Q&A: Can One Be Exact or Not?
Can One Be Exact or Not?
Question
Hello and blessings,
Following my study of the Talmudic topic of whether exact simultaneity/precision is possible or impossible, in Bekhorot 17 (and in many other places), I would be interested to know how science today, or the philosophy of science, relates to this question.
Can two things happen at exactly the same time? If not, why not?
If so, is there a way to measure and prove that they happened at exactly the same time?
All these questions also apply to measuring distances or dimensions—can it be that two things are exactly the same length? If not, why not? If so, can one measure and prove that they are exactly the same length?
And from all of the above—what does this say about the nature of time, or the nature of distance?
Thank you very much,
Yuki Meir
Answer
Hello Rabbi Yuki.
From a mathematical standpoint, there is no chance for an exact event within a set of continuous values (its probability is 0). Therefore, in science and mathematics people do not speak in that kind of language, but rather in the language of intervals: what is the probability that the difference in lengths between A and B will be up to epsilon. At a discrete point (like epsilon = 0) there is only probability density, not probability.
The probability of obtaining a certain length from a set of continuous values is described by a distribution function that describes probability densities as a function of length. The probability of a specific length is not defined (only its density is). Probability is obtained by integrating the function over the interval in question. Thus, for example, one can describe the likelihood that some length will be X by a function f(X). The probability that the value of X will be between 11 and 37 is the integral of the function from 11 to 37. At a discrete point (for example X=15) the probability is 0, and only probability density is defined there.
This means that time and distance are continuous (that is, they can take any value over a continuous interval).
Discussion on Answer
First, thank you for the quick answer.
Second—I’m not sure whether what you wrote answers my question.
From what I understood from what you wrote, time is continuous, and the probability for every point on it is 0.
But the fact that the probability of a certain event is 0 does not mean that it will not happen. If, for example, we take the axis of whole numbers and choose some number at random, then the probability of getting the number 37 is 0, and nevertheless 37 could be chosen.
Of course there is a probability of 0 squared that in two random selections the number 37 will be chosen twice, and nevertheless that could happen.
All this is when we are talking about discrete events, but as you wrote, time and distance are continuous quantities. But does that really make a difference? The question is whether there really exists a time of 3.1716 seconds, or whether that is always an approximation. If it exists, why shouldn’t it happen twice? And of course the same goes for distance.
The additional question was about the possibility of measurement—assuming that a time of 3.1716 seconds exists, can we be sure that it is indeed that.
As I write, another question comes to mind—how do we actually know that time (and length?) are continuous? Is there a proof for this, or is it an assumption?
Thanks again.
Maybe I didn’t understand, but it seems to me that that is exactly what I answered. See in my article that indeed, if you carry out such a random selection you will get a result, and still the probability is zero. So to the question of what the probability of that is, the answer is 0. That does not mean the thing will not happen. It is very tricky in the mathematical sense (because in practice you really can’t carry out such a random selection. It is not even well defined mathematically, and this is not the place to elaborate. It is also clear that nothing happens at a discrete instant of time, perhaps except for transitions like midnight or death, which I discussed in the article). Of course measurement with scientific instruments always has some error. There is no such thing as perfectly exact measurement (where the result is a single point in time).
The assumption that time and space are continuous is accepted in physics, although it has been challenged more than once (among others by Penrose, but in my view that is just chatter). You can of course insist otherwise, but I do not see what you would gain. Even if you are trying to explain some statement of the Sages, I assume you do not mean to say that the Sages knew something about space that contemporary physics does not know. By the way, there is no such thing as scientific proof. Science corroborates or refutes; it does not prove. You can always dispute any scientific claim by presenting an ad hoc thesis.
See what I wrote here on the Torah portion of Balak:
https://drive.google.com/drive/folders/0BwJAdMjYRm7IY0xlc1dmYTMweVE