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Infinite 1 > Infinite 2

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pitubo said:
...IMHO without the "connotation" of intended meanings, the discourse would also stop making sense...
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I agree with that, but that would not stop mathematics from going on. I've calculated 1st, 2nd and 3rd derivatives of far out functions that has no visibly connection to this realm, wanking the invented functions and the logic as for the sake of it, far below infinity stuff. It makes up for part of attraction in match to free ones selves of connotations, I suppose.
 
Jees said:
pitubo said:
...IMHO without the "connotation" of intended meanings, the discourse would also stop making sense...
Click to expand...
I agree with that, but that would not stop mathematics from going on. I've calculated 1st, 2nd and 3rd derivatives of far out functions that has no visibly connection to this realm, wanking the invented functions and the logic as for the sake of it, far below infinity stuff. It makes up for part of attraction in match to free ones selves of connotations, I suppose.
Click to expand...
That sort of thing (the meaningless symbolic manipulations) personally turns me off of mathematics. I'm more the kind of guy who likes to wonder which, of all the proofs of pythagoras theorem, is the most beautiful, the most elegant. To me, mathematics is the poetry of meaning.

Unfortunately, many career mathematicians actually seem to be attracted to the absence of concrete meaning. I wonder what mathematics would be like if so many mathematicians weren't like that.
 
Jees said:
Something keeps spooking in my head: despite numbering was born out of the desire to manage size, how much of numbering is solely about "size"?

Is the axioma of numbering = size?

For example if numbering indicates discrepancies in quantum states, the concept of size as we know it start to slip trough the fingers. Then the numbers start to be more connected to energy states than the literal size it occupies.

If I could squeeze out the thumb sucking reflex 'how big/much' and have them numbers cleared from connotations, they might be free to live in the abstract on their own terms. Then they don't necessarily have to make sense anymore in the real world. Then 'sensibility' can no longer be feeding a prove/disprove discourse.

Feeling puzzled about this :?:
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Numbers are not just about 'size,' although they always have at least some value associated with them. We've just been talking pretty much exclusively about cardinality, which does use numbers to represent quantities. If you want to see numbers representing something other than size, go wander over to differential calculus where numbers can represent rates, or linear algebra where numbers can represent vectors in various dimensions.

There is no axiom that numbers equal size, I don't know where you're getting that. None of the axioms of ZFC, for example, make any reference to size. We use numbers to represent sizes, in the same way we use them to represent a lot of other things. They're very useful abstractions in this regard.

I'm not sure what you mean by 'clearing numbers from their connotations.' You could easily create a formal system that makes all numbers have the same value, but you couldn't do much with it.

Blessings
~ND
 
Thank you for the numbers intel.

What I'm left with:
- is Aleph a cardinality that is all and only about size?
- can Aleph be abstracted away from size?

If numbers can express lotta thing besides size, then Aleph too?
Does aleph behave just like a number? It is shown in the vid they can be calculated with, but seem to follow another nature in the outcome than numbers do. Makes me wonder if they are anything or nothing like numbers at all. And how they relate to size.
 
Jees said:
Thank you for the numbers intel.

What I'm left with:
- is Aleph a cardinality that is all and only about size?
- can Aleph be abstracted away from size?

If numbers can express lotta thing besides size, then Aleph too?
Does aleph behave just like a number? It is shown in the vid they can be calculated with, but seem to follow another nature in the outcome than numbers do. Makes me wonder if they are anything or nothing like numbers at all. And how they relate to size.
Click to expand...
That's kind of a tricky set of questions. The Aleph numbers are not integers, they're transfinite cardinal numbers, so you can't really do arithmatic with them in the normal sense. You can have sets of them (eg Aleph-w is the upper bound on the set of all Aleph-n, where n = 1, 2, 3, 4...oo) and include them in logical operations, but asking for example, what is Aleph naught - 1 doesn't really make sense. I know it's kind of odd to think about other 'kinds' of numbers, but that's what the Aleph and Beth numbers are.

You can't really remove the Aleph numbers from the idea of size. The definition of Aleph naught is "the number of integers."

Blessings
~ND
 
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