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Mathematicians are a rounding error
What math costs
All the salaries of academic mathematicians in the United States add up to maybe $4 billion to $5 billion at most. Against the size of the economy, that is a rounding error, roughly what the country spends on bowling alleys.
What most mathematicians do
Only a few of those mathematicians do frontier research. Most are employed mainly as teachers. It is an open question whether the frontier work of even the best of them pays for itself in economic terms, since outside some cryptography and the occasional physics application, most of it has little use.
Why society pays
So the country already employs mathematicians because it considers human understanding of math worthwhile, with or without industrial applications. That reasoning survives AI discovering a vast number of new results. Most people won't care how many new results AI turns up. What counts is that some human understands them, and a society that AI has made richer can easily pay that pittance to keep professional math-understanders on the payroll.
Why do math departments cost so little?
Mathematics needs little equipment compared with a laboratory science, since the core tools are paper, a blackboard, and time. Most math departments are funded mainly through teaching, because they instruct large numbers of students in calculus, statistics, and other required courses. In the United States, research grants in the field come largely from the National Science Foundation's Division of Mathematical Sciences. Calculus alone is a required course for most engineering and science majors.
How did number theory end up securing the internet?
In 1940, the British mathematician G. H. Hardy wrote in A Mathematician's Apology that number theory had no practical use. In 1977, Ron Rivest, Adi Shamir, and Leonard Adleman published the RSA scheme, which locks messages using the product of two huge prime numbers. It works because multiplying two primes is easy, while recovering them from the product is extremely hard when they are large. Early versions of the web's SSL security protocol relied on RSA.
Why do abstract math ideas find uses later?
George Boole published his algebra of true and false values in 1854 as a study of the laws of thought. In 1937, Claude Shannon, then a master's student at MIT, showed that Boole's algebra describes electrical switching circuits. That insight became the basis of digital circuit design. Bernhard Riemann's 1850s geometry of curved spaces likewise gave Einstein the mathematics for general relativity, completed in 1915.
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