Science, fundamentally, is the pursuit of truth. Yet for a mathematical equation to have any meaningful truth value, we must first interpret it. When there are so many contradictory yet consistent interpretations, or models, how is one to discern the truth from the all the quasi-truths and coincidences?
One answer is simplicity – a principle not particularly logically founded, but one which is appealing nonetheless. Why would the universe operate under some convoluted, arbitrary rule when a straightforward intuitive one does just as well? And if this is still unconvincing, then there is the argument that a half-truth we can understand is better than a full-truth we can't. In this respect, deterministic models do rather well, which is why they are so widespread within the realm of simulation, regardless of how much (or little) belief is put into their truth. Nevertheless, there exist categories of problems for which determinism is wholly insufficient – at least for a computer – due to the unreasonably high precision required, or natural perturbations, depending on your wont.
Another answer is generality – simplicity in another direction, in a way. Rather than insisting on local simplicity – that is, the equations you are working with are quite nice – we can consider global simplicity – the property that relatively few equations to describe the entire system. Again, we have little logical basis for which to expect this, but it would certainly make our lives easier, which is always a bonus. Here, stochastic models have borne out more fruit, generalising much more widely to a range of scales (at the cost of a little more calculus). But now we lose simplicity, leading to what may seem like ridiculously over-the-top calculations for quite basic mechanics.
From here on, we attempt to provide a non-partisan overview of the current consensus models, outlining the relevant examples for and against their truth. The latter sections of the book will tend a little more philosophically, examining whether the question of truth is one that can even be asked, and how we can develop metrics for truth axiomatically. Finally, we will present the author's personal opinion as to the most truthful model, including some rebuttals to potential counterarguments.
– Introduction to 'Introduction to Scientific Model Theory' by R.H. Kuraksow (2004)
Determinism
Let $\Omega$ be a sample space and $\mathbb{P}$ a probability function on this space. Then $(G,\cdot)$ is a group if it satisfies the following axioms:
1) $G=\{x\in\Omega|\mathbb{P}(x)>0\}$
2) $\cdot$ is associative
3) $\forall g,h\in G$, $\mathbb{P}(g\cdot h)\geq\min(\mathbb{P}(g),\mathbb{P}(h))$
4) $\exists e\in G$ such that $\forall g\in G$, $e\cdot g=g\cdot e=g$
5) $\forall g\in G$, $\exists h\in G$ such that $\mathbb{P}(g)=\mathbb{P}(h)$ and $g\cdot h =h\cdot g=e$
We should interpret $\mathbb{P}(g)$ as the probability of finding the object $g$ in the group $G$. From these axioms, we can obtain through logical deduction a wide range of results with a variety of applications to the sciences. It should be stressed that the probabilistic aspect of groups does not mean that our proofs are “probably” true. We rely on chance only as an aid to the intuition.
Indeed, a mathematical theory which utilised chance in its very reasoning would be useless in its inconsistency. With unstable foundations, how can anything be built? The axiomatic approach overcomes such issues of intuition and hasty generalisation; by insisting on rigour, we have security in our knowledge.
– Excerpt from the University of Oxford lecture notes on 'Groups, Modules and Rngs'
The game of the universe is there to be played – all that is left to us is to figure out the rules.
– Baadur Jandieri, the 'Grandfather' of Transmutation
Stochasticism
We would expect that measurement be continuous, that is, if an observable is measured as taking the value $\alpha$ at time $t$, then it should – with high probability – still be $\alpha$ at time $t+\varepsilon$. Indeed, the empirical evidence aligns with this intuition. Mathematically, this means at time $t$, the wave function $\psi$ must take the form $\psi_\alpha$. We say that our observation collapses the wave-form into the state we observe.
While immediately after our observation, the wave function's behaviour is well-known, over time, it will evolve as in our earlier calculations, leading to the unpredictability we see throughout. It should be stressed that this process is truly random, there are no hidden variables here. Repeating the same initial conditions can lead to completely different results in the observations. Our very observations are entangled into the system, changing matter itself simply by being there.
Indeed, a fully pre-determined universe would be a very boring one. With each brick already decided, how can anything be built? The universe overcomes such issues by introducing the random chance we see in action here; by widening our scope, we have the possibility of future.
– Excerpt from the University of Oxford lecture notes on 'Quantum Theory'
We cannot predict how these passing seconds will fall, yet – on a good day – we may catch one.
– Tursin Rothcarrock, Doctor Emeritus of the Institute of Physics