Logs will appear no matter what, at some point in the line of reasoning. They are only held back until differentiating in this approach. I think the expression for the derivative of logn/n was much nicer to grapple with.
Seeing as we're in the integer world, I'm not sure logs need to show up ever, there's probably a proof path around unique prime factorisations. E.g. it's more or less immediately obvious that n and m would need to share the same prime factors in the same ratios.
The key thing you need is that for n >= 3 the sequence of (n^1/n) is decreasing. You can get that pretty easily without logs.
Look at two consecutive terms: n^(1/n) and (n+1)^[1/(n+1)]. The ratio of the first to the second -- we want to prove that this is >1 -- is n^(1/n) / (n+1)^[1/(n+1)]. This is bigger than 1 iff its n(n+1)th power is; that is, iff n^(n+1) / (n+1)^n > 1. We can write this as n (n/(n+1))^n; so what we need is that (n/(n+1))^n > 1/n.
(Handwavily: we know that the LHS is about 1/e, so for n>=3 this should be good. But we want a proof and we're trying to do it without nontrivial analytic machinery.)
Actually, I prefer to write this as ((n+1)/n)^n < n, or (1+1/n)^n < n. Expand this with the binomial theorem: we get sum {0<=k<=n} of (n choose k) n^-k. And we have (n choose k) = n(n-1)...(n-k+1) / k! < n^k/k! so this is strictly less than sum {0<=k<=n} of 1/k!. And for k>0 we easily have k! >= 2^(k-1) so this sum is no bigger than 1 + 1 + 1/2 + 1/4 + 1/8 + etc. = 3.
So, the function on integers n -> n^1/n is decreasing for n >= 3. Now the proof goes through as before.
(Maybe there's a more thoroughly number-theory-ish way to do it by looking at prime factorizations, but when I try it that way it always seems to end up rather a mess.)
[EDITED to add:] But elsewhere in the discussion users bustermellotron and diffeomorphism give (very similar) neat number-theory-ish proofs, either of which is definitely a better proof than the one using the calculations above.
This back-and-forth makes me wonder, is it possible to write proofs about proofs? e.g., you cannot prove this result without logarithms or there must exist a proof that involves prime factors?
I suppose at least some proofs vaguely like that are possible because Gödel's incompleteness theorem is one, although I suspect that that same theorem puts some constraints on these "metaproofs."
Yep, I took several metamathematics classes at UCLA! Mostly on proof theory, but also analyzing metamathematical results (like Godel's theorems, Henkin construction, and so on).
Agreed! I'm realizing that my comment came across as negative, but I did appreciate seeing a second path to the same place. I also agree that the expression x^(1/x) feels like a more natural place to start.
You often see this I think, in "pretty" proofs compared with the more direct approach. A clever early step or some bit of startling insight.
For the past decade there has here in Norway been a recurring news sensation surrounding the mountain Mannen[1] (literally "the man") located in Romsdal in the northernmost part of Western Norway. Geologists have found Mannen to be at high risk of a sudden and large-scale landslide, placing the local population at the base of the mountain at great peril. There have been several abrupt evacuations of the place over the years, whenever measurements indicate a sudden increase in risk, with great media exposure and public interest. However, The Man has still yet to fall as predicted.
A website was soon set up: "HarMannenFalt.no"[2] (Has the Man fallen?), displaying no more than the word "NEI" (No). The sentiment of the joke was felt by all, as many had grown tired of all this attention being directed at what was essentially, on its surface, just a mountain, standing there, doing nothing.
I was a student at the time, and remember many idle screens around campus set to this page, reeking of that sarcastic mock-worry only a student body could put on. But I suspect a lot of us were secretly waiting for the moment when the text would change into "JA".
And soon enough, something did in fact happen. A neighbouring mountain, Veslemannen (The Little Man), saw a considerable increase in daily movement in 2018, before finally there was a minor landslide in 2019. But this was not the landslide we had been promised and waited for, as The Man himself was still standing tall as ever. A "JA" was not really in order, but then perhaps neither was really a "NEI".
Instead, the website was updated to show "TJA", a Norwegian word somewhere between yes and no, with meaning closer to "Well, perhaps".
As an aside: A second website with identical design was also created: "HarDovreFalt.no"[3]. A play on the concept with allusion to the vow of our founding fathers: "Enige og tro inntil Dovre faller" (Agreeing and true until Dovre falls)[4]. The "NO" on this second site, I think, is far less likely to change any time soon.
There was a has the LHC destroyed the planet yet site created when the media kept that sensationalist lying campaign against turning it on (at the early 2000's), but I think the funniest one was the "how many days since the last Java 0-day" around the time Oracle brought Sun (around 2010, wasn't it?), just because of the sheer absurdity of keeping it static for almost an year and being correct every day.
I remember before those there was a famous "how to destroy the Earth" site, with a status page answering "has anybody done any of this yet?".
One of the silliest battles I have found myself fighting against society has been my search for comfortable yet attractive men's pants. I deeply dislike the restricting fit of modern pants, especially tighter fitting jeans, but also the semi-current fashion in suits and pants in general.
It may be that a catalyst for this conflict has been my anatomy. My thighs and butt are huge, compared to other guys of similar proportions, and so apparently they cannot be contained by modern fashions. Every time I try to put on a pair, they're either too loose in the waist, or too tight in the thighs, and always sit too low for my liking, digging into my groin if I force the belt to sit at my waist. And should i find a pair that feels acceptable at first, my optimism only lasts until I have to sit down and feel the edge of these clothes digging uncomfortably into my flesh, rekindling my hatred for these trends and their designers.
The current resolution to my woes has been Darcy Clothing's line of suspender pants[1], which I have had tailored after purchase to taper and fit my height. However, this look is quite jarring in a modern environment, so I tend to wear a sweater over the suspenders. Still, I sometimes need to take it off, and wearing something like this day in and out puts quite a label on you.
Still to this day I keep my eyes open, hoping that the pants of my desires will appear in a shop somewhere, or that trends will change, and we get style in service of comfort, instead of discomfort in service of style.
Same here. The secret I’ve found: look for athletic cut. That’s the cut many hockey players wear, who typically have huge thighs and a rumble seat because of the muscles skating works. Those are the only pants I buy now.
I even got Michael Strahan suit pants recently, since they were the only ones in the store that fit me nicely.
My solution for years is to hit the thrift store and opportunistically grab pants that are in good shape but too big, and then take them for alteration. For a few bucks plus tailoring, I can have something that will fit me better than anything off the rack.
I have similar issues. Thick thighs and meaty butt, I had some nice looking pants that used to allow me full range of movement that I had used for a while but a few weeks ago I tore a pair when squatting down. I'm now currently resigned to joggers for the time being. It feels like there are no decent pants out there for men who lift
Same here. A youth of track, soccer and skiing has bequeathed me a rear that most of the women in my life are jealous of, and yet it is a curse. Nothing fuckin fits right!
I also have the giant-thighs-and-butt-from-heavy-squats-and-deadlifts situation.
The "Athletic taper" pants from Jcrew and also the similarly named ones from Banana Republic and the Gap are great. Sometimes you can buy one size up and then have the waist taken in by a tailor to get a better thigh fit.
I too have recently adopted a suspender lifestyle while in search of comfortable trouser fits. The main annoyance I've encountered is the need to remove all top clothing layers when using a toilet. This should be less of a problem in the upcoming warmer months, though they will also provide less opportunity for suspender concealment.
I don't know if you've tried it, but there are places that can sell you custom tailored jeans, usually in the form of raw denim that -- because it will do such a great job of conforming to exactly your shape, to the point of 'remembering' your wallet -- are generally built about a half-size too large and shrunk to fit by washing.
lol, I mean that denim has the particular attribute of forming to exactly your proportions. We've all seen old denim -- its wrinkles are bespoke to the wearer, its scuffs are defined by the work that was done in them.
After enough (read: not that much, actually) usage, you'd be able to look at a pair of jeans and know which pocket their owner kept their things in, whether the things kept were a wallet vs a cell phone, etc., and exactly how they placed the wallet therein.
I bring it up largely because denim is basically the most durable thing we can buy, so if you're going to have to get custom clothing, might as well get custom clothing that will last, and not limit ourselves to only looking tailored when we're in our custom formalwear
> My thighs and butt are huge, compared to other guys of similar proportions
Have you considered looking at the women’s? Might be no better in the end, but this reads like what’s conventionally a feminine anatomy, and garment designers will generally go with conventional anatomy, if not outright prescribe completely imaginary ones you have to conform to.
I absolutely have! Didn't think it was worth the tangent so I'm glad you're bringing it up. Women's fashion is so much more varied in shape, size, style, and fit, and some of it even looks comfortable. As a matter of fact, if you search for high-rise trousers online, most of the results are precisely in the women's section. I think that will have to be my next stop, though it's a line to cross and somewhat of pill to swallow.
As an aside, I have said for a long time that I wish it was socially uncontroversial for a guy to wear a loose skirt or dress in public, and that if I found one I liked I would wear it all the time.
> As an aside, I have said for a long time that I wish it was socially uncontroversial for a guy to wear a loose skirt or dress in public, and that if I found one I liked I would wear it all the time.
Yeah it’s becoming a bit less uncommon but not by much.
Maybe you could make up a Scottish heritage and wear a kilt? Other options would be overalls (there’s overall high-fashion) you might have the same shape issue as pants but they won’t fall down thanks to built-in suspenders (which you can play with e.g. cross, halves, …), or onesies (kigurumi) if you don’t mind being seen as a weeb.
Ugh, I'd say women's clothing is way worse than men's. No standard sizing, and too many faddish items. Women's clothes typically lack usable pockets and they are not as durable. Crappy fabrics that shrink and fade, horrible zippers that break...
That being said, you might want to go to Kohl's and try on Lee women's pants. Those aren't bad, although I hate the way Kohl's marks stuff up and then puts a normal price on it and calls that a sale.
> hoping that the pants of my desires will appear in a shop somewhere
An early-web startup did custom-cut khakis. CNC cut, manually stitched in one of the last US factories doing such, and overnighted. Acquired by bigco, and death by mismanagement. Patents should be expired now, so perhaps progress can resume?
The tailoring was just for taper and length. The waist I didn't do much with so it's pretty loose. Besides, the pants are simply supposed to be worn with suspenders. I'm not sure as I haven't tried fitted high-rise trousers, but I think it would be hard to keep them in place otherwise.
As for my BMI, I'm 183cm at 85kg, which I understand nudges me just inside of "overweight".
Me too, and I was disappointed to see this comment section mostly full of people either blaming the author or explaining why their experience actually makes sense.
It's like a positive feedback-loop (though I guess morally negative) between manufacturers, developers, and users, all three dragging/pushing on the other to roll down the hill of faster hardware and heavier apps, none of them really in control or able to stop, and therefore also not directly to blame. The upgrade-train to Waste City has no breaks. Still, it seems to me at least that the manufacturers are the only ones strictly happy about the situation.
"Permitted" is a strong word for activity enabled by a protocol. It implies some moral acceptance from the ones who designed the tool. I think a better word would be "possible". Similar to how e-mail makes it possible to send all kinds of content.
You are right to point that out, but I don't think the article is making a claim that its findings compose a recipe for exceptionalism. It actually seems very careful not to make any claims at all, except that it presents "a few of the patterns that have struck [the author] after having skimmed 42 biographies".
I've been surprised to find myself making multiple similar pivots over the years. A lot of what I dismissed in my younger years as lacking of intellectual merit, I have come around to value much more highly than I ever would have guessed.
It's funny, but dancing in particular has been my greatest fear for as long as I can remember, so I've always done my best to avoid situations where I'd be forced to do it. Not only did I not see the point or appeal, but I deeply dreaded the prospect of going on the dance floor.
As fears go, however, this one is quite innocent, and also fairly simple to face. And how empowering wouldn't it be, to overcome ones biggest obstacle? So not too long ago, when I was invited to a dance class by a group of friends, I decided I'd go, just the one time, as some sort of exposure therapy.
The dance was tango. As I had expected, I was anxious, stressed, and scared through the entire class (although it was very helpful having the excuse that it was my first day). What I did not expect was how fascinating the dance would turn out to be. How beautiful and elegant, yet also technical and structured. I saw our instructors dancing some dead simple improvised steps, and somehow, despite not being planned out at all, they were totally synchronized. There had to be some brain-to-brain communication going on, a sort of body language that I could not yet perceive or understand, but that I knew I wanted to learn.
That was six months ago, and since then I've gone to some sort of tango thing multiple times a week, every week I've been able to. I'm still nervous on my way to a class, and it takes a few dances to warm up and get loose each session, but I'm starting to enjoy it and feel like I'm actually dancing. In the process, I've come to understand more deeply what people mean when they say "there's more to life than X".
That's a pretty flimsy argument. Yes, you can map mathematical ideas onto concepts within different fields of study, but that's kind of the point. Mathematics aims to abstract away any such notions of where you might encounter the natural numbers or the reals, "in reality" or elsewhere. All you've persuaded me of here is that philosophy has a use for math.
