Counting vs Measuring (Difference in Number Conception)

[gumajoi]

Had a thought/question after thinking about how I do math in my head and some confusion it caused. Hoping to get some of the brighter math minds to comment.

I guess I see numbers as a length on a number line (Unit length from 0).

So if I was adding 5 + 5 I would start at 5 on the number line and go 5 units to the right to get to 10. Now I see 10 as the length to the left of 10 on the number line so to speak (I'm sure many others see it that way too intuitively).


Now on the other hand, if I'm actually counting objects I would point to each one of 10 objects and say the integer it corresponds to as I count from 1 to 10.
So on the number line, 5 is exactly the middle of a length of 10. This makes sense because two lengths of 5 clearly make ten, meaning that 5 is the half-way point.


But if I'm counting objects 5 isn't really the middle object despite it feeling that way on the number line. 5 and 6 are both in the middle so to speak.


Does this have to do with number lines not "getting" to a number until you reach the very end of the unit length segment?


Like I guess 5 is the middle object if you think about the very "end" of the object being 5.


I guess my question is: is counting (physical objects) different from length in how we perceive it? Do we reflexively jump between the counting model and the number line model as we use numbers?


Let me know if anyone needs clarification as I realize this is a bit of an odd way to think about numbers haha but this is making me really curious as I don't think I had ever thought of it this way. Has anyone else made this distinction or is it a even logical to make this distinction?

 

[Dr.Peterson]

Had a thought/question after thinking about how I do math in my head and some confusion it caused. Hoping to get some of the brighter math minds to comment.

I guess I see numbers as a length on a number line (Unit length from 0).

So if I was adding 5 + 5 I would start at 5 on the number line and go 5 units to the right to get to 10. Now I see 10 as the length to the left of 10 on the number line so to speak (I'm sure many others see it that way too intuitively).

Now on the other hand, if I'm actually counting objects I would point to each one of 10 objects and say the integer it corresponds to as I count from 1 to 10.
So on the number line, 5 is exactly the middle of a length of 10. This makes sense because two lengths of 5 clearly make ten, meaning that 5 is the half-way point.

But if I'm counting objects 5 isn't really the middle object despite it feeling that way on the number line. 5 and 6 are both in the middle so to speak.

Does this have to do with number lines not "getting" to a number until you reach the very end of the unit length segment?

Like I guess 5 is the middle object if you think about the very "end" of the object being 5.

I guess my question is: is counting (physical objects) different from length in how we perceive it? Do we reflexively jump between the counting model and the number line model as we use numbers?

Let me know if anyone needs clarification as I realize this is a bit of an odd way to think about numbers haha but this is making me really curious as I don't think I had ever thought of it this way. Has anyone else made this distinction or is it a even logical to make this distinction?

Yes, this is all familiar. It has to do with the difference between discrete and continuous, or between inclusive and exclusive counting. Here is something I wrote long ago about a related idea. There's probably a lot more I'll think of to say, or that others can add.

 

[tkhunny]

Had a thought/question after thinking about how I do math in my head and some confusion it caused. Hoping to get some of the brighter math minds to comment.

I guess I see numbers as a length on a number line (Unit length from 0).

So if I was adding 5 + 5 I would start at 5 on the number line and go 5 units to the right to get to 10. Now I see 10 as the length to the left of 10 on the number line so to speak (I'm sure many others see it that way too intuitively).


Now on the other hand, if I'm actually counting objects I would point to each one of 10 objects and say the integer it corresponds to as I count from 1 to 10.
So on the number line, 5 is exactly the middle of a length of 10. This makes sense because two lengths of 5 clearly make ten, meaning that 5 is the half-way point.


But if I'm counting objects 5 isn't really the middle object despite it feeling that way on the number line. 5 and 6 are both in the middle so to speak.


Does this have to do with number lines not "getting" to a number until you reach the very end of the unit length segment?


Like I guess 5 is the middle object if you think about the very "end" of the object being 5.


I guess my question is: is counting (physical objects) different from length in how we perceive it? Do we reflexively jump between the counting model and the number line model as we use numbers?


Let me know if anyone needs clarification as I realize this is a bit of an odd way to think about numbers haha but this is making me really curious as I don't think I had ever thought of it this way. Has anyone else made this distinction or is it a even logical to make this distinction?

There are classified many kinds of numbers.

'Counting" normally refers to "Natural Numbers": 1, 2, 3, 4, 5, ...etc. There is no in between. You must stop on 2 or 3. You cannot take a break between them. This is NOT like a continuous Number Line. You cannot build a ruler with only Natural Numbers. There would be nothing holding it together.

Do some more homework and let's see where it leads you. Look up these terms:
* Natural Numbers (Sometimes known as "Counting Numbers")
* Whole Numbers (This collection is only slightly different from "Natural Numbers")
* Integers (This collection is only a little different from "Whole Numbers")
* Rational Numbers (Big change, here, but maybe not as big as you might think.)
* Irrational Numbers (Another big change. This time, it IS big!)
* Real Numbers (You may wonder what is included that we didn't already mention. That's an excellent thought question.)

Once you get a better handle on those types, guess what? There are even more kinds of numbers!

 

[gumajoi]

Yes, this is all familiar. It has to do with the difference between discrete and continuous, or between inclusive and exclusive counting. Here is something I wrote long ago about a related idea. There's probably a lot more I'll think of to say, or that others can add.

Hey thanks for your response. Do you mind expanding? I sort of follow but not totally.

I kind of fail to see the difference between continuous and discrete being that both are the same thing in my mind below:

[FONT=&quot]A length of 10, starting at 0, is still 10 individual unit segments just as a quantity of 10 objects is 10 units of the object.[/FONT]

[FONT=&quot]Is it just that we call the very end of a distance the number (i.e. the very end of the distance from 0 to 5 is 5 and not just the portion past 4. But when we count objects we associate the number with the object itself and not the end of the object?[/FONT]

[FONT=&quot]Like technically, if you lined up 10 objects, the middle would be the space right between 5 and 6 to the point where it's right at the "end" of 5 / "beginning" of 6?[/FONT]

 

[Dr.Peterson]

I can think of several places where the issue you are raising is important.

In statistics, we talk about a "class" of values such as 1, 2, 3, 4, 5 as discrete values. We say this class has "limits" 1 and 5, and "width" 5, because there are 5 values -- but people often want to subtract 5 - 1 = 4 and call that the width. The "class midpoint" is just what you would expect: the average of 1 and 5, (1 + 5)/2 = 3. Well, that's what you should expect; many people expect it to be just 5/2 = 2.5. That's essentially the same issue you described.

In part to make up for this, we also talk about the "class boundaries", where we replace the discrete view with a continuous, number-line view, thinking of the class as containing all numbers from 0.5 through 5.5. In effect, we are widening each value until it touches those on either side:

[FONT=courier new]    *     *     *     *     *
 |=====|=====|=====|=====|=====|
    1     2     3     4     5 [/FONT]
[FONT=courier new]0.5   1.5   2.5   3.5   4.5   5.5[/FONT]

Now the class width is in fact the difference between the upper and lower boundaries: 5.5 - 0.5 = 3. The midpoint is still the average.

A closely related idea is the median of a set of numbers. If we have 5 values, the median is the middle one; but to find which one that is, we don't divide 5 by 2; we have to add 1 first, (5 + 1)/2 = 3 (the third value in the list is the median). This can be thought of in various ways; the same idea of averaging the first and last indices, as I explained above, is a good one. Another is to realize that if you just divide by 2, you are halving the distance from 0 to 5, not from 1 to 5. Your comments on the number line measuring from zero, and counting spaces rather than individual numbers are applicable.

Here are two more discussions of related ideas: a puzzle, and a difference.

 

[JeffM]

It is late, and mine is a minority opinion in any case.

From my standpoint, the only thing that is operationally valid in the physical world is counting. Measuring in the physical world is just a form of counting. When we say something measures 7.2 inches, then what we mean physically is that we have a measuring rod marked off in tenths of inches and that we counted 72 of them.

Now most mathematicians prefer to deal with a Platonic world that involves uncountable numbers like the real numbers. In that Platonic world, measurement cannot be reduced to counting. I cannot count the number of real numbers in the interval [0, 10]. But I can say that the interval [0, 5] has the same "length" as the interval [5, 10] because 5 - 0 = 5 = 10 - 5. And thus it is perfectly reasonable to say that 5 is the midpoint of the interval [0, 10]. That is the same result that we get if we consider {0, 1, 2, 3, 4} and {6, 7, 8, 9, 10}. 5 is the only whole number that is greater than the maximum of one set and lesser than the minimum of the other set. The results are the same, but the reasoning differs.

I cannot accept Platonism as anything but mysticism. Physical cats are not unreal "shadows on the wall" of a single, necessarily unobservable real cat. But Platonism makes mathematics much easier.

 

[gumajoi]

I can think of several places where the issue you are raising is important.

In statistics, we talk about a "class" of values such as 1, 2, 3, 4, 5 as discrete values. We say this class has "limits" 1 and 5, and "width" 5, because there are 5 values -- but people often want to subtract 5 - 1 = 4 and call that the width. The "class midpoint" is just what you would expect: the average of 1 and 5, (1 + 5)/2 = 3. Well, that's what you should expect; many people expect it to be just 5/2 = 2.5. That's essentially the same issue you described.

In part to make up for this, we also talk about the "class boundaries", where we replace the discrete view with a continuous, number-line view, thinking of the class as containing all numbers from 0.5 through 5.5. In effect, we are widening each value until it touches those on either side:

[FONT=courier new]    *     *     *     *     *
 |=====|=====|=====|=====|=====|
    1     2     3     4     5[/FONT]
[FONT=courier new]0.5   1.5   2.5   3.5   4.5   5.5[/FONT]

Now the class width is in fact the difference between the upper and lower boundaries: 5.5 - 0.5 = 3. The midpoint is still the average.

A closely related idea is the median of a set of numbers. If we have 5 values, the median is the middle one; but to find which one that is, we don't divide 5 by 2; we have to add 1 first, (5 + 1)/2 = 3 (the third value in the list is the median). This can be thought of in various ways; the same idea of averaging the first and last indices, as I explained above, is a good one. Another is to realize that if you just divide by 2, you are halving the distance from 0 to 5, not from 1 to 5. Your comments on the number line measuring from zero, and counting spaces rather than individual numbers are applicable.

Here are two more discussions of related ideas: a puzzle, and a difference.

Awesome thank you. This touches on everything I was thinking about. I had never thought of it till this way recently and it threw off my entire view of numbers. Glad it's an actual thing.


Thank you!

 

[Dr.Peterson]

Hey thanks for your response. Do you mind expanding? I sort of follow but not totally.

I kind of fail to see the difference between continuous and discrete being that both are the same thing in my mind below:

A length of 10, starting at 0, is still 10 individual unit segments just as a quantity of 10 objects is 10 units of the object.

Is it just that we call the very end of a distance the number (i.e. the very end of the distance from 0 to 5 is 5 and not just the portion past 4. But when we count objects we associate the number with the object itself and not the end of the object?

Like technically, if you lined up 10 objects, the middle would be the space right between 5 and 6 to the point where it's right at the "end" of 5 / "beginning" of 6?

The difference between discrete and continuous is just what you have described. Note that the length of 10 units extends from 0 to 10, not from 1 to 10. That makes all the difference. The middle of the line from 0 to 10 is (0 + 10)/2 = 5, while the middle of the points from 1 to 10 is (1 + 10)/2 = 5.5.

I hope my second answer (written before I saw your response) made it a little clearer.

 

[gumajoi]

The difference between discrete and continuous is just what you have described. Note that the length of 10 units extends from 0 to 10, not from 1 to 10. That makes all the difference. The middle of the line from 0 to 10 is (0 + 10)/2 = 5, while the middle of the points from 1 to 10 is (1 + 10)/2 = 5.5.

I hope my second answer (written before I saw your response) made it a little clearer.

My only question (and obviously you're right since 5.5 is the median of 1 - 10):

Can't ten objects be represented on the number line as the length from 0 to 10? Like isn't 0 an implicit part of every number?
I guess what I'm saying is that 10 on the number line would represent the 10 unit length segments to the left of it which would be a one to one correspondence with the integer 1 to 10. Isn't the implication then that 5 is the middle?

Let me know where my logic doesn't hold.

Thanks,