Ok, there’s been some online debate, esp recently on Lobste.rs, about mathematical vs empirical evidence. Some people highly believe in formal proof for reasons that probably vary. Some say those methods are nonsense advocating for strictly empirical approach of experimental evidence. I’m somewhere in the middle where I want any formal method to have evidence it works in an empirical sense but I don’t see the concepts as that different. I think we believe in specific types of math/logic because they’re empirically proven. As I formulated those points & dug up related work, I found this submission that argued a lot of what I was going to argue plus some other points (esp on intuition) I wasn’t thinking about. So, enjoy!

If you’ve read it at this point, I’ll weaken that claim to mine that certain maths/logics are empirically valid by being relied on in hundreds of thousands to billions of uses. If they were wrong, we’d see massive amount of evidence in the field that the fundamental logical methods didn’t work. Matter of fact, some of these failing would make it impossible for opponents of formal methods to even write their rebuttals online given propositional logic and calculus are how mixed-signal ASIC’s work. ;)

So, I’m going to list some specific types of math with widespread, empirical confirmation as a start. Anyone else feel free to add to the list if it’s math/logic used in formal proofs that has massive level of real-world use with diverse inputs or environmental setups.

1. Arithmetic on integers and real numbers. Computers can be boiled down to glorified calculators. Arithmetic can help bootstrap trust in other things, too.

2. Algebraic laws. Logics like Maude build on these.

3. Calculus like integration and derivatives. Heavily used in engineering. Also, building blocks of analog circuits and computers implement these primitives.

4. Propositional and Boolean logic. Digital designs are synthesized, verified, and tested in this form. With all CPU’s shipped, it’s probably the most well-tested logic after arithmetic and algebra.

5. Some subset of first-order logic. There’s been a lot of experimental confirmation it works in form of Prolog and Datalog programs that do what the logic says they should. I’ll note the older Prolog work was a bit weaker since it was associated with lots of zealotry that led to AI winter. We can focus on working applications, though, of which there are many in commercial and FOSS. Although it has a niche, Prolog was pushed way out of it to implement diverse array of programs and environments. The SMT solvers also catch problems they’re supposed to on a regular basis. All problems I’ve seen investigated in these were implementation errors that didn’t invalidate the logic itself.

6. Set theory. Experiments with the principles seem to always work long as problems and solutions fit the simple modeling style. It’s pretty intuitive for people. It’s been used to model many problems, esp theoretical, whose applications and rules were reviewed by humans for accuracy. There is some use of it in programming in data structures and composition. Deployments haven’t invalidated the basic operators.

7. Basic forms of geometry. You can get really far in making stuff happen predictably if it’s modeled in or built with simple shapes. See Legos and construction in general.

8. Basic probability theory. There’s a ton of use of this in empirical research. They all seem to trust the basic concepts after their observations of the field.

One can do all kinds of hardware/software verification using 1-6 alone. I’ve seen 7 used periodically in applications such as digital synthesis and collision avoidance for guidance systems. I’ve seen 8 and similar logics used when the problem domain is imprecise. If those logics and their basic rules are empirically-proven, then the applications of them have high, empirical confidence if they’re using the rules correctly. At that point, most if not all problems would show up in the extra items with less or no empirical verification such as formal specifications of the problem and implementations/checkers for those logics. Formal methodists always note things like checkers are in TCB keeping them simple and small. Verified checkers also exist which we can further test if we want. Regardless, looking at these logics and their rules as theories with massive, empirical validation means we’d trust or watch out for the exact things the formal verification papers say we should. Everything in the TCB except the logical system and rules if it uses one of above.

The mathematical concepts themselves are as empirical as anything else. Probably more so given they’ve had more validation than techniques many empiricists use in experiments. The statisticians and experimenters arguing among themselves about the validity of certain techniques probably don’t have slightest doubt in basic arithmetic or algebra. It’s clear given they rely on math in their experiments to justify their own claims. Gotta wonder why they doubt these maths work only when the inputs are numbers representing programs.