mathematics

Paul Gordan

Paul Gordan is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Paul Gordan rather than just read about it. In short: Paul Albert Gordan (27 April 1837 – 21 December 1912) was a German mathematician known for work in invariant theory and for the Clebsch–Gordan coefficients and Gordan's lemma. He was called "the king of invariant theory".

Key takeaways

Paul Gordan belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Paul Gordan to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Paul Gordan from memory before moving on to harder problems.

Reference excerpt

Paul Albert Gordan (27 April 1837 – 21 December 1912) was a German mathematician known for work in invariant theory and for the Clebsch–Gordan coefficients and Gordan's lemma. He was called "the king of invariant theory". His most famous result is that the ring of invariants of binary forms of fixed degree is finitely generated. Clebsch–Gordan coefficients are named after him and Alfred Clebsch. Gordan also served as the thesis advisor for Emmy Noether.

Life and career Gordan was born to Jewish parents in Breslau, Germany (now Wrocław, Poland), and died in Erlangen, Germany. He received his Dr. phil. at the University of Breslau with the thesis De Linea Geodetica, (On Geodesics of Spheroids) in 1862. He moved to Erlangen in 1874 to become professor of mathematics at the University of Erlangen-Nuremberg. A famous quote attributed to Gordan about David Hilbert's proof of Hilbert's basis theorem, a result which vastly generalized his result on invariants, is "This is not mathematics; this is theology." The proof in question was the (non-constructive) existence of a finite basis for invariants. It is not clear if Gordan really said this since the earliest reference to it is 25 years after the events and after his death. Nor is it clear whether the quote was intended as criticism, or praise, or a subtle joke. Gordan himself encouraged Hilbert and used Hilbert's results and methods, and the widespread story that he opposed Hilbert's work on invariant theory is a myth (though he did correctly point out in a referee's report that some of the reasoning in Hilbert's paper was incomplete). He later said "I have convinced myself that even theology has its merits". He also published a simplified version of the proof.

Publications Gordan, Paul (1862). De linea geodetica. Retrieved 13 October 2025. Gordan, Paul (1885). Vorlesungen über Invariantentheorie. Vol. 1. Teubner. Retrieved 12 April 2014. Gordan, Paul (1887). Dr. Paul Gordan's Vorlesungen über Invariantentheorie. Vol. 2. B. G. Teubner. Retrieved 12 April 2014. Gordan, Paul (1987) [1885], Kerschensteiner, Georg (ed.), Vorlesungen über Invariantentheorie (2nd ed.), New York: Chelsea Publishing Co. or American Mathematical Society, ISBN 978-0-8284-0328-3, MR 0917266

See also Dickson's lemma Invariant of a binary form Symbolic method

References

External links

Paul Gordan at the Mathematics Genealogy Project Gordan's publication catalog: "Katalog der Deutschen Nationalbibliothek". portal.dnb.de (in German). Retrieved 25 August 2024.

Illustrations

Worked examples

Example 1 — a first encounter with Paul Gordan

Start with the simplest possible case. Write down what Paul Gordan claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Paul Gordan before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Paul Gordan ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Paul Gordan

In research: Paul Gordan appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Paul Gordan in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Paul Gordan is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1837 births, 1912 deaths, 19th-century German Jews, so understanding it makes those chapters shorter.
In everyday life: Look for Paul Gordan outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Paul Gordan in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Paul Gordan means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Paul Gordan out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Paul Gordan in simple terms?

Why does Paul Gordan matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Paul Gordan?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Paul Gordan.