Mathematics desk
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January 1

Maths - discovered or invented?

To me its amazing that inferances from the rules of mathematics can predict what happens in the real world. As a second question, have there ever been empirical studies that have led to the rules of mathematics being altered? 78.146.210.81 (talk) 11:43, 1 January 2010 (UTC)[reply]

Isn't it amazing the way the real world acts as a model for mathematics ;-) Naive set theory is a good example of some maths that was found to be wrong after a bit of study and had to be amended, see the section on paradoxes at the end. Dmcq (talk) 11:59, 1 January 2010 (UTC)[reply]

An important aspect of mathematics lies within the relationships between different branches. For instance, one could use algebra in topology, without delving deep into the subject (but of course, a deeper form of algebra may shed new light on topology, if applied). Likewise, although contradictions were found in naïve set theory approximately 100 years ago, it continues to be applied to this day. In fact, many branches of mathematics could function without reference to Russel's paradox or any deep form of axiomatic set theory. On the other hand, as I pointed out, fields such as set-theoretic topology require a very deep form of set theory; naïve set theory alone does not suffice. --PS T 12:29, 1 January 2010 (UTC)[reply]

The OP might not be that interested, and there is an overlap between who helps at the two help desks, but he or she might ask something close to what is being asked here at the Science desk and shuttle back and forth when it seems interesting if a more complex answer is desired. Part of the issue is what the 'real world' is from a scientific standpoint. There are some who will argue that it is a mathematical object when all is said and done. I wouldn't hold out too much hope for a serious reply at the Computer Science desk, but you might actually get the most interesting one there. Ultimately, the world consists of objects with rules for their behavior (it seems), and the fact that an abstraction from the real world (mathematics) would be able to feed back predictions on the behavior of the real world isn't what surprises anyone. It's that purposeless questions internal to the abstractions can be taken far from their natural sources and still find applications that is often surprising.Julzes (talk) 12:56, 1 January 2010 (UTC)[reply]

Of course there are religious/philosophical questions like whether you support the Axiom of choice or the Axiom of determinacy or go for a third way like projective determinacy. I think I read a science fiction story once where the laws of nature kept getting more complicated to cover up problems as more and more flaws were shown up in the previous laws. So relativity and quantum mechanics only became true recently as problems and paradoxes were shown in the previous simpler laws. Eventually it'll all become so complicated we'll never be able to show something is actually wrong or it might all disappear as just too paradoxical. :) Dmcq (talk) 14:09, 1 January 2010 (UTC)[reply]

Nobody has answered the first question. Does that mean the answer is "Nobody knows"? 92.24.69.222 (talk) 18:54, 1 January 2010 (UTC)[reply]

What was the first question. All I see there is "To me it is amazing that..." etc. That's not a question. Did you mean "Why does that work"? Or maybe "Am I the only one who's amazed?" Or did you maybe want an answer that would leave you unamazed? Michael Hardy (talk) 00:30, 2 January 2010 (UTC)[reply]

The first question is the title. -- Meni Rosenfeld (talk) 16:16, 2 January 2010 (UTC)[reply]

Mathematicians discover properties of natural numbers, shapes etc. and then invent notation, theories and structures to describe, investigate and explain what they have discovered, and go on to make more discoveries about what they invented. The answer has to be "both". Dbfirs 19:36, 1 January 2010 (UTC)[reply]

"Shape" can be formulated in the language of natural numbers, after employing various other mathematical (algebraic and analytical) ideas. --PS T 00:58, 2 January 2010 (UTC)[reply]

That's true, but the discoveries were made thousands of years before this formulation was invented. Dbfirs 09:57, 2 January 2010 (UTC)[reply]

Um, I'm not quite sure what the claim is supposed to mean. On its face, to talk about shape using only natural numbers, you need second-order logic; I'm not sure why you wouldn't just bring in the real numbers from the start, which are more natural to use when discussing shapes. --Trovatore (talk) 10:12, 2 January 2010 (UTC)[reply]

Not sure what kind of answer is wanted. If we had an answer that could be codified then it would be mathematical and then we'd simply have something saying it itself is applicable using methods which work only because it is applicable. Something like that which doesn't sound very sensible when applied to the real world but worked okay for Gödel in maths. That we are able to make sense of the world is only because it is reasonable. Dmcq (talk) 22:15, 1 January 2010 (UTC)[reply]

This is a classic philosophical question about the nature of mathematics. See mathematical realism for the "discovered" viewpoint, and see several alternative views on that page. Staecker (talk) 22:31, 1 January 2010 (UTC)[reply]