new maths

[michel123456]

Mathematics look like naturally emergent from reality.

1+1=2, it is an evidence.

 

I just wanted to throw an idea where 1+1 is not exactly equal to 2.

 

The concept is the following:

When stating 1+1=2, we have made an operation, called sum, in which the operator (the + sign) although "operating", has no value. I mean the result (2) has no inscription of the past operations being done in order to obtain its value.

 

Trying to explain:

2 can be the result of a very simple calculation, or the result of an incredible 20 volumes 30000 pages calculations. There is no "memory" in number 2 showing if it has been picked randomly from the number line or if it is the result of an elaborated reasoning, showing how much energy needed to obtain the 2 result.

 

 

But that can be fixed.

The idea is to give a value to each operation and implement the result with an annotation that keeps track of history.

 

If the fundamental mathematical operation is sum, we could give to the + sign a fundamental value that should be noted with the result.

 

Something like 1+1=2(+), showing that number 2 is the result of a single summation.

 

Following this idea 1+1+1=3(++).

Of course there is a notation issue.

 

To make things even more complicated, one could state that each number could be taken only once from the line number.

For example 1+1 would not be allowed because number 1 has been taken twice (really, how is it allowed to duplicate numbers and use them at will?)

 

To make 1+1, we should first "create" another 1 by operating other numbers, for example 3-2=1(-)

 

Following the above:

1 +1(-)=2(+-) in order to keep track of the history.

Which is certainly wrong because how could number 3 & 4 exist in the first place? Number 3 could come from 1+2(+-)=3(++-) and number 4 from... damned...its getting complicated. (Edit: after reading my post I wonder if it even works! Maybe with a set of 2 numbers)

 

At the end, one could insert the concept that all the operating values (++++++++++++-------------) of a simple operation gets converted in regular number and added to the result as if the operators were an energetic value able under certain circumstances to be converted in the massive result depending on the field of calculation.

 

I wonder if it could be more complicated.

Edited April 22, 2011 by michel123456

[CaptainPanic]

Starting with the last question: sure it could be more complicated. You could make numbers personal (you get to use a different number 1 than me. You get 1(Michel123456), while I get 1(CaptainPanic)... then, if we also keep a history of all numbers we use, each number in the entire world would be unique.

 

It would certainly cause all computers to come to a grinding halt, because of all the extra operations they have to do :)

 

I wouldn't want to see the result of any of the models that I build anymore, which use multiple loops of iterations (millions of calculations). Even without your suggestion, my relatively fast computer needs several minutes to arrive at an outcome... if the result would include a history of all operations, I would go nuts... a calculation would take multiple days, and the result would have such a massive history that I would need a year to read it.

 

But if you write a program, it's possible to keep a summary of the history. So, if you want to do it, it's possible. I don't really see the point though.

Edited April 22, 2011 by CaptainPanic

[insane_alien]

but a value of 2 from 1+1 and a value of 2 from a 20 page calculation have the same value. the method of arriving there doesn't affect the actual value.

 

if i give you £1 by 50p +50p and a £1 by 20p+20p+20p+20p+20p then the pounds are equally valuable.

 

It would be possible to build a numbering system that includes the history but i fail to see what possible applications this could have.

[Bignose]

Considering the wide number of possible operations, I think that this would be incredibly unwieldy very quickly.

 

Also, how would you denote the addition of a negative number?

 

In my mind, a real beauty of math is that the different operations are equal to the same thing. [math]-e^{i \pi} = 0.5 + 0.25 + 0.125 + 0.125 = 3\int_0^1 x^2 dx[/math]. In your system, they wouldn't be exactly equal because of the different notation appended to the end.

Edited April 22, 2011 by Bignose

[Xittenn]

Mathematics look like naturally emergent from reality.

1+1=2, it is an evidence.

 

I just wanted to throw an idea where 1+1 is not exactly equal to 2.

 

The concept is the following:

When stating 1+1=2, we have made an operation, called sum, in which the operator (the + sign) although "operating", has no value. I mean the result (2) has no inscription of the past operations being done in order to obtain its value.

 

 

By fundamental definition two sets are equal when the elements are the same. This means that no matter how the sets were achieved if the sets are the same they are equal. I have only come across definitions of numbers that are themselves sets, so no matter how one gets to any given number, which is a set, if two instances of the number are the same then they are equal. Again this is fundamental, I don't see any argument to the contrary as this is an agreement on definition by a general majority(I'm terrible in stating proper arguments.)

 

 

Extensionality Axiom If two sets have exactly the same members, then they are equal:

 

[math] \forall A \forall B \left [ \forall x \left ( x \in A \leftrightarrow x \in B \right ) \rightarrow A = B \right ][/math]

 

Enderton, H. "Elements of Set Theory" Academic Press, 1977; Pg. 17

 

 

I would also like to point out that in establishing the addition operator there is more than one addition operator each functions over a different field. The prescribed notation in use is merely an abbreviation of the accepted and rigorous representation of the underlying fundamentals.

 

If you wish to create a history dependent solver notation system it is sufficient to say you wish to do so. If this is the case isn't it sufficient to simply write out the steps and store them as a whole, and wouldn't this in essence be considered the proper notation under these circumstances?

Edited April 23, 2011 by Xittenn

[DrRocket]

Mathematics look like naturally emergent from reality.

1+1=2, it is an evidence.

 

I just wanted to throw an idea where 1+1 is not exactly equal to 2.

 

The concept is the following:

When stating 1+1=2, we have made an operation, called sum, in which the operator (the + sign) although "operating", has no value. I mean the result (2) has no inscription of the past operations being done in order to obtain its value.

 

Trying to explain:

2 can be the result of a very simple calculation, or the result of an incredible 20 volumes 30000 pages calculations. There is no "memory" in number 2 showing if it has been picked randomly from the number line or if it is the result of an elaborated reasoning, showing how much energy needed to obtain the 2 result.

 

 

But that can be fixed.

The idea is to give a value to each operation and implement the result with an annotation that keeps track of history.

 

If the fundamental mathematical operation is sum, we could give to the + sign a fundamental value that should be noted with the result.

 

Something like 1+1=2(+), showing that number 2 is the result of a single summation.

 

Following this idea 1+1+1=3(++).

Of course there is a notation issue.

 

To make things even more complicated, one could state that each number could be taken only once from the line number.

For example 1+1 would not be allowed because number 1 has been taken twice (really, how is it allowed to duplicate numbers and use them at will?)

 

To make 1+1, we should first "create" another 1 by operating other numbers, for example 3-2=1(-)

 

Following the above:

1 +1(-)=2(+-) in order to keep track of the history.

Which is certainly wrong because how could number 3 & 4 exist in the first place? Number 3 could come from 1+2(+-)=3(++-) and number 4 from... damned...its getting complicated. (Edit: after reading my post I wonder if it even works! Maybe with a set of 2 numbers)

 

At the end, one could insert the concept that all the operating values (++++++++++++-------------) of a simple operation gets converted in regular number and added to the result as if the operators were an energetic value able under certain circumstances to be converted in the massive result depending on the field of calculation.

 

I wonder if it could be more complicated.

 

Believe it or not 1 + 1 =2 is essentially a definition of 2. You need to read a good treatment of the development of arithmetic and number systems, through the complex numbers, starting from the Peano Axioms. A good sourrce is Foundations of Analysis by Landau. Another is Principles of Mathematical Analysis by Rudin.

 

Starting from the natural numbers, one defines addition, subtraction, multiplication and division, along the way constructing the integers, rationals, reals and finally the complex numbers.

[michel123456]

All remarks are fine.

 

then as it looks out, mathematics are not emergent. Mathematics are a construct where humans define the fundamental operations. which means that at any time one can build another system based on other fundamentals.

 

Another example is the = sign, which is called "equal", but IMHO should be called "equivalent".

 

Because by definition one thing cannot be "equal" to anything else than itself.

 

"equivalent" should mean that you can replace one side of the = by the other without further bothering (that's what mathematicians do), but not that both sides are "the same" by nature.

 

For example 1+1 = 2 should mean "you can use 1+1 instead of 2 in further mathematical operation", but

1+1 is a dynamic organization where "someone use an operator"

while

2 is a static result

 

Which is different from

 

2=1+1

 

where 2 is a static entity made of divisible parts

while

1+1 is the result of a complex dynamic analysis of the entity distributed in parts that one collates together in order to get an exact equivalent result, based on the concept that the original entity is indeed divisible in smaller parts. The premise being that operations read from left to right like western conventional text)

 

All of which are not reflected in the commutative 1+1=2

 

In your system, they wouldn't be exactly equal because of the different notation appended to the end.

 

Exactly.

Edited April 23, 2011 by michel123456

[Bignose]

biggest problem with this... how to solve for unknowns:

 

[math] 14 - x = 9 + 2 [/math] You got a - on the left hand side, and a plus on the RHS. What value of x solves this?

[DrRocket]

All remarks are fine.

 

then as it looks out, mathematics are not emergent. Mathematics are a construct where humans define the fundamental operations. which means that at any time one can build another system based on other fundamentals.

 

Another example is the = sign, which is called "equal", but IMHO should be called "equivalent".

 

Because by definition one thing cannot be "equal" to anything else than itself.

 

"equivalent" should mean that you can replace one side of the = by the other without further bothering (that's what mathematicians do), but not that both sides are "the same" by nature.

 

For example 1+1 = 2 should mean "you can use 1+1 instead of 2 in further mathematical operation", but

1+1 is a dynamic organization where "someone use an operator"

while

2 is a static result

 

Which is different from

 

2=1+1

 

where 2 is a static entity made of divisible parts

while

1+1 is the result of a complex dynamic analysis of the entity distributed in parts that one collates together in order to get an exact equivalent result, based on the concept that the original entity is indeed divisible in smaller parts. The premise being that operations read from left to right like western conventional text)

 

All of which are not reflected in the commutative 1+1=2

 

 

 

Exactly.

 

In mathematics "=" means "is".

 

Equivalence is an entirely different concept, but it is quite possible for two equivalence classes to be equal, and quite often mathematicians talk of equality when what is implicitly understood is that what is equal are equivalence classes. If one does not adopt this point of view the discussion quickly becomes pedantic.

[Xittenn]

What does the OP propose as a reasoning for such a review of the standard system? Is this an exercise in applications of anholonomic systems? Is this just an interesting observation?

[michel123456]

Believe it or not 1 + 1 =2 is essentially a definition of 2. You need to read a good treatment of the development of arithmetic and number systems, through the complex numbers, starting from the Peano Axioms. A good sourrce is Foundations of Analysis by Landau. Another is Principles of Mathematical Analysis by Rudin.

 

Starting from the natural numbers, one defines addition, subtraction, multiplication and division, along the way constructing the integers, rationals, reals and finally the complex numbers.

 

That was very profound.

 

After a night of thinking, I realised my system is not self organized.

I mean even number 1 may be the result of operations.

0,5+0,5=1 for example.

And 0,5 is also the result of operations, ad infinitum.

So the system must be set on a decided basis (say basis 1) in order to initiate.

 

 

it is even more complicated than I thought.

 

What does the OP propose as a reasoning for such a review of the standard system? Is this an exercise in applications of anholonomic systems? Is this just an interesting observation?

 

You are more advanced than me to answer. Thank you for the nonholonomics.