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PaperclipPerfector
21hr ago
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Notes -
Do you think that mathematical theorems are discovered or invented? I have a segue I want to make from this question (it involves a book that was popular w few years ago), but first I'd like to hear your opinions on that.
I would say invented, because any arguments about them being "discovered" or "already out there waiting to be found", while mostly true, apply equally to literal inventions. Was the telephone discovered or invented? Well, kind of both. The laws of physics always allowed sound waves to convert into electric signals and travel across wires to be converted back into sound waves. Someone just had to figure that out.
You can make a strong argument that invention is a subset of discovery. You can make a strong argument that mathematical theorems are a form of discovery. You can't really make arguments that they aren't inventions except by sneaking in a hidden assumption that these are mutually exclusive, when actually they're not.
I'd agree, but since having more distinct words is more useful than having fewer, I'd like to be able to say it's a proper subset, in which case there would have to be such a thing as a discovery which isn't also an invention. Then we'd be back to having a reasonable question to ask: once we fix a definition of what makes a discovery also an "invention", which mathematical discoveries are inventions and which aren't?
Could we classify non-invention-discoveries as distinct from invention-discoveries because the latter were created to solve problems and the former weren't? The laws of physics include ways to convert sound waves into electrical signals via a piezoelectric sensor, and that was an "invention" because telephones were awesome, but the same laws include ways to convert audible sound waves into seismic waves via amplifiers and a giant ground-thumping piston, and that was a "discovery" not worth noting because actually doing that would suck.
What's weird about mathematics is that so many of our pure "discoveries" keep getting hijacked and finding important applications later. Maybe some of that is selection bias, because if someone comes up with a neat mathematical game with no use cases then we only teach everyone about it if it's either closely related to something with use cases or really cool in some way or both? Maybe there are whole fields of "alien mathematicses" that we just never get into because they're really completely disconnected from anything useful?
But often even when we try to just "discover" interesting-but-useless math it turns out we're not very good at avoiding coming up with useful ideas. A couple of my favorite quotes are ironic in this way:
"I have never done anything 'useful'. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world." - G. H. Hardy, inventor (discoverer?) of math which turned out to be fundamental to population genetics and quantum physics.
Was Hardy an inventor? He certainly wasn't trying to be, but that didn't stop his discoveries from at least being the load-bearing components of inventive ideas.
"No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems unlikely that anyone will do so for many years." - G. H. Hardy again, in "A Mathematician's Apology", written at the same time that secret number theory research was becoming a decisive factor in World War II via code-making and code-breaking, and secret relativistic physics research was leading to the atom bomb.
If a discovery is finding something that you, or the people around you, didn't know about at then telling them about it, then an invention is finding a way to make something or do something, while a non-invention discovery is finding something that already existed. You discover a new plant or a ruin, you invent a lightbulb or a martial art or a programming language. The theoretical concept of how to do these things can be imagined to have existed somewhere in imagination land, but if no human being actually has this knowledge then the knowledge itself literally did not exist until you caused it to exist by putting into people's minds. And in the case of physical inventions like a lightbulb the actual thing itself also did not exist until you created one. If you had chosen to change the methods of your creation, the features of it would change. In some sense this is inventing a different thing rather than the original thing, but in many cases it's minor tweaks that don't fundamentally change its nature but are superficial (you might add a different number of coils to your lightbulb design, or you might put the steps in your mathematical proof in a slightly different order).
Meanwhile, before you discover a new plant that plant is still on the Earth doing its thing. You unambiguously did not invent the plant, it was already there before you arrived. You can't tweak the plant's discovery in minor ways to alter what it looks like (or at least, any tweaking is something you do afterwards and is not a part of the discovery process). It was already there, and you did not cause it to manifest in the real world in the way you do with an invention.
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