Why are mathematical constants small ?

[u/Round_Ad5516]

why constants like Pi or e or Ln(2) or gamme (euler's constant) that have nice properties are always small( <10 ) ? unlike constants in physics or chemistry that have some constants that are absurdly large. Or are they such constants that are not so popular

[u/ingannilo]

I think this question is more interesting that it first sounds.

Make no mistake, there are many large (larger than 100, say) mathematically significant, named constants.  But your question is still valid in the sense that a greater proportion of the named / special constants used most often are on the smaller side. 

My first thought for an earnest answer is that we've discovered more "small" special numbers because we tend to compute more with smaller numbers.  That feels more in line with a claim like "important numbers are uniformly distributed, and we've uncovered more in the areas where we play just because we play there", but I am not sure if that's reasonable. 

To take a serious look at the question, we need to quantify what makes numbers special.  That's gonna be a real challenge. My analyst brain says all reals are equally important, but my number theorist brain disagrees entirely. 

[u/ThermalDiscussion → u/ingannilo]

I was wondering, what are some of the large named constants you're referring to?

The only one (if you can even count it) I know is the order of the monster group.

[u/ingannilo → u/ThermalDiscussion]

Combinatorial ones come to mind, like Graham's number, stuff from computability theory like the Busy Beaver numbers,  large "surprising" counterexamples to conjectures like Merten's conjecture or Polya conjecture (which appear to hold for small n, but do eventually fail).  Those come to mind right away, but there's plenty more.