TobyBartels

Probably the only anglophone Toby Bartels. They/them. Mathematics instructor at a community college in Nebraska. Tolstoyan ex-Catholic. 🏴🚩🏳️‍🌈🏳️‍⚧️

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nprofile1qqstq56fmhexfgjypl3g389g9qxz9xsks6pxlhfl4fswfwyy7ywdmyspz3mhxue69uhhyetvv9ujuerpd46hxtnfduqs6amnwvaz7tmwdaejumr0dsjm5u9d

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Last Notes

2026-03-03 19:26:19 UTC

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@nprofile…ugpy
That's because bad cases don't necessarily make bad law; as you said before, it's *hard* cases that make bad law. And this case was easy!

2025-12-25 15:22:24 UTC

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@nprofile…8eua @nprofile…mdgu @nprofile…sn9y
I was taught to call it the minor second, not the diminished second. The Wikipedia page that you linked agrees with me, calling the diminished second ‘enharmonically equivalent to a perfect unison’ in equal temperament.
But the question isn't (or at least shouldn't be) whether ‘major’ and ‘minor’ are good terms for these two modes once you've picked them out, but why the two terms should be used for *those* two modes. The truly minor mode is phrygian.

2025-12-24 17:54:50 UTC

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@nprofile…u6hy
Is this a way of saying that the Sun was out for only half its average amount of time today?

2025-12-24 16:48:43 UTC

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@nprofile…8eua
In the picture, you say that Boethius used the letters A through O for the notes that a theoretical male singer could sing. Since I and J were not distinguished in his time, this is 14 letters, making two octaves. Since we didn't repeat letters, it seems strange to stop at the high G and not continue with an extra high A to finish off the scale, but perhaps he thought of scales differently (or just did not have faith in his male singers).

2025-12-10 09:52:47 UTC

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@nprofile…8eua @nprofile…stct @nprofile…9s72
Y'all need to read Curtain, the last Hercule Poirot story (not the last to be written by any means, but the one written to be the last, and published on Agatha Christie's death}. Poirot remarks that it's very suspicious if one is around a lot of murders (without an excuse like being a detective), but (without spoiling the ending) it goes a great deal further than that.

2025-12-09 19:52:27 UTC

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@nprofile…u5vz @nprofile…ztfm
Codidact and Physics Overflow, according to https://meta.mathoverflow.net/questions/4431/is-there-a-software-solution-if-we-do-cut-the-cord-from-se

2025-12-09 05:31:32 UTC

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@nprofile…ztfm
MathOverflow needs to move. Others have split from the StackExchange network before.

2025-11-28 04:08:12 UTC

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@nprofile…pklh
Here's a takedown of the entire article (5 points, only #1 is about the graphic): https://pubpeer.com/publications/E1A0A9AC672336E92673154575091E

2025-11-02 02:32:55 UTC

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@nprofile…ztfm
Why nuclear?

2025-10-31 18:08:58 UTC

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@nprofile…4x5t @nprofile…8eua @nprofile…y0ah
Wow, so not only do we only need binary functions; we only need unary functions (which may be treated as degenerate binary functions) and addition! Sort of like how you can multiply numbers by taking their logarithms, adding these, and then exponentiating.

2025-10-29 20:44:58 UTC

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@nprofile…ztfm
Aren't these union jobs? How does the math department have the authority to unilaterally cut them?

2025-10-27 12:57:27 UTC

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@nprofile…cs78 @nprofile…u6hy
Congratulations, Christian!

2025-10-05 12:58:07 UTC

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@nprofile…wn6v
Somehow the public health museum being underfunded seems appropriate.

2025-10-02 00:06:16 UTC

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@nprofile…5lc3
But is Keith's second reply also AI?

2025-09-30 13:13:14 UTC

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@nprofile…fa4k
I understand how the two valences of iron give rise to the two different colours (dull brown and bright red) of blood. But then you say that when the two are in close proximity, this can look blue, and I'm half expecting you to say that this is why veins look blue through the skin. But that seems like a little too much; it's just a coincidence, right?

2025-06-14 22:43:15 UTC

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@nprofile…6kk0
It's confusing because you have to assume that the students have already been confused in an earlier course.
But honestly, I think it really is confusing. People were doing arithmetic for millennia before Brahmagupta started doing it with 0. Aristotle mishandled syllogisms with uninstantiated terms, and this wasn't straightened out until the 19th century (and arguably still hasn't been). Large ordinal numerals in natural languages across the globe are adapted from cardinal numerals with a fencepost error. Some of this is the dead hand of tradition and authority preserving one initial mistake, but it wouldn't pop up over and over again and last so long if it wasn't legitimately harder even for some very smart people.

2025-06-14 20:36:53 UTC

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@nprofile…hszg @nprofile…6cfc
I can understand treating 0 separately if you're working with students who don't know what a permutation is and you don't want to get into that or into a discussion of what it means to multiply an arbitrary finite list of numbers, so you define n! using n−1 binary multiplications. Then that doesn't handle 0!, so it's a special case. Still, ‘by convention’ can make students (especially if they're not used to it in a math context) think that it's an arbitrary choice, so you still need to give them a reason (say by noticing that (n−1)!=n!/n for n>1 and considering how far down that can go).

2025-06-14 19:25:41 UTC

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@nprofile…6kk0
Since 0 can be confusing, it's good to point it out as a special case. This intuition among authors is sound. But it should be a theorem[*], not a definition.
[*] Probably not a numbered Theorem, just a Remark or an Example. But logically a theorem.

2025-06-14 17:21:04 UTC

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@nprofile…hszg @nprofile…6cfc
So exp(0) is necessarily undefined?
There's no ‘necessarily’ about it. You can have a subtle situation where 0⁰ is defined in some contexts and undefined in others; lots of notation is like that. But it's not necessary. It's not necessary that operations be continuous. For that matter, it's not necessary that power series be written to begin at 0; you could begin them at 1 and write the order-0 term separately.
In any case, Oscar is correct that there are textbooks out there that firstly state that 0⁰ is undefined and also write power series beginning at 0. I'm teaching from one right now. (They actually take care to write the order-0 term separately when they first introduce the concept, but later they forget.)

2025-05-28 01:36:02 UTC

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@nprofile…fynq
Thanks for the gift link! Can I share it on other platforms, or does that go too far?

2025-05-23 01:37:42 UTC

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@nprofile…59k6
I think that's □ (Halmos box).

2025-05-13 11:02:57 UTC

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@nprofile…yu7x
1845 is too early for Doyle, he did publish his own medical research 3–4 decades later, before turning to fiction.

2025-05-02 21:22:11 UTC

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@nprofile…qz4x
Doing an x-ray scan but the reflective clothing scatters the x-rays?

2025-05-02 20:26:59 UTC

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@nprofile…59k6
Is this anything special about the number 15? I notice that 15 is the largest 4-bit integer and so might in some way have the strongest signal, and yet if subtracting 15 is something that happens in the program then you can't possibly be working with only 4-bit integers even at that level of instructions, so maybe it's just a coincidence.

2025-05-01 04:21:16 UTC

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@nprofile…fa4k you might want to make a note of this.

2025-04-26 05:39:55 UTC

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@nprofile…fa4k @nprofile…2pts
I think Turdus maximus might be the winner for unintentionally great names. (It wasn't even classified in the same genus when the specific epithet was chosen, so nobody planned this.)

2025-04-25 00:06:46 UTC

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@nprofile…yu7x @nprofile…fa4k
Nice! Although to square the circle, you need a segment of length √π, not length π. (Both are impossible in finite time with straightedge and compass, since π is transcendental.)

2025-04-23 00:03:54 UTC

@nprofile…qz4x : I feel like you might have something to say about this (sorry if you did and I missed it).
https://media.mathstodon.xyz/media_attachments/files/114/384/339/573/668/271/original/2819a9aa89c07fb4.jpeg

2025-04-18 13:05:17 UTC

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@nprofile…qz4x
It's petty of me, but you've done this twice now, so: it's ‘complEments’ (pairs well with), not ‘complIments’ (says nice things about).

2025-04-15 22:17:58 UTC

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@nprofile…5k3r @nprofile…59k6
OK, they say that Cartan didn't have *any* intrinsic definition, but embedded his manifold in a larger cartesian space (which presumably is where the name ‘tangent vector’ comes from, as the vector is literally tangent to the embedded manifold). But they give no credit for the definition of vector as derivation.
And they're not exactly trying to define the tangent space here. Rather, they're starting with a curve through a point and trying to define the derivative. There seems to be no indication that the vector could be defined as an equivalence class of curves itself! In terms of Dan's sibling comment https://mathstodon.xyz/@dpiponi/114337326898112325/, they're identifying B as differential operators but still not figuring out *which* differential operators arise from differentiating curves. (Although they perhaps they do that later on.)

2025-04-14 18:34:30 UTC

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@nprofile…59k6 @nprofile…5k3r
Yes, this equivalence class is another good characterization, although notice that it gets you vectors (at the given point) rather than vector fields (not that it's hard to move between these). But there are a lot of things you could use for B here if you just want to characterize the equivalence; that is, there are a lot of choices for B and operators from A to B that have the desired equivalence kernel. What we really need is a B where the operator from A to B is surjective. And it's natural first to think of a larger B and then find a way to characterize the image. I suggest that it would be natural to think of differential operators when constructing this larger B, later restricting to derivations.

2025-04-14 16:42:22 UTC

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@nprofile…5k3r @nprofile…59k6
That's who I thought of, but I don't know.
It probably passed through the idea of differential operator. In local coordinates, a vector field (X¹,…,Xⁿ) defines a differential operator ΣᵢXⁱ∂ᵢ (where ∂ᵢ = ∂/∂xⁱ), which will get attention because it can be used in differential equations, and it doesn't take much to notice that the vector field can recovered from the operator (and even from its action only on infinitely differential functions if it occurs to anyone to consider that); but it could have been a while until someone worked out a coordinate-free description of which operators come from vector fields in this way.
This could be in Michael Spivak's Comprehensive Introduction to Differential Geometry, which has a lot of historical material.

2025-04-11 21:46:03 UTC

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@nprofile…kkxq
I'd say that this must be some kind of hilarious mistranslation, but then I saw who wrote it.

2025-04-10 02:26:00 UTC

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@nprofile…fa4k
That's just straight-up corrupt, and yet compared to everything else, it barely deserves a mention.

2025-04-10 01:27:07 UTC

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@nprofile…pmdx @nprofile…qmju
That last one at least has a reasonable explanation. The ε here is supposed to be elasticity of demand (the derivative of the logarithm of quantity demanded with respect to the logarithm of price, that is p/q⋅dq/dp), and since this is (as I have defined it) almost always negative, both this and its opposite are called ‘elasticity’ in practice. (In the applied calculus textbook that my school uses, they actually define elasticity as the absolute value, but I tell my students that this is a bad idea since elasticity of demand is not *always* negative (and elasticity of supply is not always positive); it's reasonable to swap the sign for demand so that it will usually be positive, but the sign still gives you information.) So while it's careless, it's not unheard of to say first that elasticity of demand is negative and later that it's 4 (when you really mean that it's −4).
What is unreasonable is to just make up a number out of whole cloth and pretend that it applies in every situation, just to balance out some other number that you made up out of whole cloth, all in a vain effort to retroactively justify your bullshit calculation.

2025-03-13 22:04:11 UTC

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@nprofile…7cw4 @nprofile…qz4x
‘Fascism is nothing but colonialism applied at home’ —attributed to Aimé Césaire (Martinique anti-colonial politician)

2025-03-11 05:43:41 UTC

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@nprofile…fa4k
It's an easy place to make them!

2025-03-11 05:43:10 UTC

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@nprofile…fa4k
If it actually did what it said was going to do, it could have solved the problem. There is (up to equivalence) only one monoidal category V such that the category of truth values is equivalent to the category of V-enriched categories, and that's the answer. But of course it doesn't know how to reason through that.

2025-03-11 04:13:59 UTC

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@nprofile…fa4k
But its conclusion was wrong (this is all correct until the ‘solution’, which repeats the definition of a (−1)-category instead):
https://media.mathstodon.xyz/media_attachments/files/114/141/841/412/029/851/original/ce9dc5f0ade0e96d.jpg

2025-03-11 04:10:08 UTC

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@nprofile…fa4k
My attempt started off better (this is all correct):
https://media.mathstodon.xyz/media_attachments/files/114/141/830/692/849/527/original/935165f126b17a42.jpg

2025-03-11 04:04:10 UTC

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@nprofile…fa4k @nprofile…hszg
Identity 2-morphisms?

2025-02-26 05:05:13 UTC

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@nprofile…yu7x @nprofile…fa4k
Not remotely interested, and yet I think that you took more care to make it believable and workable than they ever did. Of course it's always easier in fiction than in real life.

2025-02-26 04:42:06 UTC

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@nprofile…yu7x @nprofile…fa4k
They didn't do nearly as well as Stateless did in Distress!

2025-02-20 17:04:10 UTC

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@nprofile…fa4k @nprofile…m7f4 @nprofile…59k6
How do you get Mathstodon to do that? That's not the usual LaTeX support: \( T _ { \mathrm { i j } } \).

2025-02-20 16:26:05 UTC

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@nprofile…fa4k @nprofile…m7f4 @nprofile…59k6
For scripts, I like to copy paste the characters from the Wikipedia article on the Unicode characters https://en.wikipedia.org/wiki/Unicode_subscripts_and_superscripts . For Unicode characters more generally, if you know the name of the character then you can search for that and find web pages like https://www.compart.com/en/unicode/U+2084/.

2025-02-12 06:03:54 UTC

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@nprofile…yu7x
Apparently Google is doing that everywhere in the world except Mexico and the USA. Too insulting to do it in Mexico I guess, and full capitulation in the USA. https://blog.google/products/maps/united-states-geographic-name-change-feb-2025/

2025-02-02 02:01:36 UTC

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@nprofile…59k6 : Trying to get around Trump's executive orders, I see!

2025-02-02 01:25:36 UTC

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@nprofile…59k6 : Those genres weren't just created by Americans. Every one of them started with Black Americans.

2025-01-26 02:46:19 UTC

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@nprofile…fa4k : Re: ‘Protect the decent and hardworking members of your communities who are undocumented or whose parents are undocumented’
Protect people who are undocumented or have undocumented parents regardless of whether they're decent or hard-working. We don't need to waste time or weaken precedent deciding which are ‘the good ones’.

2025-01-24 01:09:25 UTC

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@nprofile…fa4k @nprofile…8kae : Since I didn't know that verb, I gave your sentence to Google Translate, and it even translated the noun that way. (‘The pukes make me puke’, which is certainly a thing that can happen.)

2025-01-10 18:17:14 UTC

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@nprofile…ucsf : I'm not going to understand what they mean until they convert it to cubic football fields.

2025-01-01 01:02:48 UTC

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@nprofile…fa4k : I think that I fixed the echo, and also completed a sentence (the last one before the last parenthesis) that didn't quite contain all that I'd intended.

2024-12-31 23:19:50 UTC

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@nprofile…fa4k : Seeing Planck's constant here reminds me that this was originally a course-graining parameter in statistical physics that was meant to go to zero, but Planck discovered that it doesn't quite go to zero (and classical statistical physics would lead to infinities if it did). Quantum gravity renormalization might have a similar cutoff. (Naively, thi@johncarlosbaez : Seeing Planck's constant here reminds me that this was originally a course-graining parameter in statistical physics that was meant to go to zero, but Planck discovered that it doesn't quite go to zero (and continuum statistical physics would lead to infinities if it did). Quantum gravity renormalization might have a similar cutoff. (Naively, this suggests a discrete space-time at the Planck scale, but of course that's hard to make Lorentz-invariant, as people have been discussing elsewhere in this thread.)s suggests a discrete space-time at the Planck scale, but of course that's hard to make Lorentz-invariant, as people have been discussing elsewhere in this thread.)

2024-12-28 15:55:37 UTC

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@nprofile…mqrg @nprofile…fa4k : If I had a kind of shape named after me, then I'd seriously consider it for anything I could as well! (Joke since presumably he didn't call them that. He probably just called them ‘ovals’.)

2024-11-25 15:56:54 UTC

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@nprofile…e5g0 @nprofile…fa4k @nprofile…yu7x @nprofile…j7pr :
>A free falling observer is in an accelerating frame,
This is the big difference between how Newtonian mechanics treats gravity and how General Relativity treats it: a free-falling observer is *not* in an accelerated frame. In fact, that's what makes the fall free: there are no forces on the observer (since gravity is not a force in GR), so Alice and Bob are (after they jump) in inertial frames. Only Charlie, who stayed back on the ship, is accelerating.
Standing still on Earth as I am, I might seem to be at rest, and that's how I would think of it while doing a high-school physics experiment. But even according to Newton, I'm really in a revolving frame with a cycle of a day (actually one a bit more complicated than that). Similarly, Einstein treats me as inertial in his thought experiments with trains describing Special Relativity, but he knows (since he didn't write that stuff until after he came up with GR) that in reality I am (in addition to revolving) being constantly pushed away from Earth by the floor, as if in a great space elevator (which was another of his thought experiments).
A distant observer, watching me from space, would see me revolving, but they wouldn't otherwise see me accelerating (although they could infer it from their knowledge of gravity). However, if they get closer to Earth, certainly if they're close enough to get captured by Earth's gravity themself, then they'll see me (along with the rest of Earth) rushing towards them (if they're in free fall themself), and even accelerating. So accelerating objects can appear motionless from a distance, but up close you can observe the truth.
(1/3)

2024-11-24 17:52:45 UTC

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@nprofile…e5g0 @nprofile…yu7x @nprofile…fa4k @nprofile…j7pr : If the particles are moving at exactly lightspeed, at exactly the event horizon, then Bob will see Alice hitting them exactly when Bob sees himself hitting them, but that's normal; every time Bob looks at Alice, he sees her hit a photon exactly (and I mean *exactly*) when he sees that photon hit his retina.
If the particles are moving slightly below lightspeed, accelerating slightly to maintain their distance slightly outside the event horizon, then Bob will see Alice hitting slightly different particles than he hits. Even though Alice jumped out well ahead of him, he sees her hit almost the same particles as he does, because those particles are moving so fast; the particles that she hit almost reach him before he hits slightly different ones.
For this reason, I think that it's fair to say that the event horizon is approaching Bob. But that's from Bob's perspective; you could also say that he's falling towards it.

2024-11-24 15:57:46 UTC

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@nprofile…e5g0 @nprofile…yu7x @nprofile…fa4k @nprofile…j7pr : So If I understand your issue, it's this: The ship is hovering above the black hole, and Alice and Bob jump from the same place, so they should hit the event horizon at the same place (from their point of view), albeit at different times. When Bob crosses the event horizon, he sees Alice cross the event horizon, and while he knows that he's seeing something from an earlier time (like everything we see), he also sees this happening somewhere else, unless she appears to be right in his face (which shouldn't happen). So the place the event horizon was when Alice crossed it must be somewhere different from where he is now; the event horizon has receded (or, which seems more like it to me, approached). Is this the problem?

2024-11-24 05:53:59 UTC

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@nprofile…fa4k @nprofile…yu7x : It's also on the Wayback Machine (just barely, with the first archive only 3 months before it was removed).

2024-11-24 03:17:18 UTC

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@nprofile…e5g0 @nprofile…fa4k @nprofile…j7pr : You never see your hand (or anything) the way it looks right now; you see it as it appeared in the past, when the light left it. As you fall in, hand first, your hand passes the event horizon (entering the black hole) while you're receiving the light from before it entered. Then when your head enters, you receive the light from when your hand entered. Later, you'll see the light that left your hand at the time when your head entered.

2024-11-22 08:27:21 UTC

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@nprofile…yu7x @nprofile…fa4k @nprofile…hszg : I stand corrected. And delighted but not surprised that you have this all worked out for us already.

2024-11-22 08:10:12 UTC

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@nprofile…fa4k @nprofile…hszg : The sky facing the black hole wouldn't be black, no matter how much (or how little) matter is falling in.
Above, when I compared falling into a black hole to falling into the future, that wasn't a vague analogy; the future timelike direction inside a black hole *is* towards the centre. We don't get any light from the future, but there's nowhere in our field of vision where we expect such light and get blackness instead; the same goes for the absence of light from a black hole.
If we're a little ways outside a black hole, facing it, then the light we see will be light left over from before its formation, light that almost fell in but didn't quite, light that's been moving away ever so slowly (relative to a distant observer) as it fights the gravitational pull of the collapsed star behind it, until we finally come along and see it. There's no such thing as the direction from the black hole, only the direction from a little before the black hole, and that's what we're looking at.
You can still tell that the black hole is there, but from the distortion, not from any absence.

2024-11-22 03:15:01 UTC

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@nprofile…fa4k @nprofile…hszg : And if we're falling into a black hole just at our threshold of being able to tell, how long do we have until tidal forces destroy us, and how does that compare to other end-of-the-world scenarios like the Sun engulfing the Earth in 5 billion years?
At some point, it doesn't matter if we're falling into a black hole, any more than it matters that we're falling into the future.

2024-11-21 19:46:33 UTC

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@nprofile…yu7x @nprofile…t3e7 @nprofile…fa4k : Neat! Above, when I described the Moon as ‘smaller but much closer’ than the Sun, by ‘smaller’ I meant ‹less massive›. But I could have just written ‘denser’, or maybe ‘denser (and closer, but this is almost perfectly cancelled by its smaller size)’ for more detail. (Looking up these densities on Google, it looks like their ratio is about 2.4.)

2024-09-06 06:08:42 UTC

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@npub1c9m…9xjd : I'm happy for them and hope we can too someday.

2024-08-13 23:19:01 UTC

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@npub10as…znu0 : I have seen people write \( (a_i)_{i=1}^n \). I guess that's the analogue to sigma notation.