I agree with premise (1), that there is no reason to think of infinitesimal quantities as actually part of the universe. I don’t agree with premise (2), that actual infinities imply actual infinitesimals. If you could convince me of (2), I would probably reject (1) rather than accept (3). Since an argument for (2) would be a good argument against (1), given that our universe does seem to have actual infinities.
the points on a line are of infinitesimal dimension … yet compose lines finite in extent.
No. Points have zero dimension. “Infinitesimal” is something else. There are no infinitesimal numbers on the real line (or in the complex plane, for that matter), and no subinterval of the real line has infinitesimal length, so we would have to extend the number system if we wanted to think of infinitesimals as numbers.
When I raise the same argument about an infinite set, you can’t reply that you can always make the set bigger; if I say add an element, you reply that the sets are still the same size (cardinality).
But there is a way to use an infinite set to construct a larger infinite set: - the power set. I don’t understand the rest of this paragraph.
Consider again the points on a 3-inch line segement. If there are infinitely many, then each must be infinitesimal.
Again, single points have zero length, not infinitesimal length. Note, though, that there are ways to partition a finite line segment into infinitely many finite line segmets, including the partition that Zeno proposed: 1⁄2 + 1⁄4 + 1⁄8 + … In an integration, we (conceptually) break up the domain into infinitely many infinitesimally wide intervals, but this is just an intuition. None of the formal definitions of integrals I’ve seen actually say anything about an infinitesimally wide interval.
The series comes infinitesimally close to the limit, and in this context, we treat the infinitesimal as if it were zero.
Actually, we don’t have to treat an infinitesimal as zero, we just have to treat zero as zero. If I move along a meter stick at one meter per second, then according to Zeno’s construction, I traverse half the distance in 1⁄2 second, 3⁄4 of the distance in 3⁄4 of a second, and so on. As you say, after one second, I have traversed every point on the meter stick except the very last point, because the union of the closed intervals [0,1/2], [1/2,3/4], [3/4,7/8], … is the half-open interval [0,1). So how much longer does it take me to traverse that last point? Zero seconds, because a single point has zero length. There is no contradiction, and no need to use infinitesimals.
I agree with premise (1), that there is no reason to think of infinitesimal quantities as actually part of the universe. I don’t agree with premise (2), that actual infinities imply actual infinitesimals. If you could convince me of (2), I would probably reject (1) rather than accept (3). Since an argument for (2) would be a good argument against (1), given that our universe does seem to have actual infinities.
No. Points have zero dimension. “Infinitesimal” is something else. There are no infinitesimal numbers on the real line (or in the complex plane, for that matter), and no subinterval of the real line has infinitesimal length, so we would have to extend the number system if we wanted to think of infinitesimals as numbers.
But there is a way to use an infinite set to construct a larger infinite set: - the power set. I don’t understand the rest of this paragraph.
Again, single points have zero length, not infinitesimal length. Note, though, that there are ways to partition a finite line segment into infinitely many finite line segmets, including the partition that Zeno proposed: 1⁄2 + 1⁄4 + 1⁄8 + … In an integration, we (conceptually) break up the domain into infinitely many infinitesimally wide intervals, but this is just an intuition. None of the formal definitions of integrals I’ve seen actually say anything about an infinitesimally wide interval.
Actually, we don’t have to treat an infinitesimal as zero, we just have to treat zero as zero. If I move along a meter stick at one meter per second, then according to Zeno’s construction, I traverse half the distance in 1⁄2 second, 3⁄4 of the distance in 3⁄4 of a second, and so on. As you say, after one second, I have traversed every point on the meter stick except the very last point, because the union of the closed intervals [0,1/2], [1/2,3/4], [3/4,7/8], … is the half-open interval [0,1). So how much longer does it take me to traverse that last point? Zero seconds, because a single point has zero length. There is no contradiction, and no need to use infinitesimals.