[inopinatus]

Informally, per Gödel’s method: algorithms are discovered, axioms are invented.

This may however be distinction without a difference, since it corrodes the intellectual form of neither, and the processes are indistinguishable.

[wavesonthing]

Things exist or are created. We discover an existing thing or its application.

Discovery is principally a learning process (that may entail creative effort). Invention is principally a creative process (that may entail learning effort).

[inopinatus]

The first statement is either wholehearted agreement with my remark or a rejection of Gödel numbering.

The second statement is similar but highlights the problematic distinction of the terms, since in such a framing they ascribe meaning solely and specifically to human intellectual processes, so one must also show either that we are alone in the universe, or (perhaps equally) that all cognitive entities share the same processes. This is of course merely dancing around questions of the universality of truth. Alas, only Wittgenstein knew the answer.

[wavesonthing]

I didn't attempt to delve into that, so ...

With Gödel, there is the space of the possible, the invention indeed being the axiomatic foundation that engenders the space of possibilities. If the space of possibilities is disconnected, intuitive jumps are required to arrived at unreacheable truths. So yes, here we are 'discovering' -- learning the membership set of true statements -- but these are existing things only in the context of the invention, which is why it is correct to say they are discovered in a relative sense.

[mejutoco]

> We discover an existing thing.

This is hiding the assumption in plain sight in a different form. The question then would be: was x an existing thing that we "uncovered"/discovered (it has merely hidden) or we created it by thinking it into existence?

At least in terms of Mathematics, I do not think there is an objective answer for this question.

The definitions you provided are, IMO, just moving the problem to further concepts, that are also ill-defined. "Discovery mainly a learning process, Invention mainly a creative process", could mean almost anything. Also, as a rigorous definition, "mainly" does not cut it. It needs to be 100%.

Instead of pretending I know it is my opinion to accept they are merely two ways of looking at the same thing. At least, this is my position for now, until a proper argument emerges.

P.S. I see it as arguing which of 0.5 or 1/2 is the "true" number.