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