8 ms·

Surreal Numbers – An Introduction (2019) [pdf]

110 points by ColinWright 5y ago

weatherlight 5y ago
"Finally, I hope that this paper may glorify God by showing – once again – the beauty of His world, a beauty that is demonstrated here by the wealth that springs out of pure logic." It was hard for me to get past that point.
- jan_Inkepa 5y ago
  At the risk of being slightly glib, it pales in comparison to the copy most startups put out.
- JoeyBananas 5y ago
  Should have put a trigger warning
- dvt 5y ago
  > It was hard for me to get past that point. Why does it bother you that a logician happens to be religious? I myself am a Christian, and some of the most brilliant people I know happen to be religious (Christian, Muslim, Catholic, Jewish, or otherwise). In fact some of the most brilliant logicians in the history of the world have been religious (from Avicenna, to Anselm, to Gödel, to Kripke).
  - lisper 5y ago
    Not the OP, but what bothers me is not that Knuth is relgious (but see update below), that is perfectly fine. What bothers me is that he chooses to proselytize his religion in a technical publication. That seems inappropriate to me. Imagine the reaction if instead of glorifying God, he had written instead, "Finally, I hope that this paper may glorify Allah the compassionate and merciful..." [UPDATE] I made a mistake here and assumed that Knuth had written this because I happen to know that he's a Christian, but I was wrong about that. And in retrospect I should have realized that I was wrong because Knuth is actually quite scrupulous about keeping his religious views private. But I stand by the substance of my comment subject to s/Knuth/Tøndering.
    - dvt 5y ago
      > Not the OP, but what bothers me is not that Knuth is relgious, that is perfectly fine. What bothers me is that he chooses to proselytize his religion in a technical publication. That seems inappropriate to me. First of all, the quote is not by Knuth, but by Claus Tøndering (the author of the paper linked). Second of all, the "Preface," where the quote is snipped from, is just that -- a preface -- a lot of people thank their moms in there. It's just much ado about nothing and annoying posturing. Why is why I, and I'd assume others, take issue with it.
      - lisper 5y ago
        > the quote is not by Knuth, but by Claus Tøndering Sorry, my mistake. I knew Knuth was religious so I assumed it was his quote. I stand by the substance of my comment though. But I agree with you that it's not a huge big deal.
    - Bancakes 5y ago
      Math is made by humans to be learned and applied by humans. There's no need for marketing in the form of pretext formalities. If anything, we should savour the human metadata to math - the last decades before AI takes over.
  - E-Reverance 5y ago
    There is an irony to referring to faith when discussing logic.
    - eru 5y ago
      Why? Modern math (and logic) explicitly don't make any absolute claims. They are all just "if you take these axioms as given, then you can conclude these theorems." Math doesn't care where you get your axioms from, or why you take them as given. So faith is just as good or bad a source as intuition or empirical evidence as far as logic is concerned. I'm not religious myself, but compatibility with logic is a really, really low bar to clear for any Weltanschauung. (And you can believe that Knuth and the author of the paper are smart enough to be able to reason away any inconsistencies.)
      - necovek 5y ago
        Wonderfully put! To be honest, I struggle to explain this to mathematicians even: many of them do not grasp the fact that axioms are basically the things you put your faith in. I only claim that the main difference between science and religion, which are both based on faith in unprovable statements, is that science invites you to challenge those statements (easy example is the axiom of parallelism), whereas religion doesn't.
        eru 5y ago
        In some sense, mathematics is just a game you play. Playing round with different axioms is encouraged. Another view at axioms is that they are like an API: Whenever you find a system (real world or elsewhere in math) that fits the axioms of group theory, you can apply all kinds of already proven theorems to it. Very much like building a library on top of an API spec. In this perspective, faith doesn't come into the picture at all: the axioms of a specific theory are just the shape of the socket that you need to fit your plug in to get already proven theorems for free. (A somewhat similar special case is reductions in computer science: where eg you show that some new problem is at least as hard as one we already know to be NP complete.)
  - Dylan16807 5y ago
    I'm a bit weirded out by the implication that a purely abstract number system could be possible or impossible based on the actions of any entity, even a god. That's the only way I can see for God to be glorified in the way described. Yet that would mean that it would be possible to create a universe that has free-thinking mathematicians but in which surreal numbers could not exist even as a concept. Which is quite a claim.
    - idiotsecant 5y ago
      I don't think that's so weird. An omnipotent entity controls the rules of reality. It could construct a universe in which contradictory things are true, or whatever it wanted. This is a bit like the puddle assuming the world was made for it because it fits so well into it - we take the rules of our reality for granted but that need not be the case in this thought experiment.
      - Dylan16807 5y ago
        The rules of reality apply to reality, though. It's easy for an omnipotent power to make physics do seemingly contradictory things. But pure logic is not part of reality. You could talk about how a real-life implementation of some element of math doesn't match the logical version derived from a specific set of axioms, but there's no real-life version of surreal numbers in our universe anyway so that's basically true already.
        idiotsecant 5y ago
        Are you so sure that logic is separate from reality? A great deal of logic depends on physics. Reversing the arrow of entropy, for example, pretty much breaks 'if x, therefore y' statements.
  - seventytwo 5y ago
    It all depends on which axioms you start with…
- maxbond 5y ago
  You can safely ignore it, the document is just a summary of Knuth's book. And it's a good one, I found it very helpful when I was working through the book. I found I had to read the material from several different sources to understand, in addition to working things out on paper/in code. I'm not religious myself but honestly that seems like a perfectly reasonable motivation. While there are some people who believe the Surreals could lead to a deeper form of analysis, I think the most common reason to learn them is to apprehend something beautiful without any sort of utility. But as a heads up, if that doesn't sit right with you, then Knuth's book is probably not right for you either; it's framed in the style of the old testament. On the first day, Conway created zero, and saw that it was good; things like that. This may be more to your taste. https://www.youtube.com/watch?v=mPn2AdMH7UQ https://www.youtube.com/watch?v=mPn2AdMH7UQ https://www.youtube.com/watch?v=1eAmxgINXrE https://www.youtube.com/watch?v=1eAmxgINXrE https://core.ac.uk/download/pdf/51289919.pdf https://core.ac.uk/download/pdf/51289919.pdf
  - dvt 5y ago
    > You can safely ignore it, the document is just a summary of Knuth's book. Knuth is a Lutheran, by the way (uh-oh). He even wrote 3:16 Bible Texts Illuminated[1]. [1] https://www.amazon.com/3-16-Bible-Texts-Illuminated/dp/0895792524 https://www.amazon.com/3-16-Bible-Texts-Illuminated/dp/08957...
    - maxbond 5y ago
      I didn't know that, that's interesting. Thank you. To clarify, I didn't mean, you can ignore it because it's irrelevant, as much as, you can derive the bulk of the value from the text if you ignore the introduction.
    - dhosek 5y ago
      Perhaps even more interesting is Knuth’s Things a Computer Scientist Rarely Talks About which goes into a bit more of his religious life and how he applies some mathematical techniques (particularly Monte Carlo–style random selection). The genesis of 3:16 (so to speak) is discussed as well. http://www.amazon.com/exec/obidos/ASIN/157586326X/donhosek http://www.amazon.com/exec/obidos/ASIN/157586326X/donhosek
- baggy_trough 5y ago
  What a bigoted remark.
- p1necone 5y ago
  Mentally replace "God" with "the space in which we exist" if it bothers you so much. EDIT: another commenter suggested "all that exists and the way that things turned out", I like that more. I'm very much an atheist, but I appreciate the appreciation of Gods creation from theists too, I just substitute my own more abstract idea of God when I observe the same beauty.
  - weatherlight 5y ago
    "God" with "the space in which we exist", if these are equivalent for you, so be it. I have little patiences with people definitions of what god is or isn't and the last place I expect to grapple with such whimsical notions is in a paper like such. I'm reluctant to "glorify God," "or even have the hubris of doing such in such a place or setting. I'd rather not grapple with a stranger's idea of what is (or isn't!) worthy of worship. Forgive me, I shut down.
- Bancakes 5y ago
  If you despise God, you despise love, by definition. I don't know who hurt you.
  - dang 5y ago
    Yikes, taking HN threads further into religious flamewar is not cool, and crossing into personal attack is no way to serve God, or love! Please don't do that here. https://news.ycombinator.com/newsguidelines.html https://news.ycombinator.com/newsguidelines.html
    - Bancakes 5y ago
      It's not a flame war and certainly not a personal attack. From a purely syntactic programming standpoint, without thinking about definitions, "God" implies "love". This isn't an opinion, every modern religion clearly defines "God" this way. If you deny a property of a set, you also deny the property of all its supersets. It's cold math.
      - dang 5y ago
        "I don't know who hurt you" is certainly a personal attack and an internet flamewar trope to boot. No more of that, please. Making provocative statements about God in a religious flamewar thread is obviously taking the thread further into religious flamewar. No more of that either, please. Edit: see also https://news.ycombinator.com/item?id=27371909 https://news.ycombinator.com/item?id=27371909. If you don't to be banned on HN, we're going to need you to change how you're posting to the site.
- dang 5y ago
  Please leave provocations (such as religious provocations) in place rather than opening a new loop in the HN thread. That just replaces the actual topic with a secondary flamewar, which only makes this place worse. https://news.ycombinator.com/newsguidelines.html https://news.ycombinator.com/newsguidelines.html
  - weatherlight 5y ago
    please see my response below. people always assume the worst.
    - dang 5y ago
      I certainly don't want to assume the worst, but we have to go by the comments people actually post and the known effects of such things [1], i.e. what sort of subthread they're expected to lead to [2]. Your GP comment was unsubstantive, on perhaps the most flammable of flamewar topics. That made it flamebait, and it was completely off topic to boot. I'm sure you didn't intend to start a flamewar, but that's only a necessary condition, not a sufficient condition, for not starting one. Please don't post like that to HN. [1] https://hn.algolia.com/?dateRange=all&page=0&prefix=true&sort=byDate&type=comment&query=troll%20effects%20by:dang https://hn.algolia.com/?dateRange=all&page=0&prefix=true&sor... [2] https://hn.algolia.com/?dateRange=all&page=0&prefix=true&sort=byDate&type=comment&query=%22expected%20value%22%20by:dang https://hn.algolia.com/?dateRange=all&page=0&prefix=true&sor...
tingletech 5y ago
2019
- dang 5y ago
  Added above. Thanks!
rpz 5y ago
I prefer the hyperreals
- dhosek 5y ago
  hyperreals ⊂ surreals
- dang 5y ago
  Ok, but please don't post unsubstantive comments to Hacker News.
galaxyLogic 5y ago
Don't get weirded out by somebody thanking God for something. People thank God for many things, for instance thank God it's Friday.
- dang 5y ago
  I'm sure that your comment was well-intentioned, but it's best to leave provocations (such as religious provocations) in place rather than opening a new loop in the threads. https://news.ycombinator.com/newsguidelines.html https://news.ycombinator.com/newsguidelines.html
knuthsat 5y ago
Knuth has a nice book on Surreal numbers. https://www-cs-faculty.stanford.edu/~knuth/sn.html https://www-cs-faculty.stanford.edu/~knuth/sn.html This interview is excellent https://www.youtube.com/watch?v=mPn2AdMH7UQ https://www.youtube.com/watch?v=mPn2AdMH7UQ He wrote the book in less than a week, the book was just flowing for days and after that he couldn't write a thing. Then he realized he missed a detail and had to rewrite the whole book.
ianopolous 5y ago
Here's an intro to surreal numbers I wrote a while ago: https://ianopolous.peergos.me/maths/surreal https://ianopolous.peergos.me/maths/surreal
- najarvg 5y ago
  This is a wonderful intro! Thanks for sharing..
  - ianopolous 5y ago
    Thanks! Glad you like it. I wrote it as an undergrad for the Oxford maths magazine at the time, Eureka. I'm still blown away by how beautiful Surreal numbers are.
dang 5y ago
I thought there had been more threads on this, but found only one: Donald Knuth: Surreal Numbers [video] - https://news.ycombinator.com/item?id=11987415 https://news.ycombinator.com/item?id=11987415 - June 2016 (54 comments)
- Vivtek 5y ago
  Do you literally just do this deduplication with your brain alone? Or do you have some kind of clever indexing scheme and, if the latter, have you ever written it up? Because that would be interesting to read about.
  - dang 5y ago
    It is all explained at https://news.ycombinator.com/item?id=27284079 https://news.ycombinator.com/item?id=27284079 and the links back from there.
roenxi 5y ago
One helpful intuition for the surreals - the standard numbers are basically mapped to a ruler where the 'gap' between any two consecutive numbers (eg, 4 and 5) is intuitively equivalent to any other (eg, the gap between 0 and 1). The concept of distance is abandoned in the construction of the surreals so it no longer makes sense to talk about the gap between 1 and 2 being the same as the gap between 2 and 3. It is a very pure conception of numbers and it is quite pretty that it contains everything (integers, reals, infinitesimals, infinities, concept of infinity + 1, etc.
foo92691 5y ago
Wikipedia also has a nice overview: https://en.wikipedia.org/wiki/Surreal_number https://en.wikipedia.org/wiki/Surreal_number
shannifin 5y ago
I was introduced to this concept through the Numberphile video and thought it was more about having fun with symbols / notation than contributing anything mathematically new or useful. I particularly object to the playing with infinities and infinitesimals as though they were numbers. As the pdf says on page 41: > "According to theorem 5, {Z | } is a number that is greater than all integers. Therefore, its value is infinity!" That makes no sense, nor does much of the exploration from this point. Other than playing with symbols, I'm not sure it has any value by itself. That said, "playing with symbols" can still be of great value as how we symbolize and notate things can change our understanding of things.
- ColinWright 5y ago
  When we analyse games, especially combinatorial games, we find that for a given game we can give positions "values". Then there's a concept of "Adding" game positions, and the value of the sum of positions is the sum of the values of the positions. So using "P" for position, and V(P) for the value of a position, we have V(P1+P2) = V(P1)+V(P2). (The additions here are on different types.) So things work nicely, and it feels natural. Then we find that the integers can be embedded naturally into the values we get. So the values from positions form an algebraic structure that extends the integers. Then we find that there are game positions whose values are larger than all the integers. The obvious interpretation of this is that they are infinite, but are all part of a larger structure, one where addition and subtraction of these things still makes sense, and are still well defined. When you say "That makes no sense", you're thinking of this in the limited space of counting, and ordinary numbers. But there are wider contexts where "simple counting" isn't enough, and in those cases it makes perfect sense, and is useful.
  - shannifin 5y ago
    > "Then we find that there are game positions whose values are larger than all the integers. The obvious interpretation of this is that they are infinite, but are all part of a larger structure, one where addition and subtraction of these things still makes sense, and are still well defined." But isn't this really more about notation than having "values" that are necessarily "larger than all integers"? That is, isn't this about needing a convenient way to notate something when we're already using integers for something else? And/or do you know of any books or resources that go into more detail about this? (As far as I can tell, the idea of surreal numbers still imagines a one-dimensional layout through which numbers are ordered. It's the adding of "infinity" and "1/infinity" to this line to which I object on nonsense grounds; using a non-integer symbol to represent an extension of some sort (non-spatially related) for some context still makes sense, I just don't think it implies a value "greater than all integers".)

110 points by ColinWright 5y ago

"Finally, I hope that this paper may glorify God by showing – once again – the beauty of His world, a beauty that is demonstrated here by the wealth that springs out of pure logic." It was hard for me to get past that point.

jan_Inkepa 5y ago
At the risk of being slightly glib, it pales in comparison to the copy most startups put out.
JoeyBananas 5y ago
Should have put a trigger warning
dvt 5y ago
> It was hard for me to get past that point. Why does it bother you that a logician happens to be religious? I myself am a Christian, and some of the most brilliant people I know happen to be religious (Christian, Muslim, Catholic, Jewish, or otherwise). In fact some of the most brilliant logicians in the history of the world have been religious (from Avicenna, to Anselm, to Gödel, to Kripke).
- lisper 5y ago
  Not the OP, but what bothers me is not that Knuth is relgious (but see update below), that is perfectly fine. What bothers me is that he chooses to proselytize his religion in a technical publication. That seems inappropriate to me. Imagine the reaction if instead of glorifying God, he had written instead, "Finally, I hope that this paper may glorify Allah the compassionate and merciful..." [UPDATE] I made a mistake here and assumed that Knuth had written this because I happen to know that he's a Christian, but I was wrong about that. And in retrospect I should have realized that I was wrong because Knuth is actually quite scrupulous about keeping his religious views private. But I stand by the substance of my comment subject to s/Knuth/Tøndering.
  - dvt 5y ago
    > Not the OP, but what bothers me is not that Knuth is relgious, that is perfectly fine. What bothers me is that he chooses to proselytize his religion in a technical publication. That seems inappropriate to me. First of all, the quote is not by Knuth, but by Claus Tøndering (the author of the paper linked). Second of all, the "Preface," where the quote is snipped from, is just that -- a preface -- a lot of people thank their moms in there. It's just much ado about nothing and annoying posturing. Why is why I, and I'd assume others, take issue with it.
    - lisper 5y ago
      > the quote is not by Knuth, but by Claus Tøndering Sorry, my mistake. I knew Knuth was religious so I assumed it was his quote. I stand by the substance of my comment though. But I agree with you that it's not a huge big deal.
  - Bancakes 5y ago
    Math is made by humans to be learned and applied by humans. There's no need for marketing in the form of pretext formalities. If anything, we should savour the human metadata to math - the last decades before AI takes over.
- E-Reverance 5y ago
  There is an irony to referring to faith when discussing logic.
  - eru 5y ago
    Why? Modern math (and logic) explicitly don't make any absolute claims. They are all just "if you take these axioms as given, then you can conclude these theorems." Math doesn't care where you get your axioms from, or why you take them as given. So faith is just as good or bad a source as intuition or empirical evidence as far as logic is concerned. I'm not religious myself, but compatibility with logic is a really, really low bar to clear for any Weltanschauung. (And you can believe that Knuth and the author of the paper are smart enough to be able to reason away any inconsistencies.)
    - necovek 5y ago
      Wonderfully put! To be honest, I struggle to explain this to mathematicians even: many of them do not grasp the fact that axioms are basically the things you put your faith in. I only claim that the main difference between science and religion, which are both based on faith in unprovable statements, is that science invites you to challenge those statements (easy example is the axiom of parallelism), whereas religion doesn't.
      - eru 5y ago
        In some sense, mathematics is just a game you play. Playing round with different axioms is encouraged. Another view at axioms is that they are like an API: Whenever you find a system (real world or elsewhere in math) that fits the axioms of group theory, you can apply all kinds of already proven theorems to it. Very much like building a library on top of an API spec. In this perspective, faith doesn't come into the picture at all: the axioms of a specific theory are just the shape of the socket that you need to fit your plug in to get already proven theorems for free. (A somewhat similar special case is reductions in computer science: where eg you show that some new problem is at least as hard as one we already know to be NP complete.)
- Dylan16807 5y ago
  I'm a bit weirded out by the implication that a purely abstract number system could be possible or impossible based on the actions of any entity, even a god. That's the only way I can see for God to be glorified in the way described. Yet that would mean that it would be possible to create a universe that has free-thinking mathematicians but in which surreal numbers could not exist even as a concept. Which is quite a claim.
  - idiotsecant 5y ago
    I don't think that's so weird. An omnipotent entity controls the rules of reality. It could construct a universe in which contradictory things are true, or whatever it wanted. This is a bit like the puddle assuming the world was made for it because it fits so well into it - we take the rules of our reality for granted but that need not be the case in this thought experiment.
    - Dylan16807 5y ago
      The rules of reality apply to reality, though. It's easy for an omnipotent power to make physics do seemingly contradictory things. But pure logic is not part of reality. You could talk about how a real-life implementation of some element of math doesn't match the logical version derived from a specific set of axioms, but there's no real-life version of surreal numbers in our universe anyway so that's basically true already.
      - idiotsecant 5y ago
        Are you so sure that logic is separate from reality? A great deal of logic depends on physics. Reversing the arrow of entropy, for example, pretty much breaks 'if x, therefore y' statements.
- seventytwo 5y ago
  It all depends on which axioms you start with…
maxbond 5y ago
You can safely ignore it, the document is just a summary of Knuth's book. And it's a good one, I found it very helpful when I was working through the book. I found I had to read the material from several different sources to understand, in addition to working things out on paper/in code. I'm not religious myself but honestly that seems like a perfectly reasonable motivation. While there are some people who believe the Surreals could lead to a deeper form of analysis, I think the most common reason to learn them is to apprehend something beautiful without any sort of utility. But as a heads up, if that doesn't sit right with you, then Knuth's book is probably not right for you either; it's framed in the style of the old testament. On the first day, Conway created zero, and saw that it was good; things like that. This may be more to your taste. https://www.youtube.com/watch?v=mPn2AdMH7UQ https://www.youtube.com/watch?v=mPn2AdMH7UQ https://www.youtube.com/watch?v=1eAmxgINXrE https://www.youtube.com/watch?v=1eAmxgINXrE https://core.ac.uk/download/pdf/51289919.pdf https://core.ac.uk/download/pdf/51289919.pdf
- dvt 5y ago
  > You can safely ignore it, the document is just a summary of Knuth's book. Knuth is a Lutheran, by the way (uh-oh). He even wrote 3:16 Bible Texts Illuminated[1]. [1] https://www.amazon.com/3-16-Bible-Texts-Illuminated/dp/0895792524 https://www.amazon.com/3-16-Bible-Texts-Illuminated/dp/08957...
  - maxbond 5y ago
    I didn't know that, that's interesting. Thank you. To clarify, I didn't mean, you can ignore it because it's irrelevant, as much as, you can derive the bulk of the value from the text if you ignore the introduction.
  - dhosek 5y ago
    Perhaps even more interesting is Knuth’s Things a Computer Scientist Rarely Talks About which goes into a bit more of his religious life and how he applies some mathematical techniques (particularly Monte Carlo–style random selection). The genesis of 3:16 (so to speak) is discussed as well. http://www.amazon.com/exec/obidos/ASIN/157586326X/donhosek http://www.amazon.com/exec/obidos/ASIN/157586326X/donhosek
baggy_trough 5y ago
What a bigoted remark.
p1necone 5y ago
Mentally replace "God" with "the space in which we exist" if it bothers you so much. EDIT: another commenter suggested "all that exists and the way that things turned out", I like that more. I'm very much an atheist, but I appreciate the appreciation of Gods creation from theists too, I just substitute my own more abstract idea of God when I observe the same beauty.
- weatherlight 5y ago
  "God" with "the space in which we exist", if these are equivalent for you, so be it. I have little patiences with people definitions of what god is or isn't and the last place I expect to grapple with such whimsical notions is in a paper like such. I'm reluctant to "glorify God," "or even have the hubris of doing such in such a place or setting. I'd rather not grapple with a stranger's idea of what is (or isn't!) worthy of worship. Forgive me, I shut down.
Bancakes 5y ago
If you despise God, you despise love, by definition. I don't know who hurt you.
- dang 5y ago
  Yikes, taking HN threads further into religious flamewar is not cool, and crossing into personal attack is no way to serve God, or love! Please don't do that here. https://news.ycombinator.com/newsguidelines.html https://news.ycombinator.com/newsguidelines.html
  - Bancakes 5y ago
    It's not a flame war and certainly not a personal attack. From a purely syntactic programming standpoint, without thinking about definitions, "God" implies "love". This isn't an opinion, every modern religion clearly defines "God" this way. If you deny a property of a set, you also deny the property of all its supersets. It's cold math.
    - dang 5y ago
      "I don't know who hurt you" is certainly a personal attack and an internet flamewar trope to boot. No more of that, please. Making provocative statements about God in a religious flamewar thread is obviously taking the thread further into religious flamewar. No more of that either, please. Edit: see also https://news.ycombinator.com/item?id=27371909 https://news.ycombinator.com/item?id=27371909. If you don't to be banned on HN, we're going to need you to change how you're posting to the site.
dang 5y ago
Please leave provocations (such as religious provocations) in place rather than opening a new loop in the HN thread. That just replaces the actual topic with a secondary flamewar, which only makes this place worse. https://news.ycombinator.com/newsguidelines.html https://news.ycombinator.com/newsguidelines.html
- weatherlight 5y ago
  please see my response below. people always assume the worst.
  - dang 5y ago
    I certainly don't want to assume the worst, but we have to go by the comments people actually post and the known effects of such things [1], i.e. what sort of subthread they're expected to lead to [2]. Your GP comment was unsubstantive, on perhaps the most flammable of flamewar topics. That made it flamebait, and it was completely off topic to boot. I'm sure you didn't intend to start a flamewar, but that's only a necessary condition, not a sufficient condition, for not starting one. Please don't post like that to HN. [1] https://hn.algolia.com/?dateRange=all&page=0&prefix=true&sort=byDate&type=comment&query=troll%20effects%20by:dang https://hn.algolia.com/?dateRange=all&page=0&prefix=true&sor... [2] https://hn.algolia.com/?dateRange=all&page=0&prefix=true&sort=byDate&type=comment&query=%22expected%20value%22%20by:dang https://hn.algolia.com/?dateRange=all&page=0&prefix=true&sor...

tingletech 5y ago

2019

dang 5y ago
Added above. Thanks!

rpz 5y ago

I prefer the hyperreals

dhosek 5y ago
hyperreals ⊂ surreals
dang 5y ago
Ok, but please don't post unsubstantive comments to Hacker News.

galaxyLogic 5y ago

Don't get weirded out by somebody thanking God for something. People thank God for many things, for instance thank God it's Friday.

dang 5y ago
I'm sure that your comment was well-intentioned, but it's best to leave provocations (such as religious provocations) in place rather than opening a new loop in the threads. https://news.ycombinator.com/newsguidelines.html https://news.ycombinator.com/newsguidelines.html

knuthsat 5y ago

Knuth has a nice book on Surreal numbers. https://www-cs-faculty.stanford.edu/~knuth/sn.html https://www-cs-faculty.stanford.edu/~knuth/sn.html This interview is excellent https://www.youtube.com/watch?v=mPn2AdMH7UQ https://www.youtube.com/watch?v=mPn2AdMH7UQ He wrote the book in less than a week, the book was just flowing for days and after that he couldn't write a thing. Then he realized he missed a detail and had to rewrite the whole book.

ianopolous 5y ago

Here's an intro to surreal numbers I wrote a while ago: https://ianopolous.peergos.me/maths/surreal https://ianopolous.peergos.me/maths/surreal

najarvg 5y ago
This is a wonderful intro! Thanks for sharing..
- ianopolous 5y ago
  Thanks! Glad you like it. I wrote it as an undergrad for the Oxford maths magazine at the time, Eureka. I'm still blown away by how beautiful Surreal numbers are.

dang 5y ago

I thought there had been more threads on this, but found only one: Donald Knuth: Surreal Numbers [video] - https://news.ycombinator.com/item?id=11987415 https://news.ycombinator.com/item?id=11987415 - June 2016 (54 comments)

Vivtek 5y ago
Do you literally just do this deduplication with your brain alone? Or do you have some kind of clever indexing scheme and, if the latter, have you ever written it up? Because that would be interesting to read about.
- dang 5y ago
  It is all explained at https://news.ycombinator.com/item?id=27284079 https://news.ycombinator.com/item?id=27284079 and the links back from there.

roenxi 5y ago

One helpful intuition for the surreals - the standard numbers are basically mapped to a ruler where the 'gap' between any two consecutive numbers (eg, 4 and 5) is intuitively equivalent to any other (eg, the gap between 0 and 1). The concept of distance is abandoned in the construction of the surreals so it no longer makes sense to talk about the gap between 1 and 2 being the same as the gap between 2 and 3. It is a very pure conception of numbers and it is quite pretty that it contains everything (integers, reals, infinitesimals, infinities, concept of infinity + 1, etc.

foo92691 5y ago

Wikipedia also has a nice overview: https://en.wikipedia.org/wiki/Surreal_number https://en.wikipedia.org/wiki/Surreal_number

shannifin 5y ago

I was introduced to this concept through the Numberphile video and thought it was more about having fun with symbols / notation than contributing anything mathematically new or useful. I particularly object to the playing with infinities and infinitesimals as though they were numbers. As the pdf says on page 41: > "According to theorem 5, {Z | } is a number that is greater than all integers. Therefore, its value is infinity!" That makes no sense, nor does much of the exploration from this point. Other than playing with symbols, I'm not sure it has any value by itself. That said, "playing with symbols" can still be of great value as how we symbolize and notate things can change our understanding of things.

ColinWright 5y ago
When we analyse games, especially combinatorial games, we find that for a given game we can give positions "values". Then there's a concept of "Adding" game positions, and the value of the sum of positions is the sum of the values of the positions. So using "P" for position, and V(P) for the value of a position, we have V(P1+P2) = V(P1)+V(P2). (The additions here are on different types.) So things work nicely, and it feels natural. Then we find that the integers can be embedded naturally into the values we get. So the values from positions form an algebraic structure that extends the integers. Then we find that there are game positions whose values are larger than all the integers. The obvious interpretation of this is that they are infinite, but are all part of a larger structure, one where addition and subtraction of these things still makes sense, and are still well defined. When you say "That makes no sense", you're thinking of this in the limited space of counting, and ordinary numbers. But there are wider contexts where "simple counting" isn't enough, and in those cases it makes perfect sense, and is useful.
- shannifin 5y ago
  > "Then we find that there are game positions whose values are larger than all the integers. The obvious interpretation of this is that they are infinite, but are all part of a larger structure, one where addition and subtraction of these things still makes sense, and are still well defined." But isn't this really more about notation than having "values" that are necessarily "larger than all integers"? That is, isn't this about needing a convenient way to notate something when we're already using integers for something else? And/or do you know of any books or resources that go into more detail about this? (As far as I can tell, the idea of surreal numbers still imagines a one-dimensional layout through which numbers are ordered. It's the adding of "infinity" and "1/infinity" to this line to which I object on nonsense grounds; using a non-integer symbol to represent an extension of some sort (non-spatially related) for some context still makes sense, I just don't think it implies a value "greater than all integers".)