John Mikes wrote:


"...Rectangles are not found in nature and not are numbers; both are abstractions of things we see in nature..." Pray: what things? and how are they 'abstracted into numbers? (Rectangles etc. - IMO - are artifacts made (upon/within) a system of human application). "Yet numbers and rectangles (and many other abstractions) have a suspiciously good use for modeling in nature"
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Number systems like the one asserted by the Peano axioms are abstractions of the process of counting. The box has no apples, the box has one apple, etc.. The numbers 0, 1, etc., are abstracted so that 0 can universally mean none of anything, 1 can universally mean 1 of anything, etc.. When we say 3+4=7, it is an abstraction because it universally means 3 of anything added to 4 of that anything is 7 of that anything. A rectangle traditionally is a set of points with special additional requirements. You will never find a rectangle in nature because points are smaller than particles and the edge of a rectangle is more dense than any physical arrangement. Dense meaning that between any two points there is another point in between the two. This is not true of naturally occurring arrangement of things: it is not the case that you can always find a third object between two other objects. Physical arrangements are not "infinitely fine," they are coarse even if only discernibly coarse on a very small scale.

Numbers are good models and have a use in a variety of applications such as finance and rectangles are good models for architecture and a whole lot more.
Equivalence of III + IV as VII? Or in other numbering systems (letters, etc.) used in various languages? In Bruno's example some time ago the II + I = III definitely referred to the quantity of the "I" lines. He even went up to some IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII or similar. Now in my feeble mind to construct 'symbols' for expressing /_how many "I"s there are_/ is not the other way around. "3" stands for III, the COUNTED amount of the lines and not vice versa. So: what are those _"naturally occurring"_ things that serve for being abstracted into numbers?
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Seems like the concept of number system is getting mixed up with the concept of numeral system. It does not matter if you use III, 3, three, @@@, etc. It does not matter that IIIIIII can be written 7 or seven. The numeral system is the notation and the number system are what the symbols in the numeral system point to. So while we may write III or 3 or three, what those symbols point to is a number. If you will, imagine two domains: one domain is of symbols and the other domain is what those symbols point to. Numeral systems are of the first domain and number systems are of the second domain.

Counting inspired number systems. Numeral systems are used to describe counting.
"Axioms are statements" - not controversial to what I stated. And please, do not divert into quite different topics, where you may have a point in some other aspect. We are talking about numbers, not the masculinity of the US president.
Fine, not controversial. My examples, admittedly not all drawn from mathematics, were just illustrations of my point that statements exist independently of humans. What you said was this: /_Axioms_/ however sounds to my vocabulary like inventions helping to justify our theories. Sometimes quite weird. Yet axioms exist independently of humans. What a human does is select axioms to his or her liking to momentarily assume for some purpose or another. Basically, because axioms exist independently of humans (as do all statements), they are not inventions of humans. Not inventions but a human will choose which axioms to assume momentarily for some purpose. Choose, not invent.

"Exist" is something to be identified. IMO "physical existence" is a figment pertinent to the figment of a "physical world" - quite outside of my position. I "don't permit" physical existence.
Well then perhaps numbers exist for you. I do not put the physical condition on existence; for me numbers do indeed exist.
If I may repeat: so WHAT ARE NUMBERS? (symbols for what? how do they apply them to quantitative considerations? what if another 'logic' uses them in a different math (e.g. where 17 is not identifiable as a prime number? Is it likely that more will be found - as was the zero, or are we in a "mathematical omniscience" already? Is our restriction to the 'naturals' - natural, or just a consequence of our insufficient knowledge (caabilities)? May I quote a smart person: there are no stupid questions, only stupid answers. I ask them. John Mikes
When considering number systems such as naturals, rationals, and (finite or infinite) cardinal numbers, it seems to me to not be a question with a quick answer. Division is not "possible" in all number systems, so I would have to say that in order to count (no pun intended) as a number system, there has to be a binary operation that structurally is the same as addition, meaning it has to be commutative and associative.

This opens the door for a lot of systems to be considered number systems, if that is the only requirement. This is the theory of Abelian semigroups which have a lot of research behind them. My definition that a number system must be an Abelian semigroup probably wouldn't be universal to all mathematicians. That is my requirement for what traits a number system must have, minimally. That doesn't answer the question WHAT ARE NUMBERS.

Numbers are what numeral systems are referring to.

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