
Defending Scientism: Mathematics Is a Part of Science - nickpsecurity
https://coelsblog.wordpress.com/2014/05/22/defending-scientism-mathematics-is-a-part-of-science/
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data-witcher
Math is everywhere, like it or not. If you use computer, smartphone, drive
car... all devices are either designed or using methods deeply grounded in
mathematics.

The caveat is that mathematics _isn 't_ ideal for describing the world.
Undoubtedly it is very useful in many fields (e.g. computer science, physics,
biology, chemistry ...), but requires multiple adjustments, specific to the
domain.

Take for example statistics: correlations. Start changing a single value in
one of the vectors. Pearson/Spearman correlations coefficient will start to
drop, so as the methods that are based on those measures. Eventually, they
will suggest that the vectors are not correlated. And all you did was just
adding a single outlier...

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nickpsecurity
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.

