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Comment by ghaff | Hacker News Reader

Stipulating that "understand" can cover a lot of territory, one can perhaps develop a surface and somewhat metaphorical understanding of aspects of modern physics without math. However, you're almost certainly right that any deep understanding of, say, the standard model or relativity is deeply rooted in the math.

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But math is just a bunch of ideas expressed in a very efficient notation. All those notational squiggles represent ideas, and ideas can also be represented with words. It would be much less efficient, but anything explained by math should be able to be explained by words.

Not just an efficient notation. Also very precise.

Natural languages are not precise, and they lead to misunderstandings.

The Greeks did that for a while, before the efficient notation was invented. It didn't work out too well, the explanations get very complex very quickly and it's hard to hold all the facts in your head without some convenient shorthand.

I had a girlfriend who insisted that "two plus two equals four" was a better expression than "2+2=4", but then she also had trouble with any concepts past algebra (and most of algebra for that matter). The notation is efficient and precise. English, and most other languages, are not. We may be able to express the idea of relativity or Newtonian mechanics or anything else in words, but the base concepts end up requiring us to move into a more expressive and precise language when we want to understand and apply (which today is mathematics expressed with the notations of algebra, trigonometry and calculus).

Math is a great example of how notation is sooo important. Doing math with roman numerals is.... possible, sure. But for pretty much any math that's worth anything, it's just a bad notation. It obscures what's going on and makes it harder to mentally deal with. Notation is a critical tool of thought.

Absolutely. The same is also true of programming, music, and natural languages. Probably nearly every discipline that uses its own notation and significantly divergent (from the primary natural language) nomenclature sees a major boost in understanding and expressivity by using custom notations and language.

Re programming: Some languages are simply better at expressing certain types of programs than others, C may be great, but it gets pretty hairy when you try to use it in certain parallel/concurrent coding contexts (i.e., you're probably using a lot of extra libraries and tooling).

Re music: Could you imagine expressing any piece of significant length or complexity in plain English? Even a mathematical (concise and precise) symbolic language would prove difficult, compared to the visual and multi-dimensional notation customarily used. (At least for reading, a case could possibly be made for more expressive and concise notations for writing musical pieces.)

Re natural languages: This is why we borrow so many words in English (and other languages, I'm sure) from elsewhere. A larger set of words allows us to produce more expressive and nuanced statements and questions.

The ideas are math, not the notation. So once you've converted the notation to words, you've just converted the math into a different representation.

It's still math.

In principle, yes, but in practice, no. Tools let us think previously unthinkable thoughts. See Bret Victor's Media for Thinking the Unthinkable [1], where he quotes al-Khwarizmi from the ninth century:

What is the square which when taken with ten of its roots will give a sum of thirty nine? Now the roots in the problem before us are ten. Therefore take five, which multiplied by itself gives twenty five, and amount you add to thirty nine to give sixty four. Having taken the square root of this which is eight, subtract from this half the roots, five leaving three. The number three represents one root of this square, which itself, of course, is nine. Nine therefore gives the square.

He's solving x^2+10x = 39 by completing the square.

[1] https://youtu.be/oUaOucZRlmE

You don't have to use those words. For instance, for

n >> k >> ln(n) >> 1,

he says "Specifically, we require n much greater than k much greater than log n much greater than one", and says it's unwieldy. Well, if you say it like that, then it is.

"Specifically, we require than the magnitudes of n, k, log n, and one be in decreasing order with significant distance between each."

Not a problem. Just don't try to read notation literally in the order it's written.

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