A real-world circle is not a true circle which, as far as Platos concerned, lives in the World of Forms. But isn't a fractal a mathematical form, so that does live in the the World of Forms; and as Mandelbrot pointed out, the coast of England, the shape of a cloud and the shape of a fern are fractals so they too live in the World of Forms?

Going further, isn't the standard model when expressed geometrically a form so that too lives there, and its consequences are too, so the universe itself is in the World of Forms; doesn't this then involve a infinite regress?

  • What is a mathematical form, as distinct from any other form? Jun 27, 2012 at 13:35
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    Hmm. Can you tell us a little more about the problem you're trying to solve here?
    – Joseph Weissman
    Jun 27, 2012 at 15:16
  • @Weissmann:I'm trying to understand the implications of Platos World of Forms have. Generally when I've read about this, the usual example of a Platonic form has been a circle, presumably all the other platonic solids qualify. Their qualifying attribute is a simple synthetic geometric description. I think all these can be expanded, what do mean by simple and by geometry (by synthetic, I mean not involving coordinates). The question at its most basic level - What did Plato mean by a Form. Jun 27, 2012 at 18:14
  • @Beaudrap: Good question, I'm not sure, but I'd hestitate to say all forms are mathematical. For the purposes of this question, I'm saying mathematical forms qualify as Platonic forms, going by Platos example of the circle. Jun 27, 2012 at 18:16
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    @MoziburUllah: I am not sure what books are you reading that use a circle as an example of a Platonic form; in my reading, the canonical example has always been the Horse. There's no reason to view Platonic ideals as necessarily geometrical-- I think that is only confusing matters. Jun 27, 2012 at 19:55

2 Answers 2


The coast of England cannot be a Platonic Form- at best it can be the flawed image of a Form that exists within space and time.

One can ask if there is Form that the coast of England reflects? And in this case is that Form the Form of Fractal? Would this even be a Form?

Seeing as traditional examples of Forms involve specific shapes ie- that of a Horse, a Chair, or a certain geometric shape (circle, square, cube, etc) I would argue that "Fractal" would not in and of itself be a Form as many different fractals have many different shapes. Note that there is no Form = Geometric Plane Figure. That level of abstractness appears to absent from the concept of Forms if read in the sense in which they were written.

No. The coast of England is a representation of the Form "Coast" as well it should be. A Coast being the Form that can be seen as separating Forms Land and Sea.

  • If there is no form called "geometric plane figure" (why?), then why do you not allow the coast of England correspond to a particular fractal which is a Platonic Form, which we happen not to have named? Surely the Platonic realm does have some examples of forms which are fractals, such as Cantor's Dust and the Sierpinksi Triangle. Under what conditions is some one particular fractal, or other form, recognised as a Platonic form? (For instance, is there a separate Zebra form in addition to the Horse form?) Jun 29, 2012 at 16:44
  • @Niel de Beaudrap correct. There is a Horse Form and perhaps a Zebra Form- but there is no "Fractal" form. And if the shape of the English Coast is unchanging then it would be a Form- but not of Fractal- that of the English Coast- yet it changes doesn't it? I'm just arguing as Plato- clearly there is inconstancy.
    – user151
    Jun 29, 2012 at 16:50
  • Does it matter which name I give it? Not that it matters, as it would seem clear that an appropriate name for the Platonic Form in question is "Coast Of England". Jun 29, 2012 at 16:54
  • No. the Coast of England changes- it cannot as per definition be a Form. That is clear. Read about Forms. I indicated this in my last comment- not sure how else to say it. I mean Plato is not here to argue with
    – user151
    Jun 29, 2012 at 17:03
  • No, you're talking about "the coast of England", which is indeed a changeable and imperfect representation of The Coast Of England. However, I do take your point about the fact that Plato isn't here to argue with... anything we might have to say about his forms is enitrely speculative, unless we are able to argue logically from his works what his opinion must be. Which I don't think you have. (Not that I'm trying to...) Jun 29, 2012 at 17:07

The right way to understand Mandelbrot's claim is: the coast of England can be profitably modelled as a fractal (for some purposes). It's exactly the same kind of claim as "The granma pie at Luigi's is a rectangle".

So, no: the coast of England is not literally a fractal, and it's not in the world of forms -- it's actually in England :)

  • Of course, a fractal has infinite resolution :). Jun 27, 2012 at 18:20
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    @MoziburUllah for example, yes. Also, a fractal has no spatio-temporal location, and is not made of atoms. All of these things are false of the coast of England. In short, a fractal is an abstract entity, and the coast of England is not. Talk of abstract entities is less misleading, I think, than talk of Platonic Forms (if you are more interested in metaphysics than in Plato scholarship, that is).
    – Schiphol
    Jun 27, 2012 at 18:26
  • @Martinez: Sure, but don't atoms too have a mathematical description? Of course your point still holds, since a model of some thing is not the thing itself. I'm not saying the coast of england is itself in the World of Forms; if I was being careful, rather than allowing a little leeway for rhetoric, I would have said the shape of the coast of england, in the same way I said the shape of a cloud is in the World of Forms. Jun 27, 2012 at 18:39
  • "It's exactly the same kind of claim as 'The granma pie at Luigi's is a rectangle'" --- this doesn't suffice to resolve the issue in itself. One can ask whether a particular fractal which models well England's coastline is in the Platonic realm, given that the Platonic realm is undoubtedly meant to contain several rectangles or other quadrilaterals which model well the granma pie. Jun 28, 2012 at 12:26
  • @NieldeBeaudrap Yes, I have nothing particularly insightful to say regarding what would have Plato thought of fractals. I honestly thought Mozibur was asking for the relation between abstract and concrete entities.
    – Schiphol
    Jun 28, 2012 at 13:01

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