String theory tends to end up with a lot of either infinitely large or infinitely small values for properties. If we want to calculate the property of some physical object, we would expect the calculation to give us a finite value. After all, the calculation is supposed to be a shortcut for measuring that property. And if we measure the property of something, we should certainly get a finite value, not an infinite one!
Remember, infinity is not a number. Infinity describes the fact that there are certain mathematical sequences that you can progress along until a certain point. But, at no matter which point you do stop, there is always the potential to have progressed further.
Numbers are used to measure something. However, the potential to progress in a series is not a measurement of something.
We talk more about infinity in episode fifteen of the podcast.
Suffice, to say that whatever value the measurement of a property might have, it is a finite value. And if you use mathematics to find out what that value should be, it too must be a finite value.
Of course, some string theorists seem to get this fact enough to be bothered by all these annoying infinities. So, they try to get rid of them. Often using … dubious mathematical tricks.
There is one famous sum that comes up. It is the sum of all the natural numbers (1, 2, 3 … 100 and so on). String theory claims that if you add these numbers on to infinity, this equals -1/12.
This is absurd. You cannot add an infinite number of numbers together. You can only add a finite number of numbers together.
Yes, in mathematics there is the concept of an “infinite series”.
This is a very useful concept, but it does not involve literally adding an infinite number of numbers together. It tells you that if you keep adding numbers you might get closer and closer to a certain number, the limit of that infinite series, or you might not. It makes no claims about adding an infinite number of numbers together.
If an infinite series has a limit, it simply means that as you add more and more numbers together, you get closer and closer to the limit, some number. But you never reach that number. If the series has no limit, you just keep getting a higher and higher number and that number keeps going up and up (or down and down, if you are getting smaller and smaller numbers).
But, if you add all the natural numbers together, there is no such limit. Each time you just get a number greater/smaller than the last time.
So, how does string theory propose to add an infinite number of natural numbers together?
How does it propose to have an answer? And why is the proposed answer a negative number? It uses idiotic trickery to pretend that you can somehow add an infinite number of natural numbers together, you somehow get -1/12.
How is one meant to get this? And a negative number at that? I will not bore you with the details. But one should stop and question first, how you add an infinite number of natural numbers together.
You cannot. How do you add the potential to always progress in a sequence together? That makes no sense! The potential to always be able to do something is not something that can be added together or to itself!
So, you can only add a finite number of natural numbers together. You add as many as you can before you stop and then you are done. But, at no point will you get to a negative number or one which is less than 1!
Why does string theory try this? It wants to get rid of this annoying infinity. So, it resorts to silly mathematical tricks that have no merit and which make no sense. And this is not the only example. But I won’t bore you with further examples.
And that is all we have for you today. There are many more absurdities we might have covered, but I think this covers enough of the most essential ones. If you did not think that string theory made no sense before, then you should have some idea now.
Next time we will see why the absurdity of all this should not be surprising. We shall see why it should be entirely predictable.