Yes, exactly. Information has no geometry.


Great example. As programmers, we tend to think of arrays as one-dimensional data structures -- a row (or column) of data that we access along a single dimension. e.g., array[7]. Likewise, we tend to think of a matrix as a two-dimensional array -- rows and columns of data that we access along two dimensions: matrix[3][4].

Are these not examples of "1D" and "2D" information? Well, consider how the data is actually stored in RAM.

Let's say we set matrix[0][0] = 'a' and matrix[1][2] = 'b'. We visualize the matrix looking something like this:

View attachment 208743

But to the CPU, RAM is just a long linear array of addresses, so the matrix is stored more like this:

View attachment 208746

To the CPU, the data is entirely "1D". That a programmer (or compiler) can interpret the data as being "2D" is simply the programmer/compiler applying another level of abstraction. The information itself has no such structure.

Indeed, we can create abstract structures with any number of dimensions, -- e.g., a "4D" matrix[0][1][2][4] (which would be very difficult to draw) -- that all reside on the same "1D" structure of RAM.

Note that even the "1D"ness of RAM is an abstraction of the CPU, as the information itself has no geometrical dimension.

Would you say the provisional conclusion of the above from your perspective is that every geometrical and potentially physical space “spatial” notion of dimension is a construct contrived from n-bit dimensionless strings, yes?

 

What if INFINITY had another definition, that of a stand-alone, pre-universe repository of points, energy, and information, a theoretical substance where “1” can characterize it, T that it ontologically exists, and ∞ or infinity would describe it. And it’s the year 1710. (Incidentally, I’m not making that up, this is Pythagorean reasoning, and it’s another way of saying “God”).

Humor me a second... how would this affect your reasoning about everything?

It's a category error for me. INFINITY is a number theoretic concept belonging to a domain that includes sets and natural numbers. Pre-universe repositories of points, energy, and information -- whatever that is -- have an entirely different domain.

 

It's a category error for me. INFINITY is a number theoretic concept belonging to a domain that includes sets and natural numbers. Pre-universe repositories of points, energy, and information -- whatever that is -- have an entirely different domain.

Okay, but what about tabula rasa for a moment...

 

Also what is a “number theoretic concept”?

 

But clearly numbers existed before they were made into sets. And a bit is basically an single quantity. So why can’t you say that it was simply infinite-bit?

Ancient Egyptians used the energy from the giant nuclear reactor in the sky to do lots of stuff, but they had no clue about nuclear energy. Same with the numbers in ℕ.

But if it’s infinitely long, why would you say it’s finite? Because you classify it in terms of ℝ, no?

This is ground-zero for your misconception. Pi is not infinitely long; it has zero length!

More precisely, numbers do not have a geometry and so cannot be said to have length. By mapping numbers to geometrical objects, such as points in Euclidean geometry, then -- and only then -- can we say how "long" a number is. But when we do that, we find that numbers have precisely zero length -- they are points!

Maybe this will be easier for you to grok. How many digits are in the base-10 decimal expansion of 1/3? Infinite, right? 0.3333... There are an infinite number of "3" digits in that representation; anything less than infinite "3"s and you're no longer talking about 1/3.

So, how "long" is 1/3?

 

Ancient Egyptians used the energy from the giant nuclear reactor in the sky to do lots of stuff, but they had no clue about nuclear energy. Same with the numbers in ℕ.


This is ground-zero for your misconception. Pi is not infinitely long; it has zero length!

More precisely, numbers do not have a geometry and so cannot be said to have length. By mapping numbers to geometrical objects, such as points in Euclidean geometry, then -- and only then -- can we say how "long" a number is. But when we do that, we find that numbers have precisely zero length -- they are points!

Maybe this will be easier for you to grok. How many digits are in the base-10 decimal expansion of 1/3? Infinite, right? 0.3333... There are an infinite number of "3" digits in that representation; anything less than infinite "3"s and you're no longer talking about 1/3.

So, how "long" is 1/3?

If they don’t have shape, what is this about? The man who is one of the most extensive record-breaking number cognizers, does so as shapes to do it:

https://www.ted.com/talks/daniel_tammet_different_ways_of_knowing/transcript?language=en#t-216540

(he says pi is an infinite number, it goes on forever, btw)

 

How do we “know” the length of a line then, if all points are zero length?
If between point A and point B are infinite zero-length points, what is measurement?

 

1/3 = .9 repeating = 1 = infinity (one particular incarnation). Each point has zero AND infinite length at the same time.

 

The issue is this: You exclusively define infinity as a process, and not a stand-alone element, am I right? This means you essentially insist all is finite in my estimation, no? Every number is finite to you.

Every number in the traditional number sets (N, Z, R, etc.) are 100% finite. We can rigorously define transfinite numbers (aleph, beth, etc.), but given how much conceptual difficulty we're having with strictly finite numbers, I strongly want to avoid that rabbit hole.

If infinity is not a number, what does it really have to do with numbers from your perspective?

INFINITY is only tangentially related to numbers. INFINITY is the recognition that counting is an unbounded process.

ℕ essentially terminates somewhere, no?

If you're asking if ℕ has a last element, then the answer is no.

Proof: An axiom of the natural numbers is that ℕ is closed under addition -- for any two numbers a,b ∈ ℕ, it is the case that a + b ∈ ℕ. Suppose that n is the last element in ℕ, i.e., there is no number in ℕ greater than n. However, by the axiom of additive closure, n + 1 must also be in ℕ. And since n + 1 is greater than n, it's clear that n could not have been the last element in ℕ. And since n could be any number, there cannot be a last element in ℕ. QED

Note that this does not in any way imply that there are infinite numbers in ℕ.

 

Digit 13 of pi “means” something more than digit 100, right? When we are not using it as a symbol, and breaking into various levels of resolution?

Ambiguous question. What do you mean by "means something more"? In a decimal expansion, the thirteenth digit of Pi has a greater contribution to the value of Pi than the hundredth digit. This is by definition of decimal expansion, which decomposes a number into a sum of weighted values.

 

Would you say the provisional conclusion of the above from your perspective is that every geometrical and potentially physical space “spatial” notion of dimension is a construct contrived from n-bit dimensionless strings, yes?

Pretty close. I wouldn't say that geometric notions are derived from bit strings, rather, I would say that we can characterize any geometrical notion using bit strings (which are indeed dimensionless).

The fundamental principle behind all of this is that information is dimensionless. So, whatever information we get from our perception of space or from our abstractions of geometry, we can perfectly and losslessly characterize using bit strings (which are themselves another abstraction).

 

Okay, but what about tabula rasa for a moment...

How is either INFINITY or "pre-universal repositories of points, energy, and information" tabula rasa? Both of those concepts come pre-loaded with a bunch of assumptions.

 

Also what is a “number theoretic concept”?

A concept formed from investigating the consequences of numbers and sets.

 

Every number in the traditional number sets (N, Z, R, etc.) are 100% finite. We can rigorously define transfinite numbers (aleph, beth, etc.), but given how much conceptual difficulty we're having with strictly finite numbers, I strongly want to avoid that rabbit hole.


INFINITY is only tangentially related to numbers. INFINITY is the recognition that counting is an unbounded process.


If you're asking if ℕ has a last element, then the answer is no.

Proof: An axiom of the natural numbers is that ℕ is closed under addition -- for any two numbers a,b ∈ ℕ, it is the case that a + b ∈ ℕ. Suppose that n is the last element in ℕ, i.e., there is no number in ℕ greater than n. However, by the axiom of additive closure, n + 1 must also be in ℕ. And since n + 1 is greater than n, it's clear that n could not have been the last element in ℕ. And since n could be any number, there cannot be a last element in ℕ. QED

Note that this does not in any way imply that there are infinite numbers in ℕ.

If INFINITY is ultimately outside the numeric as a definition, why can’t its adjective form be used to describe the true nature of ℕ as having infinite numbers?

 

A concept formed from investigating the consequences of numbers and sets.

So INFINITY is not a base token, like “ADDITION,” it’s triangulated, or somehow tangentially a “byproduct?“

 

Pretty close. I wouldn't say that geometric notions are derived from bit strings, rather, I would say that we can characterize any geometrical notion using bit strings (which are indeed dimensionless).

The fundamental principle behind all of this is that information is dimensionless. So, whatever information we get from our perception of space or from our abstractions of geometry, we can perfectly and losslessly characterize using bit strings (which are themselves another abstraction).

How are they characterized if they are dimensionless? Dimensionless is shapeless, so this is my contention from the get go. :—)

In an information processor, where information has no dimension, there is no innate shape of any kind. Correct? How is Daniel Tammet insisting they’re separate, and using “feel” to describe the shapes?

 

How is either INFINITY or "pre-universal repositories of points, energy, and information" tabula rasa? Both of those concepts come pre-loaded with a bunch of assumptions.

I mean, what do you feel about it... you can reckon a “beginning,” God, infinity, numbers, etc. without invoking set theory (just a general conceptualization prior to deep math...temporarily assuming “feeling” potentially separate from knowable math).

 

Btw, it’s not that I don’t understand any of the stuff... I am arguing from a different Pythagorean and Newtonian purview that infinity is a non-emergent, non-tangential, stand-alone, elemental source, involving true spatiality and tokens "feel and meaning” (which science knows little about). Perhaps transfinite has merit, but “trans” is still a modulating prefix to elemental “infinite” no matter how the cake is cut, and I do not believe it’s spurious to grok INFINITY as an innate extra-numeric concept with properties that are not mathematically “boxable.”

 

You’ll notice we’re right in the zone with my original “true cube” question... dimensionless information yielding perceived spatial dimension as an actuality. Either the dimension exists, or everything is one big discrete dimensionless array (contiguous or not).

 

If information is independent of its representation, and representation is the symbology used, then perhaps the symbology itself is tied to the word “feel” and has some kind of true spatiality? Otherwise is not representation just more dimensionless info? Why insist they’re separate? Is the representation not “2D” spatial elements describing dimensionless info that permits meaning, grokking, and feeling toward it?