Math - Discovered or created

I'm curious whether you see mathematics as something that is primarily discovered by humanity or something that is invented by humanity.  What is your reasoning?
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PaulHewsAsked:
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Dave BaldwinFixer of ProblemsCommented:
The answers to that have filled many research papers.  Counting was discovered first and then arithmetic was invented, probably to decide who gets what count.  Of apples, axes, mates... you know, the usual stuff.  

Counting comes first because it is a recognition of what's in front of them.  The count doesn't matter as long as it's just a bunch of stones that nobody wants.  But when it's a pile of apples that everyone wants, the count is used to decide who gets what... assuming they want to be 'fair'.  So names are given to different quantities, one two three four... note there is no zero.  Because in counting things like apples, zero does not matter.  Only quantities More than zero 'count'.

Arithmetic comes next to help decide how to divide things.  Mathematics comes way later because math is not about Numbers and counting but about equations and symbolic methods.

And I saw all this in a movie or a video long ago.  So counting is discovered and math is invented and arithmetic is in between the two.  No, arithmetic is not Math because math is symbolic and arithmetic is the manipulation of numbers, not symbols.
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viki2000Commented:
Here is my view:
- the things that we see now exist in an informational plane of existence. We just access that plane no matter if we speak in our language about inventions (creation) or discovery.
-in fact from information point of view we cannot create anything. It existed before. We just bringing as new thing in our world and we believe we created. And actually we do that for material forms which manifest in our 3D.
My view is based on the next fact: the functionality of our brain.
People believe that everything that is in our mind is the result of brain activity as chemical reactions. I do not believe that. The NDE and OBE experiences show that mind exists outside of the brain, when the brain does not work. The brain is only a receiver, an interface between mind and our body, our 3d world. The brain makes us aware of our material body. The brain is like a TV, projecting images which receives from spiritual world, an information dimensions accessed by our mind. Such view explains also telepathic experiences, remote viewing. There is more to say, but is enough for now.
Having these said the view over scientific discoveries or abstract thinking, mathematics is this: the information exists before and somebody just access the plane with such information. If you read about Tesla's inventions and his personal life, you will notice that he had lucid dreams about many of his inventions, the information just came in his mind.
If you read about maths or medicine or physic you may also notice that happens in the same time on earth that 2 or more independent researchers to find the same thing in almost the same time. The informational plane have openings to certain info at s certain moments.
The information is there, we just have to access it.
Of course you need the preparation for that to understand it.

To answer straight to your question: is invented for our world trough a personal discovery by accessing the information plane.
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tliottaCommented:
The various mathematics are invented. The relationships that are described with mathematics are discovered.

The different mathematics are languages, much like the usual languages we use in our everyday conversations. We made up all of the languages, including mathematics languages. We invented the symbols and the rules to manipulate them, just as invented all other letters and digits and rules of syntax and grammar.

As discoveries of natural relationships unveiled more of the world around us, we fine-tuned the symbols and the rules to more closely describe what was learned. Ideally for mathematics, the structures of the various languages are altered to match the structures of real observations as closely as we can understand what we see.

There are numerous cases where new discoveries invalidated old mathematical forms, and new ones had to be created. Would anyone think that older inaccurate or incorrect forms that have been discarded were "discovered"? If not correct, how could they have been "discovered"? Are all of our mathematics today absolutely correct? Or will they someday be superseded by some more accurate form?

Languages are always invented. But the things they represent are discovered. The numeral "1", i.e., the symbol, was invented. But the number [one] was discovered.

Tom
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PaulHewsAuthor Commented:
Maybe the term math is too broad.  Let's just focus on the concept of number.

So DaveBaldwin says that "Counting was discovered first..."
viki2000 seems to indicate that all conceptualization is discovery of an separate objective plane of reality (very Platonic.)
tliotta says 'The numeral "1", i.e., the symbol, was invented. But the number [one] was discovered.'
 

So it seems that you each think that the concept of number is discovered.  Does this mean that it has an objective existence?
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Dave BaldwinFixer of ProblemsCommented:
Does this mean that it has an objective existence?
Heavens No!  Counting like numbers and language are mostly things that we have agreed have certain meanings.  Rocks and bananas have 'objective existence' meaning (to me) without 'us'.  We have agreed amongst ourselves that certain groupings represent a certain count and we use numbers to name that count.  

One of the first things taught in Philosophy is that the word is Not the thing it refers to.  The word is Only a reference (name) we use to talk about it.  The word 'apple' only vaguely describes a large class of things.  Those things come in various colors from various places.
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tliottaCommented:
'Numbers' have abstract existence. We're nowhere close enough to understanding the nature of reality to guess if they have any kind of "objective existence".

But that does seem to be the actual point of your question.

I'll have to let it simmer a while to see if I can even decide if I can form a personal opinion. Questions of reality tend to get into circular chains that never end.

Tom
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Dave BaldwinFixer of ProblemsCommented:
The fact that numbers are different in every language should tell you everything you need to know.  But people do count the same even when they use different words to do it.
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Dave BaldwinFixer of ProblemsCommented:
If you are actually interested in this subject, you should know that a lot of other people have examined it.  This video talks about it a lot.  http://www.youtube.com/watch?v=8BbVIWHslNo
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tliottaCommented:
The fact that numbers are different in every language...

The numbers are the same in all languages I've seen. (There are some exceptions I've heard references to, but I've never thought about trying to verify them.) Their names are different. It's like objective "bananas" being the same regardless of what they're called in different languages.

Tom
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viki2000Commented:
As this is a philosophical question, my presented idea comes from old Asian thinking as:
- potential non-manifested state vs manifested reality in Tao Te Ching, few thousends years ago: http://en.wikipedia.org/wiki/Tao_Te_Ching 
- Akasha notion from various religious views: http://en.wikipedia.org/wiki/Akasha 

These are related with Maya notion http://en.wikipedia.org/wiki/Maya_%28illusion%29  , presented in similar manner in ideas from modern SF movies as Matrix or Avatar. In fact are old ideas about reality. From here the difference between what is (can be) created (invented) or discovered, no matter about what branches of science we speak.
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PaulHewsAuthor Commented:
Tom
Numbers' have abstract existence. We're nowhere close enough to understanding the nature of reality to guess if they have any kind of "objective existence".

I think maybe that the whole question might simply be a false dichotomy.  While one can make an argument that every concept is a creation--that the concept simply would not exist without us to conceive it--there is also an aspect of discovery.  The rules of number are based more on observation of reality than the rules of a game such as poker.

DaveBaldwin

Rocks and bananas have 'objective existence' meaning (to me) without 'us'.

If I think about the word "dog" I'm not thinking about a specific dog, but rather a categorization of animals.  This categorization only exists as a concept, which we created, but which has a basis in existence (discovery.)

The fact that numbers are different in every language should tell you everything you need to know.  But people do count the same even when they use different words to do it.

In isolation it seems we do not count all the same.  In some isolated groups, they do not count at all.  

http://en.wikipedia.org/wiki/Numeral_(linguistics)

At least we can say in the case where a population uses number, even if they are using "base 24 with a subbase of 6" there is a direct translation to other consistent counting systems.

There are aspects of mathematics that seem more created than others, such as the invention of imaginary and complex numbers, based on nothing other than the idea that we can take the square root of positive numbers, so there should be a corresponding concept to negative numbers.  

It's exciting for me to see the applications for those invented concepts that model the real world.
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PaulHewsAuthor Commented:
Thanks all for your thoughts.
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tliottaCommented:
The rules of number are based more on observation of reality....

Mostly agreed. Where I get confused is in cases like irrational numbers. Is there actually a physical basis for them? Or do we impose a useful description that gives us 'workarounds' for difficult problems? E.g., is there any true significance to the "Planck length"?

If there is, then perhaps there is no true significance to unending sequences of fractional digits. And if there isn't, then 'infinitely small' confuses everything at any scale; "numbers" become ultimately meaningless except on a relative basis.

Tom
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