RAGUMAGU

Bill Thurston and Humanized Mathematics

Mathematics can be notoriously dense. Check out this deceptive piece of math, trying to prove what looks like two different numbers to be the same: 

                
                          x = 0.999...
                        10x = 9.999...
                    10x - x = 9.999... - 0.999...
                         9x = 9
                          x = 1
                 ∴ 0.999... = 1

This was posted on reddit. The post asked if the above logic was correct. The response was:

It is correct that 0.999... equals 1 exactly. However, this proof is incomplete, because it is showing, that if 0.999... equals anything, then it must equal 1.

Symbols don't have meaning by themselves, until we decide on one. Addition makes sense for adding finitely many things, but if you want to extend this idea to infinities (in this case, the infinity hides behind the 3 dots in 0.999...), then you have to precisely define what you mean (when you write 0.999...).

People have agreed upon a standard definition for what it means to add together infinitely many things - they do so using the concept of a limit (usually taught in calculus). However, not all infinite sums exist (using the standard definition). We use the term converge instead of exist, but it is the same thing. For example, the sum 1 + (-1) + 1 + (-1) + 1 + (-1) ... does not converge.

To prove that 0.999... equals 1, you have to first show that it converges at all, using the precise definition of an infinite sum.

Then, once you know that it converges to something, you can call that something X and then all the steps in the proof work. More specifically, they work due to theorems we call limit laws. The limit laws let you manipulate infinite sums in the way you'd manipulate numbers, but only if all these limits exist.

u/keitamaki

There are many things to learn from this post.

Mathematics is notoriously dense. Just writing 0.999... brings in so many concepts and complexities that it can become overwhelming quite fast.

You have to precisely define every symbol, and sometimes, you will have to define numbers too. Since there is a lot of detail, mistakes can happen quite easily. You can easily fall into the trap of thinking that you know something clearly, without actually knowing it in enough detail.

Rewriting 0.999... as $$\sum_{n=1}^{∞}\frac{9}{10^n}$$ highlights the infinity part of the problem. However, most readers will read it as jargon and it won't help much by itself. An explanation in simple words and a reminder that you are dealing with infinities helps a lot more in comparison.

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The calm and clear conversational tone in the answer is soothing and it gives me comfort. On reading such answers, you know that it comes from a person passionate about mathematics, someone who cares about the reader reading and the listener listening.

Mathematics is learnt and kept alive by discussions of this kind. As Bill Thurston puts it:

Mathematics only exists in a living community of mathematicians that spread understanding and breath life into ideas both old and new. The real satisfaction from mathematics is in learning from others and sharing with others.

Bill Thurston When asked What's a mathematician to do? What can one (such as myself) contribute to mathematics? With all the tremendously clever people working so hard on mathematics, it feels like there is nothing left for someone such as myself (who would be the first to admit they do not have any special talent in the field) to do., Bill Thurston, replied with this brilliant post. I suggest you read the full post.

In most of the prescribed mathematics textbooks I had in university, paragraphs explaining the math and the history of how it came to be - were nowhere to be found. Dehumanized sentences in the form of definitions and proofs which read like computer code were plenty.

To state all related definitions and theorems and proofs in every single bit of math makes it verbose and impractical. Terse notation is quite common in math papers. Mathematicians seem to prefer concise proofs over clear notation. The reader is expected to read the notation and understand the math in context.

However, expecting a novice to just slog through symbol manipulation and hoping that they will somehow end up with reasonable math skills is naive.

Most people don't learn maths by sitting alone in a room and manipulating symbols. They learn it from/with other people, by discussion and asking questions. A few lines of mathematics can elicit a few pages of explanation and the reader might still have doubts about that piece of math. This is why math written with symbols should be accompanied by explanations in words to be effective. A balance between dense notation (to save space) and clear explanation (to make sense) is required.

Along with the skill of working with abstract symbols, the skill of explaining mathematics in simple terms, in words that are easy to understand - this has to be practiced. This allows you to connect and communicate easily, and this is crucial in mathematics.

To everyone who wishes to learn more of mathematics, remind yourself to take it in small bits and pieces.

Shrinidhi Raghunandan

Published: 30 July, 2023.