Mathematics is WRONG/RIGHT (both are CORRECT!?!)

Mathematics (maths) is BOTH wrong AND right SIMULTANEOUSLY, as it's only a whole bunch of mental constructs
used to conceptualize, model and manipulate our mind and the 'real', physical reality

[A paste by sahwar (http://twitter.com/ve4ernik), published on http://pastebin.com/uY2xggQW]

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What's tricky is that it's not actually possible to prove that different assumptions don't contradict each other.

But it is technically possible that all of Math is based on a set of contradictory rules.

The interesting thing is that Mathematics has never claimed to be fundamentally right. All theorems and formulae are derived from assumptions known as axioms (http://en.wikipedia.org/wiki/Axiom). These assumptions may or may not be accurate in the real world.
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Mathematics is just a tool for analyzing & understanding real world problems. Not an accurate description of the phenomena themselves.
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It cannot be explained or proven why we think a certain axiom makes any sense or a rule of logical deduction is justified using some meta-mathematical system.
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Mathematics, at its essence, is an exploration of mental representations.
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Well, as far as I know most, if not all, mathematicians don't care about right or wrong. The study of math is fully concentrated on what is valid and what is not, since mathematicians define things themselves. They work under postulates they assume instead of the law of nature as scientists do.
Thus there is no right or wrong in math. And yes, all arguments in math is valid.
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Math is a crazy set of imaginations gone wild, in a logical progression.  Also it consists of 2 things, a set of definitions and a set of generic proofs that facilitate the actual arithmetic.  IF you do not agree with the set of definitions, then BY DEFINITION the proofs will fail.  If you do not agree that there is a natural set of numbers, especially a set that orders 1, 2, 3,.... then tough luck proving anything.

How can anyone prove to me that natural numbers exist and they are numbered exactly that way?  You can't!  Has anyone seen a number 1 or 2 in real life?  Prove to me that 1 exist!  This would be the exact proof that God exists, because if 1/0 cannot be defined or understood, Math labels as "undefined" and theists say "God works in mysterious ways".   I don't believe you!  So if 1 doesn't exist, then how can two non-existent things be added, physically in the real world?
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YES it is *possible*!   We have no proof that any system of mathematics which includes the counting numbers as we think we understand them is logically consistent. And even if we did, it is also possible that the binary (True vs False) logic that we use in mathematics does not apply exactly to any meaningful class of statements about the physical world. (The fact that some, or even all, of us may not be able to imagine it does not make it impossible.)
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Mathematics is the science of structures, is the way I often like to think about it.. structures of thought, association, abstraction.. you name it. We observe structures, just as Newton observed the falling apple. We construct/formulate/set up our own structures and observe them, just as we observed the Michelson-Morley experiment. We formulate axioms based on observation, just like physical laws.. and we have theorems, just like predictions stemming from those laws.

To illustrate what do I mean by structures, perhaps I'll  paraphrase Hofstadter's thoughts: to him as a child, the idea of taking three 3's, the operating of 3 upon itself, seemed strange.

Show me a 3, will you? No, that's three apples, that's three buses, and that's a symbol you're writing on paper. Where-is-the-three? It's just a structure in our head..  Wrong? Probably not.. Mutable? Definitely.
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What is often overlooked is that mathematics is merely a language.

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Source: https://www.quora.com/Is-it-possible-that-mathematics-is-fundamentally-wrong