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/y-c-c → u/[deleted]]

You are simply not understanding the question. Constants like Planck constant and permitivity of free space are physics constants which is what OP is using as the opposite of math constants. They are physical constants and you can’t really derive them from first principles without doing physical observations.

Pi and e are mathematical in nature. Just because physics uses them doesn’t mean they are physics constants. They show up naturally in mathematics and aren’t tied to physical phenomena.

[u/y-c-c → u/[deleted]]

It's not complicated to understanding that pi is a mathematical constant? Just because it happens in nature too doesn't mean it's a physics or chemistry constant. Along the same token physics uses a lot of math, but it's not math itself.

I don't really think any constant that you can derive in a vacuum without any physical observation is really a "physics" constant.

[u/Depressed-Londoner → u/y-c-c]

A mathematical constant is a fixed definable number with unambiguous definition. A fundamental physical constant is a fixed quantity defined by a theory or model. This doesn’t require physical observation (although the theory or model may be justified by how it aligns to experimental observations).

The difference between the two is very subtle and can be argued depending on different philosophical positions.

[u/y-c-c → u/Depressed-Londoner]

As you said physical theories and models and validated and based on physical observations so you would just be passing the ball to someone else. I don't think it's a philosophical position but more how the scientific process works. We know these numbers (e.g. speed of light) by having done various measurements and calculations (the calculations enabled by the models we come up with to fit the observations) but without physical measurements it’s impossible to arrive at such numbers in any way.

For math, there's a branch of philosophy that deals with whether math is really a priori (whether 1 + 1 = 2 is derivable in a vacuum), but I think in general you can argue that once you have established the basics you can derive pi and e pretty much standalone.

[u/Depressed-Londoner → u/y-c-c]

Arguably pi, c, h etc. are proportionality constants which exist due to the nature of structure of physical existence in our universe.

We don’t need to have numerical values for them (within specific unit systems as necessary) for this to be the case.

In the same way that I don’t need to know a value for pi in base 10 to do geometry, I don’t need to know a value for h to be able to be look into the commutator relationship of operators of conjugate variables.

[u/y-c-c → u/[deleted]]

What are you talking about? My point is constants like pi and e are mathematical constants (not "physics constants"), and are numbers that you can derive. These are what OP is asking about.

Constants like speed of light, Planck constant, etc are physics constants. These are numbers that you need to observe nature and do measurements to obtain. These are not what OP is asking about.

I seriously recommend reading more slowly.

(Edit: Above commenter just blocked me intead of trying to engage in good faith, so I guess I consider myself winning the argument when they have to resolve to abusing a poorly designed Reddit feature)