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In school we learn about numbers through physical amounts and we take two things and put them with two other things and call it four things in total.

Is this view of numbers as amounts slightly 'old fashioned'? Before we would say 'that's two' or something like 'put two and two together', maybe 'take 2, add 2 to it', but this seems to not be correct in the way to discuss real numbers in the 'object' view.

We seem to take an object based approach and our symbols and names like 'two' or '$x$' or '2' are like proper nouns. And a mathematical statement is more like talking about an object, so in $2+2=4$ we take as a statement about objects and functions taking those objects as inputs, In this way in Mathematical logic we can actually extend this language to real objects that we use proper nouns for.

We can try to define the idea of 'quantity' through number, but when we study mechanics we find that '2' can have different meanings (such as an inherrent direction or an increase) depending on the context of what is measured, so we cannot call any number any one quantity.

However for much of school I would have taken that as Take 2(of anything) and add 2 (of anything) to it and you get 4 (in total). In this manner, thinking mentally about quantities of individualn things 'two' becomes almost a 'description' and every two things can be described as 'two' (as children and many people do in infomral situations).

Personally for me, as I learnt more algebra, logic and proofs I had to overcome the idea of viewing a number like '5' as somehow only describing 'quantity', but more as an object that exists in our mind and we discuss in a similar vain to a particular person or 'Finland'.

This issue I think comes from language we are taught for using numbers in a practical sense and the true abstract nature of objects we use.

Is there any meaningful difference here, with the 'amount' view of number as given at school/non-scientific backgrounds, or am I simply being mis-led by informal language and a lack of care to differentiate between 'concrete' and 'abstract' objects, and the use of 'numbers' in an adjective sense?

Mauro ALLEGRANZA
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    Hello, you should read a few articles about "Concrete numbers vs abstract numbers" : https://math.stackexchange.com/questions/3974144/concrete-and-abstract-numbers and https://en.wikipedia.org/wiki/Concrete_number and https://royalsocietypublishing.org/doi/10.1098/rstb.2017.0125 and https://onlinelibrary.wiley.com/doi/full/10.1111/cogs.12619 – Mateo_13 Nov 20 '22 at 14:03
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    You might have more luck with your question in a philosophy forum. There has been much written about these topics by folks like Frege, Russell, etc. – blargoner Nov 20 '22 at 14:16
  • @Mateo_13 thanks you for these references. –  Nov 20 '22 at 14:57
  • What is an object view of numbers? How can you say there is a "true abstract nature" of anything? An abstraction is a convenience for dealing with complexity and similarity. If "two of something" is in some abstract notion similar to "two" in another context, that does not change any truth about "two of something". – John Douma Nov 20 '22 at 18:16
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    I think there are really at least three different concepts in play here: the number $2$ when we show a picture of two apples on a table in an elementary school textbook; the number $2$ in higher math; and the number $2$ when it represents (for example) an increase in a force in a particular direction in physics. – David K Nov 20 '22 at 18:16
  • @DavidK But they are in essence the same 'object' conceptually, is it about different uses? –  Nov 20 '22 at 20:03
  • @JohnDouma So the fact we use the number '2' in other ways it doesnt change the fact that 'two apples' is 'two apples'? –  Nov 20 '22 at 20:04
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    @user1007028 That's how I look at it. The number "2" is a natural number but the fact that the natural numbers form a semigroup under multiplication doesn't change the meaning of "2". The number "2" is an integer but the fact that the integers form a commutative ring doesn't change what "2" is. Therefore, in my mind, the fact that the number "2" is contained in many sets that can be viewed in interesting and novel ways doesn't change anything about "2". – John Douma Nov 20 '22 at 20:09
  • @JohnDouma Just for my interest, how do you differ the idea of the number from the amount that we visualise when we talk about '2 things'? Is it a label that when combined with what is being counted, tells us how many of those things there are? –  Nov 20 '22 at 20:12
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    That's just it. That symbol stands for that amount. You may be interested in taking a course in Mathematical Logic. The symbols in a language are differentiated from the interpretation of those symbols. For example, in the language of arithmetic, we know that "+" is a binary function symbol so we can say $x+y$ is a valid expression and $0 +$ is an invalid expression but we don't define that to be addition of numbers, vectors or anything else. Those interpretations are what are called structures. – John Douma Nov 20 '22 at 20:38
  • @JohnDouma That is interesting, my only issue is that I find the 'number' can represent amounts differently depending on contexts such as with 'directions', or simply just 'ordering' where we assign a number to each position, I may simply be misunderstanding, but would you say the symbol has different meanings depending on this context? The idea of a 'label' more likely allows us to signify the amount without it being the 'amount' itself. I apologise for excessive follow-up questions this will be the last –  Nov 20 '22 at 21:03
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    @user1007028 Those aren't amounts, those are units. For example, $2$ pounds and $2$ apples are different but the difference is in what there are two of, not in two itself. – John Douma Nov 20 '22 at 21:07
  • @JohnDouma, I see what you mean thanks. –  Nov 20 '22 at 21:08
  • "Take 2(of anything) and add 2 (of anything) to it and you get 4 (in total)". Now consider the following: 2 volt cannot be added to 2 joule. 2 apples and 2 pears make ... 4 fruits, maybe. But 2 apples and 2 bags of apples? Or 2 metres plus 2 centimetres? These things are wonderfully intricate if you look closer, almost like life. – Torsten Schoeneberg Nov 21 '22 at 04:26
  • Also, related: https://math.stackexchange.com/q/865409/96384, https://math.stackexchange.com/q/494854/96384 and probably many more duplicates. This is an old philosophical question with some brilliant insights and a lot of hogwash coming out of it. What answers are what depends on perspective. – Torsten Schoeneberg Nov 21 '22 at 04:35
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    @TorstenSchoeneberg we can take all these which don't work, yet regardless $2+2=4$, I have never thought on this, it does get wonderfully intricate, and the rules of dimensional analysis make it more complex than this, I must consider that. –  Nov 21 '22 at 11:29
  • @TorstenSchoeneberg I appreciate the references. –  Nov 21 '22 at 11:29

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