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Mathematical Notation, Past and Future (2000)
11–20 of 27 posts
Re: Mathematical Notation, Past and Future (2000)
#12Alright here goes a dumb question. How much is mathematical notation contributing to the idea that "math is hard" and preventing students to think of math as lego bricks in which you can use a set of functions to build newer functions?
When someone comes up with a way to explain the same concepts using a simpler and easier to understand way, over time it becomes the new notation. But it rarely is world changing, mostly it’s small, incremental changes.
One big example of a major notation change is Grothendieck’s effect on algebraic geometry. It used to be about solutions of sets of polynomial equations. Now it’s all about sheaves and schemes and moduli spaces and representable functors and stacks and… This new notation changed the way mathematicians think. Did it simplify things and made it more accessible to lay people? I’d argue, not at all, in fact you probably need a year or two of university math education to fully understand what a scheme even is.
I guess category theory is example in the other direction: it greatly simplified and unified thinking in algebraic topology and geometry. However, it’s not that big of a help to a complete newbie: if you are at a complete loss when you encounter a cohomology theory for the first time, framing it in terms of graded functors won’t be of huge incremental help.
Point here is that notation is there to help mathematicians, not regular people, but regular people neither are able to understand all of it, nor do they care, so it’s not much of a loss.
Re: Mathematical Notation, Past and Future (2000)
#13I love math but prefer programming type syntax to express math ideas, rather than much of traditional math notation. Much of it feels too alien and non-intuitive. Too irregular and random, and not predictable. Various styles duct taped together over centuries. The names of the various symbols are hard to remember, and dont get me started on the pain from not being on the standard PC/ASCII/Western computer keyboards.
In short, a similar critique to that of Perl and PHP, except math is even worse.
Show me a Python-like notation for a math construct and I'm good to go. Even a Lisp!
Re: Mathematical Notation, Past and Future (2000)
#14Alright here goes a dumb question. How much is mathematical notation contributing to the idea that "math is hard" and preventing students to think of math as lego bricks in which you can use a set of functions to build newer functions?
This same pattern comes up a lot, whether it be mathematical notation, programming language syntax, or scientific jargon. Some people crusade against them, while others (typically practicioners of the respective field) insist that these constructs are useful.
My hypothesis is that it's a bias from limited information. You have something with inherent complexity to it: math, music theory, physics, code, you name it. In order to represent those complexities, complex jargon gets formed. An outsider attempting to understand a given field is likely to have a (in these cases correct) assumption that the field is complex: if it wasn't, then they wouldn't be putting effort into understanding it.
In their effort to understand it, they eventually run into something they don't understand immediately or easily. They then assign the majority of their perceived complexity of the system to that first thing they got stuck on.
So instead of thinking "programming is hard" they think "reading/writing source code is hard". The practicioners of the field, with the benefit of hindsight, know that picking up the syntax/jargon/notation was in fact far easier than understanding the concepts they represent. But the outsider can only see the immediate challenge, which is the symbology.
Re: Mathematical Notation, Past and Future (2000)
#15Alright here goes a dumb question. How much is mathematical notation contributing to the idea that "math is hard" and preventing students to think of math as lego bricks in which you can use a set of functions to build newer functions?
> preventing students to think of math as lego bricks in which you can use a set of functions to build newer functions Isn't this something that mathematical notation lets you be able to see at all? (I'm also not sure what you mean here exactly.) By the way, have you heard what Euclid allegedly said to Ptolemy I when the king found the Elements to be difficult? "There is no royal road to geometry." Math is inherently…
But surely notations vary a lot in quality. Think for example in roman numerals vs. positional number systems as alternative notations to work with arithmetic. One was surely a great advance over the other. So I wonder, what elements of our modern notations are akin to roman numbers, a suboptimal solution awaiting to be replaced with something vastly better. Is all perceived complexity in math inherent to it, or at least some of that is due to subpar notations?
Re: Mathematical Notation, Past and Future (2000)
#16Alright here goes a dumb question. How much is mathematical notation contributing to the idea that "math is hard" and preventing students to think of math as lego bricks in which you can use a set of functions to build newer functions?
This same pattern comes up a lot, whether it be mathematical notation, programming language syntax, or scientific jargon. Some people crusade against them, while others (typically practicioners of the respective field) insist that these constructs are useful. My hypothesis is that it's a bias from limited information. You have something with inherent complexity to it: math, music theory, physics, code, you name it. I…
The constructs are always useful, but rarely do they form a coherent whole, because they're formed by accretion. The "outsiders" are usually going to be right: throwing everything out and designing the language from scratch would allow for substantial improvements, regularization, etc. The insiders will however have adapted to the existing system and found its complexities easy enough to navigate. And that is why the USA does not use metric.
Re: Mathematical Notation, Past and Future (2000)
#17I love math but prefer programming type syntax to express math ideas, rather than much of traditional math notation. Much of it feels too alien and non-intuitive. Too irregular and random, and not predictable. Various styles duct taped together over centuries. The names of the various symbols are hard to remember, and dont get me started on the pain from not being on the standard PC/ASCII/Western computer keyboards.…
I think having 15 different ways to express multiplication is charming.
Re: Mathematical Notation, Past and Future (2000)
#18I love math but prefer programming type syntax to express math ideas, rather than much of traditional math notation. Much of it feels too alien and non-intuitive. Too irregular and random, and not predictable. Various styles duct taped together over centuries. The names of the various symbols are hard to remember, and dont get me started on the pain from not being on the standard PC/ASCII/Western computer keyboards.…
Do you have a sample comparison between an expression/statement in math syntax versus in pseudocode syntax?
Re: Mathematical Notation, Past and Future (2000)
#19Earlier quoted context omitted.
This same pattern comes up a lot, whether it be mathematical notation, programming language syntax, or scientific jargon. Some people crusade against them, while others (typically practicioners of the respective field) insist that these constructs are useful. My hypothesis is that it's a bias from limited information. You have something with inherent complexity to it: math, music theory, physics, code, you name it. I…
The constructs are always useful, but rarely do they form a coherent whole, because they're formed by accretion. The "outsiders" are usually going to be right: throwing everything out and designing the language from scratch would allow for substantial improvements, regularization, etc. The insiders will however have adapted to the existing system and found its complexities easy enough to navigate. And that is why the…
Also I just want to add... mathematicians have ⋅, ×, and ∗ to express multiplication, but decided that they also needed ADJACENCY OF CHARACTERS to express multiplication, and therefore mathematics is not allowed to use variable names longer than one letter.
Come the fuck on, mathematics.
Re: Mathematical Notation, Past and Future (2000)
#20Earlier quoted context omitted.
The constructs are always useful, but rarely do they form a coherent whole, because they're formed by accretion. The "outsiders" are usually going to be right: throwing everything out and designing the language from scratch would allow for substantial improvements, regularization, etc. The insiders will however have adapted to the existing system and found its complexities easy enough to navigate. And that is why the…
Also I just want to add... mathematicians have ⋅, ×, and ∗ to express multiplication, but decided that they also needed ADJACENCY OF CHARACTERS to express multiplication, and therefore mathematics is not allowed to use variable names longer than one letter. Come the fuck on, mathematics.
It's fine, you just add a space. It's also used for function application, like "sin x".
I'd say single-letter variable names are mostly due to ease of handwriting when doing calculations (or for giving a chalkboard talk), and authors tend to just use what they've already come up with when writing things up. Mathematicians seem to be very tolerant of ambiguous parsing, so that can't explain all of it.