Readit NewsLet's say you live in a non-flat space. You come up with the idea of a "circle" with the usual definition: the set of points on the same plane equidistant from a central point.
You then trace along the circle and measure the length, and compare it to the length of the diameter. It turns out that this ratio changes as a function of the diameter. This truth is inherent to curved spaces themselves, and is not an artifact of choosing to describe the space using a projection.
The definition of pi in this world is no longer the invariant ratio of diameter to circumference. You can still recover pi by taking the limit of this ratio as the diameter length goes to zero. But perhaps mathematicians in this world would (justifiably) not see pi as such a fundamental number.
Now back to GP's example: people living on a sphere (like us) are analogous to inhabitants of a non-Euclidean space. The surface of Earth is analogous to 3-space, and the curvature of Earth is analogous to the curvature of space.
And indeed, if you draw real-life larger and larger circles on our planet, you will find that the ratio of circumference to diameter is smaller than pi. For example, if you start at some point on the Earth (say, the North pole) and trace out a circle 100 miles distant from it, you will find that the circumference of that circle (as measured by walking around that circle) is a little bit _less_ than pi x 100 miles.
Again, we have done no "projection" here. We've limited ourselves to operations that are fairly natural from the perspective of a mathematician living in that space, such as measuring lengths.
The fact that large circle circumferences measure less than pi x diameter on Earth did not change how we developed math, likely because you only notice start to notice this effect with extremely large (relative to us) circles.
But perhaps the inhabitants of a non-Euclidean space that was much more highly curved would notice it much earlier, and it would affect their development of maths, such that the number pi is held in lower regard.
That said, you could spin this by suggesting that the mathematician living in that non-Euclidean space might also have a different perspective on numbers. If we assume pi is still constant for him, then the numbers he's always known could be shifting in value but maybe that's a stretch.
As an example, I do car restoration as a hobby. It’s a big, big task to basically dismantle a car, fix body issues, rebuild the engine and transmission, clean up all the parts, and put it back together. Looking at the entire task outside of it, seems almost impossible to do, but I almost never think about the end of the work. I just think about the next thing I need to do.
I think marathon runners do something similar, or so I’ve heard anecdotally.
Alternatively, there might be a general assumption that lower development costs equate to inferior quality, which is a flawed yet prevalent human bias.
And turns out very few people want actually free speech. We're in a forum with strong moderation and the discussion is better for it. Most communities self-enforce norms even without central moderation. There's no easy answers when you have to reckon with the real effects speech has. Germany wasn't special, they weren't even alone at the time. What folks call "fascism" naturally precipitates under the right conditions and I can't think of any time in history where it's been dealt with by the socratic method and not violence of a kind.
But once you have a word you can accuse someone of with actual repercussions folks acting in bad-faith try to fit people they don't like into the mould. We think ourselves so much better than those silly puritans accusing people of witchcraft but we just changed the words. I'm sure you could name five off the top of your head that people level without any kind of justification.
In real life on the other hand I want to be able to say stupid things and even if I might be more sensible to others' "hate speech" I would not want that to be banned.
GPT-4o is also terrifyingly good if prompted to think investigatively. I've uploaded a photo of a friend's generic backyard with no text, signs, people, sky, or anything obvious, and it nailed their specific neighborhood. I've even sent it nothing more than a photo of my bathroom (with product labels blacked out) and it still identified sufficient clues to guess a locality within 200 miles of my actual location.
A regular Sherlock Holmes with superhuman levels of world knowledge.
I would understand your argument if you were talking about services.
Most lawyers, when you ask them to write a brief, will cite only real cases.