The Little Vulgar Book of Mechanics (v0.13.0) - Models I
Last updated: March 18th 2022
Just added this section to the book: Models I
Models I #
Just because you have a model, it doesn't mean that you understand causation.
In Functions I, we saw what a function is, and I mentioned that functions are the usual mathematical way to express scientific understanding. I.e. a crucial tool for scientific modelling.
However, having a perfectly valid and mathematically rigorous function doesn't mean that we know what is causing what. This is clear from the mathematical definition of a function: A relation between sets. That's "relation," generally. Not causal relation, specifically. Not all functional relations express causation.
As an example, consider that a text file can be represented as a function \(f: N \rightarrow T\), where \(N\) is a set of natural numbers (representing line numbers, and let's say it's limited to the amount of lines in the file), and \(T\) is a set of text strings from the file. This would be a perfectly valid mathematical representation of the file.
Let me make it concrete, by running a little UNIX command to list the headings of the big file on which I'm writing this book (I know, this won't scale forever):
$ grep -irn '^## ' src/book/index.md
It basically tells the computer "show me the lines that begin with '## '." This is (part of) the output:
src/book/index.md:219:## Information I src/book/index.md:223:## Problem Solving I src/book/index.md:241:## Functions I src/book/index.md:265:## Functions II src/book/index.md:344:## Functions III src/book/index.md:441:## Models I src/book/index.md:477:## Differential Equations I
There's your function. It relates a (line) number, to some string of text.
But does it mine that each line number causes the content it is mathematically related to? Of course not. Trust me, I wrote the book. And I can tell you that it wasn't being on line 241 somehow compelled me to type '## Differential Equations I'."
The main takeaway is that a mathematical function can be helping you model the real world (e.g. the contents of a book), in a perfectly valid and rigorous way, and yet not tell you a single thing about cause and effect.
For another example, consider that you could define this function:
\[ y : A \rightarrow Y \]
Where \(A\) is the set of Cannibal Corpse album names, and \(Y\) is a set of years when Cannibal Corpse release albums. The definition would be something like:
\[y(Vile) = 1996\] \[y(Kill) = 2006\] \[y(Torture) = 2012\]
...and so on and so forth.
It's a perfectly legit function. It correctly models an aspect of the world. But it obviously doesn't say anything about causality. The year 2006 wasn't the year 2006 because Cannibal Corpse released the album "Kill." As much as I love Cannibal Corpse, I have to accept that it was the Earth's rotation around the Sun that caused the calendar year to go up from 2005 to 2006!
These examples should also reveal why the more computational intuition, or metaphor, for functions, as "a box transforming the input to the output" kinda totally breaks too. Was the Cannibal Corpse album "Kill" somehow "transformed" into the year 2006, in the real world? No. So always be careful with metaphors. Metaphors are not reality. They are, at best, temporarily helpful mind tricks. Don't get married to metaphors.
Now, you may think that this lack of causality, this "valid model that doesn't tell us anything about causes," is exclusive to the type of function I've defined so far: The simple, manual mapping from inputs to outputs. But no, later we will see functions that are defined by mathematical formulas, but also don't say anything about causality.
Bottom line: There is no reason to assume a causal relationship between the input and output of a function.
You will hear people talk about "toy models," "analytical models," "computational models," etc. What types of models are there? This will be the topic in Models II and III.
For now, take a break. And listen to my death metal track Escape Mechanics Unlocked!
See current full book's WIP here.