Sorry, the end of third paragraph should read:

formulated in 19th century were very insightful but needed an input of 20th century topology to make its the theory “bullet-proof” (the same Dirac function until Laurent Schwarz fixed it).

Dear Banjo,

I have no doubt that life and death are reality. At this time my life is reality, and my death will be reality in the future.

http://en.wikipedia.org/wiki/Gauge_theory

http://empg.maths.ed.ac.uk/Activities/STS/AdSCFT/

This is the website..for the course on the dualit.. between gauge fields and strings..delivered by Dr Carlos Núñez from Swansea University.

also note gauge fields and knots
plus gauge fields and string theory

In physics,..a gauge theory
is a type of field theory..in which the Lagrangian..is invariant
[lol]..in-varient]..under a continuous group of local transformations.

Both gauge invariance and diffeomorphism invariance
reflect a redundancy..in the description of the system.

The term gauge.refers to redundant degrees of freedom..in the Lagrangian.

The transformations between possible gauges,..called gauge transformations, form a Lie group..lol..which is referred to as the symmetry group..or the gauge group of the theory.*

Associated with any Lie group..is the Lie algebra of group generators...[lol]

For each group generator..there necessarily arises
a corresponding..vector field called the gauge field.

The field..has the property of being self-interacting
and equations of motion..that one obtains are said to be semilinear, as nonlinearities..are both with and without derivatives.

This means..that one can manage this theory..
only by perturbation theory,..with small nonlinearities.
http://en.wikipedia.org/wiki/Perturbation_theory

Perturbation theory..comprises mathematical methods
that are used..to find an approximate*..solution to a problem which cannot be..solved..*exactly>>,..[DOUBLE LOL]

..<<by starting..from the exact solution..[of a related problem] Perturbation theory..is applicable if the problem at hand..can be formulated..by adding a "small" term*..to the mathematical description..of the exactly solvable problem.

Note that the transition between "upper" ("contravariant") and "lower" ("covariant") vector or tensor components..is trivial for a indice

gauge invariance)...When such a theory is quantized,
the quanta of the gauge fields..are called gauge bosons.

If the symmetry group..is non-commutative,[lol]
the gauge theory..is referred to as non-abelian,..the usual example being the Yang–Mills theory.

Yang–Mills theory..seeks to describe the behavior*
of elementary particles..using these non-Abelian Lie groups
and is at the core..of the unification of the Weak and Electromagnetic force (i.e. U(1) × SU(2))

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Dear David,

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« I’m sure that George, OUG, you and I all see beauty in this world, all can feel love in our hearts for something, all see our own abstractions, all try to understand the world and share our humanity in many ways.”
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Yes, I do.

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“I hope we all see beauty in mathematics, and we all know mathematics at some level or other.”
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I see elementary mathematics as a tool and pure mathematics as a barrier to knowledge because I do not have access to it.

Nevertheless, the signs look quite nice but not as nice as those of music and Chinese which I don’t understand either.
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“The fertilised ova from which we all come start their development in a geometric progression – 1,2,4,8,16.. cells.”
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I thought you could not see any geometry in nature. Thanks. I’ll add it to my definition of pure mathematics.
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"Mathematics is the language in which God has written the universe." – Galileo.
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Galileo was a brilliant scientist but, in my opinion, a not so brilliant philosopher. I understand that mathematics is a tool which helps us to interpret the universe and that faith in God is not a prerequisite. According to George, mathematics is the language of physics.
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(Continued) ...

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(Continued) ...

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“G. H. Hardy, Daniel J. Cohen, George, et al.”
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A major figure of Elizabethan England was Sir Francis Bacon (1561 – 1626) who is accredited with having invented the scientific method (of which empiricism is the central concept) based on induction which proved a formidable turbo-booster for the development of abstract thought, the thought process in which ideas are distanced from reality.

Bacon was a deeply religious person. He composed and published a collection of religious meditations and theological tracts and prayers. He described “love” as the force of the instinct of primal matter, "the natural motion of the atom", "the summary law of nature, that impulse of desire impressed by God upon the primary particles of matter which makes them come together, and which by repetition and multiplication produces all the variety of nature", "a thing which mortal thought may glance at, but can hardly take in".

It’s not surprising that the natural propensity of mathematicians for abstract thought leads some of them down the same religious path as that of Bacon.
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“Since mathematics is a part of my life and I think death is oblivion there is no aftermath for me.”
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To cite one of my favourite authors:

“The boundaries of life and death are not clearly marked. Life and death are dichotomies made by human beings. What does a clam know of aftermath?”

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Dear Banjo,

>> I don’t think any reasonable person would entertain the slightest doubt as to the existence of the physical reality … – even in the absence of “adequate models of physical reality …”.<<

This is what I meant when I wrote that it is a matter of belief shared by everybody that a reality, independent of how we perceive, imagine, model or explain it, exists. The exception are solipsists, and I agree that one can call them unreasonable.

>>This is the result of an analysis of nature independently of mankind.<<

Who is doing that analysis if not a member of mankind? Some speak of God’s perspective, but then you have to admit that there is a God, and that we can understand how He does His “analysis”.

>> I could discern signs of the application of geometry in nature … David does not.<<

I am sure he does “discern signs of the application” of mathematical concepts, only they are not to be seen as part of physical reality like, e.g. horses.

It is a deeper question of what entity actually exist: Horses (in distinction to fairies) do, so do quarks (in distinction to aether) and compact manifolds (in distinction to compact linear spaces), however these are different “kinds” of existence. You can touch a horse, you cannot touch a quark but it arises as a consequence of valid theories, and neither of these two kinds applies to manifolds or other mathematical concepts, although they can be useful in explaining physical reality.

>> pure mathematics is a product of the imagination <<

Some mathematicians will agree, some will not. It certainly is not part of physical reality.

>> except as regards the operations of additions and subtractions.<<
No except here, that is all part of mathematics like any other operations and relations.

>> It’s not surprising that the natural propensity of mathematicians for abstract thought leads some of them down the same religious path as that of Bacon.<<

I do not know about mathematicians but a higher percentage of specialists in physical sciences than in life sciences are theists.