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Topic: Is logic part of mathematics - or is mathematics part of logic?
Replies: 8   Last Post: Jul 7, 2013 5:19 PM

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GS Chandy

Posts: 5,934
From: Hyderabad, Mumbai/Bangalore, India
Registered: 9/29/05
Re: Is logic part of mathematics - or is mathematics part of logic?
Posted: Jul 7, 2013 12:14 AM
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Clyde Greeno posted Jul 6, 2013 7:42 AM (GSC's remarks interspersed):
> Humbug!
> Russell so went way out on the limb of "pure
> formalism", well before the
> essence of mathematics was discerned.

I would hesitate to use the description "Humbug!" here.

It is probably true enough that Russell went "way out on the limb of 'pure formalism' well before the 'essence of mathematics' was discerned" (as you claim). But I would not describe Jonathan Crabtree's quoting from Russell's "Principles..." (or the quotation itself) as absurd or "humbug" (with or without the exclamation mark). [By the way, I disagree with you - if such happens to be your claim - that the 'essence of mathematics' has been 'fully' (or even 'adequately') discerned. I believe, in fact, that we have a very long way to go, though it is true that we have taken a few important steps in that direction].
> "Pure" mathematics entails
> descriptions of KINDS of things that are being
> attended, together with
> whatever logically substantial concepts and
> conclusions are derived from
> those descriptions.

Probably true. (Though I did once myself study and research into 'pure' mathematics, I am today not able to judge on just how true the above description may be [in detail. Broadly, your description appears more or less OK to me]. I've been otherwise occupied in trying to understand 'systems' [at a 'system level']- including 'systems of thought', for instance, math, etc, etc - and in trying to apply these ideas about systems to issues in our practical life - see below).

But is yours a 'complete' <description> of 'pure' mathematics (OR of what it entails)? Probably not. Is that an 'adequate' <description> of 'pure' mathematics? Possibly not. (See also Sidebar, below**).

No, I will NOT be able to 'complete' or 'make adequate' your description of what 'pure' math entails: as noted I am doing 'other' things - some of which could perhaps in some ways "CONTRIBUTE TO" those who are interested in working to develop the foundations of math, etc. These above are questions that would have to be decided ultimately by the 'stakeholders' in 'pure' mathematics (at each specific age, era, day and date).

[**Sidebar: In my opinion (as a keenly interested student of 'mathematical ideas' and their impact on our 'real' world), I believe that both - Russell's definition/description AS WELL AS Jonathan Crabtee's reference to it in his post that you have judged to be "Humbug!" - are indeed valid and valuable contributions to further our understanding. As to just how these may 'further our understanding', I have provided some thoughts below].

Now, who are those 'stakeholders' in 'pure' mathematics?
Well (offhand), more or less the following groups of people:

- -- 'Pure' mathematicians (obviously!) along with their associations and federations, etc; but perhaps less obviously, the stakeholders include:

- -- Logicians and philosophers;

- -- Those who apply the concepts and developments of 'pure' mathematics, namely: applied mathematicians; physicists; chemists; a whole variety of other scientists (the variety increasing all the time, as 'science' expands and as mathematics is found useful by scientists); and so on;

- -- teachers and students of mathematics - even beginning primary school learners of math are also stakeholders, though we shall need special mechanisms to get their ideas 'into the system'. However, it IS important that we do that in order to enable us to ensure that we are able to overcome the many current deficiencies in the way we currently 'teach' math at beginning levels. If we learn how to 'get it right' at the beginning level, that would "SIGNIFICANTLY CONTRIBUTE" to getting the teaching of math right at more advanced levels; (I observe that there is quite a bit of foncusion and lack of clarity about how students are 'prepared' at high-school for higher level math);

- -- many others (the stakeholders themselves should decide who they are).

One serious problem faced by the stakeholders in 'pure' mathematics and by many others in society (i.e., all of us) is that (while 'pure' mathematics develops, increases its scope, deepens, etc, etc) the 'practitioners' of pure mathematics increasingly tend to talk only to each other in an increasingly more specialised language which is generally not at all understood by the public at large - and is sometimes not understood by other practitioners of mathematics (pure as well as applied).

A couple of instances (I'm currently somewhat handicapped as I do not have immediate access to my familiar books, etc - I am therefore providing these instances below from whatever I have immediately to hand and in mind. These instances are from the life of Gottlob Frege, who would probably have died in obscurity were it not for Russell and a few others):
i) I draw attention to the quite famous story of one of Gottlob Frege's most important papers (on differential equations, if I recall rightly), having been written in such a specialised language that even other mathematicians could not understand it, till someone translated it into ordinary layperson's German!!. At that point, as I undertand, mathematicians took note of it and proclaimed it to be 'foundational'!!! (I can't now locate the referencto to this, but doubtless Math-teach readers will know all about this).

ii) Again, Russell writes of Frege:
> "In spite of the epoch-making nature of [Frege's]
> discoveries, he remained wholly without recognition
> until I drew attention to him in 1903".

Here's some more about Frege:
> "Frege's influence in the short term came through the
> work of Peano, Wittgenstein, Husserl, Carnap and
> Russell. In the longer term, however, Frege has become
> a major influence on the development of philosophical
> logic and the man who seems to have been largely
> ignored by his contemporaries has been avidly read by
> many in the second half of the twentieth century,
> particularly after his works were translated into
> English". (Both instances at ii are drawn from the
> Mactutor "History of Math" - http://www-history.mcs.st-and.ac.uk/Biographies/Frege.html ).

iii) In a few cases, 'pure' mathematicians talk only to themselves. (Frege's case noted above may have become one such if it had not been for Russell and others who recognised the fundamental nature of what he had done).

A: As observed earlier, I would hesitate to describe as "Humbug" either Russell's work in "pure formalism" or Jonathan Crabtree's allusion to Russell's definition/description of 'pure' mathematics as articulated in his "Principles of Mathematics" (quoted by Crabtree in his posting). I WOULD describe as "Humbug" the following recommendations we have often seen at Math-teach:

(These foolish ideas DO indeed constitute the most utter humbug whether they are intended as serious 'recommendations' or whether they are intended satirically or in jest: they are humbug because they accomplish precisely nothing useful, they cannot even make us smile or laugh.

Humbug, n,: 1. Something intended to deceive; a hoax or fraud.
2. A person who claims to be other than what he or she is; an impostor.
3. Nonsense; rubbish.
4. Pretense; deception.
Interjecttion, used to express disbelief or disgust).

My underlying point is that we all (including 'pure' and applied mathematicians AND including all of us participants at Math-teach) do need to learn to communicate about our ideas and our work in general more effectively and clearly than we 'normally' and conventionally do.

[Of course, as noted above, some 'pure' mathematicians "talk only to themselves" - but I believe that even they would like at least to have their profound ideas communicated to the world by others if they themselves do not want to take steps to do it. The 'rescue' of Frege by Russell and others is an instance: I am sure that Frege, were he alive today, would have been most happy that Russell rescued his ideas from obscurity].

I observe that a great many of our communications fail to communicate effectively: this is the situation in practically ALL our Forums (Parliament; Senate and House of Representatives; seminars; learned lectures; discussions here at Math-teach.

This is, I claim, because the 'language' of 'pure prose' (to which we are bound in most forums, INCLUDING Math-teach) is constitutionally incapable of enabling us to clarify the inter-relationships between factors in complex systems. It sometimes leads us to make statements such as "Humbug!" when actually we mean nothing of the sort. (This is what you've unfortunately done in your post). Or "Absurd" (as Joe Niederberger had done to one of my recent communications to him) when the fact is that we do not at all understand just what it was that we're talking about!

In several of my posts at Math-teach, I have often referred to a development that I describe as 'prose + structural graphics' (p+sg). I claim that p+sg could help us all communicate a lot more clearly on most of the issues that concern us in society - for specific instance the many issues right here at Math-teach on which we contribute much noise but throw little light.

This p+sg is actually a minor extension to our conventional language of 'pure prose' - and it develops out of the seminal contributions of the late John N. Warfield to 'systems science'.

More information about Warfield's work is available at his website http://www.jnwarfield.com and from the "John N. Warfield Collection" held at the library of George Mason University, Fairfax, VA, USA (where Warfield was Professor Emeritus) - see http://ead.lib.virginia.edu/vivaxtf/view?docId=gmu/vifgm00008.xml;query=; .

Some developments of Warfield's work that I call the 'One Page Management System' (OPMS) now enable anyone, at any level (high-school up), to apply quite sophisticated systems concepts to enhance clarity of understanding of systems of all kinds. The attachments to my post heading the thread "Democracy: how to achieve it?" (http://mathforum.org/kb/thread.jspa?threadID=2419536 ) provide brief descriptions of the OPMS and how to use it.
> That is how mankind created mathematics, and how it has
> progressively refined mathematics, to become a
> specialized *art of learning.*

Well, possibly much of the above (I don't know how much) is probably correct. However, do please understand that Russell's work in "pure formalism" is just one such part of the 'progressive refinement of math' that you acknowledge (and should laud) - though it is true that he did seem rather to go out on a limb in the development of his 'pure formalism'. (I wouldn't describe it as "outrageous", either).

We should try to understand that errors - sometimes serious ones - are often made when 'new thinking' is created (as Russell and Whitehead were doing at that time).

Many such errors are rectified over time (often by the very individuals making such errors - assuming they are honest enough to accept error, of course). I understand that Russell himself had acknowledged that much of the 'pure formalism' of 'Principia Mathematica' (PM) would NOT stand the test of time: there is a very interesting story in which Russell recounted a dream of his in which he saw St Peter (or other such archangel) take up PM in hand and hesitate on his (the archangel's) decision whether PM should be thrown into the garbage bin or whether it should be preserved.
> Outrageous formalism has done far more to
> inhibit mankind's
> mathematical progress than to facilitate it.

I disagree. 'Pure formalism' has surely failed to accomplish what no doubt Russell and Whitehead had intended it to do. So what?

"Principia Mathematica" is surely a 'great' piece of advanced thinking as part of the human endeavour.

Undoubtedly, it has in many ways led us down 'wrong paths' (in our journey to understanding 'math', 'logic', 'philosophy') - but so what? It is surely up to us to learn "how to take the right path(s)" to reach our hoped for destination (or, at least, to learn how to recognise a path is wrong when we go along it. (Incidentally, the OPMS, about which I've written above, contains practical means to enable us recognise when we may be taking 'wrong paths' in our various journeys towards enhanced understanding).

Describing Russell's work as "Humbug!" is surely much more grave an error in judgement than any error contained in "Principia Mathematica" itself. (Describing as "Humbug!" Jonathan Crabtree drawing our attention to that specific passage in the "Principles of Mathematics" is also an error - though it is not as serious an error as the previous one).

P.S.: By the way, as some of us here have failed to understand, I note that there is some sound reason why I put certain words and phrases here in ALL-CAPS, for instance, "CONTRIBUTES TO"; "INCLUDES" and the like. To explain adequately, I would need to use 'p+sg'.
>> - --------------------------------------------------
> >From: "Jonathan Crabtree"
> ><sendtojonathan@yahoo.com.au>
> >Sent: Friday, July 05, 2013 5:54 PM
> >To: <math-teach@mathforum.org>
> >Subject: Re: Is logic part of mathematics - or is
> >mathematics part of logic?
> >
> > Pure Mathematics is the class of all propositions

> of the form 'p implies
> > q' where p and q are propositions containing one or
> more variables, the
> > same in the two propositions, and neither p nor q
> contains any constants
> > except logical constants.
> >
> > And logical constants are all notions definable in

> terms of the following:
> > Implication, the relation of a term to a class of
> which it is a member,
> > the notion of such that, the notion of relation,
> and such further notions
> > as may be involved in the general notion of
> propositions of the above
> > form.
> >
> > In addition to these, mathematics uses a notion

> which is not a constituent
> > of the propositions which it considers, namely the
> notion of truth.
> >
> > Source: Principles of Mathematics Bertrand Russell

> 1903
> >
> http://archive.org/stream/principlesofmath01russ#page/
> n35/mode/2up

Message was edited by: GS Chandy

Message was edited by: GS Chandy

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