> This is a huge deal, if true. What's the potential benefit to humanity that could come about if this is true, without any other dependencies, over the next 30 years?
A deeper understanding of pure mathematics? That's like asking what benefits Michelangelo's works had for humanity.
I mean... not quite...
Unless you're talking about improvements in artistic techniques, the art itself doesn't really lead to something more practical, whereas pure mathematics actually does often lead to applications in other fields.
He's taking the running average of the partial sums: 1-1 = 0 0+1 = 1 1-1 = 0 0+1 = etc etc The partial sum would end up being equal to the final sum by definition when you're done summing all items. Since we're talking about infinite sequences you're never done summing all items so you'll have to do something else to end up with an answer. for example seeing which way the partial sum trends. In this case it trends so…
> In this case it trends solidly in the direction of 1/2 ...except that it doesn't; to me, it even looks more like it trends in the opposite direction, i.e. it is trying to stay away from 1/2.
The farthest it could get from 1/2 would be if it ran off to + or - infinity. Hovering between +1 and -1 could be compared to an orbit.
I actually believe the intent is the opposite. I think the PR people are just trying to sell Fokas as a polymath genius. "Look, he is not only a mathematician but also an MD, wow!"
Thank you, I had not considered that. I think researchers or a HN audience may perceive this differently than the average reader.
FWIW: my initial instinctive reaction to the MD part was most definitely negative.
Just pointing out that interestingly Cambridge, where Fokas is a professor, has not released anything. He is merely visiting USC so it strikes me as weird that they would claim this PR so quickly. Also Mathematician-MD somehow makes it sound like the MD means he is a lesser mathematician or not a full mathematician. Fokas is a well respected Professor at one of the top applied Maths departments in the world. A better…
When you actually read it, being a mathematician and an MD is more impressive, not less. The interesting thing about him is his cross disciplinary knowledge. I don't see the insult.
> Obviously it doesn't make sense to add all the positive integers (the series doesn't converge), but if you squint and ignore this, and just do the arithmetic a certain way, you get -1/12. I'm not a pure-maths type person but in my experience, if you get one answer by following simple, well understood maths (like "the sum of two positive integers is a positive integer") and another answer by "squinting and ignoring…
The thing impendia is referring to there is analytic continuation, which (as mentioned) is a way to extend the domain of a certain set of functions, like the zeta function. It is perfectly rigorous. His/her language was just a short-hand for "don't worry about the details here, but it does work". Mathematicians aren't stupid, and more than any other profession, they value rigor. They know what they're doing.
Rigor in logic but, it seems to me, not often rigor in description. There's a lot of willingness to say "if we change rule X to mean something totally other, then you can do this thing" and then just describe it as "you can do this thing". Sure, but "this thing" now means something different.
> Obviously it doesn't make sense to add all the positive integers (the series doesn't converge), but if you squint and ignore this, and just do the arithmetic a certain way, you get -1/12. I'm not a pure-maths type person but in my experience, if you get one answer by following simple, well understood maths (like "the sum of two positive integers is a positive integer") and another answer by "squinting and ignoring…
In mathematics, the way to the answer is essential. If you are doing /anything/ wrong on your way, your result is void. It may not be wrong, but nothing is gained.
I disagree with this. Often what is gained from a proof is not the fact that a proof is known, but more insights about the original problem. So, a small technical error may "invalidate" a proof but it does not make it meaningless. Just like a bug in software does not make it worthless.
A deeper understanding of pure mathematics? That's like asking what benefits Michelangelo's works had for humanity.
I mean... not quite... Unless you're talking about improvements in artistic techniques, the art itself doesn't really lead to something more practical, whereas pure mathematics actually does often lead to applications in other fields.
Modern mathematics is so incredibly abstract that I'd be very surprised if more than a minuscule fraction of all published theorems ever had practical applications. If you count that you should also count improvements in for example chemistry (pigments) or the theory of human perception that were initiated by art.
The problem with this first approach is that you can get arbitrary results just be reordering the series. Also, the formula you give for the geometric series only works for |x|<1.
Reordering a divergent series can indeed give you any result you want. This is not reordering though, it’s called Cesàro summation. It won’t always get you an answer, but if it does then the answer is unique and reacts nicely to sums and products. Talking about it as summation might be misleading since that’s one of those concrete terms that mathematicians like to redefine without anyone’s approval. Picture we have a…
> This is not reordering though, it’s called Cesàro summation. It won’t always get you an answer, but if it does then the answer is unique and reacts nicely to sums and products.
It's also probably worth noting that, if the series is genuinely summable, then Cesàro summation gives its sum.
Just pointing out that interestingly Cambridge, where Fokas is a professor, has not released anything. He is merely visiting USC so it strikes me as weird that they would claim this PR so quickly. Also Mathematician-MD somehow makes it sound like the MD means he is a lesser mathematician or not a full mathematician. Fokas is a well respected Professor at one of the top applied Maths departments in the world. A better…
Isn't MD for..like.. medical doctors ? If he's an MD but without a medical degree (which I guess is the case), then what's the difference betweeen MD, PHD or 'professor' ??
An MD is a medical degree. It's possible to have an MD but not practice medicine professionally in the same way someone can earn a law degree (a JD) but choose to not be a practicing lawyer. "Professor" is a academic title for people at a university or advanced teaching institution. Most professors do have terminal degrees in their respective fields like PhDs, MDs, or JDs, so you could also call a Professor with a PhD "Dr. X" instead of "Prof. X". But, "Professor" is generally considered a more prestigious title since its much rarer and harder to get a professorship than get an advanced degree.
So this goes off on a tangent but I feel it relates to noncentrality [0]. Fokas has a PhD in maths. Being an MD or having gotten an MD 40 years ago is clearly entirely non-central to his career. Calling him Mathematician-MD seems like it is meant to make him seem a lesser mathematician, e.g. by insinuating that this is just something he does part time, and that he can hence be taken less seriously. I don't know what…
I actually believe the intent is the opposite. I think the PR people are just trying to sell Fokas as a polymath genius. "Look, he is not only a mathematician but also an MD, wow!"
yes he is a polymath genius whether you accept or not