Isn't it the case that, once you can conceive of numbers, then imaginary number systems become possible? This doesn't really contradict the idea of the reality of natural numbers, it is simply an observation about the operations of rational thought.

Isn't it the case that, once you can conceive of numbers, then imaginary number systems become possible? This doesn't really contradict the idea of the reality of natural numbers, it is simply an observation about the operations of rational thought. — Wayfarer

Assuming you don't concieve something that is logically false, not correct, then yes.

And the reality of numbers? When you broaden the definition of existence to abstract, not material objects, then numbers are real. They are some of the best tools with which we can make a models of the reality around us, so the idea that it's only in our heads, that they are totally fictional, simply isn't so useful.

For the more rigorous about imaginary numbers answer I think you have to ask that from other members here.

And do note that there are many real world problems that are best modelled with using imaginary numbers. And that for me is allways the reality check: if the math, however "unrigorous" or based on non-proven assumptions has uses in the real World, it's likely to correct. Best example of this is the history of the infinitesimal: such a useful concept, but oh, the historical debate around them! However, if you have math that hasn't got any real World applications, then either the use of the math is still to be found or ...there can be something fishy about the premises of the math.

That actually comes to mind with the Cantorian infinities larger than aleph-0/aleph-1. Not much engineering or physics model use the larger aleph infinities, although I remember that someone did say there were some real world applications for them. As I have some background in economics, I am quite sceptical of the "real world" modelling of economic systems. Usually something like engineering tells it better: if the machine works which is based on some math calculations, then likely the math itself is OK.

Yes I'm sure there are some important things to be found out as yet. For me the area of interest is in divine geometry. I have developed some concepts, but they are not fleshed out yet.

Regarding absolute infinity, I feel it is an over simplification. This is not to say it is not the case, rather that it's intricacies may be beyond us at this time. The idea of a kind of transcendent continuum is interesting, but primarily within the sphere of being, rather than anything external. I have developed a thought experiment which attempts to illustrate this.

Imagine you are looking at a transcendent being, like the Christ, a bodhisattva, a god*. As you focus, you feel you are only seeing the surface, the external body of them, the real person is further back(metaphorically), under the surface. You refine your conception to peel back the layers a bit. You imagine a purer more transcendent being, on a higher plane, in a higher dimension. But equally present next to you in your plane, in the same body. It is just your conception which has changed.

So you repeat the process, imagining an even higher purer form, perhaps in a divine realm next to God. Again you are seeing exactly the same person, in the same place nothing has changed other than your perception. Then you repeat the process and begin to realise that each time you peel back a layer it curls up and folds back behind the being like a hair on his head. Then you notice there are many thousands of hairs alongside it. The being becomes in your imagination transcendent. Traversing many dimensions, even realms worlds, times. While all the time absolutely present in this one place. You realise that the dimension they inhabit is of another order another kind and for them to step into the manifest world's we find ourselves in is like dipping their toe into the water. You look again and the being is as before before you looked closer, just another person standing next to you, all along it was your conception which had changed.

* I have experienced something along these lines in person with a guru, I once new, which helped me to develop the idea.

Regarding the Hilbert's hotel illustration. It occurs to me that infinity - infinity = infinity. I say this because the Hindus define Brahman as that which is the same if you take something away from it, subtract it, it remains the same infinity.

Regarding the Hilbert's hotel illustration. It occurs to me that infinity - infinity = infinity. I say this because the Hindus define Brahman as that which is the same if you take something away from it, subtract it, it remains the same infinity. — Punshhh

But does that something refer to a finite amount? Basically I think it ( here the "infinity-infinity=?" question) is regarded in math as an indeterminate form or said to be undefined.

An otherwise inconmensurable amount is naturally equal to itself (a correct model of itself), but just how addition and substraction goes, operations which basically come from the finite realm, is not an easy question.

l am not a mathematician, so can't get involved in discussing equations. However I don't see the necessity when considering infinity.

You say an "indeterminate form", yes I can see that, but isn't an infinite quantity also indeterminate? I don't see how it is significantly different. You could add infinity to infinity and get infinity. Can't you then take that same infinity away again and leave the first infinity as it was before. I agree that addition and subtraction might come from the finite realm, so in a sense this is trying to discuss an unknown in another room, you can't see into.

Anyway, the Hindu's describe Brahman as infinitely infinite(I'll look for a reference), so there are plenty of infinites around in there for one to be subtracted without diminishing his omnipotence.

I would point out that I don't use infinity much, I find eternity much more fruitful.

I do feel sorry for Cantor, that he was not able to progress further in the direction of the continuum. I think he was lacking some transcendent insight, which might have helped.

Infinity minus infinity has the same meaning as blue minus blue - ie no meaning.

In transfinite arithmetic we have a collection of objects, called 'cardinalities', which is all the numbers, finite and infinite. Then we have a set of operations that can be performed on those cardinalities. These operations are addition, multiplication and binary exponentiation (x goes to 2^x). The first two operations take two inputs and give one output (binary operations). The last one takes one input and gives an output.

There is no operation of subtraction, so to talk of it is meaningless. Trying to talk about it is an example of inappropriate generalisation. That occurs when subset U of a set S has a property P that is not held by all members of set S, and one then asks a question that presupposes that P applies to all of S.

The subset in this case is the set of all finite cardinalities, for which we can define an operation of subtraction. We end up in confusion if we assume that means we can define an operation of subtraction for all cardinalities.

Another analogy: All primates have eyes. Primates are a subset of the kingdom of eukaryotes - organisms whose cells have nuclei. Fungi are eukaryotes. So let us ask ourselves

which is the left eye of a mushroom?

Yes I get that, but the fact that it is undefined is not really saying anything about things that might exist. The use of undefined in this maths seems to be more a case of, encountering an unspecified(undefined) value and then concluding we can't go any further because we've lost the content. Out there in the world of things that exist, you can't just deny something that exists simply because it doesn't compute.

Let's say there is an infinite amount of grains of green sand and there is also an infinite amount of grains of blue sand, both exist. We know that if we theoretically count them as one group infinity + infinity and that we will then have an infinite amount of grains of sand, which we know are green and blue, but which are still seperate, because we are only imagining them as grouped together. Now let's imagine we mix them up so that they are all randomly mixed in together, a set of an infinite amount of grains of sand, of undefined colours. Now we could theoretically sort through this set and put all the green ones in one place and all the blue in another until we are back were we started. So we have subtracted an infinite quantity from an infinite quantity.

I am by the way aware of the problems of applying infinity to existence.

Let's say there is an infinite amount of grains of green sand and there is also an infinite amount of grains of blue sand, both exist. We know that if we theoretically count them as one group infinity + infinity and that we will then have an infinite amount of grains of sand, which we know are green and blue, but which are still seperate, because we are only imagining them as grouped together. Now let's imagine we mix them up so that they are all randomly mixed in together, a set of an infinite amount of grains of sand, of undefined colours. Now we could theoretically sort through this set and put all the green ones in one place and all the blue in another until we are back were we started. So we have subtracted an infinite quantity from an infinite quantity. — Punshhh

I think that's a really good idea. Why not actually perform the experiment to prove your point? Once you have demonstrated it, mathematicians will have no place to hide, they will have to accept your proof!

If you can't find two literally infinite piles of sand of different colours, why not use the odd and even numbers instead?

If there were an infinite amount of sand, storing it might be problematical, because there wouldn't be room for anything else.....

Infinity minus infinity has the same meaning as blue minus blue - ie no meaning. - There is no operation of subtraction, so to talk of it is meaningless. — andrewk

So infinity is meaningless to you? Not something in any way used in ordinary math? Who said that infinity was now treated as a number now? Obviously infinity is incommensurable to any finite number, hence there isn't much "meaning". Yes, infinity is something that cannot be treated as finite.

In transfinite arithmetic — andrewk

You mean here Cantorian set theory and cardinals and ordinals here.

We know that if we theoretically count them as one group infinity + infinity and that we will then have an infinite amount of grains of sand, which we know are green and blue, but which are still seperate, because we are only imagining them as grouped together. — Punshhh

Nope, doesn't go like that. First, something that can be counted goes against what is infinity (hence it's even theoritically wrong), secondly, green infinite grains are as much as blue and green grains together, if they are infinite. Or much as there are green, blue, pink, purple and black and white infinite grains and so on are together... You are using finite logic here.

What Tom is talking is about Cantor's most basic findings (with even and odd numbers).

am a believer that actually infinity is a number.

Can this number be counted?

Also, I am aware how many grains there are;

blue grains = green grains = green plus blue grains = (green plus blue grains minus green grains) = infinity.

If you can't find two literally infinite piles of sand of different colours, why not use the odd and even numbers instead?

Yes, but numbers are ideas, so susceptible to human frailty. An alien, or a monkey, can count the grains of sand and can only come to the same conclusion, because they are not ideas.

Yes, perhaps if each grain is infinitely small, then they would fit on the head of a pin;)

So infinity is meaningless to you? — ssu

I didn't say that. What made you think that I did?

You mean here Cantorian set theory and cardinals and ordinals here. — ssu

Arithmetic with cardinals, not ordinals. See this wolfram page.

Arithmetic with infinite ordinals is also possible, but is different from arithmetic with cardinals. For instance cardinal addition is commutative, but ordinal addition is not.

So we have subtracted an infinite quantity from an infinite quantity. — Punshhh

We have done something, and that something has to do with the folk notion of 'taking away'. But subtraction is a much more precise notion than the folk notion of taking away. To be able to subtract two things they must be members of a set that has a binary operation, which we can call 'addition', such that the elements of that set form an Abelian Group under that operation.

Transfinite cardinals form an Abelian monoid, but not a group, under the binary operation of addition. So it is not possible to define an operation of subtraction for them.

Yes I understand this(although I'm not a mathematician), I dont see a cardinality though in the grains of sand. Unless the set of all grains of sand (green and blue) has a higher cardinality than the set of green grains?

I think this is as far as I can go in abstract mathematics, my interest is more in the direction of maths in the real world or where it is to be found, or relevant in/to existence.

Yes, but numbers are ideas, so susceptible to human frailty. An alien, or a monkey, can count the grains of sand and can only come to the same conclusion, because they are not ideas. — Punshhh

Even better, get your sand and set the monkeys to work!

I dont see a cardinality though in the grains of sand. Unless the set of all grains of sand (green and blue) has a higher cardinality than the set of green grains? — Punshhh

The set of all grains of sand, whatever colour is finite. The total number of particles in the visible universe is only 10^80!

You appeared to be discussing infinite sets earlier, which behave differently from finite sets.

I think this is as far as I can go in abstract mathematics, my interest is more in the direction of maths in the real world or where it is to be found, or relevant in/to existence. — Punshhh

If you are interested in reality, you haven't even begun to scratch the surface. Reality takes place in the continuum, where denumerable infinity is not big enough.

I was thinking of a notional infinite quantity of grains of sand. Or do you think there cannot be and infinite quantity of anything, even hypothetically?

to scratch the surface. Reality takes place in the continuum, where denumerable infinity is not big enough

I'm all ears, what is infinite about reality?

There is absolutely no way an infinite number of anything of any size can occupy a finite space.

Reality takes place in the continuum, which is Aleph1or 2^(Aleph0) if you have reason to reject the continuum hypothesis.

There is absolutely no way an infinite number of anything of any size can occupy a finite space.

Who said it was finite? I am discussing a hypothetical situation, in which there is sufficient space.

Reality takes place in the continuum, which is Aleph1or 2^(Aleph0) if you have reason to reject the continuum hypothesis.

Nice idea, but that is a mathematical form, we are talking about life and existence, where is this continuum? And how does it produce these finite things I see before me?

I did point out when I brought this up that I am aware of the difficulties in applying infinity to reality.

Who said it was finite? I am discussing a hypothetical situation, in which there is sufficient space. — Punshhh

Actually, you are not discussing an hypothetical situation, you are discussing a meaningless, impossible, unphysical, imaginary situation, in a different reality with different laws of physics.

Despite your earlier protestation:

Yes, but numbers are ideas, so susceptible to human frailty. An alien, or a monkey, can count the grains of sand and can only come to the same conclusion, because they are not ideas. — Punshhh

I think you will find the idea of "number" better defined and understood than aliens, counting monkeys, and piles of sand so large that they would create infinite black holes by now.

Nice idea, but that is a mathematical form, we are talking about life and existence, where is this continuum? And how does it produce these finite things I see before me? — Punshhh

The laws of physics take place in the continuum: they are differential equations based on the continuum.

And, there is the point of view that the integers are *not* fundamental, but the continuum is.

Are you not able to discuss a simple hypothetical, such as the grains of sand I described? I know it is probably an impossibility in physical reality, but that is not the point.

I'm interested in these ideas about an existing continuum, is this in the field of mathematics, or astrophysics?

So you are suggesting that number, i.e. Integers are not fundamental. Does this mean that there are places where 1+1 doesn't equal 2?

Are you not able to discuss a simple hypothetical, such as the grains of sand I described? I know it is probably an impossibility in physical reality, but that is not the point. — Punshhh

As I said, hypothetical alien monkeys aren't as well understood as numbers. But you refuse to discuss the odd and even numbers.

I'm interested in these ideas about an existing continuum, is this in the field of mathematics, or astrophysics? — Punshhh

The laws of physics take place in the continuum, what more do you want? You've got literally everything including astrophysics, quantum mechanics and general relativity.

So you are suggesting that number, i.e. Integers are not fundamental. Does this mean that there are places where 1+1 doesn't equal 2? — Punshhh

How does taking the continuum as fundamental entail that nonsense?

Ok let's try the odd and even numbers and see where that takes us.

Odd numbers = even numbers = odd and even numbers = ( odd numbers and even numbers munis even numbers) = infinity.

There you go, you can subtract infinity from infinity.

You still haven't either described this continuum, or said where it is, in realation to us?

Maybe you should watch this

I'm not you, you're not me, the two of us may chat about the number 2.

numbers are ideas — Punshhh

With the opening post I tried to account for abstract numbers by falling back on the concrete world.

If we speak of just 3, the abstract number, then it becomes more concrete when we speak of 3 Hollywood celebrities, 3 meters across the yard, ...
Kind of analogous to speaking of hypotheticals, if you will.

If my attempt (at falling back on concretes) works, then we could perhaps remove such numbers from Platonism?
(That was an intended discussion point. Where's Nagase?)

I don't think anyone disputes that axiomatic set theory is abstract.
I suppose there's a question of unwarranted ontologization involved somewhere as well.

I already know about Hilbert's hotel etc. What is your point again, there doesn't seem to be one?