Re: Epistemology 201: The Science of Science
From: Albert (albertwagner_at_cox.net)
Date: 03/02/05
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Date: Tue, 01 Mar 2005 21:23:30 -0600
Daryl McCullough wrote:
> Albert says...
>
>>Daryl McCullough wrote:
>
>
>>>By that, you mean that it is impossible to develop an intuition
>>>about higher dimensions. Why do you say that? It's not true.
>>
>>What does this practice consist of? How do you know if you are
>>actually practicing the right thing? What is 'intuition', as you
>>use the term mean? What does 'develop an intuition' mean?
>
>
> Well, you work a lot of examples to see which properties of geometry
> generalize from one dimension to another, and which ones don't. You
> look at a lot of special cases in which you can ignore some of the
> dimensions (through symmetry), which allows you to effectively reduce
> a 4D problem to a 3D, 2D, or 1D problem. Another trick is to take
> "slices". If you have a 1D object (that is, a line segment), and you
> take a "slice" through it, you get a 0D object (a point). If you take
> a slice through a 2D object (for instance, a filled-in circle) you get
> a 1D object. If you take a slice through a 3D object (for instance,
> a solid sphere) you get a 2D object (a circle). Similarly, if you take
> a slice through a 4D hypersphere, you get a 3D sphere. You can mentally
> investigate 4D objects by considering 3D slices through them.
Well, hell. *I* can do *that*. But, I am sorry to say that is
not intuition, but rather just logic, with possibly misleading
assumptions as you point out below. I thought that maybe a mutant
human had been born that could actually visualize a hypercube
rotating in his imagination.
> Finally,
> you can prove theorems about them.
>
> What it means to develop an intuition is to get a feel for what sorts
> of properties are true of 4D objects, and what sorts of properties are
> not, which analogies with lower dimensions hold, and which don't.
-- "Mercifully free of the ravages of intelligence" -- Time Bandits
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