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May 30

Mathematicians objecting Cantor

This quote can be found in the web:

"something on the order of 90% or so of working mathematicians accept Cantorian set theory both in theory and in practice, to some extent. (Source: The Mathematical Experience, Davis/Hersh)"

Although it's not clear what is meant by "Cantorian set theory" the 10% against set theory seems a rather high percentage. So my question is: are there notable working mathematicians not accepting traditionally accepted belief or proof techniques involving modern set theory?--Pokipsy76 (talk) 06:34, 30 May 2008 (UTC)[reply]

Yes, certainly. But they're not relevant to the crackpot arguments you're seeing bandied about. Some serious workers who are skeptical about set theory include Solomon Feferman, Saunders Mac Lane, Doron Zeilenberger. --Trovatore (talk) 07:11, 30 May 2008 (UTC)[reply]

Typo, Doron Zeilberger. Actually Shalosh B. Ekhad has always struck me as lacking in faith. JackSchmidt (talk) 04:59, 31 May 2008 (UTC)[reply]

Another one: Aleksandr Sergeyevich Yesenin-Volpin. --Lambiam 07:41, 31 May 2008 (UTC)[reply]

Particularly amusing considering his other work, Edward Nelson is another. He doubts the consistency of Peano arithmetic; cf his site and his Predicative Arithmetic. The question is not too clear.John Z (talk) 09:23, 31 May 2008 (UTC)[reply]

n > 3, n = Number Of Dimensions

I totally understand the 0th, 1st, 2nd, and 3rd dimension. I have ABSOLUTELY no idea how any other dimension above the 3rd works. I mean, let's take the simple cube 3rd dimension analagoe into the 4th dimension: the hypercube. So the net is the same as a cube's net, but instead of squares, it's cubes. So when a hypercube rotates, it looks like it's volumes are warping. How is this possible? and then, the "sides" of the hypercubes don't even look like cubes, they look have trapzoidial sides (you know what I mean?)68.148.164.166 (talk) 07:21, 30 May 2008 (UTC)[reply]

You can project a cube into a plane (in a photo or a drawing), when you do so the squares do not seems squares anymore and when it rotates the area and of the 2D picture changes. The same happens when you project a hypercube into a 3D space: cubes do not appear as cubes and when it rotates the volume of the solid picture changes.--Pokipsy76 (talk) 07:45, 30 May 2008 (UTC)[reply]

I still do not understand. Is it because my brain is to tiny? Why can't I understand?68.148.164.166 (talk) 10:16, 30 May 2008 (UTC)[reply]

Study the article fourth dimension. Bo Jacoby (talk) 07:50, 30 May 2008 (UTC).[reply]

It's hard for us, as creatures who are more or less three dimensional, to envision what a fourth dimension would be like. But then, as in Flatland, consider how hard it would be if you were two-dimensional to envision a third dimension. The easiest way to envision the 4th dimension is to just consider out 3D world as a horizontal 2D plane, and the 4th dimension being the vertical. This is what you see in all those pictures of space being warped by gravity. -mattbuck (Talk) 18:13, 30 May 2008 (UTC)[reply]

Hey, thanks for the link to Flatland, I really understand it a lot more now. But I think why I don't understand is because matter is 3 dimensional. In Flatland, that wouldn't be possible because the polygons wouldn't exist, because matter can't exist 2 dimensionally.68.148.164.166 (talk) 00:57, 31 May 2008 (UTC)[reply]

Actually, matter probably isn't three dimensional - I believe that most current theories postulate that the universe is 17-dimensional or something like that. However, there's no real reason you shouldn't have 2D matter as far as I know.

Huh, how why? I REALLY don't understand that.68.148.164.166 (talk) 04:22, 31 May 2008 (UTC)[reply]

Physicists sometimes work with toy models of physics that have fewer than three dimensions of space, and there are real models of physics (like Kaluza-Klein theory and string theory) that have more than three dimensions for various reasons. But the extra dimensions, if they exist, are nothing like the ones we see. They don't extend off to infinity but rather curl around on themselves with a very small radius. -- BenRG (talk) 05:15, 31 May 2008 (UTC)[reply]

Rather than to try to envision four dimensions, make an algebraic attempt. The coordinates of the two end points of a one-dimensional line segment are 0 and 1. The four corners of a two-dimensional square are (0,0), (0,1), (1,0), and (1,1). The eight corners of a three-dimensional cube are (0,0,0), (0,0,1), (0,1,0), (0,1,1), (1,0,0), (1,0,1), (1,1,0), and (1,1,1). Now generalize. Each point in four dimensions have four coordinates. The sixteen corners of a four-dimensional hypercube are (0,0,0,0), (0,0,0,1), (0,0,1,0), (0,0,1,1), (0,1,0,0), (0,1,0,1), (0,1,1,0), (0,1,1,1), (1,0,0,0), (1,0,0,1), (1,0,1,0), (1,0,1,1), (1,1,0,0), (1,1,0,1), (1,1,1,0), and (1,1,1,1). That was not too difficult. Bo Jacoby (talk) 20:45, 30 May 2008 (UTC).[reply]

Oh yeah, that's totally intuitive. --98.217.8.46 (talk) 00:56, 31 May 2008 (UTC)[reply]

You can't intuitively visualize what something with more then 3 spatial dimensions looks like. You can get indications of what that might look like, but since our basic visual vocabulary is based on three spatial dimensions, it's just not going to ever work out. Don't fret about it. You're not the only one who has problems visualizing it intuitively. There are little ways to try to get around this—a 4D object would have a 3D shadow, ooooh—but in the end it's just not visualizable in any sort of obvious way. A hypercube is a good example of that—the volumes do look like they are warping, and that's how we'd perceive something that was in 4D, as horribly warping and being an impossible object, because those 4D would be still only available to use in the 3D (the hypercube example actually is a little problematic because it is basically modeling a 4D object in only 2D, which makes things even worse). --98.217.8.46 (talk) 00:56, 31 May 2008 (UTC)[reply]

Why are our basic visual vocabularies based on three spatial dimensions?68.148.164.166 (talk) 04:35, 31 May 2008 (UTC)[reply]

We evolved an ability to visualize objects in three dimensions, but no more than that. Even professional mathematicians can't visualize four-dimensional objects directly. They have to visualize a three-dimensional projection or slice or analogy of the object, or else analyze it in non-visual ways. -- BenRG (talk) 05:15, 31 May 2008 (UTC)[reply]

Why have we evolved an ability to visualize objects in three dimensions, but no more than that. If that was the case, then that means 3 dimensions has more importance than 2 dimensions as well as more importance than 4 dimensions or more.68.148.164.166 (talk) 17:40, 1 June 2008 (UTC)[reply]

I've met mathematicians that claim to be able to visualise higher dimensions - I've no idea how they do it, or even if they are really doing what they claim (and not just visualising projections, like you say), but they certainly claim to be able to do it. One suggested doing it in a pitch black room so there's no 3D world to confused you. --Tango (talk) 12:09, 31 May 2008 (UTC)[reply]

We all do a similar trick all the time without even thinking about it, when we infer three-dimensional properties of objects from the two-dimensional projection that we see in our visual field. We know that the faces of a cube are squares, and the fact that they change appearance as we turn the cube and don't look like squares from most directions doesn't worry us at all. In fact, this interpretation is so ingrained that most graphic artists require a degree of practice and training to supress it in order to draw an accurate representation of what they see, instead of what they think they see. Mathematicians like the late Donald Coxeter can, through practice, develop a similar intuitive feel for inferring four-dimensional qualities by visualising a three-dimensional projection (well, really by visualising a two-dimensional projection, so there is another level of interpretation involved). The human brain is truly amazing. Gandalf61 (talk) 10:13, 31 May 2008 (UTC)[reply]