No, John no hidden agendas, no Baker's Dozens (do you have those in America?).

I was simply trying to classify things we can think of into 5 levels.

1) Those which do exist and therefore can exist.

2) Those which don't exist but we can see no reason within our rules why they couldn't.

3) Those which could exist with a specified change to our rules (including a completely new set)

4) Those which we can specifiy but break one or more of our rules to such an extent they couldn't be accomodated by level 3.

The example 3 +3 = 7 was meant to be an example of level 4.

5) Those where we specify by making a nonsensical statement (gibberish) and claiming some meaning for the statement.

I did this quickly and tried to show examples of each. My apologies to one and all if an example was flawed - but we should not loose the message because of this, just change the example.

 

You have missed the wood for the trees; I am trying to help your cause, not hinder it.

The important point is not what we call the function that the series converges to or what it might be similar to.
The point is that whatever we call it the 'function' is not continuous. It is only 'piecewise continuous' ; i.e. continuous on a series of alternate open and compact intervals.
It is easy to apply an epsilon-delta argument to the junction of these intervals, where f(x) may converge to zero in one of the adjacent intervals, but definitely something else in the neighbourhood of the junction in the other.

Actually I just noticed over my morning coffee that, although each of the terms of the saw tooth or square wave Fourier series are differentiable at a ‘discontinuity’ the series of derivatives diverges rather rapidly, so the Fourier form of the function is not differentiable at a discontinuity. I guess it helps to do the math.

Your point is well taken. The Fourier function is indeed not a member of the field that created it.

Well, on my original question, I guess I have an answer, but it is not as earth shattering as I had hoped. While I was not looking to unveil anything along the lines of the “Medusa Concept” I was hoping to reveal more than this.

But it does bring up interesting notions about the validity of abstract algebra, indeed all math. I guess Godel was right, we’re playing with broken toys.

 

1) 3 + 3 = 7

Only Ratch is allowed to split hairs and change other persons' definitions.

By the usual rules I simply meant that
3 and 7 are numbers (integers, rationals or reals as usually defined.)
If you wish to use another definition than 3 = 1+1+1 and 7 = 1+1+1+1+1+1+1, please state it.
That the operation + is defined in the normal way.
That the relation = is defined in the normal way.

I am comfortable with the four basic axioms of group theory so labelling your list G1 through G4

I note that if 3 is in your set, 7 must be in your set (G1)
thus -3 and -7 must be in the set (G3)
zero must be in the set (G2)
the result of 3 + 7 must be in the set (G1)
The result of 3 + 3 + 3 + 3 + 3......... must be in the set (repeated application of G1)




So construct your group, showing all this

Does this work?

The set {-7, -3, 0, 3, 7} and the operator + where:

+ -7 -3 0 3 7

-7 3 7 -7 -3 0
-3 7 -7 -3 0 3
0 -7 -3 0 3 7
3 -3 0 3 7 -7
7 0 3 7 -7 -3


I know, it's a bit of a stretch, but this is all just fun and games...
(Sorry about the table, can't seem to make that work)

 

No, John no hidden agendas, no Baker's Dozens (do you have those in America?).

The concept exists in Canada, but just try to get soemone to deliver on it.

 

No, John no hidden agendas, no Baker's Dozens (do you have those in America?).

I was simply trying to classify things we can think of into 5 levels.

1) Those which do exist and therefore can exist.

2) Those which don't exist but we can see no reason within our rules why they couldn't.

3) Those which could exist with a specified change to our rules (including a completely new set)

4) Those which we can specifiy but break one or more of our rules to such an extent they couldn't be accomodated by level 3.

The example 3 +3 = 7 was meant to be an example of level 4.

5) Those where we specify by making a nonsensical statement (gibberish) and claiming some meaning for the statement.

I did this quickly and tried to show examples of each. My apologies to one and all if an example was flawed - but we should not loose the message because of this, just change the example.

All 5 of these seem reasonable. 2) and 3) are the ones that usually drive our understanding of the universe forward. With 4), I'd think most of these could be accomdated in the 'theortetical only' or 'conceptual' realm provided the stucture created to accomodate them was fairly closed and self validating.

 

There in lays the problem - using all the usual rules. Look up groups and rings in abstract algebra. All you need o do is define a group where the binary operator is named addition such that 3+3=7 is valid and logical. A group in algebra is a set of elements and a binary operator such that:

1)The operator acting on two elements of the set results in an element of the set. So it is completely defined, and closed over the set.

2) One element of the set is an identity element. So if we call our operation #, there exists an element of the set e such that for any other element of the set x, e # x = x # e = x.

3) Every element of the set has an inverse element. If we take any element of the set p, there is another element q such that p # q = q # p = e.

2) The operation is associative. For any three elements of the set, (a # b) # c always equals a # (b # c).

So as a familiar example, [bold]I[/bold] is a group over [bold]normal[/bold] addition. The net of all this is that you can easily define a group where 3+3=7

I understand the concepts of groups and rings and such pretty well, having taken a year-long sequence of courses during grad school. It is true that a group can be defined any way that you want, so long as you specify the elements in the group and the operation that is defined on the group elements. The usual practice for an arbitrary group is to identify elements of the group by letters so that you don't have contradictory equations such as 3 + 3 = 6 (where 3 and 6 are in Z, the integers under addition) and 3 + 3 = 7 (where 3 and 7 are in this oddball group).

 

Does this work?

It's certainly self consistent and consistent with the rules

the result of successively adding 3s cycles round nicely.

3+3=7
7+3=-7
-7+3=-3
-3+3=0
0+3=3

etc.

Well done in constructing this group.

However the symbols 3, 7 and their inverses could be anything at all. You are not using the 'threeness' property or the 'sevenness' property any more than the 'addition' is normal addition, rather than 'operation specified by the table'

Can you also do this magic with [1+1+1} + {1+1+1} = {1+1+1+1+1+1+1} ?

I make the LHS 0 and the RHS 1 with a similar table, using -1, 0, +1

 

However the symbols 3, 7 and their inverses could be anything at all. You are not using the 'threeness' property or the 'sevenness' property any more than the 'addition' is normal addition, rather than 'operation specified by the table'

Quite true, I actually just worked it out with {-b, -a, 0, a, b} and relabelled a to 3 and b to 7. Like I said, a bit of a stretch.

Can you also do this magic with [1+1+1} + {1+1+1} = {1+1+1+1+1+1+1} ?

I make the LHS 0 and the RHS 1 with a similar table, using -1, 0, +1

I don’t think I can, not without making ‘+’ a null operator (i.e. a+b=0, a+a=0, a+0=0) which is really stretching your original intent.

Edit: Just noticed, this breaks rule 2.

 

Here is a colour mixing chart, based upon the same group table.

It has no more basis in fact than 3+3=7.


............ Red.... Blue... Green. Yellow. Black
Red....... Yellow. Black. Red.... Blue... Green
Blue...... Black... Red... Blue... Green.. Yellow
Green.... Red.... Blue... Green. Yellow. Black
Yellow... Blue... Green. Yellow. Black... Red
Black.... Green. Yellow. Black.. Red..... Blue

 

I don't think it's the same.

It might be better if it was -Back, -Yellow, Green, Yellow, Black.

If you look a the group I came up with and relabelled the elements {-2, -1, 0, 1, 2}, or a modulo 3 subgroup of Z, then the operator '+' is just the plain ordinary old addition we all kknow and love.

What you have come up with here is case of your catagory 5.

 

Perhaps I should adda sixth category

That which exists, despite our rules, and we cannot explain or resolve.

Since you introduced set theory, have you ever come across Russel's Paradox?

 

I don't think it's the same.

I constructed it to be the same, although I doubt any artist would thank me for it.

It has an operation {colour} operation {colour} makes colour

It has an identity (green)

It has an operation inverse for each colour, green being its own inverse

The table is a direct copy of yours.

There is nothing to say that a negative is required for the inverse, only that if p is a member then there is another member q such that p # q = identity. This condition is met, for instance the inverse of blue is yellow.

 

Perhaps I should adda sixth category

That which exists, despite our rules, and we cannot explain or resolve.

Since you introduced set theory, have you ever come across Russel's Paradox?

Maybe, isn't it just a Russelized version of the Liar Paradox?

 

I constructed it to be the same, although I doubt any artist would thank me for it.

It has an operation {colour} operation {colour} makes colour

It has an identity (green)

It has an operation inverse for each colour, green being its own inverse

The table is a direct copy of yours.

There is nothing to say that a negative is required for the inverse, only that if p is a member then there is another member q such that p # q = identity. This condition is met, for instance the inverse of blue is yellow.

Okay, I get your meaning and agree. It seems there is a lot to the interpretation of this seemingly simple group.

 

Maybe, isn't it just a Russelized version of the Liar Paradox?

No because the liar paradox is a way of asking two questions in one - compound question.

I think Wiki spells it out better than I can, but i am not convinced about the standard answer. After more than 100 years I think it is still a bit of a cop out.

http://en.wikipedia.org/wiki/Russell's_paradox

 

I am not convinced about the standard answer. After more than 100 years I think it is still a bit of a cop out.

You mean ZFC which solves the paradox by not letting you build the paradoxical set in the first place?

Why does this bother you? To me it seems like a valid if not aesthetically pleasing course of action. A sort of ‘constant improvement’ process where problems are eliminated by altering the process to avoid them.

But it does read as a bit of smack. Kind of like this conversation:

Student: “Professor, is the universe bounded or infinite?”
Professor: “Bounded, otherwise it would be inconsistent with observation.”
Student: “What is it bounded within?”
Professor: “That is an inappropriate question.”

 

I understand the concepts of groups and rings and such pretty well, having taken a year-long sequence of courses during grad school. It is true that a group can be defined any way that you want, so long as you specify the elements in the group and the operation that is defined on the group elements. The usual practice for an arbitrary group is to identify elements of the group by letters so that you don't have contradictory equations such as 3 + 3 = 6 (where 3 and 6 are in Z, the integers under addition) and 3 + 3 = 7 (where 3 and 7 are in this oddball group).

Right, a little slight of hand to show that 3+3=7 is not always an illogical statement. Not that it could mean much either.

All in the pursuit of mental exercise.

 

Why does this bother you?

Because it is like the conversation you mention, or this one

Mummy, are faries pink or blue?

Don't be silly dear, faries don't exist.

But, Mummy, if they did exist what colour would they be?

We could ask the same question:

If the set did exist would it be normal or non-normal?
So defining it out of existance won't wash.

It is also interesting to note the fate of the companion question:

Is the set of all non-normal sets normal or non-normal?

I see that not nearly so much fuss is made of this one. Might this be because you can 'prove' whichever you assume?

Incidentally we don't have to go to such tortuous logic to find normal sets. For instance the set of all objects within a given radius, say an inch or a mile, is itself an object within that radius and so must be a member of itself.

 

Incidentally we don't have to go to such tortuous logic to find normal sets. For instance the set of all objects within a given radius, say an inch or a mile, is itself an object within that radius and so must be a member of itself.

Very true. Actually the set Russell squirms over is a valid set too and can exist. This brings up an interesting point. Most of these logical paradoxes are the creations we need to be wary of. To me they are all in the organization of the words.

For instance, the same set that caused Russell such pain can be described “A set containing itself and all sets that do not contain them selves”.

One can always find words to create an illogical statement whether it’s in the form of a paradox or not.

One electronics analogy would be to feed the output of an inverter directly into it’s input.

 

One can always find words to create an illogical statement whether it’s in the form of a paradox or not.

One electronics analogy would be to feed the output of an inverter directly into it’s input.

All this goes to show that we are not as smart as we like to think we are (is that a song somewhere) and that Nature is smarter.

Mathematicians don't like to admit that we can't develop formulae for telling what colour pixels will be for certain cellular automata.

If you take a video camera and film its monitor screen you can generate a Chaos system. Similarly your positive feedback above can produce a chaotic output.

Those who state unequivocally that 'everything follows the known laws of physics' are as wrong as Lord Kelvin was in the late 1800s when he calculated the age of the earth to be 2500 years - according to the laws of physics.

It is a fallacy to assume that observed phenomena must be in accord with known laws even if they also obey yet uknown ones. Mostly they do, but Nature is always catching out that over arrogant animal, Man.