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Mathematics: Discovered or Invented?

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PG1995

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Hi :)

Some people are of the position that math is discovered while others say it's invented. I think in a way both parties are at extreme ends. I think math is discovered as well as invented. There are some fixed mathematical relations in nature which exist on their own such as ratio of circumference over diameter, golden ration, ratio of sides of triangle, etc. These 'natural' relations constitute part of discovered mathematics. Then, humans use these relations and invent some of their own systems to create new math which is amalgamation of 'discovered' and 'invented' mathematics. This also implies that mathematics as a whole is not a universal language. I believe the part which includes natural relations (such as ration of circumference and diameter) is universal and will be true and applicable everywhere. But the part which contains human innovations and inventions is not universal. Recently someone told me that Godel's incompleteness theorem proves this point. What is your opinion on this? Please let me know. Thank you.

Regards
PG
 
If you have interest in number theory and numbers, you should like this book:

https://www.amazon.com/Mathematics-Birth-Numbers-Jan-Gullberg/dp/039304002X/ref=cm_cr_pr_product_top

Most of what's available here can be found scattered throughout many texts, but in Gullberg, its all in one source. Chapter one is all about your query.

To summarize, mathematics is the expression and relationships of numbers, and numbers are just the notation, like letters make words, and words make sentences, but what are those sentences saying? One could say it was invented, since there were several number notations that are obscure or obsolete, and the Arabic version used today evolved from its Indo-Hindu roots, and many others. But numbers are representation of concepts, and the greater question then becomes are they the expression of the concept or the concept itself? Chances are its the former, because the concept remains regardless of the number system: you could laboriously use Roman numerals to replace Arabic in a quadratic equation, use reverse Polish syntax, change the operator symbols but the results are the same and so would the interrelationships.

What is fascinating is that the driving force was initially practical:base ten rules because of ten finger and toes, but in some cultures there is base 20, because these natives wore no shoes and added their toes to the equation! Now imagine had they used their hair to count; of what if intelligent aliens have no fingers, what base would they be thinking in? Interesting questions for pure theory, ergo, number theory.
 
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Considering mathematics is dynamic and there are many separate branches I would say it was neither, it just came about because a new language was needed, mathematics is often considered the universal language (even though it's far from universal) not an invention or something that was discovered. The basic mathematical symbols such as number and then arithmatic came to be for need of a language to quantify things with, and we got along without it for the bulk majority of human existence. Keep in mind that the concept of zero, even as a place holder wasn't used until around 700B or so. It wasn't considered a 'number' proper to be used in calculations till India got ahold of the concept in around 900AD.

Modern mathematics has developed primarily on increased technological capabilities as we are able to measure and experiment with more and more aspects of nature that we previously didn't have instruments for. There are however branches of mathematics which are pure mathematics, statistics, logic, cryptography, things like that take on such a completely different notations that people from any sufficiently specific mathematical system wouldn't be able to understand the others notations. That's basically why I consider it a language. Languages evolve as math did for it's use in basic counting systems to the advent of the modern 0 and technologies increase into our scientific measurement of nature.
 
My perspective is that it is discovered.

Mathematics isn't the numbers and symbols, it is the relationships themselves. We discover the relationship. We express and communicate that relationship with numbers and symbols. Therefore I see the numbers and symbols as a language, but not mathematics itself (although I do see mathematics as universal, and in that way I suppose could be a language). The numbers and symbols are not universal, but the relationships themselves are. For example, I believe an intelligent alien species may not understand our symbols (123456789+-*/...), but I believe they would have also discovered the relationship of circumference and diameter, or the relationships between the angles of a triangle and the lengths of a triangle. If they observed anything on our planet they would know that we have also discovered those relationships.
 
Hi,

Mathematics is an abstraction of reality, it's not the reality itself. Supposedly it originated from simple counting and probably then advanced to include comparison.
It's probably more accurate to say that it was invented because of the counting but interestingly nothing we calculate matches anything in the universe *exactly*. Everything seems to be an approximation. Even counting is an approximation. If i go to the store and buy 2 apples and you go to the store and buy two apples. we both have 2 apples, but that's only an approximation of what we really have in each of our possessions. I might have bigger apples than you, or maybe one bigger and one smaller, and your apples will be different sizes anyway.
No one on this earth knows the true relationship between a circle and its circumference, they only know approximations. Some know better ones than others, but there's not one person that can tell you the exact relationship. In fact, because of this the relationship has to be given as a symbol rather than an actual exact number.
All four sides of a square are the same length. Funny thing though, there's not one perfect square anywhere we know of in the universe. And when we go to measure something like that we have to resort to statistical means to approximate where the boundaries are, which brings us right back to yet another approximation. Measure the distance between two pool balls sitting on the pool table. What's the distance. First, how do we measure it? We get a ruler and stretch it across the table from one to the other, but we find we cant really get the end of the ruler to touch even one of the balls because at the surface the forces wont let the two touch, and that's only one of the many problems that come up, so we have to approximate again.
But surely we can calculate the time it takes the earth to make one revolution around the sun right? Here comes approximation again, and why our calendar is not accurate over many years but needs constant adjustment.

So in the end it's all only perfect in theory and theory is invented. Unless maybe we can find one of the moons of an exoplanet counting sheep before it goes to sleep :)
 
Thank you, Sig, MrAl.

@MrAl: I found your posting very useful. So, my special thanks!

Best wishes
PG
 
As in so many questions, it depends on your definitions -- which, in human language, all must wind up being circular. Yet we manage to muddle through somehow. In my opinion, mathematics is completely, totally invented, exactly the same as language. In fact, it's simply part of our language, but with a bit more rigor in the area of rules department. Language is a sequence of symbols to which we ascribe meaning. Mathematics is exactly the same thing. What is interesting is that we've found empirically and through experience that some of these sequences of symbols have utility in helping us make sense of the space-time we seem to find ourselves immersed in.

While such questions are interesting, you might reflect on answering the question "What did you do with the information?". Did it cause you to behave in any different way (or will it cause you to behave any differently in the future)? The number of possible sequences of symbols from a given set of symbols grows very, very quickly with the size of the symbol set and the length of the sequence. Only a tiny fraction of those sequences find utility. How we find those of utility is an interesting process. If you don't believe this, it's simple to write a computer program that will spit out sentences with N words per sentence picked randomly from a dictionary with M words in it. It won't take you long to realize that essentially none of the sentences make any sense. :)
 
I found this discussion very interesting. In my opinion, there are aspects of mathematics which are discovered, such as the ratio of hydrogen atoms to oxygen atoms in water. This would have remained constant whether scientists made this discovery or not.
On the other hand, there are aspects of mathematics which could be defined as invented. An example of this is, as MrAl has said, is the square. Where have you seen squares consistently occur in nature? The square is a widely accepted concept, however it is a figment of our imagination. Sure, it has been replicated everywhere in the world, but so has Santa Claus, but that doesnt make him real. Also, the very foundation of mathematics, numbers, must also be called into question. Using the same principle, the actual denotation of numbers are purely imaginary, but what they represent are always going to be the same, nomatter how we represent them.

That is my view, take from that what you will, and feel free to tell me if you agree/disagree.

I think when talking about something as open as this, there is no real correct answer. :)
 
Hi Apple,

Yes it does depend on the context to some degree, but i think the bottom line is that we cant define anything so well as to use it perfectly for anything.
For the two apples and one orange, we have three items, yet that's not enough information to know exactly what we really have. Same with a ratio, where if i say i have two apples to every one orange that doesnt say whether i have two apples and one orange or i have two truckloads of apples and one truckload of oranges. It also doesnt say how many of them are rotten :)


We seem to think we have everything down pat until a new scientific study shows we really dont. Even many aspects of light have been changing over the years and even the definition of the speed of light may change in the near future.
 
They do exist

not one perfect square anywhere we know of in the universe.

Conceptually they exist (well, the could) in the mind of anybody, like the idea of line, point or plane. And that is part of what maths deals with after all.

Relationships (loosely - and maybe wrongly named as laws) they do exist. Proof of that is that Nature runs with or without us (albeit we could do (and we do) a lot to interfere.

You invent things based on previous discoveries.

I think that the Fourier concept of a signal composed by several other maintaining a certain relationship is a discovery. The procedure of a fast Fourier transformer is pure invention.
 
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Hi atferrari,

That's interesting. My argument is partly just the opposite for the same reason :) Nature works just fine without having to do any math.

Sometimes we have new ideas that lead to new concepts.

It's interesting that we can do math about things that dont even really exist. We have math for things that do exist, but we seem to have more math than for what actually exists.

Also along the same lines, it's arguable that even time itself doesnt really exist but is another one of our inventions in an attempt to figure out what is going on with Nature.

If the Fourier breakdown of a signal seems like a discovery rather than invention, then what about the other ways of representing a signal?
A Fourier series is just an approximation too and only in theory can we have an infinite number of terms.

Agree or no?
 
I tend to agree, but...

Hi atferrari,

That's interesting. My argument is partly just the opposite for the same reason :) Nature works just fine without having to do any math.

Quoting myself: Proof of that is that Nature runs with or without us

Let me insist: Nature does not need from us at all. We could miss altogether from Earth and everything will run the same (if not better...;) ) Agreement.

Sometimes we have new ideas that lead to new concepts.

It's interesting that we can do math about things that dont even really exist. We have math for things that do exist, but we seem to have more math than for what actually exists.

Those new concepts I could say they are again recondite relationships that took you more time (and effort or just sheer luck) to grasp.

Fibonacci or Fermat or Leibnitz or Maldacena they seem to elaborate basis on previous "discoveries" (?)

For what exists or not, I have the idea, coming from Literature that whatever you name, from that very moment does exist. Be it Don Quijote, the cords or the triangle.

If the Fourier breakdown of a signal seems like a discovery rather than invention, then what about the other ways of representing a signal?

Another relationship discovered in due time as well?

You could invent any kind of machine but the physical laws applied are previous discoveries of somebody else. The same with algorithms or just the simple multiplication that my teacher taught me (oh!!) so many years ago.

My limitations in English (I tend to think that I could be much more eloquent in Spanish) make advisable to stop this rant of mine right here.

Good to see that you spent time in commenting.
 
Hi,

No that's ok, interesting to hear anyway.

Your point about Fibonacci is a good one i think. We could probably argue about that for a time much longer than i care to right now :)
 
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