Consider Mobius strips (eg. strips of paper with one or more half-twists joined together at the ends--for example, a Mobius strip with one half-twist has the interesting property of having only one side...). As anyone who has had a chance (either as a child or as an adult) to play (i.e. to cut them down the middle lengthwise for example) with such objects can verify, these objects have very interesting properties. It also seems a trivial observation that these strips of paper (or any other material that can be so twisted) so constructed constitute real mathematical objects and therefore mathematical Platonism is true. What, if anything, is wrong with this argument? Another question I find of interest regards the half-twists that give these objects their interesting properties. What sort of property are these half-twists and what are they properties of since any material that can be given half-twists and can be joined at the ends become these 'real' mathematical objects by the operations of twisting by half-twists and joining at the ends?
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4What makes you say that such bent strip of paper is a "real mathematical object"?– artmFeb 13, 2013 at 11:55
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Very simple. Take a strip of paper, give it a half-twist, and join it at the ends. Then cut it length-wise down the middle. Now take another strip of paper and simply join the two ends then cut this strip length-wise down the middle. One can easily see that the results from cutting each strip length-wise down the middle are different for each strip so constructed. This suggests to me that giving a strip of paper (as one instantiation) a half-twist and joining it at the ends has real consequences in the real world (I assume the same naive realism that informs us that jumping off– Thomas BenjaminFeb 14, 2013 at 8:22
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a tall building will lead to disastrous consequences...). This being the case, it would seem reasonable to conclude that the paper instantiation of the mathematical abstraction known as the 'Mobius Strip' is a real mathematical object because it is an instantiation in 'reality' of that mathematical abstraction.– Thomas BenjaminFeb 14, 2013 at 8:33
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1by the same token any object would be an "instantiation" of a number of formula describing some of its properties. The mathematical Möbius strip shares some properties with the paper if don't look too critically on the paper and ignore its thickness, surface quality, micro- and atomic structure etc. But each of the properties that paper Möbius strip doesn't share with the abstract one it shares with some other mathematical model. But that's the whole point of having the models, so it's neither surprising nor interesting.– artmFeb 14, 2013 at 9:17
3 Answers
Mobius strips aren't that interesting physically because they're all just rings. Everything is three-dimensional. If you took a noodle and squashed it so it was 30% wider than it was tall, and instead of doing it flat you did a half-revolution so that one edge of the bulge went from left to up to right, you'd have a "Mobius strip" of a sort. But it's a really unimpressive one because the curvature is still nearly constant as you travel around (rather than along) the noodle.
Doing it with paper just makes the aspect ratio more and more extreme. You can squash your noodle into a Y shape had have it rotate by 120 degrees also to get an interesting shape (if you forbid yourself from going over the high-curvature "edge"). I don't think any of this proves anything about the reality of objects, but it demonstrates that practical geometry is fun.
Some of your questions might be better addressed on Mathematics.SE. However, I would address the philosophical content of your question as follows.
It also seems a trivial observation that these strips of paper [...] constitute real mathematical objects and therefore mathematical Platonism is true. What, if anything, is wrong with this argument?
Just because a physical object can be described well by a model, consisting of a mathematical "object" which one might describe as having "very interesting properties", does not imply that the mathematical object is any more than an abstraction with which you can describe the physical object (or a class of similar physical objects).
This is especially clear when those very interesting properties are made with reference to imaginary physical models. What does it mean to cut the ideal Möbius strip down the middle, when you don't have manual access to it and it is in any case presumed to be eternal and unchanging? What you're doing is imagining the Platonic ideal as if it were a physical object, or at the very least imagining a model where certain details are fixed — the Platonic Möbius Strip That Is Made Of A Continuous Strip Of Some Quite Flat And Easily Cut Material — in order to express properties about physical models which you would somehow like to ascribe to the Platonic ideal.
The problem with the argument is that you equivocated between the abstraction and the real; between the map — represented by The Möbius Strip, with capital letters — and the territory, which is a single instance of a strip of paper cut and twisted so that the most natural description of it in terms of metric spaces (as opposed to material science) is as a finite non-orientable 2-manifold of a certain well-known kind which we call "Möbius strips".
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@Neil: Don't engineers make essentially the same equivocation between the abstract and the real? Isn't this equivocation between the abstract and the real the reason we observe the 'unreasonable effectiveness of mathematics' in describing the physical world? The very fact that one can construct a finite non-orientable 2-manifold known a "Mobius Strip" in the 'real world' with its associated 'real consequences' would seem to suggest that such a finite non-orientable 2-manifold is real? Feb 14, 2013 at 8:49
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1@Thomas: mathematics is effective is because we have made it so; and engineers and physicists equivocate between models is because it is convenient as shorthand. But it's no basis for ontology. That the world is orderly (often admitting succinct description) is potentially interesting; mathematics is just the name for our tool for contemplating structures similar to those we see in the world. I've equivocated between a literal map and its territory many times, but with the understanding that my fellow-travellers would understand that the map was only useful in reference to the physical world. Feb 14, 2013 at 11:10
No, not particularly. What it shows is that the idea of two sides is more subtle than one may believe without reflecting on an example as you've pointed out.
For the instrumental definition you've given it is true that the strip has one side.
But why is it confusing to someone that hasn't come across a mobius strip? This is because there's alternate definition of sides. To wit, go down through the material of the mobius strip and if you emerge again then it has another side. If you had an infinite thick strip for example, you wouldn't be able to get to the other side.
All this shows is the ambiguity of definition.
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Your alternative definition of 'sides' is a poor definition of sides, I think, because you still need to rely on the instrumental definition. Consider, If you had a very thick Mobius Strip (say, twenty feet thick) and you drilled a tunnel to 'the other side', how would you determine that the side you drilled to is really the 'other side'? Feb 14, 2013 at 9:06
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A way to do this is to mark the initial drill site with one color and the site you drilled to with another. If in fact you actually drilled to the other side you could walk around the surface back to the drill site and only see the color of the final drill site. Imagine your surprise.... Feb 14, 2013 at 9:19
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@Benjamin: It is from a geometrical point of view, perhaps. But people are not geometers, I'm describing side as would be commonly understood in terms of human psychology & perception. Why must I choose a definition that is geometrical in character? You're confusing the world of abstraction with the world-as-its-given out there in a similar way to the OP. Feb 14, 2013 at 13:16