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February 2nd, 2010, 11:03 AM   #1
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Was mathematics discovered or invented?

I am writing an essay regarding the above question. I am struggling to make up my mind. The question links much with the golden ration and pythagoras etc.
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February 2nd, 2010, 12:08 PM   #2
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Re: Was mathematics discovered or invented?

This is a question you will hear many and conflicting answers to, but here's mine:
It is invented. The abstractions and rules that are "the basis" of math were invented to describe systems we see around us (e.g., what happens when you collect a bunch of piles into a single pile?)
From there (from a set of rules), you could say the specific results are in some sense "discovered", but they are discovered in the same way new chess strategies are discovered: There is a set of rules, and following those rules-- playing with those rules-- you see a new pattern and see that this pattern is effective. In some sense, this pattern "existed" as an inevitable consequence of the rules, so you can say it was discovered, but I still think that the conscious use of this pattern is the result of an invention.

On the other hand, I don't think there are many things that were "just invented" or "just discovered". There are great stories of children stumbling upon diamonds in the river, but most discoveries require some foresight-- the require knowing where to look. Similarly, an invention has a lot of "let's try this and see if it works!" which is certainly in the realm of discovery. That is, they do the "inventing" and see if that's actually what happens, and then they do some more inventing, until the invention is what they want.
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February 2nd, 2010, 01:45 PM   #3
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Re: Was mathematics discovered or invented?

I fall into the "discovered" group. Mathematics represents eternal truth; we are explorers in this strange new world. We can fail to discover some truths, but we cannot create nor change them.

Originally Posted by cknapp
It is invented.
Of course only fools think mathematics is invented.
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February 3rd, 2010, 01:40 PM   #4
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Re: Was mathematics discovered or invented?

Thank you, this helped me alot!!
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February 3rd, 2010, 02:13 PM   #5
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Re: Was mathematics discovered or invented?

As mentioned already this is a deep question for mathematicians. Some personal thoughts.

Start with positive integers as given (we count things). Then include 0 when there is nothing there. Next we have fractions (splitting things up). Irrational numbers surface when we need to do geometry. Length of diagonal of unit square or circumference of circle of unit radius.

From now on the question of invention or discovery becomes fuzzy. For example negative numbers. Start with the concept of subtraction 0-1 (subtract 1 from 0) = -1 (negative number). Is this invention or discovery? We could have alternatively said subtraction is valid only if leads to a nonnegative value, undefined otherwise. Similarly, when handling the square root of a negative number - the choice is between undefined or talking about imaginary numbers (invention or discovery?).
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February 26th, 2010, 06:46 AM   #6
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Re: Was mathematics discovered or invented?

I think, partly invented and mostly discovered.
For the invented one - the numbers , basic calculations ,that are addition, multiplication.
For the other part, I think they are discovered
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July 9th, 2011, 09:14 PM   #7
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Re: Was mathematics discovered or invented?

What an interesting question! It seems to me that mathematics principles have been here all along, waiting to be discovered - from the Euclidian principle to Fibonacci & relativity principle and other. What we have invented is the language - the code - by which mathematicians identify math problems, reason, draw conclusion and share their results with the mathematical community.
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October 24th, 2011, 06:03 PM   #8
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Re: Was mathematics discovered or invented?

Discovered through life. There is no mathematics for a rock or a lake. The human mind saw it reasonable to abstract symmetry and repetitiveness into natural numbers. And then the miracle happened. These two simple rules themselves produced further patterns that we keep discovering till today. Just to name a few, the Pythagorean Theorem and Prime Number theorem, e, ?, ?, i and the algebraic closure, Euclidean and non-Euclidean geometries and so on. Ok, not a few but you get the point.

It might seem like this view leeds to logicism championed by Russell and opposed to formalism (i.e. mathematics is just a series of games with the goal being consistency, championed by Hilbert). But my own view is that of Platonism, also hold by Godel. This view supports mathematics is embedded into reality, we discover it because we are alive and sufficiently intricate to conceive it, we are alive because of an statistical anomally of the physical reality. Of course, if you believe in god and creation, Platonisism seems even more probable but my view is you can also believe, with good foundations, in it even by working only with the evidence you have.

For certain, it boils down to whether the axioms of mathematics are "real" or just seem real to us mortal men. Since the minimum axioms boil down to symmetry (relationships to 1) and arbitrary extensionality (in theory infinite computations with 1), we must question these. Are crown is certainly 1. This is the chief axiom. Artificial intelligence will begin to compete us when it will have "common sence" to reckognize 1 chair, 1 human, 1 self- itself. This giant step, if realised, will change the face of the world.

If we have sinned, it may be in our notion of arbitrary extensionality, or infinity. Does our universe have an end? Even more important to mathematicians - is there a point, an insanely large number after which no new theorem can be obtained. The answer amazingly is no. Think simply the never ending primes and the infinity of computations to get the "exact" ?. Can it be our abstraction of infinity be an act of genius afterall? Max Tegmark through his Mathematical universe hypothesis explains the possibility whatever entity existing in the mathemtical world occupying a physical counterpart somewhere, sometime. Of course! If the universe is unlimited this is not even a chance - it is a possibility with 100% chances of happening!

But don't let infinity fry your minds, like Cantor and Godel. We are humans and we can only live with 1. Infinity is our ally but a superior one we should use with caution. To get a foot in reality and support my Platonic view (thus math is discovered) I support symmetry is a physical law and so not just a man made axiom. See the three components of our surroundings - they end up being either the Up or Down quark, or an electron. Why is it that way? Why aren't there an infinity of different particles. The answer is because our reality is symmetrical, thus we have right to devise arithmetic.

This discussion can lead to the basic foundation of what is - or the absence of any basis. Who made symmetry? Taking a cue from some remarkable piece of proofs by Cantor, the number line is a line because of the uncountably many transcedental numbers - the numbers constructed by infinity. Our reality as it seems is not transcedental- we have finite particles afterall that support that. My conclusion is our universe is a statistical abnormallity. There exist infinitely more universes without any law, just chaos. Is there life and consciousness in those universes? Maybe we don't invent anything, we just discover...
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March 5th, 2012, 03:02 AM   #9
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Re: Was mathematics discovered or invented?

In my opinion your explanation is good it seems like that you have made a huge research on this topic but i think its partly invented and partly discovered.
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April 27th, 2012, 03:42 AM   #10
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Re: Was mathematics discovered or invented?

According to me mathematics was neither discovered nor invented .Because some things are already in this world was working according to maths and some new things were invented for new creation .
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May 28th, 2012, 06:45 PM   #11
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Re: Was mathematics discovered or invented?

Discovered. If the Pythagoreans had had the option of not "inventing" irrationals, they wouldn't exist. But they do and their existence falsified a premise of Pythagorean mathematics.

But I also think it's fair to say that inventions are really discoveries that certain things when put together certain ways can perform certain functions. Edison didn't just decide he wanted a lightbulb and then forced one to appear. He and others worked long and hard using both best available theory and lots of experimentation to FIND OUT how electricity could be used reliably to create a more of less steady amount of light. You can't just say "I have a bunch of materials x, y and z. So I'll just make them work."

It's even easier to see with simple "inventions". Wheels work the way they do because of their physical and geometrical properties, not because someone wanted them to work the way they do. The earliest wheels probably were literally discovered: portions of nearly circular tree trunks were found and found to roll in useful ways. Later, people DISCOVERED methods of wheel manufacture.

It's all about discovery.

Oh, and the old canard: "God created the integers. The rest is the work of man." That's got to be one of the stupidest things an otherwise bright person ever said. Without getting too literal over the god issue, there can be no doubt that real numbers are real. Circles have circumferences and radii. QED. If anything, I could more readily believe that humans in some sense ABSTRACTED the simple orderliness of integers from crushed together world of the reals, but I don't find even that convincing. Integers seem special because they ARE special, being the numbers with which discrete items can be counted.
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September 6th, 2012, 01:39 PM   #12
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Re: Was mathematics discovered or invented?

The truths of mathematics are always discovered, but we invent uses for them, and since the field of mathematics includes both the truths themselves and our uses for them, it is both discovered and invented, simultaneously.
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September 6th, 2012, 01:42 PM   #13
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Re: Was mathematics discovered or invented?

If its about pure mathematics, its being discovered.
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September 6th, 2012, 05:24 PM   #14
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Re: Was mathematics discovered or invented?

I would say both, you can discover 1+1 = 2 but you have to invent fractal mathematics and tools to visualize the fractal geometry.

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September 6th, 2012, 06:15 PM   #15
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Re: Was mathematics discovered or invented?

I would say that mathematicians discover the truths encompassed by the systems of mathematics that humankind invented.
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