Frank Ramsey – The Man Who Thought Too Fast
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But they grow very rapidly. Ramsey theory has usually the biggest constants found in mathematical papers.[1] There is a famous quote by Erdos:
"Suppose aliens invade the earth and threaten to obliterate it in a year's time unless human beings can find the Ramsey number for red five and blue five. We could marshal the world's best minds and fastest computers, and within a year we could probably calculate the value. If the aliens demanded the Ramsey number for red six and blue six, however, we would have no choice but to launch a preemptive attack."
I swear mathematicians are witches.
and it's companion https://waitbutwhy.com/2014/11/from-1-to-1000000.html are pretty cool.
You just get dizzy even thinking about it. The observable universe is just not really that big when talking about this stuff, there are not even 100 orders of magnitude in the universe from smallest to biggest. Anything that you can have a conception of (even stuff like the number of possible lottery tickets or possible books that could ever be written etc) is just indistinguishable from mundane everyday sizes when dealing with googology. You're trying to roar but it comes out as a whisper. You have to explicitly design these. Most of the time if you think "but wouldn't it be comparable if we counted the XYZ) the answer is no.
It's like an adult equivalent of a kid saying "I get that galaxies are big, but are they also bigger than really big houses? How about a house with its own swimming pool? Really really big mansion of a big movie star?"
Edit: Tone is difficult in writing, I don't mean to be condescending, I'm getting dizzy myself when thinking of this and remember my frustration and amazement when first hearing about these.
Why did nobody ever tell me to think of logs like this before!!?
https://math.stackexchange.com/questions/1384917/relation-be...
Fwiw, I did't read it that way at all. I thought the child's question was a great analogy.
A million seconds is 11.5 days from now. A billion seconds is 31 years, 8 months, and 15 days from now. A trillion seconds is 31,709 years from now.
Or, in other words, blowing a kiss against a nuclear blow? That absolutely hugely more relevant than taking the log of the Graham's number. I can't even say how hugely more relevant it is without an almost equally irrelevant comparison.
(And yes, I am certain you know this. It's a pedantic clarification, I hope it made the point clearer.)
On the other hand there is just maybe a page of explanation needed for a somewhat educated layperson to understand the construction of Graham's number.
So on one hand it's unexplicably big to the point where it's even unexplicable how unexplicably big it is etc, on the other hand it can be compactly described. It's an example of the power of thought and reason and deliberate design, making so much of so little.
https://en.wikipedia.org/wiki/Kurt_G%C3%B6del
https://en.wikipedia.org/wiki/Srinivasa_Ramanujan
https://en.wikipedia.org/wiki/William_James_Sidis
https://en.wikipedia.org/wiki/%C3%89variste_Galois
https://en.wikipedia.org/wiki/Niels_Henrik_Abel
In any case I hope you didn't feel attacked. It was half curiosity half critic :)
Dying early would boost DALY count.
Like von Neumann, he's famous for a reason.
And doubtless several more. By the way, not only are Scholze and Tao outstanding talents, but very nice and humble, as in ACTUAL nice and humble, and they (especially Tao) write in a crystal clear way so they are wonderful as expositors and academic writers. Makes one chuckle comparing that to so much unrestrained ego online.
[0] https://terrytao.wordpress.com/ [1] https://mathoverflow.net/users/766/terry-tao
Intuition for visualizing/imagining high-dimensional geometry: https://mathoverflow.net/questions/25983/intuitive-crutches-...
Intuition for convolution: https://mathoverflow.net/questions/5892/what-is-convolution-...
This is much more satisfying way to think of these concepts. The idea of thinking of convolution as a 'fuzzy' optical phenomena, the idea of thinking of n-dimensional space as a probability distribution.
It's interesting the way he grounds his intuition in practical applications. For convolution for example, a common application is to convolve a 2d image with a gaussian kernel to fuzz the image. I know that, but always still had a not-very intuitive, but very dry and technical understanding of convolution as a sort of dot product of two vectors representing the underlying image and the kernel. Terence Tao in contrast exploits the practical intuition of 'fuzziness' in this process to suggest thinking of convolution as a fuzzy (probablistic) addition of functions. It's a subtle step, but giving some sort of physical, or visual intuition for mathethematics like this is so helpful.
IQ doesn’t measure absolute intelligence, but rather assumes it is a normal curve and maps that to human friendly numbers: mean 100, standard deviation 15 or 16 depending who you ask.
The same thing has the curious side effect that if the number of people in comas at any given time is greater than 1, then coma patients must have an IQ > 0.
^in the sense that it measured by IQ tests, anyway. Point being it shows a real difference; deltas over 200 aren't meaningless.
IQ is total pseudoscience nonsense of zero value to anyone or anything.
It's one genuine use is as a fig leaf for the very worst kind of racism. Treat it and anyone touting IQ an indicator of anything with extreme suspicion. Nazis love IQ. Goodwin's. /Thread
Wasn’t the Nazi nuclear program severely delayed by their race-based hatred of Einstein for being Jewish?
The latter is what this thread is talking about: it's obvious that we shouldn't be studying psychometrics the way the Nazis did, but it's not obvious that we shouldn't be doing it at all because they did (as with nuclear physics).
Hilariously, even harry8's complaint that the Nazis loved IQ is precisely backwards: Hitler banned IQ testing for being "Jewish" too.
EDIT: I actually was curious about this last claim, so I checked the source that the Wikipedia article points to. While this text was written by one of the most-cited psychologists in history, there's little else out there to concretely corroborate or refute that IQ testing was _banned_ by the Nazis. The evidence indicates that their attitude was somewhere between apathy and hostility towards the tests.
IQ is loved by Nazis as proof of master race bullshit. Really.
Treat IQ with contempt. It's an indicator of idiocy in a discussion. (All of us are capable of idiocy in discussions, including me, don't fall for the IQ trap. You're above it. We all are).
I don't personally believe that racial IQ differences are significant or salient to differences in population outcomes, nor am I convinced that claims are well-founded that IQ determines individual outcomes to a significant degree. But I don't advise treating it like a magic word that shuts off the thought centers of your brain. If you think it's a useless concept, you should be able to articulate why by pointing to the science (or lack thereof), not by letting Nazis tell you what you're allowed to think critically about.
It's not only that it is used by Nazis. It's that it is used for literally nothing else of positive value to anyone. It's a disgrace.
Sorry for dancing around it not not starting it as plainly as possible.
"Anyone quoting IQ scores as justification for anything at all is talking bullshit. The end."
And they are using a technique of bullshit argument support favoured by goddamn Nazis.
The jury is back. IQ is guilty. Put it right there next to lobotomy as a gift from psychology to the world. Stamp it out. NOW!
Most of the relevant scientific community disagrees with your assertion that IQ is purely pseudoscience. From Wikipedia:
> Clinical psychologists generally regard IQ scores as having sufficient statistical validity for many clinical purposes.[25][68][69] ... "On the whole, scholars with any expertise in the area of intelligence and intelligence testing (defined very broadly) share a common view of the most important components of intelligence, and are convinced that it can be measured with some degree of accuracy." Almost all respondents picked out abstract reasoning, ability to solve problems and ability to acquire knowledge as the most important elements.
There's plenty more on the topic, and as always, Wikipedia is a great place to start your search for sources.
In what way is your blanket dismissal of the scientific consensus different from anti-vaxxers ("the only use of vaccines is support for Illuminati mind-control!!) or flat-earthers ("the only use for a round earth theory is support for, uh, the globe industry!")? Or for that matter, what makes you different from IQ essentialists like neo-Nazis, that take a nuanced scientific concept and flatten it into an all-or-nothing perfect predictor of outcomes? Hell, at least they're _directionally_ in agreement with the scientific establishment about the validity of the concept.
There is no evidence. Dig into anything that is claimed as evidence and it's clearly invalid, at best. As a statement of the overwhelimingly obvious: the field of Psychology has issues. It knows this. You've heard of the "replication crisis." There are those trying to right it and treat it like it is actually science and rip out all the pseudo-bs that psych is redolent with. Good luck to them. Maybe they'll even remove the utterly discredited stanford prison experiment from every single first year psych textbook? Or keep it as yet another example of collective delusion in the discpline caused by faked research? Yet another one. But in every textbook. Credibility?
"On the whole, scholars with any expertise in the area of intelligence and intelligence testing (defined very broadly) share a common view of the most important components of intelligence, and are convinced that it can be measured with some degree of accuracy." - Anyone who disagrees and asks for the evidence is defined as lacking expertise. It's frigging comical. But it does screw up lives and justify racism so, there's that. Get told you're stupid as a child, there's a good chance that's self-fulfilling.
There is ovewhelming evidence that vaccines work. Seen a polio case, well, ever? There is none that IQ is a useful scientific metric. See Egas Moniz's Nobel prize for how psychology can move as a discipline on mass without evidence. Or maybe butchering brains and getting nobel prizes for it isn't clear enough evidence that you can't take anything in psychology "on trust" anymore than you can physics.
IQ is a disgrace. It is already and will continue to be increasingly sighted as how bad the discpiline of psychology has been for the past 100 years. May they own their shame and do better. Your IQ score being higher than mine means absolutely nothing. My IQ score being higher than yours means nothing. Absolutely nothing. Zero. But yeah, you can convince people it's has meaning and use it mess up lives - even if you don't intend to do that. Like the altruistic motives behind lobotomy and the success they had with the Nobel committee.
A much better explanation here, from the fairly wonderful blog of cosma shalizi http://bactra.org/weblog/523.html
And most importantly, it won't be predictive.
Unless the former has taken the same test or some subset of the questions before. Which isn't unusual...
I’m struggling to understand your problem with IQ tests. If you have an IQ of between 90 and 110 as per a standard test no one really cares. You’re “normal”. Score higher maybe a bit more intelligent at spacio-temporal problems and it’s probably a good predicator that you’ll do well at mathematics.
If you score below 75 you almost certainly have an intellectual disability and need help. You can give these people any other test you like, it will make no difference. They won’t “function” in a society where the mean IQ is 100 (which it is).
Andrew Wiles, for providing a proof to Fermat's Last Theorem.
Grigori Perelman (and Richard Hamilton), solver and winner of the Millenium Prize Problem: Poincare's Conjecture.
- https://en.wikipedia.org/wiki/List_of_things_named_after_Leo...
- https://en.wikipedia.org/wiki/List_of_things_named_after_Car...
Judged by the number of multiple-surname ‘things’ on both pages, Gauss seems to have been more of a collaborator, or more of his discoveries were also made by others at about the same time.
> Lydia made the mistake of remarking, “What a beautiful tree,” presumably too casually, whereupon Wittgenstein glared and demanded, “What do you mean?” and she burst into tears.
For example consider a recently disproven long-standing problem of Ramsey theory:
Is it possible to colour the natural numbers with two colours such that no Pythagorean triple is all one colour?
This question was asked by Ron Graham many years ago with a $100 prize for the answer, assuming it would be reasonably easy to prove. It was recently solved with an absolutely enormous multi-terabyte computer proof. It turns out you can colour everything up to 7824 fine but no choice for 7825 is sufficient.
Notice the % that is content.
Wasn't he a pretty ruthless and competitive guy?
Climate change, the federal debt, increasing shareholder value over anything else, I wonder what Ramsey would have to say about the current state of the world.
And he was an academic and teacher throughout, graduating in 1923 and becoming a fellow of King’s in 1924, then lecturer, then Director of Studies.
High government spending means high taxes now, or via debt, high taxes in the future.
In general, when the government consumes a lot, the private sector can consume less of the national product.
Deficit is Government spending - Tax collected. High deficits might lead to inflation. One way to control inflation would be to increase taxes(draining money out of the economy)
> In general, when the government consumes a lot, the private sector can consume less of the national product.
This is not exactly true. When there is still "slack" in the economy(it is not at full employment, the private sector isn't investing for other reasons etc.), government spending tends to have low inflationary pressure. It is only when the economy is operating at peak productive capacity that government consumption crowds out private sector consumption.
You are right, that's why I called it another possibility. To list them all explicitly.
Higher debt now means that the future either has default, higher (than anticipated) inflation or higher taxes.
(A steady inflation doesn't do anything to debts: at the time the debt is incurred, any anticipated inflation shows up in the nominal interest rate. Similarly, getting a reputation for defaulting increases your nominal interest rate. A reputation for high taxes doesn't do anything to the interest rate directly.)
Responsible governments try to avoid default and ever increasing inflation. That leaves taxes. Or taking on less debt in the first place.
About slack:
Modern central banks target inflation. If there's slack like you describe, a competent central bank will inject more money until inflation is at the target.
Central banks wire their profits to the treasury. That includes the seigniorage income they make from propping up inflation back to the target.
That's just bog standard monetary policy. And doesn't crowd out private sector investment and consumption and is perfectly capable of picking up any slack.
(That's not to say the fiscal side of government doesn't have any effects at all. But that's more on the supply side.)
Debt help by citizens of the country is also somewhat neutral in terms of sustainability. The government needs to pay interest on it, but the money ultimately goes into the economy, so that might also be sustainable in larger amounts than might be intuitive.
Foreign-held debt is another matter, and I'm not enough of an expert to speak on the ramifications there. But I do think in general it's important to recognize that national governments which control their own monetary policy are a very different beast than a household or a business, and so their debt liabilities can't necessarily be thought of in the same way.
But government debt is a problem even if the government doesn't go bankrupt:
You are right that a government that can print its own money can hardly go bankrupt financially. But it's perhaps not ideal for the government to consume ever increasing portions of what the nation produces? Forget about money and finance for a second: given the same material resources the private sector is usually more efficient at satisfying human wants.
I think this is a very dangerous simplification. There are certainly areas where private sector does a much better job than the public sector. But there are also areas where the public sector in general is superior. Examples are things like healthcare, infrastructure and education.
The private sector tends to do poorly in any area where the returns on an investment are not captureable by the investor individual but instead accrue to society as a whole.
If you believe 'The Case Against Education' the individual captures more than 100% of the net returns from education, ie there are negative externalities. That means society consumes more education than is optimal on utilitarian grounds.
That conclusion is counter-intuitive, but the argument is simple: education is mostly a signalling arms race.
> Caplan argues that the primary function of education is not to enhance students' skill but to certify their intelligence, conscientiousness, and conformity – attributes that are valued by employers. He ultimately estimates that approximately 80% of individuals' return to education is the result of signaling, with the remainder due to human capital accumulation.
There is certainly a LOT of signaling involved in education. As a college dropout and self taught developer this is intimately clear to me.
However, I have traveled enough of the world and lived enough places to see what happens when education systems are weak or non-existent. I think you underestimate the degree to which high levels of illiteracy and innumeracy absolutely destroy the potential for economic growth and prosperity.
It is possibly true that we (the USA) consume more education than is pragmaticly optimal. However, removing public education would result in a loss of economic output that far exceeds the savings. Predicting who will need that education sufficiently accurately is imposy for both society as a whole and for individuals contemplating their future.
Also not that I hedged my statement with the weasel word 'usually'.
The lower the interest rate, the higher the safe level of debt.
Right now federal bonds have a negative interest rate, so perhaps there is no limit to the amount of debt, and in fact that there might be a minimum safe lower bound. Or maybe not; we don't have much experience in dealing with this sort of situation since it was seen to be impossible.
Putting that aside, the UK had debt around 200% of GDP after the Napoleonic Wars, and eventually paid most of it down. So the "bad & unsustainable level" appears to be well above that.
Japan has avoided financial collapse with a giant debt to GDP ratio for years now, I think the advent of QE and owning your own debt via central banks has made the whole thing seem a bit silly.
Provided the interest rates don't rise before the debt is paid off that is. Serviceable might be a better word.
If the financial system creates such quantities of debt, that the regular capital and interest repayments exceed the available money supply, that would be a hard threshold. This is quite a large number if you calculate, even with high interest rates, so highly unlike in practice. Plus bank lending has a side effect of money creation, which stops it happening within the banking system.
However government debt and company bonds are non-bank debt, and securitised loans also don't have the money creation property, so one of the issues in 2008 can be presumed to be that the expansion of the debt supply relative to the money supply probably did become an issue in some countries.
All examples I've ever read or heard of involved foreign currency debt, or to be more precise, debt in a currency that the country in question didn't control (e.g. developing countries with USD-denominated debt, Euro-zone countries, historical cases with debt denominated in gold, etc.).
That's an important qualitative difference.
> By the mid-1920s, interest on [British] government debt was absorbing 44% of all government expenditure, [...]
That definitely sounds like an issue. Without the debt, they presumably could have cut taxes in about half.
(And gold standard or not doesn't make too much of a difference, if you have a firm commitment to keep inflation in check and not to default on your debt either.)
That's not an issue if, by taking on the debt, the government allowed the economy to grow sufficiently faster.
The conservative analogy to a household budget doesn't make sense. Analogizing to a startup where you're best off taking on debt for more capital, to increase growth, also don't make sense. But it's an open question, which of these makes the least sense.
Perhaps. Though that's a big If. Most government expenditures in practice are more like consumption than investment. (In Britain's case, the expenditure was sui generis: the first World War.)
You also have to worry about crowding out (https://en.wikipedia.org/wiki/Crowding_out_(economics)): whatever resources the government uses, the private sector can't use.
So the yardstick for 'sufficiently faster' is not some absolute positive return, but the comparison to what the private sector would have done with the resources.
> The conservative analogy to a household budget doesn't make sense. Analogizing to a startup where you're best off taking on debt for more capital, to increase growth, also don't make sense. But it's an open question, which of these makes the least sense.
Please explain. As far as I can tell, making the household budget analogy work is almost the definition of competent monetary policy. So that analogy is more of an 'ought' than an 'is'.
The 'startup analogy' is perhaps more apt for small open economies. Though even there the total amount that foreigners are willing to lend the country as a whole is probably finite and limited by the country's total future earnings. Those total future earnings have to pay back the debt; either directly in the private sector, or indirectly via taxes.
Now, IF you have incompetent monetary policy, then indeed the household budget analogy breaks down. Incompetent monetary policy means that total nominal spending in the economy is not stabilized, and so fiscal stimulus MIGHT make sense.
(Interestingly, the conceptually simplest way to have competent monetary policy is to have no monetary police. Ie leave note issuing to a private laissez faire financial sector. That tends to stabilize total nominal spending in the economy. See the Canadian experience https://www.alt-m.org/2015/07/29/there-was-no-place-like-can... for a close enough approximation in the real world.)
Crowding out in the traditional sense of discouraging the private sector from investment is really a function of interest rates. For a government that runs its own, free-floating currency, interest rates are a policy variable independent from deficit and debt.
That even competent monetary policy cannot be relied on to make the household analogy work was proven empirically beyond any reasonable doubt via the famous "zero lower bound".
There is also the question of whether monetary policy should even try to make the household analogy hold. This is ultimately a political question, and especially in a democracy there a good reasons why it shouldn't.
The zero lower bond is not a problem. You can still do unlimited QE, independent of interest rates. Or you can do what Singapore does and not use interest rates as a policy instrument at all (they use foreign exchange rates; the US could use eg a basket of commodity prices, if they wanted to).
I think issues in these particular countries is related to a combination of foreign debt and trying to keep their exchange rate within a fixed band.
Inflation has a nasty tendency of sustaining itself.
I don't know much about what you're referring to, and a cursory search did not provide any evidence that the root cause of sustained high inflation in Brazil was debt at any time, much less debt in Brazil's own sovereign currency.
Nobody disputes that bad governance can lead to high inflation. The point of dispute is whether we have to reduce debt today because in the future that high debt may lead to hyperinflation, recession, or some other negative economic outcome that is so bad that its badness would outweigh the badness of austerity measures implemented today.
But yes, now that you have cited it, there aren't many mainstream media articles in English that say it. I couldn't find any either.
The US Federal debt has been continuous since soon after the revolutionary war with only brief periods of net savings.
Another amusing story about US sovereign debt is the Louisiana purchase. As the story goes Napoleon sold Louisiana to the United States to raise money to finance his wars. The US didn’t have cash on hand to pay, so they floated the Louisiana 3’s which were bonds that paid 3% to raise the money. The agent for the bond sale was Barings, and they were sold on the London and Amsterdam markets. So it was largely the British investing class who were literally financing Napoleon.
The federal debt has had no observable effect whatsoever for decades, the dollar is as strong and internationally valued as ever. Climate change has not made anything measurably worse in the 3 decades people have been screaming about it. The bit about shareholder value is too unspecific.
If you take Ramsey's ideas about intertemporal consumption and the bliss point seriously and you try to do an empirical calculation, it generally implies that the utilitarian thing is to have savings/investment rates anywhere up to 98% (!!!) vs the current savings rate of ~0%: https://plato.stanford.edu/entries/ramsey-economics/#NumeEst... See also 'turnpike theory' which builds on Ramsey and also finds that generally, the optimal thing is to grow as fast as possible to hit the limits as quickly as possible and only then start consuming: https://en.wikipedia.org/wiki/Turnpike_theory
What is the sensible/true version of this? Clearly it’s not correct as stated. You could have three pairs of two mutuals. You could have five mutuals and a singlet. Or six mutuals. Is it more a statement about probabilities?
But I suppose I will choose not to die on the hill of slightly misleading language and admit I was just wrong.
It goes in a question and answer format something like: "You knew a girl in high-school. In situation A, you find out years later that she is a stage actor and a dancer. In situation B, you find out years later that she is a stage actor. Compare how likely the second one is to the first one" In plain English there's an implied meaning of "she is a stage actor and not a dancer" in situation B. But that is not the statement. Situation B is as likely or more than Situation A but not if you use the English language description which suddenly is under-specified without real-world information.
Obviously you pick the things to play to the priors of the person you're asking so that if they think actors frequently dance you put that in there. That leads them down the false rabbit hole of whether it's "stage actor and dancer" vs. "stage actor and not dancer" that you're comparing.
In pragmatics (a subfield of linguistics) this is called "conversational implicature" - in particular it's an example of the "maxim of quantity".
Which is to say, you are correct about how the language works and mathematicians are a little weird.
Let the 6 nodes be users, and color edges between two users red if they are friends, blue if they are not. I think it's true as stated, as another comment claims.