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Mathmatics or "it's all Greek to me"

Reply #25
Thanks for the encouragement.  I think it will also be different for me simply because i'm doing it for me rather than being in school.  I had trouble with math because my mind works in little visions, not words or numbers at all.  If i ask someone to pass the salt, i see a vision of them doing so and project an emotion of 'want'. Then i take that and translate it into words.  Conversely, when someone speaks to me the words are translated back into vision form.  I don't know if you're similar but your statement of things making since because you can visualize them is a familiar sentiment.

Math is troublesome as i've only recently figured out that what i need to do to make "2+4" make any sense to my mind is to visualize myself stacking two rocks on top of four of them and count them.  I find i'm *very* fast at that and i can do simple algebra in my head in ways i never could before i started doing this about a year ago.  Calculus? I'll figure out a way to make it work i guess.


Psychologists have found that there are three main styles of thought: visual, auditory, and kinesthetic -- touch and movent. We each have a preferred sensory modality, and a secondary one. Your primary modality appears to be visual, and I'm guessing on the basis of your salt shaker example and the fact that you translate your thoughts into words, that your secondary one is kinesthetic. Also, some people are more at home in the realm of intuition, and others of the concrete. So I don't think there's anything abnormal or even unusual about the way you think, or that it precludes an understanding of advanced mathematics. The trick I think is to approach the topic from the angle with which you're most comfortable, which is, in your case, visual.

Calculus is, fortunately, almost as visual as addition. At it's essence, it's nothing more than the slope of a line on a graph, and the area under the line; that can be extended to multiple dimensions (as it is in ambisonics), but the concept is basically the same. What's more, integral calculus is basically glorified addition. You could stack those rocks of yours under a rope and count them and you'd have the area under the rope, expressed in units of rocks. You could stack those rocks in a tent and count them and you'd have the volume of the tent, again, expressed in units of rocks.

The difference between calculus and mundane kitchen measurements is that in calculus, you make the rocks vanishingly small, so you can get a perfectly accurate "analog" measurement of the area or volume. You can see intuitively that if you filled a cup with rocks and another with fine sand you'd get a more accurate measurement of the cup by counting the grains of sand than you would by counting the rocks. Newton and Leibnitz made the remarkable discovery that you could continue the process of downsizing the rocks until they were infinitely small and there were an infinite number of them, and still get results that were finite, as they would be if you used water to determine the volume of the cup rather than rocks or sand.

Mathmatics or "it's all Greek to me"

Reply #26
ask yourself questions like: If all measurements are real, why do we need imaginary numbers?

Does anyone know the answer?


1) Sometimes, the easiest way between two points on the real line is through the complex plane  . For example, the formula for the solutions of the cubic, http://planetmath.org/encyclopedia/CubicFormula.html -- it is possible to use this even when you are only interested in real roots.

2) There is lots of insight gained when used complex numbers. For example, the second-order differential equation x''=ax where a is a constant. Using complex numbers, you unify the solutions (... well, a certain nuissance for the simple case a=0 though ...), and you can make the observation that oscillations and growth are in some sense pretty much the same phenomenon.

3) "imaginary numbers" are in some sense just as real as real numbers  . A complex number is nothing more than an ordered pair (x,y) of reals, with vector addition and then this particular extension of the multiplication operator.
Memento: this is Hydrogenaudio. Do not assume good faith.


Mathmatics or "it's all Greek to me"

Reply #28
@Josh358: It's a side trip, i'm not sure why it's always been a struggle. It's not so much that i'm visual per se. I'm actually more auditory.  The visions in my head are as lifelike as dreams with full  stimulation of all 5 senses. Speaking is more like describing what i'm seeing rather than expressing a verbal idea and i sometime feel frustrated that words are not enough to convey the nuances of a thought with out taking much longer than most folks have the time for.  I'm glad not to be so unusual. I did find that graphing and geometry were my favorite math subjects in high school and that i was quite good at them and that will help.

@all: I think the way to go about this (as i think others have said) is to understand the concepts and then learn the math. Understanding the concepts will give me something to visualize when i'm going through the math and that's all i really need to get through it (plus a lot of self patience)  Also, part of me is thinking "You're 38 and you're staring this NOW?  Isn't it a little late?"  For what?  I guess i've come to realize that while i get a lot of listening to music that's a passive activity even if you're focused on it.  I want music to be an active part of my life.  I want to make it (via studio work), or bring what's already made to others (via speaker set ups or something).  I'm not setting a goal of being the next Steve Hoffman or something, but i want a piece of it.


The responses in this thread help it not look so daunting. I've got quite a few of the books mentioned on my gift list this year. and i'm working my way through Musicmathics.

Music lover and recovering high end audiophile

Mathmatics or "it's all Greek to me"

Reply #29
1) Sometimes, the easiest way between two points on the real line is through the complex plane  . For example, the formula for the solutions of the cubic, http://planetmath.org/encyclopedia/CubicFormula.html -- it is possible to use this even when you are only interested in real roots.

2) There is lots of insight gained when used complex numbers. For example, the second-order differential equation x''=ax where a is a constant. Using complex numbers, you unify the solutions (... well, a certain nuissance for the simple case a=0 though ...), and you can make the observation that oscillations and growth are in some sense pretty much the same phenomenon.


These three contradictory excuses are very fine whlie you simply apply complex analysis to ease calculus/notation. There shold be one but good reason WHY since quantum amplitudes are complex. And I'd rather questioned whether all measurements are real indeed, as you have said

3) "imaginary numbers" are in some sense just as real as real numbers .


so two observed real variables may constitute (or be derived from) some compex value.

Mathmatics or "it's all Greek to me"

Reply #30
These three contradictory excuses are very fine whlie you simply apply complex analysis to ease calculus/notation. There shold be one but good reason WHY since quantum amplitudes are complex. And I'd rather questioned whether all measurements are real indeed, as you have said


There is no contradiction. You claim otherwise, so produce your testable, verifiable evidence, please.
-----
J. D. (jj) Johnston

Mathmatics or "it's all Greek to me"

Reply #31
These three contradictory excuses are very fine whlie you simply apply complex analysis to ease calculus/notation. There shold be one but good reason WHY


Nope. Why use decimal numbers? Sometimes they are useful for this, sometimes they are useful for that. There is no need to dig up one big reason why you should always use decimal numbers -- indeed, there is none such, as there are applications where they are utterly useless.

Have a look at the Mandelbrot set (edit: just google it, you'll recognize the shape). The algebra behind is elegant in complex numbers, but I can describe it just as precise without mentioning "complex numbers" or "imaginary numbers". In reality I'd be doing precisely the same -- I would just have to define a vector-valued function of two variables, a function which does the same as the complex square. Should I or should I not invoke complex numbers?


The science of mathematics -- and crafts of applied mathematics, even -- is full of analogous examples. Adding additional machinery may or may not be useful to a given problem. Even assuming it is useful, then teaching or learning that additional machinery has an effort cost which may or may not be outweighted by its usefulness.
Memento: this is Hydrogenaudio. Do not assume good faith.

Mathmatics or "it's all Greek to me"

Reply #32
There is no contradiction. You claim otherwise, so produce your testable, verifiable evidence, please.


The first contradiction I should mention is that Porcus does provide an example of my claim

oscillations and growth are in some sense pretty much the same phenomenon.


measurements are not necessary should be considered as real. Surely you can avoid using words you don't like

The algebra behind is elegant in complex numbers, but I can describe it just as precise without mentioning "complex numbers" or "imaginary numbers".

   
but then it may be hard to interpreter your notation (e.g. what are real and imaginary parts of wavefunction?). Many physical phenomenons (e.g. existence of positrons) have been predicted just by analysis of math objects introduced in theory. And analysis of phase factor of compex valued wavefunction leaded to gauge field theory.

I used to think about decimal numbers and stuck with reading about pointless geometry, it seem to be false way though I'm quite agree that points do not exist.

What do you think?



Mathmatics or "it's all Greek to me"

Reply #33
1) Sometimes, the easiest way between two points on the real line is through the complex plane  . For example, the formula for the solutions of the cubic, http://planetmath.org/encyclopedia/CubicFormula.html -- it is possible to use this even when you are only interested in real roots.

2) There is lots of insight gained when used complex numbers. For example, the second-order differential equation x''=ax where a is a constant. Using complex numbers, you unify the solutions (... well, a certain nuissance for the simple case a=0 though ...), and you can make the observation that oscillations and growth are in some sense pretty much the same phenomenon.


These three contradictory excuses

contradictory? excuses?
Quote
are very fine whlie you simply apply complex analysis to ease calculus/notation.

It's even better than the original poster implied. Some things about real functions can /only/ be proven using complex numbers. E.g. solution to particular integrals. Sorry I don't have a more precise reference, calculus is a good few years ago now for me
Quote
There shold be one but good reason WHY since quantum amplitudes are complex. And I'd rather questioned whether all measurements are real indeed, as you have said

Quantum mechanics indeed uses complex wave functions. However, these are just /part/ of the mathematical description of the system. As soon as you try to describe taking a measurement, the outcome is real. In other words, the (complex) wave function by itself is meaningless, and as soon as you combine it with the operator to describe a (any) measurement, the resulting number (or function) is real. All measurements in physics (or exact sciences) are real, really!

By the way, why can't mathematics be a reason by itself? Mathematics is useful for other things besides describing physical measurements.

Mathmatics or "it's all Greek to me"

Reply #34
There is no contradiction. You claim otherwise, so produce your testable, verifiable evidence, please.


The first contradiction I should mention is that Porcus does provide an example of my claim

oscillations and growth are in some sense pretty much the same phenomenon.


measurements are not necessary should be considered as real. Surely you can avoid using words you don't like

The observation "oscillations and growth are in some sense pretty much the same phenomenon" is not a measurement. It is a recognition that you can describe both seemingly very different phenomena in a very similar way. "Measurement" and (mathematical or otherwise) "description" are two very different things.
Quote
The algebra behind is elegant in complex numbers, but I can describe it just as precise without mentioning "complex numbers" or "imaginary numbers".

   
but then it may be hard to interpreter your notation (e.g. what are real and imaginary parts of wavefunction?). Many physical phenomenons (e.g. existence of positrons) have been predicted just by analysis of math objects introduced in theory. And analysis of phase factor of compex valued wavefunction leaded to gauge field theory.

I used to think about decimal numbers and stuck with reading about pointless geometry, it seem to be false way though I'm quite agree that points do not exist.

What do you think?


I'm not quite sure what you're trying to argue, I find it hard to follow your English.

Mathematics uses many imaginary (not as in complex numbers) constructs. Mathematics is an application of logic, and hypothesizing about e.g. infinitely small increments is perfectly fine logically. Concepts such as natural numbers -> integers -> rational numbers -> real numbers -> complex numbers are very strictly defined and follow very strict rules for operations (e.g. addition, multiplication). This perhaps seemingly finegrained distinction between classes of numbers can actually make a difference if you're trying to solve certain mathematical problems.

With "pointless geometry" do you mean point-free geometry (http://en.wikipedia.org/wiki/Whitehead's_point-free_geometry)? You can think of that as just another imaginary construct to help solve certain problems (or in other words to find new ways to extend the logic of mathematics).

None of mathematics is real, it's all imaginary (as in perceived in the mind). Only when applied (to exact sciences or elsewhere) does it get a 'real' meaning.


Mathmatics or "it's all Greek to me"

Reply #35
Okay. You are right Maarten (also Porcus and Woodinville) there is no contradiction indeed. BearcatSandor shouldn't ask himself any questions, complex numbers are nice and useful imaginary concept (like positrons) and need not to be interpreted. Your English is very good Maarten and your numerous non-trivial examples did change my opinion.


Mathmatics or "it's all Greek to me"

Reply #37
On a side note: positrons are existing particles. They are not imaginary.

As anyone who has had a PET scan should know...

Mathmatics or "it's all Greek to me"

Reply #38
complex numbers are nice and useful imaginary concept (like positrons)


On a side note: positrons are existing particles. They are not imaginary.


And "imaginary" numbers are not really imaginary, but are perfectly real.  At least, if you use the world "real" to describe numbers. 

Actually, numbers are abstractions and as numbers have no physical reality outside our minds.  When have you ever seen the number "one" (not the numeral, the number)? 

Ed Seedhouse
VA7SDH

Mathmatics or "it's all Greek to me"

Reply #39
complex numbers are nice and useful imaginary concept (like positrons)


On a side note: positrons are existing particles. They are not imaginary.


And "imaginary" numbers are not really imaginary, but are perfectly real.  At least, if you use the world "real" to describe numbers. 

Actually, numbers are abstractions and as numbers have no physical reality outside our minds.  When have you ever seen the number "one" (not the numeral, the number)?


I think you're going to have to go some to explain the difference between the concept, the symbol, and something that is of the proper cardinality.

An 'imaginary' number is just a particular way to express a second dimension, and self-contains an operator that handles the rotations without doing 2D matrices all over the place.
-----
J. D. (jj) Johnston

Mathmatics or "it's all Greek to me"

Reply #40
The observation "oscillations and growth are in some sense pretty much the same phenomenon" is not a measurement. It is a recognition that you can describe both seemingly very different phenomena in a very similar way. "Measurement" and (mathematical or otherwise) "description" are two very different things.


On a side note: positrons are existing particles. They are not imaginary.

As anyone who has had a PET scan should know...


Thanks. Not beta plus but Alpha decay is a good example that complex wavevector can be measured. Most people here seem to follow instrumentalist interpretation: "Shut up and calculate!"

 
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