Does infinity belong to mathematics

 Bertrand Russell in the Preface to his Introduction to Mathematical Philosophy writes:

Much of what is set forth in the following chapters is not properly to be called "philosophy," though matters concerned were included in philosophy so long as no satisfactory science of them existed. The nature of infinity and continuity, for example, belonged in former days to philosophy, but belongs now to mathematics.

Russell must be the right person to make such a statement, he was both a philosopher and mathematician.

But how does he justify the claim that philosophical infinity is now mathematical infinity. 

I don't think these two can be reconciled. Philosophical infinity is something else and mathematical infinity is something else.

I'm still looking to find a satisfactory definition of the word infinite in mathematics.

In the same book, page 77, Russell writes this about infinity:

It cannot be said to be certain that there are in fact any infinite collections in the world. The assumption that there are is what we call the "axiom of infinity."

Although various ways suggest themselves by which we might hope to prove this axiom, there is reason to fear that they are all erroneous, and that there is no conclusive logical reason for believing it to be true.

At the same time, there is certainly no logical reason against infinite collections, and we are therefore justified, in logic, in investigating the hypothesis that there are such collections.

And on p. 71 Russell writes:

The method of "postulating" what we want has many advantages; they are the same as the advantages of theft over honest toil. Let us leave them to others and proceed with our honest toil.

 Very funny. As if he is referring to the axiom of infinity which postulates what mathematicians want. Russell left the business of stealing "infinity" from philosophy to Zermalo.

And this is a quote from a chatbot:

Ernst Zermelo is credited with first introducing the Axiom of Infinity as part of his set theory in 1908

This axiom is fundamental to modern mathematics, as it guarantees the existence of at least one infinite set, which is essential for constructing the natural numbers and other essential mathematical objects.

It guarantees or postulates the existence of infinite set for mathematicians to play with.