Mathematics. A large bag of tools used to discover many of the greatest kept secrets our world has to offer. Some secrets can be mined using our applicable tools, while others are unknowable even when exposed to an exhaustive collection of tools. Yes, there are unknowable ‘things’, if that is even the right word for them, yet we can prove their existence. What?! In other words, there are concrete limitations to science as a whole.
Consider Chaitin’s constant as a simple example. This is a number that is known to exist but can be shown that not a single bit of it can be computed. Sounds like trying to pin down an eggshell in water! However, if this number was known or be found it then could be used to prove a slew of theorems most notably the Riemann hypothesis, arguably the most famous unsolved problems of all mathematics!
How could this be? What is so special about this number? Let us consider the following. Suppose you have an oracle, call the Oracle Bec. The oracle tells you if something is true or not. You ask Bec, “Hey is the Riemann hypothesis true or not?” She nods her head yes. Proven. You also now inherit the million dollar award for solving this problem. Thank you Mrs. Oracle!
This is because Chaitin’s constant is connected to a certain type of computer program. This program can review any other program and see if that program will stop or not. For instance suppose we have the following program:
count=1; flag=0; while(flag==0), count=count+1; end;
This is a bad program. It continues to add one to count indefinitely, never-ending. You take your algorithm and ask Mrs. Oracle if the program will stop. She states you have a probability of zero, that is, it will not stop.
Now assume you have a program that tests a certain hypothesis of a theorem. You could then ask the Oracle if it will stop. Hence the theorem is proven or disproven. Gee, you could disprove a theorem without even constructing the counter-example, as you’ll know that one exists!
The future of science is quite mysterious, the hope of having a continuum of scientific knowledge oozing between different fields is a mere fantasy. At best our complete knowledge will still have holes for which are unknowable. They exist. And no amount of passing, fleeting time will ever encapsulate that.