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Continuous Mathematics Has Had It's Day.


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The money will be in computer science, which needs discrete maths, so if that's how you want to measure it. Or perhaps because of computers the time for advances in discrete maths is at hand.

 

I believe (and if my supervisor is to be believed) that a lot of cutting edge physics topology is described by finite group and ring theory.

 

I very much prefer finite algebra type stuff, so any future maths I do is not going to be continuous!

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Discrete math deals with numbers that are discretely separated from each other, like the integers. Logic and Finite state machines are aspects of discrete math.

 

Continuous math deals with numbers with no defining edge between them. Calculus is the core of a lot of continuous math, because it deals with an infinite number of infinitely small quantities.

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As for the orginial question: I wouldn't count continuous math out just yet. Computer science might be fuelling discrete math; however there's also growing interest in space technology, which will require engineering, and, I assume, the continuous math that goes into structural design.

 

That's just my opinion though.

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From an engineering point of view, today, if you are presented a differential equation that is not one of the 'nice' forms where a solution has already been developed, the next step is almost always to slap it into a computational solver. And, of course, all those computational methods are based on discrete math.

 

The problem with too much reliance on computational methods is that you lose some of the elegance/'music' of the solutions. Things like self-similarity and asymptotics would require a large number of simulations to identify properly, whereas often just a little analysis with the original equations would reveal some of those interesting behavior.

 

Also, if you have a large number of adjustable parameters, many, many simulations are needed to see the whole range of effects. An analytical solution is often much clearer to see how each parameter affects the final solution.

 

My guess is that ultimately both discrete and continuous analysis will have its place, both tools used in conjunction to solve problems. Discrete mathematics is hot right now since it is new and many discoveries are left. Continuous mathematics is very mature by comparison, and the work needed to get new results in continuous math is much much greater than to get a new result in discrete math.

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I would disagree that the border between the two is artificial, alan2. Discrete maths is not a simplification of continuous problems (though it can be used as such, e.g. computational methods). The study of discrete maths like graph theory, combinatorics, number theory, finite group theory, etc. are all studied for their own sake, and owe continuous maths no favours or provenance. The fact that sometimes discrete maths comes up in the study of continuous problems just goes to show that the world works in more discrete ways than we think!

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What is this all math problems solver software of which you speak?

 

FEMLAB/COMSOL is the one that springs immediately to mind, they claim you can just enter the differential equations you want to solve, and the program will solve it.

 

see http://www.comsol.com/

 

My experiences with it lead me to think that their claims are a little overstated in their ads (what else is new, eh?) but, it is still all-in-all a pretty amazing piece of software. It will attempt to solve coupled fluid flow, heat transfer, and mass transfer problems, and so long as the problem isn't too difficult, it does a pretty good job of it.

 

There are other packages out there, most of them are more specifically tailored for a specific class of problems. Like specialty fluid flow programs, specialty combustion programs, specialty solids mechanics programs, etc.

 

Like I said though, with the ease of use of software like these, all too often the engineer will turn to the computer before working with the problem a little on its own. The real danger is losing a sense of intuition about problems. By working through the solutions, you really get a feel for how the solutions behave -- the character of the solutions as it were. Computers are notorious for "garbage in, garbage out" and without any intuition about the solutions, you will have no idea when "garbage out" occurs. This is why I still think that both the computational and continuous math with each have its place.

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What is this all math problems solver software of which you speak?
MATLAB?
As for the orginial question: I wouldn't count continuous math out just yet. Computer science might be fuelling discrete math; however there's also growing interest in space technology, which will require engineering, and, I assume, the continuous math that goes into structural design.

 

That's just my opinion though.

TBH I think it's a little sad that it should be judged by where it's use lies. I agree with what you're saying though.
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Both discrete and 'continuous' mathematics will remain important in the future. Many physics problems are formulated in terms of 'continuous' mathematics. Numerical solutions of course essentially are discrete, but given high enough granularity, we may regard the outcomes as 'continuous'.

 

Just look at it like water. Water looks very continuous, but when zooming in, you finally end up with discrete entities, the molecules of water (and beyond).

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  • 3 weeks later...

Now that I learned from this thread, I would agree continuous math is going to be lost.

 

I beleive continuous came from the misinterpratation that things can obtain qualities (example energy) independently. This is obviously not so, there must be two, or that is, a compound action for any ONE measurement to be formed for each of the frames involved in that system.

 

This is exactly the thread I was looking for at this moment. It informed me with things that I needed to know.

 

I know it sounds rediculous but I a am very sure I understand how to connect relativity to quantum mechanics, and in that process expand on how quantum mechanics must behave discrete. Hopefully, taking the weirdness out of it, with a few general acceptances.

 

Now I completely suck at math so I had not known about discrete verus continuous.

 

but, yes, continuous math is dead, because relativity can be shown how it also infact works with QM, which all connects to classical.

 

It all has to do with momentum,inertia,time,and velocity

 

What occurs is duality.

 

At which there is, two types of every one thing.

Two types of Ke, two types of momentum, two types of time.

 

For now I'll take the licks untill I can learn how to express what I see in logic.

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I [believe] continuous came from the [misinterpretation] that things can obtain qualities (example energy) independently. This is obviously not so, there must be two, or that is, a compound action for any ONE measurement to be formed for each of the frames involved in that system.
I have absolutely no idea what you mean there.
but, yes, continuous math is dead, because relativity can be shown how it also in[-]fact works with QM, which all connects to classical.
Urm, classical mechanics is continuous, as is Einsteinian mechanics.
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I'm afraid I have to disagree. Discrete mathematics has an important part to play in computer science, but that certainly does not imply that the rest of mathematics is completely pointless. For instance, the study of partial differential equations and modelling of fluid flow relies totally on the Navier-Stokes equations. Whilst the only way of solving these (in general) is to indeed discretise the problem, without the study of continuous functions one could say very little about the solutions obtained, nor about any behaviour observed in a quantitative fashion.

 

Likewise, whilst quantum mechanics relies on the fact that energy is discretized, it is seldom employed when engineers need to construct bridges. In fact, most of the mathematics studied during an engineering degree will likely consist of calculus and real analysis. Both of these fields are extremely important, not only in the mathematical community, but in the scientific community in general.

 

From my (albeit limited) perspective, this is the way things are, and they are not likely to change any time in the near future.

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