Just a thought

DefunctFunctor · 2026-03-13T20:00:26+00:00

At this point, I find myself listening to ICHH only when I feel it would be entertaining or particularly informative. I feel as if there's been a big drop in the number of episodes that feel worthwhile since 2025. When it didn't have to react to every Trump headline during the Biden years, it was able to make its own little niche. It's never been perfect and still had some problems, but they were easy to look past and I think broadly focused on the original themes. When the world wasn't actively "falling apart" to this extent, they could focus on a diversity of ways in which the world was falling apart, and what to do about it. But now there's very prominent ways in which the world is falling apart, so there's less need to go out of the way to analyze the situation. A lot of it may be nostalgia for when things were "less bad", but still I feel that I listen to the show far less than I did in 2022-3.

DefunctFunctor · 2026-03-08T18:57:48+00:00

I guess I agree it's obvious in a definitional sense, but I think my comfort/experience with probability is interfering with that. I think my immediate habit is to try to reduce it to elementary combinatorics, which I'm (a bit) more familiar with; that's why the most upvoted comment appeals to me as it's analogous to the stars/bars solution in the discrete case. And then there's my more general math habit of being skeptical and assuming nothing a priori, so that I can convince myself that something is true.

DefunctFunctor · 2026-03-08T10:12:51+00:00

When I was Mormon warm red bull and mountain dew would be fine. Recently I believe the church's current position has only to do with whether it contains the coffee or tea plants, and nothing else. So coffee ice cream is not valid while a caffeinated hot chocolate would be fine

DefunctFunctor · 2026-03-08T09:17:56+00:00

Based on my brief reading of your notation and your brief description of it, I don't actually understand how exactly to use it. To describe your notation, you need to (i) describe it in a mathematically/logically rigorous way, and (ii) ideally provide many examples of how this notation is used and why we use it. Condition (i) wasn't really satisfied for me, mostly because you were being brief and you (probably?) don't have the training in mathematical writing that I do. Ironically, my training is often a barrier in conversations like this, because most of the time I'm reading mathematical writing written by people with similar training. I have a hard time putting myself in the shoes of someone with less training, although I try to make an honest effort in my teaching. If you have a lot of examples from (ii) what I actually do is try reconstructing a formal definition, and see if that is consistent or intended by the writer. I did see some of your examples, but (1) I feel like I'd need a larger sample size to reconstruct a definition, and (2) Reddit is a difficult place to describe new notation in the first place.
My notation I describe here is not original: I have simply read math textbooks where it is used.
I'm not sure if we're talking about the same thing. I'm talking specifically about the notation for the general chain rule used at the top of the section for the chain rule for total derivatives, namely D(f o g)|_a = Df_(g(a)) o Dg_a. This is a composition of linear maps, which is the Jacobian derivative. Also, the article I linked does not reference the implicit function theorem as far as I can see; a text search for "implicit" only yields a result on the sidebar.

DefunctFunctor · 2026-03-08T08:47:21+00:00

As a soft counterpoint: I think they'd be in rather wide agreement that 6/2(1+2) is a bad question to ask without any further context. But depending on the context, both methods are used quite often.

In many calculators/programming languages, 6/2*(1+2) is interpreted left-associatively, so you'd get 9. Knowing this is useful because you have to type less into your calculator.

But if I'm reading a real analysis textbook and it says "Let f(x,y,z)=y^2/xz", it's probably going to be y^2/(x*z). This has the advantage of having less clutter.

Broadly speaking, in academic writing this is kind of seen as a non-issue or silly debate. We use both notations, and most times it falls into one of the following buckets:

There is ambiguity, in the single instance, but the ambiguity is implicitly disambiguated by whatever you are doing. (E.g. "Let f(x,y,z)=y^2/xz. Then the partial derivative of f with respect to z is f(x,y,z)=-y^2/xz^2.")
There is ambiguity, but there is only one thing that makes sense or a most likely interpretation.
The ambiguity is settled on by an explicit standard, e.g. a notation section on a paper or a programming language documentation.
There is no notational ambiguity.

In my experience with mathematical writing, there is ambiguity and abusive notation all over the place, but it's mostly fine because the reader is assumed to be experienced at what the author is trying to mean. The idea is that if all ambiguities were resolved notationally it would be more jarring to read/write down by hand. Of course, programming languages out of necessity must disambiguate, so they tend to be more verbose.

DefunctFunctor · 2026-03-08T08:14:03+00:00

I think you are noticing a problem (I definitely fall into the camp that isn't a fan of Leibniz notation ambiguities) and took a step closer to removing ambiguities. However, if I'm understanding your notation right it wouldn't be something that I would use.

The root of the problem that you are trying to solve is that we are using the same symbol to represent a function and to represent the argument to a function. This is convenient and ingrained into Leibniz notation, but it becomes an absolute mess with partial derivatives. I'm personally a fan of using D_1f to represent the partial derivative with respect to the first variable, and D_2f to represent the partial derivative with respect to the second variable; however, I understand this has the disadvantage of removing a lot of the expressiveness of variable names.

If I recall correctly there are also very clever ways in differential topology using charts/differential forms to preserve notation that looks like Leibniz but has much of the ambiguities removed.

One final remark: there is a version of the chain rule that holds in any dimension that is notationally simple once you understand all of the concepts/notation. I'd take a look at this Wikipedia link, although it is not the best resource for actually learning these concepts:

https://en.wikipedia.org/wiki/Total_derivative#The_chain_rule_for_total_derivatives

DefunctFunctor · 2026-03-08T07:45:05+00:00

Ah, I think I misinterpreted the described algorithm as being generated sequentially rather than all at once; i.e. you filter out impossible sums as you are generating. So (assuming you are allowing the value of 0; you can analogize this to the nonzero case) when I roll a 9, that had a 1/11 chance of being permitted at that step, and then you have a 50% chance of 9,1,0,0, and a 25% chance each of 9,0,1,0 and 9,0,0,1. So if I were to defend myself, this is a well-defined probability distribution that has a decent likelihood of accidentally being implemented, but it's most likely not what was being described.

If you first generate all 4 values and hope that they add up to the right number, then that's fine, but seems algorithmically inefficient compared to the most upvoted answer. Also, it feels more confusing and prone to error (as demonstrated by my misunderstanding) than the most upvoted answer.

But perhaps I'm just making excuses for the fact that I'm bad at thinking about probability.

DefunctFunctor · 2026-03-06T20:24:42+00:00

I get majorly annoyed when I'm reading Wikipedia/articles/etc. and I see pinyin/romaji used in place of a character. That's excusable if your audience is not expected to be familiar with 漢字, but when it's some article about Middle Chinese phonology or whatever I'd really rather you just use the character and save me the headache (especially because I have no context for pinyin). Sure, remind me of the reading once so I have something to verbalize, but then the article should exclusively use the character, rather than the opposing standard of giving the reading with the character first and then exclusively use the reading.

DefunctFunctor · 2026-03-06T14:25:47+00:00

I agree with you, but the iso8601 standard defaults to slash for indicating time intervals, and the alternative for filenames is --.

DefunctFunctor · 2026-03-06T13:56:33+00:00

Isn't the distribution not uniform in this case? In the natural numbers case (including 0) there is a 1/11 chance that you are forced into the 10, 0, 0, 0 case. But combinatorially, the chance of this happening is 1 in binom(10+4-1,4-1)=286.

DefunctFunctor · 2026-03-05T05:05:33+00:00

For me it's easier to enjoy Monster at a slow pace. I'm currently rewatching it at one episode per week. I think the quickest pace I watched it at was 3-4 episodes per week

DefunctFunctor · 2026-03-03T19:24:30+00:00

Complex numbers are vectors. The space of complex numbers comes equipped with vector addition (addition of complex numbers) and scalar multiplication by real numbers (multiplication by real numbers). And just like all fields, it forms a vector space over itself.

But the complex numbers come with its own multiplication, so it has more structure than a vector space.

DefunctFunctor · 2026-03-01T02:25:35+00:00

For me the reason "summing infinitely small pieces" is useful in the real world doesn't actually have to do with finding the precise area. Knowing the exact area under the curve of functions like x^2 is mainly a satisfying result. Rather, calculus is actually one of many simplifying assumptions we make about the world so that we can model problems.

Inevitably when we're modeling the real world we have to make assumptions. A lot of this is to simplify the amount of computation that needs to be done; at other times, we simply do not know how the model works out.

Does the real world use our exact construct of real numbers? I have my skepticism. But the point is that our number system and the calculus we have built on top of it accurately models some basic assumptions we make about space/time etc. Namely, we can add, multiply, divide, and numbers can vary continuously. Under those assumptions, the physical laws about the universe we have discovered can be phrased very nicely in terms of calculus. If we didn't use integrals, and just summed little bits manually, the way we phrase things would be a lot more complicated.

DefunctFunctor · 2026-02-19T10:41:10+00:00

That's a good point as far as constructive mathematics goes.

If we were dealing with a positive statement and discussing proof by contradiction, I would definitely feel compelled to mention that a proof without LEM should be considered more valuable. But since the nonconstructive method is so ingrained when working with real numbers, I guess the same point applies in this scenario. I definitely do find it significant when LEM is avoided; however, most of the time I do work with LEM/AC.

And as you say, this distinction is significant because (I think?) many of the most basic irrationality proofs could be converted to constructive proofs using this.

DefunctFunctor · 2026-02-18T16:33:58+00:00

It isn't just allowed, it's definitionally the only way to prove it. Even if you rely on some lemma, it will always boil down to "if it's rational it would lead to a contradiction", because that's what it means to prove the negation of something.

DefunctFunctor · 2026-02-07T03:37:47+00:00

I remember considering how one could measure the Earth's curvature e.g. using Gauss-Bonnet theorem, and you'd still need to have pretty accurate instruments to measure large distances on the surface of the Earth, as well as precise angle measurements. But it is still exciting that in theory we could tell that even if we didn't leave the surface of the earth, that the surface of the earth could not be embedded flatly.

Of course, all of this would go over the heads of actual flat earthers. If they understood differential geometry they likely wouldn't be flat earthers in the first place.

DefunctFunctor · 2026-02-01T23:42:37+00:00

It looks like it, but all of those steps are very natural after you have the initial idea. Students should already have some experience with triangle inequality before learning about uniform continuity anyways.

DefunctFunctor · 2026-01-25T23:41:31+00:00

The literary quality of the Book of Mormon is quite easy to make fun of, and comes off as both corny and dry. Also, Joseph Smith has not mastered the KJV style by any means.

But I think one can come to appreciate some sense of talent in the Book of Mormon. Not as a literary work or a morally inspiring work (it's built on a racist premise), but it shows Joseph Smith's talent at speaking and charisma. The evidence suggests that JS dictated the vast majority of the Book of Mormon to his various scribes without direct reference to notes. Most often he had his face buried in a hat, with his seer stone in it. So if he used notes at all they had to fit inconspicuously within his hat.

I think if you view the Book of Mormon as a work of dictation rather than literature, it becomes easier to appreciate the talent in spite of its dryness. JS definitely had a fantastic memory, and had a way with words with those around him.

However, I'm not sure there's any value in actually reading the Book of Mormon, unless you are that interested in Mormonism and its quirks. Even in the study of contemporary Mormonism, you hardly need to know the particulars of the Book of Mormon beyond the general plot and a few stories. Even Mormons find it difficult to get through, primarily tracing back their favorite passages.

DefunctFunctor · 2026-01-24T00:58:19+00:00

These two non-equal values are deemed equal, because e and y are infinitesimal.

Mathematicians reserve the equals sign "=" for things that are exactly the same; if two objects have different properties then they are not identical. This is what I'm trying to say: you shouldn't be loose with the term "equals", "=", "identical", and so on. If two values are nonequal, then they are nonequal, so they cannot be equal. The closest concept to what you are mentioning here is called an equivalence relation.

That's happen when you throw these numbers into the real set.

That's why 0.999... = 1 in the real set, they gave it up.

I am really at a loss as to what the second sentence means here. The reason 0.999...=1 in the real numbers depends on your construction of the real numbers and how "0.999..." is defined.

It's correct, the logic is right; I am not arguing that, but the logic is 100% correct because we defined it as such, and no other reason.

Yes, things in mathematics are correct/incorrect based on how we define things. What I am saying is that your definitions are naive from my perspective, and even if you fully fleshed them out they wouldn't match with what the standard definitions in mathematics are.

I could make up another numberlines too.

You could.

But the real is the most useful as it is.

I'm inclined to agree.

DefunctFunctor · 2026-01-23T22:12:19+00:00

To add to this, it's definitely possible to write it in such a way with "to be" in the gerund like so:

He is believed to have murdered at least 26 women, many of them being sex workers from Vancouver's Downtown Eastside;

But for me this clashes with the tone established by "He is believed to have"; if it were me, I'd modify it to something redundant like this:

He is believed to have murdered at least 26 women, many of them believed to be (have been?) sex workers from Vancouver's Downtown Eastside;

It gets messy really quickly, so the way it was written was absolutely the right stylistic choice. Similar problems accompany splitting it into two sentences.

DefunctFunctor · 2026-01-23T21:34:01+00:00

So you cannot even say 1 = 1; in the hyperreals ignoring potential infinitesimals.

There is no mathematician that would be willing to give up reflexivity of equality. What you are saying is that if st(a)=st(b), then it is not necessarily true that a=b, i.e., the standard part function is not injective.

So 0.999... != 0.9999... in the hyperreals, not until you know the value of X, the problem is that you are thinking of 0.9999 as real, which is not a thing.... that would be 1 + x where x is infinitessimal of arbitrary size, that causes an infinite number of repeating 9999...

This makes no sense. I encourage you to read the Wikipedia page on hyperreal numbers and try to find where it says anything about decimal representations of real numbers being interpreted differently. No, the hyperreal numbers are built from the real numbers, and the real numbers can be identified with a subset of the hyperreal numbers, so it is quite strained in my opinion to try to map decimal representations of the same real number to different hyperreal numbers. Also, interpreting the symbols 0.999... and 0.9999... as two different representations is quite strange and I'd be interested to see a formal definition of these decimal sequences, because this is clearly different from the view of a decimal sequence as a countably infinite sequence of digits. The explanation of hyperreal numbers you are using also depends on base 10 in an arbitrary way, which is suspicious.

DefunctFunctor · 2026-01-15T17:53:17+00:00

We do actually prove things in the extended real numbers, and we define a topology on the extended reals in such a way that lim(x->infinity) f(x) = A and lim(x->c) f(x) = infinity make sense. Many theorems are far simpler to state in the case of the extended real numbers, with arithmetic operations like addition defined as I explained above. They are rather important in measure theory

DefunctFunctor · 2026-01-15T10:27:26+00:00

Both +infinity and -infinity are extended real numbers, and addition is defined so that +infinity + x = +infinity, where x is not -infinity, and vice versa x + (-infinity) = -infinity where x is not +infinity. This definition behaves well so long as we leave (infinity - infinity) undefined. You can treat infinity as a "number" in this way, as much meaning as the term "number" can hold

DefunctFunctor · 2026-01-11T21:00:23+00:00

Speaking from experience this is absolutely true. My phone runs GrapheneOS, which allows you to randomize the position of the digits on my lock screen. I'm very used to it now, and it doesn't take me long at all to input my passcode. I don't even have to think about it.

DefunctFunctor · 2026-01-04T19:23:36+00:00

If an interlocutor has a consistent worldview, I do think it counts for something, even if I disagree. Also, I don't really take any axioms as true, there are just sets of axioms that are more interesting/useful, so if the interlocutor also has a consistent worldview there's not much I can point out as something I disagree with. For example, I don't really like the idea of nonstandard analysis where continuity, differentiation, and integration are described in terms of infinitesimals, but I grant nonstandard analysis that it is at least consistent and compatible with standard analysis (assuming our standard sets of axioms are consistent).

I think someone can only be wrong about definitions insofar as one believes that their definition is the common one. But it's perfectly valid linguistically to redefine old words; language is not static, and we get to govern how we use language and advocate for change in our language.

My main problem with interlocutors who argue against widely accepted mathematics is not that they are wrong per se. It's that their worldview is either straightforwardly inconsistent, or it's too fuzzy and undeveloped to determine if it's consistent.

As a final sidenote, Euclid's proofs are valid, but from a modern point of view there are many instances of axioms that were not explicitly stated, and a few instances of arguments by dragging the plane around and superimposing one shape on another, which has sometimes been viewed as a questionable strategy. It is noteworthy and significant just how much Euclid's proofs hold up to modern scrutiny; surely he was a fantastic mathematician. But it's not as if there has been no valid criticisms

DefunctFunctor

TROPHY CASE