a thoughtful web.
comment by kleinbl00
kleinbl00  ·  99 days ago  ·  link  ·    ·  parent  ·  post: Teaching stuff is resolved.

First off congratulations but now I'm confused by your analogy.

So an invariant is not a constant because an invariant is subject to conditions while a constant is defined? So an invariant is kind of like a catalyst only with math?

Devac  ·  99 days ago  ·  link  ·

EDIT: More or less you can (and should) ignore this post. I'll leave it as it is for the sake of the rest of the conversation below, but I was seriously jamming words together at places.

So an invariant is not a constant because an invariant is subject to conditions while a constant is defined?

I'm going to answer "yes, but". Invariants can be constants, but not all constants are invariants. It depends on in respect to what something is invariant.

Let's make the karma example more explicit. We already have a set of 4 people. 3 of them have good karma, 1 has a bad karma. This might be an invariant depending on the rules for karma exchange we will make.

Now here are the rules for karma exchange (in pairs, interact with everyone around you):

1. A person with good karma will give good karma to a person with good karma. (Harmony)

2. A person with good karma will have to do something bad to a person with bad karma. (They get what they deserve)

3. A person with bad karma will have to do something good to a person with good karma. (They repent for what they have done)

4. A person with bad karma will do something bad to a person with bad karma. (War never changes)

Now, let's go turn by turn:

Turn 0:

``  G G``  G B``

Turn 1:

``  G <1> G  ^         ^   2         4  v         v``  G <3> B``

1. Good with Good, nothing changes.

2. Good with Good, nothing changes.

4. Good with Good, nothing changes.

So the state goes:

``  G G     G G   G G    G G    G G ``  G B     G B    G B    B G    B G``

Turn 2 (same order of operation):

``  G G    G G    B G     B G    B G``  B G    B G    G G     G G    G G``

(sorry if I made some sort of mistake along the way, I am seriously tired after last few days)

The invariant can be anything that does not changes as a holistic property when discussing examples like the one above. There will always be one Bad karma, it just passes to the next person in this case. But as a more general term, it's anything that will come as a result of a certain type of operation. Adding any number to it's opposite:

``  1 + (-1), 2 + (-2) … n + (-n)``
will be always equal to zero (for a given definition of addition ;)). While trivial, it is indeed an invariant of the operation as it will work for any object. You always have to specify in respect to what you have your invariant property.

Am I making any sense? Sorry, I'm seriously not at my best but I wanted to respond today. If that's not clear please tell me.

kleinbl00  ·  98 days ago  ·  link  ·

You're making my head hurt. Also, perhaps part of the problem is that's not how Karma works.

Here are the actual rules for Karma exchange:

1. A person with good karma will increase their good karma by helping a person with good karma.

2. A person with good karma will increase their good karma by helping a person with bad karma.

3. A person with bad karma will increase their good karma by helping a person with good karma.

4. A person with bad karma will increase their good karma by helping a person with bad karma.

The steady-state of Karma is enlightenment - we all break free of the wheel and vanish into the universe in a cloud of patchouli-scented bliss. It may not be a good example for invariance. That might be what got you in trouble - you flippantly picked a religious concept you don't understand perfectly and then used it incorrectly to make your point.

Which - even accepting your terms, even accepting your math - I still don't understand and I have an engineering degree:

While trivial, it is indeed an invariant of the operation as it will work for any object.

WHAT is indeed an invariant of the operation? What are we defining here? An integer plus its negative integer will equal zero - so is that an invariant operation? But that's "invariant" as adjective, not "an invariant" as noun.

I get the condition of invariance - the Conway glider is a great example. But in the Conway glider example, what is the invariant? This is why I gave a counter-example of catalysts - you need them for the reaction, but their mass returns to solution in the course of the process.

bfv  ·  98 days ago  ·  link  ·

WHAT is indeed an invariant of the operation? What are we defining here? An integer plus its negative integer will equal zero - so is that an invariant operation? But that's "invariant" as adjective, not "an invariant" as noun.

The invariant is a property. If you're a mathematician, a property is a thing you can define an indicator function for.

kleinbl00  ·  98 days ago  ·  link  ·

So "properties" are like "metadata" of the equation. They're not in it but they're a 2nd order quality that can be analyzed or used in other equations. They're not like catalysts, because the catalyst is actually in the equation - they're more like molarity.

bfv  ·  98 days ago  ·  link  ·

So invariants are generalized from geometry. Angles being invariant under isometries is probably the ur-example. If you have a triangle, the measure of the angle A is a property of the triangle. If you translate the triangle, the image of A has the same measure as A. Another definition of a property is a one-argument predicate; it's something you can say about an object. If you know an invariant holds before an operation, you know it holds after; that's what's interesting about them.

Something is only a catalyst in a particular reaction, so I would think it would only make sense to say "C is a catalysts" is a property of the reaction as a whole. My chemistry is beyond rusty though.

Devac  ·  98 days ago  ·  link  ·

Sorry. This is getting saved to motivate me to develop some sort "you are too sleepy to write" test.

That might be what got you in trouble - you flippantly picked a religious concept you don't understand perfectly and then used it incorrectly to make your point.

Can't argue with that, although I wouldn't say I was flippant intentionally. Regardless, I get your point.

But in the Conway glider example, what is the invariant?

Really useful frame-by-frame visualisation

Invariants:

1. Number of black squares after each step.

2. Direction of movement.

3. Distance it will move between each complete cycle.

4. There will always be only one black square that is connected to the rest of the structure by its vertex.

Some of them are invariants for a whole cycle, others work for each frame.

This is why I gave a counter-example of catalysts - you need them for the reaction, but their mass returns to solution in the course of the process.

Yes. The catalyst can be thought of as an invariant of a specific chemical reaction. Again, sorry for my more than underwhelming response earlier. I hope that this one is clearer.