## Fibonacci's sequence and Emergence.

• 2.1k
1 + 1 = 2. One apple + another is 2 apples. Easy.
This works when we count material reality. Objects. But that logic only goes so far and isn't very useful in the grand scheme of things.

What if we want to count the other parts of reality that we observe also, the other things that exist, such as the interaction between apples (the space, the distance, the motion, etc) and not just the objects themselves?

In that case we require a new formula: where 1 + 1 = 2 + 1
That is to say that in combining them, 1 + 1 = 2 (the sum) plus the addition of a new emergent conferred from the act of combination. A new phenomena or interaction. That the sum is in fact not equal to but greater than the addition of its components, as follows:

1 + 1 = 2 (1 object + 1 object = 2 objects)
2 +1 = 3 (2 objects plus their interaction with one another = 3 (a new existent relationship).
3 + 2 = 5 (the new existent relationship + 2 more of the previous objects = 5 (another existent relationship different in property to the previous) and so on...
5+3 = 8
8 + 5 = 13
13 + 8 = 21 etc.

The Fibonacci sequence. In this way 2 objects can be combined to give rise to 1 new object with new properties.
Just as hydrogen and oxygen have specific, individual properties as atoms, but when combined give rise to water (a new existent with properties that do not match the sum of its components).

And just like how individual grains of sand can be added together to give rise to a new singular object - a heap/pile of sand that has different properties to the components.
For example, a pile of sands properties are: it can be seen at a greater distance, can be stood on, can hold the form of a footprint, has a texture as it runs through your fingers, can be scooped etc. All things a single grain of sand cannot do.

It seems that when something (a grain) becomes too multiplicitous to count, when it becomes uncountable in any reasonable amount of time required to painstakingly deconstruct it grain by grain, it becomes a new existent (a heap). A pile of sand is no less a pile at 10,000 grains than it is at 100,000 or a million. So the phenomena of "pileness/heapness" is removed from the sum of its components (grains).

Water flows. Two water molecules do not flow. They are distinct existents.

Fibonacci sequence does something other counting methods do not, it does not exclude emergent phenomenon from its summation like regular object counting does. Which is why both systems start out the same: 1 +1 =2, 2+ 1= 3, but then depart from one another:

Material accountancy: 3 +1 = 4, 4 +1 =5, 6, 7, 8, 9, 10 etc

Material and phenomenological accountancy - fibonacci counting: 3 +2 =5, 5 +3 = 8, 13, 21, 34, 55 etc

It stands to reason that counting systems that consider material and behaviors/interaction both as assumed existents (previous sums) will increase in magnitude/size much faster than ones that only consider material things (as they exclude half of the reality of things - actions/behaviours).

Fibonacci sequence is seen throughout nature and can be found in virtually any system. And is tied to the golden ratio: which concerns everything from sunflower seed heads, to seashells, to the vitruvian proportions of man, to the shape of galaxies.

It is emergence as a number sequence. Not strict static objectivity as a number sequence. It is an irrational sequence (never repeating itself exactly - due to the influence of emergent phenomena, but constructed by rational numbers - finite/real objects)
• 11.7k
So, what's the point here?
• 3.2k

Putting your concept in mathematical terms involves special definitions and operations, not simple arithmetic. 1+1=2+1 no no no.
• 9.5k
Most interesting. — Ms. Marple

We start from [0, 1]
We add 1 to 0 [0. 1. 1]
We add 1 to 1 [0. 1, 1, 2]
We add 2 to 1 [0, 1, 1, 2, 3]
.
.
.

0, 1, 1, 2, 3, 5, 8, ...

Notice how it builds up in a way that's nearer to exponentiation [the (golden) ratio between fibonacci numbers gets closer and closer to 1.1816...] than addition.
• 3.2k
It is an irrational sequence (never repeating itself exactly - due to the influence of emergent phenomena, but constructed by rational numbers - finite/real objects)

Sorry, but this makes no sense. Irrationality sequence
• 1.1k

In this video Mathologer explains why the two spirals running clockwise and counter-clockwise in a sunflower head end up in ratios corresponding to the Fibonacci sequence.

There are similar sequences in nature, e.g. Lucas sequences. https://r-knott.surrey.ac.uk/Fibonacci/lucasNbs.html
• 9.5k
The Fibonacci sequence
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...

The Negative Fibonacci sequence
4, 6, 7, 9 10, 11, 12, 14, 15, ...

Could we use this to generate a (true) random number?
• 166
I'm afraid I find most of this thread unintelligible. My attention is fixated upon the proposition that in some way we could make 1 +1 = 2 + 1. What modification would we propose to the axioms of arithmetic to make this possible?

Or is Agent Smith subconsciously confusing sets and subsets? Because if we have a set of 2, the number of possible subsets which can be formed is equal to 2^2, or 4.
• 537
in combining them, 1 + 1 = 2 (the sum) plus the addition of a new emergent conferred from the act of combination.

Fitting regular hexagons together in a circle makes 6 hexagons plus 1 more emergent in the middle making 7 hexagons. But on the surface of a soccer ball 5 hexagons make 5 hexagons + 1 pentagon. It would appear that you are not entirely either right or wrong. Emergence seems to require at least one additional outside factor to guide a convergent or divergent process.
• 9.5k

From the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, ...
we get the following ordered pairs

(1, 1), (2, 3), (3, 5), (5, 8), (8, 13) ...

Is this some kinda pattern? :chin:
• 2.1k
My attention is fixated upon the proposition that in some way we could make 1 +1 = 2 + 1.

Mathematically you can't. As it's formal. But if you apply it to actual reality, if you have 2 different things, they create a third phenomenon - the transitionary state

For example if I have an apple and a background/backdrop - yes I have two different things, but I can point to the border and say that it is a third quality - the transition between the two states. Its neither the apple nor the background. But confers the definition to each.

You can't have two defined poles/opposites without the central graduation/abrupt change between the two definitions.

In essence any definition requires three things: definition of the first thing, definition of the second thing, and the information pertaining to the contrast, the parameter that distinguishes them, the change, the defining aspect itself, the boundary.

I hope that explains how 1 +1 =3.
Dark + light = 2 plus 1: the point at which they change from one state to the other. This is not strict formal mathematics, I don't pretend it to be, but rather the empirical observation written as "things/components" of the experience.

In that sense, if you have nothing, and you at one existent, you have three states, the existent, the field it exists in, and the boundary between them.

Fibonaccis sequence follows the number of "existents" rather than the number of just physical objects, as maths would.

It is a counting system that doesn't presume anything other that previous "existents"
• 166
OK, I think I'm getting a handle on this thread now. I thought we were discussing a problem in arithmetic where, as everybody knows, it is impossible to devise an argument that 1+1=2+1 without altering the axioms or redefining the inductive number line. But it appears that we are really discussing a problem in metaphysics, where of course anything is possible.

Hint: drop the sunflower seeds and Fibonacci numbers. Not helping.
• 3.2k
I hope that explains how 1 +1 =3

Well, no. But "The whole is greater than the sum of the parts" seems more or less to be what you are saying. And, yes, in a sense that's true of integers. For example, 2+3=5, but 2*3=6, so that the sum of the prime factors of 6 is less than 6.

However, I believe you are thinking of a more philosophical idea. And that's fine, just don't try to make mathematics conform to your notions. :cool:
• 2.1k
However, I believe you are thinking of a more philosophical idea. And that's fine, just don't try to make mathematics conform to your notions. :cool:

That's fair. I certainly don't intend to erode formal mathematics, it is what it is.

I was merely trying to address the gap between mathematical logic and the remaining logics we apply to reality - many of which are empirical yet don't follow/adhere to mathematic predictions. As if they did, I can hardly imagine emergent phenomenon to summate to anything qualitative more than simple mathematical combinations.

Duality seems to prevent anything from being explicitly mathematical in nature. If it were, the hard problem. If consciousness would not be so hard afterall.

However perhaps that's because we haven't developed formal computations complex enough to bridge that gap to subjective states. Maybe AI will demonstrate its possiblez maybe it would. Time will tell.

I always maintain an open mind and will happily stand corrected based on emerging evidence.
• 3.2k
However perhaps that's because we haven't developed formal computations complex enough to bridge that gap to subjective states

When I was young I never expected mathematics to become so diverse and abstract. Now, with over 26,000 pages on Wikipedia trying to guess what's coming next is nonsense. :cool:
• 2.1k
I feel you jgill. I'm the same. Originally I thought it was so straightforward, precise and non contradictory and simple. A given. Pure logic. The perfect format of explanation/demonstration. But sadly nature doesn't encourage perfect/ comprehensive systems.

I found mathematical paradoxes, in abundance, not to mention numerous unreconciled equations, and as you said, myriad Wikipedia pages and alternative mathematical descriptors up for contention/proofs and now I think it doesn't have quite the oomf/power I once believed it to have.

In any case i still admire the amount of applications maths has. It may not be the "be all-end all" but its certainly a half-way house.
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