John Mikes wrote:

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"...Rectangles are not found in nature and not are numbers; both areabstractions 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 ofhuman application)."Yet numbers and rectangles (and many other abstractions) have asuspiciously good use for modeling in nature"--- " - u s e - ". (?) -----

`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 sometime ago the II + I = III definitely referred to the quantity of the"I" lines. He even went up to someIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII or similar. Now in myfeeble mind to construct 'symbols' for expressing /_how many "I"sthere are_/ is not the other way around. "3" stands for III, theCOUNTED amount of the lines and not vice versa.So: what are those _"naturally occurring"_ things that serve for beingabstracted into numbers?*

`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. Andplease, do not divert into quite different topics, where you may havea point in some other aspect. We are talking about numbers, not themasculinity 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 afigment pertinent to the figment of a "physical world" - quite outsideof 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 theyapply them to quantitative considerations? what if another 'logic'uses them in a different math (e.g. where 17 is not identifiable as aprime number? Is it likely that more will be found - as was the zero,or are we in a "mathematical omniscience" already? Is our restrictionto the 'naturals' - natural, or just a consequence of our insufficientknowledge (caabilities)?May I quote a smart person: there are no stupid questions, only stupidanswers. 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. -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-l...@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.