12.5 Infinities Everywhere
Infinity is a concept that both astounds us and scares us at the same time. Astounds us by its sheer magnitude and scares us because of our inability to grasp it. Many philosophers in the past have outright refused that infinity as a concept exists. Nowadays, with progress in Mathematics, we at least believe that it can exist as a concept irrespective of whether there are infinite entities in the physical reality.
Zeno’s paradox
Let us start with a fun paradox. Let’s say you have to travel 2 km. To do so, you first travel 1km, then you travel 0.5km, then 0.25km, then 0.125km and so on. If you keep doing this, you will never reach the 2km mark. Zeno used this paradox to show that changes are illusions. But modern mathematics found a solution to it. It showed that even if a series is infinite, the sum of the numbers can still be finite. In this case,
1 + ½ + ¼ + ⅛ +… = 2
Thus you get the 2km though you are walking an infinite distance.
Infinite reasoning
We have tackled the concept of infinity or closer in a couple of places. Remember the Münchhausen trilemma? There are only three ways to end the cycle of proof:
The dogmatic proof
The circular proof
The regressive proof
One way to see it is that we can keep asking for proof of every statement, proof of the proof and so on, which takes us through an infinite chain of asking for proof. This is the regressive proof. In order to avoid regressive proof, we either use the dogmatic proof or a circular proof. A dogmatic proof asserts a statement as truth without any proof. It is the brute law or axioms we use in our systems.
A circular proof on the other hand, while beautiful, is considered cheating in many circumstances. For example,
An evil person is one who does evil deeds.
Evil deeds are the ones done by evil people.
It doesn’t really tell us anything about what is evil.
So, unless we say “BECAUSE I SAID SO” or use circular references like above, it is impossible to get out of the infinite regression in our quest to find the ultimate reason for everything.
Physical infinities
The other situation where we came across infinities is when we discussed the Gaps in science. Do space and time have boundaries? Are they infinitely divisible? Is matter infinitely divisible?
Maybe real physical infinities exist but humans are incapable of grasping them. Maybe they don’t exist and what you see is what you get. We may never be able to find answers to these questions.
Sometimes, we try to differentiate Actual Infinity and Potential Infinity. Actual infinity is just that - Either space or time is really infinite. Potential infinity means that even if an entity could be infinite, there is no way for us to reach it because of the limitations of the human mind.
Mental infinities
That brings us to the next question. What about the human mind? Is there an infinity? Just like we had questions about the infinite nature of multiple entities like space, time and matter, there are a couple of different issues with mental infinities.
Infinite nature of contents of the mind. Can the mind hold the idea of an infinite entity? Can the mind hold infinite entities?
What happens when the mind tries to understand itself? Does it take us into an infinite loop or is it just a self-reference problem like we saw in the last chapter?
To answer the question, let us take the list of all natural numbers. We consider them to be infinite. Can you imagine them all? We obviously could not visualise all of them in our mind. We just imagine 1, 2, 3, 4,… What does the “…” mean? We just use it as a shorthand for the infinite numbers.
How about this? We just imagine all natural numbers using their mathematical description - N is a set. It has 1. And for any number x, in the set, it also has x+1. This uses the principle of mathematical induction. The set has 1. Because it needs to have x+1 for any number x, it has 2 and because it has 2, it also has 3, and so on.
If we just visualise this rules-based description, aren’t we practically holding an infinite set in our mind? There is one problem here. The description we have here can also be used to describe other sets and hence this description is not unique to just a set of all natural numbers. The details of how to find that set is a bit technical. But Thoralf Skolem showed that any finite description of the set of all natural numbers one comes up with, there will be a different set N* that also fits the same description. This doesn’t prove that infinite thoughts are impossible. They may still be possible, but they may not be unique.
Now, what happens when the mind thinks about itself? Imagine a room that has its picture hanging on the wall. There is a small twist. The picture also shows the wall with the picture that is hanging on the wall. Now that picture also shows the wall with the picture hanging on the wall and so on.
Practically speaking, the picture obviously stops the infinite regress at some point as the size of the picture within the picture gets smaller and smaller. But theoretically, whether there can be a picture that shows an infinite regression of the picture within the picture depends on whether matter is infinitely divisible or not. If matter can be infinitely divisible, then some entity that is capable of breaking matter to smaller and smaller parts an infinite number of times could theoretically make the picture with real infinite regression.
What about the mental image of itself? If the mind is physical, like physicalists claim, then the argument for whether the mind’s reference to itself is an infinite regression depends on whether you believe matter is infinitely divisible, just like the case with the picture.
But what if the mind is not physical? There are two ways to look at this. “A mind that thinks about mind that thinks about mind that thinks about mind….” is at the end of the day just a mind that thinks about the mind and hence is simple self-reference or circular-reference. Or “A mind that thinks about mind that thinks about mind that thinks about mind….” takes us infinitely deeper into the mind and is hence an infinite regress which makes the human mind infinite.
God or the Absolute
We saw how the physical entities, mental entities and our mind itself could be either infinite or not. What about the other entity that is proposed by different philosophies?
If you skip the religious philosophies which have personal Gods, interestingly, you can find many philosophies that describe God in terms of infinities. Below is the quote by Thomas Aquinas:
Since therefore the divine being is not a being received in anything, but He is His own subsistent being… it is clear that God Himself is limitless and perfect.
St. Augustine in The City of God:
And thus, if everything which is comprehended is defined or made finite by the comprehension of him who knows it, then all infinity is in some ineffable way made finite to God
Below is a quote from Taittiriya Upanishad, part of important scriptures of Indian philosophy:
Brahman is Truth, Knowledge, Infinite.
As you have seen above, many philosophers tend to define God in terms of infinity. If we assume that God is infinite, does it have any consequences on physical and mental infinities? To understand that, let us look at the Mathematical framework of infinities built by Cantor.
Cantor showed that unlike the belief of many that all infinities are the same, there is actually a hierarchy of infinities. He categorised them into transfinities and absolute infinity. Transfinities can be calculated and placed in sets. Physical and mental infinities fall under the category of transfinities. The absolute infinity cannot be treated as a set and leads to paradoxes if we tried to. This is the Cantor’s paradox we talked about in the previous chapter.
A God who is infinite should be an absolute infinity in Cantor’s framework. What is the consequence of assuming that God exists and is the absolute infinity or just that absolute infinity exists? According to the Reflection Principle, any conceivable property of this Absolute infinity set is also a property of some other set. I won’t get into the proof or further mathematical details. But the consequence is that because we can conceive an infinite nature of the absolute, it becomes a conceivable property and hence it must be part of other sets, according to the Reflection principle. In this case, mental infinities must exist.
Also, because God is the absolute infinity of all infinities, you can’t use rational thinking to reach God. Mathematics doesn’t allow you to do that. This is where mysticism comes into picture. Mystics say that you can reach God through mystic practices.
Putting them all together:
Physical infinities
Do space or time have infinite boundaries?
Is space or time infinitely divisible?
Is matter infinitely divisible?
We don’t know
Mental infinities
Can the mind hold infinite entities?
If a mind holds an image of itself, is it an infinite reference or just a self reference?
We don’t know
God - If you believe that God is an Absolute Infinite entity
Mental infinity must exist
You cannot reach this God by rational thinking. Mystics suggest different paths to reach such a God.
Book recommendations
Infinity and the Mind by Rudy Rucker