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

Thu Dec 17 11:30:34 EST 2020

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

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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)

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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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