The nature of number

[jonb]

Is there an accepted standard definition of what a number is?
Do you think it could be defined within maths or does it have to be externally evaluated?

[Cato]

Is there an accepted standard definition of what a number is?
Do you think it could be defined within maths or does it have to be externally evaluated?

Here you go. Knock yourself out.

http://plato.stanford.edu/search/searche...ery=number

[jonb]

Is there an accepted standard definition of what a number is?
Do you think it could be defined within maths or does it have to be externally evaluated?

Here you go. Knock yourself out.

http://plato.stanford.edu/search/searche...ery=number

Yes you see this is my problem. I am not a mathematician, I am an artist and thus am trained to look. The idea that the number is a fixed point does not seem consistent with the results I am getting from my geometry. Also Russell's Paradox, it seems to me would naturally arise when both the observer and the observed are within the same system, in this case maths. As an artist I know to observe the nature of a thing I must be to some extent external. As such to understand the nature of number I would have to have an external tool which is not affected by the number.
In short I feel mathematical proofs of number cannot provide a definition of what number is.
Am I barking, or do you think there could be a scintilla of logic in my position?

[Tobie]

A simple (but not all-encompassing) definition of a (real) number is "A symbol that represents a quantity". This is almost always what they are used for. I can't think of any exceptions right now beyond when they are just used as symbols ( in codes for example ). Other kinds of number generally are conceptual ( complex/imaginary for example, seeing as we can't visualize them beyond mathematical diagrams ), constants (e for example) or ratios ( pi is a good example, because due to General Relativity it is not a constant ).

Hope this helps in some way.

[jonb]

Sorry I think that is a definition for a numeral. I'm trying to get at the nature of the number itself.

[Cato]

Sorry I think that is a definition for a numeral. I'm trying to get at the nature of the number itself.

This short Q&A may help.

http://mathforum.org/library/drmath/view/58756.html

[Categories+Sheaves]

Is there an accepted standard definition of what a number is?
Do you think it could be defined within maths or does it have to be externally evaluated?

There are more systems of numbers than you can shake a stick at--you've got natural numbers, integers, ordinals, cardinals, surreals, rationals, reals, hyperreals, p-adics (one such system for every prime p!), hyper(rationals/integers/naturals/p-adics), every integral extension of the integers, every algebraic extension of the rationals... all of these systems are given formal logical/axiomatic foundations within mathematics, (esp. modern set theory) but each of these systems is motivated by some concern outwith mathematics.

There is a tremendous plurality of ways we can interpret (and use) numbers. Which ones do you want to explore?

Yes you see this is my problem. I am not a mathematician, I am an artist and thus am trained to look.

As long as you don't feel obligated to argue 1 != 0.999... or some supposed inconsistency in modern set theory, we'll get along just fine.

The idea that the number is a fixed point does not seem consistent with the results I am getting from my geometry.

You'll have to explain that further. But for now you can enjoy this quote:

The introduction of numbers as coordinates is an act of violence.

Moving along...

Also Russell's Paradox, it seems to me would naturally arise when both the observer and the observed are within the same system, in this case maths.

Russell's paradox arises when a model of set theory (call it M) can take itself (or the totality of the sets it contains) as an set for mathematicians to fiddle with (the axiom of restricted comprehension can only protect set theory from Russell's when you don't have M contained in M). This whole observer/observed thing sounds like some watered-down pop-QM... it looks ill-founded and I'm a little leery of it.

As an artist I know to observe the nature of a thing I must be to some extent external. As such to understand the nature of number I would have to have an external tool which is not affected by the number.

You're able to observe and reflect on the way you act in everyday life, so this issue has to be at least a little more nuanced and complicated than you're making it out to be.

In short I feel mathematical proofs of number cannot provide a definition of what number is.

Well, there are two ways to take this: we have some number-intuition from handling enumerable things in our lives, and sometimes we write out a list of axioms that dictate how a certain system (in this case, numbers) should function. Do you want the axiomatization or the phenomenology of math?

Am I barking, or do you think there could be a scintilla of logic in my position?

Well, I'm game to kick this stuff around.

[jonb]

OK Let’s play. To do this I used geometry.(I am trained as an artist so if my terminology is wrong see if you can get round it and you may have to help me out).

Take a simple sum and turn that into a geometric form

1+1=2 lets express this geometrically
We have a line of an unknown length at one end a number at the other a result and between these points a place at which changes one to the other



This is the basic format of the equation


For the sake of clarity let’s name these three parts. The origin/‘O’, the function/’F’, and the result/’R’.



Ok lets play with this can we now do anything with it?
Well I have found you are not restricted to just comparing one number at a time, you can compare a series of numbers to a second series.





So to draw this up we only need the outer parameters of the two series and a function point



You will notice that the direction of the resultant series is in the opposite direction to the original series.



Any two series with a common factor or point of comparison can be matched.


For every position in the origin has a corresponding position in the result. No matter where the line going through the function is in the origin it also connects with a corresponding place in the result. So can we can draw the conclusion from that every series with a common function contains an equality of positions?



Is this actually is a problem? Pi is an irrational number yet it comes directly out of the series. We may not be able to place it exactly in the line of fractions or decimals, because it has no exact decimal or fractional position, but if we have a minimal maximal range where it could sit in the origin then that directly translates to the same area in the resultant series.



Actually it seems we only need to place three positions and their values, in the series and we automatically know where all the other positions in the coherent series are, and what their values are. It does not matter which method of counting numbers we use. It seems once you have sufficient parts, the series will generate all the rest.
Given this the supposition could be there is a single template for all series. Is this then is a problem for it seems to be in direct opposition to the conclusions academic mathematicians have drawn?

I would like to know what you think

[jonb]

Nobody want to play? Oh well just in case for future reference, if you use this form and set the function at zero the result is a set of Zeros: ie the number has width or I have found a set of numbers all of equal value however which progress towards having more or less value.
Speculation: Are numbers real? and are they composed of sub particles like more tangible things?
Is the set not composed of an infinite number of points, but rather the number is only representative of its position in the set, so is the Concept of numbers as much of the fabric of the universe as gravity and as such has to be included in any great unifying theorem. As such can we use number to explain themselves?

[Categories+Sheaves]

OK Let’s play.

Yay!

1+1=2 lets express this geometrically
We have a line of an unknown length at one end a number at the other a result and between these points a place at which changes one to the other

Never seen anyone arrange this stuff quite like that. Even the syntax of the equation is given a shape. Pretty neat... Your spatial intuition doesn't ever take a break, does it?

For the sake of clarity let’s name these three parts. The origin/‘O’, the function/’F’, and the result/’R’.

I'm so going to try to visualize stuff this way the next time I come across an endofunctor on a category of manifolds.

Ok lets play with this can we now do anything with it?

Depends on what you mean by anything, but the stuff you've brought up all seems to work.

Well I have found you are not restricted to just comparing one number at a time, you can compare a series of numbers to a second series...
...So to draw this up we only need the outer parameters of the two series and a function point...
...You will notice that the direction of the resultant series is in the opposite direction to the original series...
...Any two series with a common factor or point of comparison can be matched...
...For every position in the origin has a corresponding position in the result. No matter where the line going through the function is in the origin it also connects with a corresponding place in the result. So can we can draw the conclusion from that every series with a common function contains an equality of positions?

1. From what I can tell, the class of functions you're looking at (or at least the functions that can be represented like this) are continuous and strictly monatonic (in case you needed some jargon dumped on you )
2. I'm not sure what you mean by 'equality of positions'... are you talking about the correspondence 'y=f(x)' for each number, or the existence of some 'x' that satisfies 'x=f(x)'?
3. If your responses to #1 and #2 are something along the lines of "yes" and "the former", then this all looks good.

Is this actually is a problem? Pi is an irrational number yet it comes directly out of the series. We may not be able to place it exactly in the line of fractions or decimals, because it has no exact decimal or fractional position, but if we have a minimal maximal range where it could sit in the origin then that directly translates to the same area in the resultant series.

Nope, not a problem. This sort of approach is exactly how we're able to define stuff like "2 to the power of sqrt(2)". Because we're able to interpret exponentiation with rational numbers quite easily, we can calculate exponentiation with irrational numbers by (more or less) approximating it with rational numbers (but there's a whole bunch of delta-epsilon math-guts I'm refusing to spill here ).

Actually it seems we only need to place three positions and their values, in the series and we automatically know where all the other positions in the coherent series are, and what their values are. It does not matter which method of counting numbers we use. It seems once you have sufficient parts, the series will generate all the rest.
Given this the supposition could be there is a single template for all series. Is this then is a problem for it seems to be in direct opposition to the conclusions academic mathematicians have drawn?

Well... now it looks like you're looking at an even smaller class of functions (by 'smaller' I mean 'strictly contained in the last classification', so all the stuff I said previously still applies). My tentative answer to this is 'yes, this isn't really challenging anything in modern math' but only with the caveat that your notions of functions and series don't quite match the way these things are thought of in modern math. Depending on how you iron out the specifics of this stuff, I might have more objections later on...