[saghm]

The idea that "3" is not the same as a group of three things seems fairly straightforward, in the same way that a seeing a red ball and a red shirt next to each other and describing them as "red" is not the same as the concept of "red". The "weirdness" of numbers is that seeing 3 balls in one place and 2 balls somewhere else lets you say "3 + 2 is 5, so there are 5 balls"; we don't have a system of "composing" most descriptions of a shared category in a universal way. While we might be able to come up with a way of composing some of them, like "horizontal striped shirt plus vertical striped shirt equals plaid shirt", we don't (at least right now) have a system that makes it simple for two people to independently come up with consistent answers (does red shirt plus white shirt equal red and white striped shirt, or pink shirt, or something else entirely?)

[jacobolus]

The difference is the complexity of the concepts involved. There are predictable results when you e.g. combine and filter light spectra, stack several lenses and mirrors to make a telescope, mix two chemicals, put electric circuit components together, or perform a well-defined sequence of knitting stitches, but describing them is more difficult than whole-number arithmetic.