Science vs. Belief
Gist: If you choose to not believe any proofs, you are totally free to.
Beliefs and proofs come from orthogonal universes. Science is from the latter.
Ah, yes. Science can prove the negative. Also, as a matter of fact, it can scientifically prove that one can believe into whatever they choose to.
[In professor Farnsworth voice] Have fun!
Proof is the term from logic.
To understand what a proof is, we need to introduce the notion of a statement.
Statements are made about entities of certain domain.
For example, if domain is natural numbers — 1, 2, 3, … — then “two plus two is four” is a statement. If domain is poker, “five cards of the same suit form a flush” is a statement. You got the idea.
Some statements can be true or false.
Note that a true statement is very different from the statement of truth.
For example, we might agree that “two plus two is four” is a true statement. We would also likely agree that “one is greater than one hundred” is a false one.
Agree is the key word here. It has nothing to do with truth of beliefs. It is simply what we chose to agree upon.
Furthermore, since we can agree on some statements and build a foundation based on them, why don’t we start from very few, very simple statements, that are obviously true for you and me?
Those statements are called axioms.
Axiom is a statement that we have agreed to be true. Once again, the agreed part is essential.
Now, proofs are a formal language over statements. Logically speaking, given the set of axioms, proofs allow broadening the set of true and false statements from those.
Given a statement that is not part of axioms, we can use proofs to prove that this statement is true. Or to prove that this statement is false.
*** Or neither. But that would be rocket science! ***
Spoiler:
Arithmetic, in fact, is agreed to be defined by a very simple set of Peano axioms.
Geometry is commonly defined by Euclid’s postulates.
And set theory is commonly defined by Zermelo-Fraenkel axioms.
Logic is the discipline that studies reasoning.
Logic operates on statements and their proofs.
While statements operate on entities of some domain, the domain of entities for logic are the statements themselves.
Among other things, logic states that if A is true and B follows from A, then B is also true.
The beauty of logic is that it does not care about the domain of the statements. A and B can be from math. Or poker. Or carry legal meaning.
The “A, A → B ∴ B” rule is known as modus ponens. It is fundamental for any reasoning to make sense.
*** It also is not necessarily correct. But that would be rocket science! ***
Spoiler:
Arithmetic can neither prove nor disprove Goodstein’s theorem.
Set theory can neither prove nor disprove Continuum hypothesis.
Hyperbolic geometry is a branch of mathematics that is created by adjusting just one of Euclid’s axioms.
And Banach–Tarski paradox makes us question whether the Axiom of choice should be part of set theory axioms.
Science is about Occam’s Razor.
One way to look at what science is would be to define science as the philosophy behind choosing axioms.
In his excellent lecture “Greek versus Babylonian mathematics”, Richard Feynman points out that in the real world we do not have the luxury to definitively establish the set of axioms we would prefer to live in.
Instead, we must keep exploring the world and discovering patterns that can be generalized.
Now, remember we talked about agreeing on certain set of axioms. How do we choose what to agree upon?
I mean, if you really believe the moon is only there when someone is watching it, how can one prove you wrong? Is it even possible?
Who cares?!
*** Science does not bother messing with one’s beliefs! ***
What science focuses on is providing better justifications. And it is doing great job at this.
In order to qualify for a scientist, one has to be willing to constantly challenge the most obvious assumptions. Feynman himself speaks about the above in his other lecture, “The Scientific Method”.
The simplest hypothesis that best explains the state of the art is accepted as the working model. Working models change as new facts appear.
Albert Einstein said that “everything should be made as simple as possible, but not simpler”.
The part about challenging hypothesis is essential for the scientific method. In order to qualify for scientific, the theory should be possible to be falsified by experiment, says Karl Popper in his falsifiability criterion.
“Among competing hypotheses, the one with the fewest assumptions should be selected” is the Occam’s razor.
By no means it is seen as the absolute truth. It just presents the best described foundation, that allows logical proofs to be used to their fullest.
Ah, and science can prove the negative. It is easy to prove that no flush will be possible if the first four community cards in Texas Hold’em are of four different suits. It is also easy to prove that 25 is not a prime number, since it is divisible by five.
Spoiler:
Look at how Newton’s law of universal gravitation has been replaced by Einstein’s general relativity, which is now argued to be taken over by loop quantum gravity. None of those have been the final theories; they simply are the best to day explanations.
Truth is about belief.
Unlike science, proofs and logic, truth is subjective.
Given the same set of axioms, the conclusions will be the same. The truth is what one chooses to believe in.
Of course, logic is too powerful to disregard. I find it hard to imagine a human being, who would accept all arithmetic and still believe that two plus two is five. It would not make much sense.
In order to stay rational, one has to question the axioms, not the conclusions.
However, unlike mathematics, in real life,
- axioms are often vague enough to allow being questioned, and
- staying rational is nowhere near a requirement.
It is important to understand though, that if you and I disagree on some belief, it is entirely possible that none of us is wrong. We might just be believing into different sets of axioms.
If you and I are both rational people, we would like to get to the single most fundamental statement that we disagree upon. To handle this efficiently, we would use the scientific method, logic and proofs extensively.
At the end one of us may choose to alter our beliefs based on the arguments and evidence presented. This would have nothing to do with science; it would purely be a personal choice.
The beauty of science is that it works. When we would like to build a spacecraft, or a web service that can be used by a billion people, we would rather rely on science, than one’s beliefs.
The beauty of belief is that it does not require proofs.
Spoiler:
Gödel’s incompleteness theorems complete the picture by telling us that modus ponens can not be added as an axiom without making the theory inconsistent.
Check and mate, adepts of rationale and logical arguments!
Much like science can prove the negative, it may well be that we all live in The Matrix, or that the Moon is really only there when someone is watching.
We all make the choice about what we believe in.
Make yours wisely.
