Say t = an infinitesimally small interval of time.
How long does (t + t) take?
How about (t + t + t + t + t + t + t + t + t + t + t + t + t + t + t)?
No matter how many infinitesimally small intervals of time you string together, their combined duration is still infinitesimally small.
In your example you’re assuming that you have an infinite number of *finite *intervals. But you’re overlooking the fact that as the number of intervals becomes infinitely large, their duration becomes infinitely small. The two infinities cancel each other out and you reach your destination in a finite amount of time.
What he’s saying is that there’s no such entity as a “two” in reality. There’s a concept of two things…if you have a rock and another rock then you have two rocks, and if you have two rocks and another two rocks then you have four rocks. But where’s the two? Where’s the four? Where’s the one? You can’t hold a naked two in your hands, you can only apply the concept of two to real world objects. Therefore, “two” is a theory you have about the way the universe works. 2+2=4 is another theory. And sometimes 1 object plus another object doesn’t give you two objects, it gives you something different. What happens when you add a proton to an antiproton?
And others don’t accept your proposals as self-evident, which is why self-evidence isn’t sufficient in scientific theory. Everyone has their own standards of self-evidence; mathematics and measurements make a theoretician put his money where his mouth is, so to speak.
My point has never been that your interpretation is impossible. My point is merely that there is a wide range of possibilities, and you are not offering anything that elevates yours over any other.
straggler, all your objections seem to come down to the standard argument of “I don’t understand infinite math so I won’t accept it.” Therefore you throw out not just the math of infinities but also the calculus and the theory of limits which depend on the summation of infinite quantities. Can you explain how your mathematical system works if you toss all these fundamental elements out the window?
Remember, if you reject infinities so that space is quantized you must also then explain how every single aspect of reality that uses limits or the calculus is changed by the removal of infinities.
Suppose I (like you) assert without proof that all space is infinitely divisible (of course, you assert the opposite, but with the same quantum of evidence) and that when you cast your gaze over a item of finite extension, you observe–in the real world–the infinite summation of increasingly small fractions producing a finite number.
What defect does this argument have that your own does not?
How are you defining infinitesimal? My understanding of infinitesimal means something that is too small to be measured. Assuming there is such a thing as an infinitesimal amount of time, you could eventually sum enough of them to reach something measurable.
What do you mean by “becomes infinitely large”? Either the number of intervals is infinite or it isn’t. If they are infinite, and each one takes time, then the destination is unreachable.
This, but more importantly, I’m trying to explore and challenge your notion of “in reality.” You manipulate the symbols that represent a finite series of numbers to correctly arrive at their sum, and claim that you’re doing so in reality. Roken manipulates the symbols that represent a converging infinite series, and correctly arrives at their sum, and yet you say that is not in reality. I don’t see the distinction – for any definition of “in reality” either both are in reality, or neither is.
Yes, you’re right. I was speaking casually without considering the precise meaning of the term “infinitesimal”. Substitute “infinitely small” in my argument above.
But each one doesn’t take time. Some of the intervals you’re adding up are infinitely small.
If one step takes some time, and if all steps are sequential, and if some process requires an infinite number of steps to occur before the result can be reached, then said result is unreachable. The standard of accepting a statement as factual does not sit beyond the example I have described here. I’d sure love to see what your idea of a more robust piece of logic looks like.
Your are assuming that summing up an infinite number of things does not converge on a finite number. This is not always true. Until you buy into this or convince us otherwise there will not be a meeting of the minds.
You still haven’t explained why it should take infinite time to traverse finite space, rather than finite time to traverse infinite space for that matter. Both time and space are subject to the same infinite sums if we’re assuming continuity for the sake of disproving it, so why the distinction between the two?
Yup. When the duration of each successive subdivision becomes infinitely small you stop, having arrived at your destination in a finite amount of time.
Time and space, as I have suggested, are inextricably linked. If there has been “no change”, ie, no movement through space, then there is no time elapsed. The smallest, indivisible unit of time could be represented by the fastest possible movement through one space unit.
