Re: [peirce-l] Reply to Steven Ericsson-Zenith & Jerry Chandler re Hilbert & Peirce
CORRECTION (as usual). Sorry! I was unclear: For Peirce in those terms, matter is a Second, and so chance/spontaneity does not correspond more or less to the material cause, though it [I meant *the material cause*] seems to have a ghost of role [I meant *in the Firstness:Chance part of the trichotomy*] since matter and collections of particles so lend themselves to statistical treatment and stochastic processes. Corrected also below. - Best Ben - Original Message - From: Benjamin Udell To: PEIRCE-L@LISTSERV.IUPUI.EDU Sent: Sunday, November 27, 2011 3:09 PM Subject: Re: [peirce-l] Reply to Steven Ericsson-Zenith & Jerry Chandler re Hilbert & Peirce Irving, Jerry, Steven, list, Irving, thanks for your response, more interesting and informative than what I have to say! Irving wrote, Is there some sort of causality, Aristotelian or otherwise, in [application of] inference rules? Once again, I am at a loss here to comprehend how this issue of causality relates to the nature of axiom systems or to formalism. I suspect that Jerry has in mind causal reasoning or something like model-based reasoning. The latter is an AI subject that I don't know much about, but the simplest examples in online texts consist of causal reasoning as opposed to diagnostic reasoning, e.g., causally reasoning from stroke to confusion, as opposed to diagnostically reasoning from confusion to stroke. I am not convinced that those are just other words for predictive reasoning versus explanatory reasoning, but there seems at least some parallelism. Anyway, if one has a mathematical model of a mechanical system, and one "runs it forward," then the calculations might seem to reflect a causal process, though such model runs are often not practically feasible, and I don't know whether Newtonian mechanics, though deterministic, has been proven or disproven to be (in principle) always computable; at this point I'm thinking of digital models, while the broadest sense of 'model' could be very broad. One can expand the idea of causal reasoning to the idea of following a connection of reaction/resistance (or at least a connection of neighborhood). For example, traversal of the GW bridge from Manhattan will lead a person to be in New Jersey, or 'cause' a person to come to be in New Jersey. When one is thinking in graph-theoretical terms of the problem of the Seven Bridges of Königsberg, I'm not sure that one can still call that aspect of the reasoning 'causal' (and certainly proof of the problem's insolubility is not itself 'causal' or 'connectional' in a non-meta sense). Any deductive proof can be considered as following a 'path' but my guess is that it is indeed somewhat 'meta', be it soever fruitful, to regard every deductive proof as a 'causal' or 'connectional' reasoning about where (i.e., to what logical conclusion) the proof path leads the reasoner. If it's a meta view, then it would leave intact a distinction between causal/connectional reasoning and other kinds. And of course hovering in the background is a notion that concrete causal or connection-traversing processes are nature's own kind of inference processes, which we map with causal reasoning. At this point I tend to get confused (or more confused than I was already). Clearly my mind is wandering now, don't take this all too seriously. Is every natural process of decision or determination an inference process, and is every inference process also a decision process? I like to think that they are but in different senses, but I don't have a clear idea what senses. I'm not completely wandering. I'm thinking in terms of inference and Aristotle's four causes. Peirce somewhere said that logic is governed by final causality, and in MS 634 (Sept. 1909) quoted by Joe, Peirce says that the end does _act_ (i.e., agentially) mentally as a cause. I remember Joe Ransdell and John Collier discussing entropy's increase as a final cause, and that's how I've come to think of it, but it's a case where the final cause does not causally act in the sense of a causal agent (traditionally, 'agent cause' is the same as 'efficient cause'). In Peirce's metaphysics, the three operative principles are a 1stness-2ndness-3rdness trichotomy of (1st) chance/spontaneity, (2nd) mechanical necessity (corresponding more or less to efficient causation), and (3rd) creative love (corresponding more or less to final causation). [WITH CORRECTIONS IN BRACKETS] For Peirce in those terms, matter is a Second, and so chance/spontaneity does not correspond more or less to the material cause, though it [I meant *the material cause*] seems to have a ghost of role [I meant *in the Firstness:Chance part of the trichotomy*] since matter and collection
Re: [peirce-l] Reply to Steven Ericsson-Zenith & Jerry Chandler re Hilbert & Peirce
sation, with something else, called its _Object_. In a
word, whether physically, rationally, or otherwise directly or indirectly, its
Object, as agent, acts upon the sign, as patient." ('The Basis of
Pragmaticism', MS 283, 1905)
Traditional 3 principles Traditional 4 causes Peirce in MS 238 Peirce's
(a) operative principles, and (b) evolutionary modes,
of the cosmos and its parts
Agent. Efficient cause, agent cause. Object (a Second) as agent. (a & b)
Mechanical necessity. (Secondness.)
Patient. Matter, material cause. Sign (a First) as patient. (a)
Spontaneity, absolute chance. (b) Sporting, fortuitous variation. (Firstness.)
(Though for Peirce matter is a Second, one might note that it is in the
case of multitudes of particles that statistics becomes so important.)
Act.
(Aquinas subdivides act into action and act.) End, final cause.
(Aquinas: act as action as cause)
Form, formal cause. (Aquinas: act (not action) as cause) [Peirce doesn't
say it in the quote, but one might guess that the Interpretant (a Third) serves
as act.] (a) Creative love. (b) The law of love. (Thirdness.)
I hope I don't get myself into trouble over Aquinas here. I don't remember
where I left my copy of _The Pocket Aquinas_.
Best, Ben
- Original Message -
From: "Irving"
To: PEIRCE-L@LISTSERV.IUPUI.EDU
Sent: Sunday, November 27, 2011 1:03 PM
Subject: Re: [peirce-l] Reply to Steven Ericsson-Zenith & Jerry Chandler re
Hilbert & Peirce
Apologies for sending out the following message previously without the subject
line; the IMAP connection was temporarily broken and causing transmission and
other difficulties.
- Message from ianel...@iupui.edu -
Date: Sun, 27 Nov 2011 11:20:02 -0500
From: Irving
Reply-To: Irving
To: "PEIRCE-L@LISTSERV.IUPUI.EDU"
On 18 Nov. Steven Ericsson-Zenith wrote:
> My own interpretation may be substantively different, it may not.
> I take Hilbert's position to be that the formalism is independent of the
subject matter. That is, I take his view of formal interpretation to be
mechanistic, specifying valid transformations of the structure under
consideration, be it logical, geometric or physical. I am confused because you
use "signs" instead of "marks" here. In addition, since the formalism is
independent of the subject - as suggested by his appeal to Berkeley - a theorem
of the formalism remains a theorem of the formalism despite the subject.
> In this view, how one selects an appropriate formalism for a given subject
- if there is a fitness ("suitability") requirement as you suggest for the
"different parts of mathematics" - appears to be a mystery, unless you think
empiricism is required at this point.
And on 26 Nov., Jerry Chandler wrote:
> The separate and distinct axiom systems for mathematical structures is a
thorn in my mind as it disrupts simpler notions of the rules for conducting
calculations with numbers. While I eventually came to accept the category
theorists view of the emergence of mathematic structures as a historical fact,
the separation of formal axiom systems causes philosophical problems.
> Firstly, physicists often speak of "first principles" or "ab initio"
foundations. These terms are used in such a sense as to imply a special
connection exists between physics and the universals and to further imply that
other sciences do not have access to such "first principles".
> If such ab initio calculations were to be invoked as something more serious
than a linguistic fabrication, what mathematical structure would one invoke? In
the mid and late 20 th Century, group theory and symmetry were the popular
choice among philosophically oriented physicists and applied mathematicians.
> Philosophically, the foundations of Aristotelian causality come into play.
Philosophers abandoned material causality, substituted the formalism of
efficient causality for formal causality and summarily dissed telic reasoning
of biology. The four causes so widely discussed in medieval logic and the
trivium were reduced to one cause. Obviously, if every axiom system is a
formalism and every formalism is to be associated with efficient causality, the
concept of "ab initio" calculations is seriously diminished as every axiom
system becomes a new form of "ab initio" calculations and conclusions.
> Does anyone else see this as a problem for the philosophy of physics?
> A second question is perhaps easier for you, Irving, or perhaps more
challenging.
> You write:
>> ... the only mathematically legitimate characteristic of axiom systems
is that they be proof-theoretically sound, that is, the completeness,
consistency, and independence of the axiom system,...
> I am
Re: [peirce-l] Reply to Steven Ericsson-Zenith & Jerry Chandler re Hilbert & Peirce
Apologies for sending out the following message previously without the
subject line; the IMAP connection was temporarily broken and causing
transmission and other difficulties.
- Message from ianel...@iupui.edu -
Date: Sun, 27 Nov 2011 11:20:02 -0500
From: Irving
Reply-To: Irving
To: "PEIRCE-L@LISTSERV.IUPUI.EDU"
On 18 Nov. Steven Ericsson-Zenith wrote:
My own interpretation may be substantively different, it may not.
I take Hilbert's position to be that the formalism is independent of
the subject matter. That is, I
take his view of formal interpretation to be mechanistic, specifying
valid transformations of the
structure under consideration, be it logical, geometric or physical.
I am confused because you use
"signs" instead of "marks" here. In addition, since the formalism is
independent of the subject - as suggested by his appeal to Berkeley -
a theorem of the formalism
remains a theorem of the formalism despite the subject.
In this view, how one selects an appropriate formalism for a given
subject - if there is a fitness
("suitability") requirement as you suggest for the "different parts of
mathematics" - appears to be a mystery, unless you think empiricism
is required at this point.
And on 26 Nov., Jerry Chandler wrote:
The separate and distinct axiom systems for mathematical structures
is a thorn in my mind as it
disrupts simpler notions of the rules for conducting calculations
with numbers. While I eventually
came to accept the category theorists view of the emergence of
mathematic structures as a
historical fact, the separation of formal axiom systems causes
philosophical problems.
Firstly, physicists often speak of "first principles" or "ab initio"
foundations. These terms are used in such a sense as to imply a
special connection exists between
physics and the universals and to further imply that other sciences
do not have access to such
"first principles".
If such ab initio calculations were to be invoked as something more
serious than a linguistic
fabrication, what mathematical structure would one invoke? In the mid
and late 20 th Century, group
theory and symmetry were the popular choice among philosophically
oriented physicists and applied
mathematicians.
Philosophically, the foundations of Aristotelian causality come into
play. Philosophers abandoned
material causality, substituted the formalism of efficient causality
for formal causality and
summarily dissed telic reasoning of biology. The four causes so
widely discussed in medieval logic
and the trivium were reduced to one cause. Obviously, if every axiom
system is a formalism and
every formalism is to be associated with efficient causality, the
concept of "ab initio"
calculations is seriously diminished as every axiom system becomes a
new form of "ab
initio" calculations and conclusions.
Does anyone else see this as a problem for the philosophy of physics?
A second question is perhaps easier for you, Irving, or perhaps more
challenging.
You write:
... the only mathematically
legitimate characteristic of axiom systems is that they be
proof-theoretically sound, that is, the completeness, consistency, and
independence of the axiom system,...
I am puzzled on how to interpret the phrase,
and
independence of the axiom system,...
Many axiom systems may use the same axioms, just extend the number of
axioms in the system; the
formal axiom systems are interdependent on one another.
So, what is the sense of 'independence' as it is used in this phrase?
I would note in passing that I have attempted to write a set of
axioms for chemical relations on
several occasions and have been amply rewarded, initially with
exhilaration and subsequently with
deep remorse for having wasted my energies on such an intractable
challenge. :-) :-) :-)
I would like to take the easiest question first.
Independence, like consistency or completeness, is a model-theoretic
property of axiom systems. To put it in simplest terms, by the
independence of the axiom system, the mathematician means nothing more
nor less than that there are no axioms in the set of axioms that could
be pr oven as a theorem from the other axioms.
The issue arose when, ever since Euclid, mathematicians attempted to
determine the status of the parallel postulate. It came to a head in
1733 when Saccheri claimed to prove the parallel postulate(Vth) from
all of Euclid's axioms other than the Vth postulate, using a reductio
argument. What Beltrami showed in 1868 was that Saccheri had in fact pr
oven the independence of Euclid's Vth postulate, since in fact,
Saccheri ended up with a hyperbolic parallel postulate.
I don't know very much about Hilbert's axiomatization of physics, other
than that Leo Corry has written the most about them, including the book
David Hilbert and the Axiomatization of Physics
(1898-1918) (Dordrecht: Kluwer Academic Publishers, 2004); some of his
other work includes:
"David Hilbert and the Axiomatization of Physics", Archi
