Q&A: A Question About Infinity
A Question About Infinity
Question
Hello Rabbi,
I recently started dealing with the philosophical topic of infinity, and I came to the conclusion that unlike other abstract concepts that can be seen as having actual implementation in the physical world, like time, colors, and numbers, infinity cannot.
The first reason that made me think this is that, practically speaking, we have never seen actual infinity in physical reality, only theoretical infinity (for example, you claim there are infinitely many points on a parabola, but you never really get to all of them…)
A second reason is that I saw various paradoxes like x = x*x when x equals infinity.
A third reason comes from Zeno’s paradox, that theoretically it is impossible to traverse any space because you have to pass through infinitely many units of length (whether fixed or not, that does not really matter right now)… My answer to the paradox was that the one posing the paradox is implicitly assuming the actual existence of infinity in space, even though there is no reason to assume that, and from this I also reached the conclusion that there is no actual infinity in physical reality, otherwise we could not reach any point.
I would be glad to ask whether, in your opinion, infinity is applicable in the physical world as we know it or not?
P.S. A friend of mine told me that the Big Bang theory is not really properly grounded and is just the default because there is nothing better. Is it really not properly grounded?
Answer
1. It seems to me that in order to deal with this seriously, you need more mathematical background. I do not know what it means for infinity to have an implementation in the world. In the interval between 0 and 1 (and in any other interval—what does a parabola have to do with it?) there are infinitely many points. There are infinitely many natural numbers. Are these appearances in the world or not?
2. What paradox is there in that?
3. Zeno’s paradoxes have very simple answers. Quite a few years have passed since then.
4. It depends on what you define as existent. Is a photon an existent thing? And a person? Is he one existent thing or billions of particles? An angel?
5. It is well grounded.
Discussion on Answer
At the end of the message I got mixed up; I do not accept the thesis of an eternal universe.
You are not defining things. Of course you cannot count infinitely many things. So what? I cannot make progress in this discussion.
Infinite human stupidity has real and practical implications on the ground.
Infinity is not a number. Therefore infinity times infinity is undefined. There are indeed cases where people write infinity times infinity, but that is not an arithmetic multiplication operation. For example, in the arithmetic of infinite limits. But that is really a theorem and not actually an arithmetic operation. In calculus courses they take off points if you write infinity times infinity without quotation marks, and they take this very seriously.
Or in the sizes of sets, but there too it is not an arithmetic multiplication operation but cardinal numbers.
On the contrary, if infinity were a number, we would have a problem, because then one could prove that 0 equals 1. Just add 1 to infinity—it equals infinity—and then subtract infinity from both sides.
Therefore the second claim has no meaning.
A number that tends to infinity—its reciprocal tends to zero.
And one could say that just as zero is a number, so too infinity is a number.
That is not correct. Infinity is not a number. Nor can tending to infinity be a number. It is simply a sequence or a series (in the discrete case), or a function that from some point onward is greater than every number. There is also no such thing as one divided by infinity.
It is a theorem in the arithmetic of limits that if there is a sequence or function whose numerator tends to one and whose denominator tends to infinity, then it tends to 0. It is convenient to write one divided by infinity, but people put it in quotation marks and know that they mean a theorem. And again, if one divided by infinity equals 0, then let us multiply by infinity and get 1 = 0—a contradiction!
From here, if it is a number, then all of mathematics is worthless because there is a contradiction in the axioms, and then anything can be proved.
And if you say that infinity times 0 is undefined, then I will refer you back to the theorem that a number times 0—the additive identity—gives 0.
Most of the paradoxes floating around on the internet are not really paradoxes. They are just mathematical nonsense. Usually people take a divergent series or one that converges but not absolutely, and with illegal operations they show that it comes out to a finite number.
A sum is defined as something finite. Sometimes the definition is extended to an infinite sum or product. Then there are theorems that apply only to cases that converge to something finite. In those cases people perform operations that no theorem supports. It is simply mathematical nonsense.
Understanding convergent series also shows that Zeno’s paradoxes are not really paradoxes.
There are of course several kinds of convergence and several kinds of definitions, but there are no paradoxes here.
P.G.,
Your claim that Zeno’s paradoxes are not really paradoxes requires clarification. If you mean to say that Zeno’s conclusion—that there is no plurality and therefore no change and motion in the world—is philosophically mistaken, then you are right. The Eleatic view (of which Zeno is a part) is a kind of skepticism, and like any skeptical position, it is haunted by danger from its own hand. It is a position that undermines its own foundation.
But it seems to me that you meant to say something entirely different… Could it be that you were trying to say that one can get rid of those paradoxes by means of a mathematical procedure? If so, that is a philosophical claim which in my opinion does not stand up to critical examination (by philosophy).
What I mean by infinity being implemented in reality is actually reaching it. Between 0 and 1 there are infinitely many points only in a theoretical sense; you can never really count or use all of them. It is only an abstract concept that you theoretically impose on things.
I read in your notebook on the cosmological proof, and it seems like you also say that infinity has no implementation in reality. You call it concrete infinity.
There is an argument that the universe is eternal; it implicitly relies on the existence of concrete infinity in reality, that the abstract concept has actual implementation. It seems to me that there is no such infinity, and therefore I do not accept the thesis of infinity in physical space.
Thanks in advance.