NeatNit wrote:I don't understand how he could possibly claim that this is *actually* a paradox (at 6:12-ish), and a baffling one at that.

How much mathematics have you studied, and how deep down the rabbit hold do you want to go? Without rigorous mathematical definitions, the paradox is that an infinite process (i.e. one with no end) can be "completed."

NeatNit wrote:Welcome to the real world, where motion is continuous and doesn't work in "frames" nor "steps".

Sometimes mathematics can be used to model the real world, but mathematics is not the real world. What does the real world not moving in frames or steps have to do with the mathematical abstraction of an infinite series? Even if that abstraction was based on an observation in the real world? Moreover, I am pretty sure that many modern quantum physicists would argue that space and time are discrete (Planck length and Planck time being the smallest possible units). In that case, the whole problem is irrelevant, as these sums seem to model both space and time as infinitely divisible (i.e. continuous) objects.

NeatNit wrote:On every graph we draw, there is an infinite number of points. The graph clearly passes through all of them. This is not a paradox, it is the definition of continuous.

Actually, no it is not. One definition of continuous is

A function f is continuous at a point a if for every epsilon > 0 there exists a delta > 0 such that |f(x)-f(a)| < epsilon whenever |x-a| < delta. f is continuous on an interval (a,b) is f is continuous at c for every c in (a,b).

There are other more general definitions (the most general probably coming from topology: a function is continuous if the inverse image of an open set is open), but when considering functions that take the reals to the reals, they are all equivalent to this one.

Moreover, your definition fails in a couple of ways. For instance, I can find functions whose graphs pass through an infinite number of points, but which are continuous nowhere (consider the function that is defined to be 1 on the rationals and 0 on the irrationals). Maybe you meant to say that the function can be graphed without lifting the writing utensil. This definition is the one normally given to students until they take a rigorous class in calculus or analysis, and is implied by the formal definition, but is not, in and of itself, a complete definition (what does it even mean? how do you use it to prove anything?).

How is that baffling?

Have you ever seen the movie

*Good Will Hunting*? I am not claiming that it is a great movie, but there is a particular scene that I recall, which I think is quite appropriate. The protagonist hears some freshman philosophy undergrad waxing on about whatever theory had been taught that week, then proceeds to explain exactly what new theories this person is going to attach to over the next several years of their undergraduate career, thus demonstrating how shallow that freshman's knowledge really is. The freshman knew enough to

*think* that he was really clever, until the protagonist put him in his place.

I don't want to "put you in your place," but I think that you have a similar problem. You have studied enough to believe that you know what you are talking about, and in a few years, when you have looked into it a bit more, I suspect that you will feel differently. Or perhaps you are like my engineer students, who don't really care about the philosophy of mathematics, but only want to know how to use mathematics to make computations. If that is the case, I would suggest that you not comment on the philosophy of mathematics, because you are way out of your depth (I, also, am way out of my depth on that topic, but not quite as much as you are :P).

xander