2+2=5

[LegionOnomaMoi]

Infinity isn't a real number, it's an abstract concept.

All numbers are abstract concepts. II, 2, two, zwei, etc., all refer to the same concept, but what? Two apples, two dollars, two whatever. The concept of two is abstracted away from anything physical or concrete. Things get even more complicated when considering, for example, complex or hypercomplex numbers. In fact, while the Greek mathematicians were perfectly happy accepting the concept of irrational numbers like pi, it was some time before even negative numbers were generally accepted, because while you can point to a circle when speaking about pi, it's difficult to talk about negative 2 of this or that. Also, many numbers that are used in every day application (like pi or 1/3) are infinite in some sense.

As for "real numbers," even in high school most students don't stop with the set of real numbers. I'm not sure I see the difference between "infinity as a concept" and lots of other numbers, operations, terms, and so forth used in mathematics which are just as much "abstract concepts" as infinity. Infinity is crucial in mathematical analysis, from proofs using induction to limits (which are of course the foundation of differential and integral calculus).

There are even algebras designed to define operations with transfinite (infinite) numbers/sets. And, as I said earlier, while Godel and Cohen independently proved that the continuum hypothesis (either an infinite set is denumerable, like the integers, or it isn't, like the set of irrational numbers) can't be proven true or false, what we do know is that not all infinite sets are equal.

That is, as Cantor proved, even though there are an infinite number of integers, and there are exactly as many rational numbers as integers, there are more irrational numbers between 0 and 1 then there are integers.

 

[dust1n]

It also demonstrates a scenario where two of something (in this case, nothing) plus two of something yields five of something.

Wait... it doesn't demonstrate a scenario where two of something plus two of something yields five of something. It demonstrates a scenario where two of nothing plus two of nothing equal five of nothing... which is saying 2(0)+2(0)=5(0) or 0+0=0, not 2+2=5.

The other equation is incorrect. 2(infinity)+2(infinity)=4(infinity), not 5(infinity).

 

[LegionOnomaMoi]

The other equation is incorrect. 2(infinity)+2(infinity)=4(infinity), not 5(infinity).

Using what algebra? Groups, rings, etc., are hardly my speciality, but I'm not familiar with the definition of the relations you are using. As I understand it, 2*infinity (or aleph sub zero) +2*infinity = infinity. infinity * infinity = infinity. Any relation (addition, multiplication, etc.) from the sets and subsets of complex numbers and infinity still equals infinity. There is no difference between 4(infinity) and 5(infinity).

 

[Photonic]

All numbers are abstract concepts. II, 2, two, zwei, etc., all refer to the same concept, but what? Two apples, two dollars, two whatever. The concept of two is abstracted away from anything physical or concrete. Things get even more complicated when considering, for example, complex or hypercomplex numbers. In fact, while the Greek mathematicians were perfectly happy accepting the concept of irrational numbers like pi, it was some time before even negative numbers were generally accepted, because while you can point to a circle when speaking about pi, it's difficult to talk about negative 2 of this or that. Also, many numbers that are used in every day application (like pi or 1/3) are infinite in some sense.

As for "real numbers," even in high school most students don't stop with the set of real numbers. I'm not sure I see the difference between "infinity as a concept" and lots of other numbers, operations, terms, and so forth used in mathematics which are just as much "abstract concepts" as infinity. Infinity is crucial in mathematical analysis, from proofs using induction to limits (which are of course the foundation of differential and integral calculus).

There are even algebras designed to define operations with transfinite (infinite) numbers/sets. And, as I said earlier, while Godel and Cohen independently proved that the continuum hypothesis (either an infinite set is denumerable, like the integers, or it isn't, like the set of irrational numbers) can't be proven true or false, what we do know is that not all infinite sets are equal.

That is, as Cantor proved, even though there are an infinite number of integers, and there are exactly as many rational numbers as integers, there are more irrational numbers between 0 and 1 then there are integers.

There are an infinite number of integers.

There are an infinite amount of points between any two points on a line.

You should know this if you have ever taken anything past plane geometry.

You are correct that the value of an infinite number of points between any two defined points is not the same as the value of an infinite line.

This however is outside the context of what he is trying to prove.

 

[Viker]

2+2=5. If you can't agree to this then our friend O'Brien has just the thing for you in Room 101. You will see that 2+2 can and surely does equal 5 in the end.

I don't buy it.

 

[LegionOnomaMoi]

There are an infinite number of integers.

There are an infinite amount of points between any two points on a line.

I'm not sure who you mean by "he," but I'm guessing the one who started this thread?

But whatever the case, I think you missed my point entirely. Points, actually, as I was making more than one. A central point I was trying to make is that saying "infinitiy is an abstract concept" doesn't say much as far as I can see. Numbers are abstract concepts. What difference does it make if infinity is as well?

You should know this if you have ever taken anything past plane geometry.

I have to admit it's been a while since I've taken plane geometry. However, I do use and teach areas which deal heavily with geometry/topology. Geometric interpretations of vector space and linear algebra come up all the time for me, especially in multivariate statistics. And of course the notion of infinite subsections of lines, planes, or other "areas" in n-dimensional space is essential in modelling dynamical systems. So I don't think the fact that my last plane geometry course was probably in high school is an issue here.

You are correct that the value of an infinite number of points between any two defined points is not the same as the value of an infinite line.

Actually, as far as cardinality is concerned, it is. Another point I have either tried to make or alluded to at least twice now is that speaking of "infinity" as a singular abstract concept in mathematics doesn't really accurately reflect math much more than the equations used in the first post.

To steal (or, with your permission, borrow) your discussion of lines, let's differentiate between a line in euclidean space and the number line. On a number line, we can imagine every whole number, every integer, and every rational number as an infinite set, or a line extending infinitely in at least one direction. However, all these sets are the same size, in that they have the same number of elements.

However, other infinite sets have more elements. Hence my statement that there are infinitely more irrational numbers than integers or rational numbers.

It isn't a matter of an infinite line versus infinitely many partitions of a line segment. In coordinate geometry, both are continous functions of real numbers, and are therefore a "larger infinity" than the set of integers.

This however is outside the context of what he is trying to prove.

Perhaps. But part of the problem as I see it is a misunderstanding of 1) algebra(s) 2) infinity and 3) proof. Also, I wrote my response not just to address the original post but also because I think some subsequent posts have presented additional problems rather than addressing the original one.


As for

 

[idea]

2 + 2 = 5 is a mathematical statement of the principle of emergence. (although it is more like 2 + 2 = infinity)

Emergence - Wikipedia, the free encyclopedia

^ a b Aristotle, Metaphysics, Book H 1045a 8-10: "... the totality is not, as it were, a mere heap, but the whole is something besides the parts ...", i.e., the whole is greater than the sum of the parts.

"Although strong emergence is logically possible, it is uncomfortably like magic. How does an irreducible but supervenient downward causal power arise, since by definition it cannot be due to the aggregation of the micro-level potentialities? Such causal powers would be quite unlike anything within our scientific ken. This not only indicates how they will discomfort reasonable forms of materialism. Their mysteriousness will only heighten the traditional worry that emergence entails illegitimately getting something from nothing."(Bedau 1997)


etc. etc.


2 birds + 2 birds is more than just 4 birds...

2 birds + 2 birds and you get eggs, and baby birds, and singing, and ordered flock formations etc. etc. the interactions between the parts create more than just a mere addition of the parts.

2 + 2 = infinity
there is no upper limit to the creative power of synergy / emergence.

 

[Photonic]

I'm not sure who you mean by "he," but I'm guessing the one who started this thread?

But whatever the case, I think you missed my point entirely. Points, actually, as I was making more than one. A central point I was trying to make is that saying "infinitiy is an abstract concept" doesn't say much as far as I can see. Numbers are abstract concepts. What difference does it make if infinity is as well?


I have to admit it's been a while since I've taken plane geometry. However, I do use and teach areas which deal heavily with geometry/topology. Geometric interpretations of vector space and linear algebra come up all the time for me, especially in multivariate statistics. And of course the notion of infinite subsections of lines, planes, or other "areas" in n-dimensional space is essential in modelling dynamical systems. So I don't think the fact that my last plane geometry course was probably in high school is an issue here.



Actually, as far as cardinality is concerned, it is. Another point I have either tried to make or alluded to at least twice now is that speaking of "infinity" as a singular abstract concept in mathematics doesn't really accurately reflect math much more than the equations used in the first post.

To steal (or, with your permission, borrow) your discussion of lines, let's differentiate between a line in euclidean space and the number line. On a number line, we can imagine every whole number, every integer, and every rational number as an infinite set, or a line extending infinitely in at least one direction. However, all these sets are the same size, in that they have the same number of elements.

However, other infinite sets have more elements. Hence my statement that there are infinitely more irrational numbers than integers or rational numbers.

It isn't a matter of an infinite line versus infinitely many partitions of a line segment. In coordinate geometry, both are continous functions of real numbers, and are therefore a "larger infinity" than the set of integers.



Perhaps. But part of the problem as I see it is a misunderstanding of 1) algebra(s) 2) infinity and 3) proof. Also, I wrote my response not just to address the original post but also because I think some subsequent posts have presented additional problems rather than addressing the original one.


As for

The original concept was based so heavily on a misconception of the concept of infinity and undefined variables that it was natural that arguments also displaying some wildly variable assumptions would occur.

Either way, you did not address my point in context, which is that by the logic he is using, infinity is not a "number" you can you can simply apply without a conditional.

"For example, the set of integers is countably infinite, while the set of real numbers is uncountably infinite."

You can argue semantics as much as you want, but I'm keeping this on the topic of the OP.

Anyways, I'm an astrophysicist, not a theoretical mathematician.

 

[idea]

one more PS on Emergence - as this is a religious board - I think emergence is the key to understanding the infinite creative power of God.... At least for Christians, the big point of everything we try to do is centered on relashonships - don't kill, don't steal, don't be jealous - it's all about how we're supposed to treat one another - up to the greatest commandment to love... it's all about interactions, and interactions is where emergence happens - and where infinite creative powers exist.

sin isolates us from one another and kills emergence/synergy. Love/selflessness/kindness - this is the place where emergence/synergy thrive.... to become of one heart and one mind - united in all things - is to achieve infinite creative potential, unbounded by the conservation laws of thermodynamics.

 

[PolyHedral]

2 birds + 2 birds and you get eggs, and baby birds, and singing, and ordered flock formations etc. etc. the interactions between the parts create more than just a mere addition of the parts.

Only because you're a pattern-spotting ape that attaches significance and aesthetic value to sufficiently simple organisation. Organisation is not a thing on its own; it isn't generated by anything.

Incidentally, adding together birds is probably defined differently than adding together integers, especially since birds are Von Neumann machines.

 

[idea]

Only because you're a pattern-spotting ape that attaches significance and aesthetic value to sufficiently simple organisation. Organisation is not a thing on its own; it isn't generated by anything.

Incidentally, adding together birds is probably defined differently than adding together integers, especially since birds are Von Neumann machines.

Organization / information / intelligence is a thing....

A CD that is organized to contain music costs more than a blank CD for instance. (The blank CD and the one that is loaded with music both weigh the same amount, same matter, same energy content - but one costs more) ... information (one form of organization) is protected by laws, bought, sold - people spend their lives trying to obtain it, and create it - it is a something, not a nothing.

But then, I'm a fan of the holographic universe... that everything, at it's core, is organized information.

integers have to represent something in the real world, or they are pointless... why not count birds?

 

[LegionOnomaMoi]

integers have to represent something in the real world, or they are pointless... why not count birds?

There's nothing wrong with using integers to represent the quantity of birds. However, when you add 2 birds to a group of 2 birds, you get 4 birds. Discussiond of eggs or feathers are irrelevant. The validity of applied arithmetic or algebra is due in part to consistent application of abstract numbers to defined entities. I can take several sheets, add a mattress, a blanket, some pillows, a frame, and get one bed. But unless I represent this as something like 4a + b + c + 2d + e= 1 f, where each variable represents the different components I'm working with, I can't add all those things up and get one.

 

[PolyHedral]

integers have to represent something in the real world, or they are pointless... why not count birds?

Sure. Just keep in mind that 4 birds behave differently than 4 electrons.

(And I object that integers have to represent anything in the real world to be useful.)

 

[LegionOnomaMoi]

Either way, you did not address my point in context, which is that by the logic he is using, infinity is not a "number" you can you can simply apply without a conditional.

I'm not sure what you mean here. I quite agree that the original equations don't show what they were intended to. But what do you mean by "a conditional?" I can certainly write 2(infinity)+2(infinity)=5(infinity). It just doesn't imply that 2+2=5, any more than multiplication by zero using the same structure does. The first equation in the OP is really 0=0, and the second is infinity=infinity. The 2's and the 5 don't affect the relations and they don't matter.

 

[Photonic]

I'm not sure what you mean here. I quite agree that the original equations don't show what they were intended to. But what do you mean by "a conditional?" I can certainly write 2(infinity)+2(infinity)=5(infinity). It just doesn't imply that 2+2=5, any more than multiplication by zero using the same structure does. The first equation in the OP is really 0=0, and the second is infinity=infinity. The 2's and the 5 don't affect the relations and they don't matter.

The point is he is trying to derive a finite value out of an infinite one. I was simply pointing out that he is going about it in the wrong way.

 

[LegionOnomaMoi]

The point is he is trying to derive a finite value out of an infinite one. I was simply pointing out that he is going about it in the wrong way.

I certainly agree. I guess the way in which you pointed it out confused me. But no matter.

 

[idea]

watching Karma tubes tonight - here are a couple of synergy vids for anyone who would like to see first hand examples of the principle:

KarmaTube: Lentil As Anything

KarmaTube: Three Stages of Generosity

scarcity minded (2+2=4), vs. abundance minded (2+2=5)... keeping track of tit-for-tat and just pay for work - vs. understanding the rewards that come from living without worrying how it all adds up...

 

[idea]

Sure. Just keep in mind that 4 birds behave differently than 4 electrons.

(And I object that integers have to represent anything in the real world to be useful.)

electrons exhibit principles of emergence too.



consider what one electron is - if you are going to be adding them together, you at least need to understand what one is right? Just take one - all by itself - with nothing else around it - just one - and what do you have? If you can not use how it interact with other things - not measuring interactions (not baby birds and flocks and singing - no interactions) - just the object itself, all by itself - what it is without all the added clutter of interactions around it - what do you have?

Here is a link explaining one part of relativity (with just one electron, or one bird, or one of anything)
Relativity.

My question is pertaining to Einstein's special theory of relativity in an absolute vacuum. Consider the following: an object in space, in a complete vacuum without any boundaries or reference points. Essentially an empty universe. Let's say that the object was moving at a certain velocity, any velocity, doesn't matter what it is. Velocity by definition is, distance traveled per unit time. Now without any reference points distance cannot be calculated and conversely neither can velocity. So the thought came to me. Is there such a thing as velocity in a complete vacuum?

There is much that travels in essentially a complete vacuum. Most of matter is empty space. The electrons associated with an atom or molecule travel within a vacuum. Your question is not about vacuum, but about an empty universe, except for the one object under consideration. Wherever you are considering it from, we may regard it to have a velocity with respect to that place. If you must consider this object without existing yourself, you have an interesting mind experiment. You are, of course, imagining the impossible, and this puts us in a place where we cannot know the answers. The thought process is nonetheless a very good exercise. As I do the mind experiment, I think:
"This object could not have a velocity. This object would be the universe. The universe is 'all that is,' and this imagined object is 'all that is.' There could be no 'space' outside this object. Now I will imagine that I am the object. Where can I travel to? There is no place to go. On what road may I travel? There is no road. I am the universe; I cannot travel to non-existent places taking non-existent routes. Now if I could wiggle my fingers, then I could consider their velocity relative to other parts of me, but now I am not a single object; now I am many objects, and that was not the question. I can imagine that object splitting into two objects and separating at some velocity. Then I would have linear velocity relative to the other object. I still could not have rotational motion of one object about the other, though I could have rotation of either object about its own axis, because I could see that motion relative to the other object. With three objects, I could have two of them rotating about one another and observe that from the third object."
Space is a concept, which comes about due to the existence of "stuff," for the space to be between. Space is not a thing; it is nothing, (no thing), except a mind-made concept. The concept of "love" between two people is like this; it can exist only if the two people exist. The problem that makes "space" seem more material to us, is that we imagine it to have "dimensions." Those dimensions exist only because there is "stuff" to measure the distance between. We have a psychological need to assign dimensions to concepts (that are not things) we want to understand. A child tries to assign dimensions to love by holding his hands wide apart and declaring, "I love you this much."
Great scientists, including Albert Einstein, considered at great length the issue of space as either "nothing," or as an "ether." Science is still not completely agreed on this issue. Your question, using your phrase "complete vacuum," must be considering space as not consisting of an "ether," but consisting of exactly nothing. If there were an "ether," then we could consider your object's travel "through" that ether at some "velocity" relative to that ether.
We are used to thinking of "places" as things. I could show you an empty table and ask you, "How much space is there between the two oranges?" You would answer, "What two oranges?" But if I put two oranges on the table about three feet apart and then asked the same question, you would answer, "About three feet." Those same three feet were there without the oranges, but no "space" between oranges was recognized. We can all see that "area" of the table, but that's because we may associate the "space" with all kinds of other things - the edges of the table, for example, ourselves or the carpet and other things in the room.
You're right that velocity only exists with reference to something. We are traveling about a thousand miles per hour rotating with the Earth. Many thousands of miles per hour around the Sun, and much faster than that as we travel with the Sun around the galaxy - and I suppose again even faster than that as we travel with our Milky Way around the universe.

in short - things like velocity cease to exist for a single entity. (not undefined, but literally ceases to exist)

Everything that we use to describe an object - comes not from the object itself, but from the interactions of the object. Force / electrical charge / potentials / energies / mass - all of these quantities that describe what the one object is only exist when the object is compared to something else. All of this is from interactions - not the object itself.

We may think we are describing a single electron when we say it has a charge of −1.602×10−19 coulomb, a certain spin, or mass - but all of these things cease to exist for the single entity. They only exist relative to other particles. So when you add electrons, you are not adding the particles, you are adding their interactions with one another.... and there are infinite ways two entities can interact with one another... the only things we can talk about that really exist are interactions... no interactions / no relativity / it becomes undefined - and the thing ceases to exist.

 

[LegionOnomaMoi]

in short - things like velocity cease to exist for a single entity. (not undefined, but literally ceases to exist)

...So when you add electrons, you are not adding the particles, you are adding their interactions with one another

Without getting into the issues of your application of relativity, I don't see how your argument follows. In a constrained universe with only a single particle, then yes nothing else would exist (including velocity). But we don't live in such a universe.

The argument "if there were only a single electron, everything else (including velocity) would not exist" doesn't mean that in a universe where other things do exist we can't speak of individual entities or particles without reference to other entities or particles. The fact that three apples wouldn't exist in a universe with only one electron doesn't mean I can't talk about the apples, or add two more apples, and so on. The inability of things to exist without other things doesn't mean we can't think of or conceive of these things singularly.

.... and there are infinite ways two entities can interact with one another

On what do you base this assertion?

 

[Meow Mix]

idea, you're describing background independence as the best method to describe the cosmos (which was Mach's revolution, inherited by Einstein and Noether) -- but I fail to see what it has to do with the conversation at hand; or what it would possibly have to do with the broader question of mathematical realism that you were discussing with Poly earlier.

Can you expound a little more on where you were going with this?