And I thought I was a geek.

 

In Math 1+1 = exactly 2.

In Arithmetic (counting) one thing and another classified as similar gives us 2 things.

Arithmetic is part of Mathematics. And I do not think they can be totally isolated from each other as Mathematics started with Arithmetic.

Words are important.

Yes words are important. Everything is relatively defined.

Consider this phrase: "Before there was time"

Does it have any meaning for you?

I would reverse the question:- How can "before" be used if there was no time? "Before" is associated with time. Using "before" somehow sounds like "there was time before time". Can one understand that?

I did understand your ideology of 1 + 1 is not 2 as such, but again, to my view, this thread has run its course.

Thanks.

 

I respectfully disagree. 1 Fuji plus 1 Granny Smith equals 2 apples. Only the quantity or magnitude need be identical, and 1 = 1 per the reflexive property.

Ok so what if I have 1 Granny Smith Apple and you give me 1 Granny Smith Apple from the same branch off the same tree. Then I have 2 Granny Smith Apples.

So... 1+1=2

 

So now, provide us your definition of 1+1=2

Well Stu, over to you.

I would observe that the phrase '1+1=2' can refer to different processes, only one of which is represented in the apples and oranges examples.

Firstly we have the apples and oranges process where two distinct objects are added together and paced in a container that now has 2 objects in it. The objects are still distinct and can be separated again.

Then we have the second type where two objects are added and coalesce to form a single object, and cannot then be separated again. For example pour a glass of water into a jug. Now pour in a second glass of water.

Both these processes can be formalised in set theory.

Note I said processes. Your statement 1+1=2 refers to a process on the left and a quantity on the right. Clearly 'the process' cannot be identical to the 'quantity'. It is the result of the process that is in some way 'equal'.

Mathematicians are very careful to make sure that Mathematics accords with Natural Philosophy in matters like these. Hence my comments about equivalence classes.

To avoid the uniqueness paradox I outlined earlier, your argument would be better phrased 1+2=3, where all the players are distinct.

You should also beware the trap

A times Zero = Zero

used to negate or nullify any statement A in logical arguments.

Your question has deeper significance than perhaps you realised; you should be prepared to venture where the logic takes you as a result.

 

Well Stu, over to you.

I would observe that the phrase '1+1=2' can refer to different processes, only one of which is represented in the apples and oranges examples.

Firstly we have the apples and oranges process where two distinct objects are added together and paced in a container that now has 2 objects in it. The objects are still distinct and can be separated again.

Then we have the second type where two objects are added and coalesce to form a single object, and cannot then be separated again. For example pour a glass of water into a jug. Now pour in a second glass of water.

Both these processes can be formalised in set theory.

Note I said processes. Your statement 1+1=2 refers to a process on the left and a quantity on the right. Clearly 'the process' cannot be identical to the 'quantity'. It is the result of the process that is in some way 'equal'.

Mathematicians are very careful to make sure that Mathematics accords with Natural Philosophy in matters like these. Hence my comments about equivalence classes.

To avoid the uniqueness paradox I outlined earlier, your argument would be better phrased 1+2=3, where all the players are distinct.

You should also beware the trap

A times Zero = Zero

used to negate or nullify any statement A in logical arguments.

Your question has deeper significance than perhaps you realised; you should be prepared to venture where the logic takes you as a result.

We are on the same page.

My position was to show that,something as 'simple' as 1+1 =2 ain't necessarily so.

More importantly Mathematics is a language that may not be taken for granted.

Consider:

or equivalently,

If we define i in this way and then assume that it can be manipulated as if it were an unknown ("imagined") variable, then it follows from straightforward algebra that the second solution to the above quadratic equations is − i. It is important to realize that, although we call this construction "imaginary," and although the concept of an imaginary number is intuitively more difficult to grasp than that of a real number, the construction is perfectly valid from a mathematical standpoint.


We have a departure point between an imaginary language, Mathematics, and the real world.



BoyntonStu

P.S.

Several of my new threads have been censored by deletion.

Has this happened to you?

 

Several of my new threads have been censored by deletion.

Can't help there I'm afraid, try a PM to Dave.

Your only threads I've noticed have been one or was it two excellent puzzles.

There are lots of areas of Mathematics with no known realisation in the physical world. That trend is growing rapidly as modern maths expands.
In other threads here I have demonstrated there are also known relationships in physics, expressly forbidden by modern mathematics.

So yes, as the Bard said,

'there are more things in Heaven and Earth, Horatio, than Man has ever dreamed'

 

If we define i in this way and then assume that it can be manipulated as if it were an unknown ("imagined") variable, then it follows from straightforward algebra that the second solution to the above quadratic equations is − i. It is important to realize that, although we call this construction "imaginary," and although the concept of an imaginary number is intuitively more difficult to grasp than that of a real number, the construction is perfectly valid from a mathematical standpoint.

Several imaginary entities are REQUIRED in Real World calculations regarding properties of AC circuits. The measured results match predicted results when using imaginary numbers in equations. The same results cannot be accurately predicted without imaginary numbers. \(i=sqrt{-1}\) is the definition, and implies use of the Set containing imaginary numbers.

There are other multidimensional sets that cannot be pictured (a 5 dimension array), but are used in "everyday" calculations, giving predictable results. If 1 were not equal to 1, the entire number system and math theory which science is built on would not function. In such a case, each time a multidimensional system was calculated, the result would not be the same.

Once the set is explicitly defined, 1=1 in all cases WITHIN THAT SET. Where this is not true is when the set is undefined, or, as in your examples above, implied.

Real Numbers:
\(1 \in\mathbb{R}\)
\(\pi \in\mathbb{R}\)

Integers ("Counting Numbers"):
\(1 \in \mathbb{Z}\)
\(\pi \notin \mathbb{Z}\)

In the latter case above, ∏ could NOT be used in an equation, as it is not a counting number. If it were, we would count similar to 1,√2,2,e,3,∏,.,..

If defining the Set of Counting Numbers with the list above. The definitions of mathematical operators need to be defined as well. For the above, +1 MUST "increment set 'index'". Otherwise, 3-1 would not give e, and 2+1 would not give e.

 

The system of real numbers is not complete.
It does not contain solutions to problems like x\(^{2}\)+1=0

However, just as we remove the parallel lines axiom from euclidian geometry to move on to more general geometry,
we can remove the ordering criterion from numbering systems to move on to more general number systems.
Such a system which includes solutions to all agebraic equations are called complex numbers.
We have had to give up the ability to say that one number is greater (or less) than another to achieve this.

 

That's the complex nature of Mathematics, and it's very ingenious to have found such a way of interpreting the physical world.
Also bearing in mind that in many cases there are "assumptions" and "approximations" so as to help simplify concepts and they somehow appear to be working relatively accurate.

Thanks.

 

Several imaginary entities are REQUIRED in Real World calculations regarding properties of AC circuits. The measured results match predicted results when using imaginary numbers in equations. The same results cannot be accurately predicted without imaginary numbers. \(i=sqrt{-1}\) is the definition, and implies use of the Set containing imaginary numbers.

There are other multidimensional sets that cannot be pictured (a 5 dimension array), but are used in "everyday" calculations, giving predictable results. If 1 were not equal to 1, the entire number system and math theory which science is built on would not function. In such a case, each time a multidimensional system was calculated, the result would not be the same.

Once the set is explicitly defined, 1=1 in all cases WITHIN THAT SET. Where this is not true is when the set is undefined, or, as in your examples above, implied.

Real Numbers:
\(1 \in\mathbb{R}\)
\(\pi \in\mathbb{R}\)

Integers ("Counting Numbers"):
\(1 \in \mathbb{Z}\)
\(\pi \notin \mathbb{Z}\)

In the latter case above, ∏ could NOT be used in an equation, as it is not a counting number. If it were, we would count similar to 1,√2,2,e,3,∏,.,..

If defining the Set of Counting Numbers with the list above. The definitions of mathematical operators need to be defined as well. For the above, +1 MUST "increment set 'index'". Otherwise, 3-1 would not give e, and 2+1 would not give e.

Great post!

IOW We must first define a set before we use Math.

Most folks believe that the set is assumed and that no definition is required.

What would be the square root of a minus 1 orange?

An orange = '1' in an undefined set. (exactly = 1 is assumed)

 

Great post!

IOW We must first define a set before we use Math.

Yes. We define maths and we set the limits and rules.

Cheers.

 

Mathematics is a language and not a Science.

Rather depends upon how you define Language and Science? Yet again why cannot it belong to a third class entirely? Or even partly to Science and partly to Language?

In Math, infinite decimal point accuracy is implicit.

Nevertheless it is a fairly elementary proof in number theory to show that no matter how many decimal places you take there are real numbers you cannot represent in decimal (or any other base) notation.

By the way what do you understand by real number, since you have cocked a snoot at imaginary ones?

Consider:

or equivalently,
If we define i in this way and then assume that it can be manipulated as if it were an unknown ("imagined") variable, then it follows from straightforward algebra that the second solution to the above quadratic equations is − i. It is important to realize that, although we call this construction "imaginary," and although the concept of an imaginary number is intuitively more difficult to grasp than that of a real number, the construction is perfectly valid from a mathematical standpoint.


We have a departure point between an imaginary language, Mathematics, and the real world.

If, if, if.

But I don't define it this way.

It is not necessary (and some would say confusing) to introduce the magic i j or k to perform analysis with a complex variable.

The equal sign, equals sign, or "=" is a mathematical symbol used to indicate equality. It was invented in 1557 by Welshman Robert Recorde. The equals sign is placed between the things stated to be exactly the same, as in an equation.


Who determines what is the 'correct' way to read definition of the equal sign?

Actually I disagree. The equals sign represents two different ideas, which is why it is eschewed by some authors.

One meaning refers to the the RHS being the result of a process given in the LHS. Some authors suggest substituting a rightward pointing arrow for this meaning.
Many of the examples in this thread use this meaning.

The other meaning is the one you are referring to.

However mathematics, like other disciplines and language has moved on since the days of ye goode Sir Robert.

We now distinguish the difference between denoting members of the same equivalence class and actual identity. The correct symbol for identity is \(\equiv\), as I mentioned before.

You should realise that questions such as you are asking have been asked and resolved before. They lead to strain in the mathematical or thought system of their day and in turn to greater generalisation. You will never solve/answer them within a restricted thought system rooted in thought from several centuries ago.

 

Rather depends upon how you define Language and Science? Yet again why cannot it belong to a third class entirely? Or even partly to Science and partly to Language?



Nevertheless it is a fairly elementary proof in number theory to show that no matter how many decimal places you take there are real numbers you cannot represent in decimal (or any other base) notation.

By the way what do you understand by real number, since you have cocked a snoot at imaginary ones?




If, if, if.

But I don't define it this way.

It is not necessary (and some would say confusing) to introduce the magic i j or k to perform analysis with a complex variable.





Actually I disagree. The equals sign represents two different ideas, which is why it is eschewed by some authors.

One meaning refers to the the RHS being the result of a process given in the LHS. Some authors suggest substituting a rightward pointing arrow for this meaning.
Many of the examples in this thread use this meaning.

The other meaning is the one you are referring to.

However mathematics, like other disciplines and language has moved on since the days of ye goode Sir Robert.

We now distinguish the difference between denoting members of the same equivalence class and actual identity. The correct symbol for identity is \(\equiv\), as I mentioned before.

You should realise that questions such as you are asking have been asked and resolved before. They lead to strain in the mathematical or thought system of their day and in turn to greater generalisation. You will never solve/answer them within a restricted thought system rooted in thought from several centuries ago.

What does 1 + 1

2 mean in Nature?

Please give us an example.

BoyntonStu

 

What does 1 + 1 2 mean in Nature?

What do you mean by Nature?

You are now getting into the realms of philosophy about the meaning and validity of existence.

By the way why do you expect me to answer your question when you have left a trail of mine unanswered through this thread, including one which asked what you meant by the statement 1+1 = 2 or now 1+1 \(\equiv\) 2?
without your definitions to this last one it is impossible to answer you.

 

Great post!

IOW We must first define a set before we use Math.

Not always. In most cases, assuming "Real Numbers" is sufficient, and is the method taught from kindergarten through high school.

Most folks believe that the set is assumed and that no definition is required.

What would be the square root of a minus 1 orange?

An orange = '1' in an undefined set. (exactly = 1 is assumed)

What is "orange"? Define it as the set of fruits the are from a citrus tree? If so, it doesn't matter if they are ripe, rotten, or green, having two of such items states you have "2 oranges".

Some would define "orange" as only a ripe orange, so if one had a rotten orange and a ripe orange, they would only have "1 orange", since the rotten orange isn't in the working set.

 

In the English Parliament when they have a debate and then vote on a proposition, for or against. These are called Ayes and Noes.

In this debate the Noes would do well to stop offering examples using concrete nouns eg apples and oranges. The refutation is easier with abstract nouns, which can be manipulated to be identical.

Does the dollar exist?

How about half a dollar?

How about: one half-dollar plus one half-dollar = 2 half-dollars = one dollar ?

 

The idea of 1+1 not equal to 2 is not pure mathematics, it's philosophy of mathematics.

Cheers.

 

In the English Parliament when they have a debate and then vote on a proposition, for or against. These are called Ayes and Noes.

In this debate the Noes would do well to stop offering examples using concrete nouns eg apples and oranges. The refutation is easier with abstract nouns, which can be manipulated to be identical.

Does the dollar exist?

How about half a dollar?

How about: one half-dollar plus one half-dollar = 2 half-dollars = one dollar ?

What is the definition of a dollar?

Is there a perfect dollar?

BoyntonStu

 

Any dollar a vendor will accept is perfect enough.

If I charge you fifty cents for that Mandarin orange with the math on it, and you give me a dollar, you are going to expect fifty cents change! Or perhaps I can keep the change myself, since your dollar is less than perfect?

 

What does 1 + 1

2 mean in Nature?

Please give us an example.

BoyntonStu

I already gave you the example, but you have just ignored it. Why do you ask again? Perhaps just so you can ignore it again?