# How did we determine natural numbers?

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What are the reference points which makes the values presented by natural numbers so solid?

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Perhaps you would elaborate on what you mean by the natural numbers?

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Perhaps you would elaborate on what you mean by the natural numbers?

0,1,2,3,4,5,6,7,8,9,10.....

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OK so you mean integers.

It is interesting that you have included zero in your list.

Mathematically the set of positive integers is usually called the natural or counting numbers.

Since zero is neither positive nor negative it is usually excluded.

However some do include it in the set.

The best origination for these numbers is in the name 'counting' numbers.

The idea of a whole of complete unit precedes that of counting and the natural numbers.

Once you have a unit the numbers follow naturally from the process of counting ie adding a single unit each time to the last.

This will not help much if your objective is to study integer algebra eg Diophantine equations, but it is a start and your original post was very sparse.

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OK so you mean integers.

It is interesting that you have included zero in your list.

Mathematically the set of positive integers is usually called the natural or counting numbers.

Since zero is neither positive nor negative it is usually excluded.

However some do include it in the set.

The best origination for these numbers is in the name 'counting' numbers.

The idea of a whole of complete unit precedes that of counting and the natural numbers.

Once you have a unit the numbers follow naturally from the process of counting ie adding a single unit each time to the last.

This will not help much if your objective is to study integer algebra eg Diophantine equations, but it is a start and your original post was very sparse.

I try to understand our physical reality and the origin of our mathematical understanding. I think without our physical reality math as such would not exist. I think we started to develop math related to our observations and understanding the physically presented values (trees...apples..animals...etc)

Because of this I asked myself does mathematics and its operations has to follow the inspected physical reality to be a clear and Universal language?

I also asked myself is there any negative value in the inspected physical reality and what is the origin of the negative values we work with?

I realized that I can not provide any physical unit which would be lower than physical zero (the total absence of everything as information, space, time, energy, matter)

I realized that there is no lower value than nothing exist but there are opposite values and we present them as negative numbers.

Do you think mathematics evolving or is it a firm absolutly clear tool. The same tool which would be used by an advanced alian race 1 billion LY away from us (I quess they have evolved theirs understanding related the physical reality they observed and that reality originates from the same Laws of Nature what we experience)

The question is should mathematics follow the inspected reality and adjust its operations related to the observations (at the end of the day science is all about extracting and processing information) ?

I incoulded zero because as I understood the Universe is evolving, and if I follow this evolutionary path backwards at the end I reach to the physical zero state as the original state of the system we live in(space(time), energy, matter, information free nothing).

As such zero could be the only mathematical conception ever clearly existed.

Maybe one other mathematical unit could exist and that is the whole universe counted as One. Than everything we inspect in the system would be materialized information and could be expressed as a fractal in proportion the the whole. I also suggest that we are able to make sense of the physical reality because we exist in proportion to zero. I also suggest that we are not able to operate with zero since zero is a conception ever since anything is existing. (zero is basically a reference point)

I hope it made sense. I am a curious mind and I have difficulties to make sense why we use math as we do today.

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The meaning of reality, nothingness and non physical 'objects' have been discussed before on this forum and also the relationships between them.

Generally the discussion falls into the province of philosophy, not mathematics.

I see you are from Sweden and I know nothing of Swedish.

I can tell you that English distinguishes two types of object (noun)

Concrete nouns

Abstract nouns

This is where the difference (in English at any rate) between 'reality' and 'existence' comes in.

Objects (ideas) can exist in the abstract but not the concrete. edit clarification abstract objects like ideas can exist in the abstract without also having concrete existence.

Reality includes all concrete nouns, but only some abstract ones.

So existence has a wider remit than reality since something can exist "in your dreams" to use a phrase.

I can verify the concrete reality of an apple, but not of an atom with 300 protons and no neutrons.

There are also bordeline grey areas that are worth consideration and we return to mathematics here.

Consider a 100mm cubic block of wood.

This has a very definite physical existance, and is big enough to contain many 50mm diameter circles, I could easily cut several out of it.

But do these circles exist before I cut them out?

Since the length of one side is 100, mathematics tell me there must be a point in the wood that is 37mm from one end.

So does 37 therefore exist?

I bore a 20mm hole through the block.

Dole the hole have physical existance?

Back to counting.

There are tribes in the South Seas that do not possess our counting skills.

There number system goes 0ne, two, many.

So do numbers exist in Stockholm, but not in the Solomon Islands?

Edited by studiot

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What are the reference points which makes the values presented by natural numbers so solid?

As studiot says, the integers can be used to count things so in that sense map onto the physical world in a straightforward way. Some of their properties and the operations on them can be easily understood in similar terms (e.g. basic arithmetic operations). But you soon get to properties and operations that don't have a simple relationship to the real world (e.g. primality, exponentiation).

But the natural numbers can also be defined in purely abstract mathematical terms based on a couple of set theoretic axioms. In which case, their relationship to the real world appears more coincidental.

I can tell you that English distinguishes two types of object (noun)

Perhaps more relevantly to this thread, English also distinguishes between countable and non-countable nouns.

(Interestingly, Japanese has two sets of words for small integers: one set treats nouns as countable and the other set treats them as non-countable.)

Edited by Strange

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