# Looking for a treatise on philosophy of “smoothness”

Forgive me, I don't know what to call it. During the Enlightenment there came about an idea that the main phenomena of the universe might be represented by simple equations. For example, Netwon's second law of motion, 'F=ma' can be used to calculate trajectories.

For these simple laws to work, there has to be an assumption that an approximate model will give you an approximate prediction of real world behavior. A course model gives you a course prediction, and a fine model gives you a fine prediction. In a certain sense, the precision of the inputs to a calculation relate in a roughly linear way with the precision of the outputs.

I have described this assumption about the world as "smoothness" because it assumes that there are no discontinuous aspects of the real world. Another way to say this is that it is "not chaotic". Chaotic system (models) are one where a small change in the precision of the inputs might yield a completely different result.

I don't know what to call it, which makes it impossible to search, but I am looking for a discussion of how Newton (and other enlightenment age scientists) made an implicit assumption that an approximate model would work approximately well at explaining the world. I would like to compare this to our more modern understanding of complexity, and chaotic behavior that emerges (sometimes) from complex systems. Does anyone have a suggestion on where would I look for this and what I might call it?

• I think the word you're looking for is uniformity and here the idea being the uniformity of nature is a background assumption in science. I don't have time to write an answer now so I'll leave that to others. – virmaior Feb 12 '14 at 14:26
• You can see in Stanford Encyclopedia of Philoaophy : Laws of nature and Models in science – Mauro ALLEGRANZA Feb 12 '14 at 15:22
• Continuity? Differentiability? In those days, all the functions they were interested in were smooth, ie infinitely differentiable. These days we have a lot of different intermediate categories of smoothness. These are math concepts. I don't know what philosophical concepts correspond, if any. Back in the day they studied smooth functions because they didn't have the math to analyze functions that were fractal in nature. – user4894 Feb 12 '14 at 17:31
• Mathematically speaking, @user4894's comment is exactly right. The ultimate concept here is called "smoothness": in this case, technical jargon for "infinitely differentiable" (meaning that it is continuous, the rate at which it changes is continuous, the rate at which the rate of change itself changes is continuous, etc.) And while quantum mechanics has complicated the notion of the continuum in matter and possibly space/time, I think that complexity theory and even the Sorites paradox indicate how a new philosophy of smoothness is highly pertinent in a world fond of boolean distinctions. – Niel de Beaudrap Feb 12 '14 at 18:55
• Zalamea on Pierce's logic of continuity is very good – Joseph Weissman Feb 13 '14 at 2:01

Your question isn't about smoothness as such, which has various technical definitions in mathematics. But not generally in physics where they think of everything as being smooth enough. As virmior points out, the name of the idea that you're probing is called uniformity - which at least in ordinary language is almost synonymous with smoothness.

The idea of something being uniform is a pervasive idea in mathematics - which one understands by examining the nomenclature of the subject where adjectives such as uniform, standard, normal, smooth abound.

But as you've touched upon smoothness, and though it isn't the main intent of the question it is something that I'm deeply interested in, so I'll tackle that first:

Smoothness has a long intricate history. One might begin with Archimedes method of exhaustion which found the area of a circle by starting with a polygon and increasing the number of sides. This in hindsight is the idea of a limit which was implicit in the foundational work of Newton & Liebniz on the calculus.

In contemporary mainstream language this is now thought of in terms of smooth manifolds, and almost all of physics is written in it implicitly: the smooth spacetime manifold of Einsteins, the smooth bundles of Yang-Mills and so on.

A curve is smooth when you look at a small bit of it it looks almost straight - this is where linearity comes in. The generalisation to higher dimensions is almost obvious: The surface of a sphere is smooth when a small section of it looks almost flat, and so on. This is in fact how mathematicians & physicists think of smooth manifolds.

Until quite recently smoothness was seen as a structure additional to continuity. The Koch snowflake has a fractal boundary which is definitely not smooth. But recent work has shown that they can be conceptualised independently.

Also the category of smooth manifolds doesn't have good categorical properties, for example quotients usually give bad behaviour and pathologies. This has led to revisiting smoothness in terms of universal properties - and results in the quasi-topoi of diffeological manifolds - that is it has almost all the good properties of generalised set theories - aka topoi.

Another way to conceptualise smoothness is to take quite seriously the idea of linearity by positing axiomatically the microline - an infinitesimal line: That is it has no length and yet it is not the point but a line. To make this work one has to drop the law of the excluded middle and adopt intuitionistic logic.

One thinks of here of Deleuzes remark on the differential calculus that he made in difference and Repetition:

it is a mistake to tie the value of the symbol dx to the existence of infinitesimals; but it is also a mistake to refuse it any ontological or gnoseological value...A great deal of heart and a great deal of truly philosophical naivety is needed in order to take the symbol dx seriously...

Linearity & nonlinearity can interact in quite fruitful ways. For example one can have two functions f,g whose behaviour is fractal, but performing the sum f+g is quite-straightforward. This is how linearity enters QM, the wave-functions themselves can have, and will have, pace the simple examples of QM textbooks, exceedingly intricate behaviour.

Returning to Antiquity, one recalls that Aristotle argued against the atomists by arguing for infinite divisibility (this is consistent with his position on the infinite as being potential and not actual). One can say that he is arguing that the line cannot simply be made of points. It must have some kind of cohesion. At this point one can think of topology giving cohesion, or the microline, or Lawveres theory of cohesion in Topoi.

Smoothness generalises metaphysically into the idea of uniformity of nature. This is a background assumption made by scientists, as Virmaoir noted: Not only are manifolds smooth, they are uniform, since any little bit looks like any other little bit. Similarly one thinks that the little patch of space I'm sitting in is similar in all significant respects to some patch near the star Betelgeuse. This is an important distinction that divides Antiquity from today. Aristotle for example divided the universe into the celestial and terrestial, and it was Newton that brought them back together again.

Modelling nature is an essential part of physics. What it means to be a physical law and how accurately or how well it captures nature is a big subject in itself. The SEP carries a page on it. Induction is a crucial part of this, in fact its part of an induction and deduction dialectic. Hume quite famously pointed out that induction can have no logical validity, in the philosophical sense; it is obviously valid as it works - the question is to try to work out why. Hume himself pointed to the psychology of the human mind, and Kant probed further and put certain conceptualisations we must make to make experience even possible below the consciousness of the human mind, in fact below the unconscious, in Freuds taxonomy of the mind. He calls it the intuition.

Kuhn analysed change in scientific knowledge & culture in Marxist terms, calling the crucial event a paradigm change. In this evental history of science the originary event of science that the world is rationally explicable to some degree is associated with the rise of philosophic culture in the lands bordering Mediterrenean.

Its probably worth looking at larger questions connecting the social with the physical. Feyerabend, for example, a philosopher of science connected it to larger questions of politics. One might also note that we call it a law of nature. And laws of course were made by man to be enforced uniformly in a polis.

• Not exactly what I was looking for. But, in many ways much more than what I was looking for. What you helped clarify is that this smoothness (uniformity does not do it) was not new in the enlightenment, but may have been around for much longer. Enlightenment identified rules (formulae) for phenomena -- which might have been new -- but there is no reason to believe that earlier mode of thought expected chaos and turbulence to be the norm. I guess chaos (and fractals) are truly the new concepts. – AgilePro Aug 13 '15 at 23:33

I share an interest with you in this topic, but unfortunately I haven't been able to locate the kinds of resources on it one would expect.

The problem is that, as with most assumptions, no one notices the assumption of linearity until it is challenged. So your best bet is probably to look at references focused on nonlinearity to see how they reconceptualize "smoothness." Even with that approach, you'll need to keep in mind the fact that nonlinearity is still a new and underdeveloped discipline. Although I personally find it a very philosophically fertile topic, the philosophical literature on the subject by respected authors is almost nonexistent.

It looks like you're already familiar with chaos theory. James Gleick's "Chaos" remains a good general introduction to the field. You might also look into "complexity theory" and "catastrophe theory".

I wrote the following for an question that was deleted (author thought he was up to secretive stuff); it was an answer to what happens when you do things in smaller and smaller time-steps, or something like that. It might be helpful to you, at least for more search terms. To add one thing, it might be helpful to look for how philosophers grappled with quantization, for that might help you zero in on people who used to think about smoothness and had to change how they thought on specifically that matter.

Conservation laws require the increase of one thing to be matched by a decrease in the other. These trade-offs happen in time, and the more you zoom in on the time axis, the smaller and smaller you find the changes to be (at least up to some quantization limit). Ignoring quantization for a second, this is essentially what a delta-epislon proof is. So if physical reality has conservation laws and change happens on a very granular scale (much smaller than the human eye can see), calculus is a fantastic model for those changes. It's so great that the noise floor is often higher than the error caused by assuming continuity when there is quantization.

Conservation laws can be broken for extremely small time periods; see Heisenberg's Uncertainty Principle. We are used to the form:

∆x∆p ≥ ħ / 2

However, any two quantum operators which anti-commute have an uncertainty relation. For example:

∆E∆t ≥ ħ / 2

(Strictly speaking, there is no 'time' quantum operator, but the above may still be derived.)

What this tells us is that what seems like complete and total conservation is actually only conservation over sufficiently long time periods. In essence, things 'average out' to something well-described by calculus in the 'classical limit'. Feynman diagrams are another example of 'averaging out' craziness in a way that results in what we see. They involve summing all the different paths a subatomic particle can take, to get the 'average path'. That's not quite true, but a good enough approximation.

If you want to ask about what happens on extremely short timescales (like the Planck time), take a look at vacuum fluctuations, which are the result of exploring what that uncertainty principle between ∆E and ∆t actually means.