निरंजन replying to Skillmon

Wow! I tried unloading and loading again just to spot the difference, but sob, my eyes are not yet ready. Can you explain how you inferred that?

samcarter

![](https://cdn.fosstodon.org/media_attachments/files/113/436/574/254/849/376/original/f69b884858891639.jpg)

Ulrike Fischer replying to निरंजन

I have books but they are in german. Start with Peano Axioms, e.g. https://www.southampton.ac.uk/~wright/1001/appendix-b---peanos-axioms.html. And read Gödel, Escher, Bach from Douglas R. Hofstadter, not directly about numbers but good for the logic background of axiomatic systems.

निरंजन

@Skillmon @Ulrike Thanks for the explanations. It's really engaging to think on these lines. @Ulrike, can you suggest some references for understanding the axiomatic background of numbers? I am curious to read about it.

Ulrike Fischer replying to निरंजन

well if you imagine a number system where -infinity and +infinity are merged then you (perhaps) solve the problem with the division but destroy transitivity of ordering, so a<b and b<c no longer implies a <c. Also such an infinity merge would not solve the problem that you can't reverse the n/0 operation, for this you then need many "infinity" numbers so n/0= infinity_n, m/0=infinity_m etc, and you would have to declare (consistent) rules how this numbers add and multiply and relate to the "normal" numbers (I do not want to think what 0/0 will do here ...). You can try to develop something but be aware that there is a whole branch of mathematic dedicated to the axiomatic background of the numbers.

Skillmon replying to निरंजन

There is no wrapping around. If you let the number *x* grow larger the difference between *x* and *-x* grows larger (it's 2*x*). When *x* reaches infinity the difference between *x* and *-x* will be infinitely large. If two numbers differ they aren't the same number. Now if you pick 1/*x* and *x* is positive and the absolute value is smaller than 1 the result will be bigger than 1. If you use 1/(*-x*) the result will be smaller than -1. If you let *x* shrink towards zero the absolute value of the quotients will grow larger, but with opposing signs, so the difference between 1/*x* and 1/(*-x*) will grow larger, but at the same time the difference between *x* and *-x* grows smaller until it reaches 0 (so they are the same number). So you get the problem that 1/0 could be both +∞ and -∞.

निरंजन replying to Ulrike Fischer

Yes, I am aware of this reasoning. But, this also raises the same question. It is a theoretical explanation, but underlying the theory, there should be a semantic core to the elements of the theory, e.g., how can 0 be actualised, how can infinity be so, etc. Also, this proof runs on an assumption that 0 is at the center and the two infinities run from the center in the exact opposite directions. That's why it becomes so difficult to comprehend that the same operation can be at the two extremes of the number line. I don't have a fully constructed theory for this, but my question was what if the two infinities are at the same imaginary point? What if they are not separate at all. I had given an alternative "just works" kinda model just to change this perspective, but not sure, if it is sustainable or not..

Ulrike Fischer replying to निरंजन

think about the sequence 1/1, 1/0.1, 1/0.01, and get slowly nearer to the zero. The results gets larger and larger and so one answer is +infinity. Then do the same from the other side: 1/-1, 1/-0.1, 1/-0.01 etc. Now the answer is -infinity.

निरंजन

> In each case, the equation qb = a (where the unknown is q and b is 0) does not meet the criterion of having **one unique solution**. Thanks for the proof. I had read this explanation in a slightly different way, but yes, I still think the answer (for me) is unsatisfactory. but I think first we need to set some background for this exchange. The primary question I think should be DO mathematical numbers correlate with real world understanding of one, two, three at all? If yes, then does number zero correlate with the concept of nothingness? If the answer to any/both of these is "no", then I think we can throw away the cake. I feel most of the times mathematical numbers can be mapped to the real world understanding of numbers, but specially with zero it looks _slippery_ as you say. Mathematicians tend to explain how x/0 could destroy their existing established theory which I very well appreciate. The theoretical clarity mathematicians have is commendable, but then is the established theory buggy? Can it be bettered in order to capture some more intricate nuances?

frougon replying to निरंजन

In order to do any kind of maths, you need definitions and axioms. So, what is your definition of division? The usual definition in number contexts is: assuming a and b are “numbers” (which is already too vague for maths) a/b is “the” number such that (a/b)×b = a I put “the” in quotes because, for this to make sense, you need to ensure (by axiom or theorem) that such a “number” exists and is unique, for every “valid” (a,b) combination. How to do so depends on the particular kind of ”numbers” you are considering (integers, rationals, reals, etc.). *In other words, a/b is defined as the unique solution of equation qb = a (of unknown q), when this equation has a unique solution.* Now, with usual ”numbers”, including real and complex numbers, you can easily see why dividing by 0 (i.e., applying the definition to (a,b) pairs such that b = 0) is invalid with this definition: * if a ≠ 0, then there is no number q such that q×0 = a; * if a = 0, every number q verifies q×0 = a. In each case, the equation qb = a (where the unknown is q and b is 0) does not meet the criterion of having **one unique solution**. P.S.: sorry, no cake. :) (Or, beware that this is not maths and is slippery: you pretend that you're going to cut the cake into parts and that, after giving me zero of these parts, I'll have the full cake. No way!)

निरंजन

One mathematician managed to tell me why it is unacceptable in math. The reason, according to him, was the two orthogonal infinities (positive and negative). I was convinced that the problem is valid, but just because of that making 1/0 "unacceptable" or "invalid" was an inelegant solution according to me.

निरंजन replying to samcarter

A non mathematician here, but I have asked this question to so many math-people and always have got unsatisfactory answers. I am really interested in knowing how one should interpret n/0. When one says 1/2, it can be semantically understood as cutting one entity in two parts of equal size/measurement. What does it mean to say 1/0? Suppose I have one cake and I "divide it by two", I get two equal pieces of cake. After "dividing by zero" what will I get? What does it even mean??? I am enraging math, probably. :P

Skillmon

@निरंजन Wow, I only saw your issues on `expkv` today, no idea why I missed them for a month or two :) Feel free to share your thoughts on my remarks there, hopefully this time around I'll see your replies earlier :P

samcarter

Yearly reminder to myself: one can select different skins for ctan on https://ctan.org/user/settings (I think the gray one is the normally used one, don't use "default" as it disables scripts)

samcarter

A nice blog post about colour profiles and stuff https://balpha.de/2024/10/what-i-recently-learned-about-color/

samcarter

Some of the videos might be interesting https://fosstodon.org/@atypi@typo.social/113357945436693265

निरंजन replying to Skillmon

Thanks a lot for the explanation. Now I understood the actual referents of the _bad code_ and yeah, the `:D` makes sense too, especially because of the `\exp_not:o` example (the surprised `:o` also looks intentional now :joy:)!

Skillmon replying to निरंजन

Because you rarely want to use the primitive features of many primitives (which tend to be quite weird often for the unexperienced -- and sometimes for the experienced alike) except for low level optimised code, and hence using a standard interface that irons out those occasional weirdnesses and is readable and understandable to others turns out to be better code. For instance (this is a well known idiom, but still might baffle the beginner) the code `\unexpanded\expandafter{<stuff>}` is harder to read than `\exp_not:o` (once you know TeX and the `expl3` argument processor conventions). Why exactly does `\unexpanded\expandafter` expand the argument once? Why don't we use `\expandafter\unexpanded\expandafter` (which would work just as well, though be slower, uses more macro space and is more to type)? With other primitives things just turn out more complicated. Have you ever tried to use `\halign` directly (bad example, as there is no `expl3` equivalent, but in LaTeX there is `tabular` which wraps it)? Have you tried to use `\halign` in a `\hbox`-assignment? Try the same with `tabular` and you'll notice a minor difference (`\sbox0{\halign{#&#\cr a&b\cr c&d\cr}}` throws an error, `tabular` works fine, can you tell me why?). For those poor souls of us that venture into the primitve land there is the "solution" to use `\cs_new_eq:NN \__mymodule_<primitive>:w \tex_<primitive>:D` and then use that copy throughout the code, as that allows us to easily fix the primitive usages if there should ever be an interface added to `expl3` for that primitive featurer we needed.

निरंजन replying to Skillmon

> Also I like the fact that happy macros only show in bad code. The macros that actually made me happy were the ones I wrote (obviously). Maybe because I am yet to understand why it is _bad code_.. :P Anyways, I have finally started to enjoy LaTeX3.

Skillmon replying to निरंजन

because look up what `\tl_reverse:N` does and you'll notice that "Don't manipulate" doesn't work out. Also I like the fact that happy macros only show in bad code.

निरंजन

The deadly commands defined with `\tex_⟨name⟩:D` look so cute because of the `:D`. How is one supposed to be shooed away? Why is it not `\tex_⟨name⟩:N` (`N`: Never use) and `\seq_new:D` (`D`: Don't manipulate)? LaTeX3 would have made me smile every single time!

samcarter

https://www.atlasobscura.com/places/jet-bear > This means that Romans and bears cohabitated the island for centuries ... we already knew that from the xmas extravaganza 2022: ![roman_bear.png](/image?hash=bb40817283fdeaefe7db06ee9658f1b84c2ee90ec486592017c01ea6137e72cc)

samcarter replying to barbara beeton

I can understand the use of short form, "Mergenthaler Vocational Technical High School" is quite the name :P

barbara beeton replying to samcarter

Thank you! I spotted that, and had already set it aside to investigate. Mergenthaler emigrated to the U.S., settled in Baltimore (the city where I grew up), and the principal vocational school is named in his honor, although the name of the school has been "shortened" to Mervotech -- the Mergenthaler Vocational Technical High School. But I don't remember that typesetting is offered as a vocation to be pursued.

samcarter

@barbara The following link was shared on the Dante mailing list https://realdougwilson.com/writing/the-box-at-the-museum Would this be something for your collection of typography museums?

samcarter

Some creative names for Australian animals: https://fosstodon.org/@MatthewChat@mstdn.social/113311998424418498

samcarter replying to CarLaTeX

Another alternative for users without local installation: https://texdoc.org/index.html

CarLaTeX replying to निरंजन

Do you have TeXLive installed? You could type `texdoc package_name` on your terminal

samcarter replying to निरंजन

They might still be moving servers around. I don't know when they are finished.

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