Is infinity real? Or is it just the math nonsense you get when you divide by zero? If infinity is not real, does this mean that zero is also not real? And what does infinity appear in physics mean?
A summary of the article by Sabine Hossenfelder, a German theoretical physicist who studies quantum gravity, answers this question.
Infinity is that which is not bound.
Most of us encounter infinity (or infinity) for the first time when we learn to count, and realize that it is possible to continue counting forever. I know it’s not a terrible initial observation, but this doesn’t end the counting because you can always add one and get a larger number which is the important property of infinity. Infinity is that which is not bound. It’s bigger than any number you can think of. You could say it’s unimaginably large.
Infinity isn’t quite so simple because, as odd as it sounds, there are different types of infinity. The number of natural numbers, 1,2,3… is just the simplest kind of infinity, called “countable infinity”. And the natural numbers are in a very specific way as infinite as the other sets of numbers, because you can count these other sets using the natural numbers.
Formally, this means that a set of numbers is just as infinite as the natural numbers, if you have a one-to-one mapping from the natural numbers to that other set. If there is such a mapping, then the two sets have the same type of infinity.
For example, if you add zero to the natural numbers – so you get the set 0, 1, 2, 3, etc – then you can map the natural numbers to this by subtracting one ( – 1) from every natural number. The same goes for the set of natural numbers and the set of natural numbers plus zero are of the same type infinity.
The same is true for the set of all integers Z, which is 0, ± 1, ± 2, etc. You can assign only one natural number to each integer, so the integers are also infinite. term.
Rational numbers, i.e. the set of all fractions of integers, are also infinitely countable. A real number contains all numbers that have an infinite number of digits after the dot (.), however, uncountable is infinite. You could say it is even more infinite than the natural numbers. There are practically an infinite number of types, but these two, corresponding to natural and real numbers, are the two most commonly used.
It’s interesting to have different types of infinity these days, but it’s more appropriate to use infinity in light of the fact that most infinites are actually the same. Therefore, if you add one to infinity, the result is still infinity. And if you multiply infinity by 2, you get the same infinity again. If you divide 1 by infinity, you get a number whose absolute value is less than any value, so the number is zero. But you get the same thing if you divide 2 or 15 or root. square of 8 gives infinity. The result is always 0.
Sabine Hossenfelder, physicist and author of “Lost in Mathematics,” revolves around the theme that theoretical physicists often rely on beauty – especially simplicity and naturalness – as they develop new laws describing nature. These guidelines have had a powerful influence on the foundations of physics since the development of the standard model of particle physics, which describes all known elementary particles and explains how they interact. But this beauty can also lead scientists to a dead end.
I hope no mathematicians follow this, because technically one shouldn’t write these relations as equations. They are actually statements of the infinity type. For example the first case just means that if you add 1 to infinity, the result is the same type of infinity.
The problem with writing these relations as equations is that it can easily be wrong. For example, you could try subtracting infinity on both sides of this equation, giving you nonsense like 1 equals 0. Why is that? That’s because you forgot that infinity here really only tells you the type of infinity. It is not a number. If the only thing you know about 2 infinite numbers is that they are of the same type, then the difference between them can be anything.
It’s even worse if you do things like divide infinity by infinity or multiply infinity by zero. In this case, the result can’t just be any number, but can be any kind of infinity.
This whole infinity story sure looks like a mess, but real mathematicians know very well how to deal with infinity. You just have to be careful to keep track of where your infinity comes from.
For example, say you have a function like x squared to infinity when x goes to infinity. You divide it by an exponential function, which also goes to infinity for x. So you are dividing infinity by infinity. This sounds a bit twisted, doesn’t it?
But in this case you know how to get to infinity and so you can compute the result explicitly. In this case, the result is 0. The easiest way to see this is to plot the fraction as a function of x.
If you know where your infinity comes from, you can also subtract one infinity from another. Indeed, physicists do this all the time in quantum field theory. For example, you might have terms like 1/epsilon, 1/square epsilon, and the logarithm of epsilon. Each of these terms will give you infinity with epsilon equal to 0. But if you know that the two terms have the same infinity, so they are the same function of epsilon, then you can add or subtract them as children. number. In physics, usually the goal of doing this is to show that at the end of a calculation they cancel each other out and everything makes sense.
So mathematically, infinity is very interesting. As far as math is concerned, we know how to solve infinite problems.
But is infinity real? Does it exist?
The answer is yes, it is said to exist in the mathematical sense , in the sense that you can analyze its properties and talk about it like we just did. But in the scientific sense, infinity does not exist.
That’s because scientifically we can only say that an element of the theory of nature ” exists” if it is necessary to describe observations. And since we can’t measure infinitely, we don’t really need it to describe what we observe. In science we can always replace infinity with a very large but finite number. We don’t do this. But we can.
This is an example that demonstrates how mathematically infinite numbers cannot be measured in practice. Let’s say you have a laser pointer and you rotate it from left to right and that makes a red dot move on the wall at a long distance. What is the speed at which the dot moves on the wall?
That depends on how fast you move the laser pointer and how far away the wall is. The further away from the wall, the faster the dot moves with the rotation. Indeed, it will eventually travel faster than light. This may sound confusing, but note that the dot is not really a moving thing. It is just an image that creates the illusion of a moving object. What’s actually moving is the light from the pointer to the wall and it’s traveling only at the speed of light.
However, can you definitely observe the dot’s movement? So can we ask, can the dot move infinitely fast, and therefore we can observe something infinitely?
It seems that in order for the dot to move infinitely fast, you have to place the wall infinitely far away, which you can’t do. Wait, you can tilt the wall at an angle instead. The more you tilt it, the faster the dot moves across the wall surface as you rotate the laser pointer. Indeed, if the wall were parallel to the direction of the laser beam, it would appear that the dot would move infinitely fast across the wall. Mathematically, this happens because the value of the tangent function at pi/2 is infinity. But does this happen in practice?
In reality, the wall will never be completely flat, so there is always some point that will stick out and that will blur the dot. Also, you can’t really measure the dot at the same time on both ends of the wall because you can’t measure time arbitrarily precisely. In fact, the best you can do is show that the dot moves faster than some finite value.
This conclusion is not specific to the laser pointer example. Whenever you try to measure something infinite, the best you can do in practice is say it’s bigger than something finite you’ve already measured. And no test can show that. So infinity is unreal in the scientific sense.
However, physicists always use infinity. Take for example the size of the universe. In most contemporary models, the universe is infinitely large. But this is a statement about the mathematical properties of these models. The part of the universe that we can actually observe is only finite in size.
And the problem of unmeasurable infinity is closely related to the zero problem. Take for example the mathematical abstraction of a point. Physicists use this all the time when they deal with point particles. A point of size zero. But you would have to measure to infinite precision to show that you actually have something of size 0. So you can just display it as less than whatever the precision measures. your permission.
Infinity and 0 are everywhere in physics. Even in seemingly invisible things like space, or space-time. The moment you write down the spatial calculations, you assume there are no gaps in them. You assume it’s a smooth continuum, made up of infinitely many small points.
Mathematically, that’s a convenient assumption because it’s so easy to work with. And it seems to be working fine. That’s why most physicists don’t worry much about infinity. They just use infinity as a useful mathematical tool.
But it is possible that using infinity and 0 in physics will bring mistakes because these assumptions are not only scientifically unproven, but also scientifically unprovable. And this could play a role in our understanding of the universe or quantum mechanics. This is why some physicists, such as George Ellis, Tim Palmer, and Nicolas Gisin have argued that we should formulate physical formulas without using infinite numbers or infinitely precise numbers.