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[WalrusMcFishSr]

Adding to the confusing mathematical metaphors:

I always thought about potions being like imaginary numbers, like a separate axis or dimension for card cost. Is 2+i greater than or less than 5? Well it's not really either (unless you consider the modulus). This metaphor would be much more interesting if there were an effect that multiplied card costs...

Sir Peebles, your diagram reminds me a bit of Madelung's Rule from chemistry.

SCIENCE!!!

[Tables]

Adding to the confusing mathematical metaphors:

I always thought about potions being like imaginary numbers, like a separate axis or dimension for card cost. Is 2+i greater than or less than 5? Well it's not really either (unless you consider the modulus). This metaphor would be much more interesting if there were an effect that multiplied card costs...

Sir Peebles, your diagram reminds me a bit of Madelung's Rule from chemistry.

SCIENCE!!!

Imaginary/Complex numbers aren't really as mathematical a way to think of them as partial orders, which is literally what they are, but it's an okay approximation. The main issue with using complex numbers is that the same logic which makes "which is smaller, 3+i or 5" an unreasonable question (assuming we aren't taking argument, that is) is also that which makes "which is smaller, 3+i or 4+i" unreasonable. And that's obviously an issue.

Sir Peebles diagram is a standard Hasse diagram showing partial ordered costs. That kind of thing gets even more useful for comparing costs when you have even more variables - say there was a Mandrake cost which could be up to 3 Mandrakes on some cards, and some cards cost Coins, Mandrakes and Potions. With something like that, a Hasse diagram easily shows that something with cost $2+P+2M costs more than something costing $2+P, but is incomparable (neither more nor less than) to something costing $3+P.

[sudgy]

Also, complex numbers might be a problem because some people will say  5 > 3 + i  because some people will just use the absolute value of the number to compare complex numbers.

[ConMan]

Also, complex numbers might be a problem because some people will say  5 > 3 + i  because some people will just use the absolute value of the number to compare complex numbers.

By which logic, 5 = 3 + 4i. I can accept saying that |3 + 4i| = |5| = 5, but saying that the two are equal is stretching things a bit.

[sudgy]

Also, complex numbers might be a problem because some people will say  5 > 3 + i  because some people will just use the absolute value of the number to compare complex numbers.

By which logic, 5 = 3 + 4i. I can accept saying that |3 + 4i| = |5| = 5, but saying that the two are equal is stretching things a bit.

I wouldn't consider that to be it either, I'm just saying that some people will.  If I was forced to compare two complex numbers, I would take their absolute value.  I wouldn't usually compare two complex numbers though.

[florrat]

The only "canonical" order on the complex numbers I know is the order induced by the positive elements of the (trivial) C*-algebra C. This is a partial order, but doesn't coincide with the prices of Dominion cards. The prices of the Dominion cards correspond to the product order on N².

Sorry, couldn't resist, but this is probably the only forum not devoted to mathematics where a significant portion of the active readers will understand it.

[PSGarak]

There is no way to make the Complex Numbers into an Ordered Field (with the standard field operators), which is why there's no canonical total order. However, if you don't care about relating the ordering to the field algebra, then you're free to use whichever order you choose. I'm partial lexicographic orderings, myself. I mean, if you're not going to relate it to the field, might as go all the way.

[Riftman]

Also, complex numbers might be a problem because some people will say  5 > 3 + i  because some people will just use the absolute value of the number to compare complex numbers.

By which logic, 5 = 3 + 4i. I can accept saying that |3 + 4i| = |5| = 5, but saying that the two are equal is stretching things a bit.

It's not that !(a > b) && !(b > a) by itself implies* a = b. It's just that that relation R(a, b) = (|a| > |b|; a, b from C) is not antisymmetric, and therefore, is not a proper ordering.

*Well, if we take that > is classical "greater than", i.e. it is total, antisymmetric, anti-reflexive and transitive, then left-hand of implication
!(a > b) && !(b > a) => a = b
is trivially false, and therefore, implication is trivially true, but I don't think that's what you meant, mainly because proposed > is not classical "greater than".

[SirPeebles]

There is no way to make the Complex Numbers into an Ordered Field (with the standard field operators), which is why there's no canonical total order. However, if you don't care about relating the ordering to the field algebra, then you're free to use whichever order you choose. I'm partial lexicographic orderings, myself. I mean, if you're not going to relate it to the field, might as go all the way.

My first homework problem in my first graduate math course was to prove that 1 is positive.

[sudgy]

There is no way to make the Complex Numbers into an Ordered Field (with the standard field operators), which is why there's no canonical total order. However, if you don't care about relating the ordering to the field algebra, then you're free to use whichever order you choose. I'm partial lexicographic orderings, myself. I mean, if you're not going to relate it to the field, might as go all the way.

My first homework problem in my first graduate math course was to prove that 1 is positive.

Wait, couldn't you just say that 1 > 0, so it is positive?

[SirPeebles]

Can you prove that 1 > 0?

[sudgy]

Can you prove that 1 > 0?

I would just assume that...

[LastFootnote]

What givens were you allowed?

[Kirian]

Didn't it take like 270 pages of proofs to show that 1 + 1 = 2?

[SirPeebles]

You are just given that you have an ordered field.

A field means you have an addition and multiplication, and they obey the usual algebraic properties:  associativity, commutativity, distributivity, identities (0 and 1), and inverses (except that the additive identity 0 has no multiplicative inverse, i.e. you can't divide by zero).

An ordered field means that there is a total ordering which is compatible with the algebraic structure in the following sense:  There is a subset of positive elements.  This collection of positives is closed under both addition and multiplication, and given any element, either that element is positive, its additive inverse positive, or it is the additive identity.

tl;dr

You can add, multiply, subtract and divide as usual.  In particular, 1 and 0 do their thing.  Positive + Positive = Positive.  Positive x Positive = Positive.  Given any x, then either x is positive, x is zero, or -x is positive.

Edit:  Technically, you need to also specify that 0 is not equal to 1.

[SirPeebles]

Didn't it take like 270 pages of proofs to show that 1 + 1 = 2?

There was a book that did this, but it turns out to have been mistaken.  It turns out that the arithmetic of whole numbers is too complicated to prove its own internal consistency.

[Drab Emordnilap]

Didn't it take like 270 pages of proofs to show that 1 + 1 = 2?

There was a book that did this, but it turns out to have been mistaken.  It turns out that the arithmetic of whole numbers is too complicated to prove its own internal consistency.

O.O

This is why goko's rating formula can't be explained! I bet it's BASED on adding numbers together!

[SirPeebles]

Didn't it take like 270 pages of proofs to show that 1 + 1 = 2?

There was a book that did this, but it turns out to have been mistaken.  It turns out that the arithmetic of whole numbers is too complicated to prove its own internal consistency.

O.O

This is why goko's rating formula can't be explained! I bet it's BASED on adding numbers together!

There is strong empirical evidence that 1+1=2.

[PSGarak]

You are just given that you have an ordered field.

A field means you have an addition and multiplication, and they obey the usual algebraic properties:  associativity, commutativity, distributivity, identities (0 and 1), and inverses (except that the additive identity 0 has no multiplicative inverse, i.e. you can't divide by zero).

An ordered field means that there is a total ordering which is compatible with the algebraic structure in the following sense:  There is a subset of positive elements.  This collection of positives is closed under both addition and multiplication, and given any element, either that element is positive, its additive inverse positive, or it is the additive identity.

tl;dr

You can add, multiply, subtract and divide as usual.  In particular, 1 and 0 do their thing.  Positive + Positive = Positive.  Positive x Positive = Positive.  Given any x, then either x is positive, x is zero, or -x is positive.

Edit:  Technically, you need to also specify that 0 is not equal to 1.

I don't remember the full proof, but I do recall a few lemmas that you had to prove, and then use. One was that if x > y, then -x < -y. Since 0=-0, from this you can prove that taking a negative "swaps" from positive to negative or vice-versa. You then somehow show that the square of any number is positive (probably from the closure property). Since 1 = 1^2, 1 is positive, hence 1>0.

Then for complex numbers, -1 is the negative of 1, so it must be negative. But, it's also the square of i, so it must be positive. Contradiction, ergo the complex numbers are not an ordered field.

[Kirian]

Didn't it take like 270 pages of proofs to show that 1 + 1 = 2?

There was a book that did this, but it turns out to have been mistaken.  It turns out that the arithmetic of whole numbers is too complicated to prove its own internal consistency.

I'm not sure that Gödel's theorem show that Russell was mistaken, just that the book wasn't all that exciting or useful.

[PSGarak]

Didn't it take like 270 pages of proofs to show that 1 + 1 = 2?

There was a book that did this, but it turns out to have been mistaken.  It turns out that the arithmetic of whole numbers is too complicated to prove its own internal consistency.

I'm not sure that Gödel's theorem show that Russell was mistaken, just that the book wasn't all that exciting or useful.

Russel's goal was to make a foundation for math that would render it immune to being self-referential, because self-reference is what leads to paradoxes or unprovable propositions. I don't think Gödel exactly disproved any of what Russel had completed up to that point, but he showed that the project was doomed to failure in meeting its goals. The activity of counting is sufficient to construct self-referentiality, so either Russel's program would have failed to be able to achieve counting, or it would have eventually admitted self-reference.

[Watno]

What's the definition of 2?
I thought it was defined as 1+1.

[GendoIkari]

There is no way to make the Complex Numbers into an Ordered Field (with the standard field operators), which is why there's no canonical total order. However, if you don't care about relating the ordering to the field algebra, then you're free to use whichever order you choose. I'm partial lexicographic orderings, myself. I mean, if you're not going to relate it to the field, might as go all the way.

My first homework problem in my first graduate math course was to prove that 1 is positive.

Wait, couldn't you just say that 1 > 0, so it is positive?

Most people could; someone in a graduate math course can't.

[DG]

What's the definition of 2?
I thought it was defined as 1+1.

Indeed it is. Most of the counter examples badly define either counting or the arithmetic operator. For example, 1 river + 1 river = 1 bigger river.

[SirPeebles]

Well, the successor function is often considered to be more basic than addition, when it comes to the natural numbers (0, 1, 2, 3, ...).

A common axiomization of these numbers states, among things, that every number has a unique successor, and that every number is the successor of some unique other number, with a single exception.  That single number which is not the successor of another we name 0.  1 is the defined as the successor of 0.  2 as the successor of 1.  Addition is defined in terms of this simpler operation.