Is Math an Invention or a Natural Phenomenon?

[collinsmark]

Here's my 2 cents on the subject.

Is math invented or discovered? Let's get to the root of the question and make it even more basic.

Is the concept of prime numbers discovered or invented?

A mathematician -- a good mathematician -- might make the claim that the set of prime numbers is invented. This in turn might bring about a typical reaction from a person less well versed in mathematics,

Typical reaction to prime numbers being invented:

You got to be kiddin' me. Invented? That's about the most arrogant statement I can imagine. Do you honestly think that if we ever made contact with another species of intelligent life, with advanced technology, that they wouldn't have a concept of prime numbers? And if they did, which they assuredly would, what would you do? Throw your hands up and yell, "Thieves! Thieves! Prime numbers are man-made! The Zorgon empire stole them from us! Prime numbers were invented by humans so the Zorgon empire is just a bunch of thieves! Just like those damn, thieving cicadas! Thieves!"​

But that's not what the mathematician means by inventing prime numbers. As a matter of fact, most mathematicians actually like cicadas. More to the point, the invention of prime numbers is very different than inventing a novel recipe for a tuna sandwich on rye bread with some mayonnaise.

The mathematician is purposefully being inconclusive -- agnostic if you will -- on the subject of whether prime numbers are a crucial, fundamental part of our universe. Rather the mathematician is saying that prime numbers were at least invented by humans. If in the future we make contact with the Zorgon empire, and they also happen to have a concept of prime numbers, the mathematician would say the set of prime numbers was independently invented by both humans and Zorgons. The mathematician may very well suspect that prime numbers are in fact a fundamental part of our universe -- but he just cannot prove it. And I don't mean that we just haven't found a proof yet, but rather it seems as though the concept may not even by provable. The good mathematician, not wanting to make claims of what has not been proven, or make conjectures to what is unprovable, stays quiet on the subject.

The set of prime numbers fall out from our created axioms, such as Peano axioms which go without saying that they are obviously invented.

Then, the real crux of the issue comes from Gödel's incompleteness theorems, which when applied here, means that we can't prove anything outside our axioms. We cannot definitively prove that another technologically advanced, intelligent species would have made the same axioms.

I fully expect that if you were to take that mathematician -- or any good mathematician -- aside in private and ask him or her,

Question: If we were to encounter another form of intelligent life, of considerable technological advancement, do you think they would have a concept of prime numbers? The same set of prime numbers that we have?
Answer: Well, yeah. Of course. I mean I would really be surprised if they didn't! Well, the notation would obviously be different. Any they might include the number '1' in the set (we do not anymore). But the rest of them sure. Can you imagine! A technologically advanced society without prime numbers! That's not going to happen. So yeah, I'm pretty sure they would. But I cannot prove it.​

And similarly, although I cannot prove it, I fully suspect that with the Zorgon empire's own equivalents to Gödel's incompleteness theorems, if the zorgons were confronted with the same question, they would throw their arms up in the air and run around chanting "Qz! Qz!" where the word "qz" from the zorgon native tongue is translated as:

qz
/ kwǝz /

interjection

The affirmation that the set of prime numbers was invented by the ancient zorgonians and was independently invented by the despicable kladies of the planet Kladbor, as well as being independently invented by a species of the planet Earth know as cicadas. The term continues the affirmation that there is also another, very tasty species of Earth beings called humans that claim to have independently invented the set of prime numbers as well, but it is equally likely that the delicious humans stole the idea from the cicadas. The term has a connotation that it is welcomed and encouraged to sit down during tea and discuss the origin of prime numbers with humans over rye bread and some mayonnaise.
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noun

A sandwich consisting of humans, rye bread and mayonnaise. Sometimes served with a light spread of cicada sauce.

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[phion]

Here's an insightful clip about what Stephen Wolfram thinks:

 

[zoobyshoe]

Very interesting video, Phion. He doesn't exactly say our math is invented, but that it is one of many possible mathematics, an "artifact" of the particular route taken by human history.