@boarders@mathstodon.xyz
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boarders

@boarders@mathstodon.xyz

Interested in mathematics (homotopy theory, category theory, topos theory), programming languages and philosophy

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carapace, to random
@carapace@mastodon.social avatar

1/7

I love science, so it's strange and frustrating that healing modalities with which I have personal experience (including Reiki, Feldenkrais, NLP, EFT, hypnosis, and others) are "unreproducible" by mainstream scientists, to put it mildly.

boarders, (edited )
@boarders@mathstodon.xyz avatar

@carapace I do think science, especially as typified by “double blind studies” and in the realm of medical knowledge, can be a flawed methodology (the biggest cause of mental health is poverty, but that is too messy to causal stories about brain chemistry), however I don’t take much credence from someone saying they know something from personal experience (since people say this about astrology or other gibberish) - we are easily able to fool ourselves with fake causality and randomness (for instance, see all of the people telling you that CEOs wake up early so it must lead to success)

That said, I think it is also the case that some things are simply not easily subject to quantitative measure - how could we scientifically measure the impact of reading Dickens, perhaps one would do better on a reading comprehension exam, but one would hope what a person gets from it is far beyond that

antidote, to random
@antidote@mathstodon.xyz avatar

Everything I do academically is some combination of:

  • tiny to the point of unpublishability;
  • for someone else and not relevant to anything I’m doing;
  • wholly incorrect;
  • treading on someone’s toes.

Is it time to throw in the towel?

boarders,
@boarders@mathstodon.xyz avatar

@antidote [not in a place to offer advice as I quit academia after not getting far] a PhD, postdoc, professorship etc. is not worth hating yourself over, it’s the number one truth of academia that no one will say - none of this is worth despising who you are or what you have to offer

Moreover, the important thing to do now is to cultivate your own taste and your own sense of what is valuable - start writing up the “unpublishable results” and I bet you’ll see how once you flesh everything out and expand all the natural examples explaining what you have, it becomes a fully publishable result and one you can be happy with

MartinEscardo, to random
@MartinEscardo@mathstodon.xyz avatar

It is just me? The following definition of category hurts my categorical instincts, because it uses object equality.

A category consists of

  1. A collection of objects.

  2. A collection of morphisms.

  3. Each morphism f has two assigned objects, its source s(f) and its target t(f).

  4. For each pair of morphisms f,g such that t(f)=s(g) there exists a specified morphism g ∘ f such that [it doesn't matter what]

  5. [Some axioms are satisfied.]

It is (4) that hurts my categorical instincts.

There is no reason to have "evilness" (in the categorical sense, rather than the emotional sense) built-in in the definition of category!

This definition is, for example, adopted by Freyd.

boarders,
@boarders@mathstodon.xyz avatar

@MartinEscardo this is a definition of categories as monads in the bicategory of spans using some specific choice of construction of pullback in Set, but it is naturally a weak 2-category - so we should state it for pullbacks, but then we can’t even state the associativity law as it is a naturally a dependent equality (unless we use some kind of unbiased pullbacks)

jonmsterling, to random
@jonmsterling@mathstodon.xyz avatar

Thinking about how to teach students about isomorphisms (between sets).

It is easy to teach them about bijection (injective + surjective): you can give an example about checking that two piles are equinumerous, and walk through how you want to match each widget from one pile with one from the other without leaving any left over.

But isomorphism (in the sense of left and right inverse) is somehow much more abstract and hard to think about for someone who doesn't already get it.

Have any of you got a successful way to teach this concept without first introducing bijection?

boarders,
@boarders@mathstodon.xyz avatar

@jonmsterling some ideas but not fleshed out to usable slogans:

  • draw a function via its cograph, that is the formalization of “potato diagrams”, where we have a set A on the left, a set B on the right and an edge from any element a to where it is sent. From this we can take the mirror image diagram (reverse all edges) and ask if this is a function - the inverse function

  • draw the graph of a function and consider the reflection in the diagonal (lots of people will be used to this idea as log is the “mirror image” relation of exp or sqrt and square functions etc.) - this relation may also be functional, and is the inverse

  • more speculative: the ur-family of isomorphisms arise from the first isomorphism theorem of sets, a set quotiented by the kernel pair equiv relation is iso to the image is iso to the equalizer of the cokernel pair. I feel like there must be some way to make this idea palatable to get a deeper perspective e.g. this describes the iso between Z/nZ and [n], but this is notoriously hard to get across in an intro course

boarders, to random
@boarders@mathstodon.xyz avatar

Moderately interesting (to me!) that in mathematics a magma is basically a useless idea that you wouldn’t even define in an undergraduate degree, but the typed analog - an applicative structure - is a very useful starting point for realizability, semantics of type theory / HOL etc.

j2kun, to random
@j2kun@mathstodon.xyz avatar

Now that I have a static site with hugo, I wonder if people would like reading shorter-form thoughts, and they can be cordoned off into their own part of the site and/or their own RSS feed.

I feel like there are a lot of small mathematical tidbits I learn about that I don't have the time/energy to put into a proper article, but are still worth spreading.

boarders,
@boarders@mathstodon.xyz avatar

@j2kun please do, often at work I have a short period of time (compiling, waiting for CI etc.) in which it is great to have a supply of shorter form thoughts that are not social media or news (both of which I’d often prefer to avoid viewing in an unrestrained way) - this probably used to be fulfilled by webcomics

jonmsterling, to random
@jonmsterling@mathstodon.xyz avatar

Ultimately we have to ask ourselves if it was overall a good thing that computer science as a discipline ceased to be part of mathematics — rather than broadening the horizons of mathematics and bridging the gap between mathematics and social science. I am not speaking purely rhetorically, as there are legitimate arguments to be made on both sides.

boarders,
@boarders@mathstodon.xyz avatar

@antoinechambertloir @jonmsterling
Can't resist this immortal Girard quote:
"Theoretical Computing is not yet a science. Many basic concepts have not been clarified, and current work in the area obeys a kind of “wedding cake” paradigm: for instance language design is reminiscent of Ptolomeic astronomy — forever in need of further corrections. There are, however, some limited topics such as complexity theory and denotational semantics which are relatively free from this criticism.

In such a situation, methodological remarks are extremely important, since we have to see methodology as strategy and concrete results as of a tactical nature."

boarders,
@boarders@mathstodon.xyz avatar

@sprout @antoinechambertloir @jonmsterling I can’t say it looks from the outside like computer science is drowning in mathematicians or mathematical input. More generally, no field has ever done well out of limiting its mathematical analysis. Einstein famously came to accept this fact as a result of Weyl, and it led to our best formulation of general relativity as being about free particles following geodesics on a Lorentzian manifold which we could now teach to undergrads if we had professors that weren’t scared of mathematics

boarders,
@boarders@mathstodon.xyz avatar

@sprout @antoinechambertloir @jonmsterling John Von Neumann, Alan Turing, and Alonzo Church - three mathematicians - created computer science, maybe cool it a little on who is worthy

boarders, to random
@boarders@mathstodon.xyz avatar

So many aspects of current culture are simply different versions of: “lottery winner endorses lottery as path to success”

julesh, to random
@julesh@mathstodon.xyz avatar

Please can somebody inform MS Word that the accusative form of "who" is "who" in modern English, it's not 1924 anymore and I'm not writing in German

boarders,
@boarders@mathstodon.xyz avatar

@julesh @jer_gib also why stop here? why not use ‘thou’ and ‘thee’ if we don’t care about language as speakers actually speak it, but merely about pretending modern English is Latin

julesh, to random
@julesh@mathstodon.xyz avatar

Someone should write "mathematics for the working category theorist", to teach a bit of algebraic topology to those of us who started out in functional programming

boarders, (edited )
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@julesh Peter May already wrote two volumes of this book (‘a concise course in algebraic topology’ and ‘more concise algebraic topology’)

boarders,
@boarders@mathstodon.xyz avatar

@brokenix @julesh use a search engine

counting_is_hard, to random
@counting_is_hard@mathstodon.xyz avatar

Time to write the code: 5 minutes.
Time to figure out the types: 50 minutes.

boarders,
@boarders@mathstodon.xyz avatar

@counting_is_hard Python?

boarders,
@boarders@mathstodon.xyz avatar

@counting_is_hard makes sense. My python experience is indeed:
Time to write code: 5 mins
Time to add types to code: 50 minutes
Time to get out that last buggy behaviour: tbd

Joemoeller, to random
@Joemoeller@mathstodon.xyz avatar

This summer I'm starting a postdoc at CalTech! I'm working with the roboticist/control theorist Aaron Ames on using category theory to study stability of dynamical systems.

I got verbal confirmation a month ago, but I've been holding back on saying anything until I got bureaucratic confirmation.

boarders,
@boarders@mathstodon.xyz avatar

@Joemoeller congrats, I hope it’ll allow for writing “control theory for category theorists” instead of the converse which is more usual :)

MartinEscardo, to random
@MartinEscardo@mathstodon.xyz avatar

What is a topological space?

It is a mathematical device to define what a continuous function is, in a general setting.

  1. A topological space is a set X together with a collection of subsets of X, called open, such that finite intersections of open sets are open, and arbitrary unions of open sets are open.

  2. A function of topological spaces is continuous if inverse images of open sets are open.

What is the intuition behind (1) and (2)?

I claim that it is better to ask, instead, how mathematicians came up with (1) and (2).

1/

boarders,
@boarders@mathstodon.xyz avatar

@joshuagrochow @MartinEscardo I believe Serre noticed something like the fact that G bundles are not classified by cohomology classes, but they are if you pull back by étale maps instead

counting_is_hard, to random
@counting_is_hard@mathstodon.xyz avatar

In his famous paper, "on proof and progress in mathematics", Thurston lists 8 (and implies 29 other) ways to think of the derivative.

I was bored waiting for a bus, so I tried listing the different ways I could think of what a category is (see below). Please feel free to help me add more!

A Category is...

  1. The usual definition (omitted for space)
  2. an abstract theory of functions / arrows (or as Awodey would say "archery")
  3. a monoidoid
  4. a poset with evidence (wording stolen from Alex Kavvos)
  5. a set-enriched category
  6. an object in CAT
  7. a syntax for a programming language
  8. a maze of twisted arrows all alike
  9. a "path-complete" digraph (if there is a path x -> y there is an edge x -> y)
  10. a multicategory where every arrow has arity 1
  11. a polynomial comonad (spivak et al)

boarders, (edited )
@boarders@mathstodon.xyz avatar

@counting_is_hard
3.’. A typed monoid
12. An associative, unital deduction system (Scott, Lambek)
13. A simplicial set satisfying the segal condition
14. Monads in the category of spans
15. Universes of mathematical structures
16. A place to do semantics
17. A model of a bisorted first order theory

typeswitch, to random
@typeswitch@gamedev.lgbt avatar

the unlived life is not worth examining

boarders,
@boarders@mathstodon.xyz avatar

@typeswitch [modal logicians start typing furiously in some possible worlds]

boarders, to random
@boarders@mathstodon.xyz avatar

Applied category theory would probably be significantly better if it gave more actual examples of “compositionality” and the main proponents of it sounded less like people trying to get know-nothing VC funding - the main exception I know is Tai Danae Bradley, who has a deep appreciation for CT as a unifying concept

ColinTheMathmo, to random
@ColinTheMathmo@mathstodon.xyz avatar

Asking for a friend(*) ...

"I need to brush up on some Pure Maths for a thing, and I find I'm pretty rusty. Can any of m'Maths friends recommend a good book on rings and ideals, and that end of algebra?"

(*) No, really ...

boarders,
@boarders@mathstodon.xyz avatar

@ColinTheMathmo I strongly recommend Aluffi’s book Algebra: notes from the underground

jonmsterling, to random
@jonmsterling@mathstodon.xyz avatar

At some point, at the risk of becoming a personal knowledge management cult leader, I really need to explain how I use my forest 🌳 to manage my life… It is a life saver.

boarders,
@boarders@mathstodon.xyz avatar

@jonmsterling tools for JST

cahollenbeck, to random
@cahollenbeck@mastodon.scot avatar

I have once again had an anxiety attack in the middle of a coding interview for a company which was otherwise "very impressed" with my CV and seemed eager to give me an offer.

If the job is coding against a timer with someone staring at every keystroke I make, then yes, don't hire me. I'd be trash at it.

There has to be a better way to hire.

boarders, (edited )
@boarders@mathstodon.xyz avatar

@cahollenbeck there is a story about someone assuming the philosopher Hilary Putnam (arguably the only philosopher to be involved in the solution of one of Hilbert’s problems!) was a mathematics major, to which he said something of the form “oh god no, mathematics isn’t a race, you can’t do mathematics under time pressure - I found the idea of mathematics exams repugnant”

mjd, to random
@mjd@mathstodon.xyz avatar

New logic post on my blog: “Well, I guess I believe everything now! ”

https://blog.plover.com/math/logic/k2.html

boarders,
@boarders@mathstodon.xyz avatar

@roboguy @mjd I agree - the argument says B(exists p, Bp /\ ~p) for the belief modality, but how do we conclude exists p, ~B p, unless we know that belief commutes with connectives

brokenix, to random
@brokenix@emacs.ch avatar

> you mostly model data with sum types, which in my mind are the best way to model data

True its quite strict in Haskell though
https://blog.darklang.com/leaving-ocaml/

boarders,
@boarders@mathstodon.xyz avatar

@oantolin @brokenix they are not meaning sum types as opposed to product types but just commenting on how most languages don’t have sum types so you have to use some other way to encode data with tags or class hierarchies or etc.

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