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 proven 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 proven 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", Archive for History
of Exact Sciences
51 (1997), 83-198;
"Hilbert and Physics (1900-1915)," In Jeremy Gray (ed,), The Symbolic
Universe: Geometry
and Physics (18901930) (New York, Oxford University Press, 1999), 145187;
"On the Origins of Hilbert's 6th Problem: Physics and the Empiricist
Approach to
Axiomatization", in Marta Sanz-Solé et al (eds.), Proceedings of the
International Congress of
Mathematicians, Madrid 2006, (Zurich, European Mathematical Society,
2006), vol. 3, 1679-1718.
These, and many of Corry's other publications, are downloadable from
his web page at:
http://www.tau.ac.il/~corry/publications/hilbert.html
Speaking of Corry, I would have to say that he would agree that there
is a strong empiricism
underlying Hilbert's work, and that this is the philosophical import of
his quote from Kant's
K.d.r.V. in the Grundlagen der Geometrie: "So fängt denn alle
menschliche Erkenntnis mit
Amschauungen an, geht von da zu Begriffen und endigt mit Ideen." I
would argue, however, that
this is about how we obtain our information, and, assuming Corry is
correct, how Hilbert thought we
select the elements of our universe of discourse; but I would also
argue that it has nothing to do
with how axiomatic systems operate, which is to say, having established
the axioms, chosen the
inference rules for the system, and selected the primitives from which
theorems are constructed
from the axioms in accordance with the inference rules, is strictly
mechanical, and it does not,
working within the axiom system, whether what is being manipulated are
points, lines and planes, or
tables, chairs, and beer mugs, or integers,
, or whatever we may
require for the axiomatizing task
at hand. What matters within the system, while the calculations are
occurring, is that complex
formulas (theorems) are being constructed on the basis of the formulas
that do duty as axioms, in
accordance with the rules. (It is this distinction, of having inference
rules in place, that
renders Hilbert's systems not merely axiomatic systems, but formal
deductive systems.) Hilbert's
formalism amounts to the mechanization of these manipulations, and for
practical purposes, the
formulas are combinations of marks, and these marks become signs as
soon as an interpretation is
give, that is, a universe of discourse - - whether points, lines, and
planes, or tables, chairs and
beer mugs, or the integers. What concerns me is whether, in considering
what (else) or what
different Hilbert might have meant by his formalism, and whether or not
there was an underlying
empiricism behind this, is that we might be demanding too much of
Hilbert, who was, I understand,
concerned with mathematics and only peripherally with philosophy of
mathematics. (Having said this,
I have to also confess that I have not seen or read the contents of his
late, unpublished, lectures
on foundations, but I believe that Corry has, and it is on that basis
that Corry proposes an
empiricist epistemology behind Hilbert's formalism.)
The only other point I would make w.r.t. Hilbert on physics, is that,
at least according to Corry,
part of Hilbert's empiricism is exhibited by the requirement that his
axiomatization depends upon
his axiomatization of geometry, and that the Kantian root of geometry
is spatial intuition.
While selecting axioms is definitely nontrivial, I hardly find it to be
a mystery. When I look at a
piece of mathematics, I have a conception of what its properties are,
and what behavior to expect.
(Practically speaking, when developing a piece of mathematics, the
working mathematician starts
with what is already known and has an idea already in mind for the
theorem to be added. In effect,
before writing a proof, the mathematician is working backward, from the
theorem to the axioms that
are required, to which are added the theorems already proven. This is
not being flippant here. In
fact, the new field of reverse mathematics that emerged in proof theory
in the work of Stephen
Simpson and Harvey Friedman starting in the mid 1970s does precisely
that - - attempt to determine,
from the theorem(s) required for a particular piece of mathematics,
what the minimal axiom set would
be to enable proof of the theorem(s) in question.) We know that the
more structure a mathematical
object has, the more axioms it requires.
Consider the following example. Suppose we start with the axioms of ZF
(most mathematicians'
favorite axiomatization), with or without AC. If we add to a set an
associative binary operation
and has an identity element I in S such that for all a in S, we get the
next level of complexity,
namely monoids. To go from a monoid to a groupoid, we require
invertibility. The properties of a
group are closure, associativity, the identity property, and the
inverse property. Therefore, a
group is a set, finite or infinite, whose axioms guarantee these
properties. If we are dealing with
additive groups, we make certain that our group operator is group
addition and be careful that
association, the inverse element and the identity element are chosen
for addition; if
multiplicative groups, mutatis mutandis, being careful that the group
operator is group
multiplication, and that association the inverse element, the identity
element, are defined for
multiplication. One more axiom, commutativity, gives us abelian groups.
It is also worth noting that being careful in selecting the primitives
for the universe of discourse
can have a significant influencr on the size of one's axiom set as
well. For example, by starting
with the real line to measure lengths and angles, Birkhoff found a
shorter list minimal list of
axioms for his treatment of geometry than Hilbert's, and R. L. Moore in
1902 demonstrated that
Hilbert's Axiom II.4 was redundant (for details on the redundancy, see
William Richter's paper, at
http://www.math.northwestern.edu/~richter/hilbert.pdf
I do not know how to understand the relevance for this discussion of
causality, Aristotelian or
otherwise. Does causality have some to do with the inference rules?
This is probably an issue
better left to philosophers.
The closest I came to dealing with this - - perhaps - - is a term paper
(no longer extant) on
"What Hume Might Have Said to Peano", probably back in 1970 or 1971, in
which I noted
that Hume, in the Treatise on Human Nature, argued that a relation is
required to pass from the
addition of two units to the number two, whereas Peano did not include
among, or along with, his
axioms for natural numbers, a rule or relation that would allow him to
proceed to 2 from 1 and 1
taken together. When I wrote this paper, I did not yet own a copy of
van Heijenoort's From Frege to
Gödel, in which, as we know, van Heijenoort asserted that Peano's
axiomatic system did not contain
any inference rule that enabled passage from 1 + 1 to 2, as a
consequence Peano devised an
axiomatic system, but not a formal deductive system.
Does the presence or absence of an inference rule mean the difference
between a formal deductive
system and a [mere] axiomatic system? 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.
Irving
Irving H. Anellis
Visiting Research Associate
Peirce Edition, Institute for American Thought
902 W. New York St.
Indiana University-Purdue University at Indianapolis
Indianapolis, IN 46202-5159
USA
URL: http://www.irvinganellis.info
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