Yes, it could. The old fashioned way. Observe first, then explain it. If you can't see what something is doing, you can't really know, can you?

Multiple universes, i call that a vague concept. How do you explain going from 1 universe to multiple universes? Vaguely, and incoherently, with a bit of mathematical religion thrown in.

M-Theory, i call that an abstract idea. And lots of incoherent mathematics proving a theoretical view of what's not really happening.

Are either helping us with physics? They help me move forward with entertainment, i'll give them that.

Theory and experiment ought be cyclic and iterative. The trouble with theory is that it can continue without ever drinking at the well of reality. The trouble with experiment is that it can remain forever literal and anecdotal without generality. However you've chosen a couple of good candidates for shooting at the earliest opportunity ( in my view ).

Testing alternate universes would require ducking nextdoor to an adjacent one, and popping back to compare results, so we'll need to await either the occult or the right technology for that. See Terry Pratchett and his 'trousers of time' concept, in Night Watch say.

As for M-theory and string theories overall, I became rather discouraged after reading Lee Smolin's The Trouble With Physics where he firmly but delicately outlines the dreadful error of large swathes of physics faculties around the globe. Not one shred of data to guide them. No physical predictions emitted whatsoever. An impending risk of becoming even mathematical rubbish, as they'd not attempted to prove a conjecture in the late 80's crucial to the subsequent development of the field. The flower of an entire generation of young physicists quite wasted.

Now as for 1 + 1 = 2 here we are on solid ground. I'll relate a particular approach that I treasure. Let

0 = 'zero' = the null/empty set = {}

then

1 = 'one' = the set that contains '0' = {{}}

and

2 = 'two' = the set that contains '1' = {{{}}}

then

3 = 'three' = the set that contains '2' = {{{{}}}}

.... ad infinitum/nauseum. This easily yields the concept of a 'successor' - a set that immediately contains another. From this one can proceed as follows :

x + 1 = the set that contains 'x' = {x}

so the symbol sequence '+ 1' becomes 'take the successor of'. One still hasn't defined what 'one-ness' or 'five-ness' or 'five_hundred_and_thirty_seven-ness' etc mean, but you've converted cardinality ( associating numbers with sets ) to ordinality ( lining them up in a row ). It's simple, consistent and is built upon to produce the arithmetic we know and love. :-) :-)

Cheers, Mike.

( edit ) Dedekind had a brilliant concept of division of sets ( 'Dedekind cuts' ) to sort out the behaviour of the real numbers, especially giving exact meaning to the notions of 'supremum' and 'infimum' that are required for limit processes, and thus calculus.

( edit ) So you'd get 2 + 3 = 5 by first breaking the elements down :

here brackets are purely used as a way of grouping the symbols, even though they have other meanings added in later development. Then build back up to the answer :

If the proof survives everywhere in the universe and outside our minds and tools, I would hold on a little tighter:-)

That's exactly what Roger Penrose reckons in The Road To Reality. There's a Platonic existence of mathematics 'out there' waiting for us to discover! :-)

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter.Blaise Pascal

Nice person, prof.Penrose. He even answered a latter of mine after me having sent him a draft of a paper "The coherent brain", written in 1980, which contained ideas very similar to those of his book "The emperor's new mind". He found my text "highly interesting" and invited me to read his next book "Shadows of the mind", which is very platonic.
Tullio

Strict definition : A 1-form is a real valued function upon a set of vectors that is linear in it's arguments.

Put a vector in, get a real number out. Multiply the vector by a constant, and the real value result also is multiplied by that same constant.

It's one way to generalise the idea of a 'dot product'.

It's the best linear approximation to the change of an independent variable when it's dependent variable alters.

Take a simple one-dimensional vector analogy eg. the real number line. Suppose I have a function ( this is not the 1-form ) that goes like the square of the number. Thus

f(x) = x^2

Now I know the first derivative of this : df/dx = 2x. This essentially becomes the 1-form as

the actual functional change of [ f(x) = x^2 ] by moving along by delta_x = delta_f ~ df/dx * delta_x = ( in 'bra' and 'ket' notation )

where delta_x and delta_f are non-infinitesimal. The '*' multiplication is the one-dimensional version of the dot product. It is approximate '~' as there may be higher degrees of rates of change to apply. The full gore ( including other issues ) leads to a Taylor's expansion, but we toss away the higher terms here. The derivative used in the above is evaluated at some particular x, so the approximation holds only 'nearby' .... and of course is a better one the closer we are.

In several dimensions you get the 'grad' replacing the role of df/dx, and a vector with components like delta_x :

( various equivalent notations ). The grad components are partial derivatives - rates of change of the function f solely along the co-ordinate basis directions each in turn - evaluated at the point (x, y, z) of interest. The grad is not a number but something ( function/operator ) to be applied to what ? Yes, you guessed it, a vector. In this case v that represents a finite displacement.

the actual functional change of [ f = f(x, y, z) ] by moving along v = delta_f ~ df/dx * delta_x + df/dy * delta_y + df/dz * delta_z = [d/dx, d/dy, d/dz](f) . [delta_x, delta_y, delta_z] =

Couple of points:

- beware the 'sense' of the quantities ie. which way do things increase or decrease?

- we are not limiting the number of dimensions of the vector here. By extension we could have an infinite number of degrees of freedom.

- I say generalisation of dot product as one may not always be adding within expressions to get your real number result. So the Lorentz type metric ( working on relativistic 4-vectors ) has an opposite sign for the time co-ordinate slot compared to the spatial slots. { this has the curious property that it may evaluate to zero when the directions don't 'look perpendicular' - say when the worldline is that of a light ray, hence the phrase 'null geodesic' }

- the 1-form doesn't give a field's functional value, only how it changes with a displacement. So with one electric charge in the field of another, a 1-form applied to a displacement would give the change in the potential ( energy ) say, not the absolute value with respect to some global reference. But you could still say "I start with some field value f(P) at a point P, make a little jump along a vector v and get a new field value which is approximately f(P) + "

Example 1 : the height of grass on an otherwise bare hillside. I walk around with 2 independent degrees of freedom, with a rate of change of grass-height from one position to the next ( this of course depends also on where I am ). I use a 1-form to calculate the how the grass height changes with each step.

Example 2 : calculate a quantum mechanical outcome by applying a 'bra' function to a 'ket' vector. I get a real number representing a probability within the range zero to one.

Finally, other nomenclature and phrases :

- a 1-form is a tensor of rank 1

- the vector is contravariant ( don't ask )

- the 1-form is covariant ( don't ask )

- the vector and the 1-form are dual ( to one another )

- applying the 1-form to the vector is a tensor contraction

Cheers, Mike.

( edit ) To save you worrying about asking .... :-)

Covariant / contravariant embodies the idea of 'paradoxical motion', an ancient idea. So if I'm driving North along the highway and I look at a cow in a paddock, the cow will seem to travel southwards compared to the distant background behind it. Conversely from the cow's point of view I'll be traveling northwards compared to the background behind me.

Similiarly if I change ( say by rotating or translating ) the co-ordinate system I use in a given circumstance then I could equally say the system was static but the ( electric/magnetic/gravitational ) field I was describing rotated/translated the other way. It is essential to realise that this must be true, because we ( believe we ) are describing an entity ( the field ) that has a reality apart from our arbitrary choice of a particular layout of rulers and clocks. If not, for instance, then I could change the ( external ) field to a new value simply by rotating myself around by 360 degrees!! We don't seem to live in a universe where this happens ..... :-)

( edit ) Sharp punters will note that this co/contra-variance business implies that if I have two nearby points P and Q then a tensor contraction ( a 1-form applied to the displacement vector between the two ) is an invariant across ( sensibly related ) co-ordinate systems. So if I take a step on my grassy hillside the outcome is independent of who is looking. This is comparable to the ordinary dot product of a vector with itself giving ( the square of ) it's length, again irrespective of a particular co-ordinate arrangement.

( edit ) 'the affine connection' : I see what you were getting at LD. Yes. We like affine transformations so that points in a line with certain relative separations remain in line with the same relative displacements after the transform. Sensible transformations do this, including Minkowski/Lorentz stuff. A shear ( different scale factors applied to orthogonal directions ) is affine. Say the case of the length in the direction of motion Lorentz contracting but not in the perpendicular ones.

I have made this letter longer than usual because I lack the time to make it shorter.Blaise Pascal

If the proof survives everywhere in the universe and outside our minds and tools, I would hold on a little tighter:-)

That's exactly what Roger Penrose reckons in The Road To Reality. There's a Platonic existence of mathematics 'out there' waiting for us to discover! :-)

or

Mathematics was constructed by the brain in order to be effective in this universe. :-)

There are some who can live without wild things and some who cannot. - Aldo Leopold

Mathematics is music for the soul
Music is mathematics for the soul

Ahhhh, that'd be J.S.Bach then ! :-))

Quote:

Mathematics was constructed by the brain in order to be effective in this universe. :-)

Yup, the reason why maths seems to be 'out there' is because that's the part of the information blizzard we have evolved to match to. Those that didn't do so well at that contributed to the genetic persistence of predators instead. I don't think it is a fluke that initial explanations in physics ( historically ) relate to everyday kinematical/dynamical stuff - position, time, velocity, rates of change .....

Imagine if you could talk to a sentient bacteria ? They'd do loops around us discussing chemistry and talk your leg off on the topics of Brownian motion, diffusion gradients and crystallography. If you explained quantum mechanics to them, they'd say ( with a knowing smile and a chuckle ) "Oh that. Now, errr .... what was you're problem there? ". :-) :-)

The real joke of "Why did the chicken cross the road?" isn't the specific retorts. It's the assumption that reasoning/intent applies at all. Suppose a chicken has neither a 'road' concept nor that of 'sides'. A road shaped pattern in the visual field doesn't trigger neurology that has no need for it *.

Cheers, Mike.

( edit ) * - until recently that is. :-) :-)

I have made this letter longer than usual because I lack the time to make it shorter.Blaise Pascal

Another approach is the Peano axioms. You accept them and move on, with no deeper formal derivation being offered. But you wind up with what you want from arithmetic down the track. :-)

[pre]#1 - 1 is a natural number

#2 - To every natural number n there is another uniquely determined natural number, n', called the successor of n

#3 - 1 is not the successor of any natural number.

#4 - If two natural numbers have equal successors, they themselves are equal. Thus n' = m' implies n = m

#5 - If you have a set X consisting of some natural numbers with the properties that :

(i) k is in X, and

(ii) k is not the successor of any number in X, and

(iii) n' is in X whenever n is in X, then

the set X consists of k and all successors thereof[/pre]
Equality is meant in the sense of symbols aliasing for the same thing with reflexive ( m = m ), symmetric ( m = n n = m ) and transitive ( m = n and n = p => m = p ) properties.

Ordering then becomes an implicit consequence, so that one can proceed to unambiguously assign meaning to 'greater than' and 'less than' statements and symbology. Beware that 'greater than or equal to' ( ...etc ) is an umbrella term for two distinct cases which can't simultaneously hold - you just happen to have thrown a blanket/group label over them. The other annoyance is with inequalities and multiplying both sides by a constant : you have to reverse the inequality ( swap '>' and '<' ) if the constant is negative, and if the constant is zero as applied to a strict inequality then it doesn't follow ( zero isn't either less than or greater than zero ). This is especially relevant with purely symbolic variables like 'x', I have to consider the possible varieties of 'x' to avoid error. Anyhows 1 gets labelled as 'the first' and the Principle of Induction is locked in as #5 .....

The 'problem' with arithmetic is really zero coupled with the operation of multiplication. ANY number times zero is zero, thus the inverse ( division, or 'how many zeroes do you add together to get the number five' ) has no definition. Or put another way - if I tell you that the product of a non-zero number and zero equals zero, then you can't work back to uniquely tell me what the non-zero number must have been. It's not 'infinity', it's not more ( or less ) than any particular number you can state, it's just not a solvable problem. However, weaseling around this issue has lead to some superb stuff .... the limit process superimposed upon algebra being one path.

Cheers, Mike.

( edit ) And the other 'problems' - which number multiplied by itself gives negative one, and which ratio of integers gives the length of the longest sides of certain triangles - also have impressive solutions. :-) :-)

I have made this letter longer than usual because I lack the time to make it shorter.Blaise Pascal

## RE: Including.. 1+1 =2.

)

At least that one can be logically proven :P (of course it depends entirely on our definition of the symbols; both the syntax and semantics)

## RE: Yes, it could. The old

)

Theory and experiment ought be cyclic and iterative. The trouble with theory is that it can continue without ever drinking at the well of reality. The trouble with experiment is that it can remain forever literal and anecdotal without generality. However you've chosen a couple of good candidates for shooting at the earliest opportunity ( in my view ).

Testing alternate universes would require ducking nextdoor to an adjacent one, and popping back to compare results, so we'll need to await either the occult or the right technology for that. See Terry Pratchett and his 'trousers of time' concept, in Night Watch say.

As for M-theory and string theories overall, I became rather discouraged after reading Lee Smolin's The Trouble With Physics where he firmly but delicately outlines the dreadful error of large swathes of physics faculties around the globe. Not one shred of data to guide them. No physical predictions emitted whatsoever. An impending risk of becoming even mathematical rubbish, as they'd not attempted to prove a conjecture in the late 80's crucial to the subsequent development of the field. The flower of an entire generation of young physicists quite wasted.

Now as for 1 + 1 = 2 here we are on solid ground. I'll relate a particular approach that I treasure. Let

0 = 'zero' = the null/empty set = {}

then

1 = 'one' = the set that contains '0' = {{}}

and

2 = 'two' = the set that contains '1' = {{{}}}

then

3 = 'three' = the set that contains '2' = {{{{}}}}

.... ad infinitum/nauseum. This easily yields the concept of a 'successor' - a set that immediately contains another. From this one can proceed as follows :

x + 1 = the set that contains 'x' = {x}

so the symbol sequence '+ 1' becomes 'take the successor of'. One still hasn't defined what 'one-ness' or 'five-ness' or 'five_hundred_and_thirty_seven-ness' etc mean, but you've converted cardinality ( associating numbers with sets ) to ordinality ( lining them up in a row ). It's simple, consistent and is built upon to produce the arithmetic we know and love. :-) :-)

Cheers, Mike.

( edit ) Dedekind had a brilliant concept of division of sets ( 'Dedekind cuts' ) to sort out the behaviour of the real numbers, especially giving exact meaning to the notions of 'supremum' and 'infimum' that are required for limit processes, and thus calculus.

( edit ) So you'd get 2 + 3 = 5 by first breaking the elements down :

2 + 3 = ( 1 + 1 ) + ( 2 + 1 ) = ( 1 + 1 ) + ( ( 1 + 1 ) + 1 )

= 1 + 1 + 1 + 1 + 1

here brackets are purely used as a way of grouping the symbols, even though they have other meanings added in later development. Then build back up to the answer :

1 + 1 + 1 + 1 + 1 = ( 1 + 1 ) + 1 + 1 + 1 = 2 + 1 + 1 + 1

= ( 2 + 1 ) + 1 + 1 = 3 + 1 + 1 =

= ( 3 + 1 ) + 1 = 4 + 1 = 5

I have made this letter longer than usual because I lack the time to make it shorter. Blaise Pascal

## RE: At least that one can

)

If the proof survives everywhere in the universe and outside our minds and tools, I would hold on a little tighter:-)

There are some who can live without wild things and some who cannot. - Aldo Leopold

## RE: If the proof survives

)

That's exactly what Roger Penrose reckons in The Road To Reality. There's a Platonic existence of mathematics 'out there' waiting for us to discover! :-)

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter. Blaise Pascal

## Nice person, prof.Penrose. He

)

Nice person, prof.Penrose. He even answered a latter of mine after me having sent him a draft of a paper "The coherent brain", written in 1980, which contained ideas very similar to those of his book "The emperor's new mind". He found my text "highly interesting" and invited me to read his next book "Shadows of the mind", which is very platonic.

Tullio

## Notes on '1-forms' or

)

Notes on '1-forms' or 'differential forms'

Strict definition : A 1-form is a real valued function upon a set of vectors that is linear in it's arguments.

Put a vector in, get a real number out. Multiply the vector by a constant, and the real value result also is multiplied by that same constant.

It's one way to generalise the idea of a 'dot product'.

It's the best linear approximation to the change of an independent variable when it's dependent variable alters.

Take a simple one-dimensional vector analogy eg. the real number line. Suppose I have a function ( this is not the 1-form ) that goes like the square of the number. Thus

f(x) = x^2

Now I know the first derivative of this : df/dx = 2x. This essentially becomes the 1-form as

the actual functional change of [ f(x) = x^2 ] by moving along by delta_x = delta_f ~ df/dx * delta_x = ( in 'bra' and 'ket' notation )

where delta_x and delta_f are non-infinitesimal. The '*' multiplication is the one-dimensional version of the dot product. It is approximate '~' as there may be higher degrees of rates of change to apply. The full gore ( including other issues ) leads to a Taylor's expansion, but we toss away the higher terms here. The derivative used in the above is evaluated at some particular x, so the approximation holds only 'nearby' .... and of course is a better one the closer we are.

In several dimensions you get the 'grad' replacing the role of df/dx, and a vector with components like delta_x :

grad_f = [df/dx, df/dy, df/dz] = [d/dx, d/dy, d/dz](f)

v = [delta_x, delta_y, delta_z]

( various equivalent notations ). The grad components are partial derivatives - rates of change of the function f solely along the co-ordinate basis directions each in turn - evaluated at the point (x, y, z) of interest. The grad is not a number but something ( function/operator ) to be applied to what ? Yes, you guessed it, a vector. In this case v that represents a finite displacement.

the actual functional change of [ f = f(x, y, z) ] by moving along v = delta_f ~ df/dx * delta_x + df/dy * delta_y + df/dz * delta_z = [d/dx, d/dy, d/dz](f) . [delta_x, delta_y, delta_z] =

Couple of points:

- beware the 'sense' of the quantities ie. which way do things increase or decrease?

- we are not limiting the number of dimensions of the vector here. By extension we could have an infinite number of degrees of freedom.

- I say generalisation of dot product as one may not always be adding within expressions to get your real number result. So the Lorentz type metric ( working on relativistic 4-vectors ) has an opposite sign for the time co-ordinate slot compared to the spatial slots. { this has the curious property that it may evaluate to zero when the directions don't 'look perpendicular' - say when the worldline is that of a light ray, hence the phrase 'null geodesic' }

u = (u0, u1, u2, u3)

v = (v0, v1, v2, v3)

u.v = (-1) * u0 * v0 + (+1) * u1 * v1 + (+1) * u2 * v2 + (+1) * u3 * v3

- the 1-form doesn't give a field's functional value, only how it changes with a displacement. So with one electric charge in the field of another, a 1-form applied to a displacement would give the change in the potential ( energy ) say, not the absolute value with respect to some global reference. But you could still say "I start with some field value f(P) at a point P, make a little jump along a vector v and get a new field value which is approximately f(P) + "

Example 1 : the height of grass on an otherwise bare hillside. I walk around with 2 independent degrees of freedom, with a rate of change of grass-height from one position to the next ( this of course depends also on where I am ). I use a 1-form to calculate the how the grass height changes with each step.

Example 2 : calculate a quantum mechanical outcome by applying a 'bra' function to a 'ket' vector. I get a real number representing a probability within the range zero to one.

Finally, other nomenclature and phrases :

- a 1-form is a tensor of rank 1

- the vector is contravariant ( don't ask )

- the 1-form is covariant ( don't ask )

- the vector and the 1-form are dual ( to one another )

- applying the 1-form to the vector is a tensor contraction

Cheers, Mike.

( edit ) To save you worrying about asking .... :-)

Covariant / contravariant embodies the idea of 'paradoxical motion', an ancient idea. So if I'm driving North along the highway and I look at a cow in a paddock, the cow will seem to travel southwards compared to the distant background behind it. Conversely from the cow's point of view I'll be traveling northwards compared to the background behind me.

Similiarly if I change ( say by rotating or translating ) the co-ordinate system I use in a given circumstance then I could equally say the system was static but the ( electric/magnetic/gravitational ) field I was describing rotated/translated the other way. It is essential to realise that this must be true, because we ( believe we ) are describing an entity ( the field ) that has a reality apart from our arbitrary choice of a particular layout of rulers and clocks. If not, for instance, then I could change the ( external ) field to a new value simply by rotating myself around by 360 degrees!! We don't seem to live in a universe where this happens ..... :-)

( edit ) Sharp punters will note that this co/contra-variance business implies that if I have two nearby points P and Q then a tensor contraction ( a 1-form applied to the displacement vector between the two ) is an invariant across ( sensibly related ) co-ordinate systems. So if I take a step on my grassy hillside the outcome is independent of who is looking. This is comparable to the ordinary dot product of a vector with itself giving ( the square of ) it's length, again irrespective of a particular co-ordinate arrangement.

( edit ) 'the affine connection' : I see what you were getting at LD. Yes. We like affine transformations so that points in a line with certain relative separations remain in line with the same relative displacements after the transform. Sensible transformations do this, including Minkowski/Lorentz stuff. A shear ( different scale factors applied to orthogonal directions ) is affine. Say the case of the length in the direction of motion Lorentz contracting but not in the perpendicular ones.

I have made this letter longer than usual because I lack the time to make it shorter. Blaise Pascal

## RE: RE: If the proof

)

or

Mathematics was constructed by the brain in order to be effective in this universe. :-)

There are some who can live without wild things and some who cannot. - Aldo Leopold

## Mathematics is music for the

)

Mathematics is music for the soul

Music is mathematics for the soul

(English saying)

Therefore they commute and form an Abelian algebra.

Tullio

## RE: Mathematics is music

)

Ahhhh, that'd be J.S.Bach then ! :-))

Yup, the reason why maths seems to be 'out there' is because that's the part of the information blizzard we have evolved to match to. Those that didn't do so well at that contributed to the genetic persistence of predators instead. I don't think it is a fluke that initial explanations in physics ( historically ) relate to everyday kinematical/dynamical stuff - position, time, velocity, rates of change .....

Imagine if you could talk to a sentient bacteria ? They'd do loops around us discussing chemistry and talk your leg off on the topics of Brownian motion, diffusion gradients and crystallography. If you explained quantum mechanics to them, they'd say ( with a knowing smile and a chuckle ) "Oh that. Now, errr .... what was you're problem there? ". :-) :-)

The real joke of "Why did the chicken cross the road?" isn't the specific retorts. It's the assumption that reasoning/intent applies at all. Suppose a chicken has neither a 'road' concept nor that of 'sides'. A road shaped pattern in the visual field doesn't trigger neurology that has no need for it *.

Cheers, Mike.

( edit ) * - until recently that is. :-) :-)

I have made this letter longer than usual because I lack the time to make it shorter. Blaise Pascal

## Another approach is the Peano

)

Another approach is the Peano axioms. You accept them and move on, with no deeper formal derivation being offered. But you wind up with what you want from arithmetic down the track. :-)

[pre]#1 - 1 is a natural number

#2 - To every natural number n there is another uniquely determined natural number, n', called the successor of n

#3 - 1 is not the successor of any natural number.

#4 - If two natural numbers have equal successors, they themselves are equal. Thus n' = m' implies n = m

#5 - If you have a set X consisting of some natural numbers with the properties that :

(i) k is in X, and

(ii) k is not the successor of any number in X, and

(iii) n' is in X whenever n is in X, then

the set X consists of k and all successors thereof[/pre]

Equality is meant in the sense of symbols aliasing for the same thing with reflexive ( m = m ), symmetric ( m = n n = m ) and transitive ( m = n and n = p => m = p ) properties.

Ordering then becomes an implicit consequence, so that one can proceed to unambiguously assign meaning to 'greater than' and 'less than' statements and symbology. Beware that 'greater than or equal to' ( ...etc ) is an umbrella term for two distinct cases which can't simultaneously hold - you just happen to have thrown a blanket/group label over them. The other annoyance is with inequalities and multiplying both sides by a constant : you have to reverse the inequality ( swap '>' and '<' ) if the constant is negative, and if the constant is zero as applied to a strict inequality then it doesn't follow ( zero isn't either less than or greater than zero ). This is especially relevant with purely symbolic variables like 'x', I have to consider the possible varieties of 'x' to avoid error. Anyhows 1 gets labelled as 'the first' and the Principle of Induction is locked in as #5 .....

The 'problem' with arithmetic is really zero coupled with the operation of multiplication. ANY number times zero is zero, thus the inverse ( division, or 'how many zeroes do you add together to get the number five' ) has no definition. Or put another way - if I tell you that the product of a non-zero number and zero equals zero, then you can't work back to uniquely tell me what the non-zero number must have been. It's not 'infinity', it's not more ( or less ) than any particular number you can state, it's just not a solvable problem. However, weaseling around this issue has lead to some superb stuff .... the limit process superimposed upon algebra being one path.

Cheers, Mike.

( edit ) And the other 'problems' - which number multiplied by itself gives negative one, and which ratio of integers gives the length of the longest sides of certain triangles - also have impressive solutions. :-) :-)

I have made this letter longer than usual because I lack the time to make it shorter. Blaise Pascal