A vending machine in Cincinnati holds a delicious snack costing two dollars. If one inserts an American dollar bill and four quarters, one each commemorating Georgia, Illinois, Oregon, and Florida, how much money has one inserted? Does one have grounds to complain if no snack and no money are forthcoming from the machine?

If I have two non-identical objects, how many objects do I have? If I dispose of one of the objects, how many remain?

 

Outside of "pure math", there are units attached to each number, and their "value" is known to a great accuracy.

If this were not true, we would have been unable to orbit an object 1/4 the size of earth, which was moving 2,236 mph @ distance of 233,000 miles relative to earth, let alone land on it and come back.

 

Has any Scientist ever measured a "1"?

Equivalence is not identical.

I didn't say that equivalence implies identity.

If you really mean identity, please use the correct triple bar symbol.
The symbol you used, which is mathematical, not (english) language merely states that two members of the same equivalence class, called1, can be added to obtain a 2.

If you don't understand about equivalence classes look them up.

Similarly you need to be careful of your use of the words 'there is'; a loose language version of the more precise mathematical statement 'there exists'

In strict language terms the statement 'there is' only allows the present. The past and future are inadmissable so your comments about multiple suns are inconsistant.

 

For example, is the Sun defined for an instant in time or would it be the same Sun 1 microsecond later?

Are there exactly '2' of any of them?

"You can never step into the same river twice" - Heraclitus

You may want to avoid Zen Meditations. The battle of the philosophers against the scientists was already fought, the scientists won.

 

The examples are in response to your question whether scientists have ever measured "one", so these measurements would be of relatively short duration (static).

The example of the sun has been well defined for about 4.5 billion years and is expected to be true for the next 4 billion years. Even though the sun changes over time, it still fits into our category/definition for "sun" for a very long time.

These examples could conceivably have "two" of them also. There are "binary" star solar systems and planets with "two" moons. There are "bi"-modal optical fibers and there is the helium atom with "two" protons in the nucleus. Also, some Japanese scientists detected "two" nuclear explosions in their country at the end of WWII.

Basically, I reject your notion that things must be identical to be counted or added. Categorization of things into a fundamental class/unit is sufficient to allow us to count and add in the real world. Even the abstract mathematics of counting and addition finds roots in set theory which is fundamentally similar to the scientific approach.

"Basically, I reject your notion that things must be identical to be counted or added."

Fine, I have an orange in my left hand and an orange in my right hand.
Picture it?

OK

I have a Mathematical "one" on the left of an equal sign and a "one" on the right side of the equal sign.

I grant you that in Math, you may exchange the "one's".

In fact, you may subtract one "one" from the other "one" and what is left is zero. Nothing, nada.

Back to the oranges:

Instead of an = sign, substitute a very delicate balance scale.

For the mathematical "one's", even the most sensitive balance is not material, because the scale will always balance.

When you can exchange what you have in your left hand for what you have in your right hand and there is no way that you could detect the difference, you will have a pair of "one's".

Until that happens, 1 +1 never equals 2 in Nature.

Another way is to add the decimal accuracy positions.

Can you find anything and put it on a scale, and then add another that will exactly double the first weight?

If you cannot, do you have 2?

I mean 2.000000000000000000 times the first weight.


In Math, infinite decimal point accuracy is implicit.

In Nature, it matters.


BoyntonStu

 

If I add a po-TAY-to to a po-TAH-to, how much Stu can I make?

 

In Math, infinite decimal point accuracy is implicit.

In Nature, it matters.

Not all mathematics or science deals with a continuum. Discrete mathematics is just as valid as continuous mathematics. When we classify objects into categories, we can have precise integer units of measure. We can count the number of photons hitting a detector precisely. We can count the number of grains of sand in our hand. We can count the number of atoms in a crystal. We can count the number of fingers on our hands etc .....

In the case of scientific measurement of quantities which can not be resolved down to fundamental discrete units, there is always an implied accuracy which goes with the measurement. So, the measurement is not really a single value, but a range is used to express the likely value and the amount of possible error. Here we do not just add the numbers, but instead consider that the total also has a new likely value and a new range. This can be done is a simple-minded way, or sophisticated probability/statistical theory can be used.

In the case of the oranges in your left and right hand. You can definitely say that you are holding two oranges. However, the weight of the oranges can not be measured precisely, so you can only estimate the total weight, and provide clear upper and lower bounds on the value.

 

Continuous or Discrete measurement.

Analogue or Digital measurement.

 

Not all mathematics or science deals with a continuum. Discrete mathematics is just as valid as continuous mathematics. When we classify objects into categories, we can have precise integer units of measure. We can count the number of photons hitting a detector precisely. We can count the number of grains of sand in our hand. We can count the number of atoms in a crystal. We can count the number of fingers on our hands etc .....

In the case of scientific measurement of quantities which can not be resolved down to fundamental discrete units, there is always an implied accuracy which goes with the measurement. So, the measurement is not really a single value, but a range is used to express the likely value and the amount of possible error. Here we do not just add the numbers, but instead consider that the total also has a new likely value and a new range. This can be done is a simple-minded way, or sophisticated probability/statistical theory can be used.

In the case of the oranges in your left and right hand. You can definitely say that you are holding two oranges. However, the weight of the oranges can not be measured precisely, so you can only estimate the total weight, and provide clear upper and lower bounds on the value.

In the case of the oranges in your left and right hand. You can definitely say that you are holding two oranges. However, the weight of the oranges can not be measured precisely, so you can only estimate the total weight, and provide clear upper and lower bounds on the value.

I agree that you have orange A and orange B.

In the practical world, you have 2 oranges.

My thread was designed to delineate between the practical and the theoretical.

When I was 10 years old, I looked through a microscope and I saw the clear differences between 2 straight pins.


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.

The 2 oranges are not exactly the same and therefore 1 +1 =2 does not fit the orange situation.

BoyntonStu

 

Until that happens, 1 +1 never equals 2 in Nature.

BoyntonStu

It depends on how much importance you give something. To what extent something is replaceable.

In nature everything is unique. But you can quantify things in terms of categories/sets.

Even twins are not equal but you can identify them as two persons. Here "two" is the quantity and "person" is the category. Furthermore, e.g, a man and a cat, certainly not two persons, but you can say two living things.

Thanks.

 

The 2 oranges are not exactly the same and therefore 1 +1 =2 does not fit the orange situation.

BoyntonStu

It does if you read it the correct way. The proper statement, which is consistent with set theory, says ....

One object which has properties that allow it to be characterized as being in the class of "oranges" PLUS one object which has properties that allow it to be characterized as being in the class of "oranges" EQUALS two objects which each have properties that allow them to be characterized as being in the class of "oranges".

EDIT: I expect a mathematician can word this better by inserting the word "set" at appropriate points in the above statement.

 

1+1 = 2 is well established in Mathematics. It also depends what number system someone is using. Using base 2 and base 8 for example won't give you the same numerical answer most of the time.

In binary:- 1 + 1 = 10, 1 + 1 + 1 = 11, etc.

But in decimal:- 1 + 1 = 2, 1 + 1 + 1 = 3, etc.

Thanks.

 

EDIT: I expect a mathematician can word this better by inserting the word "set" at appropriate points in the above statement.

Let me try that again, because the phrasing is important. Lets call A the set of one orange in the your left hand, and call B the set of one orange in your right hand. The addition of both sets is the union of A and B, hence C=A+B is the set of both the left hand orange and the right hand orange. Now the sets C and A+B are precisely the same because they not only have the same number of objects, but they have the same objects in them also.

So, what are we saying when we say 1+1 = 2? Basically, we are saying that the set A with one object plus the set B with one object equals the set C which contains those exact same 2 objects. This should be consistent with your point of view, although you seem to prefer to say that the 2 oranges are 1 orange from the left hand and 1 orange from the right hand.

If you are concerned that I could make another set of 2 different oranges and say that it is equal, then I understand your point, even if I still say that the statement "2 oranges equals 2 oranges" is valid. In reality we may or may not care that they are not the same two oranges, but the two sets are not equal in some sense. Maybe one set has nice big ripe oranges and the other has rotten wormy ones.

 

It does if you read it the correct way. The proper statement, which is consistent with set theory, says ....

One object which has properties that allow it to be characterized as being in the class of "oranges" PLUS one object which has properties that allow it to be characterized as being in the class of "oranges" EQUALS two objects which each have properties that allow them to be characterized as being in the class of "oranges".

EDIT: I expect a mathematician can word this better by inserting the word "set" at appropriate points in the above statement.

"It does if you read it the correct way"


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

Sid Robert Recorde, the inventor of the equal sign have any status with his requirement that the items on both side must be exactly the same?

BoyntonStu

 

The 2 oranges are not exactly the same and therefore 1 +1 =2 does not fit the orange situation.

What exactly does fit the orange situation? What methodology would you have farmers and accountants and commodities markets adopt?

When you looked through that microscope, how did you enumerate the pins? (In your post you said "2.") What change to language and behavior are you calling for?

 

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

I'm willing to stick to a definition that the equal sign mean "exactly the same". But, I think the issue is the rest of the statement 1+1=2. How do we interpret the meaning of "1+1" and 2? My previous post above tries to correct my mis-worded statement before that. This explains how I would interpret the entire equation as applied to the left and right hand oranges.

1 means the set of one orange from either A or B
2 means the set of two oranges i.e. set C
+ means the union of the two sets A and B

Feel free to specify your interpretation of the full equation and we'll see if we are just arguing about definitions.

 

Stu,
A pity you have chosen to ignore my comments about equivalence classes (posts 14 and 23) since they are the mathematical formalisation Steve is seeking.

Steve's idea is correct.

His idea also avoids the logical inconsistency or paradox your line of reasoning leads inevitably to.

The paradox is this:

If we are to hold that each and every object in the universe is unique, the statement

one plus one equals two

cannot be substantiated because the second 'one' cannot separately exist, given that the first 'one' does exist by our very uniqueness statement!
Therefore the statement has no meaning.

The statement only gains meaning when our system of logic and language allows the possibility of a second identical object, in this case called a 'one'.

Of course, if we substitute a system in which we can select a pair of 'ones' from a bag of objects - which are equivalent for our purposes - the difficulty evaporates.

Consider this version of your statement.

1 + 1 = 2

Are you going to allow or disallow this when one '1' is red and one '1' is blue?

 

What exactly does fit the orange situation? What methodology would you have farmers and accountants and commodities markets adopt?

When you looked through that microscope, how did you enumerate the pins? (In your post you said "2.") What change to language and behavior are you calling for?

I agree.

Otherwise, it's like asking mathematicians to review the concept of numbers. And which I don't think is going to happen.

In that sense it would mean,
1 + 1 cannot be 2
likewise 1 + 1 + 1 cannot be 3
likewise 1 + 1 + 1 + 1 cannot be 4
and so on ..............................

I can see where the point is, but the thread has run its course.

2 different oranges yes, but it's 1 type of fruits. Someone can try buy "2" oranges at any local shop and argue that the "2" oranges are not the same and cannot be considered as "2" oranges as their weight, colour, chemical contents, acidic concentration, etc are all simply not the same. In that sense, give each orange its unique price and let's see if that makes life any easier. One might even ask how to put a price in the first place.

Again everything is relative. "1" is "1" relative to "2" and "2" is "2" relative to "1" etc.

Thanks.

 

I agree.

Otherwise, it's like asking mathematicians to review the concept of numbers. And which I don't think is going to happen.

In that sense it would mean,
1 + 1 cannot be 2
likewise 1 + 1 + 1 cannot be 3
likewise 1 + 1 + 1 + 1 cannot be 4
and so on ..............................

I can see where the point is, but the thread has run its course.

2 different oranges yes, but it's 1 type of fruits. Someone can try buy "2" oranges at any local shop and argue that the "2" oranges are not the same and cannot be considered as "2" oranges as their weight, colour, chemical contents, acidic concentration, etc are all simply not the same. In that sense, give each orange its unique price and let's see if that makes life any easier. One might even ask how to put a price in the first place.

Again everything is relative. "1" is "1" relative to "2" and "2" is "2" relative to "1" etc.

Thanks.

Words are what defines us.

Do you think that this distinction might help?

"Mathematics" for number theory and "Arithmetic" for counting 'things'.


In Math 1+1 = exactly 2.

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


"Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. In common usage, the word refers to a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numbers. Professional mathematicians sometimes use the term (higher) arithmetic[1] when referring to number theory, but this should not be confused with elementary arithmetic.

Words are important.

Consider this phrase: "Before there was time"

Does it have any meaning for you?


BoyntonStu

 

Words are what defines us.

Words alone do not define us, but they can completely define our mathematical symbols.

Your original post asked the following:

"Can anyone give an example in the real world where this simple mathematical relationship is perfectly correct? 1+1=2"

I used words to describe the definition of the equation 1+1=2 in the context of the right and left hand oranges. This example shows that when the set containing the left hand orange is joined (union/addition) with the set of the right hand orange, it EXACTLY equals the set made of those same two oranges. It is exact because the oranges are the same exact two oranges, even if the two oranges are not identical to each other. The sets are identical and the fundamental definitions of our mathematical symbols can be traced back to simple set theory.

This is a direct answer to your original question. As studiot pointed out, equivalence classes allow us to count, add and equate, even when things are not exactly the same, but the specific example above does not even need to resort to that.

So now, provide us your definition of 1+1=2, in the simple example provided above, and explain why this is not a valid answer to your question. The issue can't be the equal sign because i restrict myself to an example where EXACT equality is maintained. As far as I can see, you can only question the meaning of addition (the plus symbol) or the meaning of the numbers 1 and 2 and whether saying the number 2 means "the 2 objects in the set", as implied in the problem.

I think is should be remembered that mathematics is a language, but it is a very concise one. Great care must be taken to provide a valid interpretation of what is actually being said in a particular context.