Ok so mathematically you can divide any number by any other (nonzero) number and you can keep dividing that number however many times you want. Like dividing 1 by 2 and then by 2 again etc. And this is the basis of the famous paradox that mathematically, you cant really move from point a to b because first you need to get to the middle of a and b. and then to the middle of the middle. and then the middle of the middle of the middle, etc. But what if space is not continuous, but quantized? Like what if there is a smallest possible length, and you cannot be in between that length, meaning you cannot physically divide that length by 2 to get to the middle (even though mathematically you could). Wouldnt that have some serious consequences on the physical application of calculus to the real world? (maybe not when working with large bodies, but definitely with small scales?) For example the intermediate value theorem wouldnt hold true... Idk im not calculus expert (only had calc I and II nd basic physics) but this thought occurred to me and has bothered me..
No, calculus has never had the ambition to give an exact description of space. There are many problems with calculus as a description of space: space and time being discrete, the existence of points which are infinitesimal small, lines which have a length but not width, etc. Calculus should never be looked at as a complete description of our physical world, but merely as a very useful approximation. That is, when you throw a ball in the air, then its path isn't an exact parabola, but it can be approximated by parabolas. This is so with everything in physics: everything is an approximation of the real world. Exactness is never claimed. But why are our approximations so good? We don't know. This is (in my opinion) the greatest mystery of the universe. Why is math so good in approximating the universal laws?
First of all - math doesn't deal with the real space. It doesn't have to care about whether the real space is continuous or not, it does have to care about properties of the idealized space. And we define this idealized space to be as we want it to be.
But physics uses math and deals with real space...so it should care if space is continuous or not, at least when it is used to describe motion in space and watnot... Does anyone know of any literature that discusses the relationship between the possible discreteness of space and mathematics?
Good evening JessJolt. You have touched on a deep and interesting question. Shan Majid (professor of mathematics at London University) for instance offers exactly this quantisation as the reason for our difficulty in generating grand unified theories. See his essay "Quantum spacetime and physical reality" in the book he edited "On Space and Time" (Cambridge University Press) go well
However, those are questions about physics, not about Calculus or whatever other mathematics is used to model physics. If space is, in fact, discreet, it might mean that, for some questions about space, Calculus would not be the appropriate mathematical tool (which is what you perhaps meant), but it would not mean the Calculus itself was wrong.
You seem to be missing the point made by the previous posters. The math does NOT deal directly with the real world, it is an approximation. It is a damned GOOD approximation and gives us excellent answers but the real world really doesn't care what the math says and the math really doesn't have to worry about the real world. WE have to worry about the correlation between the two, which you are rightly attempting to do, but this particular "problem" that you have brought up just ISN'T one and we'll keep using the math as long as it gives good answers.
Well from my experience it seems like calculus is the main mathematical tool in describing physics... I read Majid's essay (thanks Studiot), and he describes some math that he and others developed to include the discrete nature of spacetime, although its too complex for me to understand lol, but it's interesting that in the future perhaps some more accurate mathematics will replace calculus as a main tool for describing physical reality..
Sorry, I didnt mean that calculus doesnt give good approximations nd that we should stop using it. But i still think its a problem that calculus cant describe nature at small scales if calculus assumes spacetime is continuous if it possibly isn't...my point is that there should be developed a math on the basis of discrete spacetime and to see if that math can work well with both small and large scales
But math isn't a tool for the sake of being a tool. It's a study in its own right, independent of reality. It's the fact that calculus happens to approximate the real world that makes it a tool. It wasn't designed to be a tool. Again, calculus isn't developed to describe reality. If one is worried about discrete intervals, one can just use difference quotients and Riemann sums; problem solved.
http://en.wikipedia.org/wiki/Discrete_calculus http://en.wikipedia.org/wiki/Quantum_calculus http://en.wikipedia.org/wiki/Quantum_differential_calculus The problem is that these are computationally hard.
Yes, but it is NOT a problem for CALCULUS. I guess that's what threw me about your reasoning. It is a problem for US in that we may need to find a better approximation tool if it comes to that, but as other posters have pointed out, this is NOT a flaw in calculus. I do agree w/ you that it is unfortunate that it may be that calculus, which is one of our best tools, is not applicable in situations where we might wish it to be.
You can't say Calculus is wrong unless it isn't consistent. Calculus is just a bunch of definition that turns out to be useful.
I'm not sure this is correct. It was the need to describe natural phenomena that lead to the discovery/development of calculus. So, I think it was, at least initially, designed to be a tool.
It is my understanding that Newton developed calculus as a way to describe physical stuff, is this correct? I don't really know what Liebniz was doing, but was it purely mathematical? I always thought that Liebniz's mathematical interests were motivated by solving problems. Am I wrong in this? Either way, I understand what you are saying :)
In a nutshell, as far as I know, Newton developed calculus in order to solve physical problems, Leibniz developed calculus to find the area under a curve.
An accurate title for the original post would be "If space is not a continuum then calculus is not applicable". I wonder how Zeno's paradoxes get resolved if calculus is not applicable.