Mathematical Notation, Past and Future (2000)
stephenwolfram.com
Mathematical Notation, Past and Future (2000)
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Re: Mathematical Notation, Past and Future (2000)
#2I feel like we haven't even standardized a lot of notation to express ideas in common use...talking about the future of notation is strange as people seem to just invent them as they go and there is no committee that steers this discussion.
Re: Mathematical Notation, Past and Future (2000)
#3I feel like we haven't even standardized a lot of notation to express ideas in common use...talking about the future of notation is strange as people seem to just invent them as they go and there is no committee that steers this discussion.
This is a huge struggle for me as a computational biologist with a computer science background. When I use a published method I try to have at least a cursory understanding of how the method works. However, many of the papers in my field are now statistics and machine learning heavy. The lack of a standardized notation makes the method descriptions very hard for me to follow. Occasionally the source code published alone with the papers is readable and the implementation is much simpler than I expect. But it's not uncommon for the published code to be a pile of spaghetti with no comments or sensible variable names.
Re: Mathematical Notation, Past and Future (2000)
#4Alright 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?
Re: Mathematical Notation, Past and Future (2000)
#5(2000)
Re: Mathematical Notation, Past and Future (2000)
#6Alright 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 has been discussed before a couple of times - Bret Victor keyed me into the discussion of how symbolic maths makes the discipline seem alien by pointing at two things (can't remember in which lecture though): the mathematician's lament, and a letter to leading mathematicians and physicists including Einstein asking how they work with maths (can't remember any key words to Google this). The idea being that mathematical symbols are a tool of communication, not the way you actually work with maths (if you're exploring new ground, at least).
Edit: it was in here: http://worrydream.com/#!/KillMath
Re: Mathematical Notation, Past and Future (2000)
#7For a deeper exposition on language and notation, Guy Steele[0] is the definitive authority :-) - with some discussion here[1].
Re: Mathematical Notation, Past and Future (2000)
#8Alright 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?
I think it certainly has an effect. My rationale is comparing the earliest writings on algebra and calculus to the present-day notation. Al-Khwrizmi used no symbols or numerals in his text. I can't guess at Newton because the text is in Latin, but the notation looks pretty difficult to me. At the time, you practically had to be a philosopher to understand that writing, much less do anything with it. Likewise I think Clerk-Maxwell's electromagnetic theory was impenetrable until Heaviside created a new notation for it. And we just had a HN thread talking about Mary Somerville's translation of Laplace.
Today we teach algebra and calculus to schoolchildren. I believe as bad as the notation probably still is, the improvement from the original texts is nothing short of qualifying as its own revolution in mathematics. So it's interesting to imagine it being even better.
But I also believe that we only teach one side of math in school, basically manipulation of expressions. We barely touch on other ways that people do math, through the exploration of data, computation, and theory (proofs). So we're painting a distorted picture of what math even is, making it all the harder to justify why it's important beyond just being a gatekeeper.
Re: Mathematical Notation, Past and Future (2000)
#9An interesting and often overlooked exposition by Dijkstra on notation: https://www.cs.utexas.edu/users/EWD/transcriptions/EWD13xx/E...
Re: Mathematical Notation, Past and Future (2000)
#10Alright 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 challenging, and it's doubtful that removing powerful tools would make it any less so. I think of notation as being something that makes more complicated thoughts to be easier to have -- misquoting Alan Kay, good notation is worth 80 IQ points.