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Is mathematics discovered, developed, or both?
#21
RE: Is mathematics discovered, developed, or both?
That's why I always taught my students to check their answers make sense, especially when they've been using a calculator. If they end up with a length of -5cm, they've probably hit a wrong button.

But of course, it's possible to model a real situation with an equation which then produces extra solutions that don't apply to reality. In other words, it's not a one-to-one mapping.
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#22
RE: Is mathematics discovered, developed, or both?
Humans identify things by common perceptions. An apple is round and red, with a skin and a core. Looked at another way, it's just a bunch of stuff that melds from one thing into another. We identify parts of the whole because our eyes separate things out into light and dark, boundary and substance. Our ability to recognize objects as having the same generic features or specific type lies at the foundation of mathematics. Without the ability to sort things by generality, we wouldn't have numbers. Without the mental ability to represent the world as a collection of objects, we wouldn't have math. The circle is a perfect example of how our mental representation of the world is composed of pure objects. When we see line segments lying around the periphery of a circular shape, our mind mentally connects the lines to form a circle. It is this idealized circle in our minds eye which generates the ratio Pi. So numbers, math, and geometry are built up out of lower level precepts. We didn't so much discover math as we did evolve a "mathematical mind." Our ability to separate, combine, sort, and compare are all traits which served the animals which became us in negotiating their environment. The ability to abstract things apart from our sense experience is another key feature, as is the development of robust natural languages. Mathematics was neither invented nor discovered; it evolved.
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#23
RE: Is mathematics discovered, developed, or both?
(22nd November 2016, 13:53)robvalue Wrote: That's why I always taught my students to check their answers make sense, especially when they've been using a calculator. If they end up with a length of -5cm, they've probably hit a wrong button.

At this point, I rejoice if they get cm at all, and not sqrt(kg)/cm*h
The fool hath said in his heart, There is a God. They are corrupt, they have done abominable works, there is none that doeth good.
Psalm 14, KJV revised edition

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#24
RE: Is mathematics discovered, developed, or both?
(22nd November 2016, 16:51)Jörmungandr Wrote: Humans identify things by common perceptions.  An apple is round and red, with a skin and a core.  Looked at another way, it's just a bunch of stuff that melds from one thing into another.  We identify parts of the whole because our eyes separate things out into light and dark, boundary and substance.  Our ability to recognize objects as having the same generic features or specific type lies at the foundation of mathematics.  Without the ability to sort things by generality, we wouldn't have numbers.  Without the mental ability to represent the world as a collection of objects, we wouldn't have math.  The circle is a perfect example of how our mental representation of the world is composed of pure objects.  When we see line segments lying around the periphery of a circular shape, our mind mentally connects the lines to form a circle.  It is this idealized circle in our minds eye which generates the ratio Pi.  So numbers, math, and geometry are built up out of lower level precepts.  We didn't so much discover math as we did evolve a "mathematical mind."  Our ability to separate, combine, sort, and compare are all traits which served the animals which became us in negotiating their environment.  The ability to abstract things apart from our sense experience is another key feature, as is the development of robust natural languages.  Mathematics was neither invented nor discovered; it evolved.


Our minds seem better equipped to deal with larger numbers but this parrot could handle single digit counting and a few other concepts as well.  Nice window into other minds.

https://youtu.be/ldYkFdu5FJk
Belief .. is the insistence that the truth is what one would .. wish it to be. The believer will open his mind to the truth on the condition that it fits in with his preconceived ideas and wishes. 


Faith, on the other hand, is an unreserved opening of the mind to the truth, whatever it may turn out to be.  


- Alan Watts, The Wisdom of Insecurity
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#25
RE: Is mathematics discovered, developed, or both?
Some birds can't count so you can pop an orphaned bird into their nest, and they won't realise! They'll just take care of it.

How cute Heart
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#26
RE: Is mathematics discovered, developed, or both?
Cuckoos rely on that.
What is a drop of rain... compared to the storm?

What is a thought... compared to a mind?
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#27
RE: Is mathematics discovered, developed, or both?
I think that it is discovered.  The fact that reality exists in the form of matter and energy means that things such as 2 + 2 equal 4 are fundamental truths of our existence, hence, they are discovered not developed.  They are simply brute facts; to quote Mr. Spock (during a low-budget, 3rd season episode directed by the master himself, Gene Roddenberry):


Quote:Physical reality is consistent with universal laws. Where the laws do not operate, there is no reality.

http://www.chakoteya.net/StarTrek/56.htm
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