Will science end?

[Chronos]

Geometry certainly isn't finished, because (as Euclid showed) you can use geometry to do number theory, and there are plenty of open questions in number theory.

For trigonometry specifically, last year a couple of high school students published a new proof of the Pythagorean Theorem using the Law of Sines, just because they had read in a book that it was impossible to do so.

[NichG]

That's a fun method!

I'm sort of bewildered that the 'that's impossible' claim existed in the first place. Was it just a textbook author saying it, or was there actually some formal proof that is now in conflict with this result?

[Bohandas]

That's a fun method!

I'm sort of bewildered that the 'that's impossible' claim existed in the first place. Was it just a textbook author saying it, or was there actually some formal proof that is now in conflict with this result?

I was about to ask the same thing

[Lvl 2 Expert]

Anyone publishing new results in Trigonometry these days?

Good call. Maths as a whole hasn't ended, but certain fields of math can indeed probably (I don't know much about math) be summarized as (more or less) complete ever since some old standard work, perhaps as far back as Euclid's Elements from 300 BC(E).

And those basal fields of math are also unlikely to ever be thrown out because the universe can't hold enough data, because they're important to other maths and sciences, so they won't have to be rediscovered.

Edit: Although Chronos seems to now more about math than I do, and his reaction indicates we might need to find pretty small and specific subsets of math in order to call them (more or less) complete.

Alchemy evolved into modern chemistry, I don't know if you can draw a line somewhere and say "Alchemy ended here".

At the same time alchemy is not so much complete as it is just discarded as false. It's like aether theory, the idea that light travels through a background medium called the aether and you should be able to measure differences in the speed of light based on your own movement through it. That's a field of study that’s just gone (mostly at least, I'm sure there's some dabbling) because the basic idea turned out to be wrong. The same methods and instruments and physicists were happily applied to other fields of physics, and I'm sure good things for the rest of physics were discovered or developed during the falsification of this idea, but the aether itself is a closed chapter. Does that make it complete?

[Radar]

Geometry certainly isn't finished, because (as Euclid showed) you can use geometry to do number theory, and there are plenty of open questions in number theory.

For trigonometry specifically, last year a couple of high school students published a new proof of the Pythagorean Theorem using the Law of Sines, just because they had read in a book that it was impossible to do so.

You can also do everything a quantum computer does using geometry:
https://arxiv.org/abs/quant-ph/0611279v2

There is a whole field of geometric algebra that people actively publish in.

[veti]

At the same time alchemy is not so much complete as it is just discarded as false. It's like aether theory, the idea that light travels through a background medium called the aether and you should be able to measure differences in the speed of light based on your own movement through it. That's a field of study that’s just gone (mostly at least, I'm sure there's some dabbling) because the basic idea turned out to be wrong. The same methods and instruments and physicists were happily applied to other fields of physics, and I'm sure good things for the rest of physics were discovered or developed during the falsification of this idea, but the aether itself is a closed chapter. Does that make it complete?

I haven't studied this, but my gut feeling is that there's a difference in kind between a "field of study" and a "theory". Aether was a theory, which was (AFAIK) discarded when Einstein came up with special relativity as a better alternative. But alchemy was much bigger than just one theory or even one philosophy, and it's not all "discarded as false" by any means.

What we mostly associate with alchemy today is the hilariously elaborate idea of the "great work" that culminates in the creation and use of the philosopher's stone. But that whole theory was a relatively late (Renaissance) invention, born (I'm guessing here) of some alchemists' ambition to be taken more seriously as scholars by making it seem as if they had a grand, unified plan. Long before that, people were doing "alchemy" for smaller, practical problems such as manufacturing and construction, and they did perfectly good work that's still visible in modern practices. For instance, the alchemical terms "sublimation" and "calcination" are still in use today, with their meanings basically intact.

[Quizatzhaderac]

That's a fun method!

I'm sort of bewildered that the 'that's impossible' claim existed in the first place. Was it just a textbook author saying it, or was there actually some formal proof that is now in conflict with this result?

There was no formal proof, but most of the basic rules of trigonometry depend on the Pythagorean theorem, so any proof involving them would be circular.

[NichG]

There was no formal proof, but most of the basic rules of trigonometry depend on the Pythagorean theorem, so any proof involving them would be circular.

But you can get trigonometric functions from complex numbers and the exponential, and derive sin^2 + cos^2 = 1 that way without going through geometry...

[Mordar]

Oh dear. I beg your forgiveness. Let me attend to that post haste.

My work is elegant in its simplicity. - Peelee et al.

Imean, really, your contribution is negligible, so that was a kindness.

You're in my lab. It doesn't matter if I did a lick of work!

i'd say in the 1920's that notion of science being complete was being actively crushed to bits. And this sudden opening of our worldview to the great unknown was probably fueling Lovecraft's paranoia. List of some things changing in science in the early 20th century:

Scientists didn't believe they had achieved all-knowing status...but the American public in particular was extremely impressed with how much mankind knew, and how much of it was because of the United States. This was accompanied by the science replacing religion arguments, and the prosperity of the country had people thinking the US was omnipotent and omniscient. This was the backdrop for the cosmicism that Lovecraft turned up to 11...I think it wonderfully fitting that even now we refer to his writing as the C'thulhu Mythos when C'thulhu itself was a pawn of true powers, a lieutenant at best.

- M

[Lord Torath]

Here's an interesting episode of the "Uncertain" podcast, that I thought was relevant here:
Overconfidence Can Blindside Science and Society Alike. Here's How Not to Get Fooled

Essentially, it's a reminder that science is done by humans, and as humans, we are flawed, and thus, a lot of our results end up being flawed. We have have discovered many beautiful and exciting truths using science, but there have also been many, many things that became "science" that are wrong. Everyone knows penguins have over a hundred feathers per square centimeter of skin, everyone eats a credit card's worth of plastics every week, and if you don't have 8 cups of water per day, you'll die of dehydration. And yet, those are all false.

So no, I don't think we'll ever stop learning new things, and learning that some of the things we thought were true are wrong. I think science will continue for as long as there is someone (or something) capable of doing it.

[Peelee]

You're in my lab. It doesn't matter if I did a lick of work!

You got the cite already!

Besides, if we define your lab as "my lab".......