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 ?