Mathematics is inferior to programming, in a way, because the entire hierarchy of mathematics relies on human beliefs at every stage, all the way down to its axioms. The beliefs of many highly intelligent humans, but beliefs of biased individuals nonetheless. Humans simply can't match the accuracy of a compiler when it comes to finding flaws.
On the other hand, a program's correctness doesn't depend on human beliefs. It can be proven to work perfectly on certain ranges of inputs by actually executing it on those inputs. Human subjectivity can be factored out to an increasing degree by way of automated tests. This level of concrete proof exceeds the level of social proof which mathematics relies on. The electronics upon which programs execute do not have biases as humans do. Evaluating a complex mathematical proof is as prone to errors as humans evaluating a complex computer program with their own minds. The compiler will always beat the human when determining correctness of a program for a well defined range of inputs.
Although automated tests can rarely prove universal truths, they rarely need to, as the logic they test only needs to handle a limited number of inputs and requires a limited number of guarantees. For many programs, the degree of proof that a well written automated test can provide far exceeds the degree of proof that a mathematical proof (based on human consensus) can provide.
This is why programming can accumulate complexity at a rate which is unfathomable in mathematics. With AI, programming may exceed the capabilities of mathematics to such extent that the entire field of mathematics will become a historical relic; showcasing the desperation of feeble human minds to grasp truths that are well outside of their reach.
The 'importance' of the field of knowledge comes down to the scale of the information and growth speed of the field. For all practical purposes, it seems that the programmatic, exhaustive approach will beat out the mathematical approach of trying to solve problems by uncovering universal truths.
If mathematics is so vast the human mind is feeble to it, then it’s also too vast for exhaustive search. Axioms aren’t even beliefs. They are first principles at the service of solving mathematical problems. If we want to solve other problems, we can change the axioms. That goes all the way back to Plato. We treat them as if they were true to solve the problems at hand. (It’s called as-ifism if you want to google it). Since logical and mathematical space is infinite; exhaustive search is the wrong approach.
I've read this (and similar diatribes by "real" computer scientists) before and although it's compelling - and, hell, I'd be willing to try the whole "formal methods" thing if it meant I spent less time debugging - it kind of overlooks the whole "we are successfully using computers to solve real-world problems by letting us non-mathematical 'dummies' at them" aspect. He seems to be reasoning that "yes, you're achieving results, but it would be more efficient if you spent two or three decades mastering formal methods before you produced a line of code" which sounds suspiciously to me like the way project managers reason: "yes, you can produce a program that solves the business problem if you just get started, but you would do so much more predictably if you spent 6 months 'estimating' it before you embark on a couple week's of programming".
Agreed, though mathematics and programming as we understand them are both ways for humans to express logic. As much as mathematicians want to believe it, math is not the same as logic. The former serves to express logic to other humans while the latter serves to express logic to both other humans and computers. It's not like in the business world; mathematics doesn't get to have a monopoly over logic just because it came first.
Math and computer science just happen to be two fields which concern themselves with different aspects of logic. Math being slow-moving and focused on solving universal problems and computer science being fast-moving and focused on solving concrete problems.
Once in a while, advancements like cryptography and LLMs show us that focusing on solving concrete problems can also expand our knowledge and capabilities in a radical (and useful) way.
I resent articles which try to present one as more important than the other. What does that even mean? Appeal to academic authority? Utility value? Difficulty? Degree of abstraction? They're both just languages which solve problems in different ways and which have different target audiences and have different scalability constraints.
On the other hand, a program's correctness doesn't depend on human beliefs. It can be proven to work perfectly on certain ranges of inputs by actually executing it on those inputs. Human subjectivity can be factored out to an increasing degree by way of automated tests. This level of concrete proof exceeds the level of social proof which mathematics relies on. The electronics upon which programs execute do not have biases as humans do. Evaluating a complex mathematical proof is as prone to errors as humans evaluating a complex computer program with their own minds. The compiler will always beat the human when determining correctness of a program for a well defined range of inputs.
Although automated tests can rarely prove universal truths, they rarely need to, as the logic they test only needs to handle a limited number of inputs and requires a limited number of guarantees. For many programs, the degree of proof that a well written automated test can provide far exceeds the degree of proof that a mathematical proof (based on human consensus) can provide.
This is why programming can accumulate complexity at a rate which is unfathomable in mathematics. With AI, programming may exceed the capabilities of mathematics to such extent that the entire field of mathematics will become a historical relic; showcasing the desperation of feeble human minds to grasp truths that are well outside of their reach.
The 'importance' of the field of knowledge comes down to the scale of the information and growth speed of the field. For all practical purposes, it seems that the programmatic, exhaustive approach will beat out the mathematical approach of trying to solve problems by uncovering universal truths.