![]() It is a construct of the imagination.īut the continuum is also a convenient approximation to the real world, and hence, an essential tool for doing some forms of math. It is not the analog of anything in the real world of discrete particles and quantum energy levels. We can’t even devise a system to list all the real numbers that exist between 0 and 1.īefore we get too worried, however, it helps to remember that the continuum is a purely human invention. What’s worse, Georg Cantor demonstrated that the infinity of the continuum is qualitatively different from the enumerable infinity of the counting numbers. This is the infinite range of real numbers between any two numbers, and it seems to inherently exist any time we conceive of a number line. It is not ‘that which has nothing beyond itself’ that is infinite, but ‘that which always has something beyond itself.’” (Robin Waterfield translation)Ī much thornier problem is the continuum. As Aristotle said in Book III of the Physics (which remains one of the most clearheaded discussions of the subject) “Infinity turns out to be the opposite of what people say it is. For every number you can conceive, just add 1 to get another, even if you exceed the number of particles in the universe.īecause we count on and on without reaching an end, the infinity of the counting numbers is always a potential rather than an actual quantity. Just start counting 1 2 3, and the concept of infinity raises its ugly head. So where did the notion of “infinity” even come from? From mathematics, I think. If each universe splits into two every time a wave function collapses, that’s definitely a whole lot of universes, but it’s still not infinite. But I’m not sure it’s justified in this context either. In recent years, the word “infinity” has been often used in connection with the theory of the multiverse. There is a limit to the subdividing of space. The properties of elementary particles are not continuous but discrete. ![]() Quantum theory seems to deny the existence of the infinitely small. Since space is defined by these particles, there is no infinite space either. The number of atoms in the universe is about 10 80, and while that’s certainly quite a lot, it’s still short of infinite. ![]() The Big Bang caused only a finite amount of matter and energy to come into being, and the amount can actually be estimated. I just don’t see any evidence of infinity in the real world. I hope some of these questions made sense, and maybe you could point me to some relevant information on this topic because the Wikipedia entry on infinitesimals is rather poor.For as long as I can remember, I have been skeptical about the existence of infinity. If points of an infinitesimal size exist, could the length of a line be defined as sum of the length of an infinite number of infinitesimal points? I imagine this would cause some difficulty, as the number of points between 0 and 1 are mappable to the number of points between 0 and 2, though clearly the lines are not the same length. Are there also multiple orders of infinitesimals? Can one infinitesimal value be smaller than another? It is generally agreed that there are multiple orders of infinity. ![]() If infinitesimal quantities exist, would it be reasonable to posit a point which has an infinitesimal length in every dimension? Similarly, could a one-dimensional horizontal line be seen to have an infinitesimal height (as well as width and metrics in every other dimension)? If not, is the line's height in other dimensions 0? Is there a difference between having a height of 0 and an infinitesimal height? Most mathematical definitions that I've seen for a "point" assume that a point has no size in any dimension. Does this mean, though, that the tangent line to a curve at a particular point is actually a line defined by two points separated by an infinitesimal distance? Is there a generally agreed upon, well-defined concept of infinitesimal values in mathematics? As I understand it, in differential calculus, we use the concept of the limit 1/n as n goes to infinity to define the concept of a derivative. I've recently been wondering about infinitesimal quantities, and there are a few things that I'm having difficulty wrapping my head around. ![]()
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