The nature of number

[jonb]

I really thank you I do appreciate how you are jumping through hoops for me, I will try to work out a way of conveying my meaning, but in that I am able to draw quite a bit from what you say and understand your reference points I think we maybe quite close.

[CliveStaples]

The numbers themselves are usually defined by how they function. e.g. isn't not entirely clear prima facie what it would mean for one infinite set to be 'bigger' than another, (I'm echoing Wittgenstein here) so this business with the cardinalities of sets has to be grounded in the behavior of functions between sets (esp. with bijections). These structures and the ways in which the work co-determine each other, so it's a bit wonky to talk about the 'nature' of numbers sans the way they work.

In one of the more abstract interpretations of mathematics, statements about 'numbers' (N, Q, R, C, etc.) or about any system in general are actually statements about every system that satisfies the relevant axioms.

So when we say something like "1+1=2", we're not referring to some Platonic "1" and "2". Those are just structural placeholders for any system that satisfies the Peano axioms and which therefore has an analog of "1" and "2".


So the notion of a "number" isn't even necessarily coherent, depending on how you interpret mathematical statements.

[jonb]

I want to thank every body that contributed to this thread. Yes to an extent I now think I may understand. The thing that set me off in the first place were a set of BBC documentaries explaining number, which seemed to be propagating ideas which were at odds with the knowledge I had from my own field. The problem was these documentaries were using analogies and not explaining them as being analogies, and it was not clear what the analogy was about.
I tried working through the ideas in these documentaries with friends to no avail, but your answers have enlighten us.
But if anyone comes across a good source that explains the concepts in maths to a lay audience I for one would be up for reading it.

Once again thank you all.

[Whateverist]

.. or ratios ( pi is a good example, because due to General Relativity it is not a constant ).

Really? I've never heard or thought about this. I can't think how General Relativity (or anything else) prevents pi from occupying one unique location on the number line. How can it not be a constant? I am intrigued. [But I have only begun reading this tread so perhaps this has already been discussed.]

[Categories+Sheaves]

Really? I've never heard or thought about this. I can't think how General Relativity (or anything else) prevents pi from occupying one unique location on the number line. How can it not be a constant? I am intrigued. [But I have only begun reading this tread so perhaps this has already been discussed.]

If some neighborhood of spacetime has a nontrivial curvature, any 'circles' you encounter there may not satisfy everybody's favorite equations involving pi (like the one relating radius to circumference, etc.). So in some sense it isn't very good at being an observable physical constant. But pi shows up in a bunch of equations relating to curvature anyway (exemplo gratis) so pi's existence as a fixed/abstract constant is even more important in general relativity

[Whateverist]

Hello .. anyone still here? I've had an idea about your wiggling number line mappings, jon.

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.

So we can either reverse the order of the numbers, or, if you prefer, just multiply by the opposite of each number.

Another issue is that the spacing of the whole numbers in each series (which I think of as the scaling on your number lines) is not identical. If this matters to you one way to fix it would be to move the location of the point which serves as the fulcrum for the rotating line.

If you multiply by 1, the fulcrum point is exactly midway between the two number lines. But if you multiply by 2, to maintain equal spacing between numbers you would have to move the fulcrum twice as far from the line containing the results as from the line containing the numbers which are multiplied by 2. The larger the number you multiply by, the greater the distance to the line containing the results. So I imagine placing the fulcrum point on a slider between the two lines. Now the length of a 'unit' on the starting line and results line can be equal.

With this adjustment there is no longer any problem with multiplying by zero. When multiplying by zero the fulcrum would simply be hard up against the results line at the zero point. All numbers on the starting line go only here.


[Alternatively, we can keep the fulcrum fixed midway between the two lines but allow the multiplication but allow the spacing of the numbers on the results line to increase when multiplying by numbers greater than one and shrink when multiplying by positive numbers less than one. In that case, when multiplying by zero the spacing between numbers becomes zero and any particular segment of the results line shrinks to zero too.]

[jonb]

Yes you're right, but if you think of the space theoretically, a distance of zero is still a distance, which seemed to create a theoretical width for a number, or that a width of the series could be made up by a number of numbers of the same value. Which both ideas would create a problems if as I was informed in a BBC documentary that there could be infinities of different sizes. As it would seem this would enable you to make a comparison between any infinities with a common factor and relate the numbers in them one for one.
But I was thinking only in analogue terms and the documentaries were making analogies which are not shall we say substantiated and they did not make it clear they were using analogies.
I am not good at explaining, because of my dyslexia I have problems with exact terminology, so if you still have a problem or if someone can put up a better post feel free.

[Categories+Sheaves]

The infinities you're comparing (between two lines) aren't different in the sense that your BBC documentary was talking about. The reals (numbers whose decimal/binary/whatever expansions don't have to terminate) are a "bigger set" than the integers or the rationals (numbers that eventually terminate in some base n expansion) but not in the sense that there's some real number bigger than all the rational numbers. Have you ever tried reading through Cantor's diagonalization argument? (I might dump a run-through of it later... it's really not that bad).

If you want to see these differing infinities expressed within a number system, you're asking for the ordinals. And those are kind of crazy.

[jonb]

Do it, you're our man, go on fly,
do what you do best man!

[jonb]

Until we get on to Cantor
Ok I wish to add something. It is my contention that the number is not the generator of the series, but that the number is a fraction of the series. If I am wrong please disabuse me of this. Similarly I have also noticed that a point is afected by the field or dimention it is in.
A point in 2D space

Any number of projections can be made from a point in two dimentional space and be equaly spaced around that point.

However in 3d:




Only specific numbers of projections can be made from a point in three dimensional space and be distributed equally around that point. Given this; it seems the structure of the field dictates the material within it, rather than the material creating the structure. Please help me through this.