Is Math Discovered or Invented?

[Hunter]

Is math discovered or invented? This is a question we talk about a lot here in my city with me and mine. Instead of trying to figure out the answer, I have been happy to just let it be discussion, as I think the discussion helps students understand math better.

 

But, tonight, I don't know, I saw this video pop up when watching a Euclid video, and figured this would be a good Sunday topic, for the forum.

 

ExH didn't learn much math at school, but was incredibly talented in maths. To build the things he wanted to build and to be able to write about them on paper, he sometimes invented his own math language and used symbols "wrong". So my boys grew up knowing Daddy had a different math language. That had a profound affect on their understanding of math. Daddy could build a 2 story newspaper recycling machine for a company in Japan using math that Uncle was screaming was "wrong".

 

I think this type of understanding would have gone right over older son's and my head if we hadn't lived among it. Younger son would have probably known and understood otherwise; he was incredibly amused to see people have to deal with Daddy and his math, and would sometimes toy with me a bit, using Daddy's math, and threatening to create his own. I think I was probably a bit heavy handed on insisting on him being able to communicate with others in standard math language. I was hungry and cold and figured I'd try and spare my future daughter in law and grand children the same fate if I could.

 

We also were studying Greek and Latin and sometimes did our math in Greek and Latin. And did some lessons on computer binary, and talked about a sci fi novel I read years earlier about an alien race with 8 fingers and a base 8 math system.

 

So, is math invented or discovered?

 

[Farrar]

When I first read about this controversy a few years ago, I freaked out a little. It was a big idea/debate that I'd never encountered for whatever reason and it was a bit mind boggling. Like, it's a language that's in our brains that we can discover and figure out. The fact that different thinkers worked in a vacuum and came up with the same things is sort of persuasive to me that somehow it's something we can just find and figure out. And the thing in the TEDed about how all kinds of seemingly obscure math seems to describe natural phenomena is also a point for discovery.

 

But then, what does that mean? 

[Mimm]

Interesting video. I've always thought of math as something discovered. The truths behind math still exist, even if we haven't discovered the math to express it yet. To answer the question at the end of the video, yes, that number still exists. :)

[bolt.]

I would say that math is an inverted convention (language) which discribes a true phenomenon. The phenomenon of math is a set of real attributes of the universe. Those attributes are true and the ways they can be worked with really work. So the attributes are discovered, but I definitely the language/convention that describes them is invented.

[regentrude]

That is a question we have often debated.

One could make a point to say that only integers "exist" and everything else is "invented". But certainly, fractions "exist" too, since "half an apple" exists, or "a quarter of a pizza". And then once you have agreed on a set of axioms, subsequent relationships between mathematical entities are discovered. The ration between circumference and diameter is simply pi. Pi exists. And so on...

 

But then, there are relationship between two quantities that simply exist in nature that can be described by certain functions. For example, the strength of the gravitational force is inversely proportional to the square of the distance. That is a fact that exists outside humanity and independent of the development of mathematics - but humans managed to come up with a way to describe the nature of this relationship. They did not "invent" quadratic functions - they exist in nature. They may have "discovered" that such relationships exist, and they certainly "invented" a way of notating and describing them.

Similar with lots of other functions - the relationships exist independent of humans; humans have found a way to describe them.

I don't think either term accurately describes how mathematics evolves.

 

We need to say what precisely we mean by "mathematics". The notation is completely invented, and it is a convention to use a certain standard notation. The relationships that math describes, however, exist independent of notation.

Your ExH did not use "wrong math". He used an unconventional way of notating math.

ETA: Feynman writes in his book about doing math with invented symbols, when he discovered relationships and notated them. He learned that adhering to a notation convention it is important to be able to communicate with others. none of his math was "wrong" - but his notation was only understandable by himself, which made it all rather pointless. Being smart, he decided to use standard notation.

Edited September 25, 2016 by regentrude

[Guest]

Discovered.

[Farrar]

I may be misunderstanding this a little, but one of the things that astounds me about it is that humans can apparently discover mathematical laws without knowing how they apply in nature and only later realize they do. Like, whoa. It's somehow hard wired in our brains to be able to figure this out without observational phenomena? Like, just the notation, which we have invented as a way to describe numbers and math, can lead us to understanding things that we haven't even seen yet.

[lanalouwho]

I agree that the language of math was invented in an effort to explain and communicate mathematical principles that have been discovered.

 

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[JoJosMom]

I am in the "discovered" camp.

 

And thank you for link, Hunter.  What a fascinating topic!

[Bluegoat]

I seem to be a Platonist with regard to math, (and many other things really ) so I would say discovered.

[Ausmumof3]

I reckon mathematical relationships and concepts exist and wait to be discovered but the language and symbols we use to talk about the concepts are invented and often inventing a new language or set of symbols enables us to discover more about what exists. For example massive numbers of stuff exist but until our number system was invented we had no way of describing or understanding it, planetary motion always existed but we had to reach a certain level of language and symbol to describe what we observed and as knowledge and ways of describing things increase we will be able to define and understand more and more of the natural world.

 

Or at least that's how my uneducated self understands it anyway.

[Bluegoat]

It's interesting that people with math want to differentiate the language as invented but the concepts as not.

 

I'm not sure I've ever heard anyone make that differentiation with other types of knowledge - like scientific ideas or truths found in poetry or literature.  Don't we all know that the language for describing those things is not the thing in itself, or that a scientific model of a thing (in any language) is just a model that points to the reality? 

 

I'm curious why we feel the need to say this explicitly with mathematics.

[regentrude]

It's interesting that people with math want to differentiate the language as invented but the concepts as not.

 

I'm not sure I've ever heard anyone make that differentiation with other types of knowledge - like scientific ideas or truths found in poetry or literature.  Don't we all know that the language for describing those things is not the thing in itself, or that a scientific model of a thing (in any language) is just a model that points to the reality? 

 

I'm curious why we feel the need to say this explicitly with mathematics.

 

because in my experience, most people do not distinguish between the mathematical expression "quadratic function" and the relationship of two natural quantities following a quadratic relationship. Actually, for many people the latter is not a concept at all; they see functions solely as something artificial in a math problem and never give a single thought to the existence of objective relationships that such functions describe.

[Bluegoat]

because in my experience, most people do not distinguish between the mathematical expression "quadratic function" and the relationship of two natural quantities following a quadratic relationship. Actually, for many people the latter is not a concept at all; they see functions solely as something artificial in a math problem and never give a single thought to the existence of objective relationships that such functions describe.

 

I think that's true, but I wonder if it isn't also true with a lot of scientific concepts.  I wonder how true it is of ideas more generally.  And then, why is it the case?

 

If I ask someone about, say, osmosis, or Richard III, they don't generally stop to think - these words I am using to describe this thing are really just symbols.  If you brought it up, they would of course realize that there are all kinds of languages could be used to mean the same things, and that language changes, and all the implications of that. 

 

So - why not think something similar about math?  There are  different ways of notating and talking about mathematical ideas including some that are no longer in use. We could totally change the symbols we use if that seemed like a good idea.

 

I also wonder, why isn't that an idea that I hear asked much about science?  My husband, who works as an atmospheric chemist, once told some person he was talking to that the model he was using for looking at air quality wasn't real, it was just a useful representation for his purposes.  The individual he was talking to was not at all familiar with that idea, and felt that a scientific model had to be exclusive somehow, that as it improved it should somehow become more "real".

 

Even in the humanities, I often see people talk about metaphor or allegory or poetry in a way that suggests that they think it is less "real" than a supposedly straight description, and I don't see much discussion of why that might not be the case.

 

It just seems to me that people in many cases are more likely to talk about this difference in math than other subjects, while at the same time thinking the math is more real and exact somehow.  There is a difference of perception and I wonder why?

[Farrar]

My question is where does math come from? Like, some of it is observed from natural phenomena. But other things seem to have been discovered, sometimes by different people working in isolation, that don't, at the time, describe any known natural phenomena but are "pure math." How is that possible? Where is the discovery coming from? I mean, I get that it's number and conceptual relationships and so forth. Like, is math a language that we're hard wired to learn? And then, later, sometimes those things can apparently be used to apply to natural phenomena, which, mind blown, right, people?

 

It's like in Contact how the aliens send the information all in math because it's the only guaranteed universal language. And then they tell them all (in the book anyway) to look for a message in pi. Which, yeah, so trite, but I don't know. It's just interesting.

[madteaparty]

Invented, like most human languages or constructs. Is there a "logic" or certain "rules" that are inherent to any human invention? Does the intuiting of those rules come easier to some than others? Yes and yes.

But 1. I have no idea and can't say I have spent a lot of time thinking about this or informing self. I generally have a really hard time, and this is a big limitation of mine, with the "it's all there, you just have to discover it" concepts. I'm reading quite a lot of Buddhist stuff lately, and it appeals for various reasons, but the hardest issue I have with it is the "look inward, all you need is there" type teachings. Sometimes there's not much there! ;)

[lanalouwho]

It's interesting that people with math want to differentiate the language as invented but the concepts as not.

 

I'm not sure I've ever heard anyone make that differentiation with other types of knowledge - like scientific ideas or truths found in poetry or literature. Don't we all know that the language for describing those things is not the thing in itself, or that a scientific model of a thing (in any language) is just a model that points to the reality?

 

I'm curious why we feel the need to say this explicitly with mathematics.

No one has ever asked me this question about science or literature.

 

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[daijobu]

"God created the integers.  The rest of the work of man."  --Leo Kronecker

[Bluegoat]

My question is where does math come from? Like, some of it is observed from natural phenomena. But other things seem to have been discovered, sometimes by different people working in isolation, that don't, at the time, describe any known natural phenomena but are "pure math." How is that possible? Where is the discovery coming from? I mean, I get that it's number and conceptual relationships and so forth. Like, is math a language that we're hard wired to learn? And then, later, sometimes those things can apparently be used to apply to natural phenomena, which, mind blown, right, people?

 

It's like in Contact how the aliens send the information all in math because it's the only guaranteed universal language. And then they tell them all (in the book anyway) to look for a message in pi. Which, yeah, so trite, but I don't know. It's just interesting.

 

These are the kind of questions asked in philosophy of mathematics.  If you look back to the Pythagorean philosophers, they thought that everything was numbers - so mathmatics are the basic fabric of the universe.  There are quite a few schools of thought about it.

[Guest]

My question is where does math come from? Like, some of it is observed from natural phenomena. But other things seem to have been discovered, sometimes by different people working in isolation, that don't, at the time, describe any known natural phenomena but are "pure math." How is that possible? Where is the discovery coming from? I mean, I get that it's number and conceptual relationships and so forth. Like, is math a language that we're hard wired to learn? And then, later, sometimes those things can apparently be used to apply to natural phenomena, which, mind blown, right, people?

 

It's like in Contact how the aliens send the information all in math because it's the only guaranteed universal language. And then they tell them all (in the book anyway) to look for a message in pi. Which, yeah, so trite, but I don't know. It's just interesting.

Now we are a Christian family, but we tend to answer this with math being 'discovered' because it is a reflection of the character and nature of an ordered, logical, pattern producing God who expressed this aspect of his character in the very fabric of the universe. If it wasn't designed so specifically we would expect a lot more randomness and a lot less irrefutable law than there is. My husband is fond of telling me that he loves math and science precisely because the deeper he goes in study the more he sees God's hand through it all, in ordering principle as well as expression.

 

That aside, I do think there is something to be said for the invented portion having to do with linguistics and expression - I think math is a language to describe the universe and phenomena, at its core, and in some ways the methods by which we express these concepts are invented even if the concept itself is a concrete, self evidence, or proven thing. This doesn't even touch on the times when math is more like logic or art or philosophy by virtue of what concept we are trying to express through it. Even with some of these theoretical/pure math proofs that have only recently been worked through and proven we are essentially trying to prove the existence of something we can theorize, but we don't pull the theory out of a vacuum. There is always a question or observation at the root of this and that's where I'd say the discovery aspect proves itself, even as we we inventing the process by virtue of working it successfully the very first time.

 

In my mind I see it as both - discovery from the human side and invention in how we express what we find and extrapolate further. But I'm neither artful nor precise in my language on this, so take it with a grain of salt :)

Edited September 29, 2016 by Arctic Mama

[Farrar]

Yeah, it seems indisputable to me that, like Regentrude said, mathematical notion is invented. And, in fact, has had several iterations over the centuries. I guess what's more interesting is that different people and cultures have invented notation that expresses the same things even when they aren't things observable in the natural world. Like, it's not weird that different cultures all invented words for "rocks" because that's something we can all see and observe. Or numbers (though there are cultures with just numbers for none, one, and many) but as I understand it there are concepts that weren't originally with corresponding observable phenomena that were described by mathematical notations invented by different peoples. 

 

It does feel a bit like it's one of the more compelling cases for, if not a higher power, then an ordering of the universe or what have you.

[regentrude]

My question is where does math come from? Like, some of it is observed from natural phenomena. But other things seem to have been discovered, sometimes by different people working in isolation, that don't, at the time, describe any known natural phenomena but are "pure math." How is that possible? Where is the discovery coming from? I mean, I get that it's number and conceptual relationships and so forth. Like, is math a language that we're hard wired to learn? And then, later, sometimes those things can apparently be used to apply to natural phenomena, which, mind blown, right, people?

 

The discovery comes from the ability of the human mind to think logically. Starting from a certain agreed upon starting point, the mathematician can follow a sequence of logical thoughts and arrive at a conclusion nobody else had made before.

The ability for logical thinking is hard wired in the human brain.

[Bluegoat]

The discovery comes from the ability of the human mind to think logically. Starting from a certain agreed upon starting point, the mathematician can follow a sequence of logical thoughts and arrive at a conclusion nobody else had made before.

The ability for logical thinking is hard wired in the human brain.

 

That just moves the question back a step though "Where does logic come from, why do we seem to perceive it in nature".