Bible verse used in podcast:
- John 17:1-3
- Matthew 28:18-20
- Psalms 19:1-3
- Romans 1:20-21
- Acts 17:22-25
- Godel’s incompleteness theorems
- An interesting debate about Godel’s theorem
- Probability in science
- The shift from Newton to Einstein
The limits of Truth
Before, I talked about how we can take a bunch of fundamental rules, calling them axioms, and use them to make a logistical system. From there, we used these axioms and saw what happens if they are true; exploring their consequences within this system. What becomes interesting is the idea that you can keep asking questions that relate to this logical system and learn more. But there are other things you can do. You can add more and more axioms to expand what your system can explain. So now the question is, what happens when you try to keep adding more and more axioms? How care can this go? Will your theology ever be enough? Can we ever get a complete understanding of God?
We know that on some level we can understand God. A core aspect of Christianity is the idea that we can have a relationship with the creator of the universe. He is a personal God who interacts with his creations. But how much can we know God?
1 When Jesus had spoken these words, he lifted up his eyes to heaven, and said, “Father, the hour has come; glorify your Son that the Son may glorify you, 2 since you have given him authority over all flesh, to give eternal life to all whom you have given him. 3 And this is eternal life, that they know you, the only true God, and Jesus Christ whom you have sent.John 17:1-3 ESV
Knowing God is linked to eternal life. Exactly what that means is a bit of debate but to some degree, we should be able to know God. How far exactly can this go? How much can a finite being, know an infinite one? Well, it turns out we may never be able to fully know him if we are finite beings. This might sound obvious, but how could you know that? One simple thing to point out is that God is infinite and he created this universe. For God to be “infinite” he needs to be bigger than this universe on some level. Like this legendary question, can God create a rock he can’t like? No, he can’t and it’s perfectly logical but that’s for another day. Even when you consider the idea of us going to have and getting glorified bodies (Philippians 3:21), our glorified bodies still have to be small on some level compared to God for him to be infinite. With that comes some limitations. We don’t know everything. Our knowledge is limited, or finite.
How limited is our Truth?
With that, we can come back to this question, will we ever know everything about God? About the Bible? Obviously not, but there might be a really logical reason beyond just intuition. Some time ago in Mathematics, people were trying to find a theory of everything. They were looking for a system of axioms that could be used to prove everything in math logically. Along the journey, several mathematicians stumbled across crippling issues about this. One of them is Kurt Godel, who presented the incompleteness theorems. This is a bit technical but it provides a lot of insight.
To start, you need a formal system of axioms. A system that we talked about before, a list of axioms that can be used to describe how something works and you can draw conclusions from. The formal part is that you can use the axioms to prove or disprove things in that system. If you have this and that system is finite, meaning the axioms can be written as a list that can be countable (either by a human or computer), then you will get some issues.
How do you tell if you can prove something is true or not? You can do this using a larger system to check it, often called a metalanguage. It is the same as how we have English but we critique writing using grammar. We talk about verbs, nouns, sentence structure, and the like to talk about the language on a paper. Similarly, that is what Godel did, through a complex process he created a mathematical language to talk about mathematics. This using a metalanguage, you have more rules than the first one. When you do that, you can use these extra rules to check if the first language is correct. It is like having a dictionary to check if all the words are spelled corrected or a reference book to see if the verb tenses match. In the case of Godel, what he could do is find theorems within a logical system and see whether or not you could prove them using the axioms of the system. It is a bit complicated but that is the idea in a simplistic way. Often we use logic to check things and one simple idea is the law of noncontradiction. Something can either be “a thing” or be the opposite of “that thing”. Either something is true or it is false. If you can prove something then it must be true. But what if you can’t prove it, it is always simply false?
The Incompleteness Theorems
Out of this, a discovery emerged. If your system is a finite system, it will always have things in it that are true that you can not prove. If something is false, then you should be able to prove it is false using a finite process. A finite process is a set of steps to reach the final conclusion and then if it contradicts or can’t lead to any logical conclusion then it is false. It is basically like having equations and using algebra to see if both sides of an equal side match up. If they don’t, someone messed up. But within a system, there are things that are true you can’t prove. These seem to be self-referencing things. They must be true but it is as if they point to the fact that there must be more to the system than what we know. It is similar to “I” in English or any other pronoun. Not all we can communicate on paper or even verbally is all it is to reality. The use of “I” refers to something that doesn’t exist in English, it is like we have information “the speaker” that is not inherently part of the language. You can use other words to explain who the “I” is but the speaker is not part of English, they exist independently from English. Similarly, Godel found this to be true in math that there are things that are true that can’t be proven using the axioms.
Now, what happens if you take this “unprovable thing” and make it an axiom. It turns out you will just get another system with the same problem. In this new system, there will be something (or many things) that are true, but you can’t prove. So you make another system, bigger than the previous two, and on and on.
The second theorem falls from the first: A system can not prove its own consistency and be complete. When you have a system of rules, you can ask two important questions, is it complete, is it consistent? The consistency simply refers to whether or not there are contradictions. If there are, it is inconsistent, if there aren’t it is consistent. The completeness means whether or not you have everything you need to prove things within the system. Part of Godel’s proof showed that if a system is complete, then it is inconsistent. If a system is incomplete, then it may be consistent.
This points to some staggering revelations. This does not mean that truth doesn’t exist, or that we can’t prove everything. But it shows that no matter how much we learn about a subject, we may not be able to completely explain everything with a finite list of rules. Thinking back to that pen example, you can keep adding rules to find out where the pen is on the table, but when was it on the table? What color was it? Did it change colors? And so on. We will come back to this idea later. But this does have some implications of the boundaries of what man can know. For instance, many people go to a physics class and learn about Newtonian mechanics (basic intro physics for many). Often many physics professors like to start on the first day of the lecture with a joke. They may say that everything you will learn in this course is a lie. It’s not that Newton was completely wrong, but everything he discovered, was actually an approximation that is really true. Later when Planck and Einstein came along they started general relativity and it turned out that Physics presented and understood during the days of Newton and had assumptions (or axioms) that we know today to be false. As we learn, we expand the axioms that we do know and found that in some cases what we knew before actually leads to contradictions.
For the Bible this becomes important, if we ever have a full understanding of God, we are probably wrong. But if we continue to learn, then we might be right. No man alone can fully understand God. This brings to mind the words of Paul when he talks about how the church of God is like a body that has different parts. We need each other on several levels, but one way is that we can’t know everything and we need each other to bring more understanding to the table to fully understand God together as a process.