My hope for philosophy has always been that it could provide an elucidation of some ordinary experience. In Kant, the need for such an elucidation can be seen at least here: "The empirical concept of a plate is homogeneous with the pure geometrical concept of a circle. The roundness which is thought in the former can be intuited in the latter" (KRV A137/B176). When we think a thing is round, we subsume an object under the concept of a circle.
But there is a hitch. What is the concept of a circle? Well, it is line whose every point is the same distance from another point. The length of that line is famously incommensurable with the distance from its center. The ratio of the radius to the circumference, which Kant in his own notes estimated to be 6, is actually the irrational number 2pi. So what does it mean to think something is round?
A real plate, a thing in the world, cannot be "perfectly round" (truly circular). We do not think its likeness with a geometrical figure. Rather, we imagine it spinning on the potter's wheel. (More generally, we imagine a wheel.) Such a wheel will always have a slight wobble. It will have an axel with a determinate width and there will be a small space between the axel and the hub. It is for that reason that the circumference and the radius can rationally coexist.
Thursday, May 15, 2008
Roundness
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I am not sure whether or not you are implying this – In fact, I don’t think you are. But nevertheless, it’s worth mentioning that the fact that pi (or 2*pi) is irrational has nothing to do with the fact that “real plates” are never “truly circular”.
To name a point in space by two irrational coordinates, say, (pi, pi) is just as accurate as a designation as using rational numbers, say, (6,6). The “philosophical” worry is whether or not a point designated by irrational coordinates can be said to “exist in space”. The solution (as I see it, at least) is to say that if points within a space designated by rational number can be said to “exist”, then points designated with irrational numbers within that space can be said to “exist” as well. In fact, this problem was the very first to concern Wittgenstein upon his return to philosophy in 1929. Cf. very first remarks of MS 105.
Rather, the notion of physical points (the points of “real plates”) is partly incompatible with the notion of geometrical points as such (irrational or not). This seems to be your point too.
Geometrical points is something that is APPLIED to physical points. Not something that can be said to be IDENTICAL to such-and-such an extension of physical points. Cf. Wittgenstein’s remarks in PI and RFM to the effect that the standard meter in Paris cannot be said to be not be 1 meter.
The only thing "truly circular" is my seantor boyfriend's willy. Don't tell his wifey!!!
At the MPI for Psycholinguistics we've recently been looking into the cross-cultural validity of some of those 'good Gestalt' shapes — in other words, we've tried to check whether or to what extent geometrical concepts like 'round', 'square' and 'triangle' are lexicalized in the languages of the world (see the list of our fieldsites). Not all results are in yet, but one result is that 'round' or 'roundness' seems to be more robustly attested cross-culturally than the other shapes.
Siwu, the language I work on, is a case in point: it distinguishes very neatly between giligili 'perfectly or pleasantly circular' and minimini 'perfectly or pleasantly spherical', i.e., it makes a distinction between 2d and 3d in roundness.
Not entirely sure what the link is between this and your more philosophical point, but I thought you might want to know that some of us are actually trying to verify these Kantian intuitions (which might be no more than German, or Standard Average European) cross-culturally.
I don't agree that (pi, pi) is as good a coordinate as (6, 6). It is possible to travel 6 km and stop. You can do that more or less accurately, using more or less accurate equipment to tell you when to stop moving. You can also travel 3.14159 more or less accurately, using the same sort of procedure. All rational numbers can be used to measure physical space in this way.
Pi can't. It is only be first removing the irrational aspect of pi that our procedure would be able to tell us when to stop. You can't put pi on an odometer.
For obvious perhaps reasons, I have similar issues with Cantor's proof of orders of infinity (I believe my issues are similar to Wittgenstein's, but, as you know, I am a poor scholar.) There is no way of marking out a distance in pi-sized stretches. Only in vulgar (i.e., perfectly rational) approximations of such stretches.
2pi symbolizes the physical operation of drawing a circle. You take a piece of string of a determined lenth and hold one end fast. You pull it taught and trace around...
I'm grateful for Mark's comment. I think the universality of roundness lies in obvious truth about swinging something around on the end of a tether.
All of this of course also applies to the diagonal of a square. Does that imply that there are no perfect squares in nature?
I think it does.
Circles amuse themselves.
I’m still not sure we disagree at all. If there is any disagreement, I suppose, our differences are terminological. And hopefully not due to downright technical details (None of us, I suppose, are anything but amateur mathematicians.).
The notion of a pure mathematical point that is in itself ‘inaccurate’ or ‘accurate’ is nonsensical. In pure mathematics, (√2, √2) is a fully determined point, its determination lacks nothing. In the application of geometrical/mathematical points to physical space the notion of ‘accuracy’ however slips in. A measured object might more or less accurately conform to a certain measurement or geometrical figure. And the geometrical/mathematical points might themselves be more or less accurately applied, e.g. we might have difficulties determining the exact length of something. I think we agree on these things.
If this holds, however, there seems to be no problem in saying: Walk 1 meter to the right and then walk 1 meter to the left, more or less accurately of course. Now you’ve marked out an irrational stretch, more or less accurately of course.
I liked your comment's first argument about “going” and “stopping” somewhere, but I wonder if you would also argue, that if I order you to part 74 km in 54 equal pieces, that could not be done, since the result would be 1.370370370… (infinitely recurring) km.
On the assumptions of your argument, how do I travel 1.370370370.... km?
But certainly the fraction 74/54 is a paradigm-example of a rational number. In that case, the issue is not at all about the irrationality of a number, but rather about the finiteness of its decimals. Doesn't the argument conflate these things?
In any case, I fully agree with the general point that no circle is “truly circular” and I am looking forward to seeing more of the Kantian point that can be derived from this here.
Yes, I do think we agree. I believe that the square root of 2 and pi are operations, and that in "applying" math to physical objects we need values. But, you're right, there are probably mathematicians who will instantly recognize the problem that they solved long ago.
In any case, in my opinion, if you walk one meter (more or less accurately), turn right, and then walk another meter (more or less accurately), you have not (I want to argue) "marked out an irrational stretch". That only happens once you assume (imagine) that you walked exactly 1 meter in each case, and turned exactly right.
We might also phrase our problematic like this:
I have a paper strip in front of me. I impose some numerical scale on it. And now I want to cut it in two at a place, where there is no rational number. Mathematically speaking, this is surely possible: I just have to pick a point, where there is no rational number. After all, such points are fully determined by the presence of rational numbers on the strip. But now I go on to actually making the cut, physically as it were, and now it turns out that it is impossible to make the cut. I can only make approximations of the cut. That is, I can come arbitrarily close to any non-rational cut.
But now I want to say: Surely, an irrational number still plays a vital and determinate role in my actually measuring out a cut. For what does ‘approximation’ mean here? An approximation is a rational number written in such a form that it can be compared to the irrational number.
Sorry it took me so long to continue this.
My point, I think, is that you can't "pick a point" if you can't "make the cut". The sense in which the former is "mathematically" possible is the source of philosophical problems.
I like the idea of a form that allows comparison of kinds of number. But, if I may, we can now shift our inquiry to the sense of "approximation". A number can be used to express our approximation of a quantity. But what does it mean to approximate a number?
We write down a number and use that number as a value to represent some quantity. We can't say, "This number is an approximation of another number." We have to say, "This number represent this quantity approximately."
You are suggesting two numbers, one rational and one irrational, and you are suggesting a way of "preparing" the latter for a comparison with the former. But aren't numbers irreducibly comparative?
Irrational numbers play a role in a language game called mathematics. But they are not numbers. They are more like operators or variables. (Haven't quite decide which likeness is stronger.)
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