What is the difference between a philosophical syllogism and a mathematical syllogism?

Mathematics emphasizes logic, but philosophical syllogisms emphasize truth in addition to logic. In mathematics, regardless of whether the starting point (that is, the basis for starting) is real or not, the subsequent inference process is correct, or accurate. But in philosophy, since the starting point and process of a syllogism are limited by truth, it must be discussed within a precise single category.

Russell pointed out a famous example back then: "A mountain made of gold is a mountain, and a mountain made of gold is gold, so at least one mountain is gold." It is completely consistent with Aristotle's syllogism, but it is easily proved wrong using mathematical logic. However, this does not mean that Aristotle's syllogism is wrong. Gold is matter, and mountains are forms. These are two different categories, so no correct conclusion can be drawn. Philosophically this is a change of concept. Mathematically speaking, this actually equates the status of "mountain" and "gold". Logically speaking, this reasoning is formally correct.

Different fields have different focuses. Of course, different methods must be used when dealing with different problems. In addition, mathematical proofs can ensure the basic rigor of logical reasoning and are needed in philosophy.