Re: Russell’s “Logicism” vs. Pure Logicism

Fri Dec 18 05:44:47 EST 2020

In the first chapter of the book The Life and Work of Leon Henkin (
https://www.springer.com/gp/book/9783319097183) you can read that:

In April 1, 1963, Henkin received a very interesting letter from Bertrand
Russell. In it, Russell thanks Henkin for ‘your letter of March 26 and for
the very interesting paper which you enclosed.’ Even though we do not know
for certain which paper he is talking about, we guess that it should be
Henkin’s paper entitled Are Logic and Mathematics Identical?

Right at the beginning Russell declares ‘It is fifty years since I worked
seriously at mathematical logic and almost the only work that I have read
since that date is Gödel’s. I realized, of course, that Gödel’s work is of
fundamental importance, but I was pussled by it. It made me glad that I was
no longer working at mathematical logic.’ It seems that Russell understood
Gödel's theorem as implying the inconsistency of Principia:   If a given
set of axioms leads to a contradiction, it is clear that at least one of
the axioms must be false. Does this apply to school-boys' arithmetic, and
if so, can we believe anything that we are taught in youth? Are we to think
that 2+2 is not 4, but 4.001?

I have a copy of that interesting letter given to me by Ginette Henkin, but
I don't think I can post it to the forum.

María Manzano

El vie, 18 dic 2020 a las 5:46, Oliver Marshall (<
omarshall at gradcenter.cuny.edu>) escribió:

> Following up on Timothy Chow’s message:
>
> Frege's logicism is an attempt at pure logicism: the axioms of arithmetic
> are derived from the axioms of logic together with logical definitions of
> the arithmetical primitives. This makes the theorems of Frege's derivation
> —the axioms of arithmetic— theorems of logic (as Frege understands
> "logic"), and to this extent Frege’s derivation is supposed to provide an
> analysis of arithmetic.
>
> Russell's logicism is not pure in this sense. For him, axioms can be
> logical, or non-logical (for example, at one stage R claims that Infinity
> is empirical). What makes the proposal to some extent logicist is that the
> definitions are logical and the theorems are supposed to be derived from
> the axioms by the implications (that Timothy mentions below) without appeal
> to Kantian intuition. Further, this is supposed to provide a deductive
> framework for science, in so far as you can substitute in special science
> objects as values of variables. Whether it also provides an analysis of
> arithmetic is less clear.
>
> Can anyone shed light on the sense in which Russell’s derivation is an
> analysis, given that it has an empirical element?
>
> Oliver
>
>
> Snt frmMy iPd.
> ------------------------------
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> Today's Topics:
>
>    1. Re: Russell vs Hilbert (Timothy Y. Chow)
>    2. Re: Russell vs Hilbert (Patrik Eklund)
>    3. Re: Russell vs Hilbert (Richard Kimberly Heck)
>
>
> ----------------------------------------------------------------------
>
> Message: 1
> Date: Tue, 15 Dec 2020 20:45:32 -0500 (EST)
> From: "Timothy Y. Chow" <tchow at math.princeton.edu>
> To: fom at cs.nyu.edu
> Subject: Re: Russell vs Hilbert
> Message-ID: <alpine.LRH.2.21.2012152025480.32550 at math.princeton.edu>
> Content-Type: text/plain; format=flowed; charset=US-ASCII
>
> Joe Shipman wrote:
>
> > One of my Twitter friends claims that the view that all mathematical
> > propositions are decidable from a few axioms was held by Russell prior
> > to Hilbert, and that Russell was claiming, in his books, not only to
> > have reduced all mathematical reasoning to a few logical principles, but
> > also to have claimed that those principles were sufficient to settle all
> > questions.
> >
> > This is based on his reading of the following 1903 passage:
> >
> > "THE present work has two main objects. One of these, the proof that all
> > pure mathematics deals exclusively with concepts definable in terms of a
> > very small number of fundamental logical concepts, and that all its
> > propositions are deducible from a very small number of fundamental
> > logical principles"
> >
> > In my opinion, by "propositions", Russell meant "theorems", not "true
> > statements", and therefore one may not jump to the conclusion that
> > Russell failed to achieve his object and did not prove what he claimed
> > to be trying to prove.
> >
> > Can any Russell experts shed light on this question?
>
> The quote comes from Russell's book, "The Principles of Mathematics."  In
> the same book, Russell says that pure mathematics is the class of all
> propositions of the form "p implies q."  This view, that the entire
> content of mathematics consists of assertions of implications, is usually
> called logicism.  The *conceptual* reduction of mathematics to logic is
> what Russell is concerned with in the book, not the claim that all
> mathematical truths can be *algorithmically* generated from a few axioms
> via some sequent calculus.  I think that it's anachronistic to read the
> phrase "its propositions are deducible from" in algorithmic terms; I would
> instead parse it as something like "its propositions can be reduced to" or
> "its propositions are analyzable in terms of".
>
> But you can judge for yourself by reading the preface and the first
> chapter of Russell's book.
>
> https://people.umass.edu/klement/pom/pom.pdf
>
> Tim
>
>
> ------------------------------
>
> Message: 2
> Date: Wed, 16 Dec 2020 09:28:28 +0200
> From: Patrik Eklund <peklund at cs.umu.se>
> To: Joe Shipman <joeshipman at aol.com>
> Cc: Foundations of Mathematics <fom at cs.nyu.edu>
> Subject: Re: Russell vs Hilbert
> Message-ID: <d0d4159fe8ded510854c3444ae613590 at cs.umu.se>
> Content-Type: text/plain; charset="us-ascii"; Format="flowed"
>
> A tweet shouldn't rewrite the history of the foundations of mathematics.
>
>
> Reading Grundlagen der Mathematik I, II is still recommendable, but very
> few, unfortunately, know content in those volumes.
>
> Hilbert is mostly ignored, even, funny as it is, Hilbert was the one
> speaking very strongly against "ignorabimus". That was in a certain
> sense at that time, and, little did he know that his and Bernays' GdM I,
> II would basically be ignored when decades after the rewriting of the
> history of logic began strongly influenced by Church and the way he
> controlled his journal.
>
> Patrik
>
> On 2020-12-15 23:47, Joe Shipman wrote:
>
> > One of my Twitter friends claims that the view that all mathematical
> propositions are decidable from a few axioms was held by Russell prior to
> Hilbert, and that Russell was claiming, in his books, not only to have
> reduced all mathematical reasoning to a few logical principles, but also to
> have claimed that those principles were sufficient to settle all questions.
> >
> > This is based on his reading of the following 1903 passage:
> >
> > "THE present work has two main objects. One of these, the proof that all
> pure mathematics deals exclusively with concepts definable in terms of a
> very small number of fundamental logical concepts, and that all its
> propositions are deducible from a very small number of fundamental logical
> principles"
> >
> > In my opinion, by "propositions", Russell meant "theorems", not "true
> statements", and therefore one may not jump to the conclusion that Russell
> failed to achieve his object and did not prove what he claimed to be trying
> to prove.
> >
> > Can any Russell experts shed light on this question?
> >
> > -- JS
> >
> > Sent from my iPhone
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> ------------------------------
>
> Message: 3
> Date: Wed, 16 Dec 2020 03:22:27 -0500
> From: Richard Kimberly Heck <richard_heck at brown.edu>
> To: Joe Shipman <joeshipman at aol.com>, Foundations of Mathematics
>         <fom at cs.nyu.edu>
> Subject: Re: Russell vs Hilbert
> Message-ID: <86300f25-c5cf-f054-1376-b2ff504e9fc9 at brown.edu>
> Content-Type: text/plain; charset="utf-8"
>
> On 12/15/20 4:47 PM, Joe Shipman wrote:
> > One of my Twitter friends claims that the view that all mathematical
> > propositions are decidable from a few axioms was held by Russell prior
> > to Hilbert, and that Russell was claiming, in his books, not only to
> > have reduced all mathematical reasoning to a few logical principles,
> > but also to have claimed that those principles were sufficient to
> > settle all questions.
> >
> > This is based on his reading of the following 1903 passage:
> >
> > "THE present work has two main objects. One of these, the?proof?that
> > all pure mathematics deals exclusively with concepts definable in
> > terms of a very small number of fundamental logical concepts, and that
> > all its propositions are deducible from a very small number of
> > fundamental logical principles"?
> >
> > In my opinion, by ?propositions?, Russell meant ?theorems?, not ?true
> > statements?, and therefore one may not jump to the conclusion that
> > Russell failed to achieve his object and did not prove what he claimed
> > to be trying to prove.
> >
> > Can any Russell experts shed light on this question?
>
> I'm not an expert on Russell specifically, but I do know something about
> this period, and there's a distinction I'd like to emphasize.
>
> But first: Russell lived for almost 40 years after G?del's proof of the
> incompleteness theorem. I'd be stunned if he never had something to say
> about it---and hopefully actual Russell experts will chime in on that.
> I'm hopeful, too, that, if he did have something to say, it was more
> sensible than what Wittgenstein had to say (though various apologists,
> no less than Hilary Putnam, have tried to make some sense of what he did
> say).
>
> So, first, Russell's usage of 'proposition' is notoriously ambiguous.
> Second, what Russell thought in 1903 (indeed, in some month in 1903) is
> not necessarily indicative of what he thought at any other time. His
> views changed rapidly then, and for a long time after. Third, I strongly
> suspect myself that you are right, Joe, that what Russell meant was that
> all /known/ theorems of mathematics could be derived from the axioms of
> PM. That is what he actually sets out to show. Fourth, certainly by
> /Principia/ (a decade later), Russell was aware that some /proofs/ of
> accepted results depended upon what he called the 'multiplicative axiom'
> (the axiom of choice), and he surely would have understood that this
> dependence might be essential. So, fifth, it's hard for me to see how
> Russell could not have understood, at least as an abstract possibility,
> that there might be established theorems of mathematics that could not
> be proven given the specific axioms he had identified. That is what we
> might call 'boring incompleteness'. The obvious response would be to add
> some new axioms, though it might take real work to find out which axioms
> it would make the most sense to add. (Frege considers such a
> possibility, explicitly, in "Mr Peano's Conceptual Notation and My Own",
> which was one of several off-prints that he sent to Russell in response
> to Russell's first letter to him.)
>
> What Russell surely did not anticipate was /essential/ incompleteness.
> Even G?del does not properly formulate that result in his 1931 paper, as
> he acknowledges in a famous footnote: Until Turing's work, we lack a
> good account of what a 'formal' theory is and so a good account of
> essential incompleteness.
>
> That's the crucial distinction. And I assume it must also apply to
> Hilbert. Surely he would have understood, at least as an abstract
> possibility, that the specific axioms he'd identified might be
> insufficient to decide all arithmetical questions. Sp just add some new
> ones! But what he may not have fully appreciated was the possibility
> that this insufficiency might be irresolvable. It takes some thinking to
> understand how that can be so.
>
> Riki
>
> --
> ----------------------------
> Richard Kimberly (Riki) Heck
> Professor of Philosophy
> Brown University
>
> Pronouns: they/them/their
>
> Email:           rikiheck at brown.edu
> Website:         http://rkheck.frege.org/
> Blog:            http://rikiheck.blogspot.com/
> Amazon:          http://amazon.com/author/richardgheckjr
> Google Scholar:  https://scholar.google.com/citations?user=QUKBG6EAAAAJ
> ORCID:           http://orcid.org/0000-0002-2961-2663
> Research Gate:   https://www.researchgate.net/profile/Richard_Heck
>
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