10 ms·

Math and Analogies

181 points by chrisweekly 8y ago

jacobolus 8y ago
Here’s what I wrote elsewhere about this: Why was 1997 easier for us to read than MCMDCVII? Duh... because we all have decades of experience working with the former representation and almost no experience at all working with the latter one. If you wrote a number using arbitrary different symbols and (say) base 14 instead of base 10, but still a positional number system, we would still find it incredibly difficult to interpret written numbers, because we wouldn’t be used to it. Indeed we find exactly this happens when people who are not experienced programmers try to read hexadecimal numbers written using digits 0123456789ABCDE, and that’s even in a case where we are deeply familiar with all of the symbols and their order, and in a very straightforward power-of-2 base. Romans or medieval Europeans had absolutely no problem reading numbers written like the latter example. Indeed, this is a much more straight-forward and intuitive representation, and is very easy to interpret for anyone used to working with a counting board, since it is a direct translation of the positions of counters on the counting board to symbols written on paper. [Edit to add: Note that Roman numerals were not a system for practical arithmetic – people didn’t use writing for this, instead calculating by moving counters around on a counting board — but only a written record of input data and calculated outputs.] I really hate how much people bash on Roman numerals without any significant experience working with them, and without any discussion or even acknowledgement of the historical or practical context. I would recommend anyone interested in this topic start by reading Netz’s paper “Counter Culture” about Greece 2500 years ago, http://worrydream.com/refs/Netz%20-%20Counter%20Culture%20-%20Towards%20a%20History%20of%20Greek%20Numeracy.pdf http://worrydream.com/refs/Netz%20-%20Counter%20Culture%20-%...
- deleted 8y ago
  [deleted]
- posterboy 8y ago
  This is an interesting perspective, but you have to assume that deprecating the system happened for a reason, probably due to ergonomic disdvantages. For simple counting tasks, unary tally marks are still fashionable. Edit: I never used an abakus or the like, but maybe that's what roman numerals are the analogy to, and it got overcome with long division and the like.
  - jacobolus 8y ago
    There are certainly advantages to the more abstract positional Hindu–Arabic number system (or e.g. the ancient Mesopotamian system for writing sexagesimal numbers which also positional). I am not trying to dispute that. A symbolic positional system saves space on the page. It makes recorded numbers more difficult to falsify later. Doing scratch work on paper leaves a better record of calculations (the position of a counting board is inherently ephemeral) which potentially makes it easier to spot errors. It is easier to describe pen and paper arithmetic in printed books which helps spread knowledge more easily than an oral tradition. It extends more easily to larger or (after the advent of decimal fractions) smaller numbers without needing an inordinately large number of different symbols. It is more easily extended by the use of pre-computed tables of function values, and especially by slide rules and similar devices. Once we start decimalizing our metrical system it becomes very regular to work with many different measurements in a consistent way. Perhaps most importantly the use of writing leads naturally to the development of symbolic algebra and ultimately simpler formulations of higher mathematics. But we can carefully describe the advantages and historical process of replacement etc. without unfairly disparaging the more direct representation.
    - posterboy 8y ago
      MCM requires subtraction, two directions, 1900 is additional, maybe multiplicative if you want to be strict, either way only one direction. I'm not sure what's "more direct", either is linear. I might even think of 15:45 as a quarter before 4 [1]), but I'd argue that subtraction is conceptually a level higher. Nevertheless, high level representation might be an advantage. Maybe an interesting parallel viz. cognition is the 19.95 pricing scheme that's apparently proven to be confusing to the consumer. I'm just not sure which numeral system suffers more from that. [1] which marks an isogloss between east and west germany, the latter saying "dreiviertel vier" - three quarters four; German also has a weird order for decimal place reading: 121=hundred-one-twenty
      - jacobolus 8y ago
        Well MCM is not how Romans would write this except in rare cases. That one did not become a standard form until Medieval Europe. MDCCCC is absolutely a more direct representation of the counters on a counting board than 1900. Each letter precisely represents one counter and its position. But if you want, you can make a counting board with a line down the middle and use one side for negative values, in which case MCM would be a pretty direct representation of that. (It is unclear whether this was the origin of such numerals; we don’t have a whole lot of evidence about how calculation on ancient counting boards was done, because it was largely part of oral culture.)
- gugagore 8y ago
  I agree that it's a sort of weak argument, and I'm a bit surprised by it. Usually the claim isn't that Roman numerals are hard to read and write, it's that they're hard to calculate with. And I think that's a stronger argument. Probably you can learn algorithms for doing calculations on Roman numerals, and we're familiar with our algorithms because of decades of experience. But the algorithms we learn in school actually do work marvelously for other bases.
  - jacobolus 8y ago
    People never used Roman numerals for performing written arithmetic, so that is not a fair comparison. Calculations were done mentally, with fingers, or for anything complicated or serious on a counting board.
    - JoeAltmaier 8y ago
      I believe the ancient Romans would disagree? They had complicated rules for borrowing and simplifying, but of course it had to be done or they had nothing.
      - Koshkin 8y ago
        The Romans heavily relied on the abacus. It used the 10-based positional representation of numbers.
      - jacobolus 8y ago
        > complicated rules for borrowing and simplifying [..] or they had nothing Maybe you can explain what you are referring to here? Ancient Greeks and Romans simply did not do arithmetic with pen and paper. They used a counting board with counters or pebbles (“calculi”), or for simpler problems used their fingers or did the work in their heads. Doing computation was literally called “placing pebbles”, and to settle accounts with someone you would “call them to the pebbles”. Roman numerals were not a tool for computation. They were a tool for recording data or results. The purpose of Roman numerals was to faithfully record in writing the placement of counters on a counting board. The position of the counting board (or perhaps in some contexts a pile of coins of various denominations) was the primary representation of numbers, and Roman numerals were secondary.
- bena 8y ago
  Because MCMXCVII is actually 1997. MCMDCVII is just wrong. D is 500, X is ten.
  - Reisen 8y ago
    Kind of a nice proof of his point! We're not used to the numbers, people would be far less likely to make the mistake with 1997.
  - jacobolus 8y ago
    Yes that was a typo.
netgusto 8y ago
Video giving some intuition about e^(i•PI)=-1 https://www.youtube.com/watch?v=F_0yfvm0UoU https://www.youtube.com/watch?v=F_0yfvm0UoU
- thaumasiotes 8y ago
  The intuition is as simple as it gets; r·e^{iθ} is the complex number with radius r and angle θ. pi is just the angle that puts you on the negative half of the real number line. For why that identity holds, you do simple algebra on the non-simple Taylor series of those three functions (e^x, sin x, cos x). It is, to say the least, much less intuitive.
  - chrisweekly 8y ago
    The linked article mentions how helpful it is to have the right analogy (e.g. a number line, which you reference in your explanation of its intuitivenes). That's the main point: given the right analogy, things that otherwise seem arbitrary or nonsensical become clear and even intuitive. :)
butisaidsudo 8y ago
For Roman numerals to make sense you need to remove access to convenient writing tools. In a world without paper and pens, Arabic numerals are the ones that seem unwieldy. I was on a multi-day hiking trip with a friend where we just had a pack of cards and some spiced rum to keep ourselves entertained in the evening. The first night we decided to play cribbage, but we didn't have paper or the like to keep score on. At first I tried using a stick to scratch out numbers in the dirt. It was doable, but very awkward. The low light of the camp fire made it hard to see, and I quickly ran out of undisturbed soil within arms reach. I then gathered a few twigs to shape into numbers, using the patterns you'd see on an LED clock. That worked reasonably well, but took more effort than I liked. I switched to Roman numerals thinking that the simpler shapes would be easier to work with. This turned out to be true, but I discovered it had the added benefit of being really easy to increment numbers. In most cases, incrementing is incredibly simple. To go from 0 -> 1 -> 2 -> 3, you simply add a stick each time. To go from 3 -> 4, you pinch together the bottoms of the 2nd and 3rd sticks. Then you take the first stick away, then later drop it on the other side of the V. Add another stick, cross some sticks, etc. There are a lot of things that seem poorly done at first glance, but make total sense when you understand the environment they were developed in. I've heard younger folks wonder why old TV shows are so poorly written. When you've always been able to watch on demand (or rent episodes on DVD), it's hard to understand the restrictions on writing when there was no way for your audience to watch previous episodes if they hadn't seen them when they had aired. And of course this is true for most software I've ever worked on.
- adrianmonk 8y ago
  > incrementing is incredibly simple So Roman numerals are like a decimal analogue of Gray codes?
  - butisaidsudo 8y ago
    Hadn't seen those before! Yeah I suppose they are, although could you call Roman numerals decimal? I don't know enough about the technical definition to say. I'm assuming "decimal" and "base 10" are synonymous. And the base is the number of distinct digits you have to work with. Roman numerals don't really work that way though. Man, I remember when I first learned about different bases, it took me a bit to wrap my head around the idea that base 10 is completely arbitrary. It doesn't make more sense than the other bases, it's just what we standardized on. Now I'm realizing that maybe the idea of bases themselves aren't some rule of the universe.
mesarvagya 8y ago
I was fascinated by Lambda calculus where you could think a number as the times of application of function. That is: Symbol 0 = λs.λz.z Symbol 1 = λs.λz.s(z) Symbol 2 = λs.λz.s(s(z)) and so on. Then we can find n+1 of a given number n as: Succ n = λn.λs.λz.s(ns(z)) To even think number as function, it's astonishing. Salute to Alonzo Church.
- frabert 8y ago
  This is basically Peano numbers, hardly invented by Church. Not to discredit Church for lambda calc, of course.
  - empath75 8y ago
    The church encoding goes a lot further than that encoding conditionals and booleans and much more.
- Gajurgensen 8y ago
  Learning about Church and Scott encoding was much more interesting than I thought it would be. I was expecting it to be tedious and banal, but I came out of it feeling like I had some sort of revelation. Imagining naturals, lists, trees, etc. as function application was mind-bending at first, but it eventually clicks into place. Lambda calculus is so incredible in its combination of simple semantics, expressiveness, and abstraction/compositionality. To go a step further, think about how you would implement a predecessor function on the Church encoding of the naturals. It is much more complicated than one might expect, and perfectly motivates the introduction of Scott encodings and the fixed-point function.
  - chewxy 8y ago
    Scott encoding even has a weirder implications (if you're not used to it) if you think about it 0 ≡ λf.λx.x 1 ≡ λf.λx.x (λf.λx.x) 2 ≡ λf.λx.x (λf.λx.x (λf.λx.x)) Or if you allow meta variables in your expression: 0 ≡ λf.λx.x 1 ≡ λf.λx.x 0 2 ≡ λf.λx.x 1 This implies 2 contains 1, 1 contains 0. Bam. Scott encoding also encodes some parts of set theory. Following from this, of course, all ADTs can be encoded following the Scott encoding scheme.
nabla9 8y ago
Douglas Hofstadter's Einstein lecture (Feb 4, 2018) explains how Einstein used analogy from thermodynamics to figure out that light was quantized. https://www.youtube.com/watch?v=NXdQfPrU64g https://www.youtube.com/watch?v=NXdQfPrU64g submitted few days ago: https://news.ycombinator.com/item?id=18523667 https://news.ycombinator.com/item?id=18523667
- totemandtoken 8y ago
  I actually looked at this submission because I thought it might have something to do with Fluid Concepts and Creative Analogies [0] [0] https://cogsci.indiana.edu/book.html https://cogsci.indiana.edu/book.html
- winchling 8y ago
  It's an interesting fact that argument by analogy is a fallacy unless one details what precisely is analogous to what, what the exceptions are, etc. And yet... All ideas begin life as analogies, which are gradually refined by detailing what precisely is analogous to what, what the exceptions are, etc.
  - nabla9 8y ago
    The process of thinking has no strict rules. Anything goes. Even random search. All the requirements for consistency and good form of argumentation apply only to the "final product". When you finally present your new result it should be as solid as possible.
    - winchling 8y ago
      True!
deleted 8y ago
[deleted]
skadamat 8y ago
The first 20 mins of Bret's talk (https://www.youtube.com/watch?v=agOdP2Bmieg https://www.youtube.com/watch?v=agOdP2Bmieg) discusses representations quite well in a very similar context / framing. The challenge with roman numerals was definitely computing with them and really "thinking with them". Abacuses helped, but it was still a poor UX in many ways and it took lots of training & practice to get familiar. Algebraic notation also requires training & practice but has a much quicker learning curve. Almost every literate adult uses equations in some form or another on a weekly basis. Both the medium (pencil & paper) as well as the representation (algebraic notation) were critical to this (as well as other things like education system / literacy, etc). The benefits & disadvantages of different representations is massively understudied and misunderstood. Data visualization, semiotics, and some other fields focus a bit on this but still fall short in many ways.
- Koshkin 8y ago
  > Algebraic notation ... has a much quicker learning curve Not true - many people still struggle with algebraic notation.
garmaine 8y ago
This was amazing. I have a far better understanding of complex numbers now. Where can I find more of this?
- abecedarius 8y ago
  For intuition on complex numbers, you could try playing with http://wry.me/math-toys http://wry.me/math-toys — though it’s in a hacky state I haven’t had the energy to fix. There are much better complex-function plotters, but I was aiming for a more tangible UI.
- chrisweekly 8y ago
  For starters, the rest of the betterexplained.com site! WolframAlpha is pretty great too. I'd bet heavy other HN'ers can point you to other great sites.
Tycho 8y ago
Any good analogies for eigenvectors and eigenvalues?
- quickthrower2 8y ago
  Sailing? If you are moving in a certain direction, with he sail a certain way the wind blowing won't change it. Although it may change your speed.
qwerty456127 8y ago
Betterexplained is a game changer. Understanding math was nearly impossible when I was a schoolboy, teachers could never (and hardly ever tried) explain anything the way that would make any sense other than "just write it down, memorize and use to substitute values". And now with resources like betterexplained I get everything for what I have wasted years with no success in just seconds. This ought to replace the way people are taught at schools and in colleges, the old way should be damn outlawed (;,;)
- dTal 8y ago
  The problem is that we let experts design curricula. Experts don't see a problem with the old way, because that's how they learned it, and look at them - they're experts! It's the same reason a top-down effort to use tau instead of pi can't get momentum. People who struggle with math find it a game-changer, but mathematicians just shrug and say they don't see the point. I expect a similar dynamic explains why open source software has such terrible user interfaces, and why legalese is so impenetrable: the people with the power to change it, don't see the problem.
  - qwerty456127 8y ago
    Indeed. Experts in teaching should design curricula and write textbooks, not experts in the subjects (of course experts in the subjects should be asked to check everything is valid but not how to teach). Nevertheless it has been a long time since I've last seen an open source program with a terrible user interface. I'm writing this in Chrome (which is open source too and its GUI is quite ok) on KDE5 Plasma (which gives me aesthetical and ergonomical orgasm all the way I use it although I always hated how old versions of KDE looked and felt) on Manjaro while making notes in the Atom editor (which is ridiculously resource-inefficient yet perfect aesthetically and very intuitive and comfortable to use) and managing files in Krusader (which isn't as feature-reach as Total Commander yet looks a way more eye-candy and is almost equally convenient). Also legalese is so impenetrable only in English (and, probably, some other) language speaking countries, legal English indeed seems a language distinct from what ordinary people speak but in many countries laws are readable as easily as anything else.
jamez1 8y ago
Highly recommend this book for anyone interested in this line of thinking: How mathematicians think William P. Byers

181 points by chrisweekly 8y ago

Here’s what I wrote elsewhere about this: Why was 1997 easier for us to read than MCMDCVII? Duh... because we all have decades of experience working with the former representation and almost no experience at all working with the latter one. If you wrote a number using arbitrary different symbols and (say) base 14 instead of base 10, but still a positional number system, we would still find it incredibly difficult to interpret written numbers, because we wouldn’t be used to it. Indeed we find exactly this happens when people who are not experienced programmers try to read hexadecimal numbers written using digits 0123456789ABCDE, and that’s even in a case where we are deeply familiar with all of the symbols and their order, and in a very straightforward power-of-2 base. Romans or medieval Europeans had absolutely no problem reading numbers written like the latter example. Indeed, this is a much more straight-forward and intuitive representation, and is very easy to interpret for anyone used to working with a counting board, since it is a direct translation of the positions of counters on the counting board to symbols written on paper. [Edit to add: Note that Roman numerals were not a system for practical arithmetic – people didn’t use writing for this, instead calculating by moving counters around on a counting board — but only a written record of input data and calculated outputs.] I really hate how much people bash on Roman numerals without any significant experience working with them, and without any discussion or even acknowledgement of the historical or practical context. I would recommend anyone interested in this topic start by reading Netz’s paper “Counter Culture” about Greece 2500 years ago, http://worrydream.com/refs/Netz%20-%20Counter%20Culture%20-%20Towards%20a%20History%20of%20Greek%20Numeracy.pdf http://worrydream.com/refs/Netz%20-%20Counter%20Culture%20-%...

deleted 8y ago
[deleted]
posterboy 8y ago
This is an interesting perspective, but you have to assume that deprecating the system happened for a reason, probably due to ergonomic disdvantages. For simple counting tasks, unary tally marks are still fashionable. Edit: I never used an abakus or the like, but maybe that's what roman numerals are the analogy to, and it got overcome with long division and the like.
- jacobolus 8y ago
  There are certainly advantages to the more abstract positional Hindu–Arabic number system (or e.g. the ancient Mesopotamian system for writing sexagesimal numbers which also positional). I am not trying to dispute that. A symbolic positional system saves space on the page. It makes recorded numbers more difficult to falsify later. Doing scratch work on paper leaves a better record of calculations (the position of a counting board is inherently ephemeral) which potentially makes it easier to spot errors. It is easier to describe pen and paper arithmetic in printed books which helps spread knowledge more easily than an oral tradition. It extends more easily to larger or (after the advent of decimal fractions) smaller numbers without needing an inordinately large number of different symbols. It is more easily extended by the use of pre-computed tables of function values, and especially by slide rules and similar devices. Once we start decimalizing our metrical system it becomes very regular to work with many different measurements in a consistent way. Perhaps most importantly the use of writing leads naturally to the development of symbolic algebra and ultimately simpler formulations of higher mathematics. But we can carefully describe the advantages and historical process of replacement etc. without unfairly disparaging the more direct representation.
  - posterboy 8y ago
    MCM requires subtraction, two directions, 1900 is additional, maybe multiplicative if you want to be strict, either way only one direction. I'm not sure what's "more direct", either is linear. I might even think of 15:45 as a quarter before 4 [1]), but I'd argue that subtraction is conceptually a level higher. Nevertheless, high level representation might be an advantage. Maybe an interesting parallel viz. cognition is the 19.95 pricing scheme that's apparently proven to be confusing to the consumer. I'm just not sure which numeral system suffers more from that. [1] which marks an isogloss between east and west germany, the latter saying "dreiviertel vier" - three quarters four; German also has a weird order for decimal place reading: 121=hundred-one-twenty
    - jacobolus 8y ago
      Well MCM is not how Romans would write this except in rare cases. That one did not become a standard form until Medieval Europe. MDCCCC is absolutely a more direct representation of the counters on a counting board than 1900. Each letter precisely represents one counter and its position. But if you want, you can make a counting board with a line down the middle and use one side for negative values, in which case MCM would be a pretty direct representation of that. (It is unclear whether this was the origin of such numerals; we don’t have a whole lot of evidence about how calculation on ancient counting boards was done, because it was largely part of oral culture.)
gugagore 8y ago
I agree that it's a sort of weak argument, and I'm a bit surprised by it. Usually the claim isn't that Roman numerals are hard to read and write, it's that they're hard to calculate with. And I think that's a stronger argument. Probably you can learn algorithms for doing calculations on Roman numerals, and we're familiar with our algorithms because of decades of experience. But the algorithms we learn in school actually do work marvelously for other bases.
- jacobolus 8y ago
  People never used Roman numerals for performing written arithmetic, so that is not a fair comparison. Calculations were done mentally, with fingers, or for anything complicated or serious on a counting board.
  - JoeAltmaier 8y ago
    I believe the ancient Romans would disagree? They had complicated rules for borrowing and simplifying, but of course it had to be done or they had nothing.
    - Koshkin 8y ago
      The Romans heavily relied on the abacus. It used the 10-based positional representation of numbers.
    - jacobolus 8y ago
      > complicated rules for borrowing and simplifying [..] or they had nothing Maybe you can explain what you are referring to here? Ancient Greeks and Romans simply did not do arithmetic with pen and paper. They used a counting board with counters or pebbles (“calculi”), or for simpler problems used their fingers or did the work in their heads. Doing computation was literally called “placing pebbles”, and to settle accounts with someone you would “call them to the pebbles”. Roman numerals were not a tool for computation. They were a tool for recording data or results. The purpose of Roman numerals was to faithfully record in writing the placement of counters on a counting board. The position of the counting board (or perhaps in some contexts a pile of coins of various denominations) was the primary representation of numbers, and Roman numerals were secondary.
bena 8y ago
Because MCMXCVII is actually 1997. MCMDCVII is just wrong. D is 500, X is ten.
- Reisen 8y ago
  Kind of a nice proof of his point! We're not used to the numbers, people would be far less likely to make the mistake with 1997.
- jacobolus 8y ago
  Yes that was a typo.

netgusto 8y ago

Video giving some intuition about e^(i•PI)=-1 https://www.youtube.com/watch?v=F_0yfvm0UoU https://www.youtube.com/watch?v=F_0yfvm0UoU

thaumasiotes 8y ago
The intuition is as simple as it gets; r·e^{iθ} is the complex number with radius r and angle θ. pi is just the angle that puts you on the negative half of the real number line. For why that identity holds, you do simple algebra on the non-simple Taylor series of those three functions (e^x, sin x, cos x). It is, to say the least, much less intuitive.
- chrisweekly 8y ago
  The linked article mentions how helpful it is to have the right analogy (e.g. a number line, which you reference in your explanation of its intuitivenes). That's the main point: given the right analogy, things that otherwise seem arbitrary or nonsensical become clear and even intuitive. :)

butisaidsudo 8y ago

For Roman numerals to make sense you need to remove access to convenient writing tools. In a world without paper and pens, Arabic numerals are the ones that seem unwieldy. I was on a multi-day hiking trip with a friend where we just had a pack of cards and some spiced rum to keep ourselves entertained in the evening. The first night we decided to play cribbage, but we didn't have paper or the like to keep score on. At first I tried using a stick to scratch out numbers in the dirt. It was doable, but very awkward. The low light of the camp fire made it hard to see, and I quickly ran out of undisturbed soil within arms reach. I then gathered a few twigs to shape into numbers, using the patterns you'd see on an LED clock. That worked reasonably well, but took more effort than I liked. I switched to Roman numerals thinking that the simpler shapes would be easier to work with. This turned out to be true, but I discovered it had the added benefit of being really easy to increment numbers. In most cases, incrementing is incredibly simple. To go from 0 -> 1 -> 2 -> 3, you simply add a stick each time. To go from 3 -> 4, you pinch together the bottoms of the 2nd and 3rd sticks. Then you take the first stick away, then later drop it on the other side of the V. Add another stick, cross some sticks, etc. There are a lot of things that seem poorly done at first glance, but make total sense when you understand the environment they were developed in. I've heard younger folks wonder why old TV shows are so poorly written. When you've always been able to watch on demand (or rent episodes on DVD), it's hard to understand the restrictions on writing when there was no way for your audience to watch previous episodes if they hadn't seen them when they had aired. And of course this is true for most software I've ever worked on.

adrianmonk 8y ago
> incrementing is incredibly simple So Roman numerals are like a decimal analogue of Gray codes?
- butisaidsudo 8y ago
  Hadn't seen those before! Yeah I suppose they are, although could you call Roman numerals decimal? I don't know enough about the technical definition to say. I'm assuming "decimal" and "base 10" are synonymous. And the base is the number of distinct digits you have to work with. Roman numerals don't really work that way though. Man, I remember when I first learned about different bases, it took me a bit to wrap my head around the idea that base 10 is completely arbitrary. It doesn't make more sense than the other bases, it's just what we standardized on. Now I'm realizing that maybe the idea of bases themselves aren't some rule of the universe.

mesarvagya 8y ago

I was fascinated by Lambda calculus where you could think a number as the times of application of function. That is: Symbol 0 = λs.λz.z Symbol 1 = λs.λz.s(z) Symbol 2 = λs.λz.s(s(z)) and so on. Then we can find n+1 of a given number n as: Succ n = λn.λs.λz.s(ns(z)) To even think number as function, it's astonishing. Salute to Alonzo Church.

frabert 8y ago
This is basically Peano numbers, hardly invented by Church. Not to discredit Church for lambda calc, of course.
- empath75 8y ago
  The church encoding goes a lot further than that encoding conditionals and booleans and much more.
Gajurgensen 8y ago
Learning about Church and Scott encoding was much more interesting than I thought it would be. I was expecting it to be tedious and banal, but I came out of it feeling like I had some sort of revelation. Imagining naturals, lists, trees, etc. as function application was mind-bending at first, but it eventually clicks into place. Lambda calculus is so incredible in its combination of simple semantics, expressiveness, and abstraction/compositionality. To go a step further, think about how you would implement a predecessor function on the Church encoding of the naturals. It is much more complicated than one might expect, and perfectly motivates the introduction of Scott encodings and the fixed-point function.
- chewxy 8y ago
  Scott encoding even has a weirder implications (if you're not used to it) if you think about it 0 ≡ λf.λx.x 1 ≡ λf.λx.x (λf.λx.x) 2 ≡ λf.λx.x (λf.λx.x (λf.λx.x)) Or if you allow meta variables in your expression: 0 ≡ λf.λx.x 1 ≡ λf.λx.x 0 2 ≡ λf.λx.x 1 This implies 2 contains 1, 1 contains 0. Bam. Scott encoding also encodes some parts of set theory. Following from this, of course, all ADTs can be encoded following the Scott encoding scheme.

nabla9 8y ago

Douglas Hofstadter's Einstein lecture (Feb 4, 2018) explains how Einstein used analogy from thermodynamics to figure out that light was quantized. https://www.youtube.com/watch?v=NXdQfPrU64g https://www.youtube.com/watch?v=NXdQfPrU64g submitted few days ago: https://news.ycombinator.com/item?id=18523667 https://news.ycombinator.com/item?id=18523667

totemandtoken 8y ago
I actually looked at this submission because I thought it might have something to do with Fluid Concepts and Creative Analogies [0] [0] https://cogsci.indiana.edu/book.html https://cogsci.indiana.edu/book.html
winchling 8y ago
It's an interesting fact that argument by analogy is a fallacy unless one details what precisely is analogous to what, what the exceptions are, etc. And yet... All ideas begin life as analogies, which are gradually refined by detailing what precisely is analogous to what, what the exceptions are, etc.
- nabla9 8y ago
  The process of thinking has no strict rules. Anything goes. Even random search. All the requirements for consistency and good form of argumentation apply only to the "final product". When you finally present your new result it should be as solid as possible.
  - winchling 8y ago
    True!

deleted 8y ago

[deleted]

skadamat 8y ago

The first 20 mins of Bret's talk (https://www.youtube.com/watch?v=agOdP2Bmieg https://www.youtube.com/watch?v=agOdP2Bmieg) discusses representations quite well in a very similar context / framing. The challenge with roman numerals was definitely computing with them and really "thinking with them". Abacuses helped, but it was still a poor UX in many ways and it took lots of training & practice to get familiar. Algebraic notation also requires training & practice but has a much quicker learning curve. Almost every literate adult uses equations in some form or another on a weekly basis. Both the medium (pencil & paper) as well as the representation (algebraic notation) were critical to this (as well as other things like education system / literacy, etc). The benefits & disadvantages of different representations is massively understudied and misunderstood. Data visualization, semiotics, and some other fields focus a bit on this but still fall short in many ways.

Koshkin 8y ago
> Algebraic notation ... has a much quicker learning curve Not true - many people still struggle with algebraic notation.

garmaine 8y ago

This was amazing. I have a far better understanding of complex numbers now. Where can I find more of this?

abecedarius 8y ago
For intuition on complex numbers, you could try playing with http://wry.me/math-toys http://wry.me/math-toys — though it’s in a hacky state I haven’t had the energy to fix. There are much better complex-function plotters, but I was aiming for a more tangible UI.
chrisweekly 8y ago
For starters, the rest of the betterexplained.com site! WolframAlpha is pretty great too. I'd bet heavy other HN'ers can point you to other great sites.

Tycho 8y ago

Any good analogies for eigenvectors and eigenvalues?

quickthrower2 8y ago
Sailing? If you are moving in a certain direction, with he sail a certain way the wind blowing won't change it. Although it may change your speed.

qwerty456127 8y ago

Betterexplained is a game changer. Understanding math was nearly impossible when I was a schoolboy, teachers could never (and hardly ever tried) explain anything the way that would make any sense other than "just write it down, memorize and use to substitute values". And now with resources like betterexplained I get everything for what I have wasted years with no success in just seconds. This ought to replace the way people are taught at schools and in colleges, the old way should be damn outlawed (;,;)

dTal 8y ago
The problem is that we let experts design curricula. Experts don't see a problem with the old way, because that's how they learned it, and look at them - they're experts! It's the same reason a top-down effort to use tau instead of pi can't get momentum. People who struggle with math find it a game-changer, but mathematicians just shrug and say they don't see the point. I expect a similar dynamic explains why open source software has such terrible user interfaces, and why legalese is so impenetrable: the people with the power to change it, don't see the problem.
- qwerty456127 8y ago
  Indeed. Experts in teaching should design curricula and write textbooks, not experts in the subjects (of course experts in the subjects should be asked to check everything is valid but not how to teach). Nevertheless it has been a long time since I've last seen an open source program with a terrible user interface. I'm writing this in Chrome (which is open source too and its GUI is quite ok) on KDE5 Plasma (which gives me aesthetical and ergonomical orgasm all the way I use it although I always hated how old versions of KDE looked and felt) on Manjaro while making notes in the Atom editor (which is ridiculously resource-inefficient yet perfect aesthetically and very intuitive and comfortable to use) and managing files in Krusader (which isn't as feature-reach as Total Commander yet looks a way more eye-candy and is almost equally convenient). Also legalese is so impenetrable only in English (and, probably, some other) language speaking countries, legal English indeed seems a language distinct from what ordinary people speak but in many countries laws are readable as easily as anything else.

jamez1 8y ago

Highly recommend this book for anyone interested in this line of thinking: How mathematicians think William P. Byers