Originally published at: http://boingboing.net/2017/03/10/is-mathematics-invented-or-dis.html

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# Is mathematics invented or discovered?

Is language malleable?

I believe mathematics is knowable by our human brains as a way of perceiving and understanding our universe. We can only discover what we can sense, either by sight or touch or by machine extensions of our senses.

And from another front page article, Ethics and AI:

The old stats adage goes: “All models are wrong, but some models are useful.”

If the answer is that mathematics is an evolved model that resulted from an interaction between people and the world, is that an invention or a discovery? I think probably neither.

Like so many taxonomy questions, the answer revolves around your definitions. Indeed it is the definitions that turn out to be more flexible than the understanding of the underlying process. In this case “What is the difference between inventions and discoveries and are those disjoint sets?” You could argue that inventions are discoveries that the rules of the universe permit you to make something…therefore that inventions are a subset of discoveries. This makes it easier to accept the idea that mathematics is “invented.” You “create” a mathematical proof, using principles that have already been defined to “discover” that a previous set of definitions and principles imply something that you did not heretofore know.

Depends. Pure math or accounting?

I think there is relatively little ‘inventing’ in mathematics, unless you include the invention of tools and symbols.

Let’s take the mathematics of the positive integers. The integers are an open set. Starting with a unique origin we call ‘zero’, each integer has a unique new, fixed successor. The sequence goes on forever: there are no branches and no loops. There are tree theories that branch (non-unique successor). Set theory is all loops (non-new successor). There could be mathematics where the relationships are random or change continuously (non-fixed successor), though it is rather hard to say how. But, if we stick to our unique, new, fixed sucessor and single origin, then we have all the positive integers.

So, what about negative integers? The negative integers can be made by a re-mapping of the properties of the positive integers. And yet, the use of the minus sign is an invention which allows us to handle numbers like this with greater ease, and to map concepts that can be awkward to explain - such as why minus a negative number is positive - into typographical rules. The invention of the equals sign must be one of the great inventions in mathematics.

I think most mathematicians are happy with this. The proof of Fermat’s last theorem is not a lesser thing because it is a discovery. It uncovered a property that was latent in our original assumption of the integers, but it was a huge achievement. It also ‘feels’ right: the proof is new, but it uncovered a property that was always there, implicit in the definition of the integers.

I think I know what you are trying to get at, but “open set” has an existing meaning in mathematics and ℤ is not open. It is closed.

Yes, you are right. I wanted to say ‘you could always make it one bigger’ but I didn’t want to say ‘infinite’, because that goes in all sorts of unfortunate directions. This is why mathematics uses symbols like hollow-Z rather than words, when it can. Open-ended might have been a better compromise.

Thanks.

…or statistics?

Is ontology invented or discovered?

And was inventing invented, or was that discovered?

And certainly discovery couldn’t have been discovered before discovery was discovered, so therefore discovery surely was invented.

Not necessarily. Discovery could have been discovered before discovery was understood to be discovery, I’m not sure it’s obvious that a discovering mentation necessarily precedes a discovering situation. Although I’m reasonably confident in the reverse.

I think.

If math is innate and we discover it, can the same be said of potential thoughts that can be, but have not yet been, conceived by a human mind?

Does every possible information-configuration of a system attend the creation of that system through sheer potential, or must it go to all the bother of actually existing as some configuration of brain matter before it can be considered to exist?

They should have interviewed Max Tegmark. https://en.m.wikipedia.org/wiki/Mathematical_universe_hypothesis

Most people won’t have an answer to this because they don’t know what they think ideas are to begin with. I accept that ideas are configurations of matter and energy that exist in space and thus have volume, mass, etc., and then imagine what *sort* of thing an idea would be.

So if we talk about a possible configuration of things that would constitute a thought if it happened to occur but that has not happened to occur yet, I’d say that it definitely doesn’t exist because exist means that it actually *is*. But the idea of a configuration of things that would be an idea if it happened to come about does exist.

I think you’ve got it right. Addition would still be commutative, even if no human had ever realized it. It’s the terminology and symbology that enables us to share mathematical truths which is the invention part.

That reminds me - Should an apterous fly be called a walk?

This reminds me of the “I think therefore I am” argument. It makes sense intuitively to a lot of people, but even if you don’t think it is essentially flawed you have to be *really* careful not to import a lot into the “I” part of the argument. The real conclusion of the argument is more that *something* exists than that anything like what you think of as yourself exists.

Similarly, I’d like to strip down the idea of “addition” to something that seems to exist outside of human thought. If I start from there is a way to meaningfully interpret the universe as a collection of things with locations such that a thing is still the same thing if it is moved to another location. Systems that can divide the universe into things in this way along lines of utility for the system doing the dividing (e.g. things that are food, things that are dangerous) are able to continue to exist by doing so. Because things are still the same thing if located differently, any easy-accessible arrangement in space of things that are food would be the same utility as any other arrangement if the ultimate goal was simply to consume them. That is to say, 2 apples and 2 apples is the same as 1 apple and 3 apples. So the “commutativity” of “addition” exists in the universe so long as the universe contains systems (minds?) that can meaningfully divide the universe into things and divide those things into categories that can be sorted together.

I’m left with an extremely stripped down idea of “addition” that can only be approached by metaphor as a thing that might or might not be inherent to the universe before humans came along and made it up.

So maths doesn’t exist before it is invented and it is therefore not discovered?

Some animals can count or at least work out that a>b for smallish values of a and b. So maths was around before humans if that’s true - so not invented. Discovered means no one knew about it before someone says “hey look at this!”, which is probably not entirely true either. I’d go with noticed.