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charset=UTF-8 Content-Transfer-Encoding: quoted-printable 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=3Dbws.= 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_YCE8wlRCvD7A= WTwAWzK?domain=3Ditservices.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=E2=80=AFAM Jorge Soto-Andrade wrote: > Dear colleagues. I had the same conjecture as Zoran : first category, > second category =E2=80=A6 > Best > Jorge > > Sent from Gmail Mobile > > > On Mon, 10 Jun 2024 at 11:14 Zoran =C5=A0koda 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 fiel= d >> 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 =C5=A0koda >> >> pon, 10. lip 2024. 14:28 Colin McLarty 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 co= urse >>> lots of people did that long before Ulam or Eilenberg-Mac Lane. >>> >>> On Mon, Jun 10, 2024 at 7:23=E2=80=AFAM Ross Street >>> wrote: >>> >>>> Eilenberg and Ulam were both students of Kuratowski I believe, if that >>>> is relevant. >>>> I remember Sammy talking about him but I can=E2=80=99t remember the to= pic. >>>> Ross >>>> >>>> Sent from my iPhone >>>> >>>> > On 10 Jun 2024, at 9:00=E2=80=AFPM, Noam Zeilberger < >>>> noam.zeilberger@lix.polytechnique.fr> wrote: >>>> > >>>> > =EF=BB=BF >>>> > Dear all, >>>> > >>>> > I was wondering if anyone here might be able to provide some >>>> additional historical context for a passage in Stanis=C5=82aw Ulam's >>>> autobiography "Adventures of a Mathematician", where he talks about hi= s >>>> Master's thesis at the Lw=C3=B3w 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 ca= lled >>>> Category Theory. >>>> > >>>> > Without wanting to read too much into his offhand remark, I'm curiou= s >>>> 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 cl= aimed >>>> that related ideas were current in the late 1930s in Poland.") Accord= ing >>>> to the information I found online, the Stanislaw M. Ulam Papers are st= ored >>>> in the American Philosophical Society library in Philadelphia, and inc= lude >>>> copies of both his Master's thesis in Polish titled "O operacje produk= to" >>>> 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 yo= u >>>> 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 Categorie= s >>>> mailing list group from Macquarie University. >>>> >>>> Leave group: >>>> >>>> https://url.au.m.mimecastprotect.com/s/g4LyCGv0Z6fDM7yJupaDTB?domain= =3Doutlook.office365.com >>>> >>>> >>> --000000000000d12a74061ab1abd6 Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable
Thanks everyone for the responses.=C2=A0 The hyp= othesis that Ulam was referring to categories in the sense of the Baire cat= egory theorem sounds quite plausible, but in that case it's strange tha= t he would refer to it as "what is now called Category Theory".= =C2=A0 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 (introd= uced by Baire in 1899) was by then already archaic.

Meanw= hile, I found an excerpt of Kuratowski's report on Ulam's master= 9;s thesis in a document called "Polish statisticians: biographical no= tes" (https://bws.stat.gov.pl/BWS/Archiwum/gus= _bws_67_Polish_statisticians_Biographical_notes.pdf), which makes me wo= nder whether Ulam was indeed referring to some rudiments of category theory= after all:

> The work is a study of the "product&quo= t; operation, which has been little researched until now, but is playing an= increasingly important role in mathematics. The author analyses this conce= pt in relation to problems in set theory, group theory, topology, metric sp= ace geometry, combinatorics and measure theory in connection with probabili= ty. Since the author has demonstrated complete mastery of the subject and d= ue knowledge of the relevant literature, that moreover the work contains a = number of original results, and finally that the author presents many inter= esting 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://itservices.cas.unt.edu/~maul= din/papers/no62.pdf) by Mauldin and Ulam (posthumously), although from = looking at that I don't have a good sense either way.=C2=A0 (If I under= stand correctly, the translation and description of these problems and theo= rems is due to Jan Mycielski, who is currently emeritus professor at UC Bou= lder.=C2=A0 He is 92 years old but perhaps reads his email?)
=
Of course it would be ideal to get a copy of the maste= r's thesis itself.=C2=A0 The APS library has a mechanism for requesting= digital copies, and I've made a request for the English translation.= =C2=A0 If anything comes of that I will send an update.

regards.
Noam

On Tue, Jun 11, 2024 at 12:37=E2=80=AFAM Jorge S= oto-Andrade <= sotoandrade@gmail.com> wrote:
Dear colleagues. =C2=A0 I had the sa= me conjecture as Zoran : first category, second category =E2=80=A6
Best
Jorge=C2=A0
Sent from Gmail Mobile


On Mon, 10 Jun 2024 = at 11:14 Zoran =C5=A0koda <zskoda@gmail.com> wrote:
Dear colleagues,

I was also confused (many years ago) with thi= s passage in Ulam's memoirs, but later have convinced myself that the c= onfusion is purely terminological. When Ulam refers to category theory he l= ikely means the notion of Baire's category and related theorems. Kurato= wski-Ulam theorem and a few of other related results with Ulam's contri= bution to this field show that he was interested in this problematics in hi= s early research phase.=C2=A0=C2=A0

It might be that my conclusion is not definitive but it looks a= t least very plausible.

= Zoran =C5=A0koda

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.=C2=A0 I think those people are right, who have suggest= ed that Ulam and others had used specific examples of functors and natural = transformations=C2=A0without stating any general concept of them.=C2=A0 But= of course lots of people did that long before Ulam or Eilenberg-Mac Lane.<= /div>
O= n Mon, Jun 10, 2024 at 7:23=E2=80=AFAM Ross Street <ross.street@mq.ed= u.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=E2=80=99t remember the topic.<= br> Ross

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> On 10 Jun 2024, at 9:00=E2=80=AFPM, Noam Zeilberger <noam.zeilberger@lix.polytechnique.fr> wrote:
>
> =EF=BB=BF
> Dear all,
>
> I was wondering if anyone here might be able to provide some additiona= l historical context for a passage in Stanis=C5=82aw Ulam's autobiograp= hy "Adventures of a Mathematician", where he talks about his Mast= er's thesis at the Lw=C3=B3w 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 operation= s 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 curi= ous about what he was talking about.=C2=A0 I noticed that the Wikipedia art= icle on Category theory also alludes to this but does not elaborate or give= references.=C2=A0 ("Stanislaw Ulam, and some writing on his behalf, h= ave claimed that related ideas were current in the late 1930s in Poland.&qu= ot;)=C2=A0 According to the information I found online, the Stanislaw M. Ul= am Papers are stored in the American Philosophical Society library in Phila= delphia, and include copies of both his Master's thesis in Polish title= d "O operacje produkto" as well as a 1973 English translation &qu= ot;On the operation of product".=C2=A0 But I was not able to find digi= tal copies of these.
>
> Is anyone familiar with the contents of the thesis and can suggest wha= t Ulam was referring to by "outlines" of category theory?=C2=A0 O= r do you know if Ulam wrote about category theory elsewhere, or if he had a= ny significant interactions with category theorists?
>
> this is just for curiosity, but thanks in advance for any leads!
> Noam


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