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Sitaram · Site Admin Joined: 14 Sep 2005 Posts: 1079

http://sulekha.com/chpost.asp?for...ilosophy&show=0&cid=81082

I sow exclamation points which gradually sprout into question marks.

- Sitaram

===================================

Life is not lost by dying; life is lost minute by minute, day by
dragging day, in all the thousand small uncaring ways.
- Stephen Vincent Benet

http://www.jaschke.net/cf-role-of-math.html

"Everything should be made as simple as possible, but not simpler."-
Albert Einstein

The opposite of a profound truth may well be another profound truth.
- Niels Bohr

I have always found that plans are useless, but planning is
indispensable.
- Dwight Eisenhower

"Life is a mystery to be lived, not a problem to be solved."- Soren
Kierkegaard

http://www.jefallbright.net/taxonomy/page/or/134

In the pure mathematics we contemplate absolute truths which existed
in the divine mind before the morning stars sang together, and which
will continue to exist there when the last of their radiant host
shall have fallen from heaven. Edward Everett, (1794-1865) Quoted by
E.T. Bell in The Queen of the Sciences, Baltimore, 1931.

One cannot escape the feeling that these mathematical formulas have
an independent existence and an intelligence of their own, that they
are wiser than we are, wiser even than their discoverers, that we get
more out of them than was originally put into them. Heinrich Hertz,
Quoted by ET Bell in Men of Mathematics, New York, 1937.

It is easy to come to the conclusion that circles, prime numbers,
Riemann's zeta function ... have an independent, eternal existence
and any real world objects are only imperfect resemblances of these
eternal "pure ideas". The philosophical consequence is idealism,
which was perfected by the German philosophers Kant and Hegel. In
fact, Kant uses mathematics as an example to make his point:

The science of mathematics presents the most brilliant example of how
pure reason may successfully enlarge its domain without the aid of
experience. Emmanuel Kant, (1724-1804).

A new scientific truth does not triumph by convincing its opponents
and making them see the light, but rather because its opponents
eventually die, and a new generation grows up that is familiar with
it. Max Planck (1949)

As far as the laws of mathematics refer to reality, they are not
certain; and as far as they are certain, they do not refer to
reality. Albert Einstein

If scientific reasoning were limited to
the logical processes of arithmetic, we should not get very far in
our understanding of the physical world. One might as well attempt to
grasp the game of poker entirely by the use of the mathematics of
probability. Vannevar Bush

The propositions of mathematics have, therefore, the same
unquestionable certainty which is typical of such propositions
as "All bachelors are unmarried," but they also share the complete
lack of empirical content which is associated with that certainty:
The propositions of mathematics are devoid of all factual content;
they convey no information whatever on any empirical subject matter.
Hempel, Carl G.: "On the Nature of Mathematical Truth"

Guided only by their feeling for symmetry, simplicity, and
generality, and an indefinable sense of the fitness of things,
creative mathematicians now, as in the past, are inspired by the art
of mathematics rather than by any prospect of ultimate usefulness.
Eric Temple Bell (1883-1960)

The aim of science is to seek the simplest explanations of complex
facts. We are apt to fall into the error of thinking that the facts
are simple because simplicity is the goal of our quest. The guiding
motto in the life of every natural philosopher should be, "Seek
simplicity and distrust it." Alfred North Whitehead

Practical application is found by not looking for it, and one can say
that the whole progress of civilization rests on that principle.
Jacques Hadamard

There is no branch of mathematics, however abstract, which may not
some day be applied to phenomena of the real world. Nikolai
Lobachevsky

The scientist does not study nature because it is useful; he studies
it because he delights in it, and he delights in it because it is
beautiful. Jules Henri Poincare

Bridges would not be safer if only people who knew the proper
definition of a real number were allowed to design them. Norman David
Mermin

The universe...stands continually open to our gaze, but it cannot be
understood unless one first learns to comprehend the language and
interpret the characters in which it is written. It is written in the
language of mathematics ... Galileo: Il Saggiatore (1623).

google search on: mathematics philosophical problems profound

http://www.friesian.com/goedel/chap-1.htm

Godel wrote concerning Bertrand Russel: By analyzing the paradoxes to
which Cantor's set theory had led, Russel freed them from all
mathematical technicalities, thus bringing to light the amazing fact
that our logical intuitions (i.e., intuitions concerning such notions
as: truth, concept, being, class, etc.) are self-contradictory. He
then investigated where and how these common-sense assumptions of
logic are to be corrected

It is our "logical intuitions," those concerning "truth, concept,
being, class, etc." which are "self-contradictory." And they are self-
contradictory because our "common-sense assumptions of logic" break
down

Gödel's dilemma of higher axioms may be stated as follows: Let S be
an undecidable statement of an axiom system A .

Either ( S is meaningless ) or ( S is true ) or ( S is false ) .

We do not wish to maintain that S is meaningless because of the
clarity of the concepts which express S .

Hence, ( S is true ) or ( S is false ) .

But S cannot be decided by axiom system A .

Therefore, in order to determine whether ( S is true ) or ( S is
false ) , we must adjoin a new set of axioms to A forming an
augmented axiom system, say A + B , such that from A + B , we may
deduce S , or we may deduce not-S , but not both. The augmented axiom
system, A + B , is said to decide[36 <notes.htm>] S .

Gödel argues that mathematics has always resorted to higher axioms
and new methods to resolve open questions. One of his examples
concerns number theory. Facts about integers can often only be
obtained via the methods of analytic number theory. Thus, the real
numbers constitute a higher system than elementary arithmetic. While
some elementary proofs have been found for theorems originally proved
in analytic number theory, this is not always possible. Gödel will
show that in order to obtain the answer to a question demonstrably
independent of the usual axioms of a theory, one must resort to the
addition of new axioms, not previously part of that theory. Since the
bulk of mathematics has been shown by Gödel to be both incomplete and
incompletable, Gödel's dilemma is unavoidable. However, there are
good grounds for believing that any given set of axioms constitutes
only a partial description of the theory as a whole. For example,,
one can regard integers as special kinds of real numbers. Hence there
is nothing mysterious about analytic number theory obtaining results
about integers. The fact that one chooses to work with Peano
Arithmetic does not mean that he regards the integers as a separate
system, wholly distinct from the reals, but rather, that the choice
of Peano's Axioms can be based an grounds of simplicity. Of course,
the intuitionists do not share this view. Even granting that the
integers are "distinguised" kinds of reals is not enough. Gödel
indicates that statements about integers can be proved on the
assumption of the axiom of inaccessible numbers.[37 <notes.htm>] This
gives tremendous force to Gödel's belief that there is a more
intimate relationship between the higher and lower systems than would
appear at the outset. Under this view, one introduces axioms as they
are needed, but one is not obligated to stop at Peano Arithmetic, or
even classical analysis.

http://aleph0.clarku.edu/~djoyce/hilbert/problems.html

http://www.c-parr.freeserve.co.uk/hcp/infinity.htm

THE CONCEPT OF INFINITY

The idea of infinity arises in several different contexts. Most of
the applications to which we shall refer in this paper belong to one
or other of the following six categories:
The sequence of natural numbers, 1, 2, 3, ..., is said to be
infinite.

In geometry, the number of points on a line is said to be infinite.

In mathematics, many examples occur of sequences of numbers which
tend to infinity.

It is often assumed that time is infinite, in the sense that it had
no beginning, or will have no end, or both.

Likewise, space is often assumed to be infinite in extent.

Some theories of cosmology suppose that the amount of matter in the
universe, i.e. the number of stars or of atomic particles, is
infinite.

Modern astronomers do not agree on whether or not the universe is
infinite in extent. While books on cosmology display much detailed
knowledge of the history and structure of the universe, they appear
to find the issue of infinity difficult to decide. Mathematicians
tell us that the question is closely related to the average curvature
of space, and everything depends upon whether this curvature is
positive, zero or negative. If it is positive, we are told, the
volume of space is finite, but if it is zero or negative the volume
must be infinite, and this is usually taken to imply that the number
of stars and galaxies must also be infinite. In fact this curvature
is very close to zero, making it difficult to determine by
observation or measurement which sort of universe we live in. Many of
the formulae in cosmology must therefore be given in three different
forms, so that the correct version can be chosen when we do
eventually discover whether the value of k is +1, 0 or -1.

The question of the finiteness of time seems equally uncertain; most
cosmologists now believe that the universe began with a big bang, all
its material content coming into being at a single point in a
colossal explosion, and with time itself beginning at this first
moment. But opinion is divided on whether it will end with some sort
of "big crunch", with everything finally ceasing to exist in a
mammoth implosion.

It is surprising that cosmologists do not concern themselves greatly
with these questions of finiteness. They give us the three formulae,
corresponding to the three possible values of k, and leave it at
that. Indeed some books on the subject fail to state clearly whether
particular arguments apply to an "open" or a "closed" universe, as if
it did not really matter. Some have no reference to "infinity" in
their index. Surely few questions are more significant than whether
the universe is finite or infinite.

At one time there seemed to be a strong argument against the number
of stars being infinite. A simple calculation shows that if it is,
then the whole night-sky should be ablaze with light. The surfaces of
the stars are, on average, as bright as the surface of the sun, and
if they are infinitely numerous it can easily be shown that any line
of sight will eventually terminate on a star, so that the whole sky
will shine as brightly as the sun. It can be argued that the most
distant stars might have their light dimmed by passing through gas or
dust on its way to us, but if this were the case, the gas itself
would be raised to such a temperature that it too would shine with
this same brilliance. This problem was known as "Olbers' Paradox",
after Heinrich Olbers (1758-1840). But it has now been resolved; even
if the universe were infinite, we know that its expansion would
provide an explanation for the darkness of the night-sky. Distant
stars are dimmed not because of intervening matter, but because they
are moving away from us, and the wavelength of their light is
increased, and its energy reduced, by this motion. So Olbers' effect
does not now present an obstacle to those who believe in an infinite
universe. But here again it is surprising that taking the number of
stars to be infinite is an assumption that can be adopted or
discarded at pleasure, without considering whether it should be ruled
out on logical grounds.

http://www.spaceandmotion.com/Philosophy-History.htm

It is from the more or less obscure intuition of the oneness that is
the ground and principle of all multiplicity that philosophy takes
its source. And not alone philosophy, but natural science as well.
All science, in Meyerson's phrase, is the reduction of multiplicities
to identities. Divining the One within and beyond the many, we find
an intrinsic plausibility in any explanation of the diverse in terms
of a single principle. (Aldous Huxley, The Perennial Philosophy)

The History of Philosophy is founded on the Mystical realisation
that 'All is One'. Thus the Problem for Philosophy has been to
correctly describe this One thing that exists and necessarily
connects the many things. Without this connection between the One and
the Many then it is impossible to have reason, logic and certainty
(i.e. Absolute Truth) which all require necessary connection.
Unfortunately for Philosophy, history shows that we were misled by
the ancient

Greek's 'Particle' conception of Matter, i.e., the 'particle' theory
of Matter and Reality is founded on many discrete things (which we
observe) rather than One thing (which must exist and connect / cause
these many material things). Only recently has the Wave Structure of
Matter been considered, which solves this problem by describing
Matter as Spherical Standing Waves in One thing, Space. (See below
for brief Introduction to the Wave Structure of Matter.) Following
this you will find links to over 160 Science Articles on the Wave
Structure of Matter that explain and solve many of the historical
problems of Philosophy, Physics, Metaphysics and Theology which have
caused such profound problems, not only for the Sciences, but for
Humanity herself.

Those whose hearts are fixed on Reality itself deserve the title of
Philosophers. (Plato, 380BC)

The scientist imposes two things, namely truth & sincerity, imposes
them upon himself & upon other scientists. (Schrodinger, 1967)

Speech devoted to truth should be straightforward and plain. (Seneca,
60AD)

http://www.utm.edu/research/iep/w/wittgens.htm

A proposition is a picture of reality. What can be shown, cannot be
said. The limits of my language mean the limits of my world. -
Wittgenstein

Here and elsewhere in the Tractatus Wittgenstein seems to be saying
that the essence of the world and of life is: This is how things are.
One is tempted to add "--deal with it." That seems to fit what Cora
Diamond has called his "accept and endure" ethics, but he says that
the propositions of the Tractatus are meaningless, not profound
insights, ethical or otherwise. What are we to make of this?

Most of the propositions and questions of philosophers arise from our
failure to understand the logic of our language. (They belong to the
same class as the question whether the good is more or less identical
than the beautiful.) And it is not surprising that the deepest
problems are in fact not problems at all.

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