Static typing does not change the complexity dynamics of your programming model. It's a mere tool.
If I statically type to DateTime I can still be wildy baffled to see a transaction take place in the future. Or even more baffled when at first glance the DateTime looks correct (ie in the past and recent) but to find out later when I get the support phone call that we incorrectly tallied the balance.
Static typing just restricts where I can put my shoes in my closet. If I find a need to sort all the "righties" together -- for example, to figure out how many lefties I'm short -- static typing makes things more complex -- because it will disallow this innovation.
Complexity is a count of concepts in your model, not a count of instances and not a count of state possibilities, which in any real world system are going to be endless no matter how much you waste time with static proofs.
If we take `cleanliness = number of valid states / number of total states`, then using DateTime vs String is strictly cleaner.
Yes, there are still many invalid states within a strong type and static typing is just one form of reducing complexity. I'm actually much more interested in cleanliness of design and architecture, but the String vs DateTime or Enum vs String examples are easy ways to illustrate the idea.
There's a certain size threshold for applications where a statically typed language becomes extremely valuable. Static proofs that things are "connected" properly become very valuable, even if, as you say, the benefits aren't universal. Constraints sometimes simplify things, sometimes they increase complexity, but it's hard to argue that large systems with many authors would be universally better if they were unable to use static types to constrain programs.
I've found rust really helps out with this, enforcing some strict rules around types.
I had to implement a `Lowercase` type which is just a string that can only be lowercase (for case insensitive searches). After implementing standard traits like `Into` and `Deref`, it just acted as a standard string. That way I could guarantee at compile time that things are lowercase, but still have the flexibility of strings.
Another common idiom I have seen is to compose finite state machines using enums. You can only create a particular state from another state etc..
I think the intro description introducing entropy is missing something. If the shoe closet sorted by color and the chaotic one are both composed of the same set of shoes then they have equal entropy. You can transpose any two shoes of the same color to achieve an additional microstate whether or not they're arranged by color.
I suppose it's technically correct if what you're interested in is the ways shoes can be positioned in a closet, but why bring up color then? The example sets up a comparison between some small finite number of ways to arrange shoes on a shelf and potentially infinite ways to arrange them in a pile. Better, I'd say, to stick to a grid in both cases and compare the number of microstates in the "all black" macrostate against the number of microstates in the "half brown half black" macrostate.
Right, I recognize that thermodynamic entropy can be indifferent to human-scoped definitions of order and chaos.
But if we take shoes (or number of values in a type, or number of components in a design) to be the atomic unit of measurement, then I think the concept works.
> If the shoe closet sorted by color and the chaotic one are both composed of the same set of shoes then they have equal entropy.
No, the sorted closet is more constrained and has not many micro-states. whereas _any_ shoe arrangement describes the disordered set, only a few arrangements describe the ordered set.
That's not quite right, and the author is reaching for a valid definition, but it also not quite getting it right (entropy is hard).
Entropy is defined as: "It quantifies the number of microscopic configurations that are consistent with the macroscopic quantities that characterize the system."
Where there are two levels - microscopic - which in this case doesn't necessarily mean invisible to the eye, it means the individual units of the system. In this case shoes. And macroscopic - which is the system as a whole, or in this case, the set of all shoes. A single macroscopic state is the set of shoes in some (maybe random) order.
Finally "macroscopic quantities" refers to a property of the macroscopic system. In this case the author is using the macroscopic quantity of "how fast can I find a shoe of a particular order".
In this case, entropy measures, for a given macroscopic quantity, how many microscopic configurations exist. For example, lets say the given macroscopic quantity is "I find a given color shoe in O(n) time." Then every configuration of shoes that does not attempt to order the shoes is a possible microscopic state. There are lots of those, so high entropy.
Another possible macroscopic quantity is "I find a given color shoe in O(logn) time". In this case the shoes need to be ordered, leading to far fewer states. Hence lower entropy.
If this is not clear, it is easier to see if we constrain ourselves to having one pair of shoes for each color. Then the O(logn) configuration has 1 state, while O(n) clearly has many more, so obviously the O(logn) state has lower entropy.
However the same logic applies to having >1 pair of shoes for each color. In the O(logn) case, we can only swap shoes of the same color to have the same macro state. In O(n), we can always swap any shoes of any color, as long as this swap isn't the last swap needed to achieve order.
The point of the Gibbs paradox is that exchanging two different shoes of the same color is not a new microstate, if all black shoes are considered identical. When they should be considered identical and when they shouldn't be is a bit more complicated, and the Jaynes paper goes into that.
This view is too simplistic. I've come to the realization that ultimately software is complex because it models some very complex real-world situation. E.g. how complex can tax software be? Answer: as complex as the tax code, which is not just complex, but also contains tons of unspecified behavior. Yes, sometimes you can take a simple problem and write spaghetti code for it, but IME that is often not the real problem.
Right, there is the concept of essential complexity. This is addressed in the post:
> Sometimes, our problems are essentially complex. In these cases, our solutions need some essential complexity to match. But when does essential complexity become unnecessary?
We can measure this when the number of total states dominates the number of valid states. The number of invalid states is a measure of accidental complexity.
"You should call it entropy [...] nobody knows what entropy really is, so in a debate you will always have the advantage." John Von Neumann to Claude Shannon
(also why there is clearly of market for the concept of dynamic entropy: https://xkcd.com/2318/ )
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[ 2.4 ms ] story [ 54.8 ms ] threadStatic typing does not change the complexity dynamics of your programming model. It's a mere tool.
If I statically type to DateTime I can still be wildy baffled to see a transaction take place in the future. Or even more baffled when at first glance the DateTime looks correct (ie in the past and recent) but to find out later when I get the support phone call that we incorrectly tallied the balance.
Static typing just restricts where I can put my shoes in my closet. If I find a need to sort all the "righties" together -- for example, to figure out how many lefties I'm short -- static typing makes things more complex -- because it will disallow this innovation.
Complexity is a count of concepts in your model, not a count of instances and not a count of state possibilities, which in any real world system are going to be endless no matter how much you waste time with static proofs.
Yes, there are still many invalid states within a strong type and static typing is just one form of reducing complexity. I'm actually much more interested in cleanliness of design and architecture, but the String vs DateTime or Enum vs String examples are easy ways to illustrate the idea.
If the original simple argument were true, then everyone would introduce static typing, it would be a no-brainer.
I had to implement a `Lowercase` type which is just a string that can only be lowercase (for case insensitive searches). After implementing standard traits like `Into` and `Deref`, it just acted as a standard string. That way I could guarantee at compile time that things are lowercase, but still have the flexibility of strings.
Another common idiom I have seen is to compose finite state machines using enums. You can only create a particular state from another state etc..
I suppose it's technically correct if what you're interested in is the ways shoes can be positioned in a closet, but why bring up color then? The example sets up a comparison between some small finite number of ways to arrange shoes on a shelf and potentially infinite ways to arrange them in a pile. Better, I'd say, to stick to a grid in both cases and compare the number of microstates in the "all black" macrostate against the number of microstates in the "half brown half black" macrostate.
But if we take shoes (or number of values in a type, or number of components in a design) to be the atomic unit of measurement, then I think the concept works.
No, the sorted closet is more constrained and has not many micro-states. whereas _any_ shoe arrangement describes the disordered set, only a few arrangements describe the ordered set.
Entropy is defined as: "It quantifies the number of microscopic configurations that are consistent with the macroscopic quantities that characterize the system."
Where there are two levels - microscopic - which in this case doesn't necessarily mean invisible to the eye, it means the individual units of the system. In this case shoes. And macroscopic - which is the system as a whole, or in this case, the set of all shoes. A single macroscopic state is the set of shoes in some (maybe random) order.
Finally "macroscopic quantities" refers to a property of the macroscopic system. In this case the author is using the macroscopic quantity of "how fast can I find a shoe of a particular order".
In this case, entropy measures, for a given macroscopic quantity, how many microscopic configurations exist. For example, lets say the given macroscopic quantity is "I find a given color shoe in O(n) time." Then every configuration of shoes that does not attempt to order the shoes is a possible microscopic state. There are lots of those, so high entropy.
Another possible macroscopic quantity is "I find a given color shoe in O(logn) time". In this case the shoes need to be ordered, leading to far fewer states. Hence lower entropy.
If this is not clear, it is easier to see if we constrain ourselves to having one pair of shoes for each color. Then the O(logn) configuration has 1 state, while O(n) clearly has many more, so obviously the O(logn) state has lower entropy.
However the same logic applies to having >1 pair of shoes for each color. In the O(logn) case, we can only swap shoes of the same color to have the same macro state. In O(n), we can always swap any shoes of any color, as long as this swap isn't the last swap needed to achieve order.
The point of the Gibbs paradox is that exchanging two different shoes of the same color is not a new microstate, if all black shoes are considered identical. When they should be considered identical and when they shouldn't be is a bit more complicated, and the Jaynes paper goes into that.
> Sometimes, our problems are essentially complex. In these cases, our solutions need some essential complexity to match. But when does essential complexity become unnecessary?
We can measure this when the number of total states dominates the number of valid states. The number of invalid states is a measure of accidental complexity.
(also why there is clearly of market for the concept of dynamic entropy: https://xkcd.com/2318/ )
BTW, "stringly typed" has been in use for exactly the behavior to avoid for maybe 10 years now https://wiki.c2.com/?StringlyTyped