‘In his famous essay "A Mathematician's Apology," G.H. Hardy uses the phrase "they are so" to express a sentiment about the beauty and intrinsic nature of mathematical truths. Specifically, Hardy is discussing the "eternity" and "objectivity" of mathematics, and how mathematical results exist independently of human experience.
In context, when Hardy says "they are so", he is affirming that certain mathematical truths (such as the properties of numbers or theorems) exist in a way that is absolute, self-evident, and unchangeable, regardless of human recognition or perception.
For example, Hardy writes:
"Great art is often said to be 'useless', and I believe that this is true. It is also true that mathematics is a form of art, and it is for its own sake that we pursue it. In this sense, they are so: these truths exist not because of any practical purpose but because they are simply true."
In this sense, "they are so" refers to the timeless, objective, and undeniable nature of mathematical truths.
"The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful. The mathematician does not study anything but the pure mathematics. The fact that mathematicians have not made any discoveries of value to the human race is irrelevant. [...] They are not concerned with the practical application of mathematics, but with its beauty. They are so."
Hardy sees mathematics as an art, pursued for its own beauty and intellectual elegance rather than any practical purpose. "They are so" is his way of stating that the truths of mathematics are self-evident and exist in their own right, irrespective of human utility or application.’
