I only recently heard what I think is a pretty solid answer to this question [1] that put it this way: We've pretty much got addition nailed down. We're pretty solid on multiplication. However, we are surprisingly weak mathematically when we try to put the two together; you can write down some extremely small math statements that baffled mathematicians for centuries, and for such statements that we have solutions for, they often get into very, very complicated math.
So you have things like Fermat's Last Theorum, or if you prefer, the Question: For a^n + b^n = c^n, are there any solutions with n > 2 for integers? Note how it's just multiplication and addition on the integers. There are simpler questions, but they don't get much simpler. There's no real analysis, transcendental numbers, imaginaries or higher-dimensional numbers, infinities, and so on, at least not in the question itself; it's literally so simple you can reasonably explain it to a middle schooler. It took mathematicians centuries, and the simplest version of the current proof is still PhD-level work to understand. (I suspect even a Master's student in math would have to carefully craft their entire post-doc education for the purpose of understanding this exact proof to be able to say they fully understood it, with no lemmas taken on faith, and I'm still not sure they could make it without a lot of independent study.)
You have things like "Can all of the numbers of a certain form be created via 'a^3 + b^3 + c^3'?", which sounds like it really ought to be simple. It's not too hard to eliminate 4 or 5 mod 9, but proving that it can always be done for everything else is beyond all current known math.
There's a lot of these sorts of questions in math right now. This is a particular single example of a family of very vexing problems.
It's hard to know what the practical impact of mastery over this combination would be, but I expect there probably would be some. Mastery over addition and multiplication have practical consequences that would require a series of books to explore; you have to think that mastering them in combination couldn't help but be practically useful.
[1]: I think it was a somewhat recent Numberphile video, but it may also have been a Terence Tao lecture. References solicited if you've got 'em.
Addition and multiplication (i.e. arithmetic) over the integers are all that are necessary to express first order logic and allowed Gödel to express his incompleteness theorems at such a fundamental level. The logical power of arithmetic is likely why many seemingly simple questions about arithmetic can have very complicated answers or no definitive answers at all.
I see! I'm math-dumb and always struggle with that sort of research. I need to see practical applications for what seems TO ME like ridiculous questions (I'm not implying they are, I have a feeling PhD+ level people don't usually waste time on pointless stuff). Thanks for the elaborate answer!
Questions that are simple to understand and difficult to answer will naturally attract the attention of a lot of mathematicians, because they don’t require a lot of specialized knowledge to work on.
jerf|6 years ago
So you have things like Fermat's Last Theorum, or if you prefer, the Question: For a^n + b^n = c^n, are there any solutions with n > 2 for integers? Note how it's just multiplication and addition on the integers. There are simpler questions, but they don't get much simpler. There's no real analysis, transcendental numbers, imaginaries or higher-dimensional numbers, infinities, and so on, at least not in the question itself; it's literally so simple you can reasonably explain it to a middle schooler. It took mathematicians centuries, and the simplest version of the current proof is still PhD-level work to understand. (I suspect even a Master's student in math would have to carefully craft their entire post-doc education for the purpose of understanding this exact proof to be able to say they fully understood it, with no lemmas taken on faith, and I'm still not sure they could make it without a lot of independent study.)
You have things like "Can all of the numbers of a certain form be created via 'a^3 + b^3 + c^3'?", which sounds like it really ought to be simple. It's not too hard to eliminate 4 or 5 mod 9, but proving that it can always be done for everything else is beyond all current known math.
There's a lot of these sorts of questions in math right now. This is a particular single example of a family of very vexing problems.
It's hard to know what the practical impact of mastery over this combination would be, but I expect there probably would be some. Mastery over addition and multiplication have practical consequences that would require a series of books to explore; you have to think that mastering them in combination couldn't help but be practically useful.
[1]: I think it was a somewhat recent Numberphile video, but it may also have been a Terence Tao lecture. References solicited if you've got 'em.
[2]: https://youtu.be/wymmCdLdPvM?t=209
benlivengood|6 years ago
mombul|6 years ago
empath75|6 years ago