That seems akin to programmers calling themselves 'engineers' to me - more pretension than reality. Some code will probably survive this age (some as ephemera, some because it was so elegant as to approach pure maths), but much of it is as ephemeral as the hardware it is written for, and I suspect that which does survive will arrive in the future, like the works of Archimedes, because it was recorded as part of a theory to be studied (e.g. in books teaching principles of programming), or because of happenstance and being stored with other records - as ephemera and historical curiosity rather than because it is valued. Of course some will survive, and some will be highly valued, but I suspect a limited amount.

In addition, I do think it is rather myopic of Godfrey Hardy in the above quote (or _why in the one you have cited) to view mathematics as somehow more fundamental than culture or vice versa – both are valued, and both will be preserved to some extent and translated into future languages and systems of thought. I preferred _why's more specific quote about PowerPC because it points out that much programming work will be ephemeral on the order of decades because of the nature of the systems it runs on - obsolescence is built in by the corporations who control much of our digital landscape. Some of that has changed with open source, but bit rot will doom code which can't be run any more due to missing dependencies, and when it can't be run, what's the point?

I'd say a potboiler novel from today has more chance of surviving one century intact than the code which powers most startups or runs in most of the apps we use day to day.

PS A mathematician's apology sounds interesting, I haven't read it and must look it up. Thanks for the tip.