Anyone who thinks they know is going to absolutely hate this answer.

Entropy is a measurement of how much work, with current human knowledge, can be extracted from a system. It is a measurement of all potential energy that is yet untapped, combined with a theoretical maximum of "useful" work that can be extracted from this energy. It is a useful fiction, but it is still a fiction. So is the second law of thermodynamics.

In truth, entropy has the potential to change with every change in human knowledge.

If we discover a new force - say, a subtle attractive force far weaker than gravity, but attenuating far less quickly - the entropic values of all systems change. You can use this information to extract slightly more energy from the system by building an engine that somehow takes advantage of this new source of potential energy.

In more down-to-earth terms, suppose we just discovered fire today. Suddenly the potential energy that can be extracted from coal increases enormously - previously, we could just extract whatever energy we could get from its falling, much as we extract enormous energy from the gravitational potential energy in water. We have to rewrite all of our entropy tables, which previously just concerned themselves with height relative to, say, sea level.

Fortunately for the industrial revolution, we already discovered fire, so we already had enormous amounts of potential energy to extract - although never as much as we would have liked, which produced our obsession with calculating exactly how much energy we -could- extract. Entropy is an engineer's concept which arose from that obsession.

Entropy is really just an elaborately-dressed up way of saying that time flows in one direction - that the physical processes flow in one direction. Coal burns; carbon dioxide doesn't draw heat in and convert itself into coal, raining down upon us. If the reverse were true - carbon dioxide drew heat in and converted itself into coal, it would be an endothermic, instead of exothermic, reaction - these kinds of processes do in fact exist in real life. If it did -both-, we couldn't extract any useful energy from the process, because as we "burned" coal it would re-consume the energy and re-precipitate carbon - these kinds of processes -also- exist in real life, they're called reversible reactions.

Now, it sounds like a compound like this would be really useful, and you'd be right. It's exactly what water does - absorbs heat to evaporate, then gives off heat to re-precipitate. If coal behaved as I described, you'd require a heat source to re-precipitate the carbon dioxide after you've extracted work - and thus heat - from the system, since it would require that energy to re-bond. (Actually, coal -sort of- works as I described - exposed to the right sort of energy and conditions, it -does- re-precipitate, which is part of what plants do when they convert carbon dioxide into carbon.) The difficulty is that managing the boundary conditions to make the process cyclic requires something -else- be providing useful work.

Entropy, as a concept, is made much more mysterious than it really is. In its shortest form, the second law of thermodynamics is just stating that all the laws of physics -continuously- apply. Time flows in only one direction.

The fiction is in the implication - that a given amount of energy can ever only do some finite amount of work before it is spent for good. There's nothing we have yet discovered in the laws of physics that says you can't have a perpetual motion machine, or that you can't extract an infinite amount of usable work in a closed system. There's just nothing in the laws of physics we have yet discovered which -permits- an infinite amount of usable work to be performed in a closed system.

It's an important distinction, because there's a -lot- we haven't yet discovered.

## Sunday, December 14, 2014

### Lossy Transformations and Mathematics

I'll lead with a question: What is X divided by X?

The immediate and obvious answer is "1", but this is, in fact, incorrect. The answer is, roughly, "1 except where X equals 0". This is both pedantic and important - 0 divided by 0 isn't 1, it's "Undefined". 5 times 0 is -also- 0. "Undefined" in this case really means something like "Every answer simultaneously."

Division, as it is typically defined, is a lossy transformation - you have the potential to lose information in performing the operation. So is multiplication - the equation "5 = 3" can be "made correct" by multiplying by zero, a conceptually valid operation.

Squaring numbers is too. 5^2 is 25 - but once you've done this, you can no longer determine, from the current properties of whatever it is you're working with, whether you started with five or negative five. You've lost information about your starting configuration by performing what we usually consider a perfectly valid operation. Reversing the operation doesn't give you what you started with.

The issue is one of simplification. There isn't one single zero. There are an -infinite- number of 0's. Zero apples isn't the same as zero oranges - they're different zeros.

Squares are similar; a rectangle five feet long and four feet wide has twenty square feet, but it's not the same square feet as from a rectangle ten feet long and two feet wide. A square value doesn't maintain information about its constituent parts - this information is simplified away.

Division, again, is similar; 5 / 5 equals 1. Is it the -same- 1 as provided by 4 / 4? No. They're different 1's, but once we've reduced to a single number, that information is lost to us.

This simplification is great, if you don't need that information, and terrible, if you do.

Every mathematical operation results in a loss of information. Again, this is helpful, if you're looking for a simple result, and worthless, if you end up needing that information. Knowing that the combined length of two walls is 10 doesn't tell you anything about the individual length of the individual walls - you lost that information when you added the two numbers together.

The purity, the cleanness, of mathematics is an illusion, produced by rules which encourage you not to notice the information that goes missing with every step. Mathematics, in truth, is a very messy process, the process of crossing out information until you're left only with the information you think you need. The erasure is intellectually satisfying, but it is wholly the act of hiding complexity to make the complex -seem- simple. The complexity is still there, and knowing the square footage of a room you're tiling tells you next to nothing about the number of tiles you need to cut, and how.

The immediate and obvious answer is "1", but this is, in fact, incorrect. The answer is, roughly, "1 except where X equals 0". This is both pedantic and important - 0 divided by 0 isn't 1, it's "Undefined". 5 times 0 is -also- 0. "Undefined" in this case really means something like "Every answer simultaneously."

Division, as it is typically defined, is a lossy transformation - you have the potential to lose information in performing the operation. So is multiplication - the equation "5 = 3" can be "made correct" by multiplying by zero, a conceptually valid operation.

Squaring numbers is too. 5^2 is 25 - but once you've done this, you can no longer determine, from the current properties of whatever it is you're working with, whether you started with five or negative five. You've lost information about your starting configuration by performing what we usually consider a perfectly valid operation. Reversing the operation doesn't give you what you started with.

The issue is one of simplification. There isn't one single zero. There are an -infinite- number of 0's. Zero apples isn't the same as zero oranges - they're different zeros.

Squares are similar; a rectangle five feet long and four feet wide has twenty square feet, but it's not the same square feet as from a rectangle ten feet long and two feet wide. A square value doesn't maintain information about its constituent parts - this information is simplified away.

Division, again, is similar; 5 / 5 equals 1. Is it the -same- 1 as provided by 4 / 4? No. They're different 1's, but once we've reduced to a single number, that information is lost to us.

This simplification is great, if you don't need that information, and terrible, if you do.

Every mathematical operation results in a loss of information. Again, this is helpful, if you're looking for a simple result, and worthless, if you end up needing that information. Knowing that the combined length of two walls is 10 doesn't tell you anything about the individual length of the individual walls - you lost that information when you added the two numbers together.

The purity, the cleanness, of mathematics is an illusion, produced by rules which encourage you not to notice the information that goes missing with every step. Mathematics, in truth, is a very messy process, the process of crossing out information until you're left only with the information you think you need. The erasure is intellectually satisfying, but it is wholly the act of hiding complexity to make the complex -seem- simple. The complexity is still there, and knowing the square footage of a room you're tiling tells you next to nothing about the number of tiles you need to cut, and how.

## Saturday, December 6, 2014

### Obvious Things Part 3: Entitlement

I don't think anybody -likes- an entitlement mentality, it's just that none of us can agree on exactly what this means. Loosely speaking, entitlements are just expectations to things we personally don't agree should be expected. Remember, however, that your position in society is dependent entirely upon your conformance to others' expectations - entitlements, therefore, are most injurious to the upper classes, which is exactly why entitlements are fomented by those who oppose the stratification of society.

Entitlements are fomented as a form of political brinkmanship, by either those who have power and intend to make power to difficult to aspire to, or by those who don't have power and seek to push those in power out. This is obvious if you think about it for a moment - that's exactly what political promises -are-, the setting of expectations, of entitlements.

The difficulty, however, is that this is a ratchet, a one-way process that keeps going until everything destabilizes, until the promises exceed the ability of those in power to fulfill them. I can point to several eras in history where exactly this happened - the result is never pretty, although it does generally do a pretty good job of resetting everybody's expectations.

Because there is always a benefit to some party of creating expectations - of creating entitlements - I suspect this process is inevitable, and social stability cyclical. Of course, as time has gone on, the ability of powerful people to meet promises has increased, producing longer and longer periods of stability - and with technology, it's possible the cycle may be broken already, as ever-increasing productivity ensures the ever-increasing promises of the powerful can perpetually be met.

There's thus an incentive for those seeking power to destroy the advance of technology. The question, of course, is whether they realize it. I suspect some have, given the degree of effort taken towards precisely that goal.

Consider that in a Democracy, the public is ultimately in charge. The Party Leaders are jealous of this. Consider what this means in the long term.

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