# Where to have discussions about intuition?

As a beginning math major, I'll have access to professors for seamless discussion of concepts and the underlying motivation. However, time for this will be limited. I'm wondering whether a Math.SE question, together with a "soft-question" tag, is appropriate for soliciting discussions about my understanding of concepts like the following:

"Consider $f(x) = x^2$ for all $x\in\mathbb{R}$. We call $f$ a relation (specifically, a function) because it relates elements of the domain $\mathbb{R}$ to elements of the range, a subset of $\mathbb{R}$. We have $f = \{(x,f(x)) \in\mathbb{R^2}: \forall x\in\mathbb{R}\ (f(x) = x^2)\}$. Am I understanding the motivation for relations, and did I formulate $f$ correctly?"

Should this kind of question only be asked in the chat? Physics Forums encourages more discussion, but it doesn't seem as active as Math.SE.

• math.se is not a discussion forum. So questions should be (for example) as suggested by Milo, and not asking for discussion or opinion. Mar 11, 2016 at 16:06

I don't see any reason one can't ask questions about intuition - understanding concepts intuitively is, after all, a big part of doing math. However, it's worth noting that the question

Did I formulate $f$ correctly?

Is very easy for us to answer, since we can see how you formulated $f$. However, something like

Am I understanding the motivation for relations?

Is much harder, since we don't really know how you're understanding them - and, ultimately, this question is asking about something about yourself that we can't see - and even then, it's not a very specific question, since the motivation depends heavily on the field, and other factors you haven't written.

I don't really know what you're after, but there's a number of things one can ask about intuition. One would be about applications:

I have seen that one can refer to such an $f$ as a relation because it takes elements of the domain to some elements of the range. However, it's not clear to me why we study the more general notion of a relation, when it seems that most of the result in my real analysis class regard functions only. What motivates mathematicians to study relations instead of just functions? Do they simplify or generalize the proof of any common analysis facts?

Or about the meaning of things:

I have seen homotopies defined as a continuos map $H:X\times [0,1]\rightarrow Y$ where we say two functions $h_0$ and $h_1$ are homotopic if $h_0(x)=H(x,0)$ and $h_1(x)=H(x,1)$ for some $H$. However, it's not clear to me what the condition of requiring this map $H$ between them is. What is the purpose of including the $H(x,a)$ with $0<a<1$? How can I visualize the meaning of homotopy?

The main thing is to be very specific and to make sure the question really is about mathematics rather than about yourself.

• "$\le,\, =,\, \subseteq,\, |$" are all relations. Mar 14, 2016 at 17:14
• @StevenGregory: You need to be careful about that... "$=$" and "$\subseteq$" cannot be relations in ZFC unless you want a contradiction to pop up. Mar 23, 2016 at 12:09
• @user21820 - which means that it is worthwhile to study them Mar 23, 2016 at 16:50
• @StevenGregory: Worthwhile to be more precise, you mean. They are symbols, with intended meaning, but cannot be defined in the system itself, and only described externally in the meta-system (which starts of as a natural language like English). Mar 23, 2016 at 16:56
• @user21820 “Well, art is art, isn't it? Still, on the other hand, water is water! And east is east and west is west and if you take cranberries and stew them like applesauce they taste much more like prunes than rhubarb does. Now, uh... now you tell me what you know.” - Groucho Marx Mar 23, 2016 at 20:52
• @StevenGregory: Haha sorry I don't get the joke. Mar 24, 2016 at 4:48