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There have been several recent posts about the work of N. J. Wildberger, a finitist who seems to think that mathematics should only focus on things that have some sort of "real world" connection, which excludes many infinite objects in particular. This viewpoint has some sympathy in the world of computation, for example, where algothims are a focus. Some of these post have cause some controversy and discussion over on Meta: On the closing of "Does mathematics require axioms?"

For the record, my background is traditional and formalist, with a fairly typical amount of Platoism (I think natural numbers "exist" as much as anything mathematical exist).

I'd like to take a somewhat different tack to some of these issues, hopefully in a way that satisfies this site's goal of producing actual answers. My question is:

What are some examples where using "infinite" or arguably "unreal" mathematical objects does a better job of dealing with "real" objects? I think of this concept as "idealistic approximation": when does idealizing a system make for better estimates?

Here's my example to make this question concrete: circles. Wildberger has problems with circles. I believe one of the reasons is that, in real 2-space for example, the points of a circle would have many irrational (indeed, transcendental) coordinates, and the real numbers are a problem for him. I'll go further: there are no actual circles in nature, best we can tell. Anything that appears circular must have imperfections at some level of precision. See this post for an example.

The counter to this view is that any approach short of using an idealized perfect circle is harder than a more "realistic" approach. By using unreal perfect circles, we obtain beautiful simple equations (for circumference, area, etc) that can be used for actual computations, which can be converted back to rational estimates to any desired degree of precision. To try and address circles "realistically" from the start, taking into account their imperfections, is needlessly complex: it would inhibit understanding rather than aid it.

Another example question of this type asked for real world instances of limits, here.

The point is that finitist math is already embedded inside traditional math. Finitists seem to think that the math outside of finitist hinders pedagogy. I'd like the answers to this question to provide some counterexamples to this view.

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  • $\begingroup$ Wikihammered by OP request. $\endgroup$ – Willie Wong Apr 22 '13 at 15:05
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    $\begingroup$ Your circle example reminds me of an anecdote I heard on MO: Mathematician sits in his office. Mechanical engineer walks in and asks for help. Mathematician says, "Sure, how can I help you?" Engineer: "I need an explicit general closed formed expression for the Schwarz-Christoffel mapping for a regular polygon inscribed in the unit disc." Mathematician: "Oh that would be quite complicated. Can I ask why you need it?" Engineer: "Well, I really wanted the one for the disc. But I'm afraid it is too hard. So I figure I'll ask you for the polygon case and take $n$ to infinity." $\endgroup$ – Willie Wong Apr 22 '13 at 15:15
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    $\begingroup$ @Willie Wong: Exactly. I fear that finitists would actually prefer the polygon approximation. $\endgroup$ – user452 Apr 22 '13 at 15:20
  • $\begingroup$ I have some inkling the tension really isn't about "ideal" vs "real", but is more about analytic versus synthetic reasoning, possibly with a dash of projecting onto the analytic approach one's frustrations with how difficult the subject of practical computation can be. $\endgroup$ – Hurkyl Apr 22 '13 at 16:01
  • $\begingroup$ I think this piece of Wildberger has gotten a lot more attention here than serves any good purpose. I read it -- and forced myself to read the whole thing because people discussed it here so much -- and while I hardly found anything outrageous in it, neither did it break any new ground when it comes to views of the infinite.... $\endgroup$ – Pete L. Clark Apr 25 '13 at 2:08
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I'm not sure if this point of view is of interest here, but the vast majority of physics conisists of simplifying assumptions.

Modeling fluid flow or elasticity without the fiction of a continuous media, in particular, is a clear example of a pedagogically insane approach. Of course there are errors in this approximation, due to endless small details, but these are generally best viewed as corrections to the continuous problem from the point of view of trying to understand any phenomenology at all.

More mathematically, the loss of the notion of completeness makes it very hard to distinguish between spaces where things converge and where they don't, so PDE theory probably goes out the window, along with spectral theory too.

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  • $\begingroup$ Terrific stuff, thanks! This is exactly what I would like to compile. $\endgroup$ – user452 Apr 25 '13 at 1:28
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Another example might be probability density functions, which are usually continuous real-valued functions. One could argue that most "real" applications would require discrete random variables (to be computable, e.g.). But as with the circle example, a continuous "approximation" of some complicated discrete random variable might be easier. In particular, because of the Central Limit Theorem, many discrete probability distributions (formed by taking a random sample, say) rapidly approach a normal distribution under reasonable conditions.

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One example that's close to my field is how much simpler dynamical systems theory is in continuous time than in discrete time. For example, the Poincaré–Bendixson theorem shows that continuous-time dynamical systems in less than three dimensions can never exhibit chaos; discrete-time dynamical systems can be chaotic in any number of dimensions, even just one.

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