Ulam on Category Theory?
Dear all, I was wondering if anyone here might be able to provide some additional historical context for a passage in Stanisław Ulam's autobiography "Adventures of a Mathematician", where he talks about his Master's thesis at the Lwów Polytechnic Institute:
I worked for a week on the thesis, then wrote it up in one night, from about ten in the evening until four in the morning, on my father's long sheets of legal paper. I still have the original manuscript. (It is unpublished to this day.) The paper contains general ideas on the operations of products of sets, and some of it outlines what is now called Category Theory.
Without wanting to read too much into his offhand remark, I'm curious about what he was talking about. I noticed that the Wikipedia article on Category theory also alludes to this but does not elaborate or give references. ("Stanislaw Ulam, and some writing on his behalf, have claimed that related ideas were current in the late 1930s in Poland.") According to the information I found online, the Stanislaw M. Ulam Papers are stored in the American Philosophical Society library in Philadelphia, and include copies of both his Master's thesis in Polish titled "O operacje produkto" as well as a 1973 English translation "On the operation of product". But I was not able to find digital copies of these. Is anyone familiar with the contents of the thesis and can suggest what Ulam was referring to by "outlines" of category theory? Or do you know if Ulam wrote about category theory elsewhere, or if he had any significant interactions with category theorists? this is just for curiosity, but thanks in advance for any leads! Noam You're receiving this message because you're a member of the Categories mailing list group from Macquarie University. To take part in this conversation, reply all to this message. View group files<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=files&GuestId=6bf90c14-94d1-45b7-a0b5-9dd447734d27> | Leave group<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=leave&GuestId=6bf90c14-94d1-45b7-a0b5-9dd447734d27> | Learn more about Microsoft 365 Groups<https://aka.ms/o365g>
Eilenberg and Ulam were both students of Kuratowski I believe, if that is relevant. I remember Sammy talking about him but I can’t remember the topic. Ross Sent from my iPhone
On 10 Jun 2024, at 9:00 PM, Noam Zeilberger <noam.zeilberger@lix.polytechnique.fr> wrote:
Dear all,
I was wondering if anyone here might be able to provide some additional historical context for a passage in Stanisław Ulam's autobiography "Adventures of a Mathematician", where he talks about his Master's thesis at the Lwów Polytechnic Institute:
I worked for a week on the thesis, then wrote it up in one night, from about ten in the evening until four in the morning, on my father's long sheets of legal paper. I still have the original manuscript. (It is unpublished to this day.) The paper contains general ideas on the operations of products of sets, and some of it outlines what is now called Category Theory.
Without wanting to read too much into his offhand remark, I'm curious about what he was talking about. I noticed that the Wikipedia article on Category theory also alludes to this but does not elaborate or give references. ("Stanislaw Ulam, and some writing on his behalf, have claimed that related ideas were current in the late 1930s in Poland.") According to the information I found online, the Stanislaw M. Ulam Papers are stored in the American Philosophical Society library in Philadelphia, and include copies of both his Master's thesis in Polish titled "O operacje produkto" as well as a 1973 English translation "On the operation of product". But I was not able to find digital copies of these.
Is anyone familiar with the contents of the thesis and can suggest what Ulam was referring to by "outlines" of category theory? Or do you know if Ulam wrote about category theory elsewhere, or if he had any significant interactions with category theorists?
this is just for curiosity, but thanks in advance for any leads! Noam
---------- You're receiving this message because you're a member of the Categories mailing list group from Macquarie University. Leave group: https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=leave&GuestId=6bf90c14-94d1-45b7-a0b5-9dd447734d27
I have looked into this a good bit, without finding any documentary sources. I think those people are right, who have suggested that Ulam and others had used specific examples of functors and natural transformations without stating any general concept of them. But of course lots of people did that long before Ulam or Eilenberg-Mac Lane. On Mon, Jun 10, 2024 at 7:23 AM Ross Street <ross.street@mq.edu.au> wrote:
Eilenberg and Ulam were both students of Kuratowski I believe, if that is relevant. I remember Sammy talking about him but I can’t remember the topic. Ross
Sent from my iPhone
On 10 Jun 2024, at 9:00 PM, Noam Zeilberger < noam.zeilberger@lix.polytechnique.fr> wrote:
Dear all,
I was wondering if anyone here might be able to provide some additional historical context for a passage in Stanisław Ulam's autobiography "Adventures of a Mathematician", where he talks about his Master's thesis at the Lwów Polytechnic Institute:
I worked for a week on the thesis, then wrote it up in one night, from about ten in the evening until four in the morning, on my father's long sheets of legal paper. I still have the original manuscript. (It is unpublished to this day.) The paper contains general ideas on the operations of products of sets, and some of it outlines what is now called Category Theory.
Without wanting to read too much into his offhand remark, I'm curious about what he was talking about. I noticed that the Wikipedia article on Category theory also alludes to this but does not elaborate or give references. ("Stanislaw Ulam, and some writing on his behalf, have claimed that related ideas were current in the late 1930s in Poland.") According to the information I found online, the Stanislaw M. Ulam Papers are stored in the American Philosophical Society library in Philadelphia, and include copies of both his Master's thesis in Polish titled "O operacje produkto" as well as a 1973 English translation "On the operation of product". But I was not able to find digital copies of these.
Is anyone familiar with the contents of the thesis and can suggest what Ulam was referring to by "outlines" of category theory? Or do you know if Ulam wrote about category theory elsewhere, or if he had any significant interactions with category theorists?
this is just for curiosity, but thanks in advance for any leads! Noam
----------
You're receiving this message because you're a member of the Categories mailing list group from Macquarie University.
Leave group:
https://url.au.m.mimecastprotect.com/s/5zX0CGv0Z6fDPnknhK033s?domain=outlook...
Dear colleagues, I was also confused (many years ago) with this passage in Ulam's memoirs, but later have convinced myself that the confusion is purely terminological. When Ulam refers to category theory he likely means the notion of Baire's category and related theorems. Kuratowski-Ulam theorem and a few of other related results with Ulam's contribution to this field show that he was interested in this problematics in his early research phase. It might be that my conclusion is not definitive but it looks at least very plausible. Zoran Škoda pon, 10. lip 2024. 14:28 Colin McLarty <colin.mclarty@case.edu> je napisao:
I have looked into this a good bit, without finding any documentary sources. I think those people are right, who have suggested that Ulam and others had used specific examples of functors and natural transformations without stating any general concept of them. But of course lots of people did that long before Ulam or Eilenberg-Mac Lane.
On Mon, Jun 10, 2024 at 7:23 AM Ross Street <ross.street@mq.edu.au> wrote:
Eilenberg and Ulam were both students of Kuratowski I believe, if that is relevant. I remember Sammy talking about him but I can’t remember the topic. Ross
Sent from my iPhone
On 10 Jun 2024, at 9:00 PM, Noam Zeilberger < noam.zeilberger@lix.polytechnique.fr> wrote:
Dear all,
I was wondering if anyone here might be able to provide some additional historical context for a passage in Stanisław Ulam's autobiography "Adventures of a Mathematician", where he talks about his Master's thesis at the Lwów Polytechnic Institute:
I worked for a week on the thesis, then wrote it up in one night, from about ten in the evening until four in the morning, on my father's long sheets of legal paper. I still have the original manuscript. (It is unpublished to this day.) The paper contains general ideas on the operations of products of sets, and some of it outlines what is now called Category Theory.
Without wanting to read too much into his offhand remark, I'm curious about what he was talking about. I noticed that the Wikipedia article on Category theory also alludes to this but does not elaborate or give references. ("Stanislaw Ulam, and some writing on his behalf, have claimed that related ideas were current in the late 1930s in Poland.") According to the information I found online, the Stanislaw M. Ulam Papers are stored in the American Philosophical Society library in Philadelphia, and include copies of both his Master's thesis in Polish titled "O operacje produkto" as well as a 1973 English translation "On the operation of product". But I was not able to find digital copies of these.
Is anyone familiar with the contents of the thesis and can suggest what Ulam was referring to by "outlines" of category theory? Or do you know if Ulam wrote about category theory elsewhere, or if he had any significant interactions with category theorists?
this is just for curiosity, but thanks in advance for any leads! Noam
----------
You're receiving this message because you're a member of the Categories mailing list group from Macquarie University.
Leave group:
https://url.au.m.mimecastprotect.com/s/krtZCjZ12RfBwBqxtW8I9r?domain=outlook...
Dear colleagues. I had the same conjecture as Zoran : first category, second category … Best Jorge Sent from Gmail Mobile On Mon, 10 Jun 2024 at 11:14 Zoran Škoda <zskoda@gmail.com> wrote:
Dear colleagues,
I was also confused (many years ago) with this passage in Ulam's memoirs, but later have convinced myself that the confusion is purely terminological. When Ulam refers to category theory he likely means the notion of Baire's category and related theorems. Kuratowski-Ulam theorem and a few of other related results with Ulam's contribution to this field show that he was interested in this problematics in his early research phase.
It might be that my conclusion is not definitive but it looks at least very plausible.
Zoran Škoda
pon, 10. lip 2024. 14:28 Colin McLarty <colin.mclarty@case.edu> je napisao:
I have looked into this a good bit, without finding any documentary sources. I think those people are right, who have suggested that Ulam and others had used specific examples of functors and natural transformations without stating any general concept of them. But of course lots of people did that long before Ulam or Eilenberg-Mac Lane.
On Mon, Jun 10, 2024 at 7:23 AM Ross Street <ross.street@mq.edu.au> wrote:
Eilenberg and Ulam were both students of Kuratowski I believe, if that is relevant. I remember Sammy talking about him but I can’t remember the topic. Ross
Sent from my iPhone
On 10 Jun 2024, at 9:00 PM, Noam Zeilberger < noam.zeilberger@lix.polytechnique.fr> wrote:
Dear all,
I was wondering if anyone here might be able to provide some additional historical context for a passage in Stanisław Ulam's autobiography "Adventures of a Mathematician", where he talks about his Master's thesis at the Lwów Polytechnic Institute:
I worked for a week on the thesis, then wrote it up in one night, from about ten in the evening until four in the morning, on my father's long sheets of legal paper. I still have the original manuscript. (It is unpublished to this day.) The paper contains general ideas on the operations of products of sets, and some of it outlines what is now called Category Theory.
Without wanting to read too much into his offhand remark, I'm curious about what he was talking about. I noticed that the Wikipedia article on Category theory also alludes to this but does not elaborate or give references. ("Stanislaw Ulam, and some writing on his behalf, have claimed that related ideas were current in the late 1930s in Poland.") According to the information I found online, the Stanislaw M. Ulam Papers are stored in the American Philosophical Society library in Philadelphia, and include copies of both his Master's thesis in Polish titled "O operacje produkto" as well as a 1973 English translation "On the operation of product". But I was not able to find digital copies of these.
Is anyone familiar with the contents of the thesis and can suggest what Ulam was referring to by "outlines" of category theory? Or do you know if Ulam wrote about category theory elsewhere, or if he had any significant interactions with category theorists?
this is just for curiosity, but thanks in advance for any leads! Noam
----------
You're receiving this message because you're a member of the Categories mailing list group from Macquarie University.
Leave group:
https://url.au.m.mimecastprotect.com/s/O9CWCZY146skL4WEuzg_z2?domain=outlook...
The work is a study of the "product" operation, which has been little researched until now, but is playing an increasingly important role in mathematics. The author analyses this concept in relation to problems in set theory, group theory, topology, metric space geometry, combinatorics and measure theory in connection with probability. Since the author has demonstrated complete mastery of the subject and due knowledge of the relevant literature, that moreover the work contains a number of original results, and finally that the author presents many interesting problems in
Thanks everyone for the responses. The hypothesis that Ulam was referring to categories in the sense of the Baire category theorem sounds quite plausible, but in that case it's strange that he would refer to it as "what is now called Category Theory". The first edition of the book was published in 1976, after category theory was already an established and growing field of mathematics, whereas my guess is that the "first/second category" terminology (introduced by Baire in 1899) was by then already archaic. Meanwhile, I found an excerpt of Kuratowski's report on Ulam's master's thesis in a document called "Polish statisticians: biographical notes" ( https://url.au.m.mimecastprotect.com/s/Ua3ECD1vRkCEV96BTWKZ5S?domain=bws.sta..., which makes me wonder whether Ulam was indeed referring to some rudiments of category theory after all: this work, I consider this thesis to be perfect. There is also some discussion of "Problems and theorems from Ulam's Master's Thesis of 1932" in section 1.13 of the 1987 article "Mathematical Problems and Games" (https://url.au.m.mimecastprotect.com/s/Sm_YCE8wlRCvD7AWTwAWzK?domain=itservi...) by Mauldin and Ulam (posthumously), although from looking at that I don't have a good sense either way. (If I understand correctly, the translation and description of these problems and theorems is due to Jan Mycielski, who is currently emeritus professor at UC Boulder. He is 92 years old but perhaps reads his email?) Of course it would be ideal to get a copy of the master's thesis itself. The APS library has a mechanism for requesting digital copies, and I've made a request for the English translation. If anything comes of that I will send an update. regards. Noam On Tue, Jun 11, 2024 at 12:37 AM Jorge Soto-Andrade <sotoandrade@gmail.com> wrote:
Dear colleagues. I had the same conjecture as Zoran : first category, second category … Best Jorge
Sent from Gmail Mobile
On Mon, 10 Jun 2024 at 11:14 Zoran Škoda <zskoda@gmail.com> wrote:
Dear colleagues,
I was also confused (many years ago) with this passage in Ulam's memoirs, but later have convinced myself that the confusion is purely terminological. When Ulam refers to category theory he likely means the notion of Baire's category and related theorems. Kuratowski-Ulam theorem and a few of other related results with Ulam's contribution to this field show that he was interested in this problematics in his early research phase.
It might be that my conclusion is not definitive but it looks at least very plausible.
Zoran Škoda
pon, 10. lip 2024. 14:28 Colin McLarty <colin.mclarty@case.edu> je napisao:
I have looked into this a good bit, without finding any documentary sources. I think those people are right, who have suggested that Ulam and others had used specific examples of functors and natural transformations without stating any general concept of them. But of course lots of people did that long before Ulam or Eilenberg-Mac Lane.
On Mon, Jun 10, 2024 at 7:23 AM Ross Street <ross.street@mq.edu.au> wrote:
Eilenberg and Ulam were both students of Kuratowski I believe, if that is relevant. I remember Sammy talking about him but I can’t remember the topic. Ross
Sent from my iPhone
On 10 Jun 2024, at 9:00 PM, Noam Zeilberger < noam.zeilberger@lix.polytechnique.fr> wrote:
Dear all,
I was wondering if anyone here might be able to provide some additional historical context for a passage in Stanisław Ulam's autobiography "Adventures of a Mathematician", where he talks about his Master's thesis at the Lwów Polytechnic Institute:
I worked for a week on the thesis, then wrote it up in one night, from about ten in the evening until four in the morning, on my father's long sheets of legal paper. I still have the original manuscript. (It is unpublished to this day.) The paper contains general ideas on the operations of products of sets, and some of it outlines what is now called Category Theory.
Without wanting to read too much into his offhand remark, I'm curious about what he was talking about. I noticed that the Wikipedia article on Category theory also alludes to this but does not elaborate or give references. ("Stanislaw Ulam, and some writing on his behalf, have claimed that related ideas were current in the late 1930s in Poland.") According to the information I found online, the Stanislaw M. Ulam Papers are stored in the American Philosophical Society library in Philadelphia, and include copies of both his Master's thesis in Polish titled "O operacje produkto" as well as a 1973 English translation "On the operation of product". But I was not able to find digital copies of these.
Is anyone familiar with the contents of the thesis and can suggest what Ulam was referring to by "outlines" of category theory? Or do you know if Ulam wrote about category theory elsewhere, or if he had any significant interactions with category theorists?
this is just for curiosity, but thanks in advance for any leads! Noam
----------
You're receiving this message because you're a member of the Categories mailing list group from Macquarie University.
Leave group:
https://url.au.m.mimecastprotect.com/s/g4LyCGv0Z6fDM7yJupaDTB?domain=outlook...
participants (5)
-
Colin McLarty -
Jorge Soto-Andrade -
Noam Zeilberger -
Ross Street -
Zoran Škoda