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I really need to expand this into a blog post that has been brewing in my head for years, but my overall thesis is that notation is the least difficult part to

by jordigh 10y ago

I really need to expand this into a blog post that has been brewing in my head for years, but my overall thesis is that notation is the least difficult part to learn in mathematics, yet because it's the first part that is encountered, it is also the most derided one. Sure, notation can be better, and maybe this triangle of power is a cute way to make it better. Notation changes and improves all the time, btw. Well, all the time in the mathematical scale of time, which is two or three millenia. In this scale, things like the Greek letter for the ratio of circumference to diameter are remarkably modern, merely 300 years old. Notation for linear algebra is even newer, all from the 20th century. However, I don't think better notation is where we need to focus most of our efforts in order to make our mathematics easier to understand. Logarithms and square roots are very basic things, and if keeping the mainstream notation for them straight is someone's biggest problem, then there are far bigger things that are likely to be problematic to this individual. If you start reading, say, the following mathematical discussion of neural networks, http://neuralnetworksanddeeplearning.com/chap1.html#eqtn7 http://neuralnetworksanddeeplearning.com/chap1.html#eqtn7 you're baffled because you don't know what those symbols mean, there's likely far deeper things that are unfamiliar, such as differentials, rate of change, derivatives, and the multivariable chain rule. A couple of days ago we had someone come to ##math in Freenode asking for help with this, and I tried, but the guy had never had any calculus training whatsoever. Normally going from no calculus to the multivariable chain rule as applied to differentials or as a best linear approximation takes at least three semesters in university, and I don't think this path to enlightenment could be shortened much more. I guess I am being very old school and reiterating that the royal road everyone's been looking for for the past couple millenia just doesn't exist. https://en.wikipedia.org/wiki/Royal_Road#A_metaphorical_.E2.80.9CRoyal_Road.E2.80.9D_in_famous_quotations https://en.wikipedia.org/wiki/Royal_Road#A_metaphorical_.E2....

hellofunk 10y ago
Can you recommend an approach to learn high level math on one's own? Any particular books that are good for that purpose?
deleted 10y ago
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deleted 10y ago
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pacala 10y ago
A lot of math is simple and logical. Notation is a major obstacle, because of a. the sheer amount of various notations which are rarely used by a non-expert in a particular subfield, thus easily forgotten and b. un-searchability. Reading a math text becomes an exercise in jumping around figuring out what the author meant with a particular notation, which usually leads to more notations, etc. Heck, coding CRUD apps I have access to better notation exploration tools than a mathematician. I can navigate to a definition or check the unit tests in seconds. I don't have to instantly remember the purpose of every function, I can simply look up the precise details on the fly. I can rely that the authors have not overused symbolic operators, so I can easily search StackOverflow for tips on the framework design. No more rote memorization of hundreds of tiny factoids that may or may not be germane to the problem at hand.
- jordigh 10y ago
  > A lot of math is simple and logical And I suppose you are implying a lot of math is illogical. :-) > Heck, coding CRUD apps Yeah, I hear the "bad notation" argument frequently from programmers, precisely for the reasons you describe: because you can't read a mathematical textbook with a programmer's IDE. Some mathematical texts will do you the favour of having a list of some notation that they consider to be idiosyncratic to their own text, but few would go as far as to letting you look up the definition of plus and minus. In general, though, mathematics isn't programming, despite some similarities and analogies between the two. The symbol for the partial derivative is used nearly universally to mean partial derivatives of some kind, so there's a lot of tradition that writers of mathematical texts expect you to know. Programming has similar traditions that are baffling to outsiders but as invisible to the programmers as water is to fish. For example, in programming it is understood that everyone is capable of easily handling plain text files. Have you ever seen a newcomer try to write source code in Microsoft Word? I have. Wikipedia and other online mathematical texts can help a little by hyperlinking the text to explain new notation or terminology, and sometimes texts are just plain bad in that they use idiosyncratic notation without explanation. In the last case, it takes effort to work out from context the likely meaning of a symbol. In general, though, I stand by my thesis that notation is not the biggest obstacle to mathematics just like learning to use a text editor or IDE is not the biggest obstacle to programming. The fundamental ideas behind the practices of each are much deeper than the superficial aspects of text and notation.
vlehto 10y ago
It sure doesn't help that often it's forgotten to teach notation in the beginning. Not in high school but at university level this is my experience. And it's incredibly hard to google on your own.
- yorwba 10y ago
  It's like that for other skills too, like how to write a proof. I guess it is assumed that you will simply assimilate it with time, but I have met students in their third year who still struggled with e.g. properly introducing variables.
Double_Cast 10y ago
According to the trivium [0], the path to learning consists of a sequence: grammar; dialectic; and rhetoric. In my head I think of it in terms of vocab, logic, and application (respectively). Notation may not be the hardest part. It is however (as you say) the first part. If our goal is to make math more readily accessable to the masses, perhaps the greatest immediate gains entail improving the notation to better reflect the concepts -- exactly because it is the first barrier. [0] https://en.wikipedia.org/wiki/Trivium https://en.wikipedia.org/wiki/Trivium
Cacti 10y ago
I honestly find the concepts far easier than the syntax, and I am used to weird syntax. It's not just the notation though. It's using crappy notation in place of explanation or as a short-cut. While some notation is "modern" in absolute linear terms, it's really not in any practical sense. The entire premise of most mathematical writing, exposition, and notation, is still that of thousands of years ago: writing takes significant time, writing surfaces are expensive, communication lag is long, and cultural/linguistic barriers between people are significant. That is simply not very true any more. There is no reason we should be writing, teaching, and discussing math as though it was still a thousand years ago. We have computers, we have amazing printing presses, we have relatively cheap material. There should be no reason we're still pulling this "as the reader will clearly see" crap with this massively dense short-cut notation.
sedachv 10y ago
> Sure, notation can be better, and maybe this triangle of power is a cute way to make it better. Notation changes and improves all the time, btw. Well, all the time in the mathematical scale of time, which is two or three millenia. In this scale, things like the Greek letter for the ratio of circumference to diameter are remarkably modern, merely 300 years old. Notation for linear algebra is even newer, all from the 20th century. Forget linear algebra, modern notation for regular algebra is from the 19th century. And the form of the notation absolutely does matter to learning and comprehension. Ancient Babylonians knew how to solve quadratic equations, but you basically had to be a professional mathematician to understand the way they phrased problems and work with their notation. More well known is the notoriety of Roman numerals for doing basic arithmetic. It takes more than 20 pages to explain fractions with Roman numerals: http://dmaher.org/Publications/romanarithmetic.pdf http://dmaher.org/Publications/romanarithmetic.pdf

by jordigh 10y ago

hellofunk 10y ago

Can you recommend an approach to learn high level math on one's own? Any particular books that are good for that purpose?

deleted 10y ago

[deleted]

deleted 10y ago

[deleted]

pacala 10y ago

A lot of math is simple and logical. Notation is a major obstacle, because of a. the sheer amount of various notations which are rarely used by a non-expert in a particular subfield, thus easily forgotten and b. un-searchability. Reading a math text becomes an exercise in jumping around figuring out what the author meant with a particular notation, which usually leads to more notations, etc. Heck, coding CRUD apps I have access to better notation exploration tools than a mathematician. I can navigate to a definition or check the unit tests in seconds. I don't have to instantly remember the purpose of every function, I can simply look up the precise details on the fly. I can rely that the authors have not overused symbolic operators, so I can easily search StackOverflow for tips on the framework design. No more rote memorization of hundreds of tiny factoids that may or may not be germane to the problem at hand.

jordigh 10y ago
> A lot of math is simple and logical And I suppose you are implying a lot of math is illogical. :-) > Heck, coding CRUD apps Yeah, I hear the "bad notation" argument frequently from programmers, precisely for the reasons you describe: because you can't read a mathematical textbook with a programmer's IDE. Some mathematical texts will do you the favour of having a list of some notation that they consider to be idiosyncratic to their own text, but few would go as far as to letting you look up the definition of plus and minus. In general, though, mathematics isn't programming, despite some similarities and analogies between the two. The symbol for the partial derivative is used nearly universally to mean partial derivatives of some kind, so there's a lot of tradition that writers of mathematical texts expect you to know. Programming has similar traditions that are baffling to outsiders but as invisible to the programmers as water is to fish. For example, in programming it is understood that everyone is capable of easily handling plain text files. Have you ever seen a newcomer try to write source code in Microsoft Word? I have. Wikipedia and other online mathematical texts can help a little by hyperlinking the text to explain new notation or terminology, and sometimes texts are just plain bad in that they use idiosyncratic notation without explanation. In the last case, it takes effort to work out from context the likely meaning of a symbol. In general, though, I stand by my thesis that notation is not the biggest obstacle to mathematics just like learning to use a text editor or IDE is not the biggest obstacle to programming. The fundamental ideas behind the practices of each are much deeper than the superficial aspects of text and notation.

vlehto 10y ago

It sure doesn't help that often it's forgotten to teach notation in the beginning. Not in high school but at university level this is my experience. And it's incredibly hard to google on your own.

yorwba 10y ago
It's like that for other skills too, like how to write a proof. I guess it is assumed that you will simply assimilate it with time, but I have met students in their third year who still struggled with e.g. properly introducing variables.

Double_Cast 10y ago

According to the trivium [0], the path to learning consists of a sequence: grammar; dialectic; and rhetoric. In my head I think of it in terms of vocab, logic, and application (respectively). Notation may not be the hardest part. It is however (as you say) the first part. If our goal is to make math more readily accessable to the masses, perhaps the greatest immediate gains entail improving the notation to better reflect the concepts -- exactly because it is the first barrier. [0] https://en.wikipedia.org/wiki/Trivium https://en.wikipedia.org/wiki/Trivium

Cacti 10y ago

I honestly find the concepts far easier than the syntax, and I am used to weird syntax. It's not just the notation though. It's using crappy notation in place of explanation or as a short-cut. While some notation is "modern" in absolute linear terms, it's really not in any practical sense. The entire premise of most mathematical writing, exposition, and notation, is still that of thousands of years ago: writing takes significant time, writing surfaces are expensive, communication lag is long, and cultural/linguistic barriers between people are significant. That is simply not very true any more. There is no reason we should be writing, teaching, and discussing math as though it was still a thousand years ago. We have computers, we have amazing printing presses, we have relatively cheap material. There should be no reason we're still pulling this "as the reader will clearly see" crap with this massively dense short-cut notation.

sedachv 10y ago

> Sure, notation can be better, and maybe this triangle of power is a cute way to make it better. Notation changes and improves all the time, btw. Well, all the time in the mathematical scale of time, which is two or three millenia. In this scale, things like the Greek letter for the ratio of circumference to diameter are remarkably modern, merely 300 years old. Notation for linear algebra is even newer, all from the 20th century. Forget linear algebra, modern notation for regular algebra is from the 19th century. And the form of the notation absolutely does matter to learning and comprehension. Ancient Babylonians knew how to solve quadratic equations, but you basically had to be a professional mathematician to understand the way they phrased problems and work with their notation. More well known is the notoriety of Roman numerals for doing basic arithmetic. It takes more than 20 pages to explain fractions with Roman numerals: http://dmaher.org/Publications/romanarithmetic.pdf http://dmaher.org/Publications/romanarithmetic.pdf