Could science continue even if a civilisation breaks down?
Im just wondering, would it be possible for science and progress to walk on, after democracy and peace are gone? If so how?
If you're thinking of a lawless Mad Max style apocalypse, most likely sciences will be regulated to discovering things that were already known but then lost to humanity. You could see it like the unrest after the Roman Empire fell (the Dark Ages) where lots of knowledge was lost and had to be learned again from scratch.
It is entirely possible that things will still be recognizable as a civilization even if democratic institutions are gone and even if a nation is at war.
There are pursuits of sciences under a monarchy and even authoritarian dictatorships. During time of conflict, science might focus on military technologies (which often have later civilian applications) or on technology to help pacify the population (agriculture, comfort goods, surveillance technologies).
Or you could encode knowledge in a flue-bacteria.
Assuming you're in the US or Western Europe, you need to recognize that societal collapse wouldn't look like the middle East or Venezuela, where things are a fishbowl of chaos in a stable world. In the US, just-in-time logistics means we're fragile to the point where a week of trucks not making deliveries might be sufficient to stall out the country so it can't start again. Grocery stores have minimal extra stock, but we can work around that. Vehicle repair businesses can't sustain operations for even a week before they run short on parts. Parts start breaking down and people don't show up for work, and now your power plants aren't working. From there, you need to hope there's someone who can swoop in to save our asses, because with broken down vehicles, depleted gas stations, no electric grid, and all the societal chaos that ensues, we won't be self-starting ourselves. Maybe, just MAYBE, we could plug vital electric loads into aircraft carriers and submarines long enough to restore order, but it's a race against time, because there are plenty of vital components that could go at any time.
Science would proceed in a different direction, with agriculture and botany and did preservation and hunting and animal domestication being the fields where advances are made. Backing up Wikipedia might work for a decade, if your computer and storage media hold out, and if you can defend your windmill/PV panels from armed looters, and if you have the necessary hardware to make necessary repairs to wear components. But you'll probably have more pressing matters, like eating.
And it's a long climb back from the bottom. A lot of our important industries require extreme precision and tolerate few or no material defects. In modern America, fighting off complacency to preserve top notch supply chains is no small feat. If people are worried about the next meal for anyone in their family, you can forget about all that. And then we get to the uncomfortable reality that we may not be able to sustain quite so many billion people without mechanized farming and engineered crops.
There will always be science and education, but it's hard, when living in a stable society, to recognize just how expensive it is to take large numbers of your smartest and most capable people, and allow them to produce nothing that can be utilized right now, because they're investing their time in very abstract things that may pay off in future decades. That's why smart kids from poor families so often don't go to college, too.
If there is no one here to practice science, then the word loses its meaning.( tree falling in the woods analog)
If we take the lesser of your dystopia, Civilization exists its just not democractic nor peaceful, Here I give you the example of the Klingons (Star trek nerd that I am :) )
Meaning, more strange things, less besiege weapons.