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Sat, 02 Dec 2023

Math SE report 2023-10: Peano's definition of addition is not a tautology, and what was great about Ramanujan?

Content warning: grumpy complaining. This was a frustrating month.

Need an intuitive example for how "P is necessary for Q" means "Q⇒P"?

This kind of thing comes up pretty often. Why are there so many ways that the logical expression !!Q\implies P!! can appear in natural language?

  • If !!Q!!, then !!P!!
  • !!Q!! implies !!P!!
  • !!P!! if !!Q!!
  • !!Q!! is sufficient for !!P!!
  • !!P!! is necessary for !!Q!!

Strange, isn't it? !!Q\land P!! is much simpler: “Both !!Q!! and !!P!! are true” is pretty much it.

Anyway this person wanted an intuitive example of “!!P!! is necessary for !!Q!!”

I suggested:

Suppose that it is necessary to have a ticket (!!P!!) in order to board a certain train (!!Q!!). That is, if you board the train (!!Q!!), then you have a ticket (!!P!!).

Again this follows the principle that rule enforcement is a good thing when you are looking for intuitive examples. Keeping ticketless people off the train is something that the primate brain is wired up to do well.

My first draft had “board a train” in place of “board a certain train”. One commenter complained:

many people travel on trains without a ticket, worldwide

I was (and am) quite disgusted by this pettifogging.

I said “Suppose that…”. I was not claiming that the condition applies to every train in all of history.

OP had only asked for an example, not some universal principle.

Does ...999.999... = 0?

This person is asking one of those questions that often puts Math StackExchange into the mode of insisting that the idea is completely nonsensical, when it is actually very close to perfectly mundane mathematics. (Previously: [1] [2] [3] ) That didn't happen this time, which I found very gratifying.

Normally, decimal numerals have a finite integer part on the left of the decimal point, and an infinite fractional part on the right of the decimal point, as with (for example) !!\frac{13}{3} = 4.333\ldots!!. It turns out to work surprisingly well to reverse this, allowing an infinite integer part on the left and a finite fractional part on the right, for example !!\frac25 = \ldots 333.4!!. For technical reasons we usually do this in base !!p!! where !!p!! is prime; it doesn't work as well in base !!10!!. But it works well enough to use: If we have the base-10 numeral !!\ldots 9999.0!! and we add !!1!!, using the ordinary elementary-school right-to-left addition algorithm, the carry in the units place goes to the tens place as usual, then the next carry goes to the hundreds place and so on to infinity, leaving us with !!\ldots 0000.0!!, so that !!\ldots 9999.0!! can be considered a representation of the number !!-1!!, and that means we don't need negation signs.

In fact this system is fundamental to the way numbers are represented in computer arithmetic. Inside the computer the integer !!-1!! is literally represented as the base-2 numeral !!11111111\;11111111\;11111111\;11111111!!, and when we add !!1!! to it the carry bit wanders off toward infinity on the left. (In the computer the numeral is finite, so we simulate infinity by just discarding the carry bit when it gets too far away.)

Once you've seen this a very reasonable next question is whether you can have numbers that have an infinite sequence of digits on both sides. I think something goes wrong here — for one thing it is no longer clear how to actually do arithmetic. For the infinite-to-the-left numerals arithmetic is straightforward (elementary-school algorithms go right-to-left anyway) and for the standard infinite-to-the-right numerals we can sort of fudge it. (Try multiplying the infinite decimal for !!\sqrt 2!! by itself and see what trouble you get into. Or simpler: What's !!4.666\ldots \times 3!!?)

OP's actual question was: If !!\ldots 9999.0 !! can be considered to represent !!-1!!, and if !!0.9999\ldots!! can be considered to represent !!1!!, can we add them and conclude that !!\ldots 9999.9999\ldots = 0!!?

This very deserving question got a good answer from someone who was not me. This was a relief, because my shameful answer was pure shitpostery. It should have been heavily downvoted, but wasn't. The gods of Math SE karma are capricious.

Why define addition with successor?

Ugh, so annoying. OP had read (Bertrand Russell's explanation of) the Peano definition of addition, and did not understand it. Several people tried hard to explain, but communication was not happening. Or, perhaps, OP was more interested in having an argument than in arriving at an understanding. I lost a bit of my temper when they claimed:

Russell's so-called definition of addition (as quoted in my question) is nothing but a tautology: ….

I didn't say:

If you think Bertrand Russell is stupid, it's because you're stupid.

although I wanted to at first. The reply I did make is still not as measured as I would like, and although it leaves this point implicit, the point is still there. I did at least shut up after that. I had answered OP's question as well as I was able, and carrying on a complex discussion in the comments is almost never of value.

Why is Ramanujan considered a great mathematician?

This was easily my best answer of the month, but the question was deleted, so you will only be able to see it if you have enough Math SE reputation.

OP asked a perfectly reasonable question: Ramanujan gets a lot of media hype for stuff like this:

$${\sqrt {\phi +2}}-\phi ={\cfrac {e^{{-2\pi /5}}}{1+{\cfrac {e^{{-2\pi }}}{1+{\cfrac {e^{{-4\pi }}}{1+{\cfrac {e^{{-6\pi }}}{1+\,\cdots }}}}}}}}$$

which is not of any obvious use, so “why is it given such high regard?”

OP appeared to be impugning a famous mathematician, and Math SE always responds badly to that; their heroes must not be questioned. And even worse, OP mentioned the notorious non-fact that $$1+2+3+\ldots =-\frac1{12}$$ which drives Math SE people into a frothing rage.

One commenter argued:

Mathematics is not inherently about its "usefulness". Even if you can't find practical use for those formulas, you still have to admit that they are by no means trivial

I think this is fatuous. OP is right here, and the commenter is wrong. Mathematicians are not considered great because they produce wacky and impractical equations. They are considered great because they solve problems, invent techniques that answer previously impossible questions, and because they contribute insights into deep and complex issues.

Some blockhead even said:

Most of the mathematical results are useless. Mathematics is more like an art.

Bullshit. Mathematics is about trying to understand stuff, not about taping a banana to the wall. I replied:

I don't think “mathematics is not inherently about its usefulness" is an apt answer here. Sometimes mathematical results have application to physics or engineering. But for many mathematical results the application is to other parts of mathematics, and mathematicians do judge the ‘usefulness’ of results on this basis. Consider for example Mochizuki's field of “inter-universal Teichmüller theory”. This was considered interesting only as long as it appeared that it might provide a way to prove the !!abc!! conjecture. When that hope collapsed, everyone lost interest in it.

My answer to OP elaborated on this point:

The point of these formulas wasn't that they were useful in themselves. It's that in order to find them he had to have a deep understanding of matters that were previously unknown. His contribution was the deep understanding.

I then discussed Hardy's book on the work he did with Ramanujan and Hardy's own estimation of Ramanujan's work:

The first chapter is somewhat negative, as it summarizes the parts of Ramanujan's work that he felt didn't have lasting value — because Hardy's next eleven chapters are about the work that he felt did have value.

So if OP wanted a substantive and detailed answer to their question, that would be the first place to look.

I also did an arXiv search for “Ramanujan” and found many recent references, including one with “applications to the Ramanujan !!τ!!-function”, and concluded:

The !!\tau!!-function is the subject of the entire chapter 10 of Hardy's book and appears to still be of interest as recently as last Monday.

The question was closed as “opinion-based” (a criticism that I think my answer completely demolishes) and then it was deleted. Now if someone else trying to find out why Ramanujan is held in high regard they will not be able to find my factual, substantive answer.

Screw you, Math SE. This month we both sucked.

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