A question for real mathematicians out there (or at least the math rigor curious): Do folks in #math maintain consistent distinctions between the meaning of the terms "outer product" and "tensor product" (and for bonus points throw in "Kronecker product")?
I learned these concepts mostly from physicists (which is a bit like learning manners from being raised by wolves), and there was a tendency not to use consistent terminology or draw clear distinctions, though sometimes they were being used to refer to slightly different, but related, things. I could generally follow the sense in which terms were being used in a given application by context, so I didn't worry about it too much. A cursory look online also suggests that usage is heterogeneous, but I'm curious if mathematicians are, in fact, a bit more consistent.
I was interested in the NewsFlash #RSS reader, until I saw that installing the FlatPack required 3.6 GB, whereas installing Liferea required about 3 MB. I assume this is a #FlatPack issue rather than NewsFlash being bloated. It was funny, since an article mentioned their slimmed down code base. In related news, if people have RSS feed readers they really like using on #Linux I'm open to suggestions. Some way of syncing across devices (including Android) would be a plus.
I'm sad to discover that John Clauser is a climate crackpot and continuing the unfortunate tradition of physicists wading into subject areas they don't know much about, assuming they understand them, and making a fool of themselves.
In this thread with @johncarlosbaez I was mentioning how when I first encountered Category Theory it seemed like little more than a curiosity (for my purposes, as a physicist), even though mathematicians seemed excited about it.
I had almost the opposite experience with Nonstandard Analysis (i.e. the hyperreal numbers), in the sense that I bumped into this notion, read a bit about it, and it sounded potentially quite useful. Physicists tend to talk in terms of infinitesimals anyway, so a framework where that could be done rigorously seemed useful, and I was curious if it might provide nice ways to think about other things such a path integrals or even renormalization. But the only mathematician I talked to about it dismissed it as basically a curiosity. I believe the way he put it was that it was "just a trick to avoid an extra quantifier in in proofs."
@internic - A few people have tried to use nonstandard analysis to prove theorems in analysis that haven't succumbed yet to traditional techniques. I've mainly seen this in attempts to make interracting quantum field theories rigorous - this subject is full of unsolved problems in analysis. But none of the nonstandard attempts have made much progress. And that makes sense to me, since I don't see how nonstandard analysis would help much here.
As one with a physics background who really got into both nonstandard analysis and category theory, I think the views you describe (category theory is wildly useful and NSA is a curiosity) are predominant from what I’ve seen. However, I have some fairly uncommon views that might be interesting as to why this is and how I interpret the situation.
Over the years, I have grown some strong ultrafinitist tendencies. Since we can only make finite distinctions / measurements in finite time, we can never validate infinite models. Models with continuous elements (like spacetime) are only ever “computationally useful” and not indicative of any revealed truth of verifiable / refutable “reality”. There are always many finite models available to fit any observations we may ever collect.
Because of this, I actually don’t see the bias against NSA as having any merit. Sure it’s a different model of number than the standard reals - but having many models to choose from isn’t an argument against any of them. We will never be able to choose between any of them. But we may grow fruitful ideas by knowing and considering multiple models, where a focus on a single model may cause ideas to stagnate.
And this idea of model pluralism is important in physics generally. There is a tendency in physics to want a single “right” model, and this often turns to mockery and other negative behaviors towards alternate-yet-entirely-isomorphic models. This is huge in quantum mechanics, for instance. There is a long history of proposals like Bohmian mechanics (which is just a rewrite of the evolution equation into polar form and separation into real and imaginary components to reveal a kinetic equation for worldlines), or Many Worlds (simply a reevaluation of the complex wave function into Kripke frames of possible worlds modality), or Consistent Histories, or… All are easily provable to have the exact same predictions as standard interpretations (because they are fundamentally the same equations), but have been met with horrifying abuse over the years. What happened to Everett was utterly cruel.
NSA is “just another model”, but that’s a good thing. It can offer ideas on how to formulate new theories of space and, at the same time, offer illustrative examples of where we could use some model tolerance and be less abusive in our interactions.
As for category theory, it’s similar, but there is an angle. Yes, algebra can capture all the same relationships described by categories, commutative diagrams, etc. But the key usefulness is the intensional definition behind the formalism. When you talk about a certain set of relationships, you are talking about all things that obey them. You aren’t just grabbing an object and proving things about it, you are grabbing definitions and proving things about all things that obey them. And you can move out and look at fragments of definitions and these are obeyed by larger sets. And so theorems grow in usefulness maximally, and you are led to find greater abstractions that capture the essence…
Of course, you can do that without category theory, and people have. But it was really useful to have a framework that built it in. Much of math before category theory was done extensionally, looking at specific objects like sets and building specific structures and concretizations. So there was a lot of duplication, and although much of it was acknowledged metaphorically, some was missed. The discipline of intensional definition is why so many find category theory useful. Again, though, some model pluralism is always healthy.