Thread: Einstein explains relativity to Newton.

One of the most interesting conversations I can imagine is that of Einstein explaining his theories of Special and General Relativity to Newton. If such a conversation were possible, how might it go?

Of course, Einstein would need to explain the most notable aspects of his work on relativistic effects including time dilation, length contraction, and the relation between mass and energy.

Einstein's part of the conversation is perhaps the easier half to imagine because we know how physics was advanced by Einstein's work. Einstein would know how he advanced physics too, of course, but Newton, being the forerunner, would be new to what Einstein would say about relativity.

I believe Newton would be incredulous at first. The science of his day was based on absolute time, and Einstein saying that time is relative to the observer's frame of reference might be a bit hard for Newton to accept.

Anyway, I'm sure that many of the other members here know more about the history of physics than I do. How much did physics develop between the time of Newton and Einstein? Would Newton be able to accept or even grasp the basics behind the Theories of Relativity, or would such a passage of time make it impossible for Newton to catch up and learn all the requisite physics and mathematics that formed the basis of Einstein's work?

Jagella

One of the issues I see would be with the invariance of the speed of light. Einstein would first have to get Newton to accept the wave theory of light (Newton prefered the particulate theory).

I think that the success of this would depend upon "When" this conversation took place. If it took place today, Einstein could show Newton all of the experimental evidence in support of this concept. If it took place so that only the evidence available during Newton's time could be used, it would be much more difficult.

Originally Posted by Janus

One of the issues I see would be with the invariance of the speed of light. Einstein would first have to get Newton to accept the wave theory of light (Newton prefered the particulate theory).

Had the speed of light been measured in Newton's time? Also, did Einstein believe as we do today that light is both made up of particles and is also a wave?

Originally Posted by Janus

I think that the success of this would depend upon "When" this conversation took place. If it took place today, Einstein could show Newton all of the experimental evidence in support of this concept. If it took place so that only the evidence available during Newton's time could be used, it would be much more difficult.

I was thinking that the conversation would take place between a Newton and an Einstein in which both physicists are at the pinnacle of their work and are accepting the science of their day as the best explanation for light, mass, energy and time. Einstein would have the opportunity to explain everything he knew about physics at the time he died.

Jagella

Originally Posted by Jagella

Had the speed of light been measured in Newton's time? Also, did Einstein believe as we do today that light is both made up of particles and is also a wave?

Einstein received the Nobel Prize for work showing that light is quantized.

He developed the special relativity based on a deep understanding of Maxwell's equations that govern electromagnetic waves.

I'll take a wild guess that he understood light pretty well.

Was Einstein around long enough to see (and understand) QED in its final form? If not, then I would say his understanding of light was incomplete.

Originally Posted by DrRocket

Einstein received the Nobel Prize for work showing that light is quantized.

The photoelectric affect?

Originally Posted by DrRocket

He developed the special relativity based on a deep understanding of Maxwell's equations that govern electromagnetic waves.

Newton would not have known of Maxwell's equations, of course. How much would Newton need to learn to grasp the principles behind Maxwell's work?

Jagella

Originally Posted by salsaonline

Was Einstein around long enough to see (and understand) QED in its final form?

Of course not, as you well know.

Originally Posted by salsaonline

If not, then I would say his understanding of light was incomplete.

Of course his understanding was incomplete. But that understanding did clearly include both particle-like and wave-like behavior of light, which is the subject of the question that was posed.

QED is probably incomplete also. It is just the best availablr theory.

Even our understanding of QED itself is incomplete. What it is capable of explaining in principle (most of everyday physics that does not require gravity) greatly exceeds what we can explain using it in practice (no one can explain all of chemistry using just QED).

I'd like to hear Einstein explain how he traveled back in time.

Originally Posted by Harold14370

I'd like to hear Einstein explain how he traveled back in time.

Closed timelike curve.

What Einstein knew or didn't know has been the subject of debate (historical debate). For example, it isn't clear that Einstein knew much about the Michelson-Morley experiment when he was developing special relativity. He appears to have been reticent on the matter. Also, it appears that he was initially unimpressed by Minkowski's notions on space-time and the use of tensors. This is a quote from the detailed biography of Einstein by Abraham Pais (Subtle is the Lord - The Science and Life of Albert Einstein):

Initially, Einstein was not impressed and regarded the transcriptions of his theory into tensor form as "uberflussige Gelehramkeit", (superfluous learnedness). However, in 1912 he adopted tensor methods and in 1916 acknowledged his indeptedness to Minkowski for having greatly facilitated the transition from special to general relativity.

Apparently his comment about "superfluous learnedness" was made to V. Bargmann who passed it on to Abraham Pais.

Originally Posted by Old Fool

What Einstein knew or didn't know has been the subject of debate (historical debate). For example, it isn't clear that Einstein knew much about the Michelson-Morley experiment when he was developing special relativity. He appears to have been reticent on the matter. Also, it appears that he was initially unimpressed by Minkowski's notions on space-time and the use of tensors. This is a quote from the detailed biography of Einstein by Abraham Pais (Subtle is the Lord - The Science and Life of Albert Einstein):

Initially, Einstein was not impressed and regarded the transcriptions of his theory into tensor form as "uberflussige Gelehramkeit", (superfluous learnedness). However, in 1912 he adopted tensor methods and in 1916 acknowledged his indeptedness to Minkowski for having greatly facilitated the transition from special to general relativity.

Apparently his comment about "superfluous learnedness" was made to V. Bargmann who passed it on to Abraham Pais.

I've often wondered how much Einstein developed his theories using imagination or "visualization," and how much was the result of "left-brain" or "linear" thinking using mathematical formulas.

Jagella

Originally Posted by Jagella

[I've often wondered how much Einstein developed his theories using imagination or "visualization," and how much was the result of "left-brain" or "linear" thinking using mathematical formulas.

Jagella

I'm not sure I would word my question in exactly the same way, but I think I know what you mean and it is something I have wondered about given that Einstein often conducted thought experiments and was, possibly, not in the very top rank of mathematicians.

Thought experiments, or Gedanken, were quite common in the first part of the twentieth century. They are out of fashion now, possibly because people are now more sceptical about their validity.

Originally Posted by Old Fool

Thought experiments, or Gedanken, were quite common in the first part of the twentieth century. They are out of fashion now, possibly because people are now more sceptical about their validity.

Learnt a new word today-gedankenexperiment!

[quote="Jagella"]

Originally Posted by Old Fool

I've often wondered how much Einstein developed his theories using imagination or "visualization," and how much was the result of "left-brain" or "linear" thinking using mathematical formulas.

Jagella

Most mathematicians, not all but most in my experience, think visually.

Originally Posted by DrRocket

Originally Posted by salsaonline

Was Einstein around long enough to see (and understand) QED in its final form?

Of course not, as you well know.

In his memoirs, Feynman mentions giving a presentation in which Einstein was one of the audience members. So I wasn't sure of the precise timelines of who knew what when.

Originally Posted by Halliday

Originally Posted by Jagella

[I've often wondered how much Einstein developed his theories using imagination or "visualization," and how much was the result of "left-brain" or "linear" thinking using mathematical formulas.

Jagella

I'm not sure I would word my question in exactly the same way, but I think I know what you mean and it is something I have wondered about given that Einstein often conducted thought experiments and was, possibly, not in the very top rank of mathematicians.

I understand that Einstein needed to get help from a friend of his who happened to be an astute mathematician. His friend helped him using non-Euclidian geometry.

It's interesting to note that when Newton needed mathematics to help him with his work, he invented calculus! So in the mathematics department, I'd say that Newton was the better thinker.

Jagella

[quote="DrRocket"]

Originally Posted by Jagella

Originally Posted by Old Fool

I've often wondered how much Einstein developed his theories using imagination or "visualization," and how much was the result of "left-brain" or "linear" thinking using mathematical formulas.

Jagella

Most mathematicians, not all but most in my experience, think visually.

I'd say that the "visual" part of mathematics would include geometry and its applications in trigonometry, calculus, and linear algebra. The nonvisual or linear part of mathematics comes into play in its use of abstract symbols such as variables, operation symbols (e.g. +, -, /, and X), and numerals. So when Einstein worked with his theories, he no doubt visualized ellipses that formed the orbits of celestial bodies and how these elliptical orbits may have been shaped by gravity warping the fabric of space. His famous equation, e = mc^2, is a formula that he came up with by applying nonvisual, linear thinking.

Maybe a clearer example of the two parts of mathematics would be to graph the function, f(x) = x^2. This function is abstract and nonvisual. The graph of the function in the x-y plane is, of course, visual.

Jagella

[quote="Jagella"]

Originally Posted by DrRocket

Originally Posted by Jagella

Originally Posted by Old Fool

I've often wondered how much Einstein developed his theories using imagination or "visualization," and how much was the result of "left-brain" or "linear" thinking using mathematical formulas.

Jagella

Most mathematicians, not all but most in my experience, think visually.

I'd say that the "visual" part of mathematics would include geometry and its applications in trigonometry, calculus, and linear algebra. The nonvisual or linear part of mathematics comes into play in its use of abstract symbols such as variables, operation symbols (e.g. +, -, /, and X), and numerals. So when Einstein worked with his theories, he no doubt visualized ellipses that formed the orbits of celestial bodies and how these elliptical orbits may have been shaped by gravity warping the fabric of space. His famous equation, e = mc^2, is a formula that he came up with by applying nonvisual, linear thinking.

Maybe a clearer example of the two parts of mathematics would be to graph the function, f(x) = x^2. This function is abstract and nonvisual. The graph of the function in the x-y plane is, of course, visual.

Jagella

In my experience the scenario that you suggest is probably quite wrong. If you have never done mathematical research I probably cannot convince you.

Mathematicians visualize functions and in fact infinite-dimensional spaces of functions all the time.

Originally Posted by DrRocket

Most mathematicians, not all but most in my experience, think visually.

I have to admit I don't really know how mathematicians think and will accept what you say.
But if you take, for example, the atom and its building blocks (subatomic particles) I tended to believe that individuals who tried to visualise these "objects" were those who did not possess an extensive mathematical background and who therefore sometimes felt the need to understand them in common sense terms.
Mathematics is the language of physics and the properties, such as structure, dimensions and behaviour of these objects, can only be, at least partially, understood using the tools of abstract mathematics.
So how useful is it to "think visually" in this situation?

I can only speak for myself, but when I think about physical phenomena, I usually have a wee picture of whats goin on in mind.

Likewise, with maths I try to form "images" of what's going on. I use the " " marks here deliberately as when it comes to mathematics I am not always imagining actual pictures. Sometimes in fact I will use a different scenario to understand something. For example, if I'm given a derivative of two variables, instantly in my head I pretend that im looking at the derivative of distance with time, ie velocity. I find this kind of tactic can often make things much easier to understand.

Originally Posted by salsaonline

Originally Posted by DrRocket

Originally Posted by salsaonline

Was Einstein around long enough to see (and understand) QED in its final form?

Of course not, as you well know.

In his memoirs, Feynman mentions giving a presentation in which Einstein was one of the audience members. So I wasn't sure of the precise timelines of who knew what when.

As I recall that presentation was on Feynman's non-relativistic time-symmetric treatment of electrodynamics http://en.wikipedia.org/wiki/Wheeler...bsorber_theory

I do not think that Einstein attended the 1948 Pocono conference where Feynman and Schwinger presented their formulations of QED. On the other hand. Feynman and Schwinger's work was presented there to people like Bohr, Bethe, Teller, Wheeler, etc. so Einstein must have heard of it. But this all occurred long after Einstein's most productive years. He certainly knew nothing of QED in the 1905-1915 time frame when he did his major work in quantum mechanics and relativity.

Einstein understood quantum theory. He was a major player in its original development. He just did not believe it was any more than a provisional theory, and thought that ultimately it would be replaced by something deterministic.

This debate reminds me of a question asked by a tutor when I was a student (centuries ago!). On the face of it, the question seems very simple, but there was, apparently, more to it than met the eye.

Question:

Consider a cube. The cube is divided into two equal halves by a plane parallel to one of its faces. The cube is then divided into four equal parts by another plane which is at right angles to the first. In total, how many faces are on the resultant parts?

The answer: 24

The real question: In arriving at this answer, did you you form a mental image of a cube?

According to my tutor, those who answered yes to the second question (as I did) are unlikely to be real mathematicians, who it would seem, might devise a method of thinking about the problem which does not require them to form mental images of cubes.

Blah, that just sounds too much like the No True Scotsman fallacy.

Originally Posted by Old Fool

This debate reminds me of a question asked by a tutor when I was a student (centuries ago!). On the face of it, the question seems very simple, but there was, apparently, more to it than met the eye.

Question:

Consider a cube. The cube is divided into two equal halves by a plane parallel to one of its faces. The cube is then divided into four equal parts by another plane which is at right angles to the first. In total, how many faces are on the resultant parts?

The answer: 24

The real question: In arriving at this answer, did you you form a mental image of a cube?

According to my tutor, those who answered yes to the second question (as I did) are unlikely to be real mathematicians, who it would seem, might devise a method of thinking about the problem which does not require them to form mental images of cubes.

The only conclusion thst I draw from yoir story os that your tutor is not a real mathematician. I am.

However, that does not mean that a mathematician would form a mental image and then count the faces individually.

The mental image tells me that the original cube has been divided into 4 solid rectangles, each with 6 faces.

I have a hard time believing that a good mathematician could resist visualizing such a problem.

Originally Posted by salsaonline

I have a hard time believing that a good mathematician could resist visualizing such a problem.

A friend of mine (a topologist) once had a meeting with Dennis Sullivan. Sullivan started out with "Are you a pictures guy or a numbers guy ?" He felt that he could not communicate with a "numbers guy".

I'm not sure that I even know a "numbers guy".

I am basically a "hard analyst". I think visually. Charlie Fefferman is the quintessential hard analyst, and he says that he thinks visually.

I am sure that "numbers guys" exist, but I can't come up with a specific example.

I have a hard time believing that a good mathematician could resist visualizing such a problem.

I am inclined to agree - but possibly because I tend to visualise things myself. However, there are aspects of mathematics that are much more closely related to logic than to visualising things and it is possible that visualising has little impact on those areas. I am thinking of mathematics of the kind done by Richard Dedekind who applied himself to the proof of statements such as m + n = n + m. I don't know whether Dedekind visualised things or not but, if he did, it probably wouldn't have helped much in his theory of numbers.

With regard to problems in physics, I suspect that the most significant step is developing a strategy for solving the problem and then simply remembering the bits of mathematics that enable that strategy to be implemented. I think that the importance of remembering mathematics is often understated.

Originally Posted by Old Fool

owever, there are aspects of mathematics that are much more closely related to logic than to visualising things and it is possible that visualising has little impact on those areas.

What about Venn diagrams?

Originally Posted by Old Fool

. I am thinking of mathematics of the kind done by Richard Dedekind who applied himself to the proof of statements such as m + n = n + m. I don't know whether Dedekind visualised things or not but, if he did, it probably wouldn't have helped much in his theory of numbers.

At that fundamental level which one visualizes as

So,

There is very little, if anything, in mathemstics that is not subject to some mental picture. Just what that picture might be depends on the mathematician.

What you see in a proof is a final, polished product. That often shows nothing of the discovery process and the mental pictures that led to the statement of the theorem.

Originally Posted by DrRocket

There is very little, if anything, in mathemstics that is not subject to some mental picture. Just what that picture might be depends on the mathematician..

Really? Then please provide us all with a "mental picture" (graphical representation) of the following function:

f(x,y,z,w) = x + y + z + w

If you or any other mathematician can picture five spatial dimensions, then I will be truly impressed! And I will admit I'm wrong, too, but I don't know if I need to worry about that.

Jagella

Originally Posted by Halliday

But if you take, for example, the atom and its building blocks (subatomic particles) I tended to believe that individuals who tried to visualise these "objects" were those who did not possess an extensive mathematical background and who therefore sometimes felt the need to understand them in common sense terms.
Mathematics is the language of physics and the properties, such as structure, dimensions and behaviour of these objects, can only be, at least partially, understood using the tools of abstract mathematics.
So how useful is it to "think visually" in this situation?

The model of the atom bestowed upon us by Bohr is a visual representation of that which cannot be seen, the atom. This model is very misleading, of course, if you take it too literally. The little round balls represent protons, neutrons, and electrons which are waves rather than solid objects. This model is useful for representing the position of the subatomic particles in relation to each other which is the basic structure of the atom.

In any event, we humans evolved perceptions that are largely dominated by sight. We have also evolved language which is a set of abstract symbols that can be heard in the case of speech or seen in the case of written language. The symbols making up written language have little to do with the actual appearance of what they signify. "Cat," for instance, looks nothing like the furry animal it represents! Nevertheless, such abstract symbols are very useful because they are much easier to sound and write than pictures of what they signify.

Mathematics, like English, is a written language made up largely of abstract symbols. These symbols, in most cases, look nothing like what they represent. Did you ever see a 2, for example? These abstract symbols make up the language of mathematics and are not properly what you might call "visual."

Mathematics is every bit as nonvisual as visual.

Jagella

Originally Posted by Jagella

Originally Posted by DrRocket

There is very little, if anything, in mathemstics that is not subject to some mental picture. Just what that picture might be depends on the mathematician..

Really? Then please provide us all with a "mental picture" (graphical representation) of the following function:

f(x,y,z,w) = x + y + z + w

If you or any other mathematician can picture five spatial dimensions, then I will be truly impressed! And I will admit I'm wrong, too, but I don't know if I need to worry about that.

Jagella

That is just just an ordinary 4-plane embedded in 5-space. The key is that it is co-dimension 1, so, just like a 2-plane in 3-space it is characterized by a point and a normal vector.

But nobody can provide you with a mental picture, you have to come up with them yourself. Like I said, the particular visualization depends on the particular mathematician. Learning to visualize is part of becoming a mathematician, and many people can't do it.

It is the joy of form in the highest sense that makes the geometer. -Rudolf Clebsch-

http://en.wikipedia.org/wiki/Alfred_Clebsch

I will attempt an example of a situation in which visualisation could possibly lead to an error:

If the separation of two points is zero, then the points are the same point. True or false?

(Incidentally, regarding the work of Dedekind, the whole point of his approach was to avoid geometrical explanations of concepts such as continuity. He regarded geometrical explanations as instances or useful examples, but not as proofs.)

Originally Posted by Old Fool

I will attempt an example of a situation in which visualisation could possibly lead to an error:

If the separation of two points is zero, then the points are the same point. True or false?

(Incidentally, regarding the work of Dedekind, the whole point of his approach was to avoid geometrical explanations of concepts such as continuity. He regarded geometrical explanations as instances or useful examples, but not as proofs.)

True.

But, a visualization of unit spheres in proximity, in contrast to, the answer provided by an "all n" equation, enables an understanding of the abstractness of number.
Any significant mathematical skill is unachievable without visualization.

In Minkowski geometry, does ds = 0 necessarily imply that the points (events) connected by ds are the same point (event)?

There is a related matter in one of the books by Roger Penrose. We consider a triangle ABC. In Euclidean space, AC < AB + BC for any triangle. Is this true in Minkowski space?

If the question is whether mathematicians visualize problems in hyperbolic geometry, the answer is "yes they do".

Originally Posted by Old Fool

In Minkowski geometry, does ds = 0 necessarily imply that the points (events) connected by ds are the same point (event)?

ds is a differential form an cannot "connect points".

What is true is that in Minkowski geometry there are non-zero vectors of 0 norm. Hence there are also some points that can be connected by paths of 0 length -- the integral of the differential form ds along these paths is 0. Light rays are such paths.

Originally Posted by Old Fool

There is a related matter in one of the books by Roger Penrose. We consider a triangle ABC. In Euclidean space, AC < AB + BC for any triangle. Is this true in Minkowski space?

No.

Consider the three points A=(0,0,0,0), B=(0,1,0,0), C=(1,0,0,0) in Minkowski space.

In Minkowski space (choosing the metric signature to be (+,-,-,-))

AB^2 = -1
AC^2 = 1
BC^2 = 0

The "triangle" inequality is a consequence of a positive-definite inner product. The Mimkowski inner product is not positive-definite. [/tex]

ds is a differential form an cannot "connect points".

I used the form "ds" because it is often written in that way. However, to criticise it on the grounds that it is a differential is to evade the point. One could denote it simply as s and regard it as the distance between two points in Minkowski space as calculated using the Minkowski metric. And of course it does have a physical and measureable significance - the time recorded by a clock that is moving between those points. If what is moving between the two points is a beam of light, then this distance would be zero. So we have a situation in which two points in Minkowski space have zero separation but are different points - they have different coordinates.

In general, a tendency to visualise things can't be seen as a bad thing, but it has shortcomings. We can't visualise everything.

Originally Posted by Old Fool

ds is a differential form an cannot "connect points".

I used the form "ds" because it is often written in that way. However, to criticise it on the grounds that it is a differential is to evade the point. One could denote it simply as s and regard it as the distance between two points in Minkowski space as calculated using the Minkowski metric. And of course it does have a physical and measureable significance - the time recorded by a clock that is moving between those points. If what is moving between the two points is a beam of light, then this distance would be zero. So we have a situation in which two points in Minkowski space have zero separation but are different points - they have different coordinates.

Apparently you either did not read my response or did not understand it.

A "differential form" is not a "differential". The interpretation of length as proper time applies only in the case of timelike vectors. It does not apply to spacelike vectors.

The remainder of your post is a re-statement of what I said.

So, what is your point ?

Originally Posted by old fool

In general, a tendency to visualise things can't be seen as a bad thing, but it has shortcomings. We can't visualise everything.

Apparently your ability to visualize is quite limited. Not everyone is as handicapped as are you. There are all sorts of visualizations used with regard to Minkowski space.

Apparently your ability to visualize is quite limited. Not everyone is as handicapped as are you.

DrRocket,

I regret to say that I find some of your responses too arrogant so, if you don't mind, I will ignore them in future.

Originally Posted by Old Fool

Apparently your ability to visualize is quite limited. Not everyone is as handicapped as are you.

DrRocket,

I regret to say that I find some of your responses too arrogant so, if you don't mind, I will ignore them in future.

Good idea. That might reduce the number of erroneous replies to my posts that require rebuttal.

Originally Posted by DrRocket

Originally Posted by Jagella

Originally Posted by DrRocket

There is very little, if anything, in mathemstics that is not subject to some mental picture. Just what that picture might be depends on the mathematician..

Really? Then please provide us all with a "mental picture" (graphical representation) of the following function:

f(x,y,z,w) = x + y + z + w

If you or any other mathematician can picture five spatial dimensions, then I will be truly impressed! And I will admit I'm wrong, too, but I don't know if I need to worry about that.

Jagella

Originally Posted by DrRocket

That is just just an ordinary 4-plane embedded in 5-space. The key is that it is co-dimension 1, so, just like a 2-plane in 3-space it is characterized by a point and a normal vector.

But how can we visualize five dimensions? We have only three spatial dimensions to think in and to see in. I've never seen more than these three dimensions.

In any case, I think you may be missing my point. I'm not saying that mathematicians cannot think of much of mathematics as something they might see. What I'm pointing out is that much of math is unnecessary, difficult or impossible to visualize. It is an abstract language much like English. Can anybody visualize the fourth root of two? I don't think so. The best we can do is write it as 2^(1/4). Also, visualization in mathematics is often unnecessary. As we both know, there are many techniques for solving quadratic equations. Only one of these techniques involves visualization.

I'd suggest you be careful when making all-inclusive statements. All anybody needs to do is come up with one exception to prove you wrong. Claiming that all of mathematics can be visualized or needs to be visualized can be disproved with one counter-example.

Jagella

Originally Posted by Jagella

But how can we visualize five dimensions? We have only three spatial dimensions to think in and to see in. I've never seen more than these three dimensions.

Who is we ?

I said that mathematicians tend to think visually. That applies not to just 5 dimensions, but in fact to spaces of arbitrary dimensions, including infinite-dimensional spaces. I do it all the time. So do other mathematicians. Mathematicians speak to one another in terms of visual imagery as a matter of course. It is the norm.

Originally Posted by Jagella

In any case, I think you may be missing my point. I'm not saying that mathematicians cannot think of much of mathematics as something they might see. What I'm pointing out is that much of math is unnecessary, difficult or impossible to visualize. It is an abstract language much like English. Can anybody visualize the fourth root of two? I don't think so. The best we can do is write it as 2^(1/4).

Wrong. It is a point a little to the left of 6/5 and is no more abstract or difficult to visualize than is 6/5.

Depending on the issue, I might adopt a different visual technique -- maybe as a sequence.

Originally Posted by Jagella

Also, visualization in mathematics is often unnecessary. As we both know, there are many techniques for solving quadratic equations. Only one of these techniques involves visualization.

Visualization is related to understanding. Mathematics itself is about understanding, not "solving". What you are demonstrating is one of the most common fundamental misunderstandings regarding mathematics.

Originally Posted by Jagella

I'd suggest you be careful when making all-inclusive statements. All anybody needs to do is come up with one exception to prove you wrong. Claiming that all of mathematics can be visualized or needs to be visualized can be disproved with one counter-example. Jagella

Originally Posted by DrRocket

There is very little, if anything, in mathemstics that is not subject to some mental picture. Just what that picture might be depends on the mathematician.

I'd suggest that you learn to read with comprehension, and not make strawman arguements. Particularly about a subject in which you quite clearly lack expertise.

Both salsaonline and I have told you that mathematicians visualize things that you claim are not subject to visualization. We have addressed specifics and I have provided examples. Both salsaonline and I are PhD mathematicians.

Do you also tell bumblebees that they cannot fly ?

Come-on guys, play nice.

I think all Jagella and oldfool are saying DrR, is that some people, wether mathematicians or not, use different means to understand a particular problem. Visualization is just one technique. And note that I'm not speaking for you or salsaonline; maybe a good grasp of mathematics as required for a PhD does require excellent visualization skills. I will agree with you that higher dimensional geometries and curved space/time do need some visualization to make them understandable, otherwise they are just 'playng with numbers'. You need to be able to relate to some aspects of the math to understand it. Otherwise we would still only have 3d Euclidian geometry.

Originally Posted by MigL

Come-on guys, play nice.

I think all Jagella and oldfool are saying DrR, is that some people, wether mathematicians or not, use different means to understand a particular problem. Visualization is just one technique.

I read pretty well, and that is not what either Jagella or oldfool said.

I agree with the remainder, which is quite consistent with what I said. However, it is also true that professional mathematicians, who might just be a class that understand mathematics a wee bit better than the average Joe, tend to think visually and discuss mathematics using visual imagery. So, yeah, it is one method. However, it is NOT "just one method". Most importantly, telling a former professional mathematician that he cannot do what he in fact does every damn day is the height of combined stupidity and arrogance.

I have no patience with incompetent amateurs who try to lecture professionals, and I don't fell even slightly bad about it.

Jagella and oldfool can come back and argue their case after they have completed a few years of study of mathematics in graduate school. made discoveries that constitute significant original contributions to mathematics, received a PhD, publiushed in peer-reviewed journals, lectured to professionals by invitation and put in a couple of years on the faculty of a major university.

"First, you must catch the tiger" -- Paladin

Originally Posted by DrRocket

Who is we ?

Any human who has ancestors who evolved in three spatial dimensions can only see and think in three spatial dimensions. We can use mathematics to describe more than three spatial dimensions, but we cannot visualize more than three spatial dimensions.

By the way, some physicists believe there may be as many as 11 spatial dimensions. If true, then the extra eight spatial dimensions are obviously hidden. We cannot see them although we can describe them.

Originally Posted by DrRocket

I said that mathematicians tend to think visually. That applies not to just 5 dimensions, but in fact to spaces of arbitrary dimensions, including infinite-dimensional spaces. I do it all the time.

Well, you might claim to see Bigfoot all the time too, but I don't know if I'd be too quick to believe you. I just want to know how to graphically represent four or more spatial dimensions. Two dimensions can be graphically represented using two perpendicular lines. Three dimensions can be graphically represented using three perpendicular lines (you'd fudge a bit by placing one of the lines at an angle that would show depth). Now, please explain how to use four lines to graphically represent four spatial dimensions. If you cannot do it, then concede my point that hyper-dimensions cannot be visualized.

Originally Posted by DrRocket

Wrong. It is a point a little to the left of 6/5 and is no more abstract or difficult to visualize than is 6/5.

I think I'm still right. Using a number line, 6/5 can be visualized easily enough because it is a rational number. You can just use tick marks to graphically represent fifths, and six of these marks would represent six fifths. The fourth-root of two, on the other hand, is an irrational number, and as such it would be impossible to use the above procedure to graphically represent it. All you can do is draw a point on a number line that is about where 2^(1/4) might be. An uninitiated observer would never be able to tell by looking at the point that the point you have drawn is the fourth-root of two. If you don't believe me, then just try it. I guarantee that any person, even an expert mathematician, won't he able to tell that the point is located at 2^(1/4).

Now, some irrational numbers can be graphically represented. You can use simple geometry to visualize the square-root of two. Draw a right triangle with two sides of length 1. The hypotenuse will have length sqrt(1^2 + 1^2) = sqrt(2).

Unfortunately, many other irrational numbers cannot be graphically represented.

Originally Posted by DrRocket

Visualization is related to understanding. Mathematics itself is about understanding, not "solving". What you are demonstrating is one of the most common fundamental misunderstandings regarding mathematics.

That's strange. I've been using mathematics to solve problems for decades! Earlier today, for instance, I used trigonometry and vector analysis to find the electric-field force at a point in an electric field.

Originally Posted by DrRocket

I'd suggest that you learn to read with comprehension, and not make strawman arguements.

I will be careful to do just that. Thanks for the advice.

Originally Posted by DrRocket

Particularly about a subject in which you quite clearly lack expertise.

Doc, don't you think it would be better to demonstrate your own expertise, stick to the issues, and refrain from personal attacks? That way we can advance math and science.

Originally Posted by DrRocket

Both salsaonline and I are PhD mathematicians.

That's great to hear. At which university did you get your doctorate? Did you study math education? How do you correct error(s)?

My own education involves my study of graphics. I have a diploma in graphic design that I earned at The Art Institute of Pittsburgh. I use my graphic-design skills to create diagrams and other graphics to visualize topics in physics and mathematics. As I have discovered, much of the work, such as the calculations, cannot be represented graphically. In those cases, I may type out the formulas on the graphic I'm creating.

If you have the time and the inclination, then I'd recommend that you try creating graphics this way. Adobe Illustrator and Photoshop are excellent tools for creating such work. The visual and non-visual aspects of mathematics then should be obvious to you.

Originally Posted by DrRocket

Do you also tell bumblebees that they cannot fly ?

Well, no, but I know bumblebees can fly because I've seen them fly. In the same way, I will believe four or more spatial dimensions can be visualized when I see them.

OK?

Jagella

Originally Posted by Jagella

Originally Posted by DrRocket

Who is we ?

Any human who has ancestors who evolved in three spatial dimensions can only see and think in three spatial dimensions. We can use mathematics to describe more than three spatial dimensions, but we cannot visualize more than three spatial dimensions.

By the way, some physicists believe there may be as many as 11 spatial dimensions. If true, then the extra eight spatial dimensions are obviously hidden. We cannot see them although we can describe them.

Originally Posted by DrRocket

I said that mathematicians tend to think visually. That applies not to just 5 dimensions, but in fact to spaces of arbitrary dimensions, including infinite-dimensional spaces. I do it all the time.

Well, you might claim to see Bigfoot all the time too, but I don't know if I'd be too quick to believe you. I just want to know how to graphically represent four or more spatial dimensions. Two dimensions can be graphically represented using two perpendicular lines. Three dimensions can be graphically represented using three perpendicular lines (you'd fudge a bit by placing one of the lines at an angle that would show depth). Now, please explain how to use four lines to graphically represent four spatial dimensions. If you cannot do it, then concede my point that hyper-dimensions cannot be visualized.

Originally Posted by DrRocket

Wrong. It is a point a little to the left of 6/5 and is no more abstract or difficult to visualize than is 6/5.

I think I'm still right. Using a number line, 6/5 can be visualized easily enough because it is a rational number. You can just use tick marks to graphically represent fifths, and six of these marks would represent six fifths. The fourth-root of two, on the other hand, is an irrational number, and as such it would be impossible to use the above procedure to graphically represent it. All you can do is draw a point on a number line that is about where 2^(1/4) might be. An uninitiated observer would never be able to tell by looking at the point that the point you have drawn is the fourth-root of two. If you don't believe me, then just try it. I guarantee that any person, even an expert mathematician, won't he able to tell that the point is located at 2^(1/4).

Now, some irrational numbers can be graphically represented. You can use simple geometry to visualize the square-root of two. Draw a right triangle with two sides of length 1. The hypotenuse will have length sqrt(1^2 + 1^2) = sqrt(2).

Unfortunately, many other irrational numbers cannot be graphically represented.

Originally Posted by DrRocket

Visualization is related to understanding. Mathematics itself is about understanding, not "solving". What you are demonstrating is one of the most common fundamental misunderstandings regarding mathematics.

That's strange. I've been using mathematics to solve problems for decades! Earlier today, for instance, I used trigonometry and vector analysis to find the electric-field force at a point in an electric field.

Originally Posted by DrRocket

I'd suggest that you learn to read with comprehension, and not make strawman arguements.

I will be careful to do just that. Thanks for the advice.

Originally Posted by DrRocket

Particularly about a subject in which you quite clearly lack expertise.

Doc, don't you think it would be better to demonstrate your own expertise, stick to the issues, and refrain from personal attacks? That way we can advance math and science.

Originally Posted by DrRocket

Both salsaonline and I are PhD mathematicians.

That's great to hear. At which university did you get your doctorate? Did you study math education? How do you correct error(s)?

My own education involves my study of graphics. I have a diploma in graphic design that I earned at The Art Institute of Pittsburgh. I use my graphic-design skills to create diagrams and other graphics to visualize topics in physics and mathematics. As I have discovered, much of the work, such as the calculations, cannot be represented graphically. In those cases, I may type out the formulas on the graphic I'm creating.

If you have the time and the inclination, then I'd recommend that you try creating graphics this way. Adobe Illustrator and Photoshop are excellent tools for creating such work. The visual and non-visual aspects of mathematics then should be obvious to you.

Originally Posted by DrRocket

Do you also tell bumblebees that they cannot fly ?

Well, no, but I know bumblebees can fly because I've seen them fly. In the same way, I will believe four or more spatial dimensions can be visualized when I see them.

OK?

Jagella

Ridiculous

You have simply demonstrated your total incompetence with respect to things mathematical. Better stick to graphic design.

BTW using a ruler and compass as you noted one construct the square root of 2, in principle, exactly (every schoolboy knows this). By iterating that construction twice one produces the fourth root of two exactly, which should be obvious but apparently not to you. It is no wonder that you cannot visualize mathematics, the answer is right there before your eyes but you are blind to it.

BTW there is no such thing as "the electric force at a point" only "the electric force on a charge (possibly moving) at a point", which is trivial to find if you know the electromagnetic field -- you just plug into the Lorentz force equation



So your abject incompetence extends to physics.

Do you have any more such foolish statements to make ?

Originally Posted by Jagella

I will believe four or more spatial dimensions can be visualized when I see them.

For visualize an line where each point has an embedded space.

For visualize a array where each point is an embedded space.

Visualize zooming in to the spaces as in a Serpenski triangle; http://mathforum.org/mathimages/inde...Sierp-zoom.gif.

Originally Posted by DrRocket

Ridiculous

You have simply demonstrated your total incompetence with respect to things mathematical. Better stick to graphic design.

which should be obvious but apparently not to you. It is no wonder that you cannot visualize mathematics, the answer is right there before your eyes but you are blind to it.

So your abject incompetence extends to physics.

Do you have any more such foolish statements to make ?

It is amazing what one must put up with when advancing the cause of science.

Originally Posted by DrRocket

BTW using a ruler and compass as you noted one construct the square root of 2, in principle, exactly (every schoolboy knows this). By iterating that construction twice one produces the fourth root of two exactly...

While I'm not sure how exactly I can apply the procedure you mention to find the fourth-root of two, I have thought of a very simple and elegant way to visually represent the fourth-root of two. Simply draw a square and label the area of that square as 2^(1/2). The length of each side of the square is the square-root of the square-root of two! That is, the length of each side equals 2^(1/4).

A visual representation of cube-roots can be had with drawing a cube. The length of each side of that cube is the cube-root of the volume of the cube. For instance, if the cube's volume is said to be 8, then the length of each side of the cube is 2 which is the cube-root of 8, of course.

Anyway, I hope you can see the usefulness of my approach to graphical representations of numbers. I don't just claim I can visualize these numbers—I demonstrate my ability. If it seems that such representation is impossible, then I make that assumption until proven otherwise.

Originally Posted by DrRocket

BTW there is no such thing as "the electric force at a point"...

Yes. I meant to say the electric-field intensity at a point.

Jagella

I think you're making the term visualize way too narrow. Visualization extends beyond what the average person can put on a piece of paper, or even model in clay. People only say that 4D (5D, hyperbolic, etc) space can't be visualized because they can't do it, so they assume no one can.

Originally Posted by Jagella

It is amazing what one must put up with when advancing the cause of science.

Yes. Fools like you.

Originally Posted by DrRocket

BTW using a ruler and compass as you noted one construct the square root of 2, in principle, exactly (every schoolboy knows this). By iterating that construction twice one produces the fourth root of two exactly...

Originally Posted by Jagella

While I'm not sure how exactly I can apply the procedure you mention to find the fourth-root of two,

You do it twice, as any school boy ought to be able to see.

Originally Posted by Jagella

I have thought of a very simple and elegant way to visually represent the fourth-root of two. Simply draw a square and label the area of that square as 2^(1/2). The length of each side of the square is the square-root of the square-root of two! That is, the length of each side equals 2^(1/4).

What this demonstrates is simply the relationship between a number and its square. Maybe that is why a regular rectangle is called a square. Duh.

This has absolutely nothing to do with the square root of two in particular.

What you have provided is simply another demonstration of your mathematical incompetence.

Originally Posted by Jagella

A visual representation of cube-roots can be had with drawing a cube. The length of each side of that cube is the cube-root of the volume of the cube. For instance, if the cube's volume is said to be 8, then the length of each side of the cube is 2 which is the cube-root of 8, of course.

And a regular solid rectangle is called a cube. Are you starting to see a pattern here ?

This tells you nothing about the cube root of 2 as opposed to the cube root of 8 (which a child will recognize as 2, so maybe you ought to ask a child).

(incompetence)

Originally Posted by Jagella

Anyway, I hope you can see the usefulness of my approach to graphical representations of numbers. I don't just claim I can visualize these numbers—I demonstrate my ability. If it seems that such representation is impossible, then I make that assumption until proven otherwise.

What you have demonstrated is a complete obliviousness to the principles of basic mathematics. You can't tell the difference between vision and hallucination.

Your assumption that something is impossible if it "seems impossible" to you must put quite a lot out of reach to someone with your obvious limitations. For most people that is a poor policy, but I can see why you adopt it -- the world must be very confusing to you and imposing such limits might be critical to survival.

Originally Posted by Jagella

Originally Posted by DrRocket

BTW there is no such thing as "the electric force at a point"...

Yes. I meant to say the electric-field intensity at a point.

Jagella

No, you said

Earlier today, for instance, I used trigonometry and vector analysis to find the electric-field force at a point in an electric field.

This implies that the electric field is given . Adding the word "intensity" yields nothing.

Thus utter incompetence in physics is added to equal incompetence in mathematics.

We also have

Originally Posted by Jagella

That's strange. I've been using mathematics to solve problems for decades!

Originally Posted by Jagella

My own education involves my study of graphics. I have a diploma in graphic design that I earned at The Art Institute of Pittsburgh. I use my graphic-design skills to create diagrams and other graphics to visualize topics in physics and mathematics

I don't know what you have been doing for decades, but if your posts are representative of your understanding of mathematics and science it is a wonder that you didn't starve to death. You must be very thankful for a gullible public.

This ought to be about content, not credentials, but since you choose to hang your hat on your "diploma" ----

In those decades I have completed undergraduate and graduate degrees in engineering, a PhD in mathematics, taught at three major universities, published research in engineering and mathematics at professional conferences and in peer-reviewed journals, presented invited talks, worked in the aerospace/defense industry in technical and executive management positions, participated in sending spacecraft to distant planets, worked on tactical and strategic (nuclear) weapon systems, developed advanced munitions, consulted for NASA and saved enough money to retire and try to help newbies learn something about mathematics despite the obstruction of fools like you.

I will not identify myself by naming educational institutions or companies.

salsaonline, whose statements you also seem to disregard is also a PhD mathematician with deep knowledge of physics and finance.

You are WAY out of your weight class.

Do you really mean to continue this ? Pee Wee Herman had the good sense not to climb into the ring with Mike Tyson.

Originally Posted by GiantEvil

For visualize an line where each point has an embedded space.

OK, I think I get the gist of what you're saying. You are describing a "hyper-line" that provides a fourth spatial dimension along which are an infinite number of 3D spaces. Your "line" is analogous to a line that might exist in one, two, or three spatial dimensions. Its "points" are the 3D spaces rather than the points of zero dimension that we are accustomed to and use to indicate position in space. Is that correct?

While I'm waiting for your response I can say that although this may be a useful description of 4D space, it's really not a true visualization of such a hyper-space. You are employing analogies from three dimensions to represent four dimensions. The line you describe can be represented as one of the three dimensions in three-dimensional space although it represents a fourth dimension. It's analogous to a painter's use of perspective to lend a look of depth to a painting--although smaller objects and larger objects may give a painting a sense of depth, it's still not true 3D because the canvas is flat and only has width and height. Actually, the attempt to visualize four dimensions is even more problematical than this painting analogy because although the canvas is flat, we live in a three-dimensional world and can easily visualize three dimensions and the depth the painter is representing in her painting. We have no such luxury when trying to imagine four spatial dimensions!

At any rate, I believe we may be making much ado about nothing. The power of mathematics is not limited to the physical world or what we can perceive in it. Mathematics is inherently abstract and as such can be used to describe as many dimensions as we wish. We need not necessarily concern ourselves with visualizing what we describe. It's like using a spreadsheet: although you can create charts out of your data, in many instances you can get along just fine without doing so. An accountant can calculate net income with numbers only which is the conventional practice.

Jagella

Originally Posted by MigL

Come-on guys, play nice.

You've got it!

Originally Posted by MigL

I think all Jagella and oldfool are saying DrR, is that some people, wether mathematicians or not, use different means to understand a particular problem. Visualization is just one technique.

That's essentially correct, but I would add that much of mathematics is inherently impossible to visualize. Like I have mentioned before, virtually all of mathematics, geometry included, is abstract. That which is abstract is by definition nonphysical, and if I know my optics, something needs to be physical in order to be seen. Mathematicians employ figures and diagrams to give some "look" to the concepts under study, but the visuals are merely representations of the concepts. They are in many cases rather poor representations of the concepts but are better than nothing. For instance, although the length of a side of a triangle equal to the square root of two can be illustrated, the square root of two itself cannot be visualized. We can only write it as 2^(1/2).

Good day!

Jagella

1 2 3 4 5 ...
Hey look, symbolic, visual representations of number...

Originally Posted by Jagella

That's essentially correct, but I would add that much of mathematics is inherently impossible to visualize. Like I have mentioned before, virtually all of mathematics, geometry included, is abstract. That which is abstract is by definition nonphysical, and if I know my optics, something needs to be physical in order to be seen. Mathematicians employ figures and diagrams to give some "look" to the concepts under study, but the visuals are merely representations of the concepts. They are in many cases rather poor representations of the concepts but are better than nothing. For instance, although the length of a side of a triangle equal to the square root of two can be illustrated, the square root of two itself cannot be visualized. We can only write it as 2^(1/2).
Good day!

Jagella

Bold added to emphasize self-contradictory statement.

Not only does that construction provide a concrete representation of , it revealed to the ancient world the existence of irrational numbers !

As usual, you presume to impose your personal inadequacies on the rest of humanity. Mathematicians have no such limitatiions.

What is worse, a child reading your drivel might believe it. That could stunt his/her intellectual growth. The worst case might result in a relatively bright child growing up to be as blind to real mathematics as you.

You need to get over your delusions of adequacy.

Originally Posted by DrRocket

What is worse, a child reading your drivel might believe it. That could stunt his/her intellectual growth. The worst case might result in a relatively bright child growing up to be as blind to real mathematics as you.

You need to get over your delusions of adequacy.

I am a regular reader of this forum, and although i am not a physics or mathematics expert, my studies of telecommunications engineering, specialized in electronics, gave me a good background in some aspects of boths physics and mathematics. But the most valuable thing that these years have given to me is the importance of recognizing my own limitations and lack of knowledge on this fields.
So i would suggest to Jagella (and many other alike users in this forum) to listen to DrRocket and other users who really know what they are talking about. Ok, DrRocket might not be the most "diplomatic" guy in the town :wink: , but he tells you the truth and provides you with very useful and reliable links if you are really interested in learning and that is something extremely important in physics and mathematics.
Sometimes very bad science is well camouflaged if you are not an expert. It's pretty obvious that he spends a lot of time here correcting and guiding people into the right direction and that is something most of us should be thankful for.

Originally Posted by Dani

Originally Posted by DrRocket

What is worse, a child reading your drivel might believe it. That could stunt his/her intellectual growth. The worst case might result in a relatively bright child growing up to be as blind to real mathematics as you.

You need to get over your delusions of adequacy.

I am a regular reader of this forum, and although i am not a physics or mathematics expert, my studies of telecommunications engineering, specialized in electronics, gave me a good background in some aspects of boths physics and mathematics. But the most valuable thing that these years have given to me is the importance of recognizing my own limitations and lack of knowledge on this fields.
So i would suggest to Jagella (and many other alike users in this forum) to listen to DrRocket and other users who really know what they are talking about. Ok, DrRocket might not be the most "diplomatic" guy in the town :wink: , but he tells you the truth and provides you with very useful and reliable links if you are really interested in learning and that is something extremely important in physics and mathematics.
Sometimes very bad science is well camouflaged if you are not an expert. It's pretty obvious that he spends a lot of time here correcting and guiding people into the right direction and that is something most of us should be thankful for.

Dani, you've got to be kidding! The guy is a crank, and an insulting one at that. He attacks people out of jealousy because they know more than he does. I've decided to no longer respond to him. Trolls deserve no attention.

In any case, may I suggest we get back to the issues on this thread? I'm not here to fight with egotists. I just want a chance to exchange ideas. We learn not from displaying how much we think we know but by discovering what we don't know.

Good day!

Jagella

Originally Posted by GiantEvil

1 2 3 4 5 ...
Hey look, symbolic, visual representations of number...

Giant, numerals are symbols used to describe numbers such as quantities. If you see three apples, do you see a 3? If you see a cat, do you see c-a-t? Maybe you do, but I don't! Mathematics is a language which like Spanish or Latin, uses written or spoken symbols to represent concrete entities such as three apples or a cat. These symbols in most cases look nothing like what they are supposed to represent and therefore cannot justifiably be called "visual."

So you're wrong about numerals being visual representations of numbers. Numerals are abstract symbols that we must learn through association. Elementary-school math teachers know this important fact about mathematics, and that's why they spend so much time drawing three apples and telling the kids that the number of those apples is "3." If the kids learn their lesson, then they know that the quantity of those apples is written as "3."

Got it?

Jagella

Originally Posted by Jagella

Originally Posted by GiantEvil

1 2 3 4 5 ...
Hey look, symbolic, visual representations of number...

Giant, numerals are symbols used to describe numbers such as quantities. If you see three apples, do you see a 3? If you see a cat, do you see c-a-t? Maybe you do, but I don't! Mathematics is a language which like Spanish or Latin, uses written or spoken symbols to represent concrete entities such as three apples or a cat. These symbols in most cases look nothing like what they are supposed to represent and therefore cannot justifiably be called "visual."

So you're wrong about numerals being visual representations of numbers. Numerals are abstract symbols that we must learn through association. Elementary-school math teachers know this important fact about mathematics, and that's why they spend so much time drawing three apples and telling the kids that the number of those apples is "3." If the kids learn their lesson, then they know that the quantity of those apples is written as "3."

Got it?

Jagella

I was resorting to a lame asse'd semantic argument to counter a lame asse'd semantic argument.
Seriously now, it would appear you lack the intellectual capacity or imagination to do something as simple as visualize the tangent line to a curve, or for that matter to visualize the tangent line to a curve on the background of a Cartesian grid.
Push all the symbols you want, without an active inner eye they have no power.

Originally Posted by GiantEvil

Originally Posted by Jagella

Originally Posted by GiantEvil

1 2 3 4 5 ...
Hey look, symbolic, visual representations of number...

Giant, numerals are symbols used to describe numbers such as quantities. If you see three apples, do you see a 3? If you see a cat, do you see c-a-t? Maybe you do, but I don't! Mathematics is a language which like Spanish or Latin, uses written or spoken symbols to represent concrete entities such as three apples or a cat. These symbols in most cases look nothing like what they are supposed to represent and therefore cannot justifiably be called "visual."

So you're wrong about numerals being visual representations of numbers. Numerals are abstract symbols that we must learn through association. Elementary-school math teachers know this important fact about mathematics, and that's why they spend so much time drawing three apples and telling the kids that the number of those apples is "3." If the kids learn their lesson, then they know that the quantity of those apples is written as "3."

Got it?

Jagella

I was resorting to a lame asse'd semantic argument to counter a lame asse'd semantic argument.
Seriously now, it would appear you lack the intellectual capacity or imagination to do something as simple as visualize the tangent line to a curve, or for that matter to visualize the tangent line to a curve on the background of a Cartesian grid.
Push all the symbols you want, without an active inner eye they have no power.

Well, Ok, Giant, believe what you wish. I tried to explain my position on this issue, and my position met with opposition. There's no sense in arguing.

I must say, though, that I have learned from this discussion. I have learned the following facts:

• 1. Mathematicians and scientists can be as illogical as anybody else.
  2. Many people have difficulties distinguishing between the abstract and the concrete.
  3. A lot of people don't seem to understand that mathematics is a symbolic language.
  4. I'm a heck of a lot smarter than most people.

Uh--just kidding on item #4.

Have a great day.

Jagella

Originally Posted by Jagella

I must say, though, that I have learned from this discussion. I have learned the following facts:

• 1. Mathematicians and scientists can be as illogical as anybody else.
  2. Many people have difficulties distinguishing between the abstract and the concrete.
  3. A lot of people don't seem to understand that mathematics is a symbolic language.
  4. I'm a heck of a lot smarter than most people.

Uh--just kidding on item #4.

Have a great day.

Jagella

This makes three things obvious:

1. You have learned nothing.

2. You are a slow learner.

3. You are delusional.

Originally Posted by Jagella

4. I'm a heck of a lot smarter than most people.

Not as smart as Ellatha was, but that chick/dude left in a huff. Too bad.

Jagella your definition of visualization is too narrow, you want to actually be able to see what we are discussing, and that is impossible in a lot of cases. What mathematicians and physicists do is visualise analogues, as Giant Evil did with his method of higher dimensional visualization. You take what you know and etend it to describe some of the properties of what you don't.
This doesn't just apply to math and higher dimensional geometries, but also to physics. I don't think anyone has ever seen or can visualize exactly an electron probability cloud around an atom. Or the waveparticle duality of a photon. Or the streamlines of air or fluid flow (can you see air ???).
Yet we know these things exist, and we use mental analogues or models based on everyday experiences or things to describe them. Some of us, obviously do it better than others, and get PhD.

I suggest you quit your bickering with DrR, he has no mercy and doesn't take prisoners. I suggest you listen to what he has to say, he's seldom wrong ( although I still don't agree with his description of a Faraday cage ), and you may learn quite a bit from him.

I have to agree with MigL. You are simply using a very narrow definition of visualization, to the point that you are straying into No True Scotsman territory.

With practice, I can visualize a 4D sphere pretty well. I can almost visualize a 4D cube, but rotating it around in my head is tricky. Just because I can't see it in real life, or draw it on paper (other than through various projections), doesn't mean I'm not visualizing it.

Originally Posted by MigL

he's seldom wrong ( although I still don't agree with his description of a Faraday cage ), and you may learn quite a bit from him.

????

Why don't you start a thread on the Faraday cage. If we can avoid the nut balls, they can be very easily handled with classical electromagnetics.

Originally Posted by MigL

Jagella your definition of visualization is too narrow, you want to actually be able to see what we are discussing, and that is impossible in a lot of cases. What mathematicians and physicists do is visualise analogues, as Giant Evil did with his method of higher dimensional visualization. You take what you know and etend it to describe some of the properties of what you don't.
This doesn't just apply to math and higher dimensional geometries, but also to physics. I don't think anyone has ever seen or can visualize exactly an electron probability cloud around an atom. Or the waveparticle duality of a photon. Or the streamlines of air or fluid flow (can you see air ???).
Yet we know these things exist, and we use mental analogues or models based on everyday experiences or things to describe them. Some of us, obviously do it better than others, and get PhD.

I suggest you quit your bickering with DrR, he has no mercy and doesn't take prisoners. I suggest you listen to what he has to say, he's seldom wrong ( although I still don't agree with his description of a Faraday cage ), and you may learn quite a bit from him.

Thanks for the reply, Mig.

After giving this whole issue some thought, I'm left wondering why I'm arguing about it. Is it really that important how people think about visualizing mathematical concepts? I think it's fair to say that different people literally see things differently, a fact I've learned from my graphic-design studies. Evidently some people believe they can see four-dimensional space. I suppose it isn't impossible, but I'm still wondering how it can be done. I can perceive width, height, and depth only. How additional dimensions can be perceived may be a question best left to neuroscientists.

I will take your advice and quit bickering with the trolls in this forum. They're like kids wanting attention. If they don't get that attention, then chances are they will cease and desist.

I look forward to exchanging ideas with you. A forum isn't for mischief: it's for sensible, knowledgeable people who wish to understand the relevant topics. If we cannot maintain that kind of dialogue, then I will leave.

Jagella

Originally Posted by MagiMaster

I have to agree with MigL. You are simply using a very narrow definition of visualization, to the point that you are straying into No True Scotsman territory.

With practice, I can visualize a 4D sphere pretty well. I can almost visualize a 4D cube, but rotating it around in my head is tricky. Just because I can't see it in real life, or draw it on paper (other than through various projections), doesn't mean I'm not visualizing it.

If you can perceive four spatial dimensions, then how can you see (or imagine) four, mutually perpendicular lines? As I'm sure you're aware, we can represent width, depth, and height using three mutually perpendicular lines, the x-y-z axes we are familiar with from calculus and functions of two, independent variables like z = f(x,y) = x + y. My TI-92 calculator can graph such functions, but higher-dimensional functions are not possible to display graphically.

Again, I don't wish to argue. I just want some coherent explanations.

Jagella

Originally Posted by Jagella

Originally Posted by MagiMaster

I have to agree with MigL. You are simply using a very narrow definition of visualization, to the point that you are straying into No True Scotsman territory.

With practice, I can visualize a 4D sphere pretty well. I can almost visualize a 4D cube, but rotating it around in my head is tricky. Just because I can't see it in real life, or draw it on paper (other than through various projections), doesn't mean I'm not visualizing it.

If you can perceive four spatial dimensions, then how can you see (or imagine) four, mutually perpendicular lines? As I'm sure you're aware, we can represent width, depth, and height using three mutually perpendicular lines, the x-y-z axes we are familiar with from calculus and functions of two, independent variables like z = f(x,y) = x + y. My TI-92 calculator can graph such functions, but higher-dimensional functions are not possible to display graphically.

Again, I don't wish to argue. I just want some coherent explanations.

Jagella

Simple.

Imagine any line in a 3-dimensional subspace of Euclidean 4-space and another line in the orthogonal complement that is perpendicular to it. They are orthogonal in exactly the same sense that two sides of a right angle are perpendicular on an ordinary sheet of paper.

BTW this is exactly what you attempt to do when you draw a 3-dimensional set of axes on the surface of a sheet of paper.

Originally Posted by Jagella

I will take your advice and quit bickering with the trolls in this forum. They're like kids wanting attention. If they don't get that attention, then chances are they will cease and desist.

DrRocket is not a troll.

It seems to me that you wanted to discuss whether mathematicians were able to visualize the problems they work on. I and DrRocket both asserted that visualization indeed plays an important role is mathematical problem-solving. Both of us have some experience as professional mathematicians, so we have good reason to make such an assertion.

You asked, we answered. I'm not sure there was really any more room for debate on this issue, unless you wanted to get into a philosophical discussion on the meaning of the word "visualization".

Originally Posted by Jagella

Originally Posted by MagiMaster

I have to agree with MigL. You are simply using a very narrow definition of visualization, to the point that you are straying into No True Scotsman territory.

With practice, I can visualize a 4D sphere pretty well. I can almost visualize a 4D cube, but rotating it around in my head is tricky. Just because I can't see it in real life, or draw it on paper (other than through various projections), doesn't mean I'm not visualizing it.

If you can perceive four spatial dimensions, then how can you see (or imagine) four, mutually perpendicular lines? As I'm sure you're aware, we can represent width, depth, and height using three mutually perpendicular lines, the x-y-z axes we are familiar with from calculus and functions of two, independent variables like z = f(x,y) = x + y. My TI-92 calculator can graph such functions, but higher-dimensional functions are not possible to display graphically.

Again, I don't wish to argue. I just want some coherent explanations.

Jagella

Seriously, reread that. I never said see or draw or represent graphically. I said visualize. Your definition of visualize and mine are apparently too different to have any meaningful conversation about the subject.

The point is I can hold some version of such an object in my head that's more than just an equation. I can't see it, since vision is 2D/3D, but I can mentally visualize it.

Originally Posted by Jagella

Originally Posted by Janus

One of the issues I see would be with the invariance of the speed of light. Einstein would first have to get Newton to accept the wave theory of light (Newton prefered the particulate theory).

Had the speed of light been measured in Newton's time? Also, did Einstein believe as we do today that light is both made up of particles and is also a wave?

Actually, Einstein is the one we credit with the duality. It was part of his description of the "photo-electric effect", which is the theory that actually won him his Nobel prize (he never got a Nobel prize for relativity).


Also the original experimental evidence for the invariance of the speed of light was the observation that the light emitted by binary star systems was not arriving later during the portion of the star's orbit when the star was moving away. (Otherwise their orbit would have appeared to be distorted by it.) If Einstein told Newton about that, I would think he would find it convincing. Then Newton could grab a telescope and start trying to see if it were true. (Except I'm not sure if telescopes were strong enough back then.)

Originally Posted by kojax

Originally Posted by Jagella

Originally Posted by Janus

One of the issues I see would be with the invariance of the speed of light. Einstein would first have to get Newton to accept the wave theory of light (Newton prefered the particulate theory).

Had the speed of light been measured in Newton's time? Also, did Einstein believe as we do today that light is both made up of particles and is also a wave?

Actually, Einstein is the one we credit with the duality. It was part of his description of the "photo-electric effect", which is the theory that actually won him his Nobel prize (he never got a Nobel prize for relativity).


Also the original experimental evidence for the invariance of the speed of light was the observation that the light emitted by binary star systems was not arriving later during the portion of the star's orbit when the star was moving away. (Otherwise their orbit would have appeared to be distorted by it.) If Einstein told Newton about that, I would think he would find it convincing. Then Newton could grab a telescope and start trying to see if it were true. (Except I'm not sure if telescopes were strong enough back then.)

Kojax, thanks so much for the relevant, intelligent, and polite reply. It's great to see that some of the members here do display a respect both for science and for people who wish to learn about it.

If I understand correctly, Einstein never received a Nobel Prize for relativity because much of the evidence that supports it was slow in coming. The folks in Scandinavia evidently don't wish to award prizes for work that turns out later to be less than credible.

It took me a while to understand why the speed of light is said to be "constant." I know what constant means, of course, but why is it such a big deal to say that the speed of light does not vary? Einstein realized that since the speed of light remains the same even in situations like you describe where distance does vary, then the logical conclusion is that time varies depending on the frame of reference of the observer. This insight on the part of Einstein is one of the most important points he might explain to Newton who believed that time is absolute.

I look forward to exchanging ideas with you in the future, Kojax.

Jagella

Originally Posted by salsaonline

Originally Posted by Jagella

I will take your advice and quit bickering with the trolls in this forum. They're like kids wanting attention. If they don't get that attention, then chances are they will cease and desist.

DrRocket is not a troll.

It seems to me that you wanted to discuss whether mathematicians were able to visualize the problems they work on. I and DrRocket both asserted that visualization indeed plays an important role is mathematical problem-solving. Both of us have some experience as professional mathematicians, so we have good reason to make such an assertion.

You asked, we answered. I'm not sure there was really any more room for debate on this issue, unless you wanted to get into a philosophical discussion on the meaning of the word "visualization".

If you are a "professional," then please conduct yourself as one. Defending childish mischief on the part of some of the members in this forum is not professional, in my opinion.

The discussion so far, in my view, isn't so much philosophical: it has to do with some very fundamental misunderstandings on the part of some people who call themselves mathematicians. If mathematics is your craft, then you better know what you're talking about.

Why is this point so important? I've worked a bit as a math educator in the past, and it should come as no surprise that many students struggle with math. I believe that the poor quality of math education is the reason for this problem. I see that many of the members here in this forum, including some who claim to be professional mathematicians, don't understand the difference between visual representation and association. Numbers are abstract concepts that cannot by definition be visualized. The curves used in the x-y plain to plot the values in a function are not visual representations of functions! Those curves are associations. That's a major error that some of the members here are making. This kind of error can lead to a lot of confusion and misunderstanding on the part of students.

At any rate, I plan to start another thread to discuss this important point hoping that I can get people to understand its significance.

Good day!

Jagella

Originally Posted by Jagella

Originally Posted by salsaonline

Originally Posted by Jagella

I will take your advice and quit bickering with the trolls in this forum. They're like kids wanting attention. If they don't get that attention, then chances are they will cease and desist.

DrRocket is not a troll.

It seems to me that you wanted to discuss whether mathematicians were able to visualize the problems they work on. I and DrRocket both asserted that visualization indeed plays an important role is mathematical problem-solving. Both of us have some experience as professional mathematicians, so we have good reason to make such an assertion.

You asked, we answered. I'm not sure there was really any more room for debate on this issue, unless you wanted to get into a philosophical discussion on the meaning of the word "visualization".

If you are a "professional," then please conduct yourself as one. Defending childish mischief on the part of some of the members in this forum is not professional, in my opinion.

salsaonline is indeed a professional. He is an accomplished young PhD specializing in algebraic geometry focusing on problems inspired by physics.

Your presumption to lecture him is ludicrous. You are simply a fool.

Originally Posted by Jagella

The discussion so far, in my view, isn't so much philosophical: it has to do with some very fundamental misunderstandings on the part of some people who call themselves mathematicians. If mathematics is your craft, then you better know what you're talking about.

Indeed.

Salsaonline and I are professionals. We do know what we are talking about.

You, in contrast, have no clue. Better take your own advice.

Originally Posted by Jagella

Why is this point so important? I've worked a bit as a math educator in the past, and it should come as no surprise that many students struggle with math. I believe that the poor quality of math education is the reason for this problem. I see that many of the members here in this forum, including some who claim to be professional mathematicians, don't understand the difference between visual representation and association. Numbers are abstract concepts that cannot by definition be visualized. The curves used in the x-y plain to plot the values in a function are not visual representations of functions! Those curves are associations. That's a major error that some of the members here are making. This kind of error can lead to a lot of confusion and misunderstanding on the part of students.

This is absurd.

You are woefully unqualified to be teaching mathematics to anyone.

There is a problem with poor quality of education. A former colleague, another PhD mathematician, has undertaken the task, on a statewide level in my state, to correct deficiencies in mathematics and science education.

The focus is on training teachers so that students are not exposed to people like you.

Originally Posted by Jagella

If you are a "professional," then please conduct yourself as one. Defending childish mischief on the part of some of the members in this forum is not professional, in my opinion.

The discussion so far, in my view, isn't so much philosophical: it has to do with some very fundamental misunderstandings on the part of some people who call themselves mathematicians. If mathematics is your craft, then you better know what you're talking about.

I think the real problem here is your fundamental misunderstanding of what it means to visualise a mathematical concept. Mathematicians do it all the time, and two professional mathematicians have, in this thread, described various methods they use to visualise certain mathematical concepts.

Not that he needs me to defend him, but DrRocket doesn't just call himself a mathematician - he is a fully fledged mathematician. He has a Ph.D in mathematical analysis, which means he hasn't just learned how to do maths, he has made a significant contribution to the subject of mathematical analysis itself!

There is no misunderstanding on his part in this thread, except maybe that he misunderstood your own personal definition of visualisation, and thought you wanted to know how real mathematicians visualise mathematical concepts - which is what he told you.

Don't worry about it DrR, it was when I first became a member. You were arguing that a Faraday cage can shield EM fields while I argued that fields cannot be shielded but the arrangement of the charges negates the effects of the field.

I suppose it can be viewed as semantics or differing points of view since the effect is the same.

Originally Posted by SpeedFreek

I think the real problem here is your fundamental misunderstanding of what it means to visualise a mathematical concept.

Speed, by definition a concept cannot be visualized. Can you see happiness? How about value? Concepts are abstract and as such are intangible. If I know my optics, to see something it must reflect light. Immaterial things cannot reflect light.

Originally Posted by SpeedFreek

Mathematicians do it all the time, and two professional mathematicians have, in this thread, described various methods they use to visualise certain mathematical concepts.

It's not correct to say that mathematicians visualize numbers and other concepts in mathematics. Numbers are abstract ideas. What mathematicians do is they associate concepts with pictures. Those pictures, such as curves in the x-y plane, in no way are visualizations of functions and the numbers they output. Those curves are associated with their related functions through arbitrary rules created by René Descartes in the 17th Century. If René Descartes had chosen different rules to govern his coordinate system, then the curves might look very different. That's why it is not correct to say that some mathematics can be subject to visualization.

Your position isn't hopeless, though, and I don't totally disagree with it. Mathematicians can make use of some visual representations. We can picture quantities of apples, for instance. We can draw a tall building and label its height as 500 feet. What makes visual representation possible is the application of mathematics to tangible objects. Tangible objects can be seen, of course, and it is meaningful to employ the use of visualization to such applications. That is, you can draw a picture of a car, but you can't draw a picture of ln(e).

Got it?

Jagella

Originally Posted by Jagella

That is, you can draw a picture of a car, but you can't draw a picture of ln(e).

Got it?

Jagella

ln(e)=1

One is pretty easy to visualize.

Make a fist. Extend your middle finger.

You are making a complete fool of yourself. That is pretty easy though, given the material that you have to work with.

Originally Posted by Jagella

Originally Posted by SpeedFreek

I think the real problem here is your fundamental misunderstanding of what it means to visualise a mathematical concept.

Speed, by definition a concept cannot be visualized. Can you see happiness? How about value? Concepts are abstract and as such are intangible. If I know my optics, to see something it must reflect light. Immaterial things cannot reflect light.

Originally Posted by SpeedFreek

Mathematicians do it all the time, and two professional mathematicians have, in this thread, described various methods they use to visualise certain mathematical concepts.

It's not correct to say that mathematicians visualize numbers and other concepts in mathematics. Numbers are abstract ideas. What mathematicians do is they associate concepts with pictures. Those pictures, such as curves in the x-y plane, in no way are visualizations of functions and the numbers they output. Those curves are associated with their related functions through arbitrary rules created by René Descartes in the 17th Century. If René Descartes had chosen different rules to govern his coordinate system, then the curves might look very different. That's why it is not correct to say that some mathematics can be subject to visualization.

Your position isn't hopeless, though, and I don't totally disagree with it. Mathematicians can make use of some visual representations. We can picture quantities of apples, for instance. We can draw a tall building and label its height as 500 feet. What makes visual representation possible is the application of mathematics to tangible objects. Tangible objects can be seen, of course, and it is meaningful to employ the use of visualization to such applications. That is, you can draw a picture of a car, but you can't draw a picture of ln(e).

Got it?

Jagella

Why do you insist on equating visualize and see? Are you saying you really can't concieve of anything beyond what you can see with your own two eyes? I doubt anyone who couldn't make it beyond that would make a good mathematician, or scientist, or teacher.

I'll be blunt. If your students were stuggling with math, part of the blame rests with you. It's actually a little upsetting to think of the people out there that are going to grow up hating math because of that.

Originally Posted by MagiMaster

Why do you insist on equating visualize and see? Are you saying you really can't concieve of anything beyond what you can see with your own two eyes?

I can conceive of abstract concepts, of course, but I don't normally claim to see them or imagine to see them. To do so would be misleading, in my opinion.

Originally Posted by MagiMaster

I doubt anyone who couldn't make it beyond that would make a good mathematician, or scientist, or teacher.

I seem to do just fine. I should point out that I am a graphic designer and illustrator who makes extensive use of pictures, diagrams, and schematics in my work. I just don't make them out to be what they are not. Function graphs, like I've mentioned earlier in this thread, are visual aids I use to examine the values output by functions. It is nonsense to say that the curve on the x-y plane is a “visualization” of its related function! The curve is a geometric object that is associated with its related function by arbitrary rules created by mathematicians. It is not a product of what you imagine the function to look like.

OK? I don't know why that principle is so hard to understand.

Originally Posted by MagiMaster

If your students were stuggling with math, part of the blame rests with you. It's actually a little upsetting to think of the people out there that are going to grow up hating math because of that.

Actually, most of my students did very well in their courses, or at least they improved their marks as a result of my tutoring them. I saved at least one student from failing. In most cases they asked me to tutor them because I had a good reputation for understanding the course material. In addition to math and science, I tutored economics and accounting courses.

I believe my insights along with my ability to clearly explain difficult material enabled me to help students learn.

Jagella

If what you say is true, then my only contention is that you are using a very, very narrow definition of visualize, a point which you seem to refuse to acknoledge. What would you call it when I can imagine a 4D sphere in a pseudo-visual way, if not visualizing?

Actually, other than having a different definition of visualize from everyone else, I'm not sure what your point is.

Originally Posted by Jagella

OK? I don't know why that principle is so hard to understand.

Oh, and it's hard to understand because it makes no sense. You say things like "arbitrary rules" without giving any real examples or properly defining what you mean. Again, it mostly comes down to you using a non-standard definition of visualize.

Originally Posted by Jagella

I believe my insights along with my ability to clearly explain difficult material enabled me to help students learn.

Jagella

I am quite certain that you believe that statement. That is the problem. You don't understand that you don't understand. You would be quite destructive in a classroom, without recognizing that fact.

I have been on the receiving end, in universities, of the products of "teachers" like you. It isn't pretty.

Originally Posted by MagiMaster

If what you say is true, then my only contention is that you are using a very, very narrow definition of visualize, a point which you seem to refuse to acknoledge.

Well, your idea of “narrow” is very subjective. What do you want me to acknowledge? That you think my definition of visualize is narrow?

In any case, there's a big difference between creating a picture of what can be seen and what cannot be seen. In the latter case, anything goes. Religious people might use a dove or a lamb to picture God. I don't think they believe he looks like either one, but the idea there is association rather than visualization. They associate God with animals that symbolize their beliefs. It's a somewhat similar situation for mathematics (see below).

Originally Posted by MagiMaster

What would you call it when I can imagine a 4D sphere in a pseudo-visual way, if not visualizing?

I don't know what you can imagine, but if I did know you can “ imagine a 4D sphere in a pseudo-visual way,” then I'm sure I'd be very impressed.

Originally Posted by MagiMaster

Actually, other than having a different definition of visualize from everyone else, I'm not sure what your point is.

I'm sorry if I haven't made my point clear, but my position is that it is very important for mathematicians to know their own craft especially when educating students. Students can become confused if they don't understand that the pictures mathematicians use are merely visual aids rather than actual images of what the concepts “look like.” I should know because I was one of those students. I struggled to understand math, and it wasn't until I really started thinking about the concepts that I began to understand them.

Originally Posted by MagiMaster

Oh, and it's hard to understand because it makes no sense.

What makes no sense? You don't know what association means? You don't know that mathematicians create rules to apply math to given problems? Please ask, and I promise to answer all your questions.

Originally Posted by MagiMaster

You say things like "arbitrary rules" without giving any real examples or properly defining what you mean.

The arbitrary rules I mentioned are the rules that govern the Cartesian coordinate system, the x-y plane. Positive x-values are plotted to the right of the y-axis. Why not plot positive values to the left of the y-axis? For that matter, why not plot positive y-values below the x-axis rather than above it? The x-axis is viewed as width—why not view it as depth like we do when platting values in 3D?

The answers to these questions are because mathematicians like René Descartes chose those rules! Descartes could have chosen differently, and if he did, those function curves would look very different from what we are accustomed to. So just like Christians making arbitrary choices to picture God—an intangible entity if he exists—mathematicians use arbitrary methods to picture their own intangible concepts.

Again, I don't know why I keep needing to explain all this. Is it bias, perhaps?

Jagella

Originally Posted by Jagella

Again, I don't know why I keep needing to explain all this. Is it bias, perhaps?

Jagella

The situation is quite simple.

The basics have been presented to you by several people, including two professionals.

You are apparently intellectually incapable of comprehending the facts. That would make you a menace in a classroom. Fortunately a "diploma in graphic design" (whatever that is) would not permit you to teach mathematics in most school systems. Thank God.

Was Einstein around long enough to see (and understand) QED in its final form? If not, then I would say his understanding of light was incomplete.

Originally Posted by Jagella

Originally Posted by MagiMaster

If what you say is true, then my only contention is that you are using a very, very narrow definition of visualize, a point which you seem to refuse to acknoledge.

Well, your idea of “narrow” is very subjective. What do you want me to acknowledge? That you think my definition of visualize is narrow?

In any case, there's a big difference between creating a picture of what can be seen and what cannot be seen. In the latter case, anything goes. Religious people might use a dove or a lamb to picture God. I don't think they believe he looks like either one, but the idea there is association rather than visualization. They associate God with animals that symbolize their beliefs. It's a somewhat similar situation for mathematics (see below).

Originally Posted by MagiMaster

What would you call it when I can imagine a 4D sphere in a pseudo-visual way, if not visualizing?

I don't know what you can imagine, but if I did know you can “ imagine a 4D sphere in a pseudo-visual way,” then I'm sure I'd be very impressed.

Originally Posted by MagiMaster

Actually, other than having a different definition of visualize from everyone else, I'm not sure what your point is.

I'm sorry if I haven't made my point clear, but my position is that it is very important for mathematicians to know their own craft especially when educating students. Students can become confused if they don't understand that the pictures mathematicians use are merely visual aids rather than actual images of what the concepts “look like.” I should know because I was one of those students. I struggled to understand math, and it wasn't until I really started thinking about the concepts that I began to understand them.

Originally Posted by MagiMaster

Oh, and it's hard to understand because it makes no sense.

What makes no sense? You don't know what association means? You don't know that mathematicians create rules to apply math to given problems? Please ask, and I promise to answer all your questions.

Originally Posted by MagiMaster

You say things like "arbitrary rules" without giving any real examples or properly defining what you mean.

The arbitrary rules I mentioned are the rules that govern the Cartesian coordinate system, the x-y plane. Positive x-values are plotted to the right of the y-axis. Why not plot positive values to the left of the y-axis? For that matter, why not plot positive y-values below the x-axis rather than above it? The x-axis is viewed as width—why not view it as depth like we do when platting values in 3D?

The answers to these questions are because mathematicians like René Descartes chose those rules! Descartes could have chosen differently, and if he did, those function curves would look very different from what we are accustomed to. So just like Christians making arbitrary choices to picture God—an intangible entity if he exists—mathematicians use arbitrary methods to picture their own intangible concepts.

Again, I don't know why I keep needing to explain all this. Is it bias, perhaps?

Jagella

Maybe you can understand why the rest of us also feel confused as to why we need to keep explaining this.

Yes, I wanted you to acknowledge that I had commented on the narrowness of your definition of visualize. Previously, you had completely ignored that in any of my replies.

When you say visualize, you seem to mean something much different than what most people mean when they say visualize. Despite the common root, visualize does not mean see.

That said, despite the arguments over the meaning of visualize, what you describe about understanding the concepts is more akin to what I'd call visualizing. Of course the picture on paper is just a picture. Students get hung up on things like that (and more) all the time.

Also, your comments on different ways to plot a simple 2D function suggest you are still hung up on some of these same issues. When I visualize a function like , there's no such thing as left, right, up and down; just more or less x and more or less y. And when you actually draw it on paper, it doesn't change just because you've rotated or flipped the paper.

The cartesian coordinate sysytem, BTW, isn't arbitrary.

Originally Posted by renzhee

Was Einstein around long enough to see (and understand) QED in its final form? If not, then I would say his understanding of light was incomplete.

Feynman and Schwinger presented their formulations of QED in 1948. Einstein died in 1955. I have no idea what all Einstein understood. He was a fairly bright guy.

EVERYBODY'S understanding of light is incomplete. Even Dick Feynman's. Some are just more incomplete than others.

Originally Posted by MagiMaster

The cartesian coordinate sysytem, BTW, isn't arbitrary.

You must be thinking of that Euclidean thing. Keep it VERY simple if you expect Jagella to get it.

Kinda like talking to a log, no ?

I wonder what he/she/it would do with a piece of log-log graph paper.

Originally Posted by Jagella

Those pictures, such as curves in the x-y plane, in no way are visualizations of functions and the numbers they output.

Wrong fool.

A real-valued function of a real variable is BY DEFINITION the subset of the Cartesian plane depicted by its graph.

You quite obviously do not even know the rigorous definition of a function.

Jagella,
the Collins Concise Dictionary defines visualize (visualise) as to form a mental image of (something incapable of being viewed or not at the moment visible).

So what do we have at present?

1. A reputable dictionary disagrees with you.
2. A number of fourm members with mathematical backgrounds disagree with you.
3. Individuals with substantial language skills disagree with you.

Do you think it is possible that your usage of visualise is seriously non-standard?

Originally Posted by renzhee

Was Einstein around long enough to see (and understand) QED in its final form? If not, then I would say his understanding of light was incomplete.

........This is exactly like Salsaonline's post few pages earlier on this thread lol

Originally Posted by MagiMaster

When you say visualize, you seem to mean something much different than what most people mean when they say visualize. Despite the common root, visualize does not mean see.

Well, let's take a look at the definition from Webster.com:

visualize - to see or form a mental image of

Evidently, to visualize can mean to literally see or to imagine. Even if you form a mental image, though, you still are forming a picture in your mind. If you are picturing something that has no actual way to be seen, you are associating some image with that thing wherein the image is related by some idea to the thing rather than looking like the thing. I believe there is an important distinction here that should be emphasized: things that can be seen can be visualized if only in the mind, and things that cannot be seen can only have some image related to them by some kind of association. The association may be some kind of rule developed by mathematicians.

Originally Posted by MagiMaster

Of course the picture on paper is just a picture. Students get hung up on things like that (and more) all the time.

And that's the whole problem right there. We should be careful not to mislead students with the pictures we use to explain the concepts. The pictures are merely visual aids rather than actual images of the concepts and principles that make up mathematics.

Originally Posted by MagiMaster

When I visualize a function like y = x^2, there's no such thing as left, right, up and down; just more or less x and more or less y.

What?

Originally Posted by MagiMaster

The cartesian coordinate sysytem, BTW, isn't arbitrary.

That's news to me. You mean Descartes had no choice in the matter? Is his famous coordinate system a result of some inviolable law of nature?

Jagella

Originally Posted by Jagella

Originally Posted by MagiMaster

The cartesian coordinate sysytem, BTW, isn't arbitrary.

That's news to me. You mean Descartes had no choice in the matter? Is his famous coordinate system a result of some inviolable law of nature?

Jagella

Can you think of a simpler way to represent the spatial relationship between two points in either two or three dimensions? There are, of course, different ways to represent that relationship - using a radial coordinate system for instance, but the coordinate system used should not affect the actual relationship between the two points in any way.

Sure, a straight line plotted in one coordinate system might translate into a curve when plotted using a different coordinate system, but does that affect the invariant relationship between the points in question, or does it only affect how you visualise that relationship?

If you understand the coordinate system you are using, your visualisation of that plotted relationship is just valid as when using any other coordinate system. You can visualise a relationship in a number of different ways, but they all mean the same thing.

I am by no means a mathematician, but I can, for instance, visualise the history of the expansion of the universe, using three different coordinate systems depending on what aspect of that expansion I am interested in (how's that for visualising a concept, something you cannot see?). All three different "coordinatizations" say exactly the same thing, the only difference between them is that they emphasise different parts of that relationship in greater detail.

Ooops! I thought I was in the Physics Forum, but I seem to be stuck in the Definitions Forum.

Originally Posted by Ophiolite

Do you think it is possible that your usage of visualise is seriously non-standard?

So is his definition of "function" which he asserts one cannot visualize as a graph.

For those few who may not have been exposed to mathematics beyond elementary school (like Jagella) here is the modern rigorous definition of a function:

Definition Let be sets. A function , from to is a subset such that, given , there exists a unique so that . is callled the domain of , is called the codomain of and is called the range of . If we write

So, if we consider a real-valued function of a real variable, the "curve" that represents f, called the graph of f, is precisely the subset of that is f. The modern definition of a function simply equates a function with its graph. Visualize the graph and you have visualized the function.

Originally Posted by SpeedFreek

Can you think of a simpler way to represent the spatial relationship between two points in either two or three dimensions?

No, and if I could, I suppose I'd make a name for myself. However, you didn't claim that the Cartesian Coordinate System is the simplest way to plot a function's graph: you said it is not arbitrary. How can it not be arbitrary when we can decide to plot graphs differently?

Originally Posted by SpeedFreek

Sure, a straight line plotted in one coordinate system might translate into a curve when plotted using a different coordinate system, but does that affect the invariant relationship between the points in question, or does it only affect how you visualise that relationship?

It affects the look of the graph, of course, but that's what I'm arguing. By deciding to use different graphing rules we make the function's graph look different. Hence, the picture we create depends largely on what rules we decide to employ to create the graph. The “visualization” is then very much a product not of any inherent “look” the function has but of how we decide to display it.

Originally Posted by SpeedFreek

I am by no means a mathematician, but I can, for instance, visualise the history of the expansion of the universe, using three different coordinate systems depending on what aspect of that expansion I am interested in (how's that for visualising a concept, something you cannot see?).

But I suppose you could see the expansion of the universe if you lived for the last 13 billion years. And the different views you might have with different coordinate systems just goes to show that there is no absolute perspective on the world—it depends on the observer. Isn't that what relativity and quantum mechanics have been telling us all along?

Good day!

Jagella

Originally Posted by Jagella

It affects the look of the graph, of course, but that's what I'm arguing. By deciding to use different graphing rules we make the function's graph look different. Hence, the picture we create depends largely on what rules we decide to employ to create the graph. The “visualization” is then very much a product not of any inherent “look” the function has but of how we decide to display it

All of which is completely irrelevant.

You still have not learned what a function is.

But here you are argung the visualization of something, the definition of which is apparently a complete mystery to you.

Originally Posted by Jagella

Originally Posted by SpeedFreek

Can you think of a simpler way to represent the spatial relationship between two points in either two or three dimensions?

No, and if I could, I suppose I'd make a name for myself. However, you didn't claim that the Cartesian Coordinate System is the simplest way to plot a function's graph: you said it is not arbitrary. How can it not be arbitrary when we can decide to plot graphs differently?

Well, I was replying to your statement asking whether Descartes coordinate system was the result of some inviolable law of physics, it was MagiMaster who said it was not arbitrary.

But my question is whether the simplest coordinate system is an arbitrary choice. I think not.

Originally Posted by Jagella

But I suppose you could see the expansion of the universe if you lived for the last 13 billion years. And the different views you might have with different coordinate systems just goes to show that there is no absolute perspective on the world—it depends on the observer. Isn't that what relativity and quantum mechanics have been telling us all along?

If you lived for the last 13.7 billion years, you would see the expansion of the universe from only one of those perspectives - that of proper distance. The other two coordinate systems are impossible to observe, as one involves invoking the cosmological principle in order to consider an idealised global overview using a series of observers, co-moving with the expansion but all individually at rest in relation to it, and the other involves normalising the rate of expansion at the particle horizon to c, which transforms all the curves in the other two systems into straight lines! But they all tell the same story, so they are all as valid as each other.

I am, perhaps, an example of exactly how visualisation can aid in the understanding of a concept. I can do no mathematics much higher than simple arithmetic and yet I have gained a very good understanding of a complicated and sophisticated mathematical model using visualisations of our light-cone, the co-moving worldlines of the contents of the universe, and various cosmological horizons, where they all intersect in terms of various time and distance measures.