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PEIRCE-L Digest 1269 -- January 22, 1998
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From PEIRCE-L Forum, Jan 5, 1998, [name of author of message],
"re: Peirce on Teleology"
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Topics covered in this issue include:
1) Re: slow reading: New List (paragraph 1)
by Thomas.Riese[…]t-online.de (Thomas Riese)
2) Re: Context and Continuity-Thermodynamic
by piat[…]juno.com (Jim L Piat)
3) Re: Hookway's _Peirce_:Introduction
by piat[…]juno.com (Jim L Piat)
4) Re: Peirce and the Stoics (from Douglas Moore)
by piat[…]juno.com (Jim L Piat)
5) Re: slow reading: New List (paragraph 1)
by Tom Burke
6) Re: Dedekind and the bootstrap
by Thomas.Riese[…]t-online.de (Thomas Riese)
7) Re: Context and Continuity-Thermodynamic
by Tom Anderson
8) Some more Goedel quotes
by Tom Anderson
9) Re: Context and Continuity-Thermodynamic
by piat[…]juno.com (Jim L Piat)
10) Re: Dedekind and the bootstrap
by joseph.ransdell[…]yahoo.com (ransdell, joseph m.)
----------------------------------------------------------------------
Date: Thu, 22 Jan 1998 11:22:21 +0100
From: Thomas.Riese[…]t-online.de (Thomas Riese)
To: peirce-l[…]ttacs6.ttu.edu
Subject: Re: slow reading: New List (paragraph 1)
Message-ID:
Dear Jim,
Thanks for your very kind response.
I really like your idea of 'riding a bike' very much. That's what I
intended to convey. And when you continue:
> Unfortunately, by the end of your comments I didn't know if I was
> coming or going as opposed to before I read them when I was sitting
> on the side wondering how to get started.
.. well, Jim, you said this should be 'slow reading'! Seriously: you
are right! At this stage there is already the idea of gradation and
incipient sequence but the idea of _direction_ is *not* yet born!
This will be achieved in the third paragraph! (CP 1.547) This might
seem a bit subtle a distinction at first sight but I think it is
important for the development of the text as an argument. Still
further: if I am right in my views then the third paragraph indeed
embodies the germ of the mathematical structure a lattice (though
the term, historically speaking, didn't yet exist at the time when
Peirce worked out the New List). And, at least as a figure of speech,
it would make sense to me to say that a lattice sort of works in two
opposed directions at the same ime. But that's premature and loose
speak here.
Anyway: we will not be able fully to understand the beginning until we have
seen the end. I would like to have another look at what Peirce has
achieved so far.
You wrote:
> I am reminded of your admiration for Erickson and the systems approach.
Yes, I really think that Milton Erickson was a genius in his field,
too. So Milton had the curious ability, when a patient came with pain
or paralytic symptoms, to move these symptoms about (in the body of
the patient). This might seem very 'unscientific' at first sight but
if pain is not so much a feeling as it is a sort of 'meta-feeling' as
Erickson knew from his own exploration of perceptual changes induced
by his polio then it is perhaps less improbable that we are perhaps
not able to abolish such things completely but nevertheless can
transform them (ask children how coffee or beer tastes and ask the
same persons later in life).
Perhaps it is not uninteresting, as an aside, that, if I remember right, Peirce
also wrote that pain is not a feeling, but a feeling of a feeling
(the feeling that a feeling is repelling).
Similar with paralysis (try to play the piano if you are not trained
in it).
So Milton one day had a patient and if I remember the story right,
this man's arms from time to time were paralysed without obvious physiological
reason. What Erickson did and why I tell all this stuff is: he got
his patient to move about his very real paralytic symptoms. The whole
story ended in that this man was never really healed, since finally
his fingernails were completely paralysed. And he tterly resented
this for the rest of his life.
We would expect Peirce to run into paradox with what he tries to do in the New List
and I think he was well aware of this. (In fact he was much more than
just only aware of this. In his 1865 Harvard Lectures, W1;p.204, he
writes: "Now these questions may seem trifling and puerile; but I
have no hesitation in saying that I know of none up the correct
solution to which man's happiness depends more; for the paradoxes
which beset our highest practical interests - our religion - the
puzzles of free will, of divinity, of immortality are precisely of
such a character as these.")
But if there is "a conception of gradation among those conceptions that
are universal": why then not shift paradox around a bit and place it
where it not only does not hinder us to start but even gives us the
decisive opportunity to _use_ it.
Figurativly speaking the first paragraph is such a paralysis in the
fingernails where a very uninteresting, casual text, nevertheless
fulfills a strictly _necessary_logical_ function -- purely formally.
This opens up the possibility to have both: truth _and_ logical
completeness.
Who says that truth could be established by talking alone?
Perhaps a last remark. The mechanism often reminds me of a story
of Edgar A. Poe's where a letter is hidden by placing it in the
middle of the room just in front of the eyes of the policemen who
tried to find it. It was so completely hidden *because* everybody saw
it and constantly stumbled over it. I think part of Peirce's personal
tragedy is that he , long before logicians and mathematicians saw the
problem and began to fight with paradox in logic, delivered an
extremely ingenious and elegant solution. But it is the elusive
obvious since, as Peirce put it 1906 in his 'Prolegomena to an
Apology for Pragmaticism' (CP 4.544):
"Universes cannot be described."
So you have to say what cannot be said;-)
Thomas.
------------------------------
Date: Thu, 22 Jan 1998 11:07:05 -0500
From: piat[…]juno.com (Jim L Piat)
To: peirce-l[…]ttacs6.ttu.edu
Subject: Re: Context and Continuity-Thermodynamic
Message-ID: <19980122.110810.12190.5.piat[…]juno.com>
On Wed, 21 Jan 1998 16:48:16 -0600 (CST) Tom Anderson wrote:
>You HAVE made me want to read Rotman's book, incidentally. I posted a
>bunch
>of notes in the fall using Rotman's article as a foil -- that's why I
>read
>it so carefully.
Tom, the book is in the mail - shortly. I look forward to your review.
Jim Piat
------------------------------
Date: Thu, 22 Jan 1998 10:46:55 -0500
From: piat[…]juno.com (Jim L Piat)
To: peirce-l[…]ttacs6.ttu.edu
Subject: Re: Hookway's _Peirce_:Introduction
Message-ID: <19980122.110810.12190.3.piat[…]juno.com>
On Wed, 21 Jan 1998 11:09:10 -0600 (CST) "Dr. Arthur Stewart"
writes:
>You know, I think that Professor Hookway is gonna sell a LOT of books
>if this keeps up.
>
>AFS
>
Good ! Then there's all the Peirce-Listers books to work our way through
:>)
------------------------------
Date: Thu, 22 Jan 1998 10:51:33 -0500
From: piat[…]juno.com (Jim L Piat)
To: peirce-l[…]ttacs6.ttu.edu
Subject: Re: Peirce and the Stoics (from Douglas Moore)
Message-ID: <19980122.110810.12190.4.piat[…]juno.com>
Douglas Moore,
What a joy to read your very interesting, substantial, and
straightforward post on Peirce, the Stoics and most importantly (for me)
your wide ranging comments on Russell, first classness, and the
fundamentally two sided nature of the generic. I feel you've identified
some landmarks and pointed out a path through a lot of my confusion and
conceptual underbrush from which I sometimes despair of ever seeing my
way clear. As with so many of the posts here I will read yours again and
again.
Like many folks I'd heard of object oriented programming but wondered
from afar what al the hoopla was about. It makes sense that programming
(as explicit step by step modeling) would on both a theoretical and
practical level be confronted with all sorts of philosophical conundrums
on a regular basis. In some respects (as I'm thinking about it now) the
computational sciences are the experimental basic sciences (not just
technological or applied) of the philosophy of mind/language (Searles
objections notwithstanding). I realize my opinions here on the philosophy
and computational sciences may be completely wrong headed, but I still
want to express some of the excitement, sense of freshness and new found
wealth your paper has provided this parvenu.
I hope you will consider posting this paper over at the Arisbe site and
look forward to hearing more about the implications of "first classness"
and Peirces view of signs.
Thanks,
Jim Piat
------------------------------
Date: Thu, 22 Jan 1998 12:08:43 -0500
From: Tom Burke
To: peirce-l[…]ttacs6.ttu.edu
Subject: Re: slow reading: New List (paragraph 1)
Message-ID:
At 4:24 AM -0600 1/22/98, Thomas Riese wrote:
>...
>But if there is "a conception of gradation among those conceptions that
>are universal": why then not shift paradox around a bit and place it
>where it not only does not hinder us to start but even gives us the
>decisive opportunity to _use_ it.
I agree that this kind of shift -- and especially knowing when to do it and
when not to -- is the element of genius in Peirce's triad methodology, or
James's radical empiricism, or Dewey's conception of "situations" and
"experience" in a theory of inquiry (etc etc). Often when you find (in
philosophy, science, mathematics, etc) that something is absolutely
necessary to make sense of what you are doing but is otherwise inexplicable
in terms you are already working with (e.g., something that unites the
manifold of sense, something that unites the conception and the manifold,
etc etc -- Par.2 of "New List"), the only option is to take it or something
like it as primitive (i.e., as "first class"). Of course it must be
properly formulated and truly inexplicable otherwise -- logically and
conceptually non-reducible, etc. -- not just a matter of lack of thought or
imagination on your part. Peirce's emphasis in Par.4 on the "copula" which
unites subjects and predicates in propositions is just one case of this.
>Figurativly speaking the first paragraph is such a paralysis in the
>fingernails where a very uninteresting, casual text, nevertheless
>fulfills a strictly _necessary_logical_ function -- purely formally.
>
>This opens up the possibility to have both: truth _and_ logical
>completeness.
>
>Who says that truth could be established by talking alone?
>...
>So you have to say what cannot be said;-)
Or less paradoxically: we don't have to talk about what cannot be talked
about. And for that matter we don't have to say what cannot be said. But
we have to talk about what cannot be said.
______________________________________________________________________
Tom Burke http://www.cla.sc.edu/phil/faculty/burket
Department of Philosophy Phone: 803-777-3733
University of South Carolina Fax: 803-777-9178
------------------------------
Date: Thu, 22 Jan 1998 18:44:56 +0100
From: Thomas.Riese[…]t-online.de (Thomas Riese)
To: peirce-l[…]ttacs6.ttu.edu
Subject: Re: Dedekind and the bootstrap
Message-ID:
Dear Joe Ransdell and Jim Piat,
in order to sort out the things you mention (historically, logically
and mathematically) with some strictness one would have to write a
book, a huge book :-)
In a few words I would describe Richard Dedekind's work on the
natural numbers as follows: he built a system using what he called
'Ketten' (chains) of maps where one map maps how another maps a map
etc. finally mirroring the property of the natural numbers that
(infinite) parts of them can mirror the whole by being able to be put
in one-one correspondence ("similarity") with the whole.
Thus he built a completely self-mirroring (self-referential) system.
Using the connection between transitivity and correspondence he could
then convert the whole by a mechanism similar to mathematical
induction ("complete induction") into a demonstrative consequence,
_provided_ he could find at least one existing element serving as a
premiss.
So the whole thing then depends on the following proposition
(nr.66 in Dedekind's paper):
[my translation]
Proposition:
There exist infinite systems.
Proof:
The world of my thoughts, i.e. the totality S of all things which can be objects of my thinking, is infinite.
For if s is an element of S, so the thought s' that s can be an object of my thinking, itself is an element of S.
If we consider this as the image phi(s) of the element s then the in this way determined mapping phi of S has the
property that the image S' is part of S.; and S' is a proper part (echter Teil) of S, since in S there are elements
(e.g. my own self (Ich)), which are different from each such thought s' and thus not contained in S'. So it is plain
that if a and b are distinct elements of S, their images a', b' are distinct too, so that phi is a 'similar' mapping
(aehnlich, deutlich). So it follows that S is infinite. q.e.d.
The subtle trick is that _the_whole_thing_ then has itself the form of a
simple deductive consequence. It's so to speak "self-explaining". A
really breathtaking bootstrap process indeed consisting, logically
speaking, of nothing but demonstrative consequences of demonstrative
consequences ... forming, considered in another way, nothing but one
simple consequence proving itself. Which is possible because it is
infinite and so can contain itself as a proper part being at the same
time existing. See above...
One of Peirce's comments was that he himself didn't like the idea that logic
is just a chain where the weakest element is decisive for the
stability of the whole structure. He preferred a rope where many
elements might fail without the whole thing breaking down immediately.
(Sorry, I can't find the reference. I think it is in the NEM)
I think that indeed it does not need too much mathematical
sophistication to suspect that with the above proof we are perhaps not
many steps away from some Russelian paradox or Goedelian limitation.
At least if we go and try to remove the dubiosity of the above
existence(!) proof.
But Dedekind showed that, apart from that, the whole thing really
works as it should. With all the usual arithmetical operations and all
that. Natural numbers!
And if the natural number system is fundamental for mathematics and the
natural number system itself nothing else but a bootstrap embodiment
of a simple, self-explaining demonstrative consequence, then, well,
then mathematics is nothing but an outgrowth of logic, it consists of
nothing but deductive logic.
""O Thou, All-Sufficient, Self-Sufficient, Insufficient God."" (Charles Peirce, CP 5.71)
He continues: "Now pure Self-consciousness is Self-sufficient, and if
it is also regarded as All-sufficient, it would seem to follow that it
must be Insufficient. I ought to apologize for introducing such
Buffoonery into serious lectures. I do so because I seriously believe
that a bit of fun helps thought and tends to keep it pragmatical."
And further: "It is therefore the precise analogue of pure
self-consciousness. As such it is _self-sufficient_. It is saved from
being unsufficient, that is as no representation at all, by the
circumstance that it is not _all-sufficient_, that is, is not a
complete representation but is only a point upon a continuous map."
I Peirce think gives one of the concisest versions of his own
alternative construction in his Cambridge Conferences Lectures of
1898 (Reasoning and the logic of things, Harvard U. Press 1992, pp.
156 ff.) starting with: "Although I am debarred from showing anything
in detail about the logic of relatives, yet this I may remark, that
where ordinary logic considers only a single, special kind of
relation, that of similarity, -- a relation too, of a particularly
featureless and insignificant kind, the logic of relatives imagines a
relation in general to be placed."
I think what is here meant with "relation of similarity" is clearly
Dedekind's "aehnliche Abbildung" (similar mapping).
I believe that if one wants to find out what the "technical" reason
is why Peirce was not a mathematical logicist and understand his
curious distinction between mathematics and logic one of the easiest
ways is to have a look at NEM III/2, p.837 where he discusses the
form of a transitive relation:
"Schroeder however made the curious remark that not _all_ transitive
relations are of that form [is lover of whatever is loved by...]; and
to get a form that would hold good of _all_ transitive relations, that
form must be _restricted_ (which sounds paradoxical). He shows that
every transitive relation is of the form '__is both lover of and lover
of everything loved by__.' This is a relation which not everything is
in to itself. [...] But I first analysed the matter and showed that it
depended on the following points.[...]"
For the parallel argument on the relation of correspondence see CP 3.523
To Dedekind's conception of self-similarity applies what Peirce said
about the relations of transitivity and correspondance: it has to be,
in a _seemingly_ paradoxical way, restricted in order to be expanded
and what regards the question of a logicistic view of mathematics
Peirce reasons are exactly parallel: mathematics is not just an
outgrowth of pure logic, but instead the mathematician, qua
mathematician, has to be supplied with special premisses in order
fully to be able to do his work. This is a bit paradoxical of course
in the sense in which the relation of transitivity is 'expanded by
restriction': the mathematician's job, according to Peirce then is
nothing but drawing conclusions from given premisses -- so in a sense
nothing but deductive logic in pure form. It's a curious twist. But
it is indeed sound if we wish a sound classification of the
sciences:-)
So 'bootstrapping' seems to be a good idea; the question is; which
brand of it? It seems as if a sort of "seed", something arbitrary,
were necessary for growth to take place.
On the other hand: if the relation of "similarity" is not all (the
resulting universe being all too boring) and at the same time Peirce
can expand relations by restricting them -- then this should indeed
introduce some diversity. For if relational generality
(correspondence, coordination) is not just only one-to-one or
"similar" in Dedekind's sense, then we should expect some branching
and perhaps even 'point-splitting' business to take place. I think
one should next have a look at the Dedekindian 'Cuts' with Peircean
eyes. I think Cantor then will come in, too.
Furthermore I really don't understand why our universe obviously must
be organized in such a way that if there happen interesting things
then always many at a time. I have to be on stage tomorrow and I
really don't yet know what I shall do there -- I mean even less than
is healthy. Funny feeling (think it is called "fear";-)). Please
forgive me if this message should be a bit hasty.
Thomas.
------------------------------
Date: Thu, 22 Jan 1998 13:09:10 -0800
From: Tom Anderson
To: peirce-l[…]ttacs6.ttu.edu
Subject: Re: Context and Continuity-Thermodynamic
Message-ID: <34C7B576.D230BF39[…]ix.netcom.com>
Howard Callaway wrote:
> The idea here of "experiments on infinite quantities" seems
> particularly important, and I wonder if this won't provide
> an opportunity to turn the discussion somewhat back in the
> direction of Goedel mathematical realism. In the example we
> have an infinite series defined by reference to a simple
> function, and it seems that this sort of infinity is not
> particularly problematic, in the sense that the series is
> well-defined, by reference to the function. But we come into
> a different ball game when we take up the idea, with Goedel,
> of an infinitely expandED set of axioms which would aim to
> encompass all arithmetical truth. Thinking of this list of
> axioms as like points on a line, this seems to go beyond
> Peirce, in a realist direction, since its as though all the
> points were all actually present rather than being poten-
> tialities, as in Peirce. (Potential discontinuities, as I
> like to think.)
>
I urge you to take a look at Goedel's essays in the third volume of the
collected works. I might well have misrepresented him. The hard-core
of his argument is to stress the irreducably mathematical meaning of
mathematics in the face of the positivistic position that mathematics
can be analyzed as syntax alone without reference to meaning. Along the
way, he offers some other ideas, and I'm not sure I captured them
accurately by my reference to an infinite series of axioms. What I'm a
little firmer on is his idea that mathematic concepts are prior to
axioms, and that any given set of axioms captures only a fraction of the
meaning of the mathematical concepts. He sees axioms as self-evident,
but in an 'ultimate' sense -- that is, the self-evidence of axioms
follows from an understanding of the meaning of the concepts after you
really get to know them (the concepts) well. Goedel does regard
mathematical truth as analytic, but he has an idiosyncratic way of
defining analytic -- idiosyncratic, but also very interesting -- and I
think close to some of Peirce's thinking. Goedel very explicitly
contrasts the more narrow meaning of analytic, true by virtue of the
definition of terms, with his, true by virtue of the meanings of terms.
So, Howard, I detect in your series of comments on my glosses on Goedel
that you see him as projecting some kind of implicitly mechanical
'falling out' of axioms from concepts, as if the claim that they are
implicitly there already, or that they are to the suitably informed mind
self-evident somehow implies that they don't grow or aren't the product
of human creativity. I may be misreading your comments, but that's how
I'm taking them, and I think the interpretion is just perpendicular to
what Goedel actually means to say. That is, his point is NOT one of
mechanical following out deductively at all. Rather, it's that it in
fact rather difficult to formulated deductive chains that capture the
intricacy and subtlety of the relationships among mathematical
concepts. Once it's successfully done, what's done is always partial,
will always involve systems that don't encompass all truth, and in fact
we will be able rather easily to find propositions that are evidently
true but not derivable from the system as it is.
Now as to whether augmented systems taken as a limit -- theoretically
expanded as far as they can go -- will encompass all truths -- that's
not something Goedel offered a solution to. He did think that some
questions that are undecidable might be later provable within an
augmented system, but he also believed that some of these might be true
or false but remain unprovable within augmented systems. Incidentally,
this was the topic of the doctoral dissertation that Turing wrote for
Alonzo Church -- his results were not very definitive, and later on
Solomon Feferman took up the question and I understand proved some
important results -- I have a summary somewhere, but when I read it I
confess I hadn't a clue what was going on!! My point, though, is that
the match between all potential propositions of mathematics and a
growing body of axioms is a mathematical not a philosophical question.
> Tom, you continue, addressing the idea of the infinite
> series, extended beyond any practical limits of calculation:
>
> > I think a good bit of Peirce's thinking on continuity has
> > to do with the 'inside' of that -- that is, infinitesi-
> > mals. When you talk about a relationship, you're
> > projecting the reality of how things are related, and how
> > they WOULD be related even if they never were in such and
> > such exact quantity, or even if they never will be -- if
> > x WERE a, then y WOULD BE b, and you can talk about a
> > range of possibility that just doesn't quit, and making
> > that range discrete (except in the cases where that's the
> > relationship you are projecting) is doing violence to the
> > real relationship -- and for continuous cases, projecting
> > only discrete relationships can be falsified by
> > experiment.
>
> This seems to me a good statement of the "hypothetical"
> character of mathematics in the pragmatic tradition. I'm
> somewhat puzzled by the last line here. But my main point is
> to say that the potential points on a line, in Peirce, are
> not defined by any statable function. Rather they, or some
> of them at least, seem to correspond to potential expansions
> of mathematics rather than corresponding to any finite axiom
> set for mathematics. This is to say that we will never be in
> a position to provide a function which would allow us to
> calculate all the potential points of a line, or what seems
> to be suggested by this, that axioms which allow us to
> calculate any subset of potential points on a line can be
> expanded in various alternative ways, although there is no
> way to include all the potential points on a line. This
> seems to be analogous to Goedel's incompleteness proof.
>
First, what I meant by the last line was just that if you have two
models, one discrete and one continuous, that are otherwise versions of
each other, you can test them out to see whether predictions they
generate are confirmed. I suppose that's a good part of the evidence
for quantum mechanics, isn't it?
There's an important distinction between being able to express something
and being able to compute it. We've been talking about algorithmic
information theory since Kelley proposed its relevance to understanding
Peirce on context & continuity. This theory models the information
content of a string as the minimal program that when run on a given
universal turing machine will generate the string. You can express that
functionally very easily, but you can't actually compute it for a given
string. (Do you know the busy beaver function? That's very closely
related conceptually -- it's the program with the maximum output for its
length. Also very easy to state, but non-computable beyond extremely
small numbers.) So I'm not sure what you mean -- if you mean computable
numbers vs. expressible but not producable numbers, points, etc. We can
count integers but we can't count functions of integers.
> Now, if any of this makes sense, then it might be related to
> the topic of truth and the "ultimate opinion." I'm reminded
> to Quine's complaint against Peirce on truth, from _Word and
> Object_.
>
> W&O, p. 23
>
> Peirce was tempted to define truth outright in
> terms of scientific method, as the ideal theory
> which is approached as a limit when the (supposed)
> canons of scientific method are used unceasingly
> on continued experience.
>
> Part of the complaint is:
>
> Scientific method id the way to truth, but it
> affords, even in principle no unique definition
> of truth. Any so-called pragmatic definition of
> truth is doomed to failure equally.
>
> I don't want to try to address everything involved in
> Quine's criticism. Put the point can surely be made that
> scientific methods are themselves evolving with the advance
> of knowledge, and we no more expect a final definitive
> account of scientific method than we do a final totality of
> truths. Now Peirce was, I submit, as much aware of the
> evolving character of methods, as anyone else. But reading
> Peirce sympathetically, then, this means that we should not
> see his treatment of "truth" as an attempt to state any
> "method" (which we could have all at once) to allow us to
> distinguish what is true from what is not true. Instead,
> what Peirce had in mind is more like what Tom says above:
>
> > When you talk about a relationship, you're projecting the
> > reality of how things are related, and how they WOULD be
> > related even if they never were in such and such exact
> > quantity, or even if they never will be -- if x WERE a,
> > then y WOULD BE b, and you can talk about a range of
> > possibility that just doesn't quit, ...
>
> So, when we talk about the relationship of belief to
> reality, we are talking about a relationship (which is
> mediated by inquiry), and regarding "truth" in particular,
> "you're projecting the reality of how things are related,
> and how they WOULD be related..." But this "WOULD be" is
> weaker, than anything indicated by a set of axioms or
> functions, or methods where merely practical difficulties
> might be of interest. It essentially involves the idea of
> future inquiry depending on methods suited to its own
> problems and context of inquiry, though we have no hope of
> presently stating what these might be. So (appropriately)
> vague is the idea of truth.
>
> Still, to say that we have no hope of presently predicting
> or forecasting the methods of inquiry of the future, is not
> to say that there is no "better and worse" among methods of
> inquiry for every given time or situation. So even though
> the methods of inquiry may be expected to evolve in
> unpredictable ways, we still have some purchase on "truth"
> so long as there are better and worse methods at any given
> time. Our purchase on "truth" does not depend on the idea
> that there is some complete and total best way to conduct
> inquiry, and our judgments regarding "better and worse"
> among methods must always reflect the actual details of
> situations within which inquiry may be conducted.
>
> Now, if this is anywhere near right, it is interesting in
> connection with mathematical truth. So, look at the
> functions or sets of axioms used to calculate points on a
> line as methods of defining the line. To reject Goedel's
> mathematical realism is to reject the idea that there is
> some total and complete (infinite) set of axioms which would
> finally define the points on the line (for example). Instead
> we view possible expansions of axiom systems as developments
> or augmentations of past systems which are "better or
> worse," in that they are more or less suited to solve some
> range of (non-mathematical) problems. Mathematics remains
> universal and hypothetical, though its actual expansion must
> always make reference to non-mathematical problems and
> solutions. In this way, the practical and contextual may
> enter into the growth of mathematics.
>
I agree with the point you are making, but I think that poor Goedel,
perhaps because of my inadequate summary, is getting tarred with a
caricature of platonism that doesn't represent his views. John Dawson
does this in his biography of Goedel, and part of it on his part seems
to me to be a serious misreading of Goedel's philosophy in the light of
Goedel's mental instability and paranoia that causes Dawson to project a
drive on Goedel's part to overcome his insecurity by relying on the
notion that mathematical truths are 'out there', stable and eternal.
Let me quote from a passage where Goedel is criticizing Carnap:
"Carnap, to the objection that Platonism is implied by
transfinite rules, replies (LOGICAL SYNTAX, p.114) that one may
know how to handle the transfinite concepts (in inferences,
definitions, etc.) without making any metaphysical assumptions
about the objective existence of the abstract entities concerned.
This, of course, is true in the same sense as one may also know how
to handle the concepts of physical objects without ascribing to
these objects any existence in a metaphysical sense. But
nevertheless, before one can rationally use them in science, he
must assign to them reality (or objectivity) at least in some
imminent (Kantian) sense, in order to distinguish them from wrong
(i.e. disprovable) physical hypotheses. The same, therefore,
applies to transfinite mathematical entities, whose existence can
also be disproved, namely by an inconsistency derived from them.
Hence, if mathematics is based on transfinite syntax, it is implied
that there are two different realities of equal standing, which is
exactly what the syntactical viewpoint denies. To finitary rules
evidently this argument does not apply, because the objects to
which they refer may be considered to be part of the physical world
(either existing already or producible)." ("Is mathematics syntax
of language?" version II, in ed. Francisco Rodrigues-Consuegra,
KURT GOEDEL: UNPUBLISHED PHILOSOPHICAL ESSAYS, 1995, Birkaeuser:
Boston, p 165.
> Peirce said "Once you have embraced the principle of con-
> tinuity no kind of explanation of things will satisfy you
> except that they grew" (CP 1.175, "Fallibilism, Continuity,
> and Evolution," c. 1897). Doesn't the point apply to
> mathematics as well? And, isn't it of some importance in
> understanding the special features of Peirce's philosophy
> of mathematics? In a sense the idea of (formal) "incom-
> pleteness" was already implicit in Peirce's conception of
> points on a line.
Yes, yes, & not sure.
Tom Anderson
------------------------------
Date: Thu, 22 Jan 1998 13:36:59 -0800
From: Tom Anderson
To: Peirce list
Subject: Some more Goedel quotes
Message-ID: <34C7BBFA.155E757D[…]ix.netcom.com>
I thought these passages, from the end of "Is mathematics syntax of
language?" version II, in ed. Francisco Rodrigues-Consuegra, KURT
GOEDEL: UNPUBLISHED PHILOSOPHICAL ESSAYS, 1995, Birkaeuser: Boston,
might be interesting to people who've been in the context and continuity
discussion; passages in double brackets are the editor's choice from a
variant copy:
"45. It seems to me that this whole situation viewed from
Carnap's own standpoint can only mean that the mathematical axioms
are certain irreducible, largely existential, hypotheses which are
exactly as necessary for the scientific description of reality as,
e.g., the hypotheses of the existence of elementary particles or of
some field satisfying certain equations. [[Also in both cases
these hypotheses are disprovable, in physics by deriving some wrong
proposition about sense perceptions, in mathematics by deriving
some wrong numerical equation (which, in two valued logic, is
equivalent to deriving an inconsistency.)]]
"46. Therefore, also from the empirical standpoint, THERE IS NOT
THE SLIGHTEST REASON TO ANSWER THE QUESTION OF THE OBJECTIVE
EXISTENCE OF MATHEMATICAL AND EMPIRICAL OBJECTS AND FACTS
DIFFERENTLY, WHILE IT IS THE ESSENCE OF THE SYNTACTIC VIEWPOINT TO
DISTINGUISH THE TWO EXACTLY IN THIS RESPECT. [italics in original]
"47. There is a real difference between the two that the
acceptance of mathematical axioms (unlike that of laws of nature)
so far has been based exclusively on their intuitive evidence, and
not on the success of their consequences. However, in view of the
fact that a large proportion of mathematical problems (even of the
number-theoretical problems of Goldbach type mentioned in footn.
24) may not be solvable in this fashion and that, moreover, axioms
with an incomparably greater demonstrative power might be
discovered empirically, it is not impossible that this attitude
will be given up sometime in the future. Today already, to
recognize the intutionist critique to be justified and continue to
use classical mathematics in the applications is something of this
nature.
"48. I do not want to conclude this paper without mentioning the
paradoxical fact that, although any kind of nominalism or
conventionalism in mathematics turns out to be fundamentally wrong,
nevertheless the syntactical conception perhaps has contributed
more to the clarification of the situation than any of the other
philosophical views proposed: on the one hand by the negative
results to which lead the attempts to carry it through, on the
other hand by the emphasis it puts on a difference of fundamental
importance, namely the difference between empirical and conceptual
truth, upon which it reflects a bright light by identifying it with
the difference between empirical and conventional truth. [[But it
must of course be admitted that there is a a fundamental opposition
in the character of these two kinds of truth.]]
"49. I believe the true meaning of the opposition between things
and concepts or between factual and conceptual truth is not yet
completely understood in contemporary philosophy, but so much is
clear that in both cases one is faced with 'solid facts', which are
entirely outside the reach of our arbitrary decisions." (pp.
188-189)
I think it's that last sentence that's critical -- although it only
makes sense in the context of the whole essay, one of six versions of a
critique of Carnap he drafted but never submitted to the Schillp volume
on Carnap in the living philosophers' series.
I think these passages, fragmentary as they are, give a flavor of
Goedel's subtle form of realism, a realism that's not at all mechanical
or at all meant as a comfort for the insecure or paranoid.
Tom Anderson
------------------------------
Date: Thu, 22 Jan 1998 14:26:08 -0500
From: piat[…]juno.com (Jim L Piat)
To: peirce-l[…]ttacs6.ttu.edu
Subject: Re: Context and Continuity-Thermodynamic
Message-ID: <19980122.142610.11214.0.piat[…]juno.com>
Howard, Joe, David and friends,
I haven't had this much fun since my big brother (who understood such
matters) took me to the candy store as a kid and let me get all I dared!
And as a small repayment for your uplifting exchange on authority,
rebellion and their possible synthesis in a community of peers I've
copied this favorite passage which I think speaks so powerfully to man's
paradoxical nature and some of the ideas you've all touched on.
"I am a sick man... I am a spiteful man. I am an unattractive man. I
believe my liver is diseased. However, I know nothing at all about my
disease, and do not know for certain what ails me. I don't consult a
doctor for it, and never have, though I have a respect for medicine and
doctors. Besides, I am extremely superstitious, sufficiently so to
respect medicine, any way (I am well-educated enough not to be
superstitious, but I am superstitious). No, I refuse to consult a doctor
from spite. That you probably will not understand. Well, I understand
it, though. Of course, I can't explain who it is precisely that I am
mortifying in this case by my spite: I am perfectly well aware that I
cannot "payout" the doctors by not consulting them; I know better than
any one that by all this I am only injuring myself and no one else. But
still, if I don't consult a doctor it is from spite. My liver is bad,
well- let it get worse!
I have been going on like that for a long time -twenty years. Now I am
forty. I used to be in the government service, but am no longer. I was a
spiteful official. I was rude and took pleasure in being so. I did not
take bribes, you see, so I was bound to find a recompense it that, at
least. ( A poor jest, but I will not scratch it out. I wrote it thinking
it would sound very witty; but now that I have seen myself that I only
wanted to show off in a despicable way, I will not scratch it out on
purpose!)
When petitioners used to come for information to the table at which I
sat, I used to grind my teeth at them, and felt intense enjoyment when I
succeeded in making anybody unhappy. I almost always did succeed. For the
most part they were all timid people - of course they were petitioners.
But the uppish one there was one officer in particular I could not
endure. He simply would not be humble, and clanked his sword in a
disgusting way. I carried on a feud with for eighteen months over that
sword. At last I got the better of him. He left off clanking it. That
happened in my youth though.
But do you know gentlemen, what was the chief point about my spite? Why,
the whole point, the real sting of it lay in the fact that continually
even in the moment of the acutest spleen, I was inwardly conscious with
shame that I was not only not a spiteful but not even an embittered man,
that I was simply scaring sparrows at random and amusing myself by it. I
might foam at the mouth, but bring me a doll to play with, give me a cup
of tea with sugar in it, and maybe I should be appeased. I might even be
genuinely touched, though probably I should grind my teeth at myself
afterwards and lie awake at night with shame for months after. That was
my way.
I was lying when I said just now that I was a spiteful official. I was
lying from spite. I was simply amusing myself with the petitioners and
with the officer, and in reality I never could become spiteful. I was
conscious every moment in myself of many, very many elements absolutely
opposite to that. I felt them positively swarming in me, these opposite
elements. I knew that they had been swarming in me all my life and
craving some outlet from me, but I would not let them, would not let ,
purposely would not let them come out. They tormented me till I was
ashamed: they drove me to convulsions and - sickened me, at last, how
they sickened me! Now, are not you fancying, gentlemen, that I am
expressing remorse for something now, that I am asking your forgiveness
for something? I am sure you are fancying that ...However, I assure you I
do not care if you are...
It was not only that I could not become spiteful, I did not know how to
become anything: neither a hero nor an insect. Now, I am living out my
life in my corner, taunting myself with the spiteful and useless
consolation that an intelligent man cannot become anything seriously, and
it is only the fool who becomes anything. Yes, a man in the nineteenth
century must and morally ought to be pre-eminently a characterless
creature; a man of character, an active man is preeminently a limited
creature. That is my conviction of forty years. I am forty years old
now, and you know forty years is a whole lifetime; you know it is extreme
old age. To live longer than forty years is bad manners, is vulgar,
immoral. Who lives beyond forty? Answer that, sincerely and honestly.
I will tell you who do: fools and worthless fellows. I tell all old men
that to their face, all these venerable old men, all these silver-haired
and reverend seniors! I tell the whole world that to its face! I have a
right to say so, for I shall go on living to sixty myself. To seventy!
To eighty! ...Stay, let me take breath...
You imagine no doubt, gentlemen, that I wan't to amuse you. Your are
mistaken in that, too. I am by no means such a mirthful person as you
imagine, or as you may imagine; however, irritated by all this babble (
and I feel that you are irritated) you think fit to ask me who am I -
then my answer is I am a collegiate assessor...
(The notes of this paradoxalist do not end here, however. He could not
refrain from going on with them, but it seems to us that we may stop
here.)"
Our beloved Dostoevsky 1821-1881
Jim Piat
PS: I came across a nice (short ;) passage from J.S. Mill (while looking
up stuff for the Hookway read) that I'll dig up and share later.
On Thu, 22 Jan 1998 00:29:40 -0600 (CST) David.Low[…]anu.edu.au (David W.
Low) writes:
>To Joe Ransdell, re: posting at 6.43am 21/1/97
>
>Thanks for posting your extension to Howard's comments on Peirce's
>idea of
>truth. I am intrigued by the ideas you are putting forward,
>especially in
>relation to how we might use Peirce's conception of truth to
>understand how
>and why one positions oneself in a communicational community. I
>recently
>read Peirce at CP 3.481 and was therefore able to follow the
>connections
>you make to quantifiers as selectives. Your links to the ideas of a
>self-identity and a commitment to the discovery of truth in terms of
>discourse extend this nicely and are of special interest to me.
>
>It has often perplexed me why people sometimes seem intent on creating
>an
>'oppositional identity' for themselves (environmentalists for
>example). It
>certainly yields visibility, but on the other hand, it simultaneously
>creates the situation in which a person or group appears to be
>behaving
>unreasonably. They appear unreasonable because their visible identity
>is
>built out of being in a position of opposition to the other party's
>proposition. Their grotesque appearance is made more monsterous by
>the
>fact that, if the other party is more powerful, to become visible the
>less
>powerful opponent has to appear to be very wrong in order to appear at
>all,
>which to the more powerful party seems like an entirely illogical way
>to
>behave if they are sure they are already in possession of "the truth".
> In
>this sense, the opponent embodies the hecceity of the subject-matter
>by
>opposing the proposition, but usually gets few thanks, sometimes
>worse.
>
>However, and as Peirce argues in all sorts of ways, to say someone is
>illogical is a special kind of moral argument (CP 8.191) and such a
>position will lead to arguments based in authority, which,
>fortunately, is
>a method that eventually "chokes its own stream" (CP 2.198). To
>learn, we
>must assume that our reasoning is more or less fallible - thus, to be
>wrong, and know we are, or in future may be, is the way we go about
>learning something new about the system of ideas were use to get at
>the
>truth.
>
>It seems to me that the important factor to consider in regard to
>communicational communities, therefore, is whether or not the people
>who
>create oppositional identities for themselves do so intentionally (ie,
>pragmatically), the implications of which you bring out nicely in your
>post. I think it is Roberta Kevelson who talks about 'persistent
>objectors' as people who visibly choose not to agree as a means of
>maintaining their freedom to choose. This is why I think you are
>spot-on
>when you conclude by saying :
>
>>If I can burden this message with one more implication, I perceive in
>>this the elimination in principle of all authority in such a
>>communicational community. Or to put it the other way around, the
>>insistence that there must always be authority in a communicational
>>community is the same as the refusal to acknowledge that the peer
>>relationship obtains in that community.
>
>You certainly may burden me with some more implications.
>
>Cheers
>David Low
>
>-----------------------------------------------------------------------------
>David W. Low
>Centre for the Public Awareness of Science
>Faculty of Science
>Australian National University
>Canberra 0200, Australia
>ph: +61 2 6249 2456; fax: +61 2 6249 4950
>http://www.anu.edu.au/scicom/scicom/students/low/DAVIDHM.HTM
>
>
>
------------------------------
Date: Thu, 22 Jan 1998 20:06:04 -0600
From: joseph.ransdell[…]yahoo.com (ransdell, joseph m.)
To:
Subject: Re: Dedekind and the bootstrap
Message-ID: <005301bd27a4$27d17480$151627d1[…]ransdell.door.net>
Such a fine and generous response to my question about Dedekind, Thomas!
I'll hold off for a couple of days in pursuing this further, so as not
to divert your attention from whatever your on-stage appearance tomorrow
is -- perhaps you can tell us what the occasion of the appearance is
when you get the time. I'm tacking on to this message, though, several
paragraphs from Peirce where he discuss the maps-representing-maps
metaphor in another sort of context. (I don't know if is intelligible
abstracted from the larger context, which is a review of a book by Royce
in Vol 8 of the Collected Papers. I didn't realize that Dedekind's
paper, which I haven't read but will certainly read now, made use of the
map idea in this way.)
==========Peirce quotations=====================
Peirce: CP 8.122 Cross-Ref:
122. But how, it will be asked, can the meaning of a single idea be an
entire life? An idea being a state of mind involving a purpose not fully
realized, its internal meaning being that purpose so far as it is
defined, we can understand that that purpose becomes more and more
definite, until, being a sincere purpose, free from arriŠre-pens‚e, at
the moment when it becomes in all respects determinate, it is
transformed into an act . . . . But how can it become a complete life?
The answer to this is very simple. Royce evidently thinks that a purpose
cannot be fully definite, until all the circumstances of the entire life
are taken into account; so that, however meagre the internal meaning of
an idea may be, as long as it remains general and "abstract," yet when
that internal meaning is fully accomplished by its becoming in every
respect determined, the external meaning will cover the whole life of
the individual. Certainly, it is conceivable that such might be the
result; but to prove that such would be the result, a far more exact
examination of the question would be requisite than the author attempts.
There is another difficulty which he removes very happily. How, he
supposes his reader to ask, can an idea, which is so microscopic a part
of a life, contain within its implication a distinct feature
corresponding to every feature of the entire life of which it is only a
part? Here, he resorts to Gauss's conception of an Abbild, which has
played so important a part in mathematics. That is to say, he likens the
idea representing the entire life to a map of a country lying upon the
ground in that country. Imagine that upon the soil of England, there
lies somewhere a perfect map of England, showing every detail, however
small. Upon this map, then, will be shown that very ground where the map
lies, with the map itself in all its minutest details. There will be a
part fully representing its whole, just as the idea is supposed to
represent the entire life. On that map will be shown the map itself, and
the map of the map will again show a map of itself, and so on endlessly.
But each of these successive maps lies well inside the one which it
immediately represents. Unless, therefore, there is a hole in the map
within which no point represents a point otherwise unrepresented, this
series of maps must all converge to a single point which represents
itself throughout all the maps of the series. In the case of the idea,
that point would be the self-consciousness of the idea. Since an idea is
a state of mind with a conscious purpose, it obviously must be
self-conscious.^19 Here, therefore, is a beautiful and needed, though
not complete, confirmation of the idea's really being so related to the
entire life. Singularly enough, however, for some reason, Prof. Royce
here draws back and supposes the analogy with the map to break down in
this respect . . . .
Peirce: CP 8.123 Cross-Ref:
123. It will be perceived that, according to Prof. Royce's theory, if
an idea fails of being a Self, it is only because it is general and not
fully determined. Its implicit or germinal inward meaning is a little
Self representing the entire man as its external meaning. In a similar
way, the Self of the man is perhaps included within a larger Self of the
community. On the other hand, the man's Self embraces intermediate
selves, the domestic Self, the Self of business, the better Self, the
evil spirit that sometimes possesses him. Here the author draws support
from the psychological doctrine of what he calls the "time-span," a
doctrine which, so far as it has really been placed beyond doubt,
amounts to little more than that our image of the events of the few
seconds last past is, or is very like, a direct perception, while our
representation of what happened a minute ago partakes far less of the
perceptive character.^20 The phenomenon had already been seized upon by
several idealistic writers as affording a refutation of dualism; but the
large calibre of Royce's thought cannot be better appreciated than by
comparing their style of putting the phenomenon to the service of
metaphysics with his.
Peirce: CP 8.124 Cross-Ref:
124. He imagines that greater selves will naturally have vastly longer
time-spans than lesser selves. Now a consciousness whose time-span was a
thousandth of a second or a thousand years would not ordinarily be
recognized by us, as observers of its external manifestations, as being
a consciousness, at all. The time-span of the All-seeing must cover all
time; and thus foreknowledge and freewill become more clearly
reconcilable after the fashion of Boethius, St. Augustine, and others.
Peirce: CP 8.125 Cross-Ref:
125. Every reality, then, is a Self; and the Selves are intimately
connected, as if they formed a continuum. Each one is, so to say, a
delineation, -- with mathematical truth, incongruous as the metaphor is,
we may say that each is a quasimap of the organic aggregate of all the
Selves, which is itself a Self, the Absolute Idea of Hegel, or God. It
is a flagrant offence to use this name in philosophy. It is like
inviting a man to see the body of his wife dissected. There is also a
pretension in it that the philosophy of religion can be religion. But
things shocking to right feeling are sometimes necessary in philosophy,
as they are in science. It will be observed that if the Selves did form
a continuum, each would be distinguished by its own point of
Self-consciousness. This would not generally be the same as the point of
self-consciousness of an idea within self, since each idea is
distinguished by its own exclusive self-consciousness. The systems of
delineation must be different. Here we see an inadequacy in the metaphor
of the map; for what, more than anything else, makes my ideas mine is
that they appeal to me, and are, or tend to become, represented in my
general consciousness as representations. But, of course, the
map-metaphor must be inadequate, since a map wants several of the
essential characters of the class of signs to which ideas belong. Again,
in the map the boundaries of the selves are somewhat indeterminate; each
must embrace no more nor less than a complete map of the whole surface;
but the boundary of any one can be considered to be drawn in any way
which fulfills this condition, the boundaries of the others being drawn
accordingly, just as on the Mercator's chart, which gives an endless
series of representations of the whole globe, any one line from pole to
pole may be taken as the boundary of the globe as represented in each
chart. But the boundaries between Selves are not so indeterminate,
because all that is in one Self appeals by a continuum of
representations to that Self's self-consciousness. It will be necessary,
therefore, to replace the idea of a map by that of a continuum of maps
overlying one another. A map is a section of a projection of which the
surface mapped is another section. The projection itself is a sheaf of
lines which diverge from one point. Instead of saying that a Self is a
map, a more adequate metaphor would call it a projection of the reality,
of which projection any one idea of the Self is a section. At any rate,
it is plain that the map-metaphor requires deep emendation in order to
answer the purposes of philosophy. At the same time, it is a
considerable aid even as it is; and the initiating of the introduction
of such exact ideas into philosophy is one of the momentous events in
its history.^21
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Joseph Ransdell or <>
Department of Philosophy, Texas Tech University, Lubbock TX 79409
Area Code 806: 742-3158 office 797-2592 home 742-0730 fax
ARISBE: Peirce Telecommunity website - http://members.door.net/arisbe
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-----Original Message-----
From: Thomas Riese
To: Multiple recipients of list
Date: Thursday, January 22, 1998 12:42 PM
Subject: Re: Dedekind and the bootstrap
>Dear Joe Ransdell and Jim Piat,
>
>in order to sort out the things you mention (historically, logically
>and mathematically) with some strictness one would have to write a
>book, a huge book :-)
>
>In a few words I would describe Richard Dedekind's work on the
>natural numbers as follows: he built a system using what he called
>'Ketten' (chains) of maps where one map maps how another maps a map
>etc. finally mirroring the property of the natural numbers that
>(infinite) parts of them can mirror the whole by being able to be put
>in one-one correspondence ("similarity") with the whole.
>
>Thus he built a completely self-mirroring (self-referential) system.
>Using the connection between transitivity and correspondence he could
>then convert the whole by a mechanism similar to mathematical
>induction ("complete induction") into a demonstrative consequence,
>_provided_ he could find at least one existing element serving as a
>premiss.
>
>So the whole thing then depends on the following proposition
>(nr.66 in Dedekind's paper):
>
>[my translation]
>
>Proposition:
>There exist infinite systems.
>
>Proof:
>The world of my thoughts, i.e. the totality S of all things which can
be objects of my thinking, is infinite.
>For if s is an element of S, so the thought s' that s can be an object
of my thinking, itself is an element of S.
>If we consider this as the image phi(s) of the element s then the in
this way determined mapping phi of S has the
>property that the image S' is part of S.; and S' is a proper part
(echter Teil) of S, since in S there are elements
>(e.g. my own self (Ich)), which are different from each such thought s'
and thus not contained in S'. So it is plain
>that if a and b are distinct elements of S, their images a', b' are
distinct too, so that phi is a 'similar' mapping
>(aehnlich, deutlich). So it follows that S is infinite. q.e.d.
>
>The subtle trick is that _the_whole_thing_ then has itself the form of
a
>simple deductive consequence. It's so to speak "self-explaining". A
>really breathtaking bootstrap process indeed consisting, logically
>speaking, of nothing but demonstrative consequences of demonstrative
>consequences ... forming, considered in another way, nothing but one
>simple consequence proving itself. Which is possible because it is
>infinite and so can contain itself as a proper part being at the same
>time existing. See above...
>
>One of Peirce's comments was that he himself didn't like the idea that
logic
>is just a chain where the weakest element is decisive for the
>stability of the whole structure. He preferred a rope where many
>elements might fail without the whole thing breaking down immediately.
>(Sorry, I can't find the reference. I think it is in the NEM)
>
>I think that indeed it does not need too much mathematical
>sophistication to suspect that with the above proof we are perhaps not
>many steps away from some Russelian paradox or Goedelian limitation.
>At least if we go and try to remove the dubiosity of the above
>existence(!) proof.
>
>But Dedekind showed that, apart from that, the whole thing really
>works as it should. With all the usual arithmetical operations and all
>that. Natural numbers!
>
>And if the natural number system is fundamental for mathematics and the
>natural number system itself nothing else but a bootstrap embodiment
>of a simple, self-explaining demonstrative consequence, then, well,
>then mathematics is nothing but an outgrowth of logic, it consists of
>nothing but deductive logic.
>
>""O Thou, All-Sufficient, Self-Sufficient, Insufficient God."" (Charles
Peirce, CP 5.71)
>
>He continues: "Now pure Self-consciousness is Self-sufficient, and if
>it is also regarded as All-sufficient, it would seem to follow that it
>must be Insufficient. I ought to apologize for introducing such
>Buffoonery into serious lectures. I do so because I seriously believe
>that a bit of fun helps thought and tends to keep it pragmatical."
>
>And further: "It is therefore the precise analogue of pure
>self-consciousness. As such it is _self-sufficient_. It is saved from
>being unsufficient, that is as no representation at all, by the
>circumstance that it is not _all-sufficient_, that is, is not a
>complete representation but is only a point upon a continuous map."
>
>I Peirce think gives one of the concisest versions of his own
>alternative construction in his Cambridge Conferences Lectures of
>1898 (Reasoning and the logic of things, Harvard U. Press 1992, pp.
>156 ff.) starting with: "Although I am debarred from showing anything
>in detail about the logic of relatives, yet this I may remark, that
>where ordinary logic considers only a single, special kind of
>relation, that of similarity, -- a relation too, of a particularly
>featureless and insignificant kind, the logic of relatives imagines a
>relation in general to be placed."
>
>I think what is here meant with "relation of similarity" is clearly
>Dedekind's "aehnliche Abbildung" (similar mapping).
>
>I believe that if one wants to find out what the "technical" reason
>is why Peirce was not a mathematical logicist and understand his
>curious distinction between mathematics and logic one of the easiest
>ways is to have a look at NEM III/2, p.837 where he discusses the
>form of a transitive relation:
>
>"Schroeder however made the curious remark that not _all_ transitive
>relations are of that form [is lover of whatever is loved by...]; and
>to get a form that would hold good of _all_ transitive relations, that
>form must be _restricted_ (which sounds paradoxical). He shows that
>every transitive relation is of the form '__is both lover of and lover
>of everything loved by__.' This is a relation which not everything is
>in to itself. [...] But I first analysed the matter and showed that it
>depended on the following points.[...]"
>For the parallel argument on the relation of correspondence see CP
3.523
>
>
>
>To Dedekind's conception of self-similarity applies what Peirce said
>about the relations of transitivity and correspondance: it has to be,
>in a _seemingly_ paradoxical way, restricted in order to be expanded
>and what regards the question of a logicistic view of mathematics
>Peirce reasons are exactly parallel: mathematics is not just an
>outgrowth of pure logic, but instead the mathematician, qua
>mathematician, has to be supplied with special premisses in order
>fully to be able to do his work. This is a bit paradoxical of course
>in the sense in which the relation of transitivity is 'expanded by
>restriction': the mathematician's job, according to Peirce then is
>nothing but drawing conclusions from given premisses -- so in a sense
>nothing but deductive logic in pure form. It's a curious twist. But
>it is indeed sound if we wish a sound classification of the
>sciences:-)
>
>So 'bootstrapping' seems to be a good idea; the question is; which
>brand of it? It seems as if a sort of "seed", something arbitrary,
>were necessary for growth to take place.
>
>On the other hand: if the relation of "similarity" is not all (the
>resulting universe being all too boring) and at the same time Peirce
>can expand relations by restricting them -- then this should indeed
>introduce some diversity. For if relational generality
>(correspondence, coordination) is not just only one-to-one or
>"similar" in Dedekind's sense, then we should expect some branching
>and perhaps even 'point-splitting' business to take place. I think
>one should next have a look at the Dedekindian 'Cuts' with Peircean
>eyes. I think Cantor then will come in, too.
>
>Furthermore I really don't understand why our universe obviously must
>be organized in such a way that if there happen interesting things
>then always many at a time. I have to be on stage tomorrow and I
>really don't yet know what I shall do there -- I mean even less than
>is healthy. Funny feeling (think it is called "fear";-)). Please
>forgive me if this message should be a bit hasty.
>
>Thomas.
>
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