What would our world be like if we only had knowledge of mathematics up to (and including) precalculus? Would we be able to build skyscraper? Would we be able to build planes? How different would our world be, or would it not be that much different?
I've always known that mathematics is important, but just how important is it?
Others have talked about what we could do without calculus already, so I want to pick up a different issue:
Why would we not have calculus? If your civilisation just hasn't discovered it yet - easy, look at history to see what it was like before.
But if they're at a similar point in their timeline as we are, I cannot think of any logical reason why they should discover everything up to precalculus and then just stop, especially if the rest of their society continues to develop normally without any kind of calamity causing severe setbacks in terms of science and knowledge. To me, that doesn't make much sense.
Apologies if you've already got all of this covered and it just isn't mentioned in the question. Otherwise, I think that's almost more important to figure out than the impact on the world, because it would just not be a believable starting point.
You don't need symbolic calculus to do most of the practical calculations for engineering. Numeric approximations using iterated algebra can give you the right answer -- and in fact, the majority of computer simulations of engineering problems use exactly this method.
You can apply the same techniques to most of theoretical science. True, an iterated-algebra formulation of Maxwell's equations won't be as elegant, and general relativity and quantum mechanics are probably too complex to deal with, but I'd be amazed if you couldn't get up to near-modern technology before the lack of advanced math became a problem.
We would have to do more things by trial and error. We'd have to build things as we figure out how to build them. We may not be able to build just anything we please. Mathematics gives us the power to picture something, and build it entirely in our heads (or perhaps in the computer), and then build it in one big effort from start to finish.
Grand architecture would thus have more "corrections" in it. You'd see structures whose design permits the builder to make very subtle adjustments to the structure to make up for oversights in lower levels which calculus would have caught. This would lend itself to a much more organic structure, where we spend less effort demonstrating how proudly we can trust a steel girder to support our weight, and more effort ensuring we never really have to trust the girder in the first place. I can see an architect bouncing up and down lightly on his cantilever, and making the decision to adjust the building on top because the cantilever isn't as strong as he or she might hope.
I'd say skyscrapers would be unlikely. I'm confident that, at some point, we could manage to construct them as architecture evolves. However, we also have to consider that a lack of math would affect much of the rest of our life, and we may not have the desire to push steel and glass structures into the sky.
What might be interesting is to look at what vessels can act like math in the absence of higher math. I think it'd be interesting to see a world where "plans" for a building are actually stored in a song sung by the foreman as the laborers work their own tune with the material.
Actually we can do quite a few things without using calculus. The Wright brothers didn't use calc to make their plane. We might not be making super-sonic jets or at least not as 'easily' but many calculations can be approximated with other math. I think the big thing is we would have more trial and error in many of the more math intense areas, such as nuclear bombs and power generation. In many ways Calc is a shorthand to more quickly get the same answer. kind of like multiplication is to addition. The other ways just take longer and are more prone to human error.
Well, people were able to build quite impressive structures before calculus was invented (17th century) so look at late medieval and early modern age. But I don't think that you could have a significant scientific progress without calculus. I mean, the simpliest equations for movement (s = v * t and that kind of stuff) were derived using calculus. So how can you then calculate the air flow around a curved surface (which is necessary for aircraft design) or something like that?
So my conclusion: yes it would be different, mainly in that it would be less technologically advanced - really think like going back in history, because you cannot get around mathematics, it's in all science.
Newtonian mechanics isn't possible to derive without calculus. This means no machinery of any complexity (certainly no engines), certainly no real control or mastering of electricity or magnetism.
Building a large building could still be done out of rock and cement, but not a modern steel one.
Basically it's hard to believe that the industrial revolution could ever happen. Tech would be limited to growing at a snail's pace starting in the late 1700's.
