Is mathematics invention or discovery?

Sure this topic has been around before,
This post is to refresh contemporary ideas, i.e. circa 2010.

-- Justin Johansson

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Wrong venue

Now that we have the Zen koans taken care of ;-), I would like to suggest that LtU is the wrong venue to ask this question in.

Though we often discuss mathematical issues related to PLs and computation, this isn't really a space to discuss math in general.

Is programming (computer science) invention or discovery?

Sorry if my question seemed OT for this forum; it was intended to be a lead in for my next question: Is programming** invention or discovery? which surely is quite on topic.

**I'm referring to programming in an abstract sense, i.e. computer science, algorithms, type systems etc, not day-to-day programing in a practical PL.

Still OT

Still philosophy. The fuzzy distinction between 'invention' and 'discovery' is irrelevant to PLT.

In general, I'd say invention and discovery are fundamentally the same thing. Even the most physical 'inventions' can be viewed as 'discoveries' of useful physical arrangements. And either requires a lot of hard work - to pick one subset of arrangements and distinguish it as especially useful from among a ludicrous number of permutations. I imagine it will become especially difficult to distinguish 'discovery' from 'invention' if ever desktop manufacturing printers really take off.

I remind Marc to 'be careful what you wish for'.

The perils of Plato

I remind Marc to 'be careful what you wish for'.

I never said I wished LtUers would discuss philosophy, but rather to have studied it.

People who have studied philosophy tend to get questions like this out of their system... ;-)

Is an idea discovery or invention?

Math is an idea(l) ;)





Missing the joke, you mean: ç„¡ or æ— ?

A philosophical problem used

A philosophical problem used to illustrate the interdependency of the opposites. We could discuss here rationalism vs empirism as well and start a reading group for Kants "Critique of pure reason" but then, don't you think that Marc Hamann is just right that this goes OT?

O.K. quick answer: Mu.

I would say both

I can't think of any mathematical structures that aren't in my opinion "discovered" but of course the syntax and conventions we use to describe them are invented. My experience is that generally most people just don't care one way of the other.

The way my wife and I frame the argument is: If we encountered alien intelligence would their mathematics be understandable to us and vice-versa. I would expect people, like my wife, who think mathematics is invented to answer "no" while people, like myself, who think it is discovered to answer "yes".

My question would be if you think the answer to that question has some relevance to the field of programming languages.

Of course

If algorithms are math, then algorithms are discovered rather than invented, which might make software unpatentable.

The silly thing is, if math is a language, and language are there to formulate thoughts, then no idea is invented. They are just discoveries of meaningful sequences of words.

Interesting framing

That's an interesting framing of the argument.

An answer I like (most of the time) is that some math (including the most basic structures, such as numbers, finite sets, etc.) is discovered. Beyond a certain point (maybe constructability, or computability, or maybe even a little beyond that), it's all invention. The idea of the Axiom of Choice, for example, is invented, because it's not even right or wrong! The hard thing (maybe impossible?) is to define exactly where is the line that separates them.

With your framing, this answer becomes a (somewhat) testable prediction. If we ever find aliens that "discovered" the Axiom of Choice, for example, then the answer is probably wrong (unless, by luck, they happened to invent it also).

I have no idea about the relevance of this to programming language theory, though, except this: algorithms are discovered, because you can in principle build an actual Turing Machine to simulate them, and so the laws of Physics in a sense "already contain" them.

Undoubtedly Neither - An Evolutionary Approach

Some parts of our brain are hardwired to compute "one, two, ...many" while there are other parts of our brain necessary for math beyond that. But you need them both.

So we have _evolved_ the ability to do math. We didn't invent it and we didn't discover it, but we can do it nonetheless.

Which brings a different light to the question of, if we meet "intelligent" aliens, will they use math? Will they view math the way we do? Possibly not, if their wetware is dissimilar, almost a certainty.

Indeed OT.

Indeed OT.


It doesn't have to exist. While at its core it seems to be discovery, since it follows natural principles, it quickly leads to constructions that could have been created for the first time: invention.