"Tony Thomas" <verdigris@iprimus.com.au> wrote in message
news:3fe7aefb_1@news.iprimus.com.au...
In commenting on the topic of 'infinity' competent mathematicians (like
yourself) launch themselves into the stratosphere of doctrine in order
to
escape the jaws of philosophical predators.
What do you think of this neuroscientists stratospheric launch?
10) WHAT IS INFINITY EXCEPT CONTINUITY?
Arthur and a few others have made comments about the mathematical
treatment
of infinity and infinitesimals that only make it still plainer that it is
very important to understand that the claim of discreteness is a dangerous
ontological claim. Many wrong conclusions and intuitions can easily
follow.
An excellent discussion of these matters is to be found in Robert Rosen's
recent Essays on Life Itself (Columbia, 2000). But the brief story is that
the real world is (as far as we can tell) fundamentally continuous - made
of
unbroken stuff. Even the idea of discrete particles of force or matter is
a
convenient fiction (see quantum mechanics). Mathematics is then a human
creation built on the idea of assuming the fiction of discreteness - on
treating reality as if it were discontinuous, breaking it into parts and
relationships that can then be used within computational formula to model
reality with various degrees of accuracy. Troubles then arise because
people
treat the models as the reality. Reality is assumed to be actually
discrete
because the maths seems to be working.
[Arthur]
Nevertheless, I thought I'd reply here just to mention that indeed
infinitesimals have been reintroduced into mathematics in a rigorous way
by
Abraham Robinson starting in the 50's. The field of analysis, and the
calculus, too has been rigorously based on these numbers which do just
what
you expect good infinitesimals to do
[John McCrone]
When mathematics begins to talk about infinity, you know it is attempting
to
grapple with the continuous. The number line is continuous and unbroken. A
point on the number line is a convenient fiction. The philosophically
correct position to take on infinitesimals is the one of limits - the
position of Aristotle and Zeno that infinity and infinitesimals are not
actual, instead they describe a potential, the potential to proceed in
iterative (ie: discrete!) steps without end because the mathematical goal
you are pursuing is continuous and unbroken.
Cantor eventually "tamed" infinity with set theory, simply asserting
infinity to be discrete (and deriving some useful mathematical outcomes
from
this trick). The calculus likewise "tamed" the continuity of motion by
simply asserting the infinitesimal - pretending that a point of
instantaneous velocity could exist by a procedure of iterative steps that
minimised measurement error (note that the trick of calculus works only so
long as the trajectory of change is smooth).
But inventing tricks that work some of the time is not the same as showing
discreteness to exist in the real world. As Rosen argues so eloquently,
the
realm of maths that works - for which computational tricks based on
assuming
discreteness can succeed - is actually an infinitely(!) small space of the
total possible universe of mathematical systems.
A source that more directly tackles the status of Robinson's "solution" is
Morris Kline, Mathematics: The Loss of Certainty (OUP, 1980).
[Arthur]
As someone commented recently, many mathematicians and scientists are
platonists of the literal 'the universe in JUST numbers kind'.
[John McCrone]
Or JUST information these days. Information in the Shannonesque sense of
discrete bits is the ultimate expression of a discrete-ophile ontology of
reality <grin>. But as even Shannon admitted, his was a mathematical
procedure for generating discreteness - another way of approaching
infinity.
A bit is defined as the difference that makes a difference. In other
words,
information theory is an algorithm which can produce "bits" from what may
be
a continuous reality, but which may also ultimately fail to reach that
reality - which is why there is a difference between information and
meaning. Or why Information Theory ought to have been called signalling
theory.
[Arthur]
A language of order, relation, function; differentiation, integration;
field, invariance; inside, outside, boundary, open, closed, kink, knot,
sheaf, cusp; order disorder etc can help to unify the disparate languages
with which the members of this list seem to talk past one another. Head
down
to your underlying assumptions if you can find them and see if you are
working form the same ones. If not, you can NEVER reach agreement with the
other party and can only hope to help each other develop the different
ideas
you have already.
Of course, in the year 2001, not all of us will agree with such a
foundational perspective, and I have my own middle aged shakiness on this
point. Even in the foundations of mathematics there is some hint of a
change
from a Hilbert style axiomatics to a future EXPERIMENTAL 'foundation' to
mathematics more in keeping with the style of the times. Sounds very hip
to
me.
[John McCrone]
Godel wrecked Hilbert's dream long ago - and Hilbert's was a last ditch
stab
at preserving axiomatics anyway. On the other hand, I have found huge and
unexpected hope in the work of mathematically sophisticated theoretical
biologists such as Rosen, Pattee, Grossberg and others. There is the
possibility of an exciting future indeed.
These are guys who do take a fairly topological approach. Physical reality
is a continuous manifold of "stuff" that then gets constrained though a
succession of informational controls (bottlenecks on the underlying
dynamics) to form complex structures.
And what makes a general theory of consciousness and life possible is that
there are a limited number of basic mathematical entities - what I call
causal objects - which are natural, inevitable, and universal in living
and
conscious systems. These natural shapes include agents, networks, and
hierarchies. They also probably include membranes (enclosing surfaces) and
Turing machines ("tape and gate" computing). They also include some fairly
"psychological" notions such as memory, anticipation and learning - and
even
probably subjectivity itself.
Much More in the remainder of the article
http://www.btinternet.com/~neuronaut/webtwo_features_infinoverse.htm
http://www.btinternet.com/~neuronaut/webtwo_features_reason.htm
http://www.btinternet.com/~neuronaut/webtwo_articles.html
Countless theologians, philosophers, logicians and mathematicians (note
the
order of precedence here) have spun various doctrinal webs about the
matter.
Suffice to say there are different ways of looking at things and there
is
no
necessity for these ways to be consistent with each other (remember
Kuhn).
Herein lies the advantage of philosophy over logic and mathematics;
tolerance of the inconsistent and the absurd may be considered by the
unprejudiced mind.
There is clearly (hate that word) a difference between (the
respectability
of) complex numbers and 'infinite numbers' (your quotes presumably
indicate
disapproval or disbelief) in that the former are completely integrated
into
analysis and the latter have no place there. The similarity I was
referring
to was their historical rejection by orthodoxy at some time or another,
not
their similarity as mathematical constructions. Further, the
construction
of
a logico/mathematical entity E1 and of another E2, by means of
meaningful
definitions, axioms and theorems lends them equal status from a logical
point of view regardless of whether they fit into existing, orthodox
theories. Here I'm regarding the whole of mathematics as consisting of
more
or less well constructed theories, which it probably isn't. It was this
latter question, of the consistency of mathematics that led to R & Ws
abortive Principia Mathematica.
Now, quite a lot of respectable scientists ask questions like: "is the
cosmos of finite or infinite extent" or "can an electron possess
infinite
energy." For such questions to be addressed, some precise meaning needs
to
be attached to the word 'infinite'. Since particle physicists and
cosmologists are almost completely dependent on mathematics, one would
expect to find some 'useful' theories about the meaning of infinity
there.
Furthermore, if physicists are actually forced to make calculations,
involving 'infinite' quantities, then mathematicians better come up with
something useable, just in case 'reality' does indulge itself in such
fanciful phenomena. From the point of view of an observer, largely
ignorant
of the exalted logical and mathematical theories that might bear upon
the
matter, it seems that such theories are non-existent or deficient.
The idea that there is something called 'mathematical sense' is an
intriguing one. That there is such a thing as mathematical ability seems
to
be a well established fact. However, such ability is surely enhanced by
years of study and practice in the mathematical arts. If such study
consists
in adopting existing orthodoxy holus-bolus, it is bound to induce what
might
be called Kuhnian myopia. Such, I think, is the the cause of adverse
reactions to simple statements like; 'infinite numbers can be defined in
such a way that they can be incorporated into ordinary arithmetic.'
It would be a rash person who claims to understand the boundaries
between
logic and mathematics. Once upon a time, logicians thought that
mathematicians should bow to their will and
construct everything from first principles (as defined by logicians)
using
finely stepped proof structures. I suspect that this programme was never
feasible because attention to such detail would prove so tedious that no
creative work would ever get done. If such strict procedures were ever
used,
it was likely for the purposes of validation and rationalisation after
the
event. Otherwise, there would be no room at all for 'mathematical sense'
(intuition). Any enmity between mathematicians and logicians is likely
to
be
a struggle over who should make the rules and who should obey them.
Philosophers, of course, know that this prerogative belongs to them
alone.
Returning to the question, 'are there such things as infinite numbers
and
could there be a theory about them.' The first step, it seems to me, is
to
admit that such entities can be defined (successfully or otherwise) and
that
no existing doctrine should stand in the way of this simple step. After
all,
Peano did stoop to defining finite cardinals from first principles, as
if
he
thought they did not exist prior to his work.
This leads to the annoying question as to what is meant by saying that
numbers like two 'exist'. If we define them on the basis of a certain
type
of set, the regress as to whether abstract sets exist supervenes. From
this
perspective, the existence of finite cardinals is as dubious as that of
infinite ones. The obvious objection is that sets of couples do refer to
actual occurrences in the world (states of affairs for Wittgenstein
fans)
whereas infinite numbers or sets have no such 'real' correlates.
Another approach is to say, "no one knows much about the infinite, so
I'll
define it in the way that pleases me and ignore what others have said."
This
is convenient, because it saves time on studying the works of other
crackpots. Academics frown on this approach and you have to be a genius
to
get away with it.
From a semantic point of view (with a dash of Hegel) one wonders how we
can
know what 'finite' means if we have no idea what the contrary term
'infinite' means. If there is no meaning to be attached to the term
'infinite' then 'finite' seems to be redundant, and should be removed
from
the dictionary. But this would alarm mathematicians who rely so much on
the
'infinite', even if they pretend that the never actually touch the stuff
but
only tend to inhale from a distance.
Making use of the dialectic, we can say "that which is infinite is not
finite"; a harmless enough tautology. The next step is to decide whether
the
class of infinite entities is empty or not. Since we could decide either
way, on the basis of ignorance, we can decide that there is at least one
entity which can be said to be infinite. The alternative decision is
that
there is a class of finite entities and nothing else exists beyond this
class. Then comes the silly question, "is this class inside itself or
not?'
The more serious question arises, "is this class infinite or finite?" No
answer, came the stern reply.
But all this is obfuscation. If we are have to measure the length of
God's
beard we are going to need a big number, much bigger that those old
fashioned naturals. So, why not have a new infinite unit fit for the
job.
Hang on though, all this may not be necessary at all, since arithmetic
is
independent of measures like feet inches, or God's beard length. The
number
'1' will do just as well as it is. However, this would mean we could
give
up
the distinction between the finite and infinite as far as arithmetic is
concerned, as long as we don't try and mix up finite and infinite
measurements at the same time.
So, it seems that ordinary arithmetic will do very well for measure gods
body-parts as long as we don't try and compare them with our own. The
divine
form may be comparable but the dimensions must remain incomparable.
The human mind is never satisfied, it yearns to reach beyond its last
thought, a tendency that leads us to suspect the existence of the
infinite.
This principle of transcendence ought to be enshrined in meta-logic
somewhere, particularly since it underlies that illogical operator '+'.
If only we could say something 'naive', like I + 1 > I, but that damned
madman Cantor has already convinced everybody that Ao + 1 = Ao. Oh well,
that's the end of the matter then.
Tony Thomas
"mitch" <mitchs@rcn.com> wrote in message
news:3FE67CF8.DCDEB562@rcn.com...
Tony Thomas wrote:
To say there are no infinite numbers is like saying there are no
complex
numbers. Such abstractions are mental fictions which may be invented
at
will. If one wishes to define infinite numbers, in a consistent and
meaningful way, then there is no reason they should not be accorded
a
similar status to other mental fictions (eg quaternions).
There is a big diffference between "infinite numbers" and quaternions
or
complex
numbers.
There is something called a Lindenbaum algebra that can be formed for
any
first-order language on the basis of equivalence under the deductive
calculus.
Lindenbaum algebras are Boolean algebras which, by the Stone
representation
theorem, are associated with compact Hausdorff spaces. Such spaces
are
always
associated with at least one almost periodic point.
Thus, every "logic" admits an "existence" outside of its domain.
The designs of philosophers that led to set theory and the
rationalizations for
infinite numbers conflate the checks and balances in mathematical
structure.
The number systems R, C, H, O all relate to each other in ways that
preserve the
geometric relations of conic sections. Q and Z express themselves in
R
under
interpretation of a linear order with respect to a geometric line, and
the
embedding of points in a line to orient such an association is
equivalent
to the
embedding of a quadratic field, once again introducing the constraint
of
conic
sections on the relations.
The same cannot be said of the naive approach taken to "infinite
numbers."
Worse yet, after thinking about the technical details of this matter I
am
finally turning to Aristotle only to find that much of this unsound
reasoning
can be attributed to people simply ignoring basic statements that have
served to
ground philosophy.
I am beginning to think that Bertrand Russell must have been the
biggest
idiot
in the history of philosophy. He chose to promote the ideas of a man
(Frege)
whose own biographers note a lack of preparation in the philosophy
against
which
he was arguing.
The thought that "infinite numbers" are grounded in a mathematical
sense
seems
ludicrous given the rivalry between British empiricism and Continental
rationalism that I have seen mentioned in this newsgroup. Moreover,
the
more I
look into these matters, the more I find a historical enmity of
philosophical
logicians for mathematicians. There is a great deal of "office
politics"
in the
development of "infinite numbers" that is quite atypical when compared
with
other branches of mathematical inquiry.
:-)
mitch
