How badly would a tidally locked world be affected in terms of science if they lived on the sun-facing side (because anything on the icy-side would probably freeze to death), would it just be that they'd possess the Naïve idea that their world and a select few other planets and their star because they wouldn't be able to see any stars, would it disrupt the idea of time keeping since the sundial and the like doesn't work? Exactly how scientifically affected would they be?
closed as too broad by Mołot, Hohmannfan, Azuaron, Bellerophon, JDługosz Oct 30 '16 at 0:20
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TL/DR: They probably wouldn't be too far behind us, if at all
First Point: Tidal locking of a planet wouldn't necessarily mean that you have a hot side or a cold side, if a planet has sufficient means of transporting heat from the hot Star-facing side to the cooler night-side; either through water or wind motions; then you'd find that, assuming the planet is in the habitable zone of its parent Star, the average global temperatures would be pretty good for life to exist on both sides.
Addendum: That said, plants wouldn't grow on the dark side as they need photosynthesis, so there would have to be more plants on the sunnier side or in the penumbral regions between night and day
Second Point: Our concepts of East and West are based on the rising and setting of our own Star; with a Tidally-locked planet, that wouldn't be possible; instead "East" might mean towards the Star (with the East-most point being directly underneath) and "West" might mean away from the Star; it's up to you as to how exactly your denizens would refer to it.
Third Point: I'm going to address the notion of time. If the planet is tidally-locked from the beginning of its creation; time would probably be measured on the night-side; by the motion of the stars. The Maya used the motion of the Stars in their long-count calendar to identify long periods of time; and then break those down into chunks; the same would probably work for your civilisation as well.
Fourth Point: While it's true; at least for a time that they'd believe they were stationary and the centre of the universe; the same was true of our own history. Eventually; though as people would use the dark-side of the planet and telescopes to peer out into the stars, they'd see that, in fact, they weren't the centre of the universe. This might take longer to catch on in the Star-facing side (as that would drown out all long-distance light); but I doubt any civilisation that managed to grow on a tidally-locked planet would be too far behind our current level of technology if the same time-frames as us on Earth are applied.
It is possible for a tidally-locked world to have its habitable temperature region (for its inhabitants, since that may be different from the range for humans!) extend onto its night side, but I assume you are interested in the case where it does not.
Not being able to see the stars won't have a significant effect on science for a long time; humans didn't have any information on what stars were until quite recently.
They will be able to figure that their world is round: the sun will always be overhead at the same point and lower in the sky the farther you go from that. I can't think of an obvious way to tell what direction you're going, though.
A critical question: Do they have a moon? If so, they will be able to develop the concept of orbits early on. If not, they won't have the idea until their Newton realizes what you can do with projectile paths and a round planet.
A moon would also allow them to distinguish north-south from east-west. Its period would be a measure of time.
If the planet's orbit is sufficiently elliptical, they will still have seasonal cycles and will thus have a concept of a year, though they won't have an accurate way to measure its length for much of their history.
In biological terms, living on a planet with no day/night cycle means they simply have no reason to care about measuring time in the first place. They will be just as able to make pendulums and water clocks as pre-industrial Earthlings, but at what point will they start doing something which they feel the need to time?