There are many different things called infinite in mathematics.
- There are the ordinal numbers from ω onwards.
- There are the infinite cardinals.
- There are infinitely many rational numbers, and infinitely many reals, but more reals than rationals.
- Sometimes it is convenient to add a "point at infinity" at each end of the real line.
- Sometimes it is convenient to bend it into a circle and add a single "point at infinity" that joins the ends — which actually makes it a compact topological space, which seems like a smaller sort of thing than the real line.
- In the plane one may add a single point at infinity to make the whole thing into something homeomorphic to a sphere. This is useful in projective geometry and complex analysis.
- The delta function is notionally "infinite" at zero and zero everywhere else, which makes little sense literally as stated, but can be understood as an informal way of talking about a certain distribution.
- Surreals introduce in some sense as many "infinities" as possible.
- Non-standard analysis introduces infinite numbers in yet another way; but arguments in that system can be translated back into traditional epsilon-delta forms.
When extending a space with objects whose existence would be convenient, the question to ask is not, "Does such an object really exist?" but "What properties does this thing need to have, to do what I want it to do? Can it be consistently axiomatised?" For it is said that in mathematics, existence is freedom from contradiction.
Another possible metaphor is to think of infinities as second class citizens. For example, in our world dragons don't exist, but if they existed they wouldn't be able to ride the subway as easily as humans, because that would pose practical problems for both dragons and humans. Same for infinities - in the world of numbers they don't really exist, but if they existed, it's not clear how we could extend them equal rights of addition and so on. It's up to politicians/mathematicians to imagine a world where dragons/infinities can live on equal terms with humans/numbers, and maybe such a world just can't be imagined in a way that makes sense.
Argument from lack of imagination isn't really convincing. And I do happen to think that surreal addition does make sense. Social reasons why not be interested in surreal numbers would be lack of applicability or unnaturalness reasonings. But lack of coherence really isn't one. I do agree that the onus of responcibilty of making things workable is on the mathematician.
If you did have "flying pedestrians" that would mean human piloted cars would not be adequate first-line law enforcement. But just assuming that anything flying is a brid that can't be criminally liable doesn't mean that worlds outside of that assumtion are unthinkable.
And giving an ethnicity second-class citizen status just because you are uncomfortable sitting next to them in a bus is not a defensible "practical problem".
So your intent here is to diagnose the conceptual confusion that many people have with respect to infinity yes? And your thesis is that: people are confused about infinity because they think it has a unique referant while in fact positive and negative infinity are different?
I think you are on to something but it's a little more complicated and that's what gets people are confused. The problem is that in fact there are a number of different concepts we use the term infinity to describe which is why it so super confusing (and I bet there are more).
1. Virtual Points that are above or below all other values in an ordered ring (or their positive component) which we use as shorthand to write limits and reason about how they behave.
2. The background idea of the infinite as meaning something that is beyond all finite values (hence why a point at infinity is infinite).
3. The cardinality of sets which are bijectable with a proper subset of themselves, i.e., infinite. Even here there is an ambiguity between the sets with a given cardinality and the cardinal itself.
4. The notion of absolute mathematical infinity. If this concept makes sense it does have a single reference which is taken to be 'larger' (usually in the sense of cardinality) than any possible cardinal, i.e. the height of the true hierarchy of sets.
5. The metaphorical or theological notion of infinity as a way of describing something beyond human comprehension and/or without limits.
The fact that some of these notions do uniquely refer while others don't is a part of the problem.
> people are confused about infinity because they think it has a unique referant while in fact positive and negative infinity are different?
No, that is a different point. The point is that positive infinity would be better treated as multiple different values and trying to mesh them all into one quantity leads to trouble. We differentiate between 2,4,6 and don't use an umbrella term "a lot". Should you do so you could run into trouble with claims like "a lot is divisible by 4" (proof following 4/4=1 affirms, proof following 6%4!=0 refuses).
I did a bad job of fighting ambigioity of the word infinity. Of the listed understandings 2 is closes but I am really pointing ot transfinitism that there are multiple values outside of all finites that are not equal to each other (ie a whole world to play with instead of single islands).
Ended up looking at https://www.journals.uchicago.edu/doi/10.1093/bjps/axw013
There is an explanation how a supposed paradox about flipping infinite amounts of coins is not a no-go result for infinidesimal properties if one understand the significance of using cardinals or ordinals (numerosity). The repaired that is that if you append flipping a coin and proceeding only if heads to flipping infinite amount of coins that makes the probability go down. The "broken" take is that one flip added keeps the series the same so those need to have the same probability.
That kind of logic is used in a lot of places. One place is Library of Babel. Have library with books of infinite length that are in lexiographical order using up all the letters at all the positions. A supposed paradox: Take only the books that begin with "A" then remove the leading "A" as it is not needed. you get the "full library again". This fun is stopped by the books having been of length and now being of length . But if you would use cardinality for size you could be fooled that you arrived exactly where you left off (and with numerocity you don't). There seems to be diffulty judging the infinite length to be definetely literally . But the transfinite amount that the book has should stay constant within the scenario even if the exact value of this variable transfinite amount floats between takes on the scenario.
It seemed that the same pattern is also at work in sun from pea paradox. Express each point of a sphere with infinidesimal rotations from a starting point. Chop it up into "starts with left rotation", "start with rigth rotation", "starts with up rotation", "starts with down rotation" and "do not rotate".
The group F 2
can be "paradoxically decomposed" as follows: Let S(a) be the set of all non-forbidden strings that start with a and define S(a−1), S(b) and S(b−1) similarly. Clearly,
F 2 = { e } ∪ S ( a ) ∪ S ( a − 1 ) ∪ S ( b ) ∪ S ( b − 1 )
but also
F 2 = a S ( a − 1 ) ∪ S ( a ) ,
and
F 2 = b S ( b − 1 ) ∪ S ( b ) ,
where the notation aS(a−1) means take all the strings in S(a−1) and concatenate them on the left with a.
Like adding a flip before infinite set of flips, adding a rotation before an infinite set of rotations actually gets you somewhere else. The "annihilating rotations" of actually shortens it to .
So there is an infinidesimal error going around (two or four missing points adjacent to the opposite side of the ball?). Which means in the end you get two "almost balls" which is a lot less pressing than getting two exact balls.
But it is a rather technical thing and I can't actually follow it in the needed detail. In case it ends up being a thing taking the quality of infinity to be the amount of infinity (mixing cardinal and ordinal matters) is the core of it.
I'm afraid you lost me. I don't understand why 5-7 is "inpositivity" when -2 is more precise and useful. Why do I want to "dull" a number (or number system)?
I'm pretty comfortable with my previous understanding of infinities - limits of an unbounded calculation and distinctions between different levels of cardinality. I don't see what this adds.
Yes indeed is -2 more precise than inpositivity. And so is omega + omega rather than just an infinity. There is more structure in infinity than most people give it credit for. If you already have distinctions between infinite cardinalities nothing here adds to that. But some people think of infinity as a single cardinality.
I often see the belief that infinity is a quantity, when I think it's a quality. This post introduces two qualitative notions on a field where people are likely to have a good quantitative understanding in order to show how the qualitative understanding isn't that satifying.
Assume natural numbers and then we will extend them with two new numbers. They will be suggestively named "inpositivity" and "positivity".
What these numbers are is characterised by rules involving them (I will not differentiate between axiom and theorem).
For any number a and b if a>b then b-a=inpositivity
inpositivy-a=inpositivity
inpositivity+a is undefined
inpositivity-inpositivity=inpositivity
inpositivity*a=inpositivity
inpositity*inpositivity=positivity
positivity+positivity=positivity
positivity*positivity=positivity
0*positivity=0
0*inpositivity=0
positivity+inpositivity is undefined
positivity - inpositivity = positivity
Call the system defined as the "Dulled natural numbers". Now if you compare this system with integers you might see that the qualities "positive" and "negative" have a close correspondence with these two extra dulled numbers. And this is the intention of the construction. We could consider the dulled system in it's own right. Or we can see it as talking coarsely about the more detailed world of integers.
Note that for expressions explicitly listed as undefined we can find very easily reasons why they don't have locked-in results. For example 5-3=2 and 5-7=-2 their "coarsement" analgoues would be positivity-positivity=positivity and positivity-posititivy=inpositivity. Likewise a rule like "positivity-positivity=0" could take "detailment" forms like "5-7=0" which would be clearly false.
When we are talking about infinities one of the systems we might talk about is the extended real line. It's a similar kind of system in that we add two numbers to the usual real numbers. When there we have expressions like "+infinity - +infinity" being undefined we might not be equiped to have a "detailment" argument to cast it into another system why it is so. And after all if it is good system it must be fine at it's own merits.
But when people read statements involving those kind of infinite numbers they might be reading it as "THE infinity + THE infinity". But there is the option of reading it "AN infinity + AN infinity". In the same way you do not read "THE positivity + THE positivity" but "A positive number + A positive number". When you have expressed that a number is positive you have not yet said how big it is. Likewise when you say that an amount is infinite you have expressed a quality that limits some magnitudes to be out of question but you have not actually specified its magnitude.
As an aside: boolean multiplication and addition of 1*1=1 and 1+1=1 seems to be suspiciuosly similar to positivity multiplication and addition. It might be that boolean algebra is the dulled natural numbers without the positive numbers and inpositivity. Boolean 1 does not translate to natural number 1 and this would be true even if we didn't know about positivity. We could also form other "dulled" number systems such as dulled reals or dulled rationals. Making dulling into an operation that could be applied to arbitrary number systems could be interesting but I don't know the language in which a mathematician could read it as a stand-alone idea.