Rabbi Levi ben Gershon — ר׳ לוי בן גרשון (Ralbag) (OBM) 1288–1344 | Provence, France: A Rabbi Who Advanced Mathematical Proof

Oct 14

3 min read

Rabbi Levi ben Gershon — ר׳ לוי בן גרשון (Ralbag) (OBM)

1288–1344 | Provence, France

A Rabbi Who Advanced Mathematical Proof

Rabbi Levi ben Gershon, commonly known as Ralbag or Gersonides, was a medieval Jewish scholar who made important contributions to mathematics. His mathematical investigations included algebraic identities, numerical sequences, equations involving proportions, and methods of mathematical proof.

In 1321, at approximately 33 years of age, he completed his major mathematical work, מעשה חושב — Ma'aseh Hoshev (The Art of Calculation).

The book contains 68 mathematical theorems and proofs in its first part, followed by methods of calculation and practical problems in its second part.

His Contributions to Algebra

1. Algebraic Identities

Ralbag studied general mathematical relationships that remain true for all permitted values of their variables.

For example, an algebraic identity is:

(a + b)² = a² + 2ab + b²

Unlike an equation that may be true only for a particular value of a variable, an identity is true for every permitted value.

2. Formulas for Numerical Series

Ralbag investigated formulas for finding sums of consecutive integers, squares, and cubes.

One example of the type of formula he studied is:

1 + 2 + 3 + ⋯ + n = n(n + 1)/2

This formula allows us to calculate the sum of the first (n) positive integers without adding each number separately.

3. Mathematical Induction

One of Ralbag's (OBM) most significant achievements was his early use of a method now called mathematical induction.

This method establishes that a statement is true for all positive integers by proving an initial case and demonstrating that, whenever it is true for one integer, it must also be true for the next.

Ralbag used inductive reasoning in rigorous proofs centuries before this method became widely used in European mathematics.

4. Combinations and Permutations

Ralbag investigated the number of ways objects can be selected or arranged.

For example, three different objects can be arranged in:

3! = 3 × 2 × 1 = 6

different orders.

His work helped establish mathematical methods for counting arrangements and combinations.

Why Is His Work Important?

Ralbag (OBM) demonstrated that mathematics requires more than obtaining the correct numerical answer.

A mathematical rule should be understood, justified, and, when appropriate, proven.

His work connects algebraic calculation with the logical structure of mathematics—the same principles we study in NETIVIM Algebra Alef.

Mathematical Investigation

Consider the formula:

1 + 2 + 3 + ⋯ + n = n(n + 1)/2

Exercise 1. Calculate the sum of the first 10 positive integers using ordinary addition.

Exercise 2. Calculate the same sum using the formula.

Exercise 3. Use the formula to calculate the sum of the first 100 positive integers.

Exercise 4. Explain why checking the formula for 10 and 100 does not prove that it works for every positive integer.

Exercise 5 — Challenge. Write the sum (1+2+\cdots+n) once in ascending order and once in descending order. Add the two rows term by term. Use the result to prove the formula.

Historical Sources and Further Reading

University of St Andrews, MacTutor History of Mathematics: Levi ben Gerson
Mathematical Association of America: The Mathematics of Levi ben Gershon in the Classroom
Shai Simonson, The Mathematics of Levi ben Gershon: Historical and Mathematical Study