TobyBartels

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

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npub1kpf5nh0jvj3ygrlz3zw2s2qvy2dpdp5zdlwnl2nqujugfugumkfqngr9jw

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TobyBartels@mathstodon-xyz.mostr.pub

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nprofile1qqstq56fmhexfgjypl3g389g9qxz9xsks6pxlhfl4fswfwyy7ywdmyspz4mhxue69uhhyetvv9ujumt0wd68ytnsw43qt4f8ga

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relay.mostr.pub

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npub1kpf5nh0jvj3ygrlz3zw2s2qvy2dpdp5zdlwnl2nqujugfugumkfqngr9jw

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Last Notes

2025-04-26 05:39:55

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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

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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

@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

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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

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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

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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

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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.

2025-04-10 02:26:00

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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

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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

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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

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

2025-03-11 05:43:10

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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

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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

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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

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

2025-02-26 05:05:13

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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

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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

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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

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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

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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/

2025-01-24 01:09:25

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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.)

2024-11-21 17:59:33

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@nprofile…fa4k :
>water on the sunny side of the Earth is pulled toward the Sun more than water on the night side
While the Sun does cause a tidal force as you describe, the tides that we see are mainly due to the Moon (which is smaller but much closer). So water on the moony side (if you will) is pulled more, while water on the opposite side is pulled less, and the land (which moves more as a solid chunk) is pulled some amount in the middle. So we get high tide on both the moony side (where up is towards the moon) and on the opposite side (where up is away from the moon).
You probably know this, but it seems weird to focus on the relatively minor tidal effect of the Sun, so I thought that I would clarify for other readers.

2024-09-06 06:08:42

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

2024-08-13 23:19:01

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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.

2024-08-09 08:33:33

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@npub1knz…c73p : Wikipedia says that Eudoxus's ideas were ‘revived by Tartaglia and others in the 16th century’, but it gives no details or citation for that.

2023-09-26 12:02:29

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@npub1knz…c73p : Well, it's a list of lists, right? So someone needs to create a list of mathematicians and a list of physicists, and then they can be included in this list.
Oh, and it looks like theologidon is a *theologi*an on Masto*don*.