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Where do the rules of mathematics come from?

Presumably you agree that there is a 'universal law of mathematics’ (that is, 1+1=2 whoever you are, wherever you are and this isn’t something humanity just invented) So, for example, when Pythagoras came up with his theorem he discovered a relationship between variables that already had been defined by the 'universal law of mathematics’. Our brains may have evolved to use mathematics however 1+1 would still equal 2 before life even existed. Where does this law come from?

Posted: January 7th 2008

# Eric_PK

Mathematics, though often grouped with science, is unique in that it isn’t a science.

Mathematics a human-created abstraction. 1 + 1 = 2 because it is defined to be true (or, more specifically, it is true because it follows from the axioms of that kind of mathematics).

That is why mathematics has the concept of “proof” – whether a given theory can be shown to be true under a given set of axiom (ie beginning assumptions).

Now, it’s true that mathematics is often applied to the real world. In the real world, we know that if you take one item and you add another item, you will have two items. So, there’s good correlation there.

There is also a mathematic proof that shows that the sum of the interior angles of a triangle is always 180 degrees under specific axioms.

However, if we apply that one to the real world, we find that it’s true for small triangles, but not true for large ones. That’s because one of the axioms for the original proof is that the triangle is on a planar surface, but the earth is a sphere. There is another set of theorems that cover that situation, and give a different result.

My point being that mathematics work in the real world only if you choose the right mathematics. If you choose the wrong one, they don’t.

I therefore don’t think that the word “law” applies.

Posted: January 11th 2008

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# George Locke

First I’d like to point out that math is not a completely coherent system insofar as it relies on unprovable axioms, such as “Euclid’s fifth postulate”:http://en.wikipedia.org/wiki/Euclid%27s_fifth_postulate (that parallel lines never intersect).

1+1 would still equal 2 before life even existed.

This is a case of confusing the map with the territory. Numbers are abstractions devised by humans for describing the world. No one could dispute that when you take one apple and another apple together you’ll always get two apples, regardless of whether you’re a moose or a mathematician. However, 1+1=2 is an abstract statement. It applies to the two apples, but the fact that one and one makes two doesn’t mean that numbers exist outside of consciousness.

When I say you’re confusing the map with the territory, what I mean is that the apples don’t change depending on how we describe them whereas math does change depending on our definitions. Even if there were only one way to describe the apples, that doesn’t mean the phenomenon implies its own description. You need observers and phenomena to have descriptions.

If you find that unsatisfying, then consider this: math in general is based in 2-valued (true/false) logic, but logicians have shown that n-valued logics are also tenable. Our brains work on the basis of 2-valued logic, so it is practically impossible to imagine how 3 or 4-valued thinking would map to real life. This doesn’t mean that 3-valued logic can’t describe the world, though. Another species that evolved with 3-valued logic might not understand the statement 1+1=2 at all.

Furthermore, logic itself is just one system, there may be other systems that we can’t even imagine, systems to which numbers are totally foreign.

Posted: January 11th 2008

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# John Sargeant www

Well observations about nature, the discovery of relationships that could be expressed using mathematics is a description of nature.

As such mathematics gives us a model by which we can make predictions and get results about the world.

Maths itself developed – to explain the revolution try doing simple maths using only roman numerals. Decimation does have some benefits.

The history of mathematics is actually one worth pursuing. It did develop in gradual steps; as humans we try to analyse the world we live in and mathematics is one that helps.

The rules that exist are logical constructs that have been developed. That there seems an order to it that seems incredible suggests only that we can observe the world and that possibly these rules exist because if they did not life as we know it would not exist.

Posted: January 10th 2008

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