# Irrational numbers could they be more

consistently thought of as actually emergent functions that take the desired accuracy as input?

As them being numbers would imply the apparently paradoxical concept that infinite complexity can exist in a finite volume of space.

As them being numbers would imply the apparently paradoxical concept that infinite complexity can exist in a finite volume of space.

Those words, in that order, make precisely no sense at all. They're not even wrong.
I'm not entirely sure what you're trying to say, but consider this: Almost all real numbers are irrational. If your worldview can't accommodate their existence, then your worldview is wrong.

mfb
Mentor
"more" than what?

consistently thought of as actually emergent functions that take the desired accuracy as input?
What does that mean?

As them being numbers would imply the apparently paradoxical concept that infinite complexity can exist in a finite volume of space.
Numbers are not physical objects in space.

"more" than what?

Shouldn't a number have a particular, unique, set value?

Isn't that a defining characteristic of what a number is? Isn't that intrinsic to the concept of a number?

A function doesn't have one particular, unique value.

What does that mean?

You can aproximative pi by various different methods, which close in on the value (what value? there is no value) at different rates.

So isn't there some underlying mathematical mechanism here whose output is the one merely captured by different aproximations?

Why should pi be thought of as a number (a value) instead of as a rule or function/computation method? A function that receives the desired degree of accuracy as its argument.

Numbers are not physical objects in space.

Yet that's what they're used to represent. Quantities, amounts, sizes. Distances, masses, charges etc. Scalars, vectors, tensors etc.

And isn't this their purpose? To represent amounts or concepts involving amounts (ratios, for example)?

What amount or ratio of amounts does pi represent?

But beyond that, shouldn't numbers, by definition, have a set, known value?

Do we say that a function that tends asymptotically towards a limit has that limit as its value?

Yet irrational numbers aren't even asymptotic limits.

Or do we say that a function has a particular value at all?

No, because the value of the function depends on the input.

If irrational numbers may be thought of as functions so too may real numbers with repeating decimals aren't numbers either but actually functions as well.

Those words, in that order, make precisely no sense at all. They're not even wrong.

I'm not entirely sure what you're trying to say, but consider this: Almost all real numbers are irrational. If your worldview can't accommodate their existence, then your worldview is wrong.

My world view is that space, time and everything else is discrete, quantized.

And that I have no clue how reality works or what it looks like at its fundamental scale.

Or how and why objects, particles, waves etc. skip from one discrete position in space-time to a particular other position or what the fundamental lattice of space and time looks like, what shape its quanta are (i doubt it's cubical).

But that this somehow probably accounts for quantum weirdness.

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Shouldn't a number have a particular, unique, set value?

Pi has a precise value. Why do you think it doesn't? Or do you mean its representation should be unique (i.e. that it should be represented by a unique symbol)? This is clearly untrue: 0.333... and 1/3 represent the same number, for instance.

Isn't that a defining characteristic of what a number is? Isn't that intrinsic to the concept of a number?

"Number" has no definition in mathematics; it's a colloquialism.

What amount or ratio of amounts does pi represent?

The ratio of a circle's circumference to its diameter.

Yet irrational numbers aren't even asymptotic limits.

Of course they are. I can construct a sequence of rational numbers converging to any irrational number you like (this is, in fact, where irrational numbers "come from" when constructing the reals).

If irrational numbers may be thought of as functions so too may real numbers with repeating decimals aren't numbers either but actually functions as well.

You don't seem to understand what a function is either. A function ##f:X \rightarrow Y## is a subset ##f \subset X \times Y## such that, for every ##x \in X##, there exists exactly one ##y \in Y## such that ##(x,y) \in f##.

Now tell me, how is an irrational number a function?

phinds
Gold Member
My world view is that space, time and everything else is discrete, quantized.

And that I have no clue how reality works or what it looks like at its fundamental scale.

These two statements are mutually contradictory.

There is no evidence that either space or time is quantized but you say that they ARE, then you say you don't know what things are like at the fundamental scale after having just said that you believe they are quantized. So if you don't know what they are like, why do you think they are quantized?

These two statements are mutually contradictory.

There is no evidence that either space or time is quantized but you say that they ARE, then you say you don't know what things are like at the fundamental scale after having just said that you believe they are quantized. So if you don't know what they are like, why do you think they are quantized?

1. No, I didn't say that they are. I said that's my worldview.

That is my worldview because ascribing to a continuous spacetime implies infinite complexity in a finite volume of space.

As that seems paradoxical to me I am forced to assume the alternative. Thus my worldview.

2. I said I don't know how spacetime is quantized.

I also said I don't know what movement at this scale is like or how or even why it occurs.

It might be something akin to what makes fliers in conway's game of life move and how they move.

We might be made of those fliers.

Electrons might be fliers like that, for instance.

Maybe that's also why they're waves of probability to us, looking down from our scale. Because they occupy more than a single quanta of space.

The fliers in Conway's game of life change shape as they move through the lattice. Maybe so do electrons as they orbit the nucleus.

mfb
Mentor
My world view is that space, time and everything else is discrete, quantized.
This could be true in physics (we don't know).
It is certainly wrong in mathematics.

Pi has a precise value. Why do you think it doesn't?

What is that value? Write it out in its entirety.

Or do you mean its representation should be unique (i.e. that it should be represented by a unique symbol)? This is clearly untrue: 0.333... and 1/3 represent the same number, for instance.

Reread my post, I conceded at the end that by my logic real numbers with predictably repeating decimals like 1/3 may also be regarded as functions.

It really wasn't much of a concession for me to make at all. I agree full heartedly with it.

"Number" has no definition in mathematics; it's a colloquialism.

Then this whole discussion is rather pointless, isn't it?

The ratio of a circle's circumference to its diameter.

Well for this circle that ratio seems to me to be 2.4.

Or is this not a circle?

What's the difference between a circle and this? Is it a quantitative difference or a qualitative one?

They're both enclosed, at least. A circle and my contraption above.

They're also both convex.

What other qualitative differences can there be?

Of course they are. I can construct a sequence of rational numbers converging to any irrational number you like (this is, in fact, where irrational numbers "come from" when constructing the reals).

I can't give an irrational number. Not by value, at least.

How do you pass a function by value?

You don't, you pass it by description, by outlying the algorithm contained therein.

You don't seem to understand what a function is either. A function ##f:X \rightarrow Y## is a subset ##f \subset X \times Y## such that, for every ##x \in X##, there exists exactly one ##y \in Y## such that ##(x,y) \in f##.

Now tell me, how is an irrational number a function?

An irrational number is not a value but a computation algorithm. A function. A method of computing something whose value depends on the input provided.

Computation is meaningless when no input is provided or input is never changed.

Except as a way to save on memory usage at the cost of processing time or reduce the likelihood of value corruption (by recomputing the result each time it's needed, to the degree of precision required). However the memory storing the algorithm itself may also be subject to corruption.

I must say it really does seem to me you're intent on ridiculing me or my views instead of earnestly discussing them just in case they're not self-evidently wrong, just a different.

This could be true in physics (we don't know).
It is certainly wrong in mathematics.

I think lots of what we colloquially call numbers, as number nine put it, are really functions.

I don't think it's wrong to call a function a function. I mean, it may be wrong by fiat.

I think when we're computing or manipulating expressions we're actually engaging in a sort of fundamental programming.

We are writing an algorithm for computing values depending on values received as inputs, not computing a value outright.

Writing algorithms to compute stuff.

Or rather simplifying and combining existing algorithms.

I thing a mathematical function or expression is basically an algorithm. The results of algorithms vary. They are not unique. That would be pointless. And that's why you don't call an algorithm by its value. Because it has no particular value.

This becomes more clear if and when you try to write a parser for mathematical language. You realize mathematics is actually metaprogramming.

I think there is no real reason why it would be wrong to view irrational numbers and real numbers with repeating decimals as functions.

Except for inferred dogma.

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What is that value? Write it out in its entirety.

It's value is denoted precisely by ##\pi##.

Reread my post, I conceded at the end that by my logic real numbers with predictably repeating decimals like 1/3 may also be regarded as functions.

1/3 is not a mapping between sets. It is not a function.
You are using words without understanding their meaning.

Then this whole discussion is rather pointless, isn't it?

Not at all, the terms "rational number" and "irrational" have precise definitions with respect to the reals.

Or is this not a circle?

No, it isn't. A circle is the graph of an equation of the form ##(x - a)^2 + (b - y)^2 = r^2##.

An irrational number is not a value but a computation algorithm. A function. A method of computing something whose value depends on the input provided.

Specify the sets ##X## and ##Y## as well as the mapping ##f:X \rightarrow Y## such that ##\sqrt{2} = f##.

What you're suggesting is not an "alternative explanation". It is demonstrably nonsense and suggests that you have never studied mathematics beyond a trivial high school level, and, worse, are incapable of dealing with concepts that even slightly violate your naive intuition.

I think there is no real reason why it would be wrong to view irrational numbers and real numbers with repeating decimals as functions.

You don't even understand what a function is, so clearly you have no basis to make this statement.

EDIT: After reading this thread, it is quite clear that you are a crank. This thread should be locked.

It's value is denoted precisely by ##\pi##.

You're being purposely disingenuous. That's a wildcard, not a value.

A function pointer, if you will.

Just as a or b may be regarded as pointers (or references, placeholders, if you prefer) to values or numbers.

1/3 is not a mapping between sets. It is not a function.
You are using words without understanding their meaning.

Or maybe I'm thinking of a deeper meaning that transpires subtly different but essentially the same through different definitions.

Does 1/3 not refer to the instruction to execute the algorithm known as "division" on the arguments '1' and '3'?

Is it or is it not an explicit command to perform an algorithm on specific values for arguments?

Is division not an algorithm? Is division a number?

Please take a little time to think about it before you reply as you're being needlessly argumentative and you're not really providing any new rebuttals, just rehashing your first, superficial ones.

Not at all, the terms "rational number" and "irrational" have precise definitions with respect to the reals.

No, it isn't. A circle is the graph of an equation of the form ##(x - a)^2 + (b - y)^2 = r^2##.

So that isn't a circle.

Explain why, please. Don't repeat what a circle is, explain why that one isn't.

Specify the sets ##X## and ##Y## as well as the mapping ##f:X \rightarrow Y## such that ##\sqrt{2} = f##.

What you're suggesting is not an "alternative explanation". It is demonstrably nonsense and suggests that you have never studied mathematics beyond a trivial high school level, and, worse, are incapable of dealing with concepts that even slightly violate your naive intuition.

So the square root operation is now a number as well?

I'm beginning to wonder what a number isn't.

Why? Because you can't successfully argue your assertions?

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Nugatory
Mentor
Pejeu, much of what you're saying (the emphasis on representations, algorithms, converging approximations) leads me to believe that you're coming at this from a computer science perspective. Assuming that you are, I'll offer up two thoughts:

1) The mathematical theory of numbers predates modern computing by centuries. There's an enormous body of knowledge out there that you can take advantage of. This isn't to say you should just swallow it uncritically, it's more that you cannot improve on what we already have if you don't know what we have already have.

2) Although real computers must represent numbers in a finite number of bits, and therefore cannot represent all the numbers on the number line, you should not confuse that with the problems we have finding finite-length encodings of the irrational numbers. It's true that most modern processors interpret the bit pattern 00111111100000000000000000000000 as an exact 1.0, but the CPU designer could have chosen to assign that bit pattern to the number ##\sqrt{2}## or ##\pi## instead (this would be a bad idea as the resulting design would be harder to implement and less useful, but there's nothing inherently impossible about it).

Nugatory
Mentor
Its value is denoted precisely by π.
That's a wildcard, not a value.

It is neither. As the word "denoted" suggests, it is a symbol used to name a particular member of the set of real numbers, just as is "1" or "3/5".

Integral
Staff Emeritus