The Yin/Yang of my math brain being turned on.

Right now I am struggling on the “definitions” part of my paper; I don’t know how much to define and to what precision to define it. For those are interested, I am trying to do something like this: (for those not interested, just skip this section)

imagine a solid donut in space. We call such things a “solid torus”. You can imagine one that is “not knotted”:


Now imagine, say, a simple string (circle) that goes “once around inside the donut”. You want the string to be situated in such a way that if you remove it (just the string, as if it were part of a mould), you’d get a simple thickening of 2-dimensional surface (say, imagine a really thick inner tube).

The problem is, do you get the same thing when you remove every “unknotted” string that “goes once around?” (for you experts: “goes once around” means “index one”, which is a stronger condition than having “winding number one”. )

As you can see, “unknotted, once around” curves can, at least, “look” different from one another.


It is a mathematical theorem that, at least “topologically”, one gets the same result proved the removed curve is “smooth” (has a tangent vector at every point as defined in a calculus class), is unknotted (in the case when the solid torus is unknotted) and has winding number 1. But that is a theorem that requires proof and has been rigorously proved.

So do I assume that my readers just “know” this, or do I reference this proof?

Anyway, when one writes a math article for a specialty journal, one has to make a lot of decisions of this type.

The Yin
When I get in “math mode”, it is difficult for me to have a conversation with another human being. For one, people are somewhat ambiguous and imprecise in their speech (and, frankly, in most cases, SHOULD be). And of course, most people (myself included) discuss matters that we do NOT have expert knowledge on. Unfortunately, many (most) don’t know that there IS a difference between their knowledge level and expert knowledge level; most suffer from a Dunning-Kruger effect to a degree. The upshot is that when I am in math mode, I become insufferably pedantic.

The Yang
As I said, effective human communication is often imprecise. So when people try to tell me something, I sometimes ask “ok, what do you mean by that?” It is almost as if I’ve become a high level language compiler. And, oft-times, I really did NOT know “what they meant by that”! So, in this regard, this mode is helpful.

Back to work.


March 21, 2013 - Posted by | mathematics, Personal Issues, social/political | , ,

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