taste me, as the food and drink Alice found almost said. she was cast unto a stormshorn sunderedsea. you too will fall beneath my waves in time.


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garak
@garak

By the current standards of the terms, Mathematics is ❌Not Science (approving).

tl;dr: Science makes claims of the form "X is true." Math only makes claims of the form "if X is true, then Y is true." These two can be combined fruitfully, but math on its own is not a science.


Background: I have an undergrad degree in Math, and having a hobbyist interest in its historical development. My experience is that the distinctive character of math tends to be obscured in general-audience math classes because math-as-tool looks different than math-for-math. It's only when the physicists clear out of the building after learning enough calculus for their EM equations, that the professor says "ok, now for the real stuff" and puts three axioms on the chalkboard, and you spend the entire year not being able to use any facts except those three axioms and what you can prove from them.

Science

Science is generally understood to be an empirical pursuit. The core concept of modern science is verifying beliefs against observations of the world, and being willing to reject beliefs that are contrary to empirical observations ("falsifiability"). This concept of science dates back to the Enlightenment or so, and is more-or-less considered the turning point of when "modern science" began separating from its predecessors that it has previously commingled with, like alchemy and astrology.

(Of course, the human institution of science has many failures both small and large of adhering to that principle, with a trail of human misery in its wake. But that's still basically its stated aspiration.)

Math

Mathematics is not empirical. You cannot refute a (purely) mathematical statement by making an observation of the natural word. Mathematics is not falsifiable because it doesn't make claims about the natural world in the first place.

If you make an observation about the world, and it contradicts math, that means that you picked the wrong math to describe the world, but the math itself is not wrong.

Terrestrial Example

Standard geometry, following the axioms of Euclid, says that the degrees of a triangle will always sum to 180 degrees. But if you draw a equilateral triangle on the globe with one vertex at the north Pole, and base along the Equator, its angles will sum to 270 degrees.

You will find that the axioms of Spherical Geometry apply to the surface of the Earth instead. That axiom system accurately describes the surface of the Earth. Your observations will not falsify it. So, does that mean that Euclidean geometry is "wrong"? Have you falsified Euclidean Geometry?

No!

You can still use Euclidean geometry to describe the 3-D space that the Earth lives in! And it accurately describes that space, correctly. (Well, there are some cosmological models where this is not true and space has a curvature, but my understanding is that the current status of this is that observations have not convincingly falsified this model.)

It means that you picked the wrong math to describe the surface of the Earth, but the math does describe something else. That means the math itself is not inherently or internally wrong. You have falsified the application of a mathematical model to a natural system, but not the mathematical model.

Because... Euclidean geometry doesn't make claims about the surface of the Earth. Or about space. Or about any aspect of reality or existence. It makes claims about the consequences of axioms. The claim about reality is that the Euclidean axioms apply to the surface of the Earth, and that can be falsified by looking for the consequences that math predicts from the axioms.

Mathematics as Metaphysics

The study of (pure) math is the study of self-contained, fully-abstract systems for internal consistency. And it's the study of the consequences of axioms. To an extent, it considers itself a proper sub-field of Logic. Boolean, Russelean, hard-core mechanically-separated Logic.

Mathematics does not study things that exist. That's why it can deal with things like, oh, y'know, infinity. It probably doesn't exist! The universe is finite! I couldn't give a fuck! Putting a hard-cap on things makes models irritating, so we don't. Ditto infinitesimals, ditto continuums: these are mathematical conveniences, not descriptions of reality, and if mainstream mathematicians gave even the slightest shit about fidelity to reality then we would spend more time working around them. But we don't.

Mathematics 🤝 Science

If science falsifies claims about reality, then an expressive mathematical model is a powerful tool to test claims. You know we don't actually "see" neutrons in a particle collider, right? We have statistics from macro-scale science-ass detector instruments. There's mathematical models of how particles behave, and they eventually boil down to "if I set the knob here and press the button, then twelve hours later model A says the graph will look like a hokey stick and model B says it will look like a gentle curve," and those models also disagree about the mass of the Higgs boson.

That's how we discover the structure of the universe. Extremely large mathematic models let us say that X is a logical consequence of Y. So we can do experiments that test X, and this gives us confidence about Y.

This distinction is not universal

This view of math as an ivory-tower discipline obsessed with pure-thought models largely came about in the late 19th and early 20th centuries, as a result of the influence of a prominent group of mathematicians, logicians, and philosophers like Bertrand Russel and David Hilbert. Math was literally, I am not exaggerating, described as the "manipulation of meaningless symbols according to arbitrary rules." They refuted that there are even concepts behind the axioms, that having a mental idea of a "point" or a "line" is excess ornamentation. This view is still influential, although that extreme isn't actually believed in most mathematical circles (it is, detrimentally, the basis of much grade-school mathematics education).

By contrast, in the time of Copernicus, mathematics was spelled "geometry" and was a synonym of astronomy, the field of science that studied planets and zodiac signs. Empiricism had not yet caused astronomy to cleave from astrology, and mathematics had not yet found its independence as a subject. Going further back, the ancient Greeks famously had some misgivings about polynomials: the system x^2 + x^3 is nonsensical because x^2 is an area and x^3 is a volume, and adding unlike terms is ill-founded.

So what I'm saying is, historically mathematics was much more grounded in empirical observations, and making models of a particular physical system. It's only somewhat recently that science has become an expression of Empiricism, and it's extremely recently that math has separated from science to that extent. and the results have been useful! Mathematics now has the freedom to explore and tinker with models without regards for applicability, the result is a veritable bestiary of models, some of which then find applications later on.

I'm also given to understand that some cultures (even European ones!) don't view science as so fundamentally Empiricist as we do here in the Anglosphere. I'm sure there's some academic British/Continental cultural split in the 19th Century that plays into this, but I'm unaware of the specifics.


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in reply to @NireBryce's post:

so I didn't put that much thought directly into the original comment but I think it's basically a reversal/reductio ad absurdum of arguments like "STEM is superior to the humanities/soft sciences because it has objectively right and wrong answers"—empirical observations are gathered subjectively through human experience and senses; to be truly objective you have to remove the human element, which means that science is just reading numbers off machines. and by that oversimplification, math (along with empiricism in general) involves too much human thinking to be a science.

(I think what @garak posted while I slept on this answer explains why math isn't a science in a way that I don't know enough to explain myself but had an intuition towards)

yes, I think the post answered the objection I had, which was that mathematical problems are everywhere contingent upon real-world observations—statistical observations, packing problems involving discrete objects, geometrical constructions...but the point is made that real-world observations don't falsify mathematical models that don't conform with observations.

and then this gets into issues of whether the way human beings perceive the world is in fact a function of the sort of math they're taught. suppose, because of some strange historical accident, human intellectual culture didn't converge on Euclidean geometry; others are possible, so...what if we all thought in hyperbolic terms? what would that even mean for how people interacted with basic geometric shapes? we're through the looking-glass at this point. ~Chara

i suppose it depends on the mathematician much more than it would depend on any given scientist (frauds excepted).

edit: i guess that means surgeons are engineers and doctors with any competence above checklists (a rare few) are scientists of a sort

maybe -- there's def a research aspect, but imo appendectomies are slow days but they're hired for the problem solving ability when something happens. I suppose students end up as technicians but still are surgeons, though

That's kind of the trick right? Math isn't establishing general truths about the universe in the same way that a science generally is, it's establishing truths that come with a bunch of fine print the truths follow more directly from that you have to check fits the situation before you know the truths apply

in reply to @garak's post:

ooo this gets into it far more than I expected anyone to. I was thinking about how the discovery process often is very close to the scientific method because of the history, but wasn't thinking about the fact that physics' models being continually in flux/question is more math than science here which is part of what I was comparing when thinking about it