What makes something true?
Mathematics can derive the truth,
trustworthy proof.
Science only reduces our uncertainty,
guessing nervously.
What is ‘proof’? In mathematics a proof means that we can agree upon a starting point and some determistic rules, and with those rules we can find (aka ‘prove’) statements that are true! This is also known as deductive reasoning.
For example, chess. Given a chess position, the one shown below, it is true that white can checkmate in 3 moves (if we follow the rules of chess). (Wf4->h6, Bg8->h6, Wd2->h6, Bf8->g8, Wf6->f7)
There are many other true statements that can be proven given this starting point (and following the rules of chess).
- The white queen is at f4.
- The white king can move to d1.
- Black has 7 pawns.
Similar to how we use the rules of chess to derive things that are true about a given chess game. We can agree on objects called integers, {0, 1, 2, 3, 4, …}, and agree on rules for adding and multiplying them, +, x. From these integers and their rules we can derive prime numbers.
Euclidian geometry is another example of a system of rules that can be used to derive true statements. Given the 5 postulates of Euclidian geometry, (basic equality rule, equality through operations, equality based on coincidence, universality of right angles, parallelism) we can derive the Pythagorean theorem, and other familiar ‘true’ statements, such as the sum of the angles in a triangle is 180 degrees.
The ‘truths’ proved using deduction will always be true. Euclid lived over 2000 years ago, yet his proofs are still true today, and will be true in thousands of years time. Nothing can change the truths of deductive reasoning.
But, not all logical statements have known proofs. For example, the Riemann hypothesis is a guess about the distribution of prime numbers. We have not yet found a proof for the Riemann hypothesis. Is this because the statement is false or because we have not yet imagined the correct proof?
We define this problem as a problem of bounded resources. We have not yet imagined all the possible proofs.
Going back to chess, based on the first move advantage (statistically white wins more often), we could guess that white has a winning strategy (there is some optimal set of moves where it can always win). But, to prove this, we would need to imagine all possible games of chess, and all possible responses to each move.
However, the real world is not like chess, integers or Euclidean geometry. (1.) We do not start by agreeing to any rules, (2.) and we have an imperfect view.
The universe does not come equipped with a rule book for us. We need to start by making some assumptions.
- the laws of the universe do not change with time. They are the same today as they are tomorrow, and the next day.
- the observations we made are not interfered with ( there doesn’t exist an adversary manipulating our observations )
Note that, these assumptions cannot be proven to be true, and therefore we cannot claim that science gives us the truth. We make these assumptions on the basis of our experience. We have not yet observed the laws of the universe to change, and we have not yet observed an adversary manipulating our observations. Here we are using induction, a dangerous tool.
Induction is making generalisations from incomplete observations. Applying a pattern observed in a limited set of observations to all observations.
I have not seen all swans, bespite that, based on the swans I have seen, I assume that all swans are white.
I have observed the sun rise in the east every morning for the past 30 years. Despite not knowing why, I assume that the sun will rise in the east tomorrow.
Everyday, a farmer feeds his pigs at 7am. He has done this for the past 30 weeks years. The pigs assume this is a law of the universe, that the farmer will feed them at 7am. But, one day the farmer instead loads them into a truck and takes them to the meatworks.
The next issue, we have an imperfect view of the world. We do not see everything, and what we do see is noisy.
If we imagine an imperfect view of chess it might look like the following.
You never get to observe the board directly, rather;
- through a small window, where you can only see a few squares at a time (I can see g2 square, but not the b7 square) (imperfect information)
- the window is also foggy (I’m pretty sure that’s a pawn at g2, but it might be a bishop) (measurement noise)
Imperfect information means we may be missing relevant information. For example, the existence of black swans.
Measurement noise means we may be misinterpreting the information we do have. For example,
This imperfect view is the problem science solves.
Science attempts to systematically reduce our uncertainty about the truths determined by the physical rules of our universe. However, science only reduces uncertainty (rather than eliminates it), science cannot answer ‘What makes something true?’, instead, science focuses on a different question.
What makes something likely to be true?
Consider the question; is the theory of general relativity likely to be true?
Science can tell us;
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it is more likely to be true than Newton’s theory of gravity because it reliably makes more accurate predictions about celestial mechanics (the Eddington experiment), as well as making predictions about other observations ( black holes, light, … ) ( it’s the current best hypothesis we have constructed, but there may exist others, that explain even more - ie dark matter )
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it is more likely to be true because it has generalised. It has made accurate predictions about new observations (Einstein predicted gravitational waves in 1915). This tells us that, at least in this case, we didn’t overfit to our existing observations. But, we have no clue how this theory will generalise to other unobserved phenomena.
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it is more likely to be true because it explains our observations in a ‘simple manner’ ( ie Occam’s razor ). General relativity can be derived from statement that “the laws of physics are invariant in all inertial frames of reference”.
But these successes of general relativity are not sufficient to answer is the theory of general relativity likely to be true? with much certainty.
Our estimate of likelihood is based on our history (the data we happen to have collected), our imagination (the equations and models we have constructed, so far) and some assumptions.
But, our history could have been different: imperfect view. This means that science cannot actually answer “What makes something likely to be true?”. Rather, the best science can do is;
Given what we have observed, what is the most likely thing to be true?
But, our limited imagination means we may not have imagined the most likely answer: bounded resources. There is always a chance that our search finds a new, more likely to be true, theory. So we arrive at;
Given what we have observed and the answers we have imagined, what is the most likely thing to be true?
Note that as science progresses, as new observations and new theories are imagined, our best guess at what is likely to be true should change. This is the essence of the scientific method.
Science is a best guess, nervously hoping that we have not missed something important.