If we assume that facts are only those things which we can believe in with 100% certainty then very few things are facts. That the sun rises in the east and sets in the west is not a fact. That carbon produces carbon di-oxide on burning is not a fact. The law of conservation of energy is not a fact. We only have an enormous amount of evidence in favor of all these things. We only know that thousands of experiments being done world-wide have all been consistent with the above statements, that if these statements were wrong, then the results of those experiments wouldn’t make sense. But we don’t know that these statements are true with 100% certainty. In principle it is possible that someone does an experiment some day that violates the law of conservation of energy. In principle, it is possible that someone finally builds a perpetual motion machine. It will shake the foundation of physics, it will prove that everything we thought we knew about the universe was wrong and so on. But it is still possible.
You would say that it’s depressing. It probably is. But that’s how things are. It’s impossible to make a hypothesis about something that exists in the real world and then gather evidence that establishes that the hypothesis is correct beyond doubt. If you believe in any such statement beyond doubt then there is something fundamentally inconsistent going on in your belief system and it very well deserves some repair work.
The situation is not that grave though. There are some statements whose truth is established beyond doubt. That happens when we are not dealing with real world objects. For example, the fact that there are an infinite number of prime numbers, or the fact that the square root of 9 is 3 or the fact that the ratio of the area of a circle to its circumference is exactly equal to half its radius. These are all true. The reason why it is possible to establish these statements beyond doubt is that we are the ones who have defined what a circle means or what radius means or what number, infinite, prime, square root, area or circumference mean. The definition of a circle is “a closed curve which has all its points equidistant from one given point” and not “the shape of the cross-section of coconut trees”. If circle was defined as the shape of the cross-section of coconut trees, then to check if the statement “ratio of the area of a circle to its circumference is exactly equal to half its radius” was true, one would need to cut down lots of coconut trees and measure the area, circumference and radius of the cross-section and see if the ratio was half the radius or not. But that wouldn’t at any point establish this beyond doubt, even if you checked all coconut trees on the planet and found that the statement was true for all of them. There would always be the possibility that a new coconut tree would grow up in some corner of the world that violated this statement. But with the definition that it has, the fact can be verified beyond doubt.
It is the difference between trying to figure out the rules of the game some other people are playing and trying to argue about a game whose rules have been defined by you. If some other people are playing a game whose rules are unknown to you, you can never figure them out with 100% certainty just by observing their moves. If you have designed the game yourself though, you are completely sure of what the rules are and so you can say things about the rules with 100% certainty.
But why would one define things arbitrarily and talk about them? Isn’t that a completely trivial exercise? No, it’s not. If you have defined the rules of the game, it might be easy for you to state the rules, but it may be extremely non-trivial to figure out, say, what would be the best move for the next player. For a game with rules as simple as chess, for example, we are still not capable of calculating the absolute best move for a player given the state of the board in a reasonable amount of time. And that’s just one thing that’s non-trivial. There can be several other things about those rules that are very difficult to decide. And that’s what happens in math. The whole fact that we should find it difficult to decide something about a set of rules that we have defined ourselves is fascinating itself. But very often it also happens that these things that we are trying to decide are not just difficult to decide, but are also extremely surprising. The fact that things that we define ourselves are not our slaves, but in fact, to the contrary, they are capable of confusing us and surprising us is extremely fascinating. Mathematics is full of such examples. For example, taking the number 5 and adding it 9 times is the same as taking the number 9 and adding it 5 times. Or, the fact that the rate of increase in the area under the curve representing a function is the same as the function. Or, when you take the ratio of the circumference of a circle to its diameter, multiply it with the square root of -1 and raise the product to the power of the inverse of the probability that you end up placing all letters into wrong envelopes if you do it blindly, the result that you get is exactly equal to -1.
That’s not all by the way. Surprising and confusing us about things that we have defined ourselves is not the only role mathematics has to play. Very often it so happens that one finds something in the real world that, for some reason, follows the rules of one of the games that we had designed. And that’s when all the exploration we had done, all the confusing and surprising things we had found about the set of rules come in handy. We discover that the thing in the real world doesn’t just follow the rules of this game that we had designed, but it satisfies this bunch of surprising properties as well. And that gives new insight. For example, it’s just a mathematical fact that if something is going around a circle with a constant speed then that means it’s accelerating towards the center. This can be proven with some elementary vector calculus. But since we have also empirically observed that the earth pulls things towards itself, we can use this empirical observation and the mathematical fact above to say that the reason why the moon revolves around the earth is because of earth’s gravity. We can also use these two to send communication satellites into space with that exact velocity that makes them revolve around the earth in an orbit.
So, to put it in a nutshell, you can be completely sure about the truth of a statement only if the statement is about a game whose rules are decided by you. Also, to decide whether such a statement is true or not is very often non-trivial even though you are the one who decided the rules. Besides being non-trivial, many of these statements are also very surprising and sometimes they might even play some role in broadcasting the latest episodes of your favorite sitcoms on your television.