"mathematical induction" vs "the mathematical induction"

The term “mathematical induction” is used (grammatically) in the same way as mathematical operations such as addition and subtraction. You can use the more common/familiar terms to guide your intuition on this.

When speaking about applying the technique, the definite article is typically not used.

• Prove the theorem by addition.

• Prove the theorem by (no article) mathematical induction.

• Learn about addition.

• Learn about (no article) mathematical induction.

When referencing an instance of the technique, you might sometimes use an article to indicate that you are talking about that specific instance.

• The addition used to prove the theorem involves the variable x.
• The mathematical induction used to prove the theorem involves the variable x.

The difference is whether you are referring to the technique/concept or a concrete instance of its application. More generally, mathematical induction is a noun phrase consisting of a head noun that specifies an operation of some sort (induction) and a modifier (mathematical), and follows the usual conventions regarding article usage common to such constructs.