Summer of Math Exposition

I hope I’m not being too pedantic, but there are a few formatting issues or typos that make it more confusing to parse the content.

1. “I’m more interested in it’s character as a number” (it’s -> “its”)
2. “$134 = 1 • 10^23•104•1$” (it might be clearer to add “+” signs, like so: $134 = (1 • 10^2) + (3 • 10^1) + (4 • 10^0)$)
3. “decimal/numeral form, as it shows an ignore-ance of the explicit difference” (extra hyphen)
4. “$5 \equiv 11_3$” ($5$ in base $3$ is represented as $12$, unless I misunderstood the intention) There are some other formatting issues, but I believe point #2 covered the solution.

Things I liked in the article:

• Well-expressed ideas, easy to follow
• Draws attention to a simple thing and explains it in depth
• Beautiful use of language, pleasant read
• Links different areas of math (matrices, geometry, functions, etc.), making it suitable for any audience
• Author is clearly passionate about their topic of choice
• Takes a strong philosophical approach, and encourages the reader to think over the topic themselves
• Includes the “There are 10 types of people” joke (:

Feedback from the 4 specified criteria:

Motivation: 4.5/9, lower average. The introduction alludes to the hidden beauty of even “simple” math, and is both accessible to all readers and intriguing. Clarity: 6/9, strong. The entire post was about how we define the “=“ symbol, and was a deep-dive into the concept of equivalence. Moreover, rigorous mathematical definitions (such as that of congruency) were simplified and were supported with some visuals and examples. Novelty: 4/9, just below average. While it may be possible that people do not usually ponder the concept of equivalence, it is certainly not a novel idea, with many articles on the subject of equivalence in mathematics. This article is quite good at tying together different existing approaches to understanding this concept, but does not seem to introduce many new insights. Memorability: 5.25/9, high average. An enthusiastic enough article, with inclusions of a few examples and a bit of humour.

Unfortunately, the presentation was slightly lacking, and 0.5 will be deducted from the final score for the errors in the text.

Taking into account all of the above, the final score rounds up to 4.44 (I guess it rounds up to 4.43 because I couldn’t select the 4.44 option ¯_(ツ)_/¯ )

Thank you for your submission!

it would be good if you had added a little bit of geometry as a base. Like, instead of root 2, the hypotenuse of a unit isosceles triangle. It would enhance the beauty of this article. It took me a time what the article really about was . In my opinion it was how we should not teach our students to wrap their head around notations. But while introducing (rather I would say talking) about, similarity and congruency, he could be seen defying geometrical approach too. But I would still recommend a math enthusiast teenager this post as it is not beautiful but serves a purpose to Lockhart’s dream

This is a very well written article, but it never hit me with an aha moment and risked losing my interest in a few of the more basic places.

This is a philosophical article that beautifully explains the concept of equality and its meaning. However, there are some issues in the article which I would like to point out as follows:

1. Motivation: I feel it is where the author does his/her best. He/she tries to motivate us that we have underestimated the notion of equality, despite it being the central part of mathematics.
2. Clarity: I have some issues here. There are jumps in topics without coherence, and hence, we, as readers, get lost on the main point of the article. Sometimes the article describes the supremacy of a number in comparison to its representation. Even equality in the function’s context is unclear. However, in the end, it goes to equivalence, which is close to the ‘mathematical’ treatment of equality. I say, close to, because equality is incomplete without the substitution property, which has not been covered here.
3. Novelty: The article poetically makes us feel like mathematical objects are some beings; hence, I appreciate your presentation.
4. Memorability: I guess the purpose of the article is to provide a philosophical view towards equality. However, the digression in the middle doesn’t fulfill the purpose. Hence, it didn’t make a lasting impact on me.

It’s a bit on the formal side, but it does a good job of explaining that ”=” can be interpreted in a lot of ways and that it should not be taken for granted. Which is a good lesson in my book.

If you think about it, all equalities are equally offensive ;). This entry is in the top 50 = 50 for me, but I would like to see more types of equalities like: the approximate to symbol. (Below)x = (Above) other entries.

The introduction feels a bit too pedantic, but the transition to equivalence relations is nice.

This one is hard to judge. It’s definitely too philosophical for me. For example, the way functions are explained as equalities seems too forced to follow the narrative. For me, when you write something like $f(x) = x+1$, this equal sign is not really an equality, is just an assignment, as in computer science (we define f to be this thing on the right, that accepts one input), but then I could probably also say that for things like pi or $\sqrt{2}$ or anything… I don’t know. That’s why I said this is too philosophical for me. As such, I don’t think it felt too enjoyable.

Another issue is that it seems it wants to target an elementary audience but then ends up including concepts like vector spaces and isomorphisms. Even if they’re not too important to understand in order to follow the post, their mere mention could be confusing to some readers. I see here you selected as audience high school and undergraduate. I’m not sure what your audience would be for this post.

I like your writing style though. It seems you care about writing. Because of that, I encourage you to reread the post for some proofreading to fix some typos. For example, there is at least one place where you still use ‘it’s’ where it should be ‘its’, and there are also some repeated words somewhere. Lastly, I would guess the platform you use doesn’t support inline math mode. If that’s the case then I get it. Personally I use Quarto + Github Pages for mine, and I’m very satisfied. If you can do some coding I think it’s worth.

I really liked how this began. Being pedantic about what we’re really doing when we write some mathematical notation down isn’t something I’ve really seen discussed elsewhere. I did feel like the article began to lose its way towards the end (after the bit about functions), just listing examples rather than diving into what makes these examples interesting. By the end I had certainly had my brain engaged and was made to think about what I read and what I know, which is really good. However I felt like the article didn’t really justify its title: why is $A=B$ the holy trinity of mathematics? Overall an enjoyable and thought-provoking read.

I think there could’ve been more to directly define what does equivalence mean. Also, the specific examples were nice, but there needs to be a bigger overarching theme.

Since SoME4 is a math explainer competition, I’d like to judge entries based mostly on their mathematical merit, not on the English prose. Having read the article I felt like it’s mostly a philosophical discourse on “=“. I mean was it well written from a literary standpoint? I suppose. But did I learn any new math from the article? Not really. I really wanted to see some interesting theorems or math techniques. If the subject is “=“ then for example one explainer I can think of is to explain how “=“ is rigorously defined in formal proofs so a new student in proof writing can use it confidently. If I rated this only on the math this would be a 3 or 4—math is better communicated with a more precise form of writing. Bumping up to 5 for the English I guess.