Math with the passion of a THOUSAND SUNS

[Doktor Howl]

I'm looking forward to taking more math classes, when I have time. I'll probably never get all deep into it though. I enjoy it enough that I considered a math minor at one point but then I remembered what happens to people in health sciences who have math minors, and realized it was a terrible idea.

They accidentally the elder gods?

They accidentally the fry cook.

No, worse. They end up financial or other quantitative analysts. Once you crack that door to Hell, there's no getting back into the juicier end of research.

The pay is great, which is why it's a trap. Once you get in, you can't get out.

WHY
MUST
YOU
NIGEL
SO
MUCH?

[Mesozoic Mister Nigel]

I'm looking forward to taking more math classes, when I have time. I'll probably never get all deep into it though. I enjoy it enough that I considered a math minor at one point but then I remembered what happens to people in health sciences who have math minors, and realized it was a terrible idea.

They accidentally the elder gods?

They accidentally the fry cook.

No, worse. They end up financial or other quantitative analysts. Once you crack that door to Hell, there's no getting back into the juicier end of research.

The pay is great, which is why it's a trap. Once you get in, you can't get out.

WHY
MUST
YOU
NIGEL
SO
MUCH?

I GOTTA BE ME

[Doktor Howl]

 

[Don Coyote]

Well I'm sticking with my math minor, unless I start bleeding from my face holes.

[Doktor Howl]

Well I'm sticking with my math minor, unless I start bleeding from my face holes.

You will.

Oh, you will.

[Mesozoic Mister Nigel]

Well I'm sticking with my math minor, unless I start bleeding from my face holes.

What's your major?

Make no mistake, it's a great thing to have if your end goal is employment. You will be employed.

Oh yes, you will be employed.

[Don Coyote]

Well I'm sticking with my math minor, unless I start bleeding from my face holes.

What's your major?

Make no mistake, it's a great thing to have if your end goal is employment. You will be employed.

Oh yes, you will be employed.

ARTS MEDIA AND CULTURE: COMPARATIVE ARTS TRACK!!!!!
BECAUSE I HAVE TO BE THE SPECIAL SNOWFLAKE IN ALL MY CLASSES!!!!!

[rong]

Well, that depends on the axioms you assume! In some fields of maths the definition of 2 is 1+1, for (Peano) Arithmetic it's a theorem of two axioms, they are:
(1)  ∀x∀y(x+S(y))=S(x+y)
(2)  ∀x(x+0)=x
Where 0 is our only constant^*, + is a binary function^* (intended for addition), S( ) is a unary function^* (intended to signify the successor of something), x and y are variables^*, and for a variable x, ∀x means* 'for every x'.

So 1 is defined as S(0), that is, the successor of 0, and 2 is defined as S(S(0)), that is, the successor of 1.
And we want to prove that S(0)+S(0)=S(S(0)) from axioms (1) and (2):
proof^*:
By substituting^* x=S(0) and y=0 to (1) we get:
(3)   S(0)+S(0)=S(S(0)+0)
By substituting^* x=S(0) to (2) we get:
(4)   S(0)+0=S(0)
Because S( ) is a function^*, from (4) we get:
(5) S(S(0)+0)=S(S(0))
And by deduction^* from (3) and (5) we get:
S(0)+S(0)=S(S(0))

Thanks for reminding me of that

^* of course we should have started from predicate logic, languages and theories, defined what a variable and a quantifier '(for all)' is, defined what a formula is, defined the rules that we make deductions with, defined what a proof is, defined substitution to formulas, and defined what a function is, so including all that the proof would be much longer!!

I believe you have proved S(0)+S(0)=S(S(0)), but it is not a proof that 1+1=2 until 1 and 2 are defined as S(0) and S(S(0)), respectively.  This is a bit of a different perspective for me, though.  Thanks

I meant, this all depends on what axioms you assume and how you define 1, 2, +, and = actually, not only these, but one should really start from the very beginning, and define what is the underlying logic that will be used, and how the formulas are built, for example, in this case the basic (elementary) formulas are the ones that include an equality sign, and left and right from it are numbers, which are defined here as: "0 is a number", "for every number n, s(n) is a number", and "if n and m are numbers then "n+m" is also a number. Then one can define ways of combining them (using "and", "or", "not") and a way of making formulas with variables and quantifiers ("for every x", and then a formula which has the variable x in the place of a number).

Of course this is just one logic defined in order to work with arithmetic (actually in order to formally define arithmetic too), probably because it's the most fitting to sombunall's understanding and way of arguing about arithmetic intuitively. If one thinks about arithmetic differenty, or about any different subject, they could define a completely different logic to do it!

Mathematics is all about definitions, and all mathematical statements are "if ... then..." statements!

it just dawned on me one day that i didn't know how to prove 1+1=2 and i soothed my meta-mathematical crisis by presuming there exists a number "dictionary" defining all the numbers as

0=0
1=1
2=1+1
3=1+1+1
etc.

your use of S() is basically the same thing.  it reminds me of Goedel, Escher, Bach - although I can't quite remember if it's the same thing, or just similar.

the reason i majored in math in the first place was a combination of factors.  of all key academic areas, it was the one i was worst at (so i figured i needed to "bone up" on it).  they were the only classes in college that i really seemed to enjoy.  and i also saw a poster somewhere that said,"psychology is applied biology, biology is applied chemistry, chemistry is applied physics, and physics is applied mathematics"  i figured i'd stick with math and keep my doors open until i decided what to do.

i was also always intrigued by the list in the following way: well, then - mathematics is applied _______?

i eventually took a class in symbolic logic and kind of had my eureka moment and decided that mathematics is applied logic. 

i switched majors when i started to realize that all my crazy math prof's probably didn't start out that way.

[Doktor Howl]

i switched majors when i started to realize that all my crazy math prof's probably didn't start out that way.

Physics can do that to you, too.  All 3rd year physics students are nihilists, whether they want to be or not.

[Mesozoic Mister Nigel]

Well I'm sticking with my math minor, unless I start bleeding from my face holes.

What's your major?

Make no mistake, it's a great thing to have if your end goal is employment. You will be employed.

Oh yes, you will be employed.

ARTS MEDIA AND CULTURE: COMPARATIVE ARTS TRACK!!!!!
BECAUSE I HAVE TO BE THE SPECIAL SNOWFLAKE IN ALL MY CLASSES!!!!!

Iiiiiiiii really don't know what you're going to do with that. A math minor in any science, health, or policy field is a guaranteed job, I have no idea how that fits into arts media and culture. I'm guessing it means you'll count the tills at the end of the night.

Sounds like fun though.

[Don Coyote]

Well I'm sticking with my math minor, unless I start bleeding from my face holes.

What's your major?

Make no mistake, it's a great thing to have if your end goal is employment. You will be employed.

Oh yes, you will be employed.

ARTS MEDIA AND CULTURE: COMPARATIVE ARTS TRACK!!!!!
BECAUSE I HAVE TO BE THE SPECIAL SNOWFLAKE IN ALL MY CLASSES!!!!!

Iiiiiiiii really don't know what you're going to do with that. A math minor in any science, health, or policy field is a guaranteed job, I have no idea how that fits into arts media and culture. I'm guessing it means you'll count the tills at the end of the night.

Sounds like fun though.

I decided to study what I want without having to A) go into debt to cover the parts my GI bill won't cover by going to one of the private universities nearby or B) move up to fucking Seattle and go to UW Seattle.

[Golden Applesauce]

by popular request, let's talk about

SET THEORY

I'm going to assume you are already familiar with basic logic (if "Dogs are blue" and "3 is prime" are propositions, then "Dogs are blue AND 3 is prime" is a proposition, "Dogs are blue OR 3 is prime" is a proposition, that kind of stuff. If you can tell which of those four propositions are true and which are false, you're good.) If you know how logic works and this post doesn't make sense to you, ask questions 'cuz that means I didn't explain something properly.

Earlier in this thread, when the subject of a proof for 1 + 1 equaling 2 came up, Granny immediately turned to the set theory's definition of 1, 2, and + for a proof. There's a reason for that - modern math is defined in terms of set theory. That in itself is remarkable, since set theory is a relatively young field of math, less than ~150 years old. Humans have been adding 1 and 1 together and getting 2 way further back than that - so why did mathematicians decide to rebuild everything in terms of set theory after it had been working fine for thousands of years?

Short answer: because somebody broke mathematics. Like the dwarves who mined too greedily and too deep, certain mathematicians had started from unshakable principles and layered on infallibly true logical derivations and ended up with the most frightening result possible:
EVERYTHING ALL MATHEMATICAL STATEMENTS ARE TRUE IN SOME SENSE, FALSE IN SOME SENSE, AND MEANINGLESS IN SOME SENSE.

They had conclusively and irrefutably proven that mathematics was full of shit, no better than common philosophical wankery. You've probably heard of Gödel's Incompleteness Theorem. That's a whole series of posts by itself, but the basic gist of it is that all mathematical systems are either a) So simple they're boring b) Can't prove every true statement about the system or c) Can prove all statements about the system, including statements that aren't actually true. The equivalent Dwarven Incompleteness Theorem is "No dwarf can both dig up all of the gold and live to enjoy it. If you dig deep enough to find all of the gold, you will also find the Balrog who will kill you. If you don't dig deep enough to find the Balrog, you haven't dug deep enough to get all of the gold."

The terrifying paradox that had been discovered in planted mathematics of the time firmly in the "false things can be proven true" ("the Balrog will eat you") camp, which was unacceptable. After much drama in the mathematical community, it was decided to rebuild math from the ground up with a set of foundations that pointed in the "can't prove all true statements" direction. The general idea was to assume as little as possible, so you wouldn't accidentally end up assuming two contradictory things. It wasn't as sexy as the old mathematics, but at least it wouldn't be true, false, and meaningless all at the same time.

So, onto sets. Usually people explain sets as being like physical baskets of stuff. We can have a basket with a red egg, a blue egg, and a yellow egg inside. The corresponding set would be written in math notation as:
{ red egg , blue egg , yellow egg }

We can even place sets inside of other sets. If we call the { red egg, blue egg, yellow egg } set "EGGS", then we can have a grocery cart set that contains EGGS, spinach, mushrooms, and bacon, like this:
{ EGGS, spinach, mushrooms, bacon }
We could also choose to write it as:
{ { red egg, blue egg, yellow egg}, spinach, mushrooms and bacon }
and it would be the same thing, just spelled differently.

But there are a few ways sets and grocery baskets are different. In a grocery basket, order matters. A grocery basket with a bunch of heavy metal cans on top of the eggs is much worse than one with with the fragile things on top of the sturdy things. For a set, it doesn't matter. { sledgehammer, EGGS } is the same thing as { EGGS, sledgehammer }. The mathematical motivation for defining sets that way is that order is an important mathematical property, and they didn't want to assume order works if they could prove it instead.

The other way grocery baskets are different is that grocery baskets care about how many of something you have. A basket with 1,183 cans of soup is very, very different from a basket that only has one can in it. One of them costs a lot more at checkout, for starters. Sets don't care about number of things. { can, can, can } is exactly the same set as { can } as { can, can, can, can, can, can }. The set { can, can, can } contains exactly one item, can. This is because mathematicians didn't even want to assume that numbers worked. Saying that { can x 1,183 } and { can x 1 } are different requires being able to say that the number 1,183 exists, the number 1 exists, and that they are not the same number. Instead of embedding those assumptions into the foundation of mathematics and making the whole field circular logic, sets are much simpler: a given thing is either in the set, or it is not in the set. There is no "in the set three times." can is in { can, can, can }. can is in { can }. There is no thing that is in { can, can, can } and not { can } or vice versa, therefore, { can, can, can } is just a more awkward way of writing { can }.

In general, we define the identity of sets by their elements (the things that are in the set). If sets A and B have exactly the same  elements - everything in the A also in B, and everything in B is also in A - then A and B are one and the same set, which for some reason has two different names. You can't pull any philosophical fast ones like saying that A was introduced in the 3rd word of the previous sentence and B was introduced in the 5th word, which somehow differentiates them. Qualities like that are not part of sets; the only thing that matters for them is their elements. I can say with complete accuracy that B was introduced in the 3rd word of the 2nd sentence of this paragraph, it just happened to be introduced by a different name.

While we're on the subject of counting, I want to point out that a set is one thing. { EGGS, spinach, mushrooms, bacon } is a set with four things in it:

• EGGS (aka { 'blue egg', 'red egg', 'yellow egg'} )
• spinach
• mushrooms
• bacon

The fact that EGGS itself contains three items (red egg, blue egg, yellow egg) is irrelevant. You already understand this in real life grocery baskets; if you have a carton of a dozen eggs and a gallon of milk, you can still go in the Twelve Items or Less line. The cashier doesn't say "You have 12 eggs and 128 ounces of milk, that's 140 things." It's worth mentioning explicitly for sets because the next point can be a little confusing, but if you understand this point you can demonstrate to yourself that the next point is correct.

Here's a question for you: is 'blue egg' in the set { EGGS, spinach, mushrooms, bacon } ? This is where your intuition might mislead you. For a real life container, we would say yes. A person who is buying EGGS (a pre-packaged set of three colored eggs), spinach, mushrooms, and bacon is buying a blue egg, it's just wrapped up next to some other eggs. But remember that the set { EGGS, spinach, mushrooms, bacon } has exactly four things in it: EGGS, spinach, mushrooms, and bacon. To find out if a given thing is in our set, we simply compare it to each of those four things.

• Is 'blue egg' the same thing as 'bacon' ? No, 'blue egg' is from the bird family and 'bacon' is from the mammal family. Birds and mammals don't overlap.
• Is 'blue egg' the same thing as 'mushrooms' ? No. 'blue egg' is a dead animal embryo, and food 'mushrooms' are dead adult fungi.
• Is 'blue egg' the same thing as 'spinach' ? No. 'blue egg' is blue, and 'spinach' is green.
• Is 'blue egg' the same thing as 'EGGS' ? No. 'blue egg' is an dyed organic calcium shell around a yolk and some embryonic fluid. 'EGGS' is a set that contains, among other things, 'red egg'. 'blue egg' does not contain 'red egg', which is a difference in elements between EGGS and 'blue egg'. By our earlier definition of set identity, 'blue egg' is a different thing from EGGS.

We have compared 'blue egg' to each of the four things in { EGGS, spinach, mushrooms, bacon } and it isn't any of them. 'blue egg' is not in { EGGS, spinach, mushrooms, bacon }.

Here's another question: is EGGS in EGGS? We check the same way as for { EGGS, spinach, mushrooms, bacon }.
Recall that EGGS is { blue egg, red egg, yellow egg }.

• Is EGGS the same thing as 'blue egg' ? No, as was just shown in the answer to the previous question. (Strictly speaking we only showed that 'blue egg' is not the same thing as EGGS, not the other way around. If this bothers you, you are already thinking like a mathematician!)
• Is EGGS the same thing as 'red egg' ? No. EGGS contains 'blue egg', and 'red egg' does not contain 'blue egg'. That is a difference in elements.
• Is EGGS the same thing as 'yellow egg' ? No. EGGS contains 'red egg', and 'yellow egg' does not contain 'red egg'. That is a difference in elements.

EGGS has three elements, none of which are the same thing as EGGS. Therefore, the set EGGS is not an element of itself.

That's not to say that a set can't contain itself. So far, we haven't seen anything that would make this absurd. We could define the set TYPEWRITER_MONKEYS as:
{ TYPEWRITER_MONKEYS, typewriter }.
Then TYPEWRITER_MONKEYS is a set with two things in it - a typewriter, and TYPEWRITER_MONKEYS. If TYPEWRITER_MONKEYS were a real grocery basket, some problems would occur - do we have infinite typewriters? Are there any actual monkeys? Can we say that the first monkey is next to the last typewriter? - but a set isn't a real physical basket. There's no more inherent absurdity to the idea of a set that contains itself than there is to a Möebius strip having only one side. Unintuitive, yes. Paradoxical Balrog of mathematical absurdity, no.

Next post: the paradoxical Balrog of mathematical absurdity that comes from sets that don't contain themselves.

[Freeky]

Ooo! 

[rong]

i never studied much set theory - i was unaware (or forgot?) the notion that {can,can,can} is the same as {can} - I can see the importance to make this distinction, but it seems to me there would be situations where you would want to consider these two sets as different.  perhaps a different branch of set theory or something?

i remember something fun happens when you ask the question: Does the set of all sets contain itself? but I can't remember what, exactly, that is. 

[Golden Applesauce]

i never studied much set theory - i was unaware (or forgot?) the notion that {can,can,can} is the same as {can} - I can see the importance to make this distinction, but it seems to me there would be situations where you would want to consider these two sets as different.  perhaps a different branch of set theory or something?

It turns out that we can build structures that are aware of both repeated elements and order of elements using only regular sets. If you care about repeated elements but not order, you can make a a set that looks kind of like { { can, x3 } }. The actual formulation is slightly more complicated than that to resolve ambiguities like "does { { x2, x7 } } contain two copies of the x7 multiplier or seven copies of the x2 multiplier?" but you get the general idea. We can similarly include order and repeats at the same time with a set kind of like { { 1st, Washingtion }, { 4th, Jefferson }, { 3rd, Adams }, { 2nd, Washington } }. If you just want an ordered set, usually people talk about having both a set and an ordering together. So we might say that the set { 3, 1, 4, 1, 5, 9 } under the usual ordering of numbers ( 0, +1, +2, +3, ...) is the "ordered collection" 1, 3, 4, 5, 9.

We call collections of things with both order and the possibility of repeats lists. The list [3, 2, 1] is different from the list [1, 2, 3] is different from the list [1, 2, 2, 2, 2, 2, 3, 3, 3]. Lists actually play a very important part in the definition of the Real Numbers (as opposed to the natural numbers, integers, or rationals numbers.)

Collections that care about number of things, but not order, are usually called bags, although this is less standard. I have sometimes heard them referred to as urns. They mostly come up in probability questions, like "If you take a ball out of a bag that has 4 blue balls, 5 red balls, and 1 green ball, then..."