4 ms·

My mind goes to various, unfortunate notational conventions.

xnyan 4y ago
Is there something you could say about these unfortunate notations or an article you you could point me to so that I can understand what they are?
- nextos 4y ago
  For example, calculus uses notation and terminology that predates the modern limit-based field Weierstraß and others built. It's really confusing [1]. Statistics is even worse. A mix of old tricks developed to avoid computations when these were expensive. See [2]. [1] A Radical Approach to Real Analysis https://www.davidbressoud.org/aratra/ https://www.davidbressoud.org/aratra/ [2] The Introductory Statistics Course: A Ptolemaic Curriculum? https://escholarship.org/uc/item/6hb3k0nz https://escholarship.org/uc/item/6hb3k0nz
  - aeneasmackenzie 4y ago
    Limits are not an inherent part of calculus. You can do all calculus relevant for the physical world just fine with nilpotent infinitesimals if you but give up excluded middle.
    - jqgatsby 4y ago
      I've heard of this constructivist approach to calculus, but hadn't made the connection with nilpotents. that's really interesting, could you explain why nilpotenxy and forgoing the law of the excluded middle relate to each other?
      - enriquto 4y ago
        You can use nilpotents with classical logic and the excluded middle. This is called dual numbers and it's already a good model for "calculus without limits". They are like complex numbers, but instead of x^2=-1 you set x^2=0. However, if you want to get really serious about that, you'll need that zero plus an infinitessimal be equal to zero. This is impossible in classical logic due to the excluded middle (which forces each number to be either equal to zero or non-zero).
        Qem 4y ago
        Can you recommend a introductory calculus book that builds it up from dual numbers?
        zozbot234 4y ago
        You could try Calculus made easy by S. P. Thompson.
        enriquto 4y ago
        The Silvanus P. Thompson book suggested by the sibling comment is lovely and very clear. For a more algebraic treatment, and its important applications to automatic differentiation, I'd suggest starting with the relevant wikipedia articles: https://en.wikipedia.org/wiki/Dual_number https://en.wikipedia.org/wiki/Dual_number https://en.wikipedia.org/wiki/Automatic_differentiation https://en.wikipedia.org/wiki/Automatic_differentiation
    - nextos 4y ago
      My point is not that limits are an inherent part of calculus. My point is that calculus as currently taught mixes infinitesimal-like notation that predates limits with limit-based calculus.
WoahNoun 4y ago
Such as? Many branches of mathematics have their own mutually unintelligible dialects of notation. Many longer papers or Ph.D thesis will just create notation just for the context of the paper.
- User23 4y ago
  An obvious one is that traditional mathematical convention requires single glyph variable names due to the unfortunate decision to save paper by denoting the product by juxtaposition. That’s an admittedly trivial example though, even though the higher order consequences are considerable. Computing science is when notation came into its own. Younger mathematicians have taken those lessons to heart, but as the old saying goes, progress comes one funeral at a time. Being forced to mechanically parse and interpret a syntax has a way of really bringing out any ambiguity.
  - worthless-trash 4y ago
    > Being forced to mechanically parse and interpret a syntax has a way of really bringing out any ambiguity. This is absolutely beautifully said user23. I as a programmer often struggle with understanding notations used in some papers.
  - ithinkso 4y ago
    > that traditional mathematical convention requires single glyph variable names due to the unfortunate decision to save paper by denoting the product by juxtaposition. It's not to save paper or because of the product. You don't know the solution to the problem you are working on from the beginning and most of the time is spent writing and writing and writing in a scratchpad trying to solve what you need. Anything longer than a single glyph for variables would be too tedious so everyone evolved to use single letters. And then the papers are written with the same convention since it's natural. You have variable names though with the use of subscripts with the added benefits that it can be (and is) used to elegantly group relevant variables together giving you some sort of abstraction I once wrote a comment about it here on HN - language in maths is not a programming language used to tell a computer how to go from A to B, but a natural language used to talk about maths between peers. Every natural language have idioms, inconsistences and other quirks. Polish will not change for you so it's easier for you to learn it, it will change in the way that let's polish people communicate better with each other which also include a lot of historical and cultural happenstances. Same with maths There are attempts like Esperanto and other artificial languages like that and I think any attempts at 'codification' of maths to use some programming language has the same chance of success of wide adoption
kazinator 4y ago
So, like, inequality against people with visual disabilities and dyslexics?

Is there something you could say about these unfortunate notations or an article you you could point me to so that I can understand what they are?

nextos 4y ago
For example, calculus uses notation and terminology that predates the modern limit-based field Weierstraß and others built. It's really confusing [1]. Statistics is even worse. A mix of old tricks developed to avoid computations when these were expensive. See [2]. [1] A Radical Approach to Real Analysis https://www.davidbressoud.org/aratra/ https://www.davidbressoud.org/aratra/ [2] The Introductory Statistics Course: A Ptolemaic Curriculum? https://escholarship.org/uc/item/6hb3k0nz https://escholarship.org/uc/item/6hb3k0nz
- aeneasmackenzie 4y ago
  Limits are not an inherent part of calculus. You can do all calculus relevant for the physical world just fine with nilpotent infinitesimals if you but give up excluded middle.
  - jqgatsby 4y ago
    I've heard of this constructivist approach to calculus, but hadn't made the connection with nilpotents. that's really interesting, could you explain why nilpotenxy and forgoing the law of the excluded middle relate to each other?
    - enriquto 4y ago
      You can use nilpotents with classical logic and the excluded middle. This is called dual numbers and it's already a good model for "calculus without limits". They are like complex numbers, but instead of x^2=-1 you set x^2=0. However, if you want to get really serious about that, you'll need that zero plus an infinitessimal be equal to zero. This is impossible in classical logic due to the excluded middle (which forces each number to be either equal to zero or non-zero).
      - Qem 4y ago
        Can you recommend a introductory calculus book that builds it up from dual numbers?
        zozbot234 4y ago
        You could try Calculus made easy by S. P. Thompson.
        enriquto 4y ago
        The Silvanus P. Thompson book suggested by the sibling comment is lovely and very clear. For a more algebraic treatment, and its important applications to automatic differentiation, I'd suggest starting with the relevant wikipedia articles: https://en.wikipedia.org/wiki/Dual_number https://en.wikipedia.org/wiki/Dual_number https://en.wikipedia.org/wiki/Automatic_differentiation https://en.wikipedia.org/wiki/Automatic_differentiation
  - nextos 4y ago
    My point is not that limits are an inherent part of calculus. My point is that calculus as currently taught mixes infinitesimal-like notation that predates limits with limit-based calculus.

WoahNoun 4y ago

Such as? Many branches of mathematics have their own mutually unintelligible dialects of notation. Many longer papers or Ph.D thesis will just create notation just for the context of the paper.

User23 4y ago
An obvious one is that traditional mathematical convention requires single glyph variable names due to the unfortunate decision to save paper by denoting the product by juxtaposition. That’s an admittedly trivial example though, even though the higher order consequences are considerable. Computing science is when notation came into its own. Younger mathematicians have taken those lessons to heart, but as the old saying goes, progress comes one funeral at a time. Being forced to mechanically parse and interpret a syntax has a way of really bringing out any ambiguity.
- worthless-trash 4y ago
  > Being forced to mechanically parse and interpret a syntax has a way of really bringing out any ambiguity. This is absolutely beautifully said user23. I as a programmer often struggle with understanding notations used in some papers.
- ithinkso 4y ago
  > that traditional mathematical convention requires single glyph variable names due to the unfortunate decision to save paper by denoting the product by juxtaposition. It's not to save paper or because of the product. You don't know the solution to the problem you are working on from the beginning and most of the time is spent writing and writing and writing in a scratchpad trying to solve what you need. Anything longer than a single glyph for variables would be too tedious so everyone evolved to use single letters. And then the papers are written with the same convention since it's natural. You have variable names though with the use of subscripts with the added benefits that it can be (and is) used to elegantly group relevant variables together giving you some sort of abstraction I once wrote a comment about it here on HN - language in maths is not a programming language used to tell a computer how to go from A to B, but a natural language used to talk about maths between peers. Every natural language have idioms, inconsistences and other quirks. Polish will not change for you so it's easier for you to learn it, it will change in the way that let's polish people communicate better with each other which also include a lot of historical and cultural happenstances. Same with maths There are attempts like Esperanto and other artificial languages like that and I think any attempts at 'codification' of maths to use some programming language has the same chance of success of wide adoption

kazinator 4y ago

So, like, inequality against people with visual disabilities and dyslexics?