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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.

[SirPeebles]

As for proving that 1 is positive.

We know that either 1 is positive, 1 is zero, or -1 is positive.  Let's suppose that 1 is not positive, and look for a contradiction.  Since 1 is not positive, either 1 is zero or -1 is positive.  But 1 cannot be zero by hypothesis.  So all we need to do is rule out the possibility that -1 is positive.

Suppose that -1 were positive.  Well, positive x positive = positive, so then -1 x -1 is a positive number.  But I claim that -1 x -1 = 1.  Once we see why, then we will have our contradiction, since we are currently supposing that 1 is not positive.  To check that a number is equal to 1, it suffices to check that its sum with -1 is 0.

(-1 x -1) + -1 = (-1 x -1) + (-1 x 1) = -1 x (-1 + 1) = -1 x 0

Above we first used that 1 is the multiplicative identity, then the distributive law, and then the fact that -1 is the additive inverse of 1.

What is left is to show that -1 x 0 = 0.  In fact, a x 0 = 0 for any number a, as I'll now show:

a = a x 1 = a x (1 + 0) = (a x 1) + (a x 0) = a + (a x 0).

Now add -a to both sides:

-a + a = -a + ( a + (a x 0))
0 =  (-a + a) + (a x 0)
0 = 0 + (a x 0)
0 = a x 0.

Thus above we had that (-1 x -1) + -1 = 0, so -1 x -1 = 1.  This gives us our contradiction.

[Kirian]

As for proving that 1 is positive.

We know that either 1 is positive, 1 is zero, or -1 is positive.  Let's suppose that 1 is not positive, and look for a contradiction.  Since 1 is not positive, either 1 is zero or -1 is positive.  But 1 cannot be zero by hypothesis.  So all we need to do is rule out the possibility that -1 is positive.

Suppose that -1 were positive.  Well, positive x positive = positive, so then -1 x -1 is a positive number.  But I claim that -1 x -1 = 1.  Once we see why, then we will have our contradiction, since we are currently supposing that 1 is not positive.  To check that a number is equal to 1, it suffices to check that its sum with -1 is 0.

(-1 x -1) + -1 = (-1 x -1) + (-1 x 1) = -1 x (-1 + 1) = -1 x 0

Above we first used that 1 is the multiplicative identity, then the distributive law, and then the fact that -1 is the additive inverse of 1.

What is left is to show that -1 x 0 = 0.  In fact, a x 0 = 0 for any number a, as I'll now show:

a = a x 1 = a x (1 + 0) = (a x 1) + (a x 0) = a + (a x 0).

Now add -a to both sides:

-a + a = -a + ( a + (a x 0))
0 =  (-a + a) + (a x 0)
0 = 0 + (a x 0)
0 = a x 0.

Thus above we had that (-1 x -1) + -1 = 0, so -1 x -1 = 1.  This gives us our contradiction.

This looks suspiciously like the specious proof that 1 = 2 (you know, the one that divides by zero a couple times and obfuscates it).

[Watno]

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.

Isn't addition defined like this in that model?

0 + n := n
Succ (m) + n := Succ(m + n)

So in that case it is obvious that 1+1= Succ(0)+Succ(0)=Succ(0+Succ(0))=Succ(Succ(0))=2 ?

[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.

Isn't addition defined like this in that model?

0 + n := n
Succ (m) + n := Succ(m + n)

So in that case it is obvious that 1+1= Succ(0)+Succ(0)=Succ(0+Succ(0))=Succ(Succ(0))=2 ?

Probably something like that.  Then it is a bit of work to show that addition is associative and commutative.  In other words, that (x + y) + z = x + (y + z)  and that x + y = y + x.

[Watno]

I don't think I used associativity and commutativity?

[SirPeebles]

Sure, I didn't mean to imply that you did.  I was just drawing attention to the fact that commutativity and associativity are non obvious from this formal definition.  As opposed to the intuitive idea of 2+3 means put two apples into a bag, then three apples, and count how many you have.

[WanderingWinder]

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.

From this, I logically deduce that entering a graduate math course makes you less intelligent



Seriously, this kind of thing is what I hate about some of this formal math - you come up with some obscure, strange 'proof' that 1>0, but what it rests on is not at all more intuitive than the conclusion 1>0 which it is supposedly providing a basis for.

[Watno]

But what is it based on in significantly less.

[AJD]

Seriously, this kind of thing is what I hate about some of this formal math - you come up with some obscure, strange 'proof' that 1>0, but what it rests on is not at all more intuitive than the conclusion 1>0 which it is supposedly providing a basis for.

Really what's being proven in these case is these axioms are sufficient to describe a system in which 1 > 0; we don't need to stipulate that 1 > 0 a priori in order to state the fundamental properties of the number system (and therefore there isn't some other possible system which shares all those other properties but not the property that 1 > 0).

[clb]

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.

This reminds me of my friend's father. He asserts that a sandwich cut into two (probably triangles) is not two half-sandwiches, but rather two entire (albeit admittedly smaller) sandwiches. However, a single piece of bread folded over [stuff put on sandwiches] is a half sandwich. It is frustrating that it almost makes sense, but then doesn't.

[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.

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

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

From this, I logically deduce that entering a graduate math course makes you less intelligent



Seriously, this kind of thing is what I hate about some of this formal math - you come up with some obscure, strange 'proof' that 1>0, but what it rests on is not at all more intuitive than the conclusion 1>0 which it is supposedly providing a basis for.

It is true that we already have a strong intuition.

Two points:

1)  This is really about showing that in every ordered field, the number 1 is a positive element.  You can think of this either as proving an interesting fact about ordered fields, or as evidence that the concept of an ordered field really is matching our intuition.  There are ordered fields which come up which are not just subsets of the real numbers.

2)  Perhaps even more important historically, ordered fields show up in the axiomization of the real numbers.  Somehow, we humans have quite a bit of intuition for some rather bizarre numbers.  Not just whole numbers, but fractions, the square root of two, pi, the cosine of a 72 degree angle, and so forth.  We can list lots of "obvious" facts about real numbers:  any number times zero is zero, the product of two negative numbers is positive, every positive number has a square root, every cubic equation has at least one solution, there is no largest number, between any two distinct numbers there is a third number, ... .  Lots and lots of facts.  If I want to prove something about real numbers, how many of these intuitive facts do I need to know?  In physics, if you can't solve a problem, sometimes you need to go observe the physical world to discover new truths.  Is it the same for the study of real numbers?  Well, it turns out that our intuition informs us that the real numbers form a complete ordered field, and then we can prove mathematically that there is only one complete ordered field.  Thus, the basic axioms of a complete ordered field must be all that one needs to begin a comprehensive study of the real numbers.  In particular, one is able to recover intuitive facts such as those I listed above.

[WanderingWinder]

But what is it based on in significantly less.

I think what you're trying to say is "But what it is based on is significantly less", which I can say, uh, no, no it isn't. Or I guess, define "less", and I'm sure you're doing it in a way which is different from mine.

Seriously, this kind of thing is what I hate about some of this formal math - you come up with some obscure, strange 'proof' that 1>0, but what it rests on is not at all more intuitive than the conclusion 1>0 which it is supposedly providing a basis for.

Really what's being proven in these case is these axioms are sufficient to describe a system in which 1 > 0; we don't need to stipulate that 1 > 0 a priori in order to state the fundamental properties of the number system (and therefore there isn't some other possible system which shares all those other properties but not the property that 1 > 0).

Okay... but the problem is, (well first of all, who cares, we don't live in some fairy world of different mathematics), but more importantly, if there WERE some contradiction, it would show the axioms wrong rather than showing that 1>0 is possibly false in some universe or number system or whatever. i.e. you're proving things backwards.

Okay, so you can prove that these things logically prove the other thing. Why is this considered an achievement? It would be like asking to prove, given that I have typed this, that I exist.

[WanderingWinder]

1)  This is really about showing that in every ordered field, the number 1 is a positive element.  You can think of this either as proving an interesting fact about ordered fields, or as evidence that the concept of an ordered field really is matching our intuition.  There are ordered fields which come up which are not just subsets of the real numbers.

Can I think of it as someone coming up with a definition to match common understanding then patting themself on the back for having that match common intuition?



Seriously, the logical argument that "Ordered fields match our intuition about real numbers" -> "Ordered fields have applications beyond the real numbers" -> "Ordered fields can thus provide us with an understanding which surpasses our intuition" is really bad. It just IS your intuition and/or you actually don't have any evidence at all, because you haven't established its applicability beyond the realm which you originally verified it on.

[sudgy]

I think one of the points of math is to start with as few axioms as possible, and prove anything else from that.  If we can prove 1 > 0, then we don't have to postulate it.

I remember reading a proof (that was really simple) that with any line segment, you can create an equilateral triangle with sides that long.  Now, the conclusion is obvious, but if we can prove it with other axioms, we don't need to make that an axiom.

[WanderingWinder]

I think one of the points of math is to start with as few axioms as possible, and prove anything else from that.  If we can prove 1 > 0, then we don't have to postulate it.

Why? Shouldn't the point be to not have as numerically few axioms as possible (which also brings up the disturbing question of what counts as separate axioms), but rather to have things rest on as sure a set of 'givens' as possible? I mean, this isn't a Math question at all, it's an epistemology question.

I remember reading a proof (that was really simple) that with any line segment, you can create an equilateral triangle with sides that long.  Now, the conclusion is obvious, but if we can prove it with other axioms, we don't need to make that an axiom.

But this doesn't actually gain us anything. Why did we have to prove it in the first place? What is wrong with making something obviously true an axiom? If there is something wrong with making something obviously true an axiom, how does that not invalidate the whole process of making axioms?
Didn't you take it as an additional axiom (and one which actually seems most probably wrong to me) that it is better to have fewer axioms?

[sudgy]

Well, as soon as we know that a theorem can be proven with our chosen set of axioms, we can act like the theorem is an axiom.  The reason to have a few axioms is that we don't need to list some insane number of axioms.  Axioms are supposed to be the bare minimum of points required to show everything.

Also, we need to have some things that are proved.  Certain theorems, though true, don't make much intuitive sense (and some to certain people more than others).  Where do you draw the line of what needs to be proven and what not?

[SirPeebles]

It really is quite an accomplishment that the theory of real numbers can be reduced to a finite number of fundamental properties.

Perhaps to gain an appreciation for this utterly nonobvious fact, it would be nice to look at an example where it doesn't happen.  Perhaps the most famous example was in geometry, regarding Euclid's postulates.  Eucild attempt to reduce all of geometry to just a short list of axioms.  The plan was to have a short list of basic assumed facts that essentially all humans would agree are true.  Now yes, what is obvious to one person may not be obvious to another.  But that's the point really.  Euclid began from observations like "for any two points, there is a straight line which joins them"  rather than something about inscribed angles.  But ultimately he included one postulate which seemed not quite as evident, the parallel postulate.  For centuries mathematicians tried to reduce the parallel postulate to more intuitive statements, but failed.  Eventually the 18th and 19th centuries, people discovered that there were geometries -- models of the original four postulates -- where the parallel postulate did not hold.  This led to the discoveries of spherical, hyperbolic, and ultimately Riemannian geometry.  20th century advances in physics has revealed that in the actual universe, the parallel postulate isn't obeyed -- space is locally curved.

Another instance is in set theory.  Mathematicians began studying set theory in earnest in the 19th century.  Many fundamental axioms have been agreed upon:  the empty set exists, the union of two sets exists, ...  But then there are statements like the Continuum Hypothesis.  This statement is too complicated for most people to believe that it is obviously true or obviously false.  So you could try to prove or disprove it from the agreed upon axioms.  The trouble is, mathematicians have been able to show that it is independent from the communally agreed upon axioms.  That is, they constructed one model of set theory where the Continuum Hypothesis is obeyed, and a different model of set theory where the Continuum Hypothesis is violated.

This does not happen in the theory of real numbers.  Any two complete ordered fields are isomorphic.  It's not as though there is going to be one complete ordered field where every 17th degree polynomial has a root and another complete ordered field where there is a 17th degree polynomial without a root, thereby requiring us to go back to our evolved reason and intuition to divine which one of these two properties is valid for the true real numbers.  I can build a fancy-pants complete ordered field where every 17th degree polynomial has a root.  The true real numbers is the only complete ordered field.  So I have proved something for the true real numbers.  You can't argue that your personal conception of the real numbers might have a 17th degree polynomial with no roots unless you are willing to reject that the real numbers form a complete ordered field -- that is, unless you are willing to reject one of a short list of "obvious" facts.

[TWoos]

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.

In certain graduate level math classes you can't.  It's perfectly valid in others.  It all depends on how fundamental the class gets.

[TWoos]

I think one of the points of math is to start with as few axioms as possible, and prove anything else from that.  If we can prove 1 > 0, then we don't have to postulate it.

Why? Shouldn't the point be to not have as numerically few axioms as possible (which also brings up the disturbing question of what counts as separate axioms),

What counts as a separate axiom is what you define as a separate axiom. There's nothing disturbing about it.  Axioms are specified.

I remember reading a proof (that was really simple) that with any line segment, you can create an equilateral triangle with sides that long.  Now, the conclusion is obvious, but if we can prove it with other axioms, we don't need to make that an axiom.

But this doesn't actually gain us anything. Why did we have to prove it in the first place? What is wrong with making something obviously true an axiom?

Because when you create a new system of numbers (and yes, you can do this, and yes, there are reasons you might want to), you need to prove that every axiom that applied to the old system applies to the new system in order to use the proofs derived from those axioms.  The fewer axioms you used, the less you have to prove.

If there is something wrong with making something obviously true an axiom, how does that not invalidate the whole process of making axioms?
Didn't you take it as an additional axiom (and one which actually seems most probably wrong to me) that it is better to have fewer axioms?

No, it is not an axiom that it is better to have fewer axioms.  That's planning for future laziness, and mathematicians are all about being lazy.

[PSGarak]

Intuition has limits--it's of no use when we're faced with situations dissimilar to what we have faced before. Logic has different limits--it only works on things that are well-defined, because otherwise you get caught in meaningless semantic arguments. The strength of one approach covers for the weakness of the other, if we can direct both at a single problem.

Axiomatizing systems is an approach for taking what we know from intuition, and using logic to extend that knowledge to cover things we don't have experience for yet. There are other approaches, but they're fuzzier.

Axiomatic systems have two major weaknesses: They're fragile, and they can be unwieldy. By fragile, I mean that classical logic doesn't deal with consistency well. Like, at all. Like, a single inconsistency, no matter how small, renders the entire system invalid. By unwieldy, I mean that systems with large numbers of axioms are difficult to reason about, because they have more stuff in them.

Both of these problems can be alleviated by reducing the number of axioms you're dealing with. The more that you can reduce "intuitive axioms" to "theorems" by proving them from other axioms, the simpler your system becomes. This makes it less likely to be inconsistent, and easier to work with because there's less facts to keep straight.

Simpler systems also make it easier to test things that you don't know if they're true or not. You can try to test if a fact is true by seeing if it interacts with other axioms in a way that creates a contradiction. In simpler systems, it's much easier to pinpoint which actual a contradiction arose from, and then you can evaluate which of the two axioms you want to discard.

Even in an example as synthetic as the real number line, this gets important when e.g. you want to start talking about infinities. Infinity is inherently a non-intuitive concept, so any attempt to intuit how it ought to behave is going to lead to contradictions. If your foundation is a giant pile of "intuitive facts" about how numbers behave, it's really hard to add infinity to the mix and keep them all straight. You'd have to gut and replace like half your damn system in order to work in infinity, and even then you won't be sure you've done a complete job. When your system is built on like 5 rules, it's easy to change one rule to admit infinity, and then study the consequences in a logical fashion.

[SirPeebles]

i = sqrt(-1) = sqrt(1/-1) = 1/sqrt(-1) = 1/i

Since we've shown i = 1/i, we conclude that i^2 = 1

Thus 1 = -1. 

Whoops, guess 1 is negative after all.

[sudgy]

i = sqrt(-1) = sqrt(1/-1) = 1/sqrt(-1) = 1/i

Since we've shown i = 1/i, we conclude that i^2 = 1

Thus 1 = -1. 

Whoops, guess 1 is negative after all.

Argh, where is the contradiction?

[AJD]

i = sqrt(-1) = sqrt(1/-1) = 1/sqrt(-1) = 1/i

Since we've shown i = 1/i, we conclude that i^2 = 1

Thus 1 = -1. 

Whoops, guess 1 is negative after all.

Argh, where is the contradiction?

I think the contradiction is that the operator sqrt() isn't well-defined; sqrt(–1) can evaluate to either i or –i. So

i = 1/sqrt(–1)

is true, but only if the sqrt(–1) on the right side there is –i, not i.

[AJD]

Seriously, this kind of thing is what I hate about some of this formal math - you come up with some obscure, strange 'proof' that 1>0, but what it rests on is not at all more intuitive than the conclusion 1>0 which it is supposedly providing a basis for.

Really what's being proven in these case is these axioms are sufficient to describe a system in which 1 > 0; we don't need to stipulate that 1 > 0 a priori in order to state the fundamental properties of the number system (and therefore there isn't some other possible system which shares all those other properties but not the property that 1 > 0).

Okay... but the problem is, (well first of all, who cares, we don't live in some fairy world of different mathematics), but more importantly, if there WERE some contradiction, it would show the axioms wrong rather than showing that 1>0 is possibly false in some universe or number system or whatever. i.e. you're proving things backwards.

Well, again, the real numbers aren't the only thing mathematicians are interested in. What this shows is that any system that resembles the real numbers in conforming to the axioms that define an ordered field also resembles it in having 1 > 0.

[WalrusMcFishSr]

[PSGarak]

i = sqrt(-1) = sqrt(1/-1) = 1/sqrt(-1) = 1/i

Since we've shown i = 1/i, we conclude that i^2 = 1

Thus 1 = -1. 

Whoops, guess 1 is negative after all.

Argh, where is the contradiction?

I think the contradiction is that the operator sqrt() isn't well-defined; sqrt(–1) can evaluate to either i or –i. So

i = 1/sqrt(–1)

is true, but only if the sqrt(–1) on the right side there is –i, not i.

The problem is definitely with the Square-Root operation and the fact that it can produce more than one result. Since mathematicians don't like the phrase "isn't well-defined," we've cooked up a bunch of technicalities that make it not only well-defined, but so that everything works out to the correct answer.

The usual approach involves something called "branch cuts"; basically, the input to the function is not the normal complex plane, but rather a pair of complex planes "sewn together" along a seam (the result looks kinda like spiral-cut ham). Then, rather than one input having two possible results, we say that there are two possible inputs (one from each plane), each of which has its own result.

Having a one-to-one mapping of input and output ("injectivity") makes the function perfectly reversible, which prevents the paradox above. However, the the details of this are a giant pain the ass to work through because you have to do a lot of bookkeeping of which plane you're on. The details also depend on where exactly you place your branch cut. Also don't forget to define how the function f(z) = 1/z interacts with your branch cut, since that's sort of a big part of the paradox as well (this probably involves more branch cuts).

[ftl]

Okay... but the problem is, (well first of all, who cares, we don't live in some fairy world of different mathematics),

I think you're trying to think about the mathematics in too applied a fashion.

Mathematics is/(can be) about abstract concepts, not about "the real world" which you're contrasting to the "fairy tale world of different mathematics". I think geometry is a fine common example of how that makes sense.

Euclidean geometry is a self-consistent geometry. You can prove lots of stuff! There's a whole geometrical world of interesting things! But there also non-Euclidean geometries. Which have axioms which differ, and thus you can derive entirely different things. There isn't a need to pick one of those sets of axioms and declare it "right" and the others "wrong". We live in a universe where you can formulate both euclidean and non-euclidean geometries and have them each be internally consistent, though obviously not in agreement with each other. They'll describe different abstract concepts. They may or may not be useful for describing the physical world. Both Euclidean and non-euclidean geometries are reasonable fields of study.

(As it turns out, in practice, our universe is fundamentally non-euclidean, but of course euclidean geometry is still useful.)

So if you could make up a different number system which was internally consistent and which violated some obvious "intuitive" properties while keeping others, that would probably be an interesting endeavor! And knowing how mathematics goes, it would stay an esoteric branch of math with everybody looking at it askance and thinking "Why the F does anyone care about this weird thing?" until 100 years later it ends up being the cornerstone of some field of physics or something.

Here's another example - modular arithmetic. Imagine, if you will, a different set of axioms about numbers - ones that say the number line wraps around. 0,1,2,...,N,0,1,2,... I mean, that certainly fits your description of "a fairy world of different mathematics" - if you take 6 apples and add more apples to them, you'll never end up with 0 apples. But yet you can make a number system like that, and prove a lot of things about its properties, how addition and multiplication and division work in this weird system. None of which would be at all useful for counting apples, but so what? Since when is apple-counting so important?

...and then you find that this system is incredibly useful in the modern world. It's how all computers do arithmetic at a fundamental level. Also it's the foundation of all of cryptography. And a bunch of other things, which I don't have the math expertise to understand.

The history of mathematics is filled with ideas which seemed weird and counterintuitive and "The real world doesn't work like that so why do you even care?" which turn out, in retrospect, to be super-important and super-useful.  There's nothing wrong with formulating a self-consistent system of axioms and just seeing what it turns out to be, even if it's not immediately evident how the axioms relate to apple-counting.

[sudgy]

So if you could make up a different number system which was internally consistent and which violated some obvious "intuitive" properties while keeping others, that would probably be an interesting endeavor! And knowing how mathematics goes, it would stay an esoteric branch of math with everybody looking at it askance and thinking "Why the F does anyone care about this weird thing?" until 100 years later it ends up being the cornerstone of some field of physics or something.

Actually, I've realized that also some things the universe does isn't intuitive (you mean this particle is here and there at once?!?), so maybe the fundamental math behind it isn't either.

[ftl]

Well, all of quantum mechanics does basically run on imaginary numbers.

[sudgy]

Well, all of quantum mechanics does basically run on imaginary numbers.

Well, imaginary numbers aren't really intuitive...

[SirPeebles]

Sometimes I see people think that studying the geometry of a five dimensional curved space is ridiculous, since the physical space in which we move around is so clearly Euclidean and 3-dimensional (or absolutely close enough for most anyone's daily life).  As though mathematicians are taking unwarranted precautions in case it is revealed that we're living on a Klein bottle.

Suppose we would like to study the motion of single small object.  With three numbers, we are able to specify it location.  With three more numbers, we are able to specify its momentum (or velocity, if you prefer).  So in sum, to understand all of the possible states of motion for this object, we need to track these six numbers.  These numbers can each take any value (here I'm brushing aside issues of the speed of light or finite extent of the universal.  I'm trying to keep this down to earth).  So you can think of each possible state of this object as being the coordinates of some point in 6-dimensional Euclidean space.

But now, suppose you add in some sort of constraint, like tether your object to some point by means of either a rigid pole or a loose rope.  Each of those deform the region of this 6-dimensional space whose points correspond to admissible states.  I haven't studied these situations since 2004, so I can't think of a particularly compelling example on the spot, but this is where curved higher dimensional spaces often come up:  the points in the space correspond to "good" or "admissible" states of data subject to certain constraints.  Understanding the geometry can help you predict how the data will change, whether the data can be changed from one state to another without passing through the inadmissible region, and how this could be done most efficiently.

[PSGarak]

The most accessible example of non-Euclidean spaces is just the surface of the Earth. It lives in three dimensions, but the surface is effectively (locally) two-dimensional because you only need two coordinates to specify it (say, lat & long). This is a curved space living inside a normal one.

And, as you say, finding optimal paths isn't entirely obvious. Airplanes want to fly in the shortest possible path, because fuel is stupid expensive. We're used to straight lines being the shortest path, but that doesn't work here. A straight line on a map isn't close to being the shortest path on a globe, and a straight line in three-dimensional space tunnels underground, which airplanes are currently unsuited for. But by studying the surface of the globe as a non-Euclidean space, we can find the geodesics that minimize fuel costs.

It gets cooler when you start modeling things like e.g. cooler temperatures further north, so that one mile in one location costs more than a mile in another location. Now space is stretched in addition to curved.

[Schneau]

Re: Alchemy with the other sets numbers

[sudgy]

Re: Alchemy with the other sets numbers

FTFY

[Watno]

I thought the OP meant sets of numbers?

[PSGarak]

I thought the OP meant sets of numbers?

Absolutely. Alchemy with ℕ, Alchemy with ℤ, Alchemy with ℝ, Alchemy with ℂ. Unfortunately, Dominion doesn't seem to enable the construction of transfinite numbers, but it doesn't explicitly disallow them either.

[Witherweaver]

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.

This is why Analysis is way cooler than Algebra.

Edit: By the way, why isn't this trivial?

Let x be a positive element in the field. (Such an element must exist, because a field must have a zero element {0} and a multiplicative identity {e}.  Either e is positive or its additive inverse (-e) is positive.)  Let e be the multiplicative identity element.  Then ex = x.  If e were negative, then ex=x would be negative as well, which contradicts the positivity of x.

Or is it not clear that (negative)*(positive)=(negative)?  This should be clear since x being negative means -x is positive.  Suppose we have a negative x and a positive y.  We want to show -(xy) is positive.  Now -(xy) is the additive inverse of (xy), but so is (-x)y by the distributive law: (xy) + (-x)y = (x+-x)y = 0y = 0.  Thus, -(xy) = (-x)y.  As -x is positive and y is positive, (-x)y = -(xy) is positive, and so xy is negative.

[Witherweaver]

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.

From this, I logically deduce that entering a graduate math course makes you less intelligent



Seriously, this kind of thing is what I hate about some of this formal math - you come up with some obscure, strange 'proof' that 1>0, but what it rests on is not at all more intuitive than the conclusion 1>0 which it is supposedly providing a basis for.

This seems like it's missing the point entirely.  The "1" and "0" are the multiplicative and additive identities of the ordered field.  They are only your usual counting 0 and 1 in the case of the field of real numbers (under the usual operations).  This need not have anything to do with real numbers, but you still want to have your collection of things that you're dealing with be well defined and have structure.  So that's why it has an axiomatic definition.  It's a set of stuff with certain properties.  With those properties you can prove more things,  yes, and you could have taken those things in the definition, but why do so?

I don't do anything with algebra, but maybe someone has an example of an ordered field where the ordering is not intuitive and the fact that a certain element is positive is important yet not intuitive.  But even if there isn't such an example, these things are still important because often you deal with these structures in a non-concrete  way.  You may need to use properties of a field in your argument, or deal with a generic one, so you can't just rely on what you find intuitive about real numbers.

Edit: Another reason not to include a lot of provable properties in definitions is because those properties may not be universal to generalized objects.  There are a bunch of things that are true about commutative groups, so you could make the definition of a commutative group be the usual definition plus all those things.  But then the definition of a group and a commutative group are vastly different.  One has a list of things it satisfies, and one has a much larger list.  It's more palatable to define them in a more minimal way and focus on the key difference being ab=ba holds for one and not the other. 

[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.

This is why Analysis is way cooler than Algebra.

Edit: By the way, why isn't this trivial?

Let x be a positive element in the field. (Such an element must exist, because a field must have a zero element {0} and a multiplicative identity {e}.  Either e is positive or its additive inverse (-e) is positive.)  Let e be the multiplicative identity element.  Then ex = x.  If e were negative, then ex=x would be negative as well, which contradicts the positivity of x.

Or is it not clear that (negative)*(positive)=(negative)?  This should be clear since x being negative means -x is positive.  Suppose we have a negative x and a positive y.  We want to show -(xy) is positive.  Now -(xy) is the additive inverse of (xy), but so is (-x)y by the distributive law: (xy) + (-x)y = (x+-x)y = 0y = 0.  Thus, -(xy) = (-x)y.  As -x is positive and y is positive, (-x)y = -(xy) is positive, and so xy is negative.

That would be another valid proof, although you didn't explain why 0y=0.

If you consider all of that together to be trivial, then yes it is trivial.  I mean, it was problem 1 of assignment 1 of real analysis 1.  It isn't supposed to be a huge strain.

[Witherweaver]

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.

This is why Analysis is way cooler than Algebra.

Edit: By the way, why isn't this trivial?

Let x be a positive element in the field. (Such an element must exist, because a field must have a zero element {0} and a multiplicative identity {e}.  Either e is positive or its additive inverse (-e) is positive.)  Let e be the multiplicative identity element.  Then ex = x.  If e were negative, then ex=x would be negative as well, which contradicts the positivity of x.

Or is it not clear that (negative)*(positive)=(negative)?  This should be clear since x being negative means -x is positive.  Suppose we have a negative x and a positive y.  We want to show -(xy) is positive.  Now -(xy) is the additive inverse of (xy), but so is (-x)y by the distributive law: (xy) + (-x)y = (x+-x)y = 0y = 0.  Thus, -(xy) = (-x)y.  As -x is positive and y is positive, (-x)y = -(xy) is positive, and so xy is negative.

That would be another valid proof, although you didn't explain why 0y=0.

If you consider all of that together to be trivial, then yes it is trivial.  I mean, it was problem 1 of assignment 1 of real analysis 1.  It isn't supposed to be a huge strain.

Oh I see, given the amount of discussion that followed I had thought at first that it was something major   And yeah I guess 0y=0 isn't part of the definition since it follows from 0=e-e and ey=y.

By the way, did you go for your PhD?  If so, what focus?

[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.

This is why Analysis is way cooler than Algebra.

Edit: By the way, why isn't this trivial?

Let x be a positive element in the field. (Such an element must exist, because a field must have a zero element {0} and a multiplicative identity {e}.  Either e is positive or its additive inverse (-e) is positive.)  Let e be the multiplicative identity element.  Then ex = x.  If e were negative, then ex=x would be negative as well, which contradicts the positivity of x.

Or is it not clear that (negative)*(positive)=(negative)?  This should be clear since x being negative means -x is positive.  Suppose we have a negative x and a positive y.  We want to show -(xy) is positive.  Now -(xy) is the additive inverse of (xy), but so is (-x)y by the distributive law: (xy) + (-x)y = (x+-x)y = 0y = 0.  Thus, -(xy) = (-x)y.  As -x is positive and y is positive, (-x)y = -(xy) is positive, and so xy is negative.

That would be another valid proof, although you didn't explain why 0y=0.

If you consider all of that together to be trivial, then yes it is trivial.  I mean, it was problem 1 of assignment 1 of real analysis 1.  It isn't supposed to be a huge strain.

Oh I see, given the amount of discussion that followed I had thought at first that it was something major   And yeah I guess 0y=0 isn't part of the definition since it follows from 0=e-e and ey=y.

By the way, did you go for your PhD?  If so, what focus?

Most crucial in proving 0y=0 is the distributive law.  You see, 0y=0 is rather special because you are asserting a multiplicative property about 0, which is itself defined by an additive property.  Thus you absolutely must invoke distributivity since that is the only field axiom which imposes any relationship between addition and multiplication.

I wrote my dissertation on homological algebra.

[Witherweaver]

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.

This is why Analysis is way cooler than Algebra.

Edit: By the way, why isn't this trivial?

Let x be a positive element in the field. (Such an element must exist, because a field must have a zero element {0} and a multiplicative identity {e}.  Either e is positive or its additive inverse (-e) is positive.)  Let e be the multiplicative identity element.  Then ex = x.  If e were negative, then ex=x would be negative as well, which contradicts the positivity of x.

Or is it not clear that (negative)*(positive)=(negative)?  This should be clear since x being negative means -x is positive.  Suppose we have a negative x and a positive y.  We want to show -(xy) is positive.  Now -(xy) is the additive inverse of (xy), but so is (-x)y by the distributive law: (xy) + (-x)y = (x+-x)y = 0y = 0.  Thus, -(xy) = (-x)y.  As -x is positive and y is positive, (-x)y = -(xy) is positive, and so xy is negative.

That would be another valid proof, although you didn't explain why 0y=0.

If you consider all of that together to be trivial, then yes it is trivial.  I mean, it was problem 1 of assignment 1 of real analysis 1.  It isn't supposed to be a huge strain.

Oh I see, given the amount of discussion that followed I had thought at first that it was something major   And yeah I guess 0y=0 isn't part of the definition since it follows from 0=e-e and ey=y.

By the way, did you go for your PhD?  If so, what focus?

Most crucial in proving 0y=0 is the distributive law.  You see, 0y=0 is rather special because you are asserting a multiplicative property about 0, which is itself defined by an additive property.  Thus you absolutely must invoke distributivity since that is the only field axiom which imposes any relationship between addition and multiplication.

I wrote my dissertation on homological algebra.

Right, I was using the distributive law.  That's a good point that the distributive property connects the two operations.

I did my dissertation on nonlinear boundary value problems.  I wonder if there are a lot of math people on this forum.  I have a feeling Dominion would attract more of the combinatorics and algebra types.

[Watno]

There are.

[Drab Emordnilap]

This is all alchemy to me. 

[sudgy]

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.

This is why Analysis is way cooler than Algebra.

Edit: By the way, why isn't this trivial?

Let x be a positive element in the field. (Such an element must exist, because a field must have a zero element {0} and a multiplicative identity {e}.  Either e is positive or its additive inverse (-e) is positive.)  Let e be the multiplicative identity element.  Then ex = x.  If e were negative, then ex=x would be negative as well, which contradicts the positivity of x.

Or is it not clear that (negative)*(positive)=(negative)?  This should be clear since x being negative means -x is positive.  Suppose we have a negative x and a positive y.  We want to show -(xy) is positive.  Now -(xy) is the additive inverse of (xy), but so is (-x)y by the distributive law: (xy) + (-x)y = (x+-x)y = 0y = 0.  Thus, -(xy) = (-x)y.  As -x is positive and y is positive, (-x)y = -(xy) is positive, and so xy is negative.

That would be another valid proof, although you didn't explain why 0y=0.

If you consider all of that together to be trivial, then yes it is trivial.  I mean, it was problem 1 of assignment 1 of real analysis 1.  It isn't supposed to be a huge strain.

Oh I see, given the amount of discussion that followed I had thought at first that it was something major   And yeah I guess 0y=0 isn't part of the definition since it follows from 0=e-e and ey=y.

By the way, did you go for your PhD?  If so, what focus?

Most crucial in proving 0y=0 is the distributive law.  You see, 0y=0 is rather special because you are asserting a multiplicative property about 0, which is itself defined by an additive property.  Thus you absolutely must invoke distributivity since that is the only field axiom which imposes any relationship between addition and multiplication.

I wrote my dissertation on homological algebra.

Right, I was using the distributive law.  That's a good point that the distributive property connects the two operations.

I did my dissertation on nonlinear boundary value problems.  I wonder if there are a lot of math people on this forum.  I have a feeling Dominion would attract more of the combinatorics and algebra types.

I'm working my way towards that, I'm doing advanced algebra now and am doing Calculus next year.

[WalrusMcFishSr]

Since you guys apparently know way more about this than I...

Would it be correct to describe the poset of coins and potions as a lattice?

Is it possible to describe the order type of a poset using an extension of the ordinal numbers? Like maybe some sort of sweet ordinal matrix or something?

I apologize in advance if these questions are stupid.

[florrat]

Since you guys apparently know way more about this than I...

Would it be correct to describe the poset of coins and potions as a lattice?

Is it possible to describe the order type of a poset using an extension of the ordinal numbers? Like maybe some sort of sweet ordinal matrix or something?

I apologize in advance if these questions are stupid.

Warning: complicated math ahead (but if you need this warning, why are you still in this thread?)

Yes, the costs of cards is a lattice, because you can take suprema and infima of costs. For example the supremum of 3P and 6 is 6P and the infimum is 3. Taking suprema and infima of card costs is never used in Dominion (because apart from alchemy it would just be taking maxima/minima). Actually the order is the product order of N and N (or N and the 2-element lattice, if you prefer), and the product order of lattices is always a lattice.

Your other question is interesting, and my answer is that I have no idea. It depends on what you exactly mean. Of course you can take the class of all posets, and call two posets equivalent if they are isomorphic. Then you can take the equivalence classes of this relation (it is possible to do this on a proper class, but one should be careful). Then this class contains all order types of all posets, in particular of all ordinal numbers. You can now try to define some operations from the ordinals on this class, but I have no idea how interesting this is (I make this up on the spot, so I have no idea if this idea is silly).

You can define addition of two posets by taking the disjoint union and then define the lexicographic order on that (putting one poset above the other).
You can probably also define an order on this class of posets. I'm not sure what the "correct" definition of this would be, but maybe one of the following:
* A <= B iff there exists an injective order-preserving function from A to B.
* A <= B iff there exists an injective order-preserving function f from A to B satisfying: for all a in im(f) and b in B we have b<=a implies b in im(f).
The first one is simple, but the second one states that the image is a "initial segment", which is often useful for well-orders. Both relations will turn the class of posets into a (proper) poset (not a linearly ordered one, of course).

If you're interested in extensions of the ordinal numbers, make sure to check out the Surreal numbers. This is an extension of the ordinals, containing all real numbers, infinitesimals, and which is in fact an ordered field! The addition and multiplication on the surreal numbers are of course different than those on the ordinal numbers, because they have to be commutative. But the fact that such an extension exists really blowed my mind. Think of things like 1/omega, omega-1 or the square root of omega (where omega is the smallest infinite ordinal). They are all defined in the surreal numbers!

Hopefully somebody came this far through my post. Please ask questions (or use wikipedia, wikipedia is great for definitions in mathematics, imho) if something is not clear.

[SirPeebles]

Some of you may find Tarski's High School Problem interesting.

[Witherweaver]

Some of you may find Tarski's High School Problem interesting.

Sweet, I have a friend that teaches high school algebra/calc.  I'll get him to assign this for the midterm!

[florrat]

Some of you may find Tarski's High School Problem interesting.

Sweet, I have a friend that teaches high school algebra/calc.  I'll get him to assign this for the midterm!

Poor students...

[Warfreak2]

Absolutely. Alchemy with ℕ, Alchemy with ℤ, Alchemy with ℝ, Alchemy with ℂ. Unfortunately, Dominion doesn't seem to enable the construction of transfinite numbers, but it doesn't explicitly disallow them either.

Cellar says "Discard any number of cards". It doesn't have to be a natural number!

[Warfreak2]

We need a broken fan-card "When you discard this, put it in your hand", so that you can play Cellar discarding ω cards, and drawing your deck*.

*For this to work, you have to forget that Cellar's discarding is atomic.

[Polk5440]

I think the main reason your first problem in graduate math (or doing similarly simple problems in undergraduate real analysis, abstract algebra, etc.) is "prove 1 is positive" is NOT because the result itself is insightful or meaningful for the vast majority of even math PhDs, but because learning how to write proofs correctly is an incredibly important skill (especially the importance of not letting hidden assumptions leak into your reasoning). And you have to start simple.

[sudgy]

I realized, talking about having a wrapping number system and such, our time system is weird.

It's that wrap-around type that wraps at 12 or 24, whichever you feel like, then it has a decimal in base 60, which also has a decimal in base 60 (which can have decimals like normal numbers).

WHO MADE THIS SYSTEM?!?

[Warfreak2]

(which can have decimals like normal numbers).

Normal numbers have to have decimals!

[SirPeebles]

I realized, talking about having a wrapping number system and such, our time system is weird.

It's that wrap-around type that wraps at 12 or 24, whichever you feel like, then it has a decimal in base 60, which also has a decimal in base 60 (which can have decimals like normal numbers).

WHO MADE THIS SYSTEM?!?

The Babylonians had a base 60 number system.  You need to understand that well before decimal point notation was invented, everyone used fractions.  A number like 60 is easily divided up among several people, such as 1, 2, 3, 4, 5, or 6 people.  In fact, 60 is the least common multiple of these first six numbers.  That makes it a particularly natural choice.

Edit:  Oh right, and 12.  Well, 12 is similarly divisible.  It is smallest number which is divisible by each of those numbers except 5.

[Kirian]

I realized, talking about having a wrapping number system and such, our time system is weird.

It's that wrap-around type that wraps at 12 or 24, whichever you feel like, then it has a decimal in base 60, which also has a decimal in base 60 (which can have decimals like normal numbers).

WHO MADE THIS SYSTEM?!?

You mean you don't use metric time yet?  It's 03.80 here right now.

[Witherweaver]

I realized, talking about having a wrapping number system and such, our time system is weird.

It's that wrap-around type that wraps at 12 or 24, whichever you feel like, then it has a decimal in base 60, which also has a decimal in base 60 (which can have decimals like normal numbers).

WHO MADE THIS SYSTEM?!?

The decimal isn't exactly base 60; we don't use a true base-60 expansion.  We don't write 1 hour and 25 minutes as 1.<the 25th symbol we pick, say O>.  That would be really confusing.  Rather we have three main subdivisions, hours, minutes, seconds.  Each subdivision is computed modulo* (12 or 24 for hours, 60 for minutes and seconds), but is expressed in base 10.  Also you can say 1.5 hours and it will be taken as 1h30m not 1h5m.

But yes, the system is stupid.  I really hate how on a microwave timer, 1:00 is less than 0:99. 

*Actually, each subdivision is only computed modulo on a clock.  There's no reason you can't talk about, e.g., 64 minutes, and sometimes we do.

[XerxesPraelor]

I realized, talking about having a wrapping number system and such, our time system is weird.

It's that wrap-around type that wraps at 12 or 24, whichever you feel like, then it has a decimal in base 60, which also has a decimal in base 60 (which can have decimals like normal numbers).

WHO MADE THIS SYSTEM?!?

You mean you don't use metric time yet?  It's 03.80 here right now.

In all seriousness, I've been trying to get myself to think in terms of Metric time for quite a while (since June). The centid (~15 minutes) is a really useful time segment. Instead of having 43 minute class periods like I had a year ago, you can have 3 centid ones. It's really cool. BTW, the time would be 380, not 03.80. You don't need the extra 0, which is another benefit.

[Asper]

Could somebody move this thread? I like reading the rules threads, but whenever i see the letters turn dark and think there's a new rules question that i could help answering or learn something from, i have to find it's just some new, incomprehensible hyper-math mumbo jumbo that has nothing to do with dominion anymore. No offense.

Edit: Ah, i see it's about other kinds of metrics now...

[GendoIkari]

Could somebody move this thread? I like reading the rules threads, but whenever i see the letters turn dark and think there's a new rules question that i could help answering or learn something from, i have to find it's just some new, incomprehensible hyper-math mumbo jumbo that has nothing to do with dominion anymore. No offense.

Edit: Ah, i see it's about other kinds of metrics now...

Agreed. I like the math stuff too, but this thread doesn't belong in the rules forum.

[GendoIkari]

Hmm, all the recent "spring cleaning" but this is still here. Theory, can we get a split on this? Thanks!

[Kirian]

Hmm, all the recent "spring cleaning" but this is still here. Theory, can we get a split on this? Thanks!

I'm sure one of the mathematicians here can find the most efficient splitting algorithm.

[theory]

Done!

[ehunt]

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.

I dont understand,  the positive elements for the trivial C* algebra on C are just the positive reals, right?

[WanderingWinder]

FWIW, if it were phrased as 'prove that the multiplicative identity is greater than the additive identity for any ordered number field', I wouldn't have nearly as much of a problem with it. (I would have said this just boils down to how you define greater than, but then we at least get an interesting interaction between our intuitive understanding of greater and the formal, axiomatic one). Certainly, this would be a fine (if not too difficult) thing to prove. It's just that this isn't what I would mean when saying that I'm proving 1 is positive. For me, 1 is more fundamentally defined as the successor to 0 and predecessor to 2 in the counting operation, in which case it being greater than 0 is trivial. Of course, anything showing that the multiplicative identity must be the additive identity in a counting function (SHOWS it rather than just postulates) would also be interesting.

Cheers.

[blueblimp]

Edit:  Oh right, and 12.  Well, 12 is similarly divisible.  It is smallest number which is divisible by each of those numbers except 5.

Yeah, all-around 12 would be a better base for our number system than 10 is. 12 = lcm{2,3,4}, while 10 = lcm{2,5}. Both are divisible by two primes, and being divisible by a smaller prime (3) is more useful than a larger prime (5), all other things being equal. Plus 12 is divisible by 4 as a bonus.

The main downside of large bases is the increase in multiplication table size, but it's already common to memorize the multiplication table up to 12x12.

[qmech]

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.

I dont understand,  the positive elements for the trivial C* algebra on C are just the positive reals, right?

So x > y if x - y is a positive real: you can only compare things with the same imaginary part.

[werothegreat]

I realized, talking about having a wrapping number system and such, our time system is weird.

It's that wrap-around type that wraps at 12 or 24, whichever you feel like, then it has a decimal in base 60, which also has a decimal in base 60 (which can have decimals like normal numbers).

WHO MADE THIS SYSTEM?!?

The Sumerians!

And we should also switch to the Hex Clock anyway: http://www.intuitor.com/hex/hexclock.html

[ehunt]

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.

I dont understand,  the positive elements for the trivial C* algebra on C are just the positive reals, right?

So x > y if x - y is a positive real: you can only compare things with the same imaginary part.

I don't understand how the language of C* algebras is helping. Anyway, this suggests 3P isn't more than 2

[pingpongsam]

For an ever present, massive, life impacting numerology that has no solid basis in mathematics at all look no further than your calendar.

Have fun answering the question, "Why are there 7 days in a week?". It's a great history lesson and you need very little math at all because really, math has nothing to do with it!

It makes the clock's numerology look downright astute.

[Kirian]

For an ever present, massive, life impacting numerology that has no solid basis in mathematics at all look no further than your calendar.

Have fun answering the question, "Why are there 7 days in a week?". It's a great history lesson and you need very little math at all because really, math has nothing to do with it!

It makes the clock's numerology look downright astute.

Sure, but that also came from Babylon originally!

[sudgy]

For an ever present, massive, life impacting numerology that has no solid basis in mathematics at all look no further than your calendar.

Have fun answering the question, "Why are there 7 days in a week?". It's a great history lesson and you need very little math at all because really, math has nothing to do with it!

It makes the clock's numerology look downright astute.

Year's length makes sense and a month's length makes kind of sense (it started as moon cycles), so it isn't as weird as you would think.

[pingpongsam]

Right, 7 day weeks make perfect sense because there is this natural phenomenon it is based on called Easter.

[WanderingWinder]

Right, 7 day weeks make perfect sense because there is this natural phenomenon it is based on called Easter.

Not only is this a clear slam on (certain) religion(s), it's actually an absurd one - nobody (well, there are some strange people, so almost nobody) is going to say that the week is based on Easter. If you're trying to go for a "crazy religious" argument, you would say something like "because God made the universe/earth in seven days". However, I think you'll find most religious people won't really buy that, either.

I mean, there are a few reasons, basically practical. There is some way in which having a day off now and then can help, it is easy to do this on a basis with some periodicity, and seven seems roughly correct. And then, the bigger thing is that it's roughly one fourth of a moon cycle. Also, a month being moon-based is probably correct but a little bit fishy, because months have irregular lengths which are (except February) all longer than the actual lunar cycle.

[SirPeebles]

Right, 7 day weeks make perfect sense because there is this natural phenomenon it is based on called Easter.

Not only is this a clear slam on (certain) religion(s), it's actually an absurd one - nobody (well, there are some strange people, so almost nobody) is going to say that the week is based on Easter. If you're trying to go for a "crazy religious" argument, you would say something like "because God made the universe/earth in seven days". However, I think you'll find most religious people won't really buy that, either.

I mean, there are a few reasons, basically practical. There is some way in which having a day off now and then can help, it is easy to do this on a basis with some periodicity, and seven seems roughly correct. And then, the bigger thing is that it's roughly one fourth of a moon cycle. Also, a month being moon-based is probably correct but a little bit fishy, because months have irregular lengths which are (except February) all longer than the actual lunar cycle.

Month is clearly based on the lunar cycle.  Hell, even the word "month" is etymologically related to the word "moon".  The reason that the length of a month does not quite match the length of a lunar cycle is because our calendar is primarily based on solar cycles, and they don't evenly divide.  The Romulus calendar consisted on 10 months (hence our final months being named September, October, November, December) and the remaining 50 of the year were not assigned a month.  It was later reformed to add in two additional months to divide the solar year into 12 months.

One astronomical feature seen in the seven days of the week is that they correspond to the seven "planets" that were known in the ancient world, and you can still see this is you look at the names in both English and French:

Sunday - Sun
Monday/Lundi - Moon
Mardi - Mars
Mercredi - Mercury
Jeudi - Jupiter
Vendredi - Venus
Saturday - Saturn

[WanderingWinder]

Right, 7 day weeks make perfect sense because there is this natural phenomenon it is based on called Easter.

Not only is this a clear slam on (certain) religion(s), it's actually an absurd one - nobody (well, there are some strange people, so almost nobody) is going to say that the week is based on Easter. If you're trying to go for a "crazy religious" argument, you would say something like "because God made the universe/earth in seven days". However, I think you'll find most religious people won't really buy that, either.

I mean, there are a few reasons, basically practical. There is some way in which having a day off now and then can help, it is easy to do this on a basis with some periodicity, and seven seems roughly correct. And then, the bigger thing is that it's roughly one fourth of a moon cycle. Also, a month being moon-based is probably correct but a little bit fishy, because months have irregular lengths which are (except February) all longer than the actual lunar cycle.

Month is clearly based on the lunar cycle.  Hell, even the word "month" is etymologically related to the word "moon".  The reason that the length of a month does not quite match the length of a lunar cycle is because our calendar is primarily based on solar cycles, and they don't evenly divide.  The Romulus calendar consisted on 10 months (hence our final months being named September, October, November, December) and the remaining 50 of the year were not assigned a month.  It was later reformed to add in two additional months to divide the solar year into 12 months.

One astronomical feature seen in the seven days of the week is that they correspond to the seven "planets" that were known in the ancient world, and you can still see this is you look at the names in both English and French:

Sunday - Sun
Monday/Lundi - Moon
Mardi - Mars
Mercredi - Mercury
Jeudi - Jupiter
Vendredi - Venus
Saturday - Saturn

Well, I think you misunderstood my point - it wasn't that "ah, this month relationship isn't well-founded", it was more "it doesn't line up exactly, so don't expect it to". But I do appreciate the specific information.

[XerxesPraelor]

Right, 7 day weeks make perfect sense because there is this natural phenomenon it is based on called Easter.

Not only is this a clear slam on (certain) religion(s), it's actually an absurd one - nobody (well, there are some strange people, so almost nobody) is going to say that the week is based on Easter. If you're trying to go for a "crazy religious" argument, you would say something like "because God made the universe/earth in seven days". However, I think you'll find most religious people won't really buy that, either.

I mean, there are a few reasons, basically practical. There is some way in which having a day off now and then can help, it is easy to do this on a basis with some periodicity, and seven seems roughly correct. And then, the bigger thing is that it's roughly one fourth of a moon cycle. Also, a month being moon-based is probably correct but a little bit fishy, because months have irregular lengths which are (except February) all longer than the actual lunar cycle.

Well, causually, even if you don't think it's a good reason, the beginning of Genesis is one reason why we have seven-day weeks. The jewish tradition that christians took and carried into the middle ages.

[pingpongsam]

Right, 7 day weeks make perfect sense because there is this natural phenomenon it is based on called Easter.

Not only is this a clear slam on (certain) religion(s), it's actually an absurd one - nobody (well, there are some strange people, so almost nobody) is going to say that the week is based on Easter. If you're trying to go for a "crazy religious" argument, you would say something like "because God made the universe/earth in seven days". However, I think you'll find most religious people won't really buy that, either.

I mean, there are a few reasons, basically practical. There is some way in which having a day off now and then can help, it is easy to do this on a basis with some periodicity, and seven seems roughly correct. And then, the bigger thing is that it's roughly one fourth of a moon cycle. Also, a month being moon-based is probably correct but a little bit fishy, because months have irregular lengths which are (except February) all longer than the actual lunar cycle.

Whatever, dude. I have no issues with any religions and I'm not trying to slam anything.
When writing some code that attempted to parse dates I started to understand that our dates are meaningless mathematically. Parsing dates, specifically date ranges and gaps, is a recurring programming issue with no right answer and trade-offs for any given approach. In coming to understand why I discovered it was completely religious based. The fact is the Gregorian Calendar was specifically designed with Easter as the basis for calculation. It is not the sole reason for the 7 day week, that preceded Gregory, but the 7 day week is still a religious overtone and our modern calendar is designed expressly around timing the Easter date.

[Kirian]

Right, 7 day weeks make perfect sense because there is this natural phenomenon it is based on called Easter.

Not only is this a clear slam on (certain) religion(s), it's actually an absurd one - nobody (well, there are some strange people, so almost nobody) is going to say that the week is based on Easter. If you're trying to go for a "crazy religious" argument, you would say something like "because God made the universe/earth in seven days". However, I think you'll find most religious people won't really buy that, either.

I mean, there are a few reasons, basically practical. There is some way in which having a day off now and then can help, it is easy to do this on a basis with some periodicity, and seven seems roughly correct. And then, the bigger thing is that it's roughly one fourth of a moon cycle. Also, a month being moon-based is probably correct but a little bit fishy, because months have irregular lengths which are (except February) all longer than the actual lunar cycle.

Well, causually, even if you don't think it's a good reason, the beginning of Genesis is one reason why we have seven-day weeks. The jewish tradition that christians took and carried into the middle ages.

Except that, again, the Hebrews were not the originators of the seven day week.  The Babylonians divided the month into four parts, which meant each part had to be seven days, as the lunar cycle is about 29.6 days.  They added the extra days at the end of the month and defined the new month on either first crescent (new moon) or full moon, I don't remember which.

The Hebrews decided that keeping the worship cycle as 7 days every week was easier and kept that part, allowing the lunations to start in the middle of the week.

[SirPeebles]

Right, 7 day weeks make perfect sense because there is this natural phenomenon it is based on called Easter.

Not only is this a clear slam on (certain) religion(s), it's actually an absurd one - nobody (well, there are some strange people, so almost nobody) is going to say that the week is based on Easter. If you're trying to go for a "crazy religious" argument, you would say something like "because God made the universe/earth in seven days". However, I think you'll find most religious people won't really buy that, either.

I mean, there are a few reasons, basically practical. There is some way in which having a day off now and then can help, it is easy to do this on a basis with some periodicity, and seven seems roughly correct. And then, the bigger thing is that it's roughly one fourth of a moon cycle. Also, a month being moon-based is probably correct but a little bit fishy, because months have irregular lengths which are (except February) all longer than the actual lunar cycle.

Well, causually, even if you don't think it's a good reason, the beginning of Genesis is one reason why we have seven-day weeks. The jewish tradition that christians took and carried into the middle ages.

Genesis is probably a good reason for why the seven day week has persisted over time in Europe.  But I suspect that the seven day week in Genesis was based on Jewish tradition, and not vice-versa.  Maybe not though.

Also, I think it is probably worth considering that in the olden days less value would have been placed on a single, standard unit of measurement that would be used ubiquitously through society.  Some parts of society used seven day weeks, maybe other parts didn't bother subdividing months at all.  In the end, what we refer to now as having been that society's calendar is really just the calendar of the rulers, priests, or scholars.

I think it is interesting that both lunar and solar calendars have been used.  It makes sense I suppose.  The moon and sun are by far the two most obvious astronomical bodies.  This is speculation, but I would imagine that lunar calendars were more common and were generally developed earlier, if only because the lunar cycle is so much more obvious.  I mean, the visual appearance of the moon in the night sky morphs over the course of a month from being a bright disk to disappearing entirely, which all the grades in between.  The solar cycle is less obvious from the sun itself; probably the largest difference is the drifting position of sunrise/sunset each day.  But in the end, the solar cycle has much more practical significance.  It helps to predict the weather, which in turn influences such long term decisions as when to plant crops, when to begin construction, and when to mobilize armies.

[pingpongsam]

If it were your job to figure out, which would you rather look at on a daily basis to make your detailed positioning records, the moon or the sun?

[SirPeebles]

Right, 7 day weeks make perfect sense because there is this natural phenomenon it is based on called Easter.

Not only is this a clear slam on (certain) religion(s), it's actually an absurd one - nobody (well, there are some strange people, so almost nobody) is going to say that the week is based on Easter. If you're trying to go for a "crazy religious" argument, you would say something like "because God made the universe/earth in seven days". However, I think you'll find most religious people won't really buy that, either.

I mean, there are a few reasons, basically practical. There is some way in which having a day off now and then can help, it is easy to do this on a basis with some periodicity, and seven seems roughly correct. And then, the bigger thing is that it's roughly one fourth of a moon cycle. Also, a month being moon-based is probably correct but a little bit fishy, because months have irregular lengths which are (except February) all longer than the actual lunar cycle.

Whatever, dude. I have no issues with any religions and I'm not trying to slam anything.
When writing some code that attempted to parse dates I started to understand that our dates are meaningless mathematically. Parsing dates, specifically date ranges and gaps, is a recurring programming issue with no right answer and trade-offs for any given approach. In coming to understand why I discovered it was completely religious based. The fact is the Gregorian Calendar was specifically designed with Easter as the basis for calculation. It is not the sole reason for the 7 day week, that preceded Gregory, but the 7 day week is still a religious overtone and our modern calendar is designed expressly around timing the Easter date.

If the calendar was expressly designed to compute the date of Easter, then is was one of the greatest fails in history.  I'm pretty sure that there were lots of practical concerns that took priority over Easter in the construction of the Gregorian calendar.

[Kirian]

Right, 7 day weeks make perfect sense because there is this natural phenomenon it is based on called Easter.

Not only is this a clear slam on (certain) religion(s), it's actually an absurd one - nobody (well, there are some strange people, so almost nobody) is going to say that the week is based on Easter. If you're trying to go for a "crazy religious" argument, you would say something like "because God made the universe/earth in seven days". However, I think you'll find most religious people won't really buy that, either.

I mean, there are a few reasons, basically practical. There is some way in which having a day off now and then can help, it is easy to do this on a basis with some periodicity, and seven seems roughly correct. And then, the bigger thing is that it's roughly one fourth of a moon cycle. Also, a month being moon-based is probably correct but a little bit fishy, because months have irregular lengths which are (except February) all longer than the actual lunar cycle.

Whatever, dude. I have no issues with any religions and I'm not trying to slam anything.
When writing some code that attempted to parse dates I started to understand that our dates are meaningless mathematically. Parsing dates, specifically date ranges and gaps, is a recurring programming issue with no right answer and trade-offs for any given approach. In coming to understand why I discovered it was completely religious based. The fact is the Gregorian Calendar was specifically designed with Easter as the basis for calculation. It is not the sole reason for the 7 day week, that preceded Gregory, but the 7 day week is still a religious overtone and our modern calendar is designed expressly around timing the Easter date.

No, the modern calendar is quite obviously not based around Easter, or Easter would be on the same day each year.  In the Hebrew calendar, which is lunisolar, the religious holidays do fall on the same days each year.

The conversion to a civil solar calendar caused no end of difficulty to early Christians, who spent something like 200 years before they settled the calculated date of Easter, because Easter was taken from the Hebrew Pesach holiday, which of course moved around on a solar calendar.

Now, you're correct that the 7 and 30-ish day cycles have little meaning now, except as tradition.

[pingpongsam]

Why did Pope Gregory see a need to amend the Julian calendar and what was the primary basis for the Calendar Reform of 1575 as proposed by Luigio Lillio?

[WanderingWinder]

The Gregorian Calendar IS based on Easter maintaining its place in the year seasonally. Go Google it or Wikipedia it. Easter moves around because it's the 1st Sunday after the 1st full moon after the beginning of Spring (by the equinox), which is a complicated thing that goes back to Passover, which itself was probably tied with some kind of harvest/planting thing. The 7 day week, like the timing of Easter, predates both the Gregorian Calendar and Christianity by hundreds of years, at least.

pingpongsam, if you truly don't want to slam religion, then you should not call a religious observance a "natural phenomenon".

[pingpongsam]

Being tongue in cheek is not the same as being disparaging. The sarcasm was rooted in frustration at something so numerically important being so mathematically byzantine as to basically have no standard method of calculation at all. The context of natural phenomenon is that all of the other methods of calculating time use one. That the Gregorian calendar is designed to cause Easter to occur at or near the same time of the year is analogous to making the Easter holiday a type of "natural phenomenon" much like the solstice, equinox and moon phase.

[SirPeebles]

The Gregorian calendar updated the Julian calendar by further refining the leap year system.  A solar year is on average about 365.25 years, so a strict 365 day calendar would "be off" by about a day very four years, which accumulates over time such that winter's starting date will drift by nearly a month each century.  Even the Romans recognized that this was too much, so the Julian calendar added an extra day once every four years to improve stability.

That still isn't a perfect fix, since the average solar year is actually a little less than 365.36 days.  Over millennia, the error accumulates to a few weeks.  The fix was the remove 3 leap days per 400 year cycle, which is roughly a week per millennium.

In the long term this was a good fix to make, although it does appear that the reason it had become a pressing matter in Gregory's time is because Easter was gradually slipping out of the agreed upon time of year.  So as a one time measure, Pope Gregory skipped ahead by ten days to retroactively counteract the error which had accumulated and return Easter to the proper time frame.

So yes, Easter was a big part of the political impetus to correct the problem, but it was a problem which already existed, and which really ought to have been addressed eventually.  I suppose that skipping ten days wasn't strictly necessary apart from Easter.

[Qvist]

WalrusMcFishSr, what have you done!?

[WalrusMcFishSr]

I didn't know Qvist! I...DIDN'T...KNOW!!!

[DStu]

I didn't know Qvist! I...DIDN'T...KNOW!!!

... you should have ...