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Comment by YorkshireSeason | Hacker News Reader

I'd be interested to learn about results in the "hard sciences" that I work in (mathematics, logic, program verification, typing systems), what papers "everyone in the niche knows is false", but "fears to speak out about".

Sure, mistakes happen, my own PhD contained a major blunder (discovered several years after submission -- at least somebody read it ...), but I'm extremely open about it.

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In my view, the hard sciences aren't really directly comparable. We doubtlessly have our own problems, but not necessarily the same problems as the health and behavioral sciences.

I think that the hard sciences aren't necessarily so dependent on papers derived from isolated results, or even replication of those results, because there is more of a "web" of knowledge. An idea is hopefully supported by an experiment, but possibly also by theory and multiple different kinds of experiments, sufficient to be usable by non-experts and laypeople. For instance the basic theory of electromagnetism is good enough for virtually any kind of engineering, and it would take some monumental discovery to prove it false, even if a portion of the supporting work fails upon replication.

Another thing is, perhaps because of the precision of physics, we get proven wrong every day, and it ceases to be an embarrassment. The road is littered with the wreckage of failed theories and experiments.

FWIW, your examples aren’t hard sciences per se. The term hard science is referring to natural world subjects like physics, chemistry, biology and astronomy. It’s not a difficulty or value judgement, it’s just a term with a specific categorical meaning.

I wonder sometimes whether math isn't a natural science by accident. If 2+2=4 wasn't applicable to the real world, I don't believe we would have defined math the same way

That’s an open (and interesting) question, I think, whether there are absolute physical truths in math, and whether our basic math rules can be defined any other way. But certainly there are huge swaths of math that don’t correspond to the physical world, and there are parts of math that are decided rather than natural consequences.

Ultimately though, the math that is natural science is just called “physics”, right? ;) The reason math can’t be a hard science is because we can’t prove abstract math by physical experiment. That’s all “hard science” means - things you can show by physical experiment. We can prove math theorems by showing they’re consistent under all the other math theorems, but generally not by conducting physical tests.

Not all mathematics formalises physics. Nowadays, much math formalises other maths, e.g. set theory, category theory etc.

Right, exactly. Physics already is the subset of math that is a hard science, the rest of math isn’t.

Yet

"I will begin by saying that there is probably less difference between the positions of a mathematician and a physicist than is generally supposed, and that the most important seems to me to be this, that the mathematician is in much more direct contact with reality." -- G.H. Hardy (1922)

I suspect that to the ancient Greeks, math and natural science were not yet separate disciplines. "Geometry" means something like measuring the world. They had to be aware of the relationship between their math and the ability to function as a minimally technological society, surveying land, navigating, and so forth. As I understand it, they believed that math was telling them something deep about reality.

I think turning math into a purely abstract game came later, and we owe our fun to the discipline's more humble origins.

I think it would be more accurate to say that they were far more separate disciplines anciently, and far more rarely mixed. See for example, Euclid vs the the contents of this work of Aristotle: https://en.wikipedia.org/wiki/Physics_(Aristotle)

Now you're getting into philosophy. As a materialist, I do believe everything comes from the natural world. I take the view that if every time we combined two objects with two other objects we ended up with something other than four objects, we would not be saying "2+2=4".

There are of course other viewpoints that math has a Platonic metaphysical reality in and of itself, independent of the natural world.

Mathematics isn't a science because it doesn't provide us with empirical answers. That's precisely the difference between physics and mathematics: the former tells us what is, while the latter only tells us what must be, given certain specific assumptions.

Or more succinctly: mathematics doesn't give us truths, it gives us consequences. The statement, "2 + 2 = 4" is not an empirical fact intrinsic to the universe. It's a consequence of our definition of natural numbers. It's considered a useful definition because (among other things) of its application as a foundation for engineering and the sciences. But many other useful areas of mathematics have essentially no basis in reality.

> Mathematics isn't a science because it doesn't provide us with empirical answers.

Then physics isn't a science either since only parts of the field fits this criteria. I think you will realize that the distinction between physics and math isn't as clear cut as you imagine, it is not uncommon for physicists to publish in math journals and vice versa.

The distinction is pretty clear cut. The two disciplines are philosophically different by category, not degree. Just because much of theoretical physics is applied math and physicists can publish in math journals doesn't mean the two occupy the same domain. One is ostensibly guided by empiricism (even if it's not always possible), the other is not.

>Sure, mistakes happen, my own PhD contained a major blunder (discovered several years after submission -- at least somebody read it ...), but I'm extremely open about it.

So what was it?

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