Because we know what the number 1 is, does that mean we have "complete" knowledge of every real value between 0 and 1? On the real number line? I don't think it does.

What is the real number line continuous on, or with respect to? Since continuity requires that it be relative or in respect of something, as some seem to think?

The real number line is continuous, because because you can devided into infinitly many numbers. If you take a subset of the line, say bewteen 1 and 2, this subset hold an infinite number of real numbers, 1.00000000000000..........1, 1.0000000000000.........2 and so on.

This is my undeerstanding anyway (I dont have my old algebra text book at hand :)

So the real number line, which, by definition, extends indefinitely from both ends, is continuous?
I'm sure I remember a lecture about that. So there's no "on", or "with respect", it's just a line?

The first question I asked in the OP, is there an answer? Is it a question?

I am not sure I understand the question. By complete knowledge do you mean complete knowledge of all possible numbers or...

The real number line is not continuous. Continuity is a property of functions, not sets. You are using the wrong terms again. Why am I being so picky? Mathematicians need to be precise, so their use of language is precise. More importantly, your misuse of mathematical terms is leading to your misunderstanding.

The terms you want are dense and complete. The real numbers are dense (http://en.wikipedia.org/wiki/Dense_set) in the sense that for any real number, one can find some other real number that is arbitrarily close to the first real number. The integers obviously are not dense. The rational numbers do form a dense set. What distinguishes the reals from the rationals is that the reals are complete. The square root of 2 is irrational. One can construct a sequence of rationals that tend toward the square root of 2. The limit of the sequence is not a member of the set from which the terms are constructed. The rationals are not complete (http://en.wikipedia.org/wiki/Completeness_%28topology%29). The real numbers do form a complete set.

I disagree. 8 is a positive number, and furthermore, it is a subset of 1,2,4 and 8 - which says that 8 is an even number... but 8 is a natural number, a whole number an INTEGER and even a rational number of 8/1, and a real number 8.0.

Strictly speaking, 8 (the integer), 8/1 (the rational) and 8.0 (the real number) are different things because they come from different sets.

The real number line is not continuous. Continuity is a property of functions, not sets. You are using the wrong terms again. Why am I being so picky? Mathematicians need to be precise, so their use of language is precise. More importantly, your misuse of mathematical terms is leading to your misunderstanding.

The terms you want are dense and complete. The real numbers are dense (http://en.wikipedia.org/wiki/Dense_set) in the sense that for any real number, one can find some other real number that is arbitrarily close to the first real number. The integers obviously are not dense. The rational numbers do form a dense set. What distinguishes the reals from the rationals is that the reals are complete. The square root of 2 is irrational. One can construct a sequence of rationals that tend toward the square root of 2. The limit of the sequence is not a member of the set from which the terms are constructed. The rationals are not complete (http://en.wikipedia.org/wiki/Completeness_%28topology%29). The real numbers do form a complete set.

As I said, I did not have my old algebra text book at hand. Be precise, yes, but I think I answerd his question. But, the correct term is dense, youre right.

Strictly speaking, 8 (the integer), 8/1 (the rational) and 8.0 (the real number) are different things because they come from different sets.

But...

8=8/1=8.0

But...

8=8/1=8.0

You're not thinking abstractly enough.

The real number line is not continuous. Continuity is a property of functions, not sets. You are using the wrong terms again. Why am I being so picky? Mathematicians need to be precise, so their use of language is precise. More importantly, your misuse of mathematical terms is leading to your misunderstanding.Yes, I was about to make the same point. But I think you are being a little harsh; the continuity of the real line was a term once widely used, and still is, at least at elementary level. If our friend used the term, you may perhaps blame me, as I used it in another thread, in full knowledge that I was abusing language, but in the hope the intuitive import would be clear.

(Real numbers is when you clump all the irrational numbers and rational together - in other words, any number that can be expressed as A REPEATING OR A TERMINAL, that is ANY NUMBER EXPRESSED AS A DECIMAL.)

If anybody gets really angry about someone else's opinions on this thread, the Middle East becomes not simply more understandible and banal, but even more, dare I say it, rational - speaking of rational numbers.

So, the real numbers are not, strictly speaking, continuous.
Instead, they're complete. Also dense.
So what is the point of describing a line, extending arbitrarily, when talking about the reals? Why is there no "continuous with respect to", except with respect to the line itself? I mean, like teachers do, and calculus lecturers...

Why, IOW, is it important that something is continuous "on" something? The real number line isn't described as being "on" anything..., at least I don't recall being told about how important it is that the reals are "on" anything, except this line?

The point is that every point on the line (coupled with some arbitrary point on the line chosen as the origin) can be mapped to exactly one real number, and vice-versa. There is a one-to-one and onto mapping between the points on the line and the real numbers. In other words, they are one and the same thing. (Absane - help please: Name of this theorem. I think the theorem is named after Dedekind and someone else.)

Continuity is a very important concept in mathematics. For one thing, a function is differentiable at a point only if the function is continuous at that point.

Right. I got that years ago myself (but what can be said about any other potential observer?), Functions is a way of describing continuity. Functions have a domain and range, either of which can be continuous or not on R, I think.
But how many functions map the same domain and range? Continuity is the idea of extension or projection.
With math, you use other ideas to explain or constrain the meaning of some principle. Continuous intervals and functions and limits is the standard approach to the real numbers, but usually a basic understanding of a continuous line is the model. I recall some proof of a step function being discontinuous, when we looked at continuity and limits.

Because we know what the number 1 is, does that mean we have "complete" knowledge of every real value between 0 and 1? On the real number line? I don't think it does.

I really don't want to get into it, but you can construct the real numbers just from the integers. I suppose that you first start out with 0 and 1. You establish basic rules of addition. 0 + 1 = 1, 0 + 0 = 0. Then you DEFINE 2 to be 1 + 1. Then you keep going. Soon you have all the integers greater than 0. You soon start asking about inverses. Well, we define x - y = k such that k + y = x. You find it easy to do 10 - 2 = 8. But, what about 2 - 10? Well, we can proceed the same way, but call it "-8" instead of 8 (since 8 + 10 = 12, not 2). After ALL this you simply define multiplication. Naturally, you ask about inverse operations of multiplication. Using "The Division Algorithm" (it's an actual theorem) you can get a sense of what decimal numbers look like. Once you fool around with this for a while, you ask questions like "find a rational such that a^2 = 2." You find that you cannot describe the solution with our current definitions. What do you do then? Define "a" (and all others like it) to be irrational. Blah blah blah.... define imaginary numbers.. blah blah blah.

My point? It all depends on your definitions, assumptions, and intelligence.

And the reals are complete for two reasons: Euclidean metric space and the definition of completeness. Based on these two reasons, there are no "holes" in the real numbers.

And the reals are complete for two reasons: Euclidean metric space and the definition of completeness. Based on these two reasons, there are no "holes" in the real numbers.OK. So you should be able to extend each point on the real number line orthogonally, i.e, extend the line "sideways", and create a surface, with no holes in it? You might even call it a continuous surface?

P.S. I recall exams where we were allowed to assume a continuous, real number line extending from + to - infinity.

Sure. The gadget for doing this is called "Cartesian product" R \times R, which is usually, and perhaps confusingly, written R^2. This is pronounced R-two, and not R-squared. It's called the plane.

The points in the plane are called "ordered pairs", so that for each x and each y in R, (x,y) defines a unique point in the plane. They are called ordered to emphasize the fact that (x,y) and (y,x) are different points.

Sure. The gadget for doing this is called "Cartesian product" R \times R, which is usually, and perhaps confusingly, written R^2. This is pronounced R-two, and not R-squared. It's called the plane.

The points in the plane are called "ordered pairs", so that for each x and each y in R, (x,y) defines a unique point in the plane. They are called ordered to emphasize the fact that (x,y) and (y,x) are different points.

More generally, A \times B = \{ (a,b) \ : \ a \in A \ and \ b \in B \}

Notice the order of the elements.

OK. So you should be able to extend each point on the real number line orthogonally, i.e, extend the line "sideways", and create a surface, with no holes in it? Exactly. If you have a sheet of matter that is a two-dimensional graph, and every real number (between zero and your arbitrary upper limit defining the size of the sheet) is represented by an actual point of matter, the sheet will not leak.

Why, IOW, is it important that something is continuous "on" something? The real number line isn't described as being "on" anything..., at least I don't recall being told about how important it is that the reals are "on" anything, except this line?

I can't understand what you mean by "on" something. Can you tell me one example of something being continuous "on" something?

I can't understand what you mean by "on" something. Can you tell me one example of something being continuous "on" something?Try to think of something called a "background".
If you wanted to draw a continuous line, with a pen or a pencil, say, you could use a piece of paper, or a desktop, which would represent a "background".
Any line drawn without lifting the pen, should be a continuous line, drawn on the paper or the desktop, at least from the start point to the endpoint, it should be continuous (assume there is enough ink in the pen, or the pencil has sufficient lead in it); you could also scratch a line in the paper with something.

The real number line is "background independent". It's something that represents the set of real values (rational and irrational), extending indefinitely, and bidirectionally, to "infinity".

In the real world, you might need to consider something like a piece of elastic string, or wire that can be drawn out indefinitely. Unfortunately, real physical bits of string and wire, even if they are elastic, are made of real physical atoms. Atoms are discontinuous; therefore, at some scale a wire or elastic string will reach a limit of extension, beyond which discrete atoms will separate (it will break), and no longer be a continuous "line".

A fairly accurate lay definition of a continuous function is that a graph of the function can be sketched on paper without lifting the pen -- exactly what you described.

Let me demonstrate why using the word "continuous" for the real number line is not a good idea. I am going to disingenuously prove that the rationals are continuous.

One way to render the real number line as a function is the equation y(x)=0\forall x\in\bb R. The graph of this curve coincides with the real number line. The function is trivially continuous because |f(x)-f(a)|=0\forall a,x\in\bb R. So, in a lay sense, the real number line is continuous.

We can also do the same thing with this function defined over the rationals: y(x)=0\forall x\in\bb Q. This function, too is continuous, either using the same definition of continuity as used above or using the metric space definition and using the L1 norm (which maps \bb Q\times \bb Q to \bb Q). The rational number line is continuous!


The term you want is that the real number line is complete: no "holes".

Returning to the OP, which includes the question: do we have complete knowledge of all the values between 0 and 1?

I don't think we can possibly (ever) have this. Just like we can't have complete knowledge of any system.

Another view, using a 2-d surface (a Cartesian product): if we had to specify every point, along both axes, by writing it down and then drawing it, then instead of a continuous surface (with no holes), there would be gaps, or if only one of each x,y pair was known, a pattern would emerge...
even if you used a fast machine to "find" all the points along both axes.

Returning to the OP, which includes the question: do we have complete knowledge of all the values between 0 and 1?
In what sense? We do know the reals are complete. Any other system you can invent that is complete and has operations analogous to addition, multiplication, is homeomorphic to the real numbers. For this reason the reals are also called the complete ordered field. However, most of the real numbers are not computable. (In fact, the computable numbers are a countable subset of the reals). So in this sense, we cannot have complete knowledge of the reals.

Just like we can't have complete knowledge of any system.
That's not true. You are thinking of Godel's theorems. These theorems apply only to systems of some inherent complexity. They do not apply to simple systems.

For this reason the reals are also called the complete ordered field. Professor Tim Gowers likens it to a soccer chant,

"There's only one complete ordered field! There's only one complete ordered field!"

Much higher brow than actual ones, more along the lines of

"You're going home in a ****ing ambulance!"

You are thinking of Godel's theorems. These theorems apply only to systems of some inherent complexity. They do not apply to simple systems.Define "simple system"...?
The reason we cannot have complete knowledge of any system is because of observation: you cannot have any knowledge (you can imagine as many complete sets of numbers as you like), without observation.

I agree.

Here is the answer.

Numbers can only get so big...before they cease to be numbers.
This also applies to the lengths of decimals.
They can only get so small...before they cease to be numbers.

Stupid Question....Stupid Answer!!!

Define "simple system"...?The answer is obviously found in Godel's work. Might I suggest looking it up?

http://en.wikipedia.org/wiki/Godel

"[Godel's] more famous incompleteness theorem states that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (Peano arithmetic), there are true propositions about the naturals that cannot be proved from the axioms."
The reason we cannot have complete knowledge of any system is because of observation: you cannot have any knowledge (you can imagine as many complete sets of numbers as you like), without observation.You're mixing up the notions of physics and mathematics.

Let me define field \mathbb{F}_{2}. It has two elements, 0 and 1 and the FULL set of all possible combinations of those elements is :

0+0 = 0
1+0 = 1
1+1 = 0
0*1 = 0
0*0 = 0
1*1 = 1

It is a field, by definition of a field (http://en.wikipedia.org/wiki/Field_(mathematics)#Definition_3)

There is nothing else to know about this field. It is very uninteresting. We have total knowledge of it. Even simpler would be the trivial ring. It has one element, 0 and the full set of all algebraic combinations is

0+0 = 0
0*0 = 0

If something is 'trivial' it generally means it lacks any interesting structure. This entity is insufficent to describe the natural numbers and so, I suspect, falls within the bounds of 'simple', as Godel's notion describes.
Numbers can only get so big...before they cease to be numbers.False.

Claim : There is no largest integer

Proof : By the Peano axioms, if n is a number, σ(n) is a number too. \sigma(n)\equiv n+1 and thus if n is a number, so is n+1. Suppose there is a largest integer, N. However, since σ(N) = N+1 > N is a number, N is not the largest integer. Thus there is no largest integer.

Therefore numbers have no upper bound.
They can only get so small...before they cease to be numbers.False.

Claim : There is no smallest number

Proof : The Reals are an ordered field. A field implies that for each N there exists M such that N*M=1, ie M = 1/N. Ordered implies that N+1>N is well defined. This also implies that if N+1>N then 1/N > 1/(N+1). Suppose that 1/N is the smallest number. That implies that N>'all other numbers'. By previous proof, such an N does not exist. Therefore 1/N being the smallest number is false. Therefore there is no smallest positive number.

Gee wiz....

Does that mean that that PROFESSOR (I'm too tipsy to remember his name) WITH A PHD IN MATHMATICS WAS LYING TO ME?



'Numbers can only get so large'

False.

Claim : There is no largest integer

Proof : By the Peano axioms, if n is a number, σ(n) is a number too. \sigma(n)\equiv n+1 and thus if n is a number, so is n+1. Suppose there is a largest integer, N. However, since σ(N) = N+1 > N is a number, N is not the largest integer. Thus there is no largest integer.

Therefore numbers have no upper bound.

'Numbers can only get so small'

False.

Claim : There is no smallest number

Proof : The Reals are an ordered field. A field implies that for each N there exists M such that N*M=1, ie M = 1/N. Ordered implies that N+1>N is well defined. This also implies that if N+1>N then 1/N > 1/(N+1). Suppose that 1/N is the smallest number. That implies that N>'all other numbers'. By previous proof, such an N does not exist. Therefore 1/N being the smallest number is false. Therefore there is no smallest positive number.

M is P is Q is WTF...Blah Blah Blah...

Can you make more sense?

Yeh mate. Alphanumeric is right.

So Physics and Math shouldn't be "mixed up"...? That's kind of amusing, to say the least.

Physics involves "making" observations. Math involves imagining things. Like I said, you can imagine as many sets of numbers as you like, you cannot observe all information, just like you can't draw a continous pattern using continuous lines - at some scale, it won't be continuous, and it takes time to draw a line, or compute the location of some point on a surface.

This thread was "derived" from another one about the nature of time. It turned into an "argument", about continuity, and was about (I thought), what the physical world would appear like at small scales. IF we could see all the detail, then we would see a continuous (complete, dense, or whatever - in the same sense I was allowed to assume a continuous line of real values during a calculus exam) pattern.

That got sidetracked to here.

He didn't prove didily squat!!!

No wait..."Suppose that 1/N is the smallest number."
I get it now!!! That proves everything!!! We are basing this 'proof'
on supposed numbers!!! Its now so clear!!!

Didn't prove didily squat!!!

Well... he is right in saying there is both neither an upper limit to any number, or a lower limit... You can count into infinity +, as much as you can count to a lower infinity -.

Well... he is right in saying there is both neither an upper limit to any number, or a lower limit... You can count into infinity +, as much as you can count to a lower infinity -.

The only thing we know is that we can never know the limit!!!

I'll clarify this later...I gotta do something!

Don't get me wrong. There are limits according to physical equations... but essentially, there is no limit without some kind of pointer.

Such as the Planck Length... 1.616 x 10^-33... this is a specified limit, but it has been shown that there could be smaller things called ''solitons.''

The reason we cannot have complete knowledge of any system is because of observation: you cannot have any knowledge (you can imagine as many complete sets of numbers as you like), without observation.You're mixing up the notions of physics and mathematics.So Physics and Math shouldn't be "mixed up"...? That's kind of amusing, to say the least.
Frud, there is a vast difference between mathematics and science. Your original post is about mathematics, not science. This thread is, per your original post, about mathematics, not science. You are conflating the two very distinct concepts by talking about observation. Observation belongs in the realm of science, not math.

So what is the difference between math and science? Proof is the backbone of mathematics. Once a mathematical theorem has been proven, it remains proven for all time. (Finding a flaw in some purported mathematical proof does disprove the theorem, but in that case the theorem was never really proven in the first case.) New developments in mathematics do not suddenly disprove existing concepts.

While science does have some theorems, for the most part science relies on theories. Scientific theories cannot be proven to be true. Scientific theories rely on observation to bolster the theories. Observation is not proof. To say otherwise is the logical fallacy of affirmation of the consequent (http://www.fallacyfiles.org/afthecon.html). Observing thousands of black crows only bolsters the hypothesis that all crows are black. On the other hand, observing one single white crow will disprove the hypothesis that all crows are black. Scientific theories are falsifiable but are not provable.

From http://en.wikipedia.org/wiki/Theorem#Relation_with_scientific_theories:
Theorems in mathematics and theories in science are fundamentally different in their epistemology. A scientific theory cannot be proven; its key attribute is that it is falsifiable, that is, it makes predictions about the natural world that are testable by experiments. Any disagreement between prediction and experiment demonstrates the incorrectness of the scientific theory, or at least limits its accuracy or domain of validity. Mathematical theorems, on the other hand, are purely abstract formal statements: the proof of a theorem cannot involve experiments or other empirical evidence in the same way such evidence is used to support scientific theories.

Frud:

Because we know what the number 1 is, does that mean we have "complete" knowledge of every real value between 0 and 1?

What does it mean to have "complete" knowledge of a number?

What is required for me to have complete knowledge of the number 1, for example?

I have a sneaky suspicion he is referring to the UP... But i may be wrong.

Forb instance... superposition allows a 0.50 factor to exist.

Mathematical theorems...are purely abstract formal statements: the proof of a theorem cannot involve experiments or other empirical evidence in the same way such evidence is used to support scientific theories.That's all very interesting. The assumption about the OP being a math question, is just that, an assumption.

Physics and Math are definitely related, at least, I'm pretty sure I was told this.
I've said something like this a few times too: "at some scale, reality would look like a continuous pattern". Does anyone else at this forum have a clue what I'm saying with this? There have been a few objections, some to do with what the word "continuous" means, someone said that a continuous pattern has to be continuous with respect to something, and so on. The first "objection" might have been resolved (hard to say, especially here at 'forum). The latter objection might be a non-issue: I'm sure a continuous line, for example, isn't continuous with respect to anything except itself, it isn't continuous "on" anything either.

Does the idea of a continuous pattern have anything at all to do with mathematical ideas?
What does it mean to have "complete" knowledge of a number?
What is "complete knowledge", you mean?
]What is required for me to have complete knowledge of the number 1, for example?Well, I asked a while ago, about how many real values are between 0 and 1, and if complete knowledge of them was possible. Perhaps if I rephrase the question: "can we observe every real value between 0 and 1, or if we observe the value 1, is that equivalent to observing every real value between 0 and 1?".

Again, I don't think this is possible: we can't observe every value, so we can't have complete knowledge--except in an imaginary (mathematical) sense.
Try drawing a continuous pattern with a pen; the only continuous part is any continuous lines that get drawn. It will take time to draw adjacent lines, that appear to be part of any continuous pattern.

That's all very interesting. The assumption about the OP being a math question, is just that, an assumption.
You talked about numbers in the first post and in terms close to those used in algebra. Looking back, the first inkling that this thread was about something more than math didn't appear until post #23 or #25, (but still very vague).

While we can't read your mind, but we can read what you typed, and what you typed looked like a math question. Since the original post is supposed to be what drives a thread, I logically assumed this was a mathematics thread.

I've said something like this a few times too: "at some scale, reality would look like a continuous pattern". Does anyone else at this forum have a clue what I'm saying with this?
Not really. I can't read your mind, which is why I have asked you to be precise. Precision in language is very important in science and math. It looks to me like you have read an article or two on fractals and are misconstruing terminology.

Does the idea of a continuous pattern have anything at all to do with mathematical ideas?
It looks like you mean self-similarity, not continuous here. The seminal paper on this subject is by Benoît Mandelbrot titled "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension", published in the May 1967 issue of Science. This paper is so important that an entire Wikipedia article is dedicated to it, link here (http://en.wikipedia.org/wiki/How_Long_Is_the_Coast_of_Britain%3F_Statistical_Se lf-Similarity_and_Fractional_Dimension).


Well, I asked a while ago, about how many real values are between 0 and 1, and if complete knowledge of them was possible. Perhaps if I rephrase the question: "can we observe every real value between 0 and 1, or if we observe the value 1, is that equivalent to observing every real value between 0 and 1?".
This is what got Absane going and made him comment about mixing physics and math. Mathematicians don't observe things. Given that, please read about computable numbers. You'll have to google that phrase; my wife want's to go out to eat NOW.

Mathematicians don't observe things. Given that, please read about computable numbers.That's really interesting. "Mathematicians don't observe things".
"Math is disjoint from science, and has nothing to do with it". "Math is about numbers and abstraction, not reality". That isn't what I learned at Uni...?

I wonder how they manage to think about all that, er, math? Seeing as they don't observe anything, this must mean that any ideas they have must be carefully and studiously ignored. This seems to be a bit of a tactic with some of the posters here, too. Maybe it works?

Another almost completely fallacious argument about what numbers are, and what a continous "thing" is. I can't say anything about drawing a line, or stretching a metal wire out, because this has nothing to do with abstract mathematical ideas.
What a load of bullshit, man. You're just looking for any excuses to diss my argument.
It's a simple enough idea: there's this pattern; if you get close, the pattern disappears, because it looks "continuous" (i.e. the same in all directions).
Tear it apart, by all means, but use logic, not specious arguments about what a number is. I know what a number is, for god's sake.

Having studied a little math, a bit of physics (and even biology), I think I can safely say (to myself, not to you) that mathematicians, who have the same biology and live under the same physical conditions as biologists and physicists, certainly DO observe things. I cannot see what the point of your post might be, sorry.

P.S. you appear to have conceded that a continous pattern is a possibility. But this idea - linking the concept of continuous, and "pattern", seems to be a logical bridge too far? It's just too complicated a concept for the math-wits: a pattern which looks continuous...
If it was continuous (completely self-similar) there would be NO pattern (instead, you might see a vague, beige-coloured background in all directions). I used "self-similar", or "with respect to itself", in previous posts, did you not see these?

What a load of bullshit, man. You're just looking for any excuses to diss my argument.

Tear it apart, by all means, but use logic, not specious arguments about what a number is. I know what a number is, for god's sake.

Having studied a little math, a bit of physics (and even biology), I think I can safely say (to myself, not to you) that mathematicians, who have the same biology and live under the same physical conditions as biologists and physicists, certainly DO observe things. I cannot see what the point of your post might be, sorry.Frud: Please make your comments about the arguments, not about the people. This is a place of science, not a talk show.

I myself have often pointed out that math is a tool of science (and of other disciplines) but not a science unto itself, since it deals with pure abstractions and reasoning and does not need empirical data. For this reason mathematical theories can be proven true, unlike scientific theories.

The fact that math can be used to analyze empirical observations, derive logical theories, and predict future events makes it a very useful tool for studying reality. But the principles of math do not require a context of reality. We know that Lobachevskian geometry is "true," but we may never encounter an environment in which it can be applied.

the principles of math do not require a context of reality.You mean, in more or less the same sense that a real continuous line, that is an abstraction of real values (the real number line), is something that does not require a "background". Or as I keep saying does not need to be continuous on anything?

Physics doesn't require the existence of maths, either. Conservation of energy doesn't "need" a frame of reference, we apply a reference for our benefit, not because it's a requirement of Nature. The math is a tool.

There is, nonetheless, a reality, that math is a part of. It deals in abstractions, but also deals in real things (like the economy, physical processes, something we call time, etc). Math has an existence, and there is a lot of math that is extremely useful in science (possibly there is a lot more that isn't "useful", because it doesn't model any real-world process). There, I said it.

P.S. Logical arguments, I have no problem with.
I do have problems with quibbling over terminology, though (and will continue to). Words like "continue", and "pattern", shouldn't have to be argued over.
People who quibble over terminology, which they see as having some explicit, restricted meaning, and who insist that you are abusing its use, are simply trying to derail a discussion. I just made a direct accusation.
I bet everyone knows what "derail" means, huh?

"Math is disjoint from science,
That is correct. I have one suggestion. READ rather than rant.

and has nothing to do with it".
I did not say that. Obviously math has influenced science significantly, and vice versa. Once again, READ rather than rant.

I'll keep ranting, thanks. maybe I'll read that reference in some possible future, I really can't say just now, however.

This post can safely be ignored by anyone who understands math.

If you have a piece of fabric, woven out of threads, it looks like a continuous surface, from a distance.
Up close, you can see gaps in it, at small scales it looks discontinuous.

This is kind of the inverse notion behind my earlier post about the nature of reality.
I tried getting in to this again from the point of view of numbers, but I now think that ploy needs to be abandoned.
I'll go with a simpler model, and one that the math-heads will not be able to say anything about with their math-lingo.

Reality looks like a pattern (visually). This pattern changes a lot. If we could see it up close, like looking at a surface, it would look continuous, the pattern would "become" a continuous, same-looking, surface.

This is the essential idea behind certain approaches to quantum gravity theories, and how the "emergence" of features is explained; also how time "appears". Even in these theories, there is still no kind of explanation for the "one-way" behaviour of physical processes that involve energy conservation, or the nature of time and the "arrow" of entropy and expansion.

Reality looks like a pattern (visually). This pattern changes a lot. If we could see it up close, like looking at a surface, it would look continuous, the pattern would "become" a continuous, same-looking, surface.

This is the essential idea behind certain approaches to quantum gravity theories, and how the "emergence" of features is explained; also how time "appears". Even in these theories, there is still no kind of explanation for the "one-way" behaviour of physical processes that involve energy conservation, or the nature of time and the "arrow" of entropy and expansion.

Good stuff!!! Yeah the patterns within patterns. There are patterns everywhere, in everything, are we seeing the pattern here? What makes it difficult is that there is so much pattern to see, it is a bifurcated array if subtlety. A Whole Universe that springs out of a Singularity? Could there be a pattern in singular structures that is connected to this?

Which is odd...since a singularity by itself is not a pattern. Is it even there?
1??? The idea of it is great...we have done so much with this idea...is one just an idea as some have told me when I started another thread "what is the meaning of numbers?" If so, then might the whole universe be an idea too, a product of the mind? So far I have never seen anyone isolate and show me a One. One what?

This post can safely be ignored by anyone who understands math ...
In other words, "this post is pure pseudoscientific garbage and lacks any meaningful content".

Reality looks like a pattern (visually). This pattern changes a lot. If we could see it up close, like looking at a surface, it would look continuous, the pattern would "become" a continuous, same-looking, surface.
Please. This is indeed pure pseudoscientific garbage. Is there anything useful or testable that falls out of this model? What new predictions can you make of the physical world with this model?

Please. This is indeed pure pseudoscientific garbage. Is there anything useful or testable that falls out of this model? What new predictions can you make of the physical world with this model?

While the first quote you responded to was a little strange.

The second quote is a miss. There has been a lot of knowledge that came out of the study of patterns and what he is talking about is seriously related to the very legimate Science of Chaos. If you put this Science down...then you are a crappy scientist!!! It has done more to improve modern technology then most care to admit. Without the Knowledge of how Chaos Operates we could throw away most all of our most modern techs...like fast speed internet, cellular phones, artificial hearts, the latest satellite developments, and lots more.

Just because it doesn't fit into your limited view of science does not make it unscientific to observe the phenomena Frud11 is talking about. Patterns have everything to do with everything. We should understand how they work, what their limits are, and anything else we can about them; as these observations have already proven themselves many times over to be of great significance.

Frud11...you will make leaps and bounds and none of these people could challenge the scientific value of your observations with any ease if you Study up the science of Chaos more. It is serious stuff and anyone who says the contrary is reviewing in the most obvious manner how ignorant they are of science. I recommend the book: Chaos, by James Gleick...but there is a good chance you already are familiar with it or books on Chaos. If not, you will find it of incredible value!!!

As for this thread...I don't think we should trash it to pseudo unless someone can give a good reason why Chaos Science is invalid...which really can not be done!!! (With the exception of those who are blatantly ignorant and clueless about the advanced science of chaos!!!)

If you put this Science down...then you are a crappy scientist!!!
I don't put this science down per se. That said, chaos theory has contributed a lot less than extreme amount of hype about it in the lay press. What I do object to is empty statements. You interpreted Frud's post as talking about chaos theory. He doesn't actually say that.

It has done more to improve modern technology then most care to admit. Without the Knowledge of how Chaos Operates we could throw away most all of our most modern techs...like artificial hearts, the latest satellite developments, and lots more.
Note: I deleted the terms "fast speed internet, cellular phones" because chaos theory has indeed made limited contributions in the arena of communications. Please describe what chaos theory has done for artificial hearts and "the latest satellite developments" other than KAM theory (which was usurped into chaos theory because it actually does something useful).

Chaos theory is extremely heavy on the math. If you think linear system dynamics are hard, try non-linear system dynamics. Chaos theory also addresses a fairly narrow realm. It does not address linear systems because we have perfectly good mechanisms for that. It does not address random systems. Chaotic systems lie on the boundary between predictability and randomness, and after years of research, people are finding that boundary to be pretty thin.

The purported prototypical example of a chaotic system is the weather and climate. This is where the term "butterfly effect" originates. Meteorologists for the most part do not use chaos theory at all. They instead use massive simulations that rely on good old fluid dynamics.

I will have to get back to you on that...I'm feeling a little lazy because it is my weekend.

By the way...I think this might be best debated in another thread, unless Frud11 himself ties in the Science of Chaos to his thread. As to you D H...prepare yourself...I have more then enough artillery myself. You are getting into a debate you simply cannot win. I will have no difficulty bringing up the information you want...just not now...I am tired.

Chaos Theory??? What the??? I am talking about Chaos Science...its not some finished and polished theory, as if there is such a thing as an EXACT theory.

As to the math behind Chaos...yeah I am well aware of it. So? I will ask you one question so you have ample space to prepare for a hard spanking...why do you have rings or spirals on your fingertips and why does a human grow outward in a spiral shape? You had better prepare your mental magic real good for this one!!! Good Luck!!!

Chaos Theory??? What the??? I am talking about Chaos Science
The word "theory" has multiple meanings, one of them being "a body of knowledge". This word in the phrase "chaos theory" has the same sense as when used in terms like "music theory", "string theory", and "set theory". Most practitioners in the field call it chaos theory, not chaos science, as does Wikipedia (http://en.wikipedia.org/wiki/Chaos_theory).

Chaos theory is not a "science". It is more properly a branch of applied mathematics. Like other branches of mathematics, it touches on many different fields of science. Chaos theory comprises a set of tools, some quite powerful, that scientists can apply to their fields of study. To call chaos theory a science by itself is more than a bit disingenuous and vexing. To those who study hammers exclusively, everything in the world looks like a nail.

Full disclosure here: I did my senior physics project on one of the chief precursors of chaos theory and was accepted into a PhD physics program where that is what I would have studied. I decided to take my life on a different path.

Full disclosure here: I did my senior physics project on one of the chief precursors of chaos theory and was accepted into a PhD physics program where that is what I would have studied. I decided to take my life on a different path.

Sorry if I can't resist a bit of trolling (http://www.urbandictionary.com/define.php?term=Bitch+Slap).

"Math is disjoint from science"

That is correct. I have one suggestion. READ rather than rant.This still intrigues me, I have to say.

Maybe you could take up your own suggestion, and read up on why mathematics is disjoint (it's in a separate set of "axioms", or whatever "disjoint" relation it has with science), and point out the reason you agree with my conjecture (which is based on stuff you have posted here). I have a couple or so questions about this, which, seeing how it's so obvious to a mathematically-aware person such as yourself, you might be able to answer:

If mathematics is disjoint from science:

1) Does this mean math is not a science?

2) Is the math used in science, therefore disjoint from that science (i.e. physics equations are disjoint from Physics, even though they explain "behaviour", and the physical world appears mathematical; Biology's use of algebra and arithmetic, though it can analyse genetics, DNA fingerprinting, and a load of other stuff, is separate from that)?

3) Why are there so many "Mathematical Sciences" departments, in so many universities around the world?

P.S. I don't personally believe this "math is disjoint" argument (I made it up), and I think you would have quite a job finding many scientists who agree it is even a valid conjecture, let alone some kind of principle or truth. But don't let that stop you...

P.P.S. I don't actually care very much about what you have to say about your expertise and qualifications (you could just be yanking my chain, maybe you really just deliver pizzas, or something). What Jozen-Bo had to say is a perfectly legitimate connection to make. Although, as you so readily point out, I didn't mention chaos, patterns and chaos are related; he hasn't exactly made a wild grab.

Maybe you really do have degrees in adv. math. Big deal.
So you and the math-wits can have a good chuckle at us kids who are so obviously confused, and struggling with all sorts of misconceptions (chuckle, tee-hee).
On the other hand, us kids can have a good laugh, too (guffaw, har har), at the potato-heads who think they know stuff, but say absolutely ludicrous things that just don't make much sense (unless you happen to have a degree in adv. math maybe). I refer, of course, to your agreeing with a statement I made (that I don't think you will be able to demonstrate is valid, or even a "reasonable" thing to say): "math is disjoint from science". The hell it is.

Is mathematics a science?

The distinction between math and science is a bit fuzzy, and growing fuzzier with the widespread use of computers to aid in mathematical discovery.

What is meant by the term "science"? Webster has several definitions, starting with
"the state of knowing", Bzzzt, wrong. This definition is void of meaning. Next.
"a department of systematized knowledge as an object of study". Better, but this definition still makes just about any intellectual endeavor a science. Next.
"knowledge or a system of knowledge covering general truths or the operation of general laws especially as obtained and tested through scientific method".
Better! I will the bolded phrase as the characteristic that distinguishes whether or not somet body of knowledge is a science. Broadly speaking, we categorize sciences today into two classes: the natural sciences (physics, chemistry, biology, ...) and the social sciences (economics, psychology, sociology, anthropology, ...). The scientific method got its start in and remains the backbone of the natural sciences. The social sciences use the scientific method as well. Both of these broad classes of science truly are science based on my scientific test of what constitutes a science.

What about mathematics? None of the mathematicians that I know consider themselves scientists. To them, mathematics is a branch of applied logic, and logic in turn is a branch of philosophy. Until very, very recently, mathematicians have not used the scientific method. This recent trend in which some mathematicians do use the scientific method does make the distinction between science and math a bit fuzzier.

Is mathematics a science, part II

As noted in the previous post, the scientific method forms the backbone of those intellectual endeavors we call "science". Scientific theories, the fruit of the scientific method, must be consistent with established scientific facts and must have some rationale explanation. A new hypothesis must also add something to the field of which it is a part before it is deemed to be a theory. In particular, it must predict some measurable result that will distinguish it from extant scientific theories.

Suppose a new hypothesis does predict some behavior measurably different from the existing body of knowledge in some field, and observations agree the new hypothesis and disagree with predictions based on existing theories. Others perform the experiments and replicate the results. While these surprising results most certainly do disprove the existing theories, they do not prove the new hypothesis. To say otherwise is to be guilty of the logical fallacy of affirmation of the consequent. Nonetheless, scientists do say that this "proves" the new theory. We also know that new evidence can disprove it in an instant.

In a trial of law, we do not ask for absolute proof that a person is guilty of committing some offense. Such proof cannot exist. Instead, we ask for proof "beyond a reasonable doubt". Scientific proof works on the same basis. For this reason, we usually try not to do anything drastic in the law. Some new evidence might arise that proves that a guilty result was incorrect. Science operates much like the law. It does not ask for absolute proof for the simple reason that absolute proof does not exist.

In comparison, mathematics uses theorems, in which proof is indeed absolute. The 2500 year old Pythagorean theorem was true 2500 years ago and will remain true 2500 years into the future. Mathematical theorems are based on pure logic. Mathematics is applied logic, not a branch of science.

"Math is applied logic". Sure, and numbers only exist inside our heads. Mathematical proof is a logical, not a physical principle.

Math is not disjoint from science, because it is a science (it has research centres, it has theories that predict things). If it is a science, a science cannot be disjoint from itself.

Benjamin Peirce called it "the science that draws necessary conclusions". Other practitioners of mathematics maintain that mathematics is the science of pattern, and that mathematicians seek out patterns whether found in numbers, space, science, computers, imaginary abstractions, or elsewhere--wikipedia.org

P.S. to those wondering about a model: I haven't presented any model.

All I did a couple posts back was say something about the nature of reality, that has taken all this (and the previous thread that's the source of the OP in this one). Quite a job, just getting it out without pedantic objections from left and right.
I'm still waiting for any potential objectors to say their thing about the idea, before I return to it. So far, there's been only one, which I can ignore, for the straightforward reason that it isn't a model, it's just an idea.

"Math is applied logic". Sure, and numbers only exist inside our heads. Mathematical proof is a logical, not a physical principle.

You sure did start on the right foot. The rest of your post was typical Frud drivel, but I will ignore that.

Math is indeed applied logic, and logic is indeed the cornerstone of mathematical proof.

Because we know what the number 1 is, does that mean we have "complete" knowledge of every real value between 0 and 1? On the real number line? I don't think it does.

What is the real number line continuous on, or with respect to? Since continuity requires that it be relative or in respect of something, as some seem to think?

How could anyone or anything ever have complete knowledge of th ENTIRE continuity of the full value between 0 and 1? It is a fathomless impossibility to any logical conclusion. Who is going to write out the length of all those numbers beyond the decimal and get to the end of the thread...err last digit?

Writing that out, we would spend our life's in total vain, only to discover at an old age we didn't even get close, not a even a fraction. So...does the old man pass on the tedious curse of writing down more numbers for a single value, when there are so many other values as well, with numbers as potentially long as the one being counted?

Even if we were absurd enough to try, passing the task on from one generation to the next, over the next billion years we would of made no real progress to reaching that last digit. We would be drown and surrounded with notes of numbers. They would invariably get tweaked somehow in the storage process and the whole number would be lost. Then there would be a lot of garbage to clean up!

So that answers the first question. No, we will never have complete knowledge of every real value between 1 and 0, never ever...the end.

"What is the real number line continuous on, or with respect to?"

This is a tricky question. We can't even begin to do justice to given it an answer unless we have a very solid idea what a real number line is. I think I wait on answering this one...or was it somewhere between 0 and 1?

"Since continuity requires that it be relative or in respect of something, as some seem to think?"

It seems that is would actually be somehow Relative to our being here, that the pattern that connects the Total Value is somehow a Fundamental Element of Existence itself. I say seems, because I can't draw any conclusions at the moment.

What is the Pattern within Continuity itself? Wouldn't it be somehow evasive?
As we look in we find more space to measure. Our tools break down, they can take us so far before they cease to measure more. We move over to deduction, in an effort to get past this. The Essence of Continuity escapes us. Can we pin it down?

Two features come to my mind, it continues and continues. Almost like where a movement continues as a movement and a steadiness remains steady. It seems like it does both! Am I describing a pattern, at least an important part of it? All there is in reality from what I see are patterns or a pattern, I think the two are the same.

The entire pattern of existence would somehow be wrapped up in that Continuity. How much can we extract from it? So there is Infinite value between 0 and 1, which is why it is possible to have any value in the first place. I have often stated that 0 and 1 are the same thing ultimately, and that they are the same as Infinity, how could this be false if we are talking about a Continuity?

How could this possibly not be science? Looking for the patterns is all science is, in a sense of the word. We are trying to make an accurate and precise observation of what reality really consists of. If we are not doing this, then science is not progressing.

No one is talking about ghost here, there are no ufo's nor big foot. Valuating and extrapolating the fundamental realm where pattern becomes possible and deriving a solid observation is looking for key features that would give us much greater abilities to put those patterns to work for us.

I have no idea what you guys talking about

Patterns are difficult to discuss, because they're mathematical objects.

The thing that struck me about Bernard Peirce's comment is that we've developed new math any time new observations turn up on the horizon. Would we be able to make the observations (build the equipment), without the math?

Any time some new observation about the patterns we see, especially when things like light and charge are constrained, because of our ability to constrain them, in vaccum tubes, and prisms and lenses, recently in more exotic kinds of materials, and in wires and so on gets made, we adjust the logic "radar", and apply the science of numbers (math) to see if a pattern emerges from the numbers.

A fabric is also a mathematical structure. The internet is a fabric. This forum is available to everyone because of things called switching fabrics. An integrated circuit is a fabric (it gets fabricated, too).

Just for example.

Such as the Planck Length... 1.616 x 10^-33... this is a specified limit, but it has been shown that there could be smaller things called ''solitons.''Solitons are something completely different to 'smaller than Planck length displacements'. You can get solitons in everyday life. Infact, the first notion of such a wave was when a British academic saw a barge on some canal produce a displacement wave which didn't disperse. It was another 30+ years before someone explained what he'd seen in terms of equations (ie the KdV equation).

He didn't prove didily squat!!!

No wait..."Suppose that 1/N is the smallest number."
I get it now!!! That proves everything!!! We are basing this 'proof'
on supposed numbers!!! Its now so clear!!!

Didn't prove didily squat!!!I suggest you learn how logic and proofs work before knocking them.

I assumed the existence of a smallest positive number, which was inline with what you'd said (since eventually, by your claim, numbers stop being numbers) and then proved that supposition to be logically inconsistent.

I assumed the existence of a smallest positive number, which was inline with what you'd said (since eventually, by your claim, numbers stop being numbers) and then proved that supposition to be logically inconsistent.

:huh:

Could you elaborate on what you just said? I can't make sense out of it?

How did you prove it? Where exactly does the inconsistency lie...please make it crystal clear so that even an idiot like myself can follow. As to talking about logic, maybe it would be logical to make your points easier to absorb. Logic is not always in agreement. What one says is logical another says is absurd. Happens all the time, that is one of the reasons we have wars. Or is this not logical?:scratchin:

The assumption that there is a smallest number? I believe you misunderstood the implication of what I said. There is NO smallest number, things get fuzzy on the way. We can never actually find the smallest number, because it does not exist.

Another implication of what I said was that the numbers don't go infinitely in the sense that we can follow them. If you were paying attention you might have read that I wrote there are infinite variables in the Continuity of 0 and 1. And that being continous, it is not separate from anything, thus 1 and 0 and Infinity are all the same and at the same time different.

This is what you call abstract logic, it is abstract meaning it is not easy to grasp. The problem in the first place was assuming, you should first make sure you clearly understand what I am saying before assuming. I do it, you do it, everyone does it. So what? I am trying to understand you, please believe this.

I will try to clear this up a little more, There is no such thing as a smallest number and numbers cannot continue forever in a universe that is closed. There isn't enough space to write them down. My claim that numbers stop being numbers is here in the physical realm were we are now. Not theory land.

If we follow them...that is, if we can figure out how to follow and extract information from the Continuity between 0 and 1 all we have is a pattern to describe. We don't actually have .111, .1111, .123122, .124235243, .1246584, .00000000009563, etc etc.

Numbers can't even go for ever in theory, or try this little experiment...think of the biggest number you can and write it down. Then I will think of a bigger one and write it down...now now...no using the Infinity symbol, that's a cop out. If you do that then you don't have a number anymore, but a Continuity which is nothing more then an idea you are using to describe a pattern.

Are you starting to follow my logic Mr. Spock?:shrug:

Or is this not logical?
No.

Another implication of what I said was that the numbers don't go infinitely in the sense that we can follow them.
What does this mean? Numbers do go on forever. There is no largest integer.


If you were paying attention you might have read that I wrote there are infinite variables in the Continuity of 0 and 1.
While there are rational numbers and real numbers between 0 and 1, there are no variables between 0 and 1.

And that being continous, it is not separate from anything, thus 1 and 0 and Infinity are all the same and at the same time different.

This is what you call abstract logic
This is what I call woo-woo logic. This forum is not the place for that. Take it to pseudoscience or the cesspool. Please.

I will try to clear this up a little more, There is no such thing as a smallest number and numbers cannot continue forever in a universe that is closed.
The existence of the real numbers (or any other numbers) has nothing to do with the shape or extent of the universe.

D H,
You have just made a bunch of empty statements without even putting forth any effort to back your points up with anything. Huffy puffy claims based on your opinion without any substance behind them. You are going to have to do much better then that!!!

I am tired...I am going to sleep now!!!
I have work to do and don't have time at this moment to go over the flaws of your argument in detail. I will get on in later. You really aren't saying anything other then how you feel...please back your self up a little stronger, cause that is a bunch of hot air!

Woo-woo logic??? Are you saying you are the dictator of logic now? I get back on this later!!!

Again with the "has nothing to do with". Mathematics is (a) science. It isn't disjoint from itself (despite what a certain "logic-meister" has to say about it).

Numbers, would be a lot stranger if they "had nothing to do" with reality.

Numbers aren't anything more than a logical construct, that we invent (other primates and some other species may have this ability too).
It helps to explain stuff, and the idea of unitarity is probably fairly important (a single individual, a single "thing", etc).
Without the concept of unity ("one-ness"), there might be a bit of a "logic problem" dealing with non-unity ("many-ness").

Languages have words for "one thing", and "many things". This concept connects with the idea of selection (inclusion and exclusion). Numbers themselves are not "real", but there is a real physical limit at which numbers start to lose their meaning (perhaps this is what Jozen-Bo means to say). We also use numbers to represent time, so time is a logical construct, therefore. Time becomes "meaningless" at very small intervals.

Once more with feeling.

Books, paintings, etc all convey information, which we filter through our value selection process, by comparing it with other knowledge we have, or by feeling what it says.
Information in the Shannon, von Neumann, etc sense does not "say" anything, as such; it's devoid of meaning; it's "mathematical", a pattern.

Things like uncertainty and expectation are usually considered in value terms.
In courses where Shannon's theories and the ideas behind "communication" are studied (at least when I went to school), an example of uncertainty is often given which uses this very notion:
there is less information in the message: "there are some clouds up in the sky", than in: "there are some strange-looking objects up in the sky", though this has the idea of an expected/unexpected "message", the messages convey meaning or value. Unexpected meaning is similar to a "probability", but probability and meaning are not commutative, in any physical or mathematical science.

In Information Theory, both messages would only differ by the number of bits used to encode them (in whatever alphabet).
So how do we conceive of the information-theoretic (non-physical) notions of uncertainty and expectation?
(Hint: it's to do with the concept of a pattern, and recognition - something a machine can do).

Or is this not logical?

"-No."

What a great argument!!! "No." Thats all. Well put!!! Can you explain why? Or is it too difficult and so all you do is a 'no'. What a great way to make your point? Lets all start talking like this!!! Yes...yes...no...no...no...yes...we are now overdoing it with content!!! (Can you sense my Irony?)

Another implication of what I said was that the numbers don't go infinitely in the sense that we can follow them.

"-What does this mean? Numbers do go on forever. There is no largest integer."


Thank you for asking a good question!!! Do numbers go on forever. I do believe I already gave an answer. But I will elaborate: We can never have a list of every number there is. And it means as well that logically, yes, there is a Continuity of every possible number between 0 and 1. All we can do at best is look for the best way to describe the pattern and its implications to our own existence!

If you were paying attention you might have read that I wrote there are infinite variables in the Continuity of 0 and 1.

"-While there are rational numbers and real numbers between 0 and 1, there are no variables between 0 and 1."

I was tired and assumed that you would understand that I meant variety of different numbers by the word variables (as in .33, .34, .35, .353, etc). I used the word in the sense 'variations of of values'. My english is a little rusty from speaking so often in another language, as my paycheck depends on me doing this.This time I made the assumption and it quickly led to a misunderstanding. As I said before, we all make assumptions at one point in time or another!

And that being continous, it is not separate from anything, thus 1 and 0 and Infinity are all the same and at the same time different. This is what you call abstract logic.

"-This is what I call woo-woo logic. This forum is not the place for that. Take it to pseudoscience or the cesspool. Please."

This is what I call TROLLING!!! Don't make me get the big billy goat out!!! You haven't even used any logic at all to make your point. You are merely waving your handsd around and saying childish words like "Woo-woo". Or is Woo-woo all one needs to say in order to make a good argument?
In that case: Your argument is a woo-woo!!!

Before you can hope to succeed in your Troll mission, you need to ground your statements. Believe it or not I am trying to help you: if you can throw tougher questions and make better points, then we have to work harder to bring clarity to the topic as well; thus, we will be able to get somewhere even quicker!

If you start accusing others of having woo-woo brains, be prepared for a rebuttal!

I will try to clear this up a little more, There is no such thing as a smallest number and numbers cannot continue forever in a universe that is closed.

"-The existence of the real numbers (or any other numbers) has nothing to do with the shape or extent of the universe."

I cannot beleive you just said that! Truly Absurd!!! You are embarrassing yourself at this point! Numbers (real or not) have nothing to do with shapes? OMG!!! Are you pretending to be something you are not? Did you even really go to school? Let me ask you a question...How many corners does a cube have? Oh..wait...we can't answer that because numbers have nothing to do with shapes!!! Wake up!!!

I will try to clear this up a little more, There is no such thing as a smallest number and numbers cannot continue forever in a universe that is closed.
The existence of the real numbers (or any other numbers) has nothing to do with the shape or extent of the universe.
I cannot beleive you just said that! Truly Absurd!!! You are embarrassing yourself at this point! Numbers (real or not) have nothing to do with shapes? OMG!!! Are you pretending to be something you are not? Did you even really go to school? Let me ask you a question...How many corners does a cube have? Oh..wait...we can't answer that because numbers have nothing to do with shapes!!! Wake up!!!
Stop putting words in my mouth. I said "The existence of the real numbers (or any other numbers) has nothing to do with the shape or extent of the universe," not "numbers have nothing to do with shapes".

Of course we use numbers to describe shapes. That is the subject of algebraic geometry, differential geometry, and several other branches of mathematics.

Let me give you a quick primer on numbers. I'll start with the Peano postulates. I'll start with a zeroth postulate: There exists a set of things we call the natural numbers, \mathbb N. The first four postulates define the concept of equality of natural numbers: Equality is reflexive, symmetric, transitive, and closed.

The next four postulates define the natural numbers starting with some base natural number (some use zero and others use one as this root of all natural numbers) and the "successor function" S(x). The successor function defines the natural number that immediately follows some other natural number. For example, the successor of zero is one, and two is the successor of one. The basis number is not the successor of any natural number, and distinct natural numbers have distinct successors. The last postulate defines the concept of mathematical induction.

The operation of addition can be defined based on these axioms. a+b is simplying the bth successor of a. Subtraction is the inverse of addition. The difference between two natural numbers, a-b, is the number, if it exists such that adding b to a-b yields a. There is a problem: the difference of two natural numbers isn't always defined. For example, 1-2 is not defined for the natural numbers. So define it. This leads to a new set, the set of all integers \mathbb I. The integers are the closure of subtraction.

The operation of multiplication can be defined similarly to the way in which addition was defined. Division is similarly the inverse of multiplication. Just as subtraction provides the means for constructing a new set based on the natural numbers, division provides the means for constructing a new set based on the integers, in this case the rationals \mathbb Q.

The next branch along the road of "numbers" are the real numbers \mathbb R. The real numbers are the set of all convergence Cauchy sequences in the rationals.

Where in any of this does the concept of shape appear? (Answer: Nowhere.)

Things like algebraic geometry, differential geometry, etc. need the concept of number as a starting point. Numbers are not a consequence of shape. It's the other way around. We use numbers to help in the analysis shapes.

Stop putting words in my mouth. I said "The existence of the real numbers (or any other numbers) has nothing to do with the shape or extent of the universe," not "numbers have nothing to do with shapes".

Of course we use numbers to describe shapes. That is the subject of algebraic geometry, differential geometry, and several other branches of mathematics....

I appreciate your giving a much more solid answer!!! I misunderstood your final statement. My apologys. You also took the bait, which is good for us both, as the information you provided can be used to build on. This was part of my intention. I will have to go over your statement a second time later before commenting on it fully, as I have a bit to do at the moment.

I will add one bit for now. We both agree that numbers have much to do with shapes and geometry. As I see it, geometric shapes are a fundamental feature of our universe and all that is within it. One of the biggest questions of science is what is the shape of our universe? Is it closed? Is it open? No one has been able to provide adequate proof which. This is one of the big questions. Thank you again, for valid thoughts and comments regarding this topic.

I will add one bit for now. We both agree that numbers have much to do with shapes and geometry.
NO!! Numbers are a result of algebra, period. Algebra and geometry are distinct branches of mathematics. Yes, algebra and geometry are used together in the field of algebraic geometry. That does not mean numbers have "much to do with shape and geometry." It does means that our concept of number has a very wide range of uses, analyzing geometric shapes being one of them. Numbers exist independent of geometry.

You are intentionally distorting my words. You completely overlooked the key question near the end of my previous post, repeated here, this time in bold: Where in any of the development of the concept of numbers does the concept of shape appear? (Answer: nowhere.)

As I see it, geometric shapes are a fundamental feature of our universe and all that is within it.
No. Geometry exists independent of the shape of the universe. Whether the universe is open or closed has nothing to do with the validity of Euclidean geometry, for example. Euclidean geometry will remain a valid concept regardless of the ultimate shape of the universe.

One of the biggest questions of science is what is the shape of our universe? Is it closed? Is it open? No one has been able to provide adequate proof which.
Mathematical theorems are the backbone of mathematics. "Proof" is a mathematical concept, not a scientific concept. Mathematical hypotheses become mathematical theorems by means of "proof". Scientific theories are the backbone of science. "Proof" is not a part of the scientific method. Scientific theories are not provable. Scientific hypothesis become scientific theories after analysis shows the hypothesis is reasonable and after observation shows that predicted results agree with the hypothesis. Tons and tons of confirming evidence does not "prove" a scientific theory to be true. Logicians and mathematicians have a phrase for this: the logical fallacy of affirming the consequent (http://en.wikipedia.org/wiki/Affirming_the_consequent). We use different words to describe these things (theory versus theorem) because they represent different concepts.

Why this harping on semantics? Because it points to a basic misunderstanding of mathematics and science. Whether the universe is open or closed has absolutely nothing to do with the nature of the real numbers between 0 and 1.

How did you prove it? Where exactly does the inconsistency lie...please make it crystal clear so that even an idiot like myself can follow. As to talking about logic, maybe it would be logical to make your points easier to absorb. Logic is not always in agreement. What one says is logical another says is absurd. Happens all the time, that is one of the reasons we have wars. Or is this not logical?:scratchin:Are you aware of how "Proof by contradiction" works? You wish to prove something so you assume it's compliment and show that such an assumption leads to inconsistencies. I wished to prove that there's no largest integer. So I assumed there was 'the largest integer' and then disproved that assumption. Since either there is or isn't a largest integer, if I disprove the existence of a largest integer, I prove the initial thing, that there's no largest integer.

And your comment about wars demonstrates you have no studied maths. Maths starts with a series of axioms and seems what logic can derive from that.

For instance :

Statement 1 : Bob is French
Statement 2 : All French people are European

I can logically deduce from this that Bob is European. That's using logic. Given statements A, B etc, what do they imply? What can I prove from them?

Wars are something completely different. Do we have 'axioms of life'? Nope. You invent a completely irrelevent example and think it somehow supports your case.

Have you ever studied basic mathematical proofs? It seems you haven't but then you get arsey when you don't understand methods people use. Are you suprised you don't understand something you've never tried to learn?

The notion that shape doesn't appear anywhere when "developing" a concept of numbers might look nice & logical, but it's also disingenuous.

Now I admit, disingenuity can be useful, when trying to explain concepts in simple terms (i.e. without using "big" words); I also admit that I resort to this myself, I've done this in this thread and others. I refer to my earlier post about using a "model" that the "math-heads" wouldn't be able to comment on, using "math-lingo". This is, of course, complete rubbish. Nothing is beyond mathematical description (I've already pointed out that patterns and fabrics are mathematical, or maybe someone can present some proof that they aren't?).

It's disingenuous to say that shape is not connected to numbers. I would say plenty of mathematicians and physicist would agree.

What shape is a number, like, say 1, or the cube root of 11?
Does the number line (of naturals, integers, reals, etc), have a shape? Does R have a shape, or does any Cartesian product? A surface, flat or otherwise, looks like a shape to me. But then I don't understand this math stuff that well...

Numbers are polymorphic; they have classifications; they have a base, an index (the index of 1 is zero), etc. There are automorphs, homomorphs, mappings, graphs. Numbers definitely have a shape, even if it's an "imaginary" shape.

lol

lol

I know the feeling. I feel no obligation to jump in since someone else is doing a fine job.

http://www.playa.info/playa-del-carmen-forum/images/smilies/popcorn.gif

Ah yes, intelligent commentary...

Such a hoot, you should tell us more...

So what shape (or form, if you prefer), is a number? Pick a number and tell me the shape. Does it feel smooth or lumpy. Does it have any bits sticking out, or any projections? Can you even get near a number, or is it "stuck" in something? Is an interval on some line or other, a collection of numbers (forms), or a single shape?

Getting back to a thing I said about how a line could be represented by an elastic string.

Elasticity is a fundamental thing that you see everywhere, collisions are inelastic and elastic (in terms of momentum), and things have a tension, or springiness.

Anyway, an elastic string can be inextensible along its length (I think there's some word for this), but still be elastic in an orthogonal sense. You can have a string that doesn't stretch lengthways, but stretches sideways.
A projection from some point along this elastic string can go in any direction (and both ways), or the projections (with definite lengths, say like vectors), in series, might only have a slight, directional rotation, say, to the left; a ribbon-like structure emerges, with "elastic" edges, that is also twisted. I think there's some model around, courtesy of a Leonard Euler, that describes something similar.

The reals are like a string that has a "sideways" elasticity, because it can be stretched indefinitely, between any two points (say, 0 and 1), to "map" every real value on the string between the points.

NO!! Numbers are a result of algebra, period. Algebra and geometry are distinct branches of mathematics. Yes, algebra and geometry are used together in the field of algebraic geometry. That does not mean numbers have "much to do with shape and geometry." It does means that our concept of number has a very wide range of uses, analyzing geometric shapes being one of them. Numbers exist independent of geometry.

Numbers are a result of algebra? What is that suppose to mean? That we cannot learn about numbers until we reach the math level of Algebra? Gee...wiz...I think I started to learn things like 2+2 in elementary school and that that was a prelude to algebra.

Can you make your mind up? First you say "no, numbers have nothing to do with shapes", then "well off course they do", then "That does not mean numbers have 'much' to do with shape and geometry---Numbers exist independent of geometry". Please be consistent if you can. Make up your mind! Does a cube (a geometric shape!!!) have six corners or not?

You are intentionally distorting my words. You completely overlooked the key question near the end of my previous post, repeated here, this time in bold: Where in any of the development of the concept of numbers does the concept of shape appear? (Answer: nowhere.)

Intentionally??? No, I am having difficulty following you, because you keep changing your mind! I am unintentionally misunderstanding where you stand!
I will get back to your bold question in a bit...it looks like fun!

No. Geometry exists independent of the shape of the universe. Whether the universe is open or closed has nothing to do with the validity of Euclidean geometry, for example. Euclidean geometry will remain a valid concept regardless of the ultimate shape of the universe.

Does that mean our universe has no shape? Huh? What does Euclidean geometry have to do with the question of the shape of the universe? You are bringing things in that are either unconnected or you are failing to make the connection obvious!

Stating that the Universe has no shape is absurd. Even nothing has a dinstinguishing quality relative to shape. I am setting this one up...it looks like
I am saying something funny...the statement in this paragraph is bait for the fish...be careful!!!

Mathematical theorems are the backbone of mathematics. "Proof" is a mathematical concept, not a scientific concept. Mathematical hypotheses become mathematical theorems by means of "proof". Scientific theories are the backbone of science. "Proof" is not a part of the scientific method. Scientific theories are not provable. Scientific hypothesis become scientific theories after analysis shows the hypothesis is reasonable and after observation shows that predicted results agree with the hypothesis. Tons and tons of confirming evidence does not "prove" a scientific theory to be true. Logicians and mathematicians have a phrase for this: the logical fallacy of affirming the consequent (http://en.wikipedia.org/wiki/Affirming_the_consequent). We use different words to describe these things (theory versus theorem) because they represent different concepts.

Oh...wow...news to me!!! Ehh...don't you think I don't already know this? What is your point in stating the obvious? I can say something too, with out making the connection clear...women have babies and men don't!!! My point is that your failed to tie your point into a valid statement regarding the topic. You are just stating the obvious with out saying why or how this adds to your own point.

Why this harping on semantics? Because it points to a basic misunderstanding of mathematics and science. Whether the universe is open or closed has absolutely nothing to do with the nature of the real numbers between 0 and 1.

Heh???????? Are you kidding me? Can you give any proof once and for all to deal the death blow? That is, proof it!!! What is a REAL number, then? Have fun with this one...take your time and give us a REAL answer! Thanks!

Are you aware of how "Proof by contradiction" works? You wish to prove something so you assume it's compliment and show that such an assumption leads to inconsistencies. I wished to prove that there's no largest integer. So I assumed there was 'the largest integer' and then disproved that assumption. Since either there is or isn't a largest integer, if I disprove the existence of a largest integer, I prove the initial thing, that there's no largest integer.

Did I ever say there is a largest Integer? Maybe you should study the English Language a little more!!! You obviously haven't been paying attenton to a word I said...your too busy trying to look smart! I am carefully looking at your words...word for word...and searching for the flaws if they occur.

And your comment about wars demonstrates you have no studied maths. Maths starts with a series of axioms and seems what logic can derive from that.

For instance :

Statement 1 : Bob is French
Statement 2 : All French people are European

I can logically deduce from this that Bob is European. That's using logic. Given statements A, B etc, what do they imply? What can I prove from them?

Wars are something completely different. Do we have 'axioms of life'? Nope. You invent a completely irrelevent example and think it somehow supports your case.

??? Please learn English!!! The point I was making was that misunderstanding occurs frequently due to the way logic works differently from one mind to the next. You fail to understand how my mind perceives logic, thus you are unable to convince me of hardly anything. If you can get into my way of thinking...understand why I see logic where you don't...you stand a chance to correct me IF I am wrong.

Have you ever studied basic mathematical proofs? It seems you haven't but then you get arsey when you don't understand methods people use. Are you suprised you don't understand something you've never tried to learn?

Dor dee dor...Mee noot knuww matmateacs...wat it porof? Dora dee dor!!

Yes...I went to school and got many A's in math (Especailly Algreba!!!). I completed Algebra and left off at Trig. I understand how proof works. I could of probably go much, much further...but I have a busy life (maybe later). My father is a math wiz...he is so freaky good at it that he could of gone to any school in the world based on his work in math...solving a problem no one thought could be solved. I think I get that from him.

Once he made me learn a whole school year of math in 5 days...5 DAYS!!! I was pounding out problems 8-10 hours a day at full speed...head throbbing and all. It was...ALGEBRA!!! The schools tested me the next week and jumped me up a level so as to not waste my time learning stuff I already knew.

Maybe you are getting arsey when you fail to make your logic clear in a way that has no weakness. What we are talking about is both simple and complex (sorta like an equation with two or more answers). I am not getting arsey...I am teasing you for overlooking the obvious!!!

Can you make your mind up? First you say "no, numbers have nothing to do with shapes", then "well off course they do", then "That does not mean numbers have 'much' to do with shape and geometry---Numbers exist independent of geometry". Please be consistent if you can. Make up your mind! Does a cube (a geometric shape!!!) have six corners or not?
I have consistently said that numbers per se have nothing to do with shape. Yes, we do use numbers in analyzing shapes. That does not mean numbers per se have anything to do with shape. We also use numbers to count the number of sheep in a flock (one of the first uses of numbers). Numbers per se have nothing to do with sheep either, or with dollars in a bank account, or with any other application where we use numbers. Numbers are a very useful tool. So are hammers. Don't confuse the tool (a number or a hammer) with the things we build with those tools (analytic geometry or a house). Your insistence on confusing the tool with the use of the tool is one cause of your difficulties in understanding.

No, I am having difficulty following you, because you keep changing your mind!
You are the one who insists on numbers having something to do with shape, so you keep putting words in my typing that are not there.

Here is an example of you putting words in my typing:
As I see it, geometric shapes are a fundamental feature of our universe and all that is within it. One of the biggest questions of science is what is the shape of our universe?No. Geometry exists independent of the shape of the universe. Whether the universe is open or closed has nothing to do with the validity of Euclidean geometry, for example. Euclidean geometry will remain a valid concept regardless of the ultimate shape of the universe.Does that mean our universe has no shape? Huh? What does Euclidean geometry have to do with the question of the shape of the universe?
Where did I ever say our universe doesn't have a shape. Stop putting words in my typing. Perhaps I misunderstood you when said "geometric shapes are a fundamental feature of our universe" and I am likewise guilty of putting words in your typing. I took you to mean that the shape of our universe has influence on geometry.

Stating that the Universe has no shape is absurd. Even nothing has a dinstinguishing quality relative to shape. I am setting this one up...it looks like
I am saying something funny...the statement in this paragraph is bait for the fish...be careful!!!
I never said the universe has no shape. You did.

What is a REAL number, then? Have fun with this one...take your time and give us a REAL answer! Thanks!
First, an elementary explanation: the reals are the set of all numbers expressed in decimal notation (http://en.wikipedia.org/wiki/Decimal).

I already did answer your question in post #73, repeated here: The real numbers are the set of all Cauchy sequences (http://en.wikipedia.org/wiki/Cauchy_sequence) of the rationals. Another short answer: The real numbers are the set of all Dedekind cuts (http://en.wikipedia.org/wiki/Dedekind_cut) on the rationals. Yet another answer: The real numbers are the complete ordered field (http://www.math.louisville.edu/~lee/RealAnalysis/ra_sect02.pdf).

That one can say something as bold as the complete ordered field is quite deep. That all four answers are equivalent is also quite deep. For example, it takes 450 pages of development in my algebra text (post-calculus algebra) before they even start delving into the reals. I am not going to write a 500+ page exposition for you.

I gave you some links. I suggest you read them. These web pages will only take you so far. There are some old-fashioned things called libraries, bookstores, and colleges where you can get these deep results.

My "view" of the shape of numbers goes like this: 1 is an important number (a primary shape), and so is 0.

"One" is smooth (like a sphere), zero has no shape, or it's the shape of a circle or sphere with no radius.
Two is the first number with "pointy bits"; a line (wire segment), or a shape with two "ends" (two circles or spheres, connected by projecting the circumference of one, is what Two looks like (it isn't necessarily a nice straight shape, it might be curved, like a horseshoe shape, or something).
Three has three projections or points, a triangle comes to mind. You can construct a triangle with line segments (Twos), but you need three of them; you could use just two (three minus one), like a wire with a sharp bend, or two bits of (not necessarily straight) wire joined at an apex, so it looks something like a "v"
Four is a square, or three "line segments", with two joins, like a square with a missing edge. Five is a pyramid, with perhaps equal area faces, or four segments, like a "w". And so on. These are just some of the shapes; the more points a number has, the more kinds of shape (geometry) are possible.

None of these shapes is necessarily "flat" (a sphere certainly isn't, a circle is). All numbers have a dimensionality, a form, or a shape (at least I think they do, but then I am kind of crazy, or something).

My "view" of the shape of numbers goes like this: 1 is an important number (a primary shape), and so is 0.

"One" is smooth (like a sphere), zero has no shape, or it's the shape of a circle or sphere with no radius.
Two is the first number with "pointy bits"; a line (wire segment), or a shape with two "ends" (two circles or spheres, connected by projecting the circumference of one, is what Two looks like (it isn't necessarily a nice straight shape, it might be curved, like a horseshoe shape, or something).
Three has three projections or points, a triangle comes to mind. You can construct a triangle with line segments (Twos), but you need three of them; you could use just two (three minus one), like a wire with a sharp bend, or two bits of (not necessarily straight) wire joined at an apex, so it looks something like a "v"
Four is a square, or three "line segments", with two joins, like a square with a missing edge. Five is a pyramid, with perhaps equal area faces, or four segments, like a "w". And so on. These are just some of the shapes; the more points a number has, the more kinds of shape (geometry) are possible.

None of these shapes is necessarily "flat" (a sphere certainly isn't, a circle is). All numbers have a dimensionality, a form, or a shape (at least I think they do, but then I am kind of crazy, or something).

Daniel Tammet: http://www.boingboing.net/2007/03/21/daniel-tammet-amazin.html

http://www.youtube.com/watch?v=7bVVQ0FZeys

Truthfully, I think Daniel has a better grasp of numbers than you do. The way you visual numbers has a STRONG basis on learned concepts and memorized facts.

Is this how you see Pi?

http://www.techrivet.com/content/binary/pi.JPG

Of course, this is a product of the mind. Similarly with numbers. By themselves, numbers have no meaning. Meaning must be assigned to them to do anything useful.

Maybe someone will say that because of the nature of numbers, 2 apples added to 3 apples is always 5 apples. This isn't because of numbers but because of how we look at it. If I shift my mental paradigm a bit, I can make myself see only one apple instead of five.

It all depends on how you look at things. It's not like God gave Moses, Pythagoras, or anyone else a tablet with THE rules of mathematics and the universe. Therefore, we are free to explore the results and consequences of the rules we choose. The beauty of mathematics is that we can make the rules so basic that they describe everything, both realistic and unrealistic. Just how realistic the results are depend on the accuracy of your assumptions. Gabriel's Horn (http://en.wikipedia.org/wiki/Gabriel's_Horn), for example. Totally unrealistic. But the math that gives us GH is the same that gives us space exploration (among other things).

Numbers are a result of algebra? What is that suppose to mean? That we cannot learn about numbers until we reach the math level of Algebra? Gee...wiz...I think I started to learn things like 2+2 in elementary school and that that was a prelude to algebra.

It takes a lot of mathematics study to understand what DH means by numbers coming from algebra.

Mathematics is sort of taught in a backwards manner, similar to the way mathematics was developed. We started out by counting and eventually doing things similar to high school algebra and basic calculus and number theory. However, we would always run into problems or have questions we could not answer. For example, we can easily calculate 2^4. What do we do for 2^{2.4}? What about 2^\pi? Or even i^i? What we do is develop axioms and work old problems again, this time with better rules and assumptions.*** From these axioms we CONSTRUCT basic arithmetic. With abstract algebra, we can construct our own number systems. So, if you like, we can use abstract algebra to create numbers (or set-theoretic definition). It's a pain in the ass though because you have to define what "+" means and what "*" means. Subtraction and division are definitions, depending on addition and multiplication respectively.

Just imagine if we started out first graders with basic set theory. It's the same in college mathematics. In many schools (mine included) they teach linear algebra with just real numbers (and sometimes polynomials). They don't address the field. Why? In general, linear algebra is easier than abstract algebra so they teach linear algebra first. Once you understand both concepts, you can go back and generalize it even more.

And this is how research is conducted, too. Researchers start off very basic. They lay out the general framework. Later on, they work on tweaking it and turning specific statements into general ones.

I think I rambled on long enough. I'm supposed to be studying.


***Referring back to the part about running into problems and changing our assumptions. Many people say that multiplication is just repeated addition. 2*3 = 3 + 3 = 6. But what about (-1)*(-3)? You can't do this knowing only first grade math. To PROVE it equals 3, you must know two rules: distribution and additive inverse cancellation. We add the latter rule to our list of axioms when we ask questions like what x gives 2 + x = -3? Of course, we must have already discovered negative numbers, too. This is an example of.. what? Oh right, algebra. We aren't using arithmetic and basic counting to equate (-1)*(-3).

Is this how you see Pi?That would be more like a way someone might think "seeing" a number would "look". Here's the thing though: I don't "see" pi, at least not the way you seem to imply with a picture.

It takes a lot of mathematics study to understand what DH means by numbers coming from algebra.I disagree, algebra is actually quite straighforward, or the basic ideas are. I don't see any problem with approaching algebra via geometry and through combinatorics at the same time. Doing it in small steps is a good idea, just where to start looks like the problem. Like I say, numbers have a shape. The integers aren't just marks on bits of paper or a whiteboard. Introduce the notion of numbers having "extended" bits, or connections to other numbers, and to "spaces", rather than being discrete members of some set, should start a lot earlier than it does now, I reckon. Also numeration and denomination should be explored more than just "division" and "repeated subtraction".

To PROVE (-1) x (-3) equals 3, you must know two rules: distribution and additive inverse cancellation. We add the latter rule to our list of axioms when we ask questions like what x gives ? Of course, we must have already discovered negative numbers, too. This is an example of.. what? Oh right, algebra. We aren't using arithmetic and basic counting to equate Multiplication of two "negative" numbers, is multiplication of two "same kind" of things. Explaining positive and negative (reverse) counting is simple enough.
You can explain what roots are and what i is using (2) x (2) and (-2) x (-2), pretty simply. Two points are, or define a line, and convey the idea of points on a line, and extension or projection. Algebra isn't a separate math from other math, why treat it like it is?

The idea of counting single things comes straight from the idea of "one" (things) and "not-one". Is there a simpler connection?
Increase and decrease, things we see happening all around, explain "real" or continuous variables. Why does a youngster need years of math to understand these basic ideas?
The symbology and rigid theorems take some of the fun out, but that doesn't have to happen either. Like most things, mathematics is based on essentially simple ideas, or rules, but a lot of complexity emerges (like in a chess game).
By themselves, numbers have no meaning.Numbers aren't "by themselves". This is what young people get told, maybe, but it's something someone made up. The problem with it is you need to explain what it means for numbers to be "by themselves", and why meaning is important.
Despite all the attempts and hundreds of pages of theorems, lemmas and proofs, numbers are not a "fixed" shape. Not even a little bit, except when we think about them.

It occurs to me that education is largely a series of encounters with ideas.
First you get told the "easy" stuff, then they tell you to forget all those ideas you just learned, because actually the story is so-and-so; you get to high school, and they tell you there's a "lot" to learn, and some of the stuff you know already is actually wrong, it's really so-and-so. At uni, they say things like "all that stuff you learned about so-and-so, isn't actually correct so you need to forget all about what you know", and so on.

So the dogma seems to be that you never actually learn the "story", because the teachers have to tell porkies, and half-truths (so young minds can grasp the difficulties).
If kids were told about QM, would it be "over their heads"? Do they really have to be led gently down the path of knowledge, and get told at each step that they haven't been told the "real story" yet? Why the need to instil notions of "difficulty" and "hard to understand", into children's minds? Isn't science supposed to be about essentially simple ideas?

To be quite honest, the universe will only have a shape if it is observed to have a shape. This could be a number of things... from a flat-like universe, to some spherical shape. Or perhaps, just as i have speculated, some shape with lobes sticking out.

If kids were told about QM, would it be "over their heads"?
Kids are introduced to basic concepts of QM such as wave-particle duality in elementary school. They aren't taught what that really means because (a) they don't have the requisite mathematical background to do so, and (b) even the teachers at that level don't have the requisite mathematical background.

Do they really have to be led gently down the path of knowledge, and get told at each step that they haven't been told the "real story" yet?
Yes. How is this different from any other human endeavor? This isn't even true of animals, let alone humans. Wolf cubs have to be taught, in stages, how to hunt. A wolf cub does not instantly go from chasing its own tail to taking down a moose. A human cub similarly cannot instantly go from one+one=two to the Schrödinger wave equation or the eightfold way. These things must be done delicately and in stages.

Why the need to instil notions of "difficulty" and "hard to understand", into children's minds?
Maybe because things like the Schrödinger wave equation and the eightfold way are both difficult and hard to understand, even for quantum physicists? To quote Feynman, "I think I can safely say that nobody understands quantum mechanics."

Isn't science supposed to be about essentially simple ideas?
No. Science is about understanding the nature of the universe. "Essentially simple ideas" are the domain of philosophy and religion. Quoting Feynman again, "Philosophers say a great deal about what is absolutely necessary for science, and it is always, so far as one can see, rather naive, and probably wrong."

A human cub similarly cannot instantly go from one+one=two to the Schrödinger wave equation or the eightfold way. These things must be done delicately and in stages.Nor would hunters give spears and clubs to children and tell them to go and get dinner for the tribe. Usually this happens somewhere near puberty, as an initiation into the "adult" world, but that's culture and anthropology, or something.

My point is, despite all the "tricky" mathematical ideas, the world looking complicated, and full of strange things; fundamentally there are simple "explanations" for much of it.
Mathematical games (board games, e.g. go, chess, draughts) are based on a handful of simple rules. Einstein tried to show "everyone" that the relation between matter and energy is simple (not necessarily easy to understand, though). You can explain his ideas with math and symbology, or with words and pictures. Words are symbols too; math is a language that's constrained, exacting, precise (as you've pointed out, you can construct hundreds of pages of equations, just to connect algebra with "numbers").

I don't see that kids need to understand hundreds of pages of math, or the entire contents of an encyclopedia, to "get the hint". I think we give them a lot less credit for insightfulness than we should, for starters.

My point is, despite all the "tricky" mathematical ideas, the world looking complicated, and full of strange things; fundamentally there are simple "explanations" for much of it.
Mathematical games (board games, e.g. go, chess, draughts) are based on a handful of simple rules.
Quantum mechanics is not draughts. The quantum mechanical world is very strange and definitely is not simple. Physicists try to explain at least parts of quantum mechanics in simple words, such as "wave-particle duality". For those who lack the requisite mathematical understanding, these words remain little different from fairy tales. If you understand the math it is quite wonderful, but it is not simple. If you want simple answers, go to philosophy or religion. They have simple answers. Wrong, but simple.

Quantum M is a statistical theory at best.

Please. Quantum electrodynamics, quantum field theory, the eightfold way are far from statistical theories. They are very, very deep -- and very, very accurate. If QM were merely "a statistical theory at best" it would be chock full of ad-hoc relationships and chock full of magic numbers. Instead, there are but fourteen magic numbers (aka fundamental constants) for which we currently do not have an explanation. Fourteen fundamental constants to describe everything to the incredible accuracy that QM attains is not a "statistical theory at best." The LHC is expected to reduce the number of requisite magic numbers by means of the Higgs.

But it is full of ad-hoc theories, despite it's own predictions. Dr. Balentine shares my own opinions.

You take a particle from the past t=0... it's past rallies on past variables that can only be summed statistically. If we could map put all of reality, we would have a set of binary digits that described everything.

Besides... the Higgs will never be found. I can assure you of that; If it is, you can hold me wrong.

Nothing said?

Tell me... due to UP, can you map out a single particle>?

No.. is the answer.

Notwithstanding that QED is "the most accurate theory we have to date", Richard Feynman was fairly categorical (in the lectures I went to at Waikato) about the probabilistic nature of the world (the behaviour of light, was his favourite example).
I'd say I agree with what he said (except the bits I don't really understand). The world and quantum behaviour is statistical, unless someone can come up with a non-statistical meaning for "probability-amplitude", we're stuck with it.

I didn't say QM is draughts, I said mathematical games (or call them sets of transforms on a 2-d surface, maybe), have simple rules; complexity nonetheless emerges. This is still a guiding principle in science, despite thousands of math equations and a lot of complex systems that still defy analysis (but we're "getting there").