Well, it can be proven given the standard definition of the real numbers. I'm sure the people who argue that they are not equal have carefully pondered the construction of the real numbers, its advantages and disadvantages, and reject the construction in favor of some alternative number system that they construct rigorously from a self-consistent set of axioms, and are not just arguing for inequality because they don't know what they're talking about. . .

John D Barrow in his book "The Book of Nothing" has a chart called the "Structure of Modern mathematics". It is wider than it is tall, but across the top he has:
1) Lie Groups
2) Lie Algebras
3) Differential Operators
4) Manifolds with Tensor Fields
5) Metric Manifolds

I sometimes have trouble with very abstract areas of math. I like to have a mental picture of what I'm doing, and that gets hard when you do things at different/transformed levels. (Nyquist plots to solve PID control problems, for example.)
Something I'm good at is writing numerical solvers for various problems. I see an integral and think t=0; Int = 0; while(t<tend) {Int+=f(bla)*dt; t+=dt;}. Derivatives, gradients, laplacians, are similarly easy. I have a very literal idea of what is happening numerically. Searching solution spaces for maximums/minimums/intercepts is just a matter of applying an algorithm that evaluates points within the solution space.

I was wondering if mathematicians are good at generating these mental pictures, or if, at their level, it is all just abstract manipulation to them. How do you guys think of your problems?

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I'm pure algebra and some geometry. If I come up with a problem I can't figure out an equation for, I just plug the numbers into U-Basic and let the computer do the work. I try different values for all of the variables, along the range from lowest possiblities to the highest, and find a relationship from the results. I then test them with a few mid-range values. Used to be, I would sit for days and use up notebook after notebook trying to work out some of the most complex formulas. I can't even comprehend how I did that considering I have no patience for being stuck, and when I came up with a wrong answer I would spend just as long finding where I erred or simply starting over. Thank goodness for U-Basic!

By the way, I have come up with a way to find the formula for any summation to any value of n (number of summations) that do not include geometrical progressions, or irrationational or complex numbers, in the summation or as a limit. This probably doesn't leave much, but I have been looking for a challenge by which to test it. I can be given the progression for the summation itself or just a few of the results of the series to some value of n, but I have to know n as well (and every summation has to be performed in the same way). Any takers?

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Let's put together the pieces of The Grand Puzzle . (website)

"Let's define another operator, Sz, which we won't pay any attention to."
"This transformation will automatically make zero equal zero."
"It may be true that zero equals zero -- and that is certainly an equality -- but I don't want to go into the details at this time."

It depends on the problem. Some are just manipulation. Some are helped by mental picutes. But, then, most of the problems I work with are in the geometric world, so the mental pictures can help, for me anyway.

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I work with partial differential equations, and when I have nice clear picture forming, I know I am close to the solution. . .

All the above, and more...

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There isn't one as such, but catergory theory or soemthing simlair is probably the most genralized area of maths.

how about this question:


what are the three most recent maths discovered. Calculus was hundreds of years ago.

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-Cross
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Those that lack education have a hard time understanding its value. - Cross

My niece's checkbook because it is multi-variable, time shifted, missing data, and appears to call on metaphysics for resolution.

Thats one.

what does he have across the bottom?

There is a lowest math level, which is simple addition (or subtraction) of integers. We start maths study with them. The highest levels maths would correpond to increasingly higher levels of abstraction.

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I was wondering:

Since one definition of multiplication (M*N) could be
K = 0; for(I = 0; I < N; I++; ) K = K + M;
and another definition for powering could be (M^N)
K = 0; for(I = 0; I < N; I++; ) K = K * M;
and so on for higher levels of this series which aren't often used
(M^^N): K = 0; for(I = 0; I < N; I++; ) K = K ^ M;
(M^^^N): K = 0; for(I = 0; I < N; I++; ) K = K^ ^ M; (Reaaly big numbers for any good value of N)

could you have some level of this series preceeding addition?
(M+N):
K = 0; for(I = 0; I < N; I++; ) K = K ?? M;

?

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For addition, say this instead
(M+N):
K = M; for (I = 0; I < N; I++) K = K + 1;
or
K = 0; for (I = 0; I < N; I++) K = K + 1;
for (I = 0; I < M; I++) K = K + 1;

I don't know if your progressive series can be applied in this case.

Fred

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K = M; for(I = 0; I < N; I++; ) K = K ++;

As noted, the "level" of math is not really quantifiable. Nevertheless, HERE is a listing of graduate level mathematics courses, which should give you some idea....

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Everyone is entitled to his own opinion, but not his own facts.

In terms of major branches? Numerical analysis has seen a resurgence since the advent of the computer (though perhaps Jacobi would argue that it's been around for much longer). Fractal theory/chaos theory is pretty recent, at most a hundred years or so old. Non-linear dynamics (which some argue is the same thing) -- heck, non-linear anything -- is pretty active now that we have computers, just because results were hard to come by before that.

Part of the problem here is that it's difficult to pin down exactly when a branch of math was discovered, because they all kind of merge. Sometimes, you get a big jump in development, like with Newton and Liebnitz in calculus, but the Greeks used some of the same concepts thousands of years earlier. It's just that the popular founders were the first to really sit down and delve deeply (it turns out that it doesn't always matter who publishes first ).

Computing theory is relatively recent... you could maybe attribute the start to the time of Babbage and Lovelace, but Godel, Turing, et al. made it what it is today. Topology is in the last hundred and fifty years, as is set theory.

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"It's turtles all the way down."

Well, there's the inverse power tower function. It's pretty much the slowest-growing function of them all. http://en.wikipedia.org/wiki/Power_tower

By definition, all numbers are constructed (usually) from the positive integers. So, learning to count and order numbers is the most basic math there is, sort of. You can do a fair amount of math without even that. But in a sense, there's no point in asking which series precedes or follows the other: they're all the same (the mathy term is isomorphic). We had a big argument about this somewhere in the general science thread a while back.

Here's an interesting way of counting, by the way: 3, 5, 7, 9, ..., 2*3, 2*5, 2*7, 2*9, ..., 2^2*3, 2^2*5, ..., 16, 8, 4, 2, 1.

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"It's turtles all the way down."

And what about work done at the very "lowest" level (the logical foundations of math)? Most definitely not for the faint of heart.

Ditto, the branch of metamathematics that studies the consistency of the axioms of various parts of math (and which lead to the entirely unexpected results of Cantor, on proof)? Is this a "low" level, or one of the most advanced "levels"?

If a result can be simply stated, simply understood, and simply proved, does that mean it is (automatically) a "low" level thing? Example: Russell's 'set of all sets'