Please allow me to preface this post by declaring that I am not a math guy, but I smell a lot of brains here, so I come to you all in hopes that some one will be able to help me figure out what I have stepped in.
I will attempt to lay this out as clearly as possible and perhaps not sound entirely mad.

Here are the propositions:
1) numbers are the language we use to comprehend reality
2) the number line is in fact a geometrical artifact representing one dimensionless point expanding in two directions to infinity.
3) the tic marks below the numbers of the number line express that the numbers are integral or are the line

Assuming the above points are true, here is the root of my problem:
Shouldn't one be able to put a dot on a piece of graph paper and by use of the inherent value of that origin generate the rest of the number line mathematically?

If one of my kind and insightful readers can provide such a method I would greatly like discourse on the deeper meaning that the answer denotes.

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GS

I have no idea if I understood your post but....
How would you derive the lines direction from the dot?

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I think you need a formula (eg, y=x) to describe a line ...

not all lines continue to infinity - some have a set origin (eg, 0,0); some describe closed shapes (eg, a circle of radius n) ...

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Heya,

If your describing a one dimensional line is it necessary to determine direction?

Dee

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It appears that mind, as manifested by the capacity to make choices, is to some extent inherent in every atom. The universe as a whole is also weird, with laws of nature that make it hospitable to the growth of mind. I do not make any clear distinction between mind and God. God is what mind becomes when it has passed beyond the scale of our comprehension. ~ Freeman Dyson

In the Pythagorean school, as far as I know, the only constructions allowed were with (circle-drawing) compass and unmarked straightedge. I don't think they considered it valid to "pick a random point" other than what was "given". So, the only points that could be constructed from a finite number of beginning points were intersections of circles centered about a point, and lines through a pair of points, where the points were either those starting points, or already-constructed points.

Thus, in the Pythagorean world, the only numbers that "existed" (as distances--angles, areas, etc. are a different story) were real numbers constructible from the integers and the operations: add, subtract, multiply, divide, and square root.

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My apologies, the number line that I was talking about is the one from the chalk board in seventh or eighth grade math, when it was just, math. A point on a infinite line, first year geometry , the line is then graduated into equal arbitrary lengths with the point falling on one of these tics we will call this point the origin and we will assign it a value: zero. Along this line, outwards from the origin in each of two directions the other tics will be valued with the integer number set in ascending order. Positive integers to the right and negatives to the left just as tradition teaches.

My poser, to clarify, is this: If you were to take a piece of paper and draw a number line on it, you would most likely draw a dot first. then a line to intersect. Perhaps as two rays emanating outwards from your dot with little arrow heads on them like I do. Then you would make a series of divisions and number them, etc... If you were to think of what you have there in terms of an origin and rays, that is as a source and its emission; is there a mathematical process by which five comes from four from three and so on.
Hope that Helps
GS

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GS

right, what do you think?
GS

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GS

Nope. I always draw the line first and mark the dots from there. For one, my artistic skills are such that I've more of a shot of getting a dot on a line than crossing a dot with a line.

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+1 ?

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lol
so, then logically, one should be able to work that in reverse. 5-1 = 4-1 = 3-1 = 2-1 = 1-1= 0
therefor reversing again:
0+1 = 1+2 = 3+1 = 4+1 = 5...
by what means the initial 1? or any of the others?
for 0 to become one via addition 1 must be added to it. I am looking for a way for the number line to boot strap its way into existence. starting with an initial value and then using that value to produce the next value etc... 1 1 3 5 8 13 21
that sort of thing. preferably in harmony with conservation

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GS

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The theory I have is this: if one takes a value for the origin of a number line to be the absolute value of one instead of zero, then it is possible by polarization for the origin to expand outwards in a fib type sequence building the number line out to infinity pointing to some interesting concepts relative to cosmology if taken as a parallel to the expansion of a singularity.

I am seeking help with some of the deeper areas of this as I am over my head when it comes to things like field theory. I have the proofs out to thirty five or so on disk and 100 on paper, That is where I found connections to phi, and other related irrationals. It seems like what I am looking at is like a zero dimensional point being acted on by an unknown outside influence to produce a one dimensional figure. this 1d figure then is acted upon to give rise to a 2d existence etc...



where 7.b.1 is the origin/ singularity / abs 1 polarizing to emit +1/-1
7.b.2,3 & 5 are the fib seq
7.b.4 is the tribonacci seq
by repeating this process, I maintain that the full and infinite set of integers can be generated, and further more that this process is an analog of how our universe is self induced (dimensionally, not electrically) from and by a singularity.

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GS

Then I have some questions about your theory...

Why is it preferable to define the origin as the absolute value of one, instead of zero?
What do you compute the the absolute value of one to be?
How do you achieve "polarization for the origin"?
How does your number line relate to cosmology?

Those'll do for starters.

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???? By placing the dot on a piece of graph paper, of course, you have inserted it into an existing number matrix, so I guess it depends on where you want to go from there. The only way to make your question meaningful (if I understand it at all) is to ask: if you create an isolated Universe that consists only of the dimensionless dot (we'll observe more rigorous niceties and say "point"), is it possible to generate the rest of the number line from that? Is the big bang all about the numbers?

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"To complete the picture all the photons can be seen to be synchronising friction on and off throughout the overall cone which itself is synchronised to the equal and opposite reaction of equilateral triangulation"... by a scientificator in ATM, too priceless to be lost to posterity.

Maybe, or it could be the bautwurst I had last night1 1 2 3 5 8 13 21 ... but I know you knew that, it was just a typo It gets deeper than field theory! As near as I can tell, without knowing the exact details about your process, you can get integers and rational numbers from this process. Phi is a limit of the ratio of the Fibonacci numbers, so it will appear only in approximation, as you get larger and larger values. Is that what you are seeing?

Thanks! I think. Do we smell bad? Do you smell anything else, here, such as wit, sarcasm, or poor attempts at humor?

Well, judging by it's smell...

I would argue that we comprehend by means of conceptions, and that mathematics is merely one construct (a precise one, to be sure) whereby we may grasp various concepts. Numbers are merely a descriptive language that's more precise than saying "no, a little bit more... more... slowly... there - hold it, and hammer that nail home."

The tic marks are arbitrary. I could just as easily invent my own yardstick and say, "mark the line every Muglong."

No. A dot is defined by two things:

1. A set of three values, whether they be cartesian (x, y, z) or polar (r, θ, φ). Other coordinate systems exist, as well, but in three-dimensional space you need, naturally, three values.

2. A known reference point. Center of the Earth, the Sun, or the SW corner of one's backyard are always handy. Anything, really, so long as it's position is known, and you're referencing all measures off the common reference point.

The combination of these two comprise a coordinate system. I think if you'll take a look at this last link, you'll find we've come a long, long way since Pythagorus.

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If we halt the ISS, all versions of Ares, and transport Orion and Altair aboard DIRECTv3's Jupiter family of Shuttle-Derived Launch Vehicles, we just might make it back to the Moon by 2020.

Minor point, but I don't entirely agree. The marks on the line share a relationship between themselves, and with their environment. Only the labeling is arbitrary. If the marks are evenly spaced, there is a realtionship of interval; likewise if they vary in a regular progression. If you want to get really bothersome about it, assuming the marks have a finite width and length, there is also a specific proportional relationship between the width/length of the mark and the interval between marks, etc. Those proportions are inherent and invariant (in an idealized set); we simply provide defined units with which to describe them. I have a book somewhere that describes mathematics as man-made. I couldn't disagree more. Mathematical relationships are inherent in just about everything that is measurable. It's an observational and logical discipline, not an artificial one.

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Time is a great teacher, but unfortunately, it kills all its pupils. Hector Berlioz

"To complete the picture all the photons can be seen to be synchronising friction on and off throughout the overall cone which itself is synchronised to the equal and opposite reaction of equilateral triangulation"... by a scientificator in ATM, too priceless to be lost to posterity.

Certain mathematical concepts are indeed man-made. The base ten number system and our various mathematical symbology are prime examples. Other mathematical concepts are inherent in the natural systems they describe, such as the derivative, which describes the slope of a curve at a specific point.

Whether or not something mathematical is man-made or natural depends entirely upon what that something is.

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If I set the budget, we'd have Ares and more. Unfortunately, I don't set the budget, and Ares is just too expensive and too far out for us to accomplish our goals within the budget we were given.

If we halt the ISS, all versions of Ares, and transport Orion and Altair aboard DIRECTv3's Jupiter family of Shuttle-Derived Launch Vehicles, we just might make it back to the Moon by 2020.

Still don't quite agree. Base ten and symbology are descriptive tools, and those are created. But the relationships they describe are real. You can choose to use base 2, base 3, or base 25 systems if you wish, and create whatever system of symbols you wish, so long as they are consistently descriptive of observation and logical thought, but in the end, they describe the same real relationships. I guess I should have been more clear, but the whole point is that I disagree with any suggestion that mathematics is definable in terms of symbology instead of relationship. The relationships are real, and the same regardless of the system used, whereas the symbolic tools (labels) we use are arbitrary.

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Time is a great teacher, but unfortunately, it kills all its pupils. Hector Berlioz

"To complete the picture all the photons can be seen to be synchronising friction on and off throughout the overall cone which itself is synchronised to the equal and opposite reaction of equilateral triangulation"... by a scientificator in ATM, too priceless to be lost to posterity.

Numbers in and of themselves are a pure abstraction - they are meaningful to reality only when we assign them as values in a process of measurement.

The mathematics of measurement is the language we use to comprehend reality, not the numbers themselves.

Yes...

...and no. A dimensionless point has no defined size, so cannot expand.

Be careful here. The number line represents a portion of the set of real numbers, and the ticks represent regular incremental intervals. The numbers that you assign to the ticks merely indicate the location of your number line in the (already infinite) real number field.

Numbers are abstract - they don't form nice lines by themselves, or any other shape for that matter. Numbers are not physical things. We use devices like the number line for convenience as they are useful when we try to understand the relationships between them and when we make measurements of our physical world.

When you draw a number line on a piece of paper and assign numbers to the ticks, you are in effect making a measurement. This is where counting becomes geometry. However numbers by themselves are not geometry.

Yes of course, assuming that your graph paper represents an abstract space based on a given number field, and that you have assigned a number (i.e. value) to the point on the graph paper. You don't even need graph paper, algebra will do just fine

Personally, I prefer base 77, as it's the sum of the first 8 primes, and, as such, is easily represented in binary. But that's beside the point...

I'm pretty sure we're on the same sheet of music, here, aside from the terms each of is using to describe it.

The base 77 comment was a joke, by the way...

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If I set the budget, we'd have Ares and more. Unfortunately, I don't set the budget, and Ares is just too expensive and too far out for us to accomplish our goals within the budget we were given.

If we halt the ISS, all versions of Ares, and transport Orion and Altair aboard DIRECTv3's Jupiter family of Shuttle-Derived Launch Vehicles, we just might make it back to the Moon by 2020.

I thought possibly so as well.

I dunno...has interesting possibilities. I've been trying to think of the sort of being that might fall into such a system naturally--the mathematics of the intelligent centipede, maybe....? Or would a 77-segmented creature go naturally to base 154?

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Time is a great teacher, but unfortunately, it kills all its pupils. Hector Berlioz

"To complete the picture all the photons can be seen to be synchronising friction on and off throughout the overall cone which itself is synchronised to the equal and opposite reaction of equilateral triangulation"... by a scientificator in ATM, too priceless to be lost to posterity.

Even base ten might not have been so "normal" for us.

Maybe we could have instead gravitated to base 6, with one hand for the "6's column" and one for the "1's column", instead of just ten fingers for the "1's column".

That lets us count to 55_{base 6} ( = 35_{base 10} ) with the very simple counting of fingers for each digit.

(Which I think would be more natural than using binary (base 2) where we'd be able to count to 1111111111_{base 2} ( = 1023_{base 10} ) using each finger for a binary digit (no pun intended).)

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Hmmm....Which brings us to the potential for a different thread (since I don't want to drag this one too far afield) on the human psychology of numerical intuition: which (6 or 10) is the more intuitive organization, and do we use base 10 just because of the way our brains are wired (or is it because some Neanderthal counting sheep was too dumb to make the other connection?--Rhetorical question only--especially regarding the N word, which I toss out with tongue fully in cheek, claiming no lineage whatsoever).

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Time is a great teacher, but unfortunately, it kills all its pupils. Hector Berlioz

"To complete the picture all the photons can be seen to be synchronising friction on and off throughout the overall cone which itself is synchronised to the equal and opposite reaction of equilateral triangulation"... by a scientificator in ATM, too priceless to be lost to posterity.

Well, with a base-77 system, it would be easy to subdivide it into it's constituent primes: 77=2+3+5+7+11+13+17+19

Thus, it may be that the creature works in base-77, but if need be, he/she/it can break problems apart into base 2, 3, etc. up to 19, as required - or any sum of any of those numbers, for that matter, such as base 38 (2+3+3+17+13).

Why? I dunno. Perhaps it's brain would be wired such that it would have eight computing centers, each one "tuned" to one of the constituent primes. Perhaps communicating in code is a requisite for survival, and it's prime-based system allows it to do just that with lightening rapidity.

As for legs, it's an arachnid. Eight each of: eyes, legs, brains.

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If I set the budget, we'd have Ares and more. Unfortunately, I don't set the budget, and Ares is just too expensive and too far out for us to accomplish our goals within the budget we were given.

If we halt the ISS, all versions of Ares, and transport Orion and Altair aboard DIRECTv3's Jupiter family of Shuttle-Derived Launch Vehicles, we just might make it back to the Moon by 2020.

How about base 64? There's already a standard representation for it (it can be represented in ASCII characters, and makes a reasonably compact encoding for binary data in ASCII text).

I would not call positional number systems an invention, though their application to everyday use might be one. They simply follow from the fact that any number can be represented by a polynomial with terms of the form a*b^n, where a < b and a, b, and n are integers. In the case of the decimal system, a is one of 0 to 9, b is 10, and n is the column, 0 for the 1's column. I would expect any alien mathematician to be aware of the encoding, whether they use it or not. If they lack a formal understanding of such things, their computer sciences would be in a truly sorry state, and I doubt they'd be up to interstellar communications...

Personally, I like balanced ternary. +, -, and 0, with a base of 3. From -5 to 5:
-++, --, -0, -+, -, 0, +, +-, +0, ++, +--

Compact, naturally represents negative numbers, and is easy to represent electronically.

Pythagoras forbade his followers to eat beans. Maybe he was on to something after all!

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The bar has been lowered for the promotion of ATM ideas; the bar for the acceptance of ATM ideas must remain high.

That's the key question I'd have. It seems like some ideas about mathematics. OK, but what does it have to do with cosmology?

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It is not terribly clear precisely what you are trying to do here, but I gather that you are trying to construct the real numbers from something like first principles. I have also looked at many of the other posts in the thread, and it appears that some sort of geometrical approach, not necessarily a ruler and compass construction but similar ideas are being used.

First, you will not be able to develop the real numbers using a simple constructive process. That is because there are, in a sense that can be made precise, too many real numbers to permit that sort of thing. There are infinitely many integers, infinitely many rational numbers, and infinitely many real numbers. It turns out that the infinity that corresponds to the integers and the rational is the same infinity. The infinity that corresponds to the reals is bigger. This can be made precise with the idea of cardinal numbers -- see Naive Set Theory by Paul Halmos for a nice discussion.

It is fairly easy to accept the natural numbers and the integers, they are the "tic marks" on the number line. If you can accept the natural numbers without proof, then you should also be able to accept what are called the Peano Axioms, which are just a set of assumptions that say that there exists a set that behaves the way we intuitively believe the natural numbers should behave. From those axioms one can quite literally construct the integers, the rationals, the associated operations of addition, subtraction, multiplication and division. Next comes the harder part.

The rationals provide a framework for the number line which is the real numbers. The basic problem is that there a lot of "holes" in the rational that need to be filled in. This problem is not algebraic in nature but is topological. Topological methods are needed to complete the rationals. this can be done using what are called "Dedekind Cuts". It is a rather involved process, and is described in Halmos's book or quite succintly in Foundations of Analysis by Landau. In fact both books start with the Peano Axioms and go through the construction of the real numbers.

Both books referenced are quite thin and relatively easy reading if you enjoy mathematics. Landau is a bit dry and his style is telegraphic, but he gets the job done quickly and efficiently.

Only after you have built the real numbers in this fashion can you reall proceed to develop the necessary concepts for calculus, field theory and such.

on a number line each integer has a single nextinteger[]. nextinteger[] is the same as +1. the inverse of the next[] function gives the previous[] function. previous[] is the same as -1. the number line is therefore a specific type of graph. in a more general graph each node can have many nextnode[]'s and many previousnode[]s. to determine whether 0 or 1 is more fundamental you might want to look at how graph theory would look at a number line.

geometry is just arithmetic in more than 1 dimension with an extra axiom for determining the length of the hypotenuse.