The difference between counting and measuring is that counting is in whole units like one two three four five..
Measurement is a continuous thing so your waist can be 24 inches 24 and a bit more a bit more and a bit more and you can measure all that but the numbers you’re getting are not all whole numbers or even fractions like the square root of 2 is an infinite non-recurring decimal
So you have the concept of continuity which is a very difficult concept in some ways.
But with the digital age we have gone back to counting and you only need two numbers to count zero and one.
It’s rather like the digital radio stations as opposed to the FM radio stations which do you prefer?
I prefer FM.
But I think if we taught mathematics in schools starting when the children are old enough to talk about how numbers happened and these two ways that they happen and then it’s much easier for children or students to learn the technical side of it because they will see some kind of context and some kind of meaning to it
I have yet to find any reasonable way of telling people the ought to learn how to do quadratic equations using a formula that they’re going to understand.
We spend too much time in school learning things we are told when we’re useful later on but it’s very hard for some children to believe it’s going to be useful later on.
Yet if people like to gamble on the horses they soon learn what odds mean even when they can’t do arithmetic at school
How my heart sings 
There are Maths and Physics – the real (physical) world tends to interpose new factors and allows pragmatic solutions.
For example, we think we know what an apple is. We could go to a shop and ask for 10 apples but we often don’t – we ask for (say) a kilo of apples. This raises a new problem. In the old days (in UK) we would have asked for a pound of apples and the shopkeeper would put several in a bag. (S)he would see what they weighed and then look at the whole pile of apples and see that they were of different sizes. (S)he would select another one or two apples of the appropriate size (measurement ‘by eye’) to bring the total weight as near as (s)he could to one pound. Nowadays, however, the apples might be priced at £3 per kilogram and they will be weighed at the checkout. If our selection weighs 943 grams, we will have 3 x 0.943 pounds debited from our credit card. Here are two ‘real world’ pragmatic solutions to a problem.
Thank you for that interesting comment.
I think that what I was thinking about was that mathematicians rather be with him interested in the numbers themselves and their properties over a long period of time and discovered that there are far more irrational numbers than there are rational ones for example.
When you say how can there be more when they are both infinite it’s also been discovered or proved that there are different orders of infinity.
Yes there are different possibilities in thinking about such problems and it’s led to the development of machinery such as weighing machines and more and more accurate way machines they mention of credit cards and otherwise have paying for your apples so we see so much that connected sometimes to more abstract thought sometimes to reality as we know it but the main thing is there are two completely different ways that numbers come into our world counting is one way and that is a discrete way and then there is measuring. Even two fields of the same area do not have the same length of size necessarily and he needed the size is meant to be the same it’s very hard to be accurate in the real world whereas on a piece of paper or a computer one can be as accurate as is needed
Because outside of thought 100% accuracy is impossible anywhere and even the words of the apples and the scale will differ by half an ounce or even 100th of an ounce so we only accurate to the degree that is necessary.
If you are feeling children you know you have to make amount on their p plates look the same but you do not get out a scientific measuring tool to weigh them all
The dinner will go cold if you do that although sadly they may be some people with OCD dl2 who might and so common sense has to come in and it’s usually fairly obvious when you need to be very accurate such as making baby food and when you just giving sausage and mash to several people for their dinner
Perhaps one could say that there are different routes to infinity and some are longer than others.
No it’s not quite that as I understand it.
The number of integers is infinite but it is countable infinity as it were
The infinity of the real numbers is not equal to the infinity of the whole numbers.
These are referred to as
Aleph null
Aleph 1
It is unproven as to whether there is another infinite number between those two.
I am not sure where you could read about this but I think it’s very interesting and it may be another way of saying what you say I don’t I might be suffering some cognitive decline.😭
Of course we may we confusing infinity and God here