Ulam on Category Theory?

Ulam on Category Theory?

[Noam Zeilberger]

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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 (It is unpublished to this day.)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


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Re: Ulam on Category Theory?

[Ross Street]

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


----------

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Re: Ulam on Category Theory?

[Colin McLarty]

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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.office365.com
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Re: Ulam on Category Theory?

[Zoran Škoda]

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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.office365.com
>> <https://url.au.m.mimecastprotect.com/s/krtZCjZ12RfBwBqxtW8I9r?domain=outlook.office365.com>
>>
>

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Re: Ulam on Category Theory?

[Jorge Soto-Andrade]

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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.office365.com
>>> <https://url.au.m.mimecastprotect.com/s/O9CWCZY146skL4WEuzg_z2?domain=outlook.office365.com>
>>>
>>

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Re: Ulam on Category Theory?

[Noam Zeilberger]

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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.stat.gov.pl,
which makes me wonder whether Ulam was indeed referring to some rudiments
of category theory after all:

> 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
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=itservices.cas.unt.edu)
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.office365.com
>>>> <https://url.au.m.mimecastprotect.com/s/g4LyCGv0Z6fDM7yJupaDTB?domain=outlook.office365.com>
>>>>
>>>

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