Author Topic: Math with the passion of a THOUSAND SUNS  (Read 11092 times)

Golden Applesauce

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Math with the passion of a THOUSAND SUNS
« on: May 25, 2013, 10:22:06 am »

analysis
calculus
topology
graph theory
game theory
abstract algebra (math over fields of functions or other weird objects that aren't normally thought of as numbers.)
vector spaces
predicate logic
lambda calculus
regular expressions
computability
set theory <-- actually probably the best starting point for learning math, even easier than algebra
number theory

I know a little about fractals and chaos theory, and next to nothing about knot theory, but I might be able to help with articles intended for a lay audience about them.

Stats are useful and therefore not real math, and outside the scope of this thread.

I swear to god do not even talk about quantum physics. Quantum physics math is bullshit. I don't mean the goofy physical interpretations that imply magic not-particles or whatever. I mean physicists make up axioms as they go along. "Okay, we have a raising and lowering operator. But we must have a bottom rung of our energy ladder or else we'd get particles with negative energy, and that would be silly. So a step down from the bottom step must have a non-normalizeable square integral, which means it [lower of]a0 is the constant zero function. Working backwards we can derive all the permissible energy states... yes GA? Well, mathematically yes, there are a lot of other functions that aren't normalizably square integrable but those don't happen in physics. Now, we return to our square well, where we have PE(x) = 0 for x in [0, 1] and PE(x) = Infinity for x not in [0, 1]...."

And then the very next chapter they are suddenly completely okay with a particle having negative spin, but negative energy is so preposterous that they can't even think about it. But they have no problem with "infinite" energy, and will happily tell you that the integral of f(x) = Infinity if x = 0, 0 otherwise, is 1. Like, the number one. They just integrated a point discontinuity at infinity and got 1. That's not even on the real number line anymore. If your target spaces is "The reals + a number larger than all of the reals" then every nice algebraic property you're used to explodes. a + b = a + c doesn't imply that b = c any more, for starters. Infinitely wide sin waves are a well-defined square integrable function that doesn't break math at all, and are sometimes allowed in quantum (e.g., as part of an orthogonal base) and sometimes not okay. ( sin(x) is a perfectly good replacement for constant 0 in the bottom rung argument, except that then you don't derive the right things so unnnnnnnnnnngh we declare that the world doesn't work that way.)

Two thirds of the way through the semester, I finally figured out that all of these "wavefunctions" in "Hilbert space" (the thing physicists call "Hilbert space" is one particular Hilbert space with a whole bunch of goofy extra rules, but they don't care that they're forking nomenclature) are not functions at all, but Cauchy series of equivalence classes of functions under some kind of strange distance metric that I think was degenerate for most pairs of equivalence classes. Which is a fine, if unusual, space for a mathematician to work in -- you don't have functions that map from from the reals to the reals anymore so it's a bit Twilight Zone-ish. But that's apparently "too abstract" for physicists, so they turn around and pretend that the limit of a Cauchy series of equivalence classes of functions is itself a function. Maddening and in defiance of all sense, I tell you.
« Last Edit: May 25, 2013, 10:24:51 am by Golden Applesauce »
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GrannySmith

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Re: Math with the passion of a THOUSAND SUNS
« Reply #1 on: May 25, 2013, 01:19:05 pm »

Stats are useful and therefore not real math, and outside the scope of this thread.
And a pure maths thread, double

I swear to god do not even talk about quantum physics. Quantum physics math is bullshit. I don't mean the goofy physical interpretations that imply magic not-particles or whatever. I mean physicists make up axioms as they go along. "Okay, we have a raising and lowering operator. But we must have a bottom rung of our energy ladder or else we'd get particles with negative energy, and that would be silly. So a step down from the bottom step must have a non-normalizeable square integral, which means it [lower of]a0 is the constant zero function. Working backwards we can derive all the permissible energy states... yes GA? Well, mathematically yes, there are a lot of other functions that aren't normalizably square integrable but those don't happen in physics. Now, we return to our square well, where we have PE(x) = 0 for x in [0, 1] and PE(x) = Infinity for x not in [0, 1]...."

And then the very next chapter they are suddenly completely okay with a particle having negative spin, but negative energy is so preposterous that they can't even think about it. But they have no problem with "infinite" energy, and will happily tell you that the integral of f(x) = Infinity if x = 0, 0 otherwise, is 1. Like, the number one. They just integrated a point discontinuity at infinity and got 1. That's not even on the real number line anymore. If your target spaces is "The reals + a number larger than all of the reals" then every nice algebraic property you're used to explodes. a + b = a + c doesn't imply that b = c any more, for starters. Infinitely wide sin waves are a well-defined square integrable function that doesn't break math at all, and are sometimes allowed in quantum (e.g., as part of an orthogonal base) and sometimes not okay. ( sin(x) is a perfectly good replacement for constant 0 in the bottom rung argument, except that then you don't derive the right things so unnnnnnnnnnngh we declare that the world doesn't work that way.)

Two thirds of the way through the semester, I finally figured out that all of these "wavefunctions" in "Hilbert space" (the thing physicists call "Hilbert space" is one particular Hilbert space with a whole bunch of goofy extra rules, but they don't care that they're forking nomenclature) are not functions at all, but Cauchy series of equivalence classes of functions under some kind of strange distance metric that I think was degenerate for most pairs of equivalence classes. Which is a fine, if unusual, space for a mathematician to work in -- you don't have functions that map from from the reals to the reals anymore so it's a bit Twilight Zone-ish. But that's apparently "too abstract" for physicists, so they turn around and pretend that the limit of a Cauchy series of equivalence classes of functions is itself a function. Maddening and in defiance of all sense, I tell you.

Woah now i want to read MORE of qm maths!! I changed from set theory to stats recently (btw strangely quite a common change!) and yes - it does seem like absolute nonesense what applied mathematicians do sometimes, but you gotta respect that they can at least approximately describe something "real" with it (as opposed to playing with models of mathematics and extremely hugely infinite numbers )

Anyway - i love the thread already! And to start, do you know of any theorems about partitions of the natural numbers in finitely many colours? Or a book i should look into? My books are all useless on finitely many colours... (and i might want to use something like this for stats - or shouldn't i have said that? )
« Last Edit: May 25, 2013, 01:34:10 pm by GrannySmith »
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Golden Applesauce

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Re: Math with the passion of a THOUSAND SUNS
« Reply #2 on: May 25, 2013, 04:55:47 pm »
To start, do you know of any theorems about partitions of the natural numbers in finitely many colours? Or a book i should look into? My books are all useless on finitely many colours... (and i might want to use something like this for stats - or shouldn't i have said that? )

I found Schur's Theorem ( Wikipedia | Proofwiki ) which says that given any finite r colors, there is an initial set of the naturals [1, 2, 3, ... n] such that every r-coloring of it has a triplet x, y, and z such that x+y=z and x, y, z are all the same color.

Things involving the http://en.wikipedia.org/wiki/Ramsey_Number are always related to finite colorings and can often be related to the naturals.

If you're talking about a finite coloring off all of the naturals... I think most interesting results would be about the min/max number of colors to ensure that you have a coloring with a specific property, with generally falls under Ramsey Theory.
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Nephew Twiddleton

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Re: Math with the passion of a THOUSAND SUNS
« Reply #3 on: May 25, 2013, 05:59:59 pm »
I'm doomed.
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Golden Applesauce

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Re: Math with the passion of a THOUSAND SUNS
« Reply #4 on: May 25, 2013, 06:38:20 pm »
I'm doomed.

Don't worry! There is fun math that's a lot less complicated. Complicated math is actually just a bunch of thin layers of simple math, except humans can't think that many layers deep at a time. The trick is that if you practice each layer until it becomes mental muscle memory - like the way you know that 3 + 7 = 10 without having to count - you can easily learn the next layer at a "competent" level. But if you only know a layer at the "competent" level instead of the "automatic" level, you can only learn the next layer at the "basic" level and the layer after that will be basically impossible. If you've ever read math and thought that it made sense to you at the time, but then couldn't explain it two days later, you've had the experience of trying to read at three levels above your automatic zone.

It's especially depressing if you're trying to learn math to do one specific thing, like quantum or stats or engineering, because you think you can build a thin tall tower that reaches to exactly the point you want to get to and save some time / effort, but that doesn't work. Towers of math knowledge collapse if they're more than two levels above the foundation. If you want to build your math up to where you can do quantum or stats or whatever, you have to lay thin layer after thin layer of foundation until you're almost there and then stick one layer of building on top. People think they suck at math because they can't build skyscrapers, but the truth is that the guys doing high level math cant build mental skyscrapers either - they're sitting on a step ladder that just happens to be perched on a foundation that's 40 stories tall.

The natural tendency is to practice one layer until it becomes easy and then move on to the next, but that's wrong. You can work at the easy level and even work ahead a level or two if you have enough timeand whiteboards, but you can't advance until the "easy" part becomes mind numbiingly boring.

I'll try to post some gentle foundational predicate logic and set theory later, since it's a three day weekend and if I leave my head stuck in sixth grade that long I shall go insane.
« Last Edit: May 25, 2013, 06:42:57 pm by Golden Applesauce »
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Mesozoic Mister Nigel

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Re: Math with the passion of a THOUSAND SUNS
« Reply #5 on: May 25, 2013, 07:18:05 pm »
I find that repetition is absolutely vital for learning math, in the sense that if I preview the chapter, then attend lecture, then do the practice exercises, then do the homework twice, I will retain the new information.

I should be doing homework right now, actually.
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rong

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Re: Math with the passion of a THOUSAND SUNS
« Reply #6 on: May 26, 2013, 06:55:50 am »
I really loved math until I figured out what I really love is logic.  I still love math, though.

I believe there is no actual proof that 1+1=2, but rather, 1+1=2 is actually a definition.  Discuss?
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GrannySmith

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Re: Math with the passion of a THOUSAND SUNS
« Reply #7 on: May 26, 2013, 12:43:30 pm »
I really loved math until I figured out what I really love is logic.  I still love math, though.

I believe there is no actual proof that 1+1=2, but rather, 1+1=2 is actually a definition.  Discuss?
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!!

Golden Applesauce, i find what you said to El Twid, really one of the best descriptions for how it is/should be to learn maths!
I'll try to post some gentle foundational predicate logic and set theory later,
Looking forward to that!
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GrannySmith

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Re: Math with the passion of a THOUSAND SUNS
« Reply #8 on: May 26, 2013, 12:55:53 pm »
I found Schur's Theorem ( Wikipedia | Proofwiki ) which says that given any finite r colors, there is an initial set of the naturals [1, 2, 3, ... n] such that every r-coloring of it has a triplet x, y, and z such that x+y=z and x, y, z are all the same color.

Things involving the http://en.wikipedia.org/wiki/Ramsey_Number are always related to finite colorings and can often be related to the naturals.

If you're talking about a finite coloring off all of the naturals... I think most interesting results would be about the min/max number of colors to ensure that you have a coloring with a specific property, with generally falls under Ramsey Theory.

Right, I should have described it/thought about it better before i asked, i need theorems that talk not about the homogeneous (one colour) subsets, but the complete opposite, the subsets that contain elements of pairwise different colours. And "colourings of sets of natural numbers" is enough i guess. Hm - maybe this doesn't fit in this thread anyway, if I understand it right you're intending this as a mini course on foundations of mathematics?
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Golden Applesauce

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Re: Math with the passion of a THOUSAND SUNS
« Reply #9 on: May 26, 2013, 08:55:48 pm »
I figured that either people would ask questions that I didn't know, in which case I would have to Do Research and Learn Stuff, or if not then I'd get to try my hand at figuring out the best way to present / teach math.

Teaching math is a bit of a fetish for me and a lot of other math people. The idea of a "pons asinorum" (bridge of fools) has been around since the Academy. The original Pons Asinorum was a basic geometry proof that a lot of students struggled with. It became a barrier to entry, a "You must be this good at math to learn geometry" marker. If you could cross the Bridge of Fools, you were intelligentsia material; if not, you were a Fool and should probably go back to farming pigs. As we've opened up more fields of mathematics we've found more humps that portions of the population apparently just can't get over. A doctor once told me that there are two kinds of smart people: those who can do calculus and those who can't. Those who can make good scientists and engineers, and those who can't go into medicine or law or some other prestige field that requires lots of intelligence but no advanced math. My grandfather is an example of that - he wanted to be a civil engineer, but after failing calculus three times he gave up and got a PhD in medicine instead. The Greeks didn't have a problem with the idea that some people are just fated to suck at math, but that's deeply offensive to a modern egalitarian. The only other explanation is that we - over two thousand years of mathematicians - suck at teaching math.
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Don Coyote

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Re: Math with the passion of a THOUSAND SUNS
« Reply #10 on: May 26, 2013, 09:06:32 pm »
I figured that either people would ask questions that I didn't know, in which case I would have to Do Research and Learn Stuff, or if not then I'd get to try my hand at figuring out the best way to present / teach math.

Teaching math is a bit of a fetish for me and a lot of other math people. The idea of a "pons asinorum" (bridge of fools) has been around since the Academy. The original Pons Asinorum was a basic geometry proof that a lot of students struggled with. It became a barrier to entry, a "You must be this good at math to learn geometry" marker. If you could cross the Bridge of Fools, you were intelligentsia material; if not, you were a Fool and should probably go back to farming pigs. As we've opened up more fields of mathematics we've found more humps that portions of the population apparently just can't get over. A doctor once told me that there are two kinds of smart people: those who can do calculus and those who can't. Those who can make good scientists and engineers, and those who can't go into medicine or law or some other prestige field that requires lots of intelligence but no advanced math. My grandfather is an example of that - he wanted to be a civil engineer, but after failing calculus three times he gave up and got a PhD in medicine instead. The Greeks didn't have a problem with the idea that some people are just fated to suck at math, but that's deeply offensive to a modern egalitarian. The only other explanation is that we - over two thousand years of mathematicians - suck at teaching math.

I suspect it's the bolded.
For some reason I think that some of the concepts in calculus could be taught taught alongside geometry in middle school or high school.
Because I am all like, "hey this is cool that we just used 4 boards to use calculus to create the formula to calculate the volume of a cylinder but it would have blown my mind when I was a kid."
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Golden Applesauce

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Re: Math with the passion of a THOUSAND SUNS
« Reply #11 on: May 26, 2013, 09:18:26 pm »
I found Schur's Theorem ( Wikipedia | Proofwiki ) which says that given any finite r colors, there is an initial set of the naturals [1, 2, 3, ... n] such that every r-coloring of it has a triplet x, y, and z such that x+y=z and x, y, z are all the same color.

Things involving the http://en.wikipedia.org/wiki/Ramsey_Number are always related to finite colorings and can often be related to the naturals.

If you're talking about a finite coloring off all of the naturals... I think most interesting results would be about the min/max number of colors to ensure that you have a coloring with a specific property, with generally falls under Ramsey Theory.

Right, I should have described it/thought about it better before i asked, i need theorems that talk not about the homogeneous (one colour) subsets, but the complete opposite, the subsets that contain elements of pairwise different colours. And "colourings of sets of natural numbers" is enough i guess.

I guess I don't understand enough about what you're doing to see what's interesting about it. You're coloring some numbers and then interested in rainbow subsets - those that don't repeat any colors. But a singleton counts as a one-colored rainbow, and those are super boring. Do you require that a rainbow subset on a n-coloring exhibit all n colors? Then the set of rainbow subsets given a given coloring ends up being the Cartesian product of all the colored partitions, so each coloring implies a specific set of n-dimensional vectors. Then you could ask about the structure of the rainbow vectors. You'll never get a nice vector space because you don't have enough zeroes to go around, but maybe there's something interesting there?
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Golden Applesauce

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Re: Math with the passion of a THOUSAND SUNS
« Reply #12 on: May 26, 2013, 09:29:57 pm »
I figured that either people would ask questions that I didn't know, in which case I would have to Do Research and Learn Stuff, or if not then I'd get to try my hand at figuring out the best way to present / teach math.

Teaching math is a bit of a fetish for me and a lot of other math people. The idea of a "pons asinorum" (bridge of fools) has been around since the Academy. The original Pons Asinorum was a basic geometry proof that a lot of students struggled with. It became a barrier to entry, a "You must be this good at math to learn geometry" marker. If you could cross the Bridge of Fools, you were intelligentsia material; if not, you were a Fool and should probably go back to farming pigs. As we've opened up more fields of mathematics we've found more humps that portions of the population apparently just can't get over. A doctor once told me that there are two kinds of smart people: those who can do calculus and those who can't. Those who can make good scientists and engineers, and those who can't go into medicine or law or some other prestige field that requires lots of intelligence but no advanced math. My grandfather is an example of that - he wanted to be a civil engineer, but after failing calculus three times he gave up and got a PhD in medicine instead. The Greeks didn't have a problem with the idea that some people are just fated to suck at math, but that's deeply offensive to a modern egalitarian. The only other explanation is that we - over two thousand years of mathematicians - suck at teaching math.

I suspect it's the bolded.
For some reason I think that some of the concepts in calculus could be taught taught alongside geometry in middle school or high school.
Because I am all like, "hey this is cool that we just used 4 boards to use calculus to create the formula to calculate the volume of a cylinder but it would have blown my mind when I was a kid."

Definitely. I think part of the problem is that we teach math in order, which is stupid. You don't need algebra or even arithmetic to learn set theory and you don't need to derivatives to learn second-order functions, but we for some reason we refuse to teach math except in arithmetic > algebra > geometry > trigonometry > calculus > formal logic > everything else order.
Q: How regularly do you hire 8th graders?
A: We have hired a number of FORMER 8th graders.

rong

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Re: Math with the passion of a THOUSAND SUNS
« Reply #13 on: May 28, 2013, 04:18:24 pm »
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
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Doktor Howl

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Re: Math with the passion of a THOUSAND SUNS
« Reply #14 on: May 28, 2013, 04:26:47 pm »
I figured that either people would ask questions that I didn't know, in which case I would have to Do Research and Learn Stuff, or if not then I'd get to try my hand at figuring out the best way to present / teach math.

Teaching math is a bit of a fetish for me and a lot of other math people. The idea of a "pons asinorum" (bridge of fools) has been around since the Academy. The original Pons Asinorum was a basic geometry proof that a lot of students struggled with. It became a barrier to entry, a "You must be this good at math to learn geometry" marker. If you could cross the Bridge of Fools, you were intelligentsia material; if not, you were a Fool and should probably go back to farming pigs. As we've opened up more fields of mathematics we've found more humps that portions of the population apparently just can't get over. A doctor once told me that there are two kinds of smart people: those who can do calculus and those who can't. Those who can make good scientists and engineers, and those who can't go into medicine or law or some other prestige field that requires lots of intelligence but no advanced math. My grandfather is an example of that - he wanted to be a civil engineer, but after failing calculus three times he gave up and got a PhD in medicine instead. The Greeks didn't have a problem with the idea that some people are just fated to suck at math, but that's deeply offensive to a modern egalitarian. The only other explanation is that we - over two thousand years of mathematicians - suck at teaching math.

I suspect it's the bolded.
For some reason I think that some of the concepts in calculus could be taught taught alongside geometry in middle school or high school.
Because I am all like, "hey this is cool that we just used 4 boards to use calculus to create the formula to calculate the volume of a cylinder but it would have blown my mind when I was a kid."

Definitely. I think part of the problem is that we teach math in order, which is stupid. You don't need algebra or even arithmetic to learn set theory and you don't need to derivatives to learn second-order functions, but we for some reason we refuse to teach math except in arithmetic > algebra > geometry > trigonometry > calculus > formal logic > everything else order.

I gotta say, that method worked for me.
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