The n-Category Café

Perhaps he felt that if he’d accepted he’d have been obliged to receive it in Madrid and give a plenary talk. But then surely most people would have been happier had he accepted but made his excuses for Madrid. Perhaps any intrusion from outside his world, even recognition from afar, was to be blocked. I wonder how common it is for mathematicians to enter the mathematical world as a form of escape from the everyday world.

Altogether not very Platonic. After having the eye of your soul opened to the forms by studying mathematics for ten years, and a period studying dialectic, the philosopher king of The Republic takes on administrative duties, returning to the cave to deal with those still shackled there.

d wrote:

I love the MMORG analogy precisely because it makes clearly the arbitrary, socially constructed nature of rules like ‘success’, ‘worthy of a chair’, ‘worthy of publication’ and so on.

You might mean two things by this.

1. However the mathematical community organised itself, it would be arbitrary and socially constructed which pieces of research got recognised as important, and which people got elected to chairs.
2. The way the mathematical community is organised now is such that it is arbitrary and socially constructed which pieces of research get recognised as important, and which people get elected to chairs. But things need not be like this.

These are very different proposals. The first is more in accord with what I, after MacIntyre, have termed a ‘genealogical’ theory of enquiry. The second with what I have termed ‘traditional’.

David Corfield wrote:

If I could press you a little harder on this. Are you implying that:

1. No piece of mathematics is of any greater intrinsic value than any other piece.
2. Some pieces are of greater value, but however the mathematical community organises itself, it will not be able recognise this value, at greater than chance levels, ever.
3. Some pieces are of greater value, but however the mathematical community organises itself, it will not be able recognise this value, at greater than chance levels, in time to reward the relevant researchers, although it has a good idea of the intrinsic value of older mathematics.

Sorry, none of the above. I am implying that value is time and reader dependent, and that the tenure-weighted average across the community moves, too. I am not saying this is a good thing or a bad thing, just that like any semiotics, ‘mathematical value’ is context dependent. Just as relativising ‘proof’ to ‘proof within the internal language of the topos I happen to be working in today’ is liberating and useful, so relativising ‘value’ to ‘value within the society of mathematicians I respect’ is liberating too.

We might think that we can agree on general principles of value, and perhaps you and I can, but they would be hardly recognisable by mathematicans of 100 years ago, let alone 200. Not only do we not have a Platonic ideal of ‘value’, we don’t need one.

I’m slightly wary of following you down your riff. It’s not that I don’t find it interesting, but the topography of the philosophy of mathematics is such that within minutes of your uttering the word ‘proof’, you tumble headlong into a deep canyon, where the only sound you’ll hear are echoed voices chanting about the rational warrant for our belief in propositions of arithmetic.

Agreed, but don’t abandon your good empiricist tendencies. ‘Proof’, as Alice might say, having learnt her lesson, means what I want it to mean today. Or rather what the community wants it to mean. Just (in the bath, if need be - you needn’t write it down) speculate on what mathematical praxis would be like if ‘published proof’ meant something slightly different.

From virtual reality to neutrosophic rings in only 8 comments - we’re sinking fast!

I’d be willing to make a devil’s-advocate case that almost any mathematical entity, studied with sufficient persistence and cleverness, could rival modular forms in interest (though I don’t actually believe this).

But when I tried to learn what a neutrosophic ring is, so I could take a stab at this devil’s-advocacy, here’s the first thing I saw:

… a neutrosophic group in general does not have group structure. We also define yet another notion called pseudo neutrosophic groups which have group structure.

The authors then go on, not to define neutrosophic groups, but to define a specific example, the “neutrosophic group associated to a group”.

We see here then not a competing mathematical concept so much as a competing style of doing mathematics, which begins by breaking three currently widespread rules:

1. Unless there’s a darn good reason, an adjective should limit the scope of a term.
2. Exception to rule 1: the adjective “pseudo” can extend the scope of a term.
3. When introducing a new term, that term should be defined.

Most mathematicians (e.g. myself) will classify math that starts by breaks too many of such rules as “bad”, and lose interest in it and stop reading.

One can imagine a variant of d’s riff where we study how these rules were developed - and what math might be like if some of these rules were dropped, or others were added.

In particular, math does not always increase in rigor: when the Roman Empire took over Greece, we saw a massive decrease in rigor as people wrote simplified textbooks based on simplified textbooks based on the original Greek stuff. This may have led to a massive loss of mathematical knowledge. This could happen again.

I don’t think we’re on the brink of a severe decline in mathematical knowledge, but it’s worth noting that we always have to work to pass on good rules of thumb, a few of which I listed above. Making these rules more explicit and explaining their value might help.

As for David’s fear that talking about “proof” will make us

… tumble headlong into a deep canyon, where the only sound you’ll hear are echoed voices chanting about the rational warrant for our belief in propositions of arithmetic.

I don’t think we need to worry about that here! We’re not analytic philosophers, after all.

d was proposing a much more interesting exercise, namely to study the effects of different standards of proof on the growth of mathematics:

What if we valued intuitive but wooly ideas more than we do today? Or less?

For example; how should we respond to the wave of wooly but fascinating ideas entering mathematics from physics, like Feynman path integrals? Should we take a “buccaneering” attitude as Atiyah suggests, or a more cautious attitude, as Mac Lane would have us do? Are “physicist’s proofs” useful, or even a substitute for mathematician’s proofs - or are they just dangerous? These are questions real mathematicians are grappling with now, and the philosophers of real mathematics should be interested too.

In the Harris paper you mention, we read:

“The case is exceptional, however; “certifying knowledge,” in Rosental’s sense, is as such relatively unimportant to mathematicians, and I suspect Perelman’s close readers would describe what they are doing as attempting to understand his proof rather than “certifying” it as knowledge (for the sake of the community, or a generous benefactor, or philosophers or sociologists).”

This is a useful comment. For myself, when I read mathematics, I find myself trying to understand. A good deal of learning and of practising mathematics is involved in such experiences, it would seem. It would also seem that the process of understanding at least implicitly involves an understanding that a piece of mathematics is correct, or incorrect. After all, in the case of a new result, one needs to know whether it can be used.

So I don’t see that here there is rift between the experience of understanding and the experience of certifying knowledge, although the latter might be more restricted than the former.

David Corfield wrote:

We and our descendents may never achieve this, or may achieve it but not know we have achieved it, but still a notion of rational enquiry presupposes its possibility.

I’m not sure that particular diagram possesses a colimit, let alone an ω-filtered one. Or, to be less irritatingly arch, I’m not sure there is one telos. It is purely taste, but I find Deleuze’s* mathematics as a history of problematizations a much more satisfying ontology than thinking of it as a discourse refining some essential truths.

(* He wrote ‘knowledge’ rather than ‘mathematics’, but the sense is clear.)

One of the virtues required of the MacIntyrean practitioner is a readiness to acknowledge the current weaknesses or apparent resourcelessness of their mathematical position. You will sooner find whether you are on the right path this way.

Of course: this is clearly a good thing.

One good way of stepping in the Aristotelian direction would be to make available research proposals and referees comments. They may not all be of the quality of Grothendieck’s Esquisse, but their publication would be of tremendous value.

That’s a good idea. Or how about making all referee’s reports public, after the fact. Those shabby little supervisor reviewing student paper arrangements one sometimes comes across should be exposed…

David writes:

You can believe that what has been taken to be of central importance in mathematics has changed greatly over the centuries and will continue to do so, while still believing there is a something towards which mathematics is moving. For Aristotle, this something is a perfected understanding. We and our descendents may never achieve this, or may achieve it but not know we have achieved it, but still a notion of rational enquiry presupposes its possibility.

I don’t think the notion of rational understanding presupposes some concept of “perfected understanding”. On the contrary, I think the concept of a “perfected” understanding is a red herring that confuses discussions about rational enquiry. One can make perfect sense of the word “better” without there being a “best”. Or, in math jargon: you can have a partially ordered set without a maximal element.

So far, it seems to me, we’re able to continually improve our understanding of even the simplest issues without ever running into some final “perfect” understanding. I think it’s an empirical question, and an open one, whether we’ll ever decide we understand some things “perfectly”.

I think this “perfection” idea in Aristotle had to do with some distinctively Greek distaste for the apeiron - the unlimited. They didn’t seem to get the idea of partially ordered sets with infinite chains. If they had, Zeno’s paradoxes wouldn’t have seemed paradoxical. And the whole First Cause Argument, or “Cosmological Argument”, seems to be based on a distaste for infinite chains. As summarized at Wikipedia, it goes like this:

1. Every effect has a cause(s).
2. Nothing can cause itself.
3. A causal chain cannot be of infinite length.
4. Therefore, there must be a first cause; or, there must be something which is not an effect.

Which assumption seems the least self-evident? To me it’s number 3. But maybe Aristotle found it convincing.

Mathematical digression: you can easily turn the above argument into a proof that a partial ordered set with no infinite chains has a maximal element. But, there could be lots of maximal elements if the partially ordered set fails to be linearly ordered - as indeed causal relations typically do. So, this argument for god doesn’t rule out polytheism!

Similarly, as d points out, it doesn’t rule out the possibility of lots of distinct perfected understandings.

Another funny thing: in a linearly ordered cosmology, the “first cause” comes before all the rest, while the “perfected understanding” comes after all the rest. There’s some sort of duality between “efficient cause” and “final cause” (telos) going on here.

But now I’m getting a bit slap-happy. I don’t actually think we can get far in philosophy by reducing everything to partially ordered sets, though I think they’re a handy way to make precise certain naive assumptions. My serious point is just that we can - and often should - talk about improvement without the concept of perfection.