161–170 of 229 posts
I suggest that should one strive to 'fine tune' notation N to possess the above two qualities for a family of objects in X, other categories of objects in X will become opaque and difficult to express, i.e. a domain specific notation.
Earlier quoted context omitted.
I don't think this is true in general. It may be true for a novice, who needs all the available help to keep them rigorous (but even then, there is definitely room for reading-to-build-intuition), but symbols definitely slow you down while you translate them.
As a math PhD I disagree. I can read math notation far faster and more accurately than ambiguous English. We don’t translate symbols. If anything, when reading English we have to translate into symbols. For example, reading 5-7, I don’t have to translate the - symbol to the word “subtract”. I know this is -2. And I don’t translate the - in that to the word “negative,” and certainly not to the word “subtract”. And it’…
Earlier quoted context omitted.
Are you sure it’s not the opposite? Plenty of students can solve a set of equations. But when you start asking them about trains speeding in the night...
I think this is actually proof that students are treating math as a set of rules applied to a process, and not having any real understanding of how the variables and equations they’re mechanistically applying rules to have any connection to the real world. If they did, it would be a lot easier to talk about trains speeding through the night!
A person can wonder if it would be better to start with understanding. (Personally, I only briefly taught that age but my oracle for how to teach it, Mr Barton, does say that experience shows you have to have down the class's ability to mechanically solve before you introduce the trains. The other order, apparently, doesn't work when you try it with actual students in practice.)
Is it just me, or does probability theory in general have fairly terrible notation? Ambiguity between random variables and their distributions because of them simply being distinguished by being upper-case or lower-case, writing likelihood functions alternatively with an L() or p(), and using p() (with different arguments) to refer to different probability distributions. Perhaps I'm just having such a difficult time…
Earlier quoted context omitted.
That's not a genuine example. Most academic texts I've read will use something like s = d/t And sometimes not even explain what the components mean, because obviously 't' stands for time. That's what all their lecturers used, so why bother explain it. Some will even make up their own notation: s = d(t) And somewhere will say "f(x) in this article describes an inverse multiplicative relation", without explaining that…
>s = d/t Slightly worse than that. That equation is always written as v = s/t. v represents velocity and s represents distance, for some reason.
Earlier quoted context omitted.
That's not a genuine example. Most academic texts I've read will use something like s = d/t And sometimes not even explain what the components mean, because obviously 't' stands for time. That's what all their lecturers used, so why bother explain it. Some will even make up their own notation: s = d(t) And somewhere will say "f(x) in this article describes an inverse multiplicative relation", without explaining that…
>s = d/t Slightly worse than that. That equation is always written as v = s/t. v represents velocity and s represents distance, for some reason.
Earlier quoted context omitted.
Do you really think > speed = distance / time is less clear than > speed is the ratio of distance over time ? To me the first equation is genuinely easier to read for the purposes of understanding, not just for formal manipulation. For larger equations the difference is only more stark, not less. I have a maths PhD so your comment about symbols being for novices doesn't apply. I suppose the beauty of the first equati…
But that example is a little bit artificial, isn't it? A lot of mathematical concepts are more complex than that and sometimes symbols are not the best option. Say, for example the definition of Hausdorff space, in words and symbols: - Any two distinct points in the space have disjoint neighbourhoods. - ∀x,y ∈ X with x ≠ y, ∃U,V ⊂ X s.t. x ∈ U, y ∈ V and U ∩ V = ∅. Another example would be Navier-Stokes equations, wh…
Mathematical notation is great at facilitating formal manipulations. This is its critical feature, and without it we would get stuck at the level of ancient mathematics. This is the reason it was invented a few hundred years ago in the first place. That said, I find that notation is often abused in texts as a mere substitute for the normal human language which, while allowing to compress the text, does in fact nothin…
Math symbols are a minor issue for me. What confuses me the most are descriptions of mathematical concepts. For example, Wikipedia describes a 'field' like this: "In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do." It doesn't make sense to me. What does it mean if an operation 'is defined…
In the first place, what you quoted is not a formal, precise definition; it is not a substitute for such a definition, nor is it intended to be one. The Wikipedia page you mention has a precise definition further down the page.
So what, then, is the purpose of the description you quoted? Why include it at all?
Because it's how mathematicians conceptualize of what a field is. It is the peg we hang our hat on; it is what we remember. A mathematician who has seen fields would be able to fill in the details; and if not, they would know to look up the precise definition in a textbook.
In short, these definitions are how we keep track of the forest at the same time as the trees.
I should note that taste differs among mathematicians, and you can find different styles of exposition in math books. Some are very formal and precise; whereas others are more informal and have lots of handwavy statements along the lines of the one you quoted.
Earlier quoted context omitted.
But that example is a little bit artificial, isn't it? A lot of mathematical concepts are more complex than that and sometimes symbols are not the best option. Say, for example the definition of Hausdorff space, in words and symbols: - Any two distinct points in the space have disjoint neighbourhoods. - ∀x,y ∈ X with x ≠ y, ∃U,V ⊂ X s.t. x ∈ U, y ∈ V and U ∩ V = ∅. Another example would be Navier-Stokes equations, wh…
You are comparing apples to oranges. Your English definition is missing the definition of neighborhood. To actually be an accurate comparison you'd need to add ", where a neighborhood is a set that contains an open ball that contains the point." But now, you're also missing the definition of open ball...
Difficulties, if any, perceived or real, arising in connection with notation, are usually incomparably smaller than those presented with the subject itself. (Personally, I only wish mathematical notation were better integrated with software in general and programming languages in particular.)
If programming languages strive for the same elegance via notation, you end up with something like Perl.