructive argument to begin with. Nonetheless, most mathematicians finally agreed with him. Gordan himself conceded gracefully, stating, "I have convinced myself that theology also has its merits."
Return to Main Article
Permanent URL: http://unstructuredfootnotes.blogspot.com/2004/08/hilberts-basis-theorem-footnote.html
Adapted from "Hilbert", by Constance Reid, Springer-Verlag, 1996.
Hilbert's Basis Theorem was his solution to Gordan's Problem, a major unsolved problem in invariant theory at the time. It would take too long to describe exactly what everything means, so allow me to just gloss over most of the technical issues. 'Invariants' are certain mathematical objects, each with a finite number of variables, and they were well understood at the time; it was well known that there were an infinite number of invariants. The question, however, was whether one could find a finite collection of invariants from which all the other invariants with the same number or fewer variables could be "constructed" through easily understood processes; this is called a "finite basis" for the invariants.
In 1868 Gordan had managed to show that this was the case for the simplest type of invariants, the binary invariants (two variables), by explicitly constructing such a finite collection. For twenty years, people had struggled with the next step, the ternary invariants (three variables), attempting to produce an explicit finite basis.
Rather than attack the problem directly, Hilbert rephrased the question and asked a seemingly harder question: If an infinite system of forms is given, expressed in a finite number of variables, under what conditions will there be a finite basis for them?
What Hilbert managed to prove was that there would always be a finite basis... but he did not do so by explicitly exhibiting such a basis. Rather, the proof proceded by induction on the number of variables, and by contradiction: assuming that no such finite basis existed, a contradiction was obtained. From this contradiction, one deduces that a finite basis must exist, even though we still do not know what that basis is.
This came as a surprise to most, who of course had been thinking along the lines of a constructive proof. Cayley, who 50 years earlier had laid the foundation of the theory, at first was politely dismissive, saying that the argument had a very interesting idea, but he was not sure Hilbert had actually managed to solve the problem; two weeks later, however, he acknowledged in a letter to Hilbert that his difficulty was an a priori one rather than one with the argument, and now he was convinced that Hilbert had solved the problem. Klein and Minkowski quickly hailed the new method.
Others were not so polite or forgiving: Lindemann called the method "unheimlich" (unconfortable, sinister, weird). Gordan himself dismissed the entire argument in a famous aphorism: "Das ist nicht Mathematik. Das ist Theologie."
Eventually, Hilbert returned to the subject again; based on his argument that a basis did indeed exist, he was able to produce an algorithm that would, in principle at least, construct such a basis. The proof that the algorithm worked, however, depended heavily on his nonconstructive argument to begin with. Nonetheless, most mathematicians finally agreed with him. Gordan himself conceded gracefully, stating, "I have convinced myself that theology also has its merits."
Return to Main Article
Permanent URL: http://unstructuredfootnotes.blogspot.com/2004/08/hilberts-basis-theorem-footnote.html