7 ms·

Hermann Grassmann and the nature of abstractions

95 points by eigenvalue 2y ago

eigenvalue 2y ago
I find this guy endlessly fascinating to read about and wanted to share some of my thoughts that developed as I learned more about his life and work.
- rhelz 2y ago
  Very nice write-up. Can you expand a bit on one of the phrases you used: tensors as essentially matrices of matrices? Also, the connection between the wedge product and tensors is still pretty fuzzy for me.
  - eigenvalue 2y ago
    Thanks! "Tensors as matrices of matrices" isn't particularly deep; basically you can think of a 3D tensor of being a "cube" of entries, where each cross section in the direction going into the page is just a regular 2D matrix. For example, if you wanted to store the density of discrete voxels in a region of space, you could use such a tensor. This is in fact how MRI scan data is stored, as a bunch of cross sections that are glued together to make a 3D structure. You can then extend this to 4 dimensions, by taking an array of those 3D tensors, and so on. Understanding the link between wedge products and tensors, I think the key thing is the anti-symmetry of the wedge product, where if you reverse the order, the sign changes-- this is related to how the determinant is an "oriented" measure of area, and can take on negative values. The results of combining elements with the wedge product are called differential forms, and these can be thought of as "antisymmetric tensors." I agree that it's all a bit confusing. You can read more here: https://www.physicsforums.com/threads/how-do-tensor-and-wedge-products-relate-in-differential-geometry.150289/ https://www.physicsforums.com/threads/how-do-tensor-and-wedg... And ChatGPT can also do a good job explaing and answering questions about it. Hope that helped!
    - nsingh2 2y ago
      I have a rudimentary understanding of differential geometry, but aren't tensors a bit more special than nd-arrays? I recall a tensor being an object that changes under coordinate transforms according to the jacobian of the transformation.
      - eigenvalue 2y ago
        Yes, you're right, I was talking more about how to visualize a tensor. They also do have special transformation rules under coordinate changes where you have to compute the Jacobian to account for how the tensor "warps" regular Euclidean space locally.
    - 082349872349872 2y ago
      for antisymmetry via oriented measures, compare Stolfi, Primitives for Computational Geometry (1989): https://bitsavers.org/pdf/dec/tech_reports/SRC-RR-36.pdf https://bitsavers.org/pdf/dec/tech_reports/SRC-RR-36.pdf
      - vinnyvichy 2y ago
        Pluecker coords is the connection to Grassmann coords. (Aside: wheat in the heat, compare brevity in wit/spice in brevity) More gore less mess: Kidney in the Whitney
        082349872349872 2y ago
        aha, Plücker is how one gets Grassmann back down to dot/cross memory traffic? K is developer Whitney's, or Wit is threefold Halving, dedicatedly? I am disappointed there is no Spice Girls schlager/cover band named Die Würze in der Schürze. (although if I'm browsing YT for 90s covers, I guess that strongly implies my stage name should be "Old Spice") March: when the elite need to seed their wheat to beat the heat. (no more chill? get the drill!) EDIT: Sheaves, stalks, epis, germs, and fields — it just occurred to me that your noon demon in the rye might be metaphorical? (what is temperature in mathematics? the ratio of entropy delta, which makes sense, to energy delta, which requires a suitable def'n...)
082349872349872 2y ago
> ...partly because there isn't a whole lot of benefit to his approach if you're just doing things in 2D and 3D— in those cases, the alternative approaches are in a sense simpler and more straightforward This is the crux: Grassmann had discovered and explored several mountain passes while others had not yet settled the valleys which they connected? Even if he hadn't been an outsider, he seems to have been (from a career standpoint) "too early". (compare https://news.ycombinator.com/item?id=40312021 https://news.ycombinator.com/item?id=40312021 )
- eigenvalue 2y ago
  Yes, agreed. But he could have done a better job outlining and explaining the valleys to people first so they would be in a position to appreciate the mountain passes! Luckily, credit in academia tends to reward such people better than the commercial market, even if it ends up happening long after death. Look at Galois for example.
  - 082349872349872 2y ago
    True. Even were it to have been making a virtue of necessity, he did note "some day these ideas, even if in an altered form, will reappear and with the passage of time will participate in a lively intellectual exchange. For truth ... remains even if the garments in which feeble men clothe it fall into dust."
    - vinnyvichy 2y ago
      Thanks! Probably approximately no citation needed.
      - vinnyvichy 2y ago
        Since you are possibly not terrible at unearthing obscure but Critical rocksnacks, exactly why, Archangelsk Governorate? I mean, if besides that there it being harder to tell noon from midnight, e.g. neighboring Finnics having obnoxiously dissimilar mythology ) https://en.wikipedia.org/wiki/Lady_Midday https://en.wikipedia.org/wiki/Lady_Midday
frogeyedpeas 2y ago
Being too early is a privilege. Not everyone CAN invent things that are valuable but ahead of their time. If you can then on the one hand don't expect tremendous accolade but also do continue working on what you're doing if it feels obvious to you that this is important. Today with the pace at which technology is advancing people tend to receive credit in their lifetime. In Grassman's day that was just NOT the case. If more people took risks building out ideas they "felt" were the right thing to do the world would be a substantially better place.
- levzettelin 2y ago
  If people took risks building out ideas they "felt" were the right thing, the world would be a better place for you; but not for those people.
eigenvalue 2y ago
Freeman Dyson did a nice job of summing up the situation with Grassmann: "In the year 1844 two remarkable events occurred, the publication by Hamilton of his discovery of quaternions, and the publication by Grassmann of his “Ausdehnungslehre.” With the advantage of hindsight we can see that Grassmann’s was the greater contribution to mathematics, containing the germ of many of the concepts of modern algebra, and including vector analysis as a special case. However, Grassmann was an obscure high-school teacher in Stettin, while Hamilton was the world-famous mathematician whose official titles occupy six lines of print after his name at the beginning of his 1844 paper. So it is regrettable, but not surprising, that quaternions were hailed as a great discovery, while Grassmann had to wait 23 years before his work received any recognition at all from professional mathematicians. When Grassmann’s work finally became known, mathematicians were divided into quaternionists and antiquaternionists, and were spending more energy in polemical arguments for and against quaternions than in trying to understand how Grassmann and Hamilton might be fitted together into a larger scheme of things." (Source: "Missed opportunities")
- deleted 2y ago
  [deleted]
- vinnyvichy 2y ago
  Yep, and this might be why Grassmmann was more optimistic about intertemporal exchanges.
  - 082349872349872 2y ago
    Speaking of intertemporal exchanges, I need to reread Twirlip of the Mists' comments.
    - vinnyvichy 2y ago
      "Six-legs matter"? https://en.wikipedia.org/wiki/The_Ungoverned https://en.wikipedia.org/wiki/The_Ungoverned
      - 2y ago
peter_d_sherman 2y ago
>"...these ideas were totally revolutionary, and he ended up collecting them all several years later in his groundbreaking work, Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (https://dn790002.ca.archive.org/0/items/dielinealeausde00grasgoog/dielinealeausde00grasgoog.pdf#page=13.00 https://dn790002.ca.archive.org/0/items/dielinealeausde00gra...), published in 1844 when he was 35. Grassmann essentially came up with a totally original and new conception of linear algebra. Instead of the more traditional development of systems of linear equations, vectors, matrices, determinants, etc., Grassmann had a more general and abstract conception of the subject, which focused on what is now known as an exterior algebra, also known as the "wedge product." Unlike in the traditional development, where we use simpler operations like the dot product or cross product, in Grassmann's presentation of the subject, everything is couched in terms of the wedge product, which is a bit more abstract and harder to explain. In a nutshell, the wedge product of two vectors can be thought of as the region spanned by the vectors in space using the traditional parallelogram rule; it's basically what you probably already know as the determinant, but instead of representing the quantity of the area spanned by this space, it's the space itself. Basically, the wedge product generalizes the determinant to higher dimensions and different contexts."

95 points by eigenvalue 2y ago

I find this guy endlessly fascinating to read about and wanted to share some of my thoughts that developed as I learned more about his life and work.

rhelz 2y ago
Very nice write-up. Can you expand a bit on one of the phrases you used: tensors as essentially matrices of matrices? Also, the connection between the wedge product and tensors is still pretty fuzzy for me.
- eigenvalue 2y ago
  Thanks! "Tensors as matrices of matrices" isn't particularly deep; basically you can think of a 3D tensor of being a "cube" of entries, where each cross section in the direction going into the page is just a regular 2D matrix. For example, if you wanted to store the density of discrete voxels in a region of space, you could use such a tensor. This is in fact how MRI scan data is stored, as a bunch of cross sections that are glued together to make a 3D structure. You can then extend this to 4 dimensions, by taking an array of those 3D tensors, and so on. Understanding the link between wedge products and tensors, I think the key thing is the anti-symmetry of the wedge product, where if you reverse the order, the sign changes-- this is related to how the determinant is an "oriented" measure of area, and can take on negative values. The results of combining elements with the wedge product are called differential forms, and these can be thought of as "antisymmetric tensors." I agree that it's all a bit confusing. You can read more here: https://www.physicsforums.com/threads/how-do-tensor-and-wedge-products-relate-in-differential-geometry.150289/ https://www.physicsforums.com/threads/how-do-tensor-and-wedg... And ChatGPT can also do a good job explaing and answering questions about it. Hope that helped!
  - nsingh2 2y ago
    I have a rudimentary understanding of differential geometry, but aren't tensors a bit more special than nd-arrays? I recall a tensor being an object that changes under coordinate transforms according to the jacobian of the transformation.
    - eigenvalue 2y ago
      Yes, you're right, I was talking more about how to visualize a tensor. They also do have special transformation rules under coordinate changes where you have to compute the Jacobian to account for how the tensor "warps" regular Euclidean space locally.
  - 082349872349872 2y ago
    for antisymmetry via oriented measures, compare Stolfi, Primitives for Computational Geometry (1989): https://bitsavers.org/pdf/dec/tech_reports/SRC-RR-36.pdf https://bitsavers.org/pdf/dec/tech_reports/SRC-RR-36.pdf
    - vinnyvichy 2y ago
      Pluecker coords is the connection to Grassmann coords. (Aside: wheat in the heat, compare brevity in wit/spice in brevity) More gore less mess: Kidney in the Whitney
      - 082349872349872 2y ago
        aha, Plücker is how one gets Grassmann back down to dot/cross memory traffic? K is developer Whitney's, or Wit is threefold Halving, dedicatedly? I am disappointed there is no Spice Girls schlager/cover band named Die Würze in der Schürze. (although if I'm browsing YT for 90s covers, I guess that strongly implies my stage name should be "Old Spice") March: when the elite need to seed their wheat to beat the heat. (no more chill? get the drill!) EDIT: Sheaves, stalks, epis, germs, and fields — it just occurred to me that your noon demon in the rye might be metaphorical? (what is temperature in mathematics? the ratio of entropy delta, which makes sense, to energy delta, which requires a suitable def'n...)

082349872349872 2y ago

> ...partly because there isn't a whole lot of benefit to his approach if you're just doing things in 2D and 3D— in those cases, the alternative approaches are in a sense simpler and more straightforward This is the crux: Grassmann had discovered and explored several mountain passes while others had not yet settled the valleys which they connected? Even if he hadn't been an outsider, he seems to have been (from a career standpoint) "too early". (compare https://news.ycombinator.com/item?id=40312021 https://news.ycombinator.com/item?id=40312021 )

eigenvalue 2y ago
Yes, agreed. But he could have done a better job outlining and explaining the valleys to people first so they would be in a position to appreciate the mountain passes! Luckily, credit in academia tends to reward such people better than the commercial market, even if it ends up happening long after death. Look at Galois for example.
- 082349872349872 2y ago
  True. Even were it to have been making a virtue of necessity, he did note "some day these ideas, even if in an altered form, will reappear and with the passage of time will participate in a lively intellectual exchange. For truth ... remains even if the garments in which feeble men clothe it fall into dust."
  - vinnyvichy 2y ago
    Thanks! Probably approximately no citation needed.
    - vinnyvichy 2y ago
      Since you are possibly not terrible at unearthing obscure but Critical rocksnacks, exactly why, Archangelsk Governorate? I mean, if besides that there it being harder to tell noon from midnight, e.g. neighboring Finnics having obnoxiously dissimilar mythology ) https://en.wikipedia.org/wiki/Lady_Midday https://en.wikipedia.org/wiki/Lady_Midday

frogeyedpeas 2y ago

Being too early is a privilege. Not everyone CAN invent things that are valuable but ahead of their time. If you can then on the one hand don't expect tremendous accolade but also do continue working on what you're doing if it feels obvious to you that this is important. Today with the pace at which technology is advancing people tend to receive credit in their lifetime. In Grassman's day that was just NOT the case. If more people took risks building out ideas they "felt" were the right thing to do the world would be a substantially better place.

levzettelin 2y ago
If people took risks building out ideas they "felt" were the right thing, the world would be a better place for you; but not for those people.

eigenvalue 2y ago

Freeman Dyson did a nice job of summing up the situation with Grassmann: "In the year 1844 two remarkable events occurred, the publication by Hamilton of his discovery of quaternions, and the publication by Grassmann of his “Ausdehnungslehre.” With the advantage of hindsight we can see that Grassmann’s was the greater contribution to mathematics, containing the germ of many of the concepts of modern algebra, and including vector analysis as a special case. However, Grassmann was an obscure high-school teacher in Stettin, while Hamilton was the world-famous mathematician whose official titles occupy six lines of print after his name at the beginning of his 1844 paper. So it is regrettable, but not surprising, that quaternions were hailed as a great discovery, while Grassmann had to wait 23 years before his work received any recognition at all from professional mathematicians. When Grassmann’s work finally became known, mathematicians were divided into quaternionists and antiquaternionists, and were spending more energy in polemical arguments for and against quaternions than in trying to understand how Grassmann and Hamilton might be fitted together into a larger scheme of things." (Source: "Missed opportunities")

deleted 2y ago
[deleted]
vinnyvichy 2y ago
Yep, and this might be why Grassmmann was more optimistic about intertemporal exchanges.
- 082349872349872 2y ago
  Speaking of intertemporal exchanges, I need to reread Twirlip of the Mists' comments.
  - vinnyvichy 2y ago
    "Six-legs matter"? https://en.wikipedia.org/wiki/The_Ungoverned https://en.wikipedia.org/wiki/The_Ungoverned
    - 2y ago

peter_d_sherman 2y ago

>"...these ideas were totally revolutionary, and he ended up collecting them all several years later in his groundbreaking work, Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (https://dn790002.ca.archive.org/0/items/dielinealeausde00grasgoog/dielinealeausde00grasgoog.pdf#page=13.00 https://dn790002.ca.archive.org/0/items/dielinealeausde00gra...), published in 1844 when he was 35. Grassmann essentially came up with a totally original and new conception of linear algebra. Instead of the more traditional development of systems of linear equations, vectors, matrices, determinants, etc., Grassmann had a more general and abstract conception of the subject, which focused on what is now known as an exterior algebra, also known as the "wedge product." Unlike in the traditional development, where we use simpler operations like the dot product or cross product, in Grassmann's presentation of the subject, everything is couched in terms of the wedge product, which is a bit more abstract and harder to explain. In a nutshell, the wedge product of two vectors can be thought of as the region spanned by the vectors in space using the traditional parallelogram rule; it's basically what you probably already know as the determinant, but instead of representing the quantity of the area spanned by this space, it's the space itself. Basically, the wedge product generalizes the determinant to higher dimensions and different contexts."