Remove the Mysticism from Mathematics!

[Chas]

And this means its useful when used as a guide to facilitate our understanding the natural world rather than as a prescription of what that world ought to be.  It's a tool not immune to being misused.

That inability to distinguish between the prescriptive and the descriptive is not unique to math.

Yeah, you're describing a prescription for failure.  

[Thumpalumpacus]

And this means its useful when used as a guide to facilitate our understanding the natural world rather than as a prescription of what that world ought to be.  It's a tool not immune to being misused.

That inability to distinguish between the prescriptive and the descriptive is not unique to math.

Yeah, you're describing a prescription for failure.  

I've had this discussion with fellow musicians over the years, regarding music theory. I see music theory as descriptive, meaning that after I'm done playing, say, a jazz improv, if I'm recording it I can listen back and say "Yeah, I'm substituting a iim7b5 for a domV7 here" ... but many others seem to think that if you don't resolve a progression a certain way it simply isn't musical, because music theory says the "best resolution is" chord X, and if I resolve it to a different key-center then I'm doing it wrong.

I reckon that if the song works, it's good, and theory is useful in analyzing it after-the-fact so that I can cadge a trick to put into other places. So much pop music over the years is theoretically conventional, and absolutely boring.

[skyking]

My sister ran off to study in Hawaii on a lark in the mid 60ś, and music theory was the class that just kicked her ass at the university.
She had a hole in the schedule, played piano and read music and thought it would be easy!

[mordant]

My stepson is into 20th and 21st century classical music, which is to say, dissonant, un-lyrical stuff like serialism which plays by entirely different rules than your standard-issue Western music. There is definitely no "one right way" to do anything in music. Indeed, recently with easier and faster to use tools for digitally disassembling and reassembling various tonalities, musicians are inventing whole new musical forms. So ultimately music theory and analysis just describes what it sees and what is commonly done; it is not some kind of arbiter of "correctness".

That said, I am eternally grateful that my stepson uses head phones to listen to his music library ;-)

[BensonBear]

Aleph-null is the infinite set of all computable numbers,

No, aleph-0 is the set of all counting numbers.

from the integers to calculatable transcendentals like pi and phi;

No, phi is not transcendental, it is irrational but algebraic.

Aleph-one is the infinite set of the continuum, which includes the numbers between the members of the aleph-null set.

No, aleph-1 is the infinite set of all countable ordinals, conceived of as a cardinal, which makes it the cardinal immediately larger than aleph-0.  (Hence its name).    The continuum, if it is a well defined set, should be equal in size to some cardinal, but it is not known which cardinal that would be.   Many, probably most, people who believe in this sort of thing do not believe the cardinality of the continuum is aleph-1.

which includes the numbers between the members of the aleph-null set.

No, you appear to be thinking of the irrationals (the bulk of the continuum when conceived of in the usual sense as the real numbers) which I suppose in some sense are "between" the rationals (or the algebraics, or the computables -- all a meager part of the continuum of total size 0) although that is not really right since of course between any two rationals there are an infinite number of rationals, also.   Indeed you can develop the irrationals by "cutting" the rationals into a lower part and an upper part, but the cut that must be performed is not between any two rationals.   This is one reason people have trouble getting their heads around the meaningfulness of these mathematical objects.

The proof of this is in what's called Cantor's diagonal argument, which is quite accessible even without mathematical training.  And again, it has not been refuted. 

No, Cantor's diagonal argument merely shows that there is no surjection from the set aleph-0 to the set of all subsets of aleph-0.   And while this proof cannot be refuted, what it actually amounts to is less than clear. 
 

Extreme Finitism is flatly wrong in his video.

Wrong yes, but flatly, probably not.   His wrongness is more twisty.

[SYZ]

Uh... All above my pay grade Benson.   

[BensonBear]

Uh... All above my pay grade Benson.   

Not really, it's all pretty simple so far and you could easily master it.  

But it doesn't address your own question that was not really answered by anyone, especially not "Extreme Finitism":

And is this correct?       ∞ + 1 = ∞ 

It depends what you mean by "∞".
That symbol is not normally used in mathematics.

The most basic kind of infinity is an ordinal, and this sentence would be incorrect for such infinities.   If you add one to such an infinity you get the next infinity in the ordering of infinities.   

It is different, but it is not larger.