Expressing large ammounts of power to regular people

[Darth Wong]

Expecting people who hated science and math in high-school to comprehend scientific figures is a Quixotic enterprise. You're speaking to them in a language they don't understand, about a subject they don't care about.

[Utsanomiko]

I was talking to my mother about the amount of power an asteroid impact would cause, and about 5 minuets through the conversation, it hit me, most people cannot relate large ammounts of powwer to their ways of thinking. As I went back to my room, and started doing some math, it hit me.. Expressing power in trems of food. calories, or Kilo calories are common, so why not us a common food, such as big macs to express the ammount o power in something. it would be a nice way to scae down "asplodes a planet" to a regular units. There are 560 000 calories in a Big Mac. There are 4.184 joules in a calorie, so when you mulpy the numbers, you get 2 343 040 millon joules in a big mac. Placing that in sciencetific notation, thats 2.34304*10^6 or 2.34304E6 joules. Now it takes according to the death star page 3.7E32 joules to explode a planet at bare minum. the divison sugests that it takes 1.579145042E26 Big Macs to explode a planet. In normal numbers thats 15 791 450 420 000 000 000 000 000 or... 15 octillion big macs to explode a planet...

What the hell brought that conversation up? "Hey mom, can I have some Cheerios?"

"Sure."

"Oh by the way, do you know about the amount of power associated with asteroids that could hit the earth?"

She was probably thankful he at least stopped talking about trains for five minutes.

[brianeyci]

I just finished reading my chapter on Integration in Higher Dimensions for my Advanced Calculus class and I stand corrected. Visualizing something is critically important in advanced mathematics, especially for true understanding of the material rather than memorizing meaningless notation. Ready for some crazy shit?

Okay to integrate on let's say R^2, that is R x R, R being real numbers, in other words the x-y plane. That is, finding the area of a surface in the x-y plane. At first let's consider a rectangle in the x-y plane (we'll consider oddly shaped surfaces later.) How do we find the area of such a rectangle? Well at first define R such that

R = [a,b] x [c,d] = { (x,y) | x belongs to [a,b] and y belongs to [c,d] }

Does that notation make any sense to you? Not only does it not make any sense, it's got a lot of fucking letters in it, variables. Oh no, how the hell do I remember it. Well visualize it. It's a rectangle in the x-y plane. Now a is the smallest value of x and b is the largest value of x. And c is the smallest value of y and d is the largest value of y. This would be a lot more intuitive with a picture, but basically imagine a rectangle with the verticies (a,c), (b,c), (a,d), (b,d).

Now what we need to do is divide the rectangle into smaller chunks. So we define a partition of P to be

P = {x0, ... , xJ, y0, ... yK}

such that a = x0 < ... < xJ = b, c = y0 < ... < yK = d. Now how the hell do I remember the definition of partition again. It's simple. Imagine a grid on top of the arbitrary rectangle we created. This grid is basically splitting up the rectangle into smaller chunks like we wanted, and that's exactly what we're doing with this partition. Now what have we created? We've created the following,

Rjk = [x(j) - x(j-1)] x [y(j) - y(j-1)]

What the hell is this? This is basically the decomposition of the partition into a zillion tiny rectangles. In other words, visualize that rectangle in the x-y plane again, visualize the grid on top of that rectangle, and that is what the above notation says, an arbitrary cartesian product of x and y defining a rectangle.

Now, let's calculate the area of these zillions of rectangles. Area of a rectangle is its length multiplied by its width, so if we wanted the area of the first rectange we'd say A1 = (x1 - x0)(y1 - y0). So let's define the arbitrary area of a rectangle to be

deltaAij = (x(j) - x(j-1))(y(k) - y(k-1))

where j belongs to J and k belongs to K, J and K being real numbers. Now if you have problem understanding the above notation, just go back to my example of A1 and visualize. We're splitting up the rectangle into a zillion smaller rectangles, so we need some notation to arbitrarily define one of those smaller rectangles. That's exactly what we're doing here.

Now do you remember what the infinium and supremum are? Very quickly and loosely, the infinimum is just the largest lower bound for a set S and the supremum is the smallest upper bound for a set S. In short, the infinimum is the smallest number that belongs to a closed set and the supremum is the largest number that belongs to a closed set. I could define this much more rigorously with episilons and deltas but this will suffice for our purposes. How the hell do I remember what infinimum and supremum are? Simple, just imagine a real number line, and a subset of the real number line. The infinimum is the smallest x belonging to that set, and the supremum is the largest x belonging to that set, again assuming a closed set (we're considering closed rectangles here.) More visualization.

Now define the following.

mij = inf{f(x,y) | (x,y) belongs to Rjk} and Mij = sup{f(x,y) | (x,y) belongs to Rjk}

What the hell is this? Do I just remember the notation for the test? Of course not, again, visualize what the notation is trying to say. Given a partition I defined above, take the smallest value belonging to a rectangle and the largest value belonging to a rectangle which I defined above as well, and define that to be the infinimum and supremum. In other words, imagine a rectangle. We're taking the bottom left corner of the rectangle and the upper right corner of the rectangle. In other words, the smallest value of the rectangle and the largest value of the rectangle. Remember our goal here: we're working with rectangles, but eventually we want to work with any arbitrary surface f on the x-y plane.

Now, define the lower and upper Riemann sums. Basically we're taking the sum of the area of all the smaller rectangles in the larger rectangle we're trying to find the area of.

spf = sum(j = 0 to J) sum (k = 1 to K) (mij)(x(j) - x(j-1))(y(k) - y(k-1))

is the lower Riemann sum. Are you still following me? We're taking the sum of all the rectangles in the larger rectangle, but the smallest value of that sum. This took me awhile to wrap my head around, because how the hell can I have the smallest value of an area, an area is an area right. But imagine a rectangle, and imagine splitting up a rectangle into smaller rectangles. Each one of those rectangles can be split into smaller rectangles. In that way, it makes sense to define the "smallest area" and the "largest area." Similarly, define the upper Riemann sum to be

Spf = sum(j = 0 to J) sum (k = 1 to K) (Mij)(x(j) - x(j-1))(y(k) - y(k-1))

Now, at last, we can define the integral. Obviously, there is a lower and upper integral corresponding to the lower and upper sums we used above. Now again, to understand the definition I have to visualize. The integral is basically the area of a surface in the x-y plane, which for now we're considering only rectangles. Now I have the lower Riemann sum and the upper Riemann sum. Now the lower integral, it makes sense to define the lower integral as the greatest lowest value of all lower Riemann sums. Similarly, for the upper integral, it makes sense to define the upper integral as the smallest largest value of all upper Riemann sums. Remember we considered an arbitrary partition of R, so therefore there's an infinite number of ways we can partition the rectangle. So define the lower integral to be:

I_(f) = supp{spf}

and the upper integral to be:

_I(f) = infp{Spf}

These are the lower and upper integrals of the function f on R.

I_ and _I represent the idea I explained above, that is taking the largest of the lower Riemann sums and the smallest of the upper Riemann sums. Now, if you've been following me so far you'll understand what I say next. The function f is said to be Riemann integrable (or simply integrable) if I_ and _I coincide. In other words, I_ equals _I. Again, this is hard to wrap your head around unless you use a geometric interpretation. Suppose you have a rectangle in R^2. That rectangle is integrable. But what if you want to integrate a surface in R x R x R or R^3, or R x R x R x ... x R n times, an n-dimensional shape. The upper and lower integrals may not be equal, simply because there are large non-trivial (that is they have content or they are not zero content) holes or gaps in the shape. Does that mean we can't integrate them? As a whole, yes, but in those cases we simply split that larger shape into a lot of smaller rectangles, take the smaller rectangles and integrate them individually. Then I_ = _I for each of these smaller rectangles, and later we can add up for the area of the larger volume.

Anyway, I am done. I won't go any further, but basically to integrate non-rectangular shapes, you enclose the function f in a rectangle, define f to be 1 inside the rectangle and 0 outside the rectangle. You now have a far more rigorous definition of integration than in single variable Calculus, or even most second year Calculus courses... this is bordering on third year material, or advanced second year material (engineers eat your heart out.) The only way I remember or even understand any of this at all is to be able to visualize everything, "see" the rectangles, see myself splitting up the rectangles into smaller chunks, intuit the sum of infinitely many infinitesimal rectangles. So for understanding of higher mathematics, you need a geometric or intuitive understanding of what you're doing. Do you think I sit there for hours and hours memorizing notation? I recalled all of this from just the one or two hours I spent imagining shit in my head, and the notation was just a side-effect of really understanding what I was doing.

So in short, if you didn't read any of the above or barely understand it, a geometric interpretation or visualization of very large numbers may be helpful... in fact, required for understanding of advanced mathematics. Otherwise you'll be left memorizing notation for tests and not really understanding what it means at all. By the way all of the above... is not in the textbook. The definition of multi-dimensional integration is condensed in a single page. The intuitive understanding I had to work out on my own, after reading that notation over and over and over, perhaps a hundred times, and letting my brain wrap itself around it for hours, days, arguably years since we've been trained since high school to understand the building blocks of Calculus.

None of this means that memorizing SI prefixes means you're intelligent. But it does mean that people who can see things easier in their heads are a little better off (I would say a lot better off) in mathematics than someone who can't. Without a geometric foundation, there is no fucking way in hell I could keep all the j's and k's and x's and y's straight.

[Surlethe]

So in short, if you didn't read any of the above or barely understand it, a geometric interpretation or visualization of very large numbers may be helpful... in fact, required for understanding of advanced mathematics.

Concrete geometric interpretation or visualization is definitely helpful, but certainly not necessary: it's simply not possible to concretely visualize something like a group, or a line integral over a complex function, like you can see partitions of a rectangle in the plane, which is simply a really, really, really nice example. It's difficult enough to visualize what one of the trig functions -- say sin(z) -- does to the plane (and this is in undergraduate complex analysis) let alone trying to find some geometric interpretation or visualization for something as abstract as some weird topology over a weird set*.

*As an example, let R be a ring with 1. Define Spec(R) = {P: P is a prime ideal in R}. Now, given an ideal I in R, define V(I) = {P: P contains I}. Show that the V(I)s form a topology on Spec(R). Can you describe with a concrete visualization of this topology, for any R?

[brianeyci]

Concrete geometric interpretation or visualization is definitely helpful, but certainly not necessary: it's simply not possible to concretely visualize something like a line integral over a complex function, or a general group. It's difficult enough to visualize what one of the trig functions -- say sin(z) -- does to the plane (and this is in undergraduate complex analysis) let alone trying to find some geometric interpretation or visualization for something as abstract as some weird topology over a weird set.

So you're one of the purists huh. I disagree. It's not possible to visualize n-dimensional objects either, but I reduce it down to two dimensions. I can barely visualize surfaces, but I reduce it down to a box. Geometric interpretation is absolutely essential, and I hate the current trend of textbook authors who rip out all the pictures from math books. I would like to know how many definitions you can remember, because without some intuition to fall back on it becomes memorization. As for the visualization of large numbers, if someone cannot wrap his head around the sum of an infinity of infinitesimal terms, he will never truly understand Calculus.

[General Zod]

Concrete geometric interpretation or visualization is definitely helpful, but certainly not necessary: it's simply not possible to concretely visualize something like a line integral over a complex function, or a general group. It's difficult enough to visualize what one of the trig functions -- say sin(z) -- does to the plane (and this is in undergraduate complex analysis) let alone trying to find some geometric interpretation or visualization for something as abstract as some weird topology over a weird set.

So you're one of the purists huh. I disagree. It's not possible to visualize n-dimensional objects either, but I reduce it down to two dimensions. I can barely visualize surfaces, but I reduce it down to a box. Geometric interpretation is absolutely essential, and I hate the current trend of textbook authors who rip out all the pictures from math books. I would like to know how many definitions you can remember, because without some intuition to fall back on it becomes memorization. As for the visualization of large numbers, if someone cannot wrap his head around the sum of an infinity of infinitesimal terms, he will never truly understand Calculus.

That's probably my biggest problem with mathematics and why I was never able to get much farther than Algebra. None of my teachers seemed to explain how the numbers actually fucking related to real world work. Geometry is relatively simple because I can visualize what it's supposed to be used for, like figuring out angles, volumes and trajectories. But give me something like a complex calculus equation and I'll be stumped because i don't have the slightest fracking clue what the symbols do, or what the formula is used for in general.

[Darth Servo]

That's probably my biggest problem with mathematics and why I was never able to get much farther than Algebra. None of my teachers seemed to explain how the numbers actually fucking related to real world work. Geometry is relatively simple because I can visualize what it's supposed to be used for, like figuring out angles, volumes and trajectories. But give me something like a complex calculus equation and I'll be stumped because i don't have the slightest fracking clue what the symbols do, or what the formula is used for in general.

Didn't your math teachers ever give you word problems?

You're solving quadratic equations, factoring polynomials all the time in freshman chemistry, to solve equliibrium equations.

[General Zod]

That's probably my biggest problem with mathematics and why I was never able to get much farther than Algebra. None of my teachers seemed to explain how the numbers actually fucking related to real world work. Geometry is relatively simple because I can visualize what it's supposed to be used for, like figuring out angles, volumes and trajectories. But give me something like a complex calculus equation and I'll be stumped because i don't have the slightest fracking clue what the symbols do, or what the formula is used for in general.

Didn't your math teachers ever give you word problems?

You're solving quadratic equations, factoring polynomials all the time in freshman chemistry, to solve equliibrium equations.

I don't really remember that much about math classes in high school except that the teachers basically gave us page numbers, lectured at us for a bit and expected to do them right off the bat. I'm fairly sure part of it was due to me mostly zoning out in Math. But most of my HS classes are a blur nowadays and I can't recall any of the explanations ever really being satisfactory to someone that's a visual thinker.

[brianeyci]

I don't really remember that much about math classes in high school except that the teachers basically gave us page numbers, lectured at us for a bit and expected to do them right off the bat. I'm fairly sure part of it was due to me mostly zoning out in Math. But most of my HS classes are a blur nowadays and I can't recall any of the explanations ever really being satisfactory to someone that's a visual thinker.

Sure they could've given you word problems, but I bet anything that teachers who aren't really math majors who don't really have an intuitive understanding of Calculus contribute greatly to many people's difficulties. Without the lecture, the word problem could just be the 10% of the test most students ignore as a "hard problem" and meanwhile the teacher doesn't really understand the material himself and stinks at it so he just assigns drill problems and formulas to memorize. Derivative is just the instantaneous rate of change. Doesn't make sense? How about I explain it as the acceleration of a car? How about I draw a velocity time graph, where time is on the x axis and velocity is on the y axis. The derivative with respect to t, dv/dt, is exactly how fast the car is accelerating at a specific time, say t = 1 seconds. I would like to see anybody explain from the ground up derivative without a picture to someone at high school level and see if they understand it. I would especially like to see if anybody can understand a rigorous definition of differentiability with the episilon and deltas without a picture easily. Math is beautiful, it doesn't have to be hard.

That being said, a lot of people are just lazy so blaming teachers only goes so far (not saying you specifically.)

[Wyrm]

So you're one of the purists huh. I disagree. It's not possible to visualize n-dimensional objects either, but I reduce it down to two dimensions. I can barely visualize surfaces, but I reduce it down to a box. Geometric interpretation is absolutely essential, and I hate the current trend of textbook authors who rip out all the pictures from math books. I would like to know how many definitions you can remember, because without some intuition to fall back on it becomes memorization.

Since when is intuition == visualization? And since when is geometric interpretation essential? I didn't come to understand the proposition that any infinite subset of compact metric spaces must have a cluster point through visualizing it — it was a pure deductive exercise, and if anything the geometrical interpretation only delayed my understanding. There are quite a few mathematical concepts that I came to terms with without visualization... some because I couldn't visualize through lack of imagination, others when the visualization was actually confusing me.

And, no, it's not because I'm a "purist," thank you. It's because I have trained to be able to think abstractly when needed. Visualization is a tool to understanding, but it's not "essential" to understanding by any means.

I think you're confusing "sufficiency" with "necessity". You understood multivariable integration by visualizing it geometrically by way of chopped-up rectangles, and that's a perfectly fine way of understanding it. But it's not the only way, and it's best not to let it be you're only way to understanding.

As for the visualization of large numbers, if someone cannot wrap his head around the sum of an infinity of infinitesimal terms, he will never truly understand Calculus.

Since when does the understanding of large numbers have anything to do with calculus? Your calculus might show that the Death Star must pump in ~1e38 J of energy to destroy Alderaan, but it doesn't tell you that that's the energy content of a Big Mac about the size of the Earth.

[Darth Wong]

Everyone seems to be missing the point, which is that the fundamental difference between the way laypeople and scientists/engineers approach math is simple: laypeople would never allow their perceptions of reality to be influenced by numbers and calculations. Scientists and engineers are required to do so.

That is why laypeople appear to be incapable of grasping huge numbers. It's not that they literally can't grasp the idea of a 1 with a shitload of zeroes behind it. It's that they won't allow their notions of reality to be altered by calculations.

Scientist/engineer's hierarchy of knowledge:
1) Direct empirical observations
2) Calculations based on direct empirical observations and well-tested scientific theories
3) Intuition

Ordinary person's hierarchy of knowledge:
1) Direct empirical observations
2) Intuition
3) Pop culture science

Stupid person's hierarchy of knowledhge:
1) Intuition
2) Direct empirical observations

Religious person's hierarchy of knowledge:
1) Religious intuition
2) Direct empirical observations
3) Religious scriptures, retconned to make #1 and #2 appear to coexist
4) Pop culture science
5) Calculations, but only if they confirm #3

[His Divine Shadow]

I'd like to see anyone here comprehend 100,000 first, of anything macroscale from a car to a person. That number is itself a massive figure to put into practical terms, even if it's just a number of steps you take a week or kilometres you drive every few years.

I kinda do mental excersizes now and then, trying to imagine stuff like this on a large, large scale. I find it has given me a different perspective on life actually.

[Darth Wong]

I'd like to see anyone here comprehend 100,000 first, of anything macroscale from a car to a person. That number is itself a massive figure to put into practical terms, even if it's just a number of steps you take a week or kilometres you drive every few years.

I kinda do mental excersizes now and then, trying to imagine stuff like this on a large, large scale. I find it has given me a different perspective on life actually.

100,000 is pretty easy to comprehend. A large football stadium holds that many people.

[Kuroneko]

[Note: Apologies for the tangent.]

Brianeyci, what you've desribed is the Darboux integral rather than the Riemann integral Int_a^b[ f dx ], although since f is Riemann-integrable iff f is Darboux-integrable, a lot of authors erase the distinction. The Riemann-Stieltjes integral Int_a^b[ f dφ ] replaces [x_k - x_{k-1}] by [φ(x_k) - φ(x_{k-1})] in one dimension (we then need something like dφdψ in two), and the much more powerful Lebesgue integral (with Lebesgue measure) is analogous to partitioning the range rather than the domain of the integrand.

Again, this is hard to wrap your head around unless you use a geometric interpretation. Suppose you have a rectangle in R^2. That rectangle is integrable.

Sets are measurable; functions are integrable. The outer measure of a set in R^n is defined in pretty much the same manner--cover the set with countable many closed intervals, take the sum of the volumes of the intervals, and then take the infimum over all such interval coverings. The Lebesgue measure is a little bit stricter; it's equal to the outer measure if a certain condition is met. In R^n, a closed interval is the Cartesian product of n closed intervals in R.

By the way all of the above... is not in the textbook. The definition of multi-dimensional integration is condensed in a single page.

It's particularly nice in terms of measure: the integral of a non-negative real-valued function f:E→R, E a measurable subset of R^n, is the (n+1)-dimensional Lebesgue measure of the set {(x,f(x)): x in E}. Intuitively, for each x in the domain E, make a line segment of height f(x) into the (n+1)-st dimension, and then take measure of the resulting set. This gives integrals of all dimensions using one general notion of Lebesgue measure. Arbitrary real-valued functions can be split into positive and negative parts, f+(x) = max{0,f(x)}, f-(x) = max{0,-f(x)}.

So in short, if you didn't read any of the above or barely understand it, a geometric interpretation or visualization of very large numbers may be helpful... in fact, required for understanding of advanced mathematics.

Well, that depends on the area of mathematics. Intuitive understanding is essential everywhere, but intuition, or even visualization, is not the same thing as geometrization. An trivial example is, say, [q→r]→[pvq → pvr], which can be taken as an axiom (it is in some systems). You're correct in that it does no good to memorize symbols, but the intuitive understand is that when one has an implication, one can make a disjunction out of both the antecedent and consequent with the same sentence. It then becomes much more agreeable, but there is no actual geometry involved, and the only visualization is rather superficial.

---

It's difficult enough to visualize what one of the trig functions -- say sin(z) -- does to the plane (and this is in undergraduate complex analysis) let alone trying to find some geometric interpretation or visualization for something as abstract as some weird topology over a weird set.

(That seems familiar... did you already ask this?) The transformation w = sin z can be thought of as either mapping the upper-halfplane into the upper-halfstrip (width π) or "smudging" the origin into a line segment, with circles centered at origin turned into ellipses with foci at the endpoints of that segment; more physically, it transforms a 2-D electric field of a charge into the electric field of a uniform charged line segment. It's very much related to the elliptic coordinates (suppressing z); there, if z = x + iy and w = u+iv, then z = a cosh(w) = a sin(iw+π/2). But your general point stands.

As an example, let R be a ring with 1. Define Spec(R) = {P: P is a prime ideal in R}. Now, given an ideal I in R, define V(I) = {P: P contains I}. Show that the V(I)s form a topology on Spec(R). Can you describe with a concrete visualization of this topology, for any R?

Heh. Did you mean to take the complements of V(I)'s? This construction can also be broken for non-commutative rings. But overall, it's not too pathological, as it still has points (even though they might not be topologically distinguishable). A real mind-bender of an example is taking the correspondence between commutative C*-algebras and locally compact Hausdorff spaces seriously, and then asking what corresponds to non-commutative C*-algebras. These spaces may have no points at all (this happens if the spectrum of the C*-algebra is empty), and yet still have meaningful structure. And that's not even completely pure mathematics, since having such a "non-commutative space" is a way of sidestepping the need for renormalization in quantum field theories. (This was in fact Heisenberg's original proposal.)

[Surlethe]

(That seems familiar... did you already ask this?)

Perhaps I used it as an example in a previous question regarding complex transformations.

The transformation w = sin z can be thought of as either mapping the upper-halfplane into the upper-halfstrip (width π) or "smudging" the origin into a line segment, with circles centered at origin turned into ellipses with foci at the endpoints of that segment; more physically, it transforms a 2-D electric field of a charge into the electric field of a uniform charged line segment. It's very much related to the elliptic coordinates (suppressing z); there, if z = x + iy and w = u+iv, then z = a cosh(w) = a sin(iw+π/2). But your general point stands.

This does make sense. Thank you.

As an example, let R be a ring with 1. Define Spec(R) = {P: P is a prime ideal in R}. Now, given an ideal I in R, define V(I) = {P: P contains I}. Show that the V(I)s form a topology on Spec(R). Can you describe with a concrete visualization of this topology, for any R?

Heh. Did you mean to take the complements of V(I)'s?

The example is actually from a homework problem where we were given the definition of topology in terms of closed sets. I understand it's the same as open sets, except for open sets, unions are infinite and intersections are finite.

This construction can also be broken for non-commutative rings. But overall, it's not too pathological, as it still has points (even though they might not be topologically distinguishable).

Well, yes. Of course it won't be too bad, if it's only being asked in an undergraduate algebra class. The point, naturally, is that it's very difficult to visualize the topology of V(I)s on Spec(R). And it gets worse when one starts considering how unity-preserving homomorphisms from R -> S induces a function from Spec(S) to Spec(R). Showing continuity, that surjectivity -> injectivity, and that injectivity -> surjectivity were real bitches.

A real mind-bender of an example is taking the correspondence between commutative C*-algebras and locally compact Hausdorff spaces seriously, and then asking what corresponds to non-commutative C*-algebras. These spaces may have no points at all (this happens if the spectrum of the C*-algebra is empty), and yet still have meaningful structure. And that's not even completely pure mathematics, since having such a "non-commutative space" is a way of sidestepping the need for renormalization in quantum field theories. (This was in fact Heisenberg's original proposal.)

Hell, it's difficult enough to wrap your head around a Hausdorff space at first glance; what does it even mean to have no neighborhoods separating points? (Actually, now that I think about it, are the integers a Hausdorff space? Is that question even meaningful?)

I'd think that visualization or intuition simply means there's something going on in the math that corresponds roughly to something that could have occurred on the African plains as we evolved: we're as unprepared for quantum physics as we are for hardcore topology. Conversely, classical mechanics is probably the nicest sort of math you can come up with in terms of visualization, since it pretty much sums up our evolved rules of thumb.

[Surlethe]

So you're one of the purists huh. I disagree. It's not possible to visualize n-dimensional objects either, but I reduce it down to two dimensions. I can barely visualize surfaces, but I reduce it down to a box. Geometric interpretation is absolutely essential, and I hate the current trend of textbook authors who rip out all the pictures from math books.

"Purists"? What the fuck are you talking about? I'm not talking about ripping out pictures from math books, as you well know; I'm contesting your assertion that geometric interpretation of concepts is "absolutely essential" to understanding math. So, for example, what's your geometric interpretation of the topological problem I gave above? Integrating over a rectangle is, like I said, a really, really, really nice example; you ought to know better than to assume circumstances surrounding a nice, concrete example holds generally.

I would like to know how many definitions you can remember, because without some intuition to fall back on it becomes memorization. As for the visualization of large numbers, if someone cannot wrap his head around the sum of an infinity of infinitesimal terms, he will never truly understand Calculus.

I hate to break it to you, buddy, but nobody can really visualize an infinite sum of infinitesimal terms. All we can do is make representative pictures, which are helpful. Topologically, for example, the best visualizations are sets in RxR, but things like the Jordan Curve theorem hold there that don't hold in more abstract terms. At some point, like it or not, requiring geometric interpretations will hold back your understanding.

[Kuroneko]

This does make sense.

Just remember that the two interpretations are not compatible--think of the first operationally as mapping horizontal lines into flow lines of an ideal fluid flowing down on one side of the half-strip, turning around once a certain distance from the bottom is reached, and going upward to infinity on the other side. The other interpretation is what happens if you treat (u,v) as "polar", in which case it's a coordinate transformation.

The example is actually from a homework problem where we were given the definition of topology in terms of closed sets. I understand it's the same as open sets, except for open sets, unions are infinite and intersections are finite.

Ah, yes. A topology is usually defined as open sets, and a set is open iff its complement is closed.

And it gets worse when one starts considering how unity-preserving homomorphisms from R -> S induces a function from Spec(S) to Spec(R).

A unit-preserving homomorphism φ:R→S always has the property that if J is an ideal (or prime ideal) in S, the preimage φ'(J) is an ideal (or prime ideal) in R. One can work with that without too much difficulty, since a function is continuous iff the preimages of closed sets are closed.

Hell, it's difficult enough to wrap your head around a Hausdorff space at first glance; what does it even mean to have no neighborhoods separating points?

Non-Hausdroff, you mean? A space is Hausdorff iff for every pair of distinct points (x,y), there exist closed sets F_1 containing x but not y and F_2 containing y but not x such that their union is the whole space. In particular, if X is Hausdorff, then X is also Fréchet, meaning all singletons are closed.

(Actually, now that I think about it, are the integers a Hausdorff space? Is that question even meaningful?)

That depends on the topology, of course. The indiscrete topology {{},X} would not be Hausdorff except when X is a singleton. If the integers Z inherit the topology from the reals (i.e., Z considered a subspace of the reals), then this space is Hausdorff, as the topology is simply the power set P(Z). Generally, if the topology is induced by some metric, as it is in this case, then the space is Hausdorff; the converse does not hold.

If you intended to ask about Spec Z with the Zariski topology, then it's a fairly interesting question. The prime ideals of Z are Spec Z = {pZ: p = 0 or p prime}. The closure of the singleton {0Z} is the intersection of all closed sets that contain the ideal 0Z, which in this case is just V(0Z) = Spec Z, as for any other ideal I, {0Z} is not a subset of V(I). Hence {0Z} is not closed, Spec Z is not Fréchet and hence not Hausdorff. The Zariski topology on X = Spec Z is actually the cofinite topology, meaning all closed sets other than X are finite (the name actually comes from the open sets being the complements of finite sets, or empty).

Q: Do you see the correspondence between the closed sets and radical ideals?

[drachefly]

I hate to break it to you, buddy, but nobody can really visualize an infinite sum of infinitesimal terms. All we can do is make representative pictures, which are helpful.

What is visualization, but making a representative picture, which is helpful?