### Mathematical Robustness

#### Posted by David Corfield

In his paper The Ontology of Complex Systems William Wimsatt explains how he chooses to approach the issue of scientific realism with the concept of *robustness*.

Things are robust if they are accessible (detectable, measureable, derivable, defineable, produceable, or the like) in a variety of independent ways.

The robustness of Jupiter’s moons was precisely up for debate when Galileo let leading astronomers of his day look towards the planet through his telescope. Even if telescopes had proved their worth on Earth, allowing merchants to tell which ship was heading towards port beyond the range of the naked eye, this did not completely guarantee its accuracy as an astronomical instrument. How do we know that light travels and interacts with matter in the same way in the superlunary realm as down here on Earth? How could we trust this device when what appeared to the eye to be a single source of light was split in two in the telescope’s image?

Now we have robustness for the moons established, and can send probes close to their surfaces to report back on phenomena such as the Masubi Plume on Io. And we have an array of means to tell us that many stars are binary, so know that Galileo’s telescope was reliable.

Does anything like robustness happen in mathematics?

Well, let’s see what Michiel Hazewinkel has to say in his paper Niceness theorems:

It appears that many important mathematical objects (including counterexamples) are unreasonably nice, beautiful and elegant. They tend to have (many) more (nice) properties and extra bits of structure than one would a priori expect…

These ruminations started with the observation that it is difficult for, say, an arbitrary algebra to carry additional compatible structure. To do so it must be nice, i.e., as an algebra be regular (not in the technical sense of this word), homogeneous, everywhere the same, … . It is for instance very difficult to construct an object that has addition, multiplication and exponentiation, all compatible in the expected ways.

He lists five phenomena:

A. Objects with a great deal of compatible structure tend to have a nice regular underlying structure and/or additional nice properties: “Extra structure simplifies the underlying object”.

I suppose we saw this with our discussion of the real numbers as the unique irreducible locally compact topological group with no compact open subgroups.

B. Universal objects. That is mathematical objects which satisfy a universality property. They tend to have:

- a nice regular underlying structure
- additional universal properties (sometimes seemingly completely unrelated to the defining universal property)

Hazewinkel’s ‘star example’ is **Symm**, the ring of symmetric functions in an infinity of indeterminates.

Symmis an object with an enormous amount of compatible structure: Hopf algebra, inner product, selfdual (as a Hopf algebra), PSH, coring object in the category of rings, ring object in the category of corings (up to a little bit of unit trouble), Frobenius and Verschiebung endomorphisms, free algebra on the cofree coalgebra over Z (and the dual of this: cofree coalgebra over the free algebra on one element), several levels of lambda ring structure, … .The question arises which ones of these have natural interpretations in the other nine incarnations occurring in the diagram (and whether the isomorphisms indicated are the right ones for preserving these structures).

The last three phenomena are:

C. Nice objects tend to be large and inversely large objects of one kind or another tend to have additional nice properties. For instance, large projective modules are free.

D. Extremal objects tend to be nice and regular. (The symmetry of a problem tends to survive in its extremal solutions is one of the aspects of this phenomenon; even when (if properly looked at) there is bifurcation (symmetry breaking) going on.)

E. Uniqueness theorems and rigidity theorems often yield nice objects (and inversely). They tend to be unreasonably well behaved. I.e. if one asks for an object with such and such properties and the answer is unique the object involved tends to be very regular. This is not unrelated to D.

Might it be that all of A-E are ‘not unrelated’?

A key question is whether we should adopt the Zeilberger approach to these phenomena, invoking “our human predilection for *triviality*, or more politely, *simplicity*”, since “human research, by its very nature, is not very deep”. Or whether we take these phenomena in a Wimsattian way as robustness indicating reality.

## Pythagorus, Wigner, Tao; Re: Mathematical Robustness

Wigner said “reality” but wondered why. Pythagorus said so too, but we don’t trust cults as much these days.

Article in the New York Times, and maths education

Terry Tao:

“…surprisingly often the pursuit of one goal can lead to unexpected progress on other goals (cf. Wigner’s ‘unreasonable effectiveness of mathematics’). See also my article on ‘what is good mathematics?’.”