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charset="utf-8"; format=flowed Content-Transfer-Encoding: quoted-printable Just do it, Jean-Pierre. It will be very interesting to see it. Generally speaking, we do such things all too little. When we wrote our boo= k on quantales we came across a few such historical pathways, one of which = was the monad and e.g. also the star composition of natural transformation.= We spent a "reasonable portion of time" unravelling that history, and it c= omes down to less than one page, so we could have done more. We also had a = chapter on finite machines and modules, and we look back on finite machines= , where CS fixes the "origin" to some paper or papers, but when we looked a= t it, these "original papers" were quite sparse in their bibliography, whic= h kind of "boosted their originality". For a while I've been looking into the categorization of foundations, i.e.,= not the foundations of category theory. For decades we have had a formal t= erm monad which has shown to be useful for categorization of formulas, enta= ilment, proof rules and proofs, etc. Categorical constructivity is very dia= metral to the traditional "putting symbols one by one in sequence" approach= in logic and foundations, which leads to antinomies and many other strange= situations. This work, which is all too much for me personally, and I can = do only fractions, involves many historical pathways, e.g. Schr=C3=B6der tp= Peano, Peano to Principia; Hilbert's lectures 1917-1922; deduction by Hert= z and Post; Bernays; Hilbert's and Ackermann's logic foundations; G=C3=B6de= l 1931 and just before; Hilbert-Bernays Grundlagen 1934/39; Church, Turing,= lambda calculus and computing; etc. just to name a few. Category theory ex= isted at that time, because of set theory, even if it wasn't "discovered" j= ust by the end of this "mathematical foundations pathway". So I'm fascinate= d by the thought that that categories would have been "discovered" already = before Principia. Principia said that "logic must be freed from the burden = of algebra", and foundations never recovered from that attitude. Bernays an= d G=C3=B6del wrote letters in the early 1960's about "something new called = category theory" and "there was a guy speaking about something in Poland" (= it was Mac Lane), but they simply thought a bit on foundations of such cate= gories rather than such categories being used in constructions in their fou= ndations, that had puzzled them for almost half a century. My point here is= that "tracing far back" is very important. Categorization of foundations (= Hilbert wasn't even close, but he could have been) connects subpathways in = this history and e.g. makes criticism against antinomies (like e.g. G=C3=B6= del 1931) quite precise. So, Jean-Pierre, just do it. Or anyone else thinking similarly about "deep = tracing", just do it, because nobody else will, most probably. We need to s= trengthen the culture of "deep tracing", rather that desperately seeking fo= r "this was the moment, this was the breakfast, these were the persons arou= nd that table at that moment in time, and that person must be named as the = originator, for that person we must create a bust". "Deep tracing" opens up= more research, "creating busts" does the opposite. Best, Patrik On 2023-12-18 22:08, Jean-Pierre Marquis wrote: I guess it is a paper like this one that justified Eilenberg & Mac Lane's c= hoice of terminology. It would be interesting to trace how far the terminol= ogy goes back, who used it and where. Jean-Pierre De : Evgeny Kuznetsov Date : lundi, 18 d=C3=A9cembre 2023 =C3=A0 14:43 =C3=80 : Jean-Pierre Marquis Cc : Wesley Phoa , Michael Barr, Prof. , categories@mq.edu.au Objet : Re: Modification of what I said Here is a copy of the paper by Hassler Whitney of 1938 titled "Tensor produ= cts of abelian groups" On Mon, Dec 18, 2023, 23:32 Jean-Pierre Marquis > wrote: Most likely. Whitney uses the terms 'natural isomorphism' and 'natural homomorphism' as = well as the terms 'natural topology' and 'natural neighborhood' at many dif= ferent places in the paper. But these terms are never explicitly defined. Cheers, Jean-Pierre De : Wesley Phoa > Date : lundi, 18 d=C3=A9cembre 2023 =C3=A0 14:18 =C3=80 : Michael Barr, Prof. > Cc : categories@mq.edu.au > Objet : Re: Modification of what I said Was he referring to the paper "Tensor products of abelian groups", cited in= this discussion? https://mathoverflow.net/questions/287869/history-of-natu= ral-transformations I don't have access to it either, but it's on Scribd: https://www.scribd.co= m/document/172981416/Hassler-Whitney-Tensor-Products-of-Abelian-Groups The terms "natural isomorphism" and "natural homomorphism" are used on page= s 500-501, and these do turn out to be natural transformations, but it's no= t obvious that he intended to explicitly define a new formal concept. Wesley Sent from my iPad On Dec 18, 2023, at 10:00=E2=80=AFAM, Michael Barr, Prof. > wrote: Peter Freyd claims that Hassler Whitney defined natural transformation in a= 1938 paper. I no longer have access to Math. Reviews (except by going to = McGill, which I have done only once in the last four years) so I cannot sup= ply a reference. Michael You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message. View group files | Leave group | Learn = more about Microsoft 365 Groups You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message. View group files | Leave group | Learn = more about Microsoft 365 Groups You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message. View group files | Leave group | = Learn more about Microsoft 365 Groups --=_6c2d2f409dfa80bdc2da372e3e9bf859 Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset=UTF-8

Just do it, Jean-Pierre. It will be very interesting to see it.

Generally speaking, we do such things all too little. When we wrote our = book on quantales we came across a few such historical pathways, one of whi= ch was the monad and e.g. also the star composition of natural transformati= on. We spent a "reasonable portion of time" unravelling that history, and it comes down to less than one= page, so we could have done more. We also had a chapter on finite machines= and modules, and we look back on finite machines, where CS fixes the "= ;origin" to some paper or papers, but when we looked at it, these "original papers" were quite sparse in th= eir bibliography, which kind of "boosted their originality".

For a while I've been looking into the categorization of foundations, i.= e., not the foundations of category theory. For decades we have had a forma= l term monad which has shown to be useful for categorization of formulas, e= ntailment, proof rules and proofs, etc. Categorical constructivity is very diametral to the traditional "= ;putting symbols one by one in sequence" approach in logic and foundat= ions, which leads to antinomies and many other strange situations. This wor= k, which is all too much for me personally, and I can do only fractions, involves many historical pathways, e.g. Schr= =C3=B6der tp Peano, Peano to Principia; Hilbert's lectures 1917-1922; deduc= tion by Hertz and Post; Bernays; Hilbert's and Ackermann's logic foundation= s; G=C3=B6del 1931 and just before; Hilbert-Bernays Grundlagen 1934/39; Church, Turing, lambda calculus and computing; etc. ju= st to name a few. Category theory existed at that time, because of set theo= ry, even if it wasn't "discovered" just by the end of this "= mathematical foundations pathway". So I'm fascinated by the thought that that categories would have been "discovered"= already before Principia. Principia said that "logic must be freed fr= om the burden of algebra", and foundations never recovered from that a= ttitude. Bernays and G=C3=B6del wrote letters in the early 1960's about "something new called category theory" and "th= ere was a guy speaking about something in Poland" (it was Mac Lane), b= ut they simply thought a bit on foundations of such categories rather than = such categories being used in constructions in their foundations, that had puzzled them for almost half a century. My point her= e is that "tracing far back" is very important. Categorization of= foundations (Hilbert wasn't even close, but he could have been) connects s= ubpathways in this history and e.g. makes criticism against antinomies (like e.g. G=C3=B6del 1931) quite precise.

So, Jean-Pierre, just do it. Or anyone else thinking similarly about &qu= ot;deep tracing", just do it, because nobody else will, most probably.= We need to strengthen the culture of "deep tracing", rather that= desperately seeking for "this was the moment, this was the breakfast, these were the persons around that table at that moment= in time, and that person must be named as the originator, for that person = we must create a bust". "Deep tracing" opens up more researc= h, "creating busts" does the opposite.

Best,

Patrik


On 2023-12-18 22:08, Jean-Pierre Marquis wrote:

I gue= ss it is a paper like this one that justified Eilenberg & Mac Lane's ch= oice of terminology. It would be interesting to trace how far the terminolo= gy goes back, who used it and where.

 = ;

Jean-= Pierre

 = ;

De=  : Evgeny Kuznetsov <jenkakuznecov@gmail.com>
Date : lundi, 18 d=C3=A9cembre 2023 =C3=A0 14:43
=C3=80 : Jean-Pierre Marquis <jean-pierre.marquis@= umontreal.ca>
Cc : Wesley Phoa <doctorwes@gmail.com>, Michael= Barr, Prof. <barr.michael@mcgill.ca>, categories@mq.edu.au <categ= ories@mq.edu.au>
Objet : Re: Modification of what I said

Here is a copy of the paper by Hassler Whitne= y of 1938 titled "Tensor products of abelian groups"

 

 

 

On Mon, Dec 18, 2023, 23:32 Jean-Pierre Marquis &l= t;je= an-pierre.marquis@umontreal.ca> wrote:

Most likely.

 

Whitney uses the terms 'natural isomorphism' and 'natural homomorphis= m' as well as the terms 'natural topology' and 'natural neighborhood' at ma= ny different places in the paper. But these terms are never explicitly defi= ned.

 

Cheers,

 

Jean-Pierre

 

 

De : Wesley Phoa <doctorwes@gmail.com>
Date : lundi, 18 d=C3=A9cembre 2023 =C3=A0 14:18
=C3=80 : Michael Barr, Prof. <barr.michael@mcgill.ca>
Cc : categories@mq.edu.au <categories@mq.edu.au>
Objet : Re: Modification of what I said

Was he referring to the paper "Tensor products of abelian groups"= , cited in this discussion? https://mathoverflow.net/questions/287869/history-= of-natural-transformations

 

 

The terms "natural isomorphism" and "natural homomorphism&qu= ot; are used on pages 500-501, and these do turn out to be natural transfor= mations, but it's not obvious that he intended to explicitly define a new f= ormal concept.

 

Wesley

 

Sent from my iPad

 

On Dec 18, 2023, at 10:00=E2=80=AFAM, Michael Barr, Prof. <barr.michael@mcgil= l.ca> wrote:

 

Peter Freyd claims that Hassler Whitney defined natural transformati= on in a 1938 paper.  I no longer have access to Math. Reviews (except = by going to McGill, which I have done only once in the last four years) so I cannot supply a reference.

 

Michael

 

 

 

You're = receiving this message because you're a member of the Categories mailing li= st group from Macquarie University. To take part in this conversation, reply all to this message.

 

View group files   |   Leave group   |   Learn more about Microsoft 365 Groups

 

 

 
 
You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message.
 
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