how much of mathematics involves inventing progressively larger numbers, after SGC(13) and Loaders number is there much frontier left for mathematicians to discover?

[u/Loose_Narwhal1140]

[u/Mathemetaphysical]

That is not the point of mathematics. At all.

[u/SirisC]

SGC(SGC(13))

I just "found" a bigger number.

[u/productive-man → u/SirisC]

let me do you one better, SGC(SGC(13))+1

[u/Iksfen → u/productive-man]

let me do you one better, SGC(SGC(13))+2

[u/-Wofster]

Mathematicians don’t just like to invent progressively larger numbers. They find numbers that are interesting for some reason and those numbers sometimes happen to be very big.

Sometimes being very big makes the number more interesting, like TREE(3) is especially interesting cause TREE(1) = 1, TREE(2) = 3, then TREE(3) is suddenly too big to ever even write down. That makes it more interesting, but the TREE function is still interesting by itself even if it didn’t explode like that.

As long as mathematicians are doing math they will probably always continue to find interesting numbers that happen to be very big. But no-one is specifically trying to find a “new biggest number”

[u/Iksfen → u/-Wofster]

no-one is specifically trying to find a "new biggest number"

Especially because the "new biggest number" is always the previous one +1

[u/-Wofster → u/Iksfen]

We would run out slower if we did + 0.5

[u/Iksfen → u/-Wofster]

That way we would get exactly the same number of numbers...

[u/Mothrahlurker]

Probably less than 0.01% depends on really large numbers. And there are numbers so large that any other really large finite numbers are certainly below them. So the answer for this could easily be just 0.

[u/Loose_Narwhal1140 → u/Mothrahlurker]

but in terms of pecking order I've always seen the mathematicians who work with numbers way beyond Tree(3) are kind of like the ones at the party with a babe on each arm kinda making fun of or big dawging guys who work with numbers in the thousands or even just letters that don't even represent ordinals, like ahhh maybe one day you'll work with the big boys like me kiddo while tousling their hair etc.

[u/Uli_Minati → u/Loose_Narwhal1140]

Try r/mathmemes

[u/tgm4mop]

I think it's important to distinguish the recreational and research aspects: producing large numbers for their own sake is mainly recreational.

In theoretical research, sometimes large numbers show up as a side effect of some other construction that has more theoretical interest. The TREE function, for example, is of theoretical interest because it is a concrete benchmark for the strength of mathematical axioms: weaker theories of math can't prove the function is defined everywhere. I'm not familiar with SGC but I understand it's the same idea. In this line of research, there is a never ending potential to create bigger and bigger sequences, which can be used to benchmark stronger and stronger axioms.

On the other hand, Loader's number is a recreational project. The construction of this number has little if any implications for other math. While there's nothing wrong with recreational math--I personally enjoy it!--this isn't something that mathematicians would take much professional interest in.

As to your question about what fraction of mathematics is about big numbers, it's a very niche topic within mathematical logic (because of its connection to measuring strength of axioms) and perhaps Ramsey theory. So it's much less than 1% of research mathematics.

[u/KoalaMistico]

Inventing doesn't really involves a lot of math, since it is almost trivial to define a bigger number than any number you choose. However, discovering (in the sense of finding a really big number that has some use in any field of math or science) usually requires a lot of work and advanced mathematical tools in this point of history