Thanks to reader Victor Domin for pointing us to the paper “On teaching mathematics” by V.I. Arnold. Here’s an extended quote on models (do read the whole thing).
At this point a special technique has been developed in mathematics. This technique, when applied to the real world, is sometimes useful, but can sometimes also lead to self-deception. This technique is called modelling. When constructing a model, the following idealisation is made: certain facts which are only known with a certain degree of probability or with a certain degree of accuracy, are considered to be “absolutely” correct and are accepted as “axioms”. The sense of this “absoluteness” lies precisely in the fact that we allow ourselves to use these “facts” according to the rules of formal logic, in the process declaring as “theorems” all that we can derive from them.
This follows along our recent weight of evidence post, and the crucial distinction (discussed in loving detail here) between local and universal or necessary truths. Local truths are when the evidence used to prove the truth of a proposition is itself not true or known to be true. Universal or necessary truths proof propositions based on evidence that is itself a necessary truth. Mathematical, logical, metaphysical, and revelational truths fall into this category.
Probabilities or propositions deduced from ad hoc models, or models which are tentative in the sense Arnold states, are true. But they are local truths. That doesn’t mean they aren’t useful, but usefulness is not the same as truthfulness. Mistaking these things accounts for nitwits running around screeching “Denier!” Scidolatry, of course, is to blame.
It is obvious that in any real-life activity it is impossible to wholly rely on such deductions. The reason is at least that the parameters of the studied phenomena are never known absolutely exactly and a small change in parameters (for example, the initial conditions of a process) can totally change the result. Say, for this reason a reliable long-term weather forecast is impossible and will remain impossible, no matter how much we develop computers and devices which record initial conditions.
This sensitivity to initial conditions has the official name of chaos theory. Giving it a name does not take away from Arnold’s correct conclusion, however. There are some tricks to not improve guesses from these models not in an absolute sense, but to better nail the uncertainty. These are to be admired. But they can’t do what can’t be done.
In exactly the same way a small change in axioms (of which we cannot be completely sure) is capable, generally speaking, of leading to completely different conclusions than those that are obtained from theorems which have been deduced from the accepted axioms. The longer and fancier is the chain of deductions (“proofs”), the less reliable is the final result.
This exactly fits into the weight of probability discussion. There is only probability, or logic, but it makes sense to discuss how influential evidence is.
It’s not that the length of the chain of deductions matters per se, but the more crap you stick on the right hand side of the logic/probability equation, the greater the chance (based on experience) one of those premises won’t itself be true. One weak link is all takes to turn a necessary truth into local truth, i.e. a falsity.
Complex models are rarely useful (unless for those writing their dissertations).
The mathematical technique of modelling consists of ignoring this trouble and speaking about your deductive model in such a way as if it coincided with reality. The fact that this path, which is obviously incorrect from the point of view of natural science, often leads to useful results in physics is called “the inconceivable effectiveness of mathematics in natural sciences” (or “the Wigner principle”).
That has to be because the universe is designed.
“Dude, that makes you a creationist.”
Yep. It do. But consider that most people misunderstand this, thinking creation applies only to the physical word, which supposedly “evolution” and “laws of physics” own. Creation does apply to that, for the “laws” of physics (which is a metaphor) must themselves have been created. They cannot have come from nowhere.
Forget that and think of this: creation also applies to mathematics itself. There must be a reason math is the way it is. Math could not have created itself. Math has a cause, all those theorems have a cause.
So it’s no wonder physics often aligns with mathematics the closer physics gets to its fundaments. This is because both must have the same creator.
There is no escaping this conclusion.
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