Numbers, what are they?

[cicero117]

As I was solving my calculus homework, a simple question popped into my mind..

What exactly is a number? How can something define our world like so?

Is it real? because well..we can't see them, we can only calculate them,
or rather is it only a system of some kind so that we can make sense of the things around us?

It's a "basic" ontological problem, but I'm interested in what others think

[commonsense]

A number is a concept that can be used to answer the question of how many.

[Scott Mayers]

As I was solving my calculus homework, a simple question popped into my mind..

What exactly is a number? How can something define our world like so?

Is it real? because well..we can't see them, we can only calculate them,
or rather is it only a system of some kind so that we can make sense of the things around us?

It's a "basic" ontological problem, but I'm interested in what others think

Set theory (with expected background in symbolic logic) expresses numbers without using number. They do it by asserting the 'cardinal' count of members of any set, regardless of what they are. So, the cardinal count of {burgers, X, vaccines} is 'equal' by cardinality to {0,1,2} and represents the meaning of 'three'.

Another way of thinking it, imagine that you can express all specific groups of "three" things. This would take a continuity of infinities to express. As such, the meaning of the number we symbolize as '3' is represented as a FORM(ula). It is a form using open sentences where you 'close' it by filling in specifics. For example, using the set example, we might imagine using the way we learn in elementary school, blank lines to represent the 'open' variable possible constants as,

{____, ____, ____} means 'three'. Just fill in the blanks with any unique finite element. The meaning is not what you fill in as constants but to the form itself.

[Skip]

They're symbols.
Letters are symbols for sounds. We dissect the spoken word down to its component sounds and make a written symbol for each sounds. These then become the building blocks of written language.
Numbers are verbal and visual representations of quantities. They can be manipulated, like letters, to show the relation of one quantity to another, to communicate in the language of mathematics.

[commonsense]

They're symbols.
Letters are symbols for sounds. We dissect the spoken word down to its component sounds and make a written symbol for each sounds. These then become the building blocks of written language.
Numbers are verbal and visual representations of quantities. They can be manipulated, like letters, to show the relation of one quantity to another, to communicate in the language of mathematics.

You’re describing numerals, not numbers.

[Skip]

Yes. And the difference between a diagram of a triangle and an actual triangle is....?

[commonsense]

Yes. And the difference between a diagram of a triangle and an actual triangle is....?

Do you mean a diagram as a symbol of a concept called a triangle?

[Skip]

Yes. And the difference between a diagram of a triangle and an actual triangle is....?

Do you mean a diagram as a symbol of a concept called a triangle?

I mean the word "triangle" as a verbal symbol for the concept of a figure with three straight sides, and the diagram as a visual symbol for the same concept: both are tools for communicating that idea from one human intelligence to another.
In the same way, the word "number" is a collective for the verbal symbols, one, two, etc., and for numerals, which are the visual symbols, for communicating the concept of quantity.
Once the symbols have been invented and adopted, human intelligence can manipulate them: arrange them in sequences, patterns, formulae; can arrange and rearrange them to depict different relationships, changes, configurations and interactions.
Neither the number nor the geometric figure exist in nature as independent entities: they manifest only as qualities of natural entities.
There is no three/3/III but you can have three goats, and there is no triangle, but you can build a three-sided pen for those goats.
In the same way, there is no such things as brown or hunger or dark, but a bear with dark brown fur can be very hungry indeed.
So, lock up your goats!

[commonsense]

Thanks.

[jayjacobus]

Numbers are a frame of reference for amounts. Without numbers amounts are imprecise.

Using numbers in a sentence is an example, not a definition. Triangles is an example.

Saying that numbers are words, which are symbols, is not a definition because all words are symbols, not just numbers.

A mathematician uses numbers in geometry, algebra. calculus, etc. but without a way to specify amounts each subject would be meaningless.

[Immanuel Can]

As I was solving my calculus homework, a simple question popped into my mind..

What exactly is a number? How can something define our world like so?

Here's another question: How did our world end up being the kind of place that corresponds to numbers? In other words, why is it a place where rationality is not only possible, but also even seems to "unlock" things to human understanding?

If it was created by chaos plus chance, it ought to be a place of randomness, confusion and irrationality. Instead, it seems to be composed of patterns and describable in figures; and most marvellously of all, we seem to be exactly the kinds of little "calculators" that are able to unpack all those numbers.

That, if it is a product of chance, is by far the least probable thing one can possibly expect. And the fact that we, odd creatures that we are, can recognize the oddness of it is just one example of that mystery.

[Skip]

Numbers are a frame of reference for amounts.

amount = quantity

Without numbers amounts are imprecise.

You mean, if you didn't know numbers, three goats might really be four goats, or two and half goats, and you couldn't tell?
No, I'm pretty sure that amounts/quantities were the same before humans named them and will still be same when humans are extinct.

Using numbers in a sentence is an example, not a definition.

Unless the sentence is a definition, i.e. : A number is a verbal symbol for a quantity. A numeral is a visual symbol for a quantity. Mathematical signs are visual representations (symbols) of mathematical operations that express interactions and relationships between quantities. The verbal counterpart of these symbols are 'plus' 'times' 'square root', etc.

Saying that numbers are words, which are symbols, is not a definition because all words are symbols, not just numbers.

I already said that. Words are verbal symbols for real things, places, persons, processes, relationships and ideas. The numbers are the words we use for quantities.

A mathematician uses numbers in geometry, algebra. calculus, etc. but without a way to specify amounts each subject would be meaningless.

This is absolutely true. Also true: if humans had not invented the language of mathematics, there could be no disciplines of algebra, geometry, calculus, etc. And then where would the mathematicians be, eh? They might have to go herd goats and never be quite sure how many they had.

[Belinda]

As I was solving my calculus homework, a simple question popped into my mind..

What exactly is a number? How can something define our world like so?

Here's another question: How did our world end up being the kind of place that corresponds to numbers? In other words, why is it a place where rationality is not only possible, but also even seems to "unlock" things to human understanding?

If it was created by chaos plus chance, it ought to be a place of randomness, confusion and irrationality. Instead, it seems to be composed of patterns and describable in figures; and most marvellously of all, we seem to be exactly the kinds of little "calculators" that are able to unpack all those numbers.

That, if it is a product of chance, is by far the least probable thing one can possibly expect. And the fact that we, odd creatures that we are, can recognize the oddness of it is just one example of that mystery.

That is your best argument for the existence of God. However that argument does not imply God intervenes in history.

[Immanuel Can]

As I was solving my calculus homework, a simple question popped into my mind..

What exactly is a number? How can something define our world like so?

Here's another question: How did our world end up being the kind of place that corresponds to numbers? In other words, why is it a place where rationality is not only possible, but also even seems to "unlock" things to human understanding?

If it was created by chaos plus chance, it ought to be a place of randomness, confusion and irrationality. Instead, it seems to be composed of patterns and describable in figures; and most marvellously of all, we seem to be exactly the kinds of little "calculators" that are able to unpack all those numbers.

That, if it is a product of chance, is by far the least probable thing one can possibly expect. And the fact that we, odd creatures that we are, can recognize the oddness of it is just one example of that mystery.

That is your best argument for the existence of God. However that argument does not imply God intervenes in history.

It's not actually my best argument, but I think it's a very good one.

And you're quite right: it's the kind of argument that takes us only so far as kind of grateful Deism. But it does not show -- and does not even attempt to show -- what the nature of the relationship is, beyond that God's been very kind to make us the kind of entities that inhabit a universe which they have been gifted with a remarkable ability to understand.

There are other arguments that deal with the specifics of God's intervention in history. And they would, of course, have to be empirical-historical rather than empirical-mathematical, like this one.

[Belinda]

Skip wrote:

You mean, if you didn't know numbers, three goats might really be four goats, or two and half goats, and you couldn't tell?
No, I'm pretty sure that amounts/quantities were the same before humans named them and will still be same when humans are extinct.

Then order is eternally true. That is what Immanuel Can claimed in his post that immediately preceded yours.