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CoronavirusIt's great, isn't it? If you keep readjusting your model as you go along, you can do pretty much anything. You can make your model linear, or logarithmic, or... why even settle on one function? You can make up a new one for each iteration.
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CoronavirusApparently, it was originally China's idea. — Baden
Any bets on whether Trump will start calling chloroquine The Chinese Cure? -
Coronavirusis fatal only among the already very compromised — Hanover
Yeah, those sick old people had it coming - good riddance!
(I hear this "point" surprisingly often from people who argue that the threat has been blown out of proportion.)
Italy had 475 deaths yesterday. With population of 60.5 million and annual mortality rate of 10.2 per 1000, that's 25% increase of mortality rate (even if we allow that, say, 10% of those who died from coronavirus would have died from other causes within the same period). -
The Diagonal or Staircase ParadoxOK, I didn't notice that you said continuously differentiable in a later post, so sorry about that. But the staircase function is worse than not continuously differentiable - it is not differentiable at all in some points. In any case, no one has actually tried to show that not being continuously differentiable is even a sufficient criterion for the non-convergence of the length difference, and in fact it is easy to construct counterexamples. So all this arguing is kind of a waste of time.
As for solving any of them. You'll need to do so relative an axiomatic system. If it's Euclidean geometry — boethius
I am not sure why you keep talking about Euclidean geometry, which, as you admit, doesn't even have the notion of a limit. You may as well be talking about group theory. Yes, I think it's pretty obvious that we are talking in the context where limits and such are defined; real analysis will do for the purpose.
For any continuous function like whose arclength for a <= x <= b is greater than b-a, its scaled down versions will still have the same ratio of arclength to b-a. So just about any continuous function at all that's not a constant. — Daz
No, if the arc length decreases any faster than in the examples that have been considered so far, it will converge to the length of the diagonal, as we intuitively expect. This is easy to do with any function whose distance from the diagonal can be scaled. For example, take the half-circle function and scale the peaks down by a factor of n - it will converge like a champ.
But yes, it is evident that whether the length error is constant, converging to zero or growing without bound is pretty precarious. You have to work to make sure that you get the "intuitive" result, because a lot of the times you will get something else entirely.
You can obtain the result of the other "paradox" by drawing a symmetrical sawtooth graph on [0,1] that collapses as n increases, and whose length increases without bound. I leave this as an exercise for those interested. — jgill
Yep, if your sawtooth graph doesn't have this property out of the box, you can easily make it so by multiplying it by some uniformly increasing function of n.
The reason this looks very counterintuitive to me is because if we put aside analysis and just look at what we get in the limit, every point on the staircase converges towards a matching point on the straight line - which of course has the length of the straight line. So what gives? Well, the formal answer is that the limit towards which the sequence is converging is not an element of the sequence: the limit points do not themselves lie on any staircase curve. This is not so unusual; for example, most converging rational number sequences do not converge to rational numbers.
Still, it just looks... wrong :) -
The Diagonal or Staircase ParadoxMy main purpose, as mentioned, was just to explain the definition of "discontinuous" and that normal calculus concepts may not apply. — boethius
Er, your terminology is all over the place. A continuous function has left and right limits converging to each of its points. The staircase function is, strictly speaking, discontinuous as it is pictured in the OP, but that is just an artefact of the coordiate system. If you tilt the X-Y axes, it will become continuous.
A differentiable (or smooth) function has the first derivative at each point; the half-circle function is differentiable (again, modulo axis orientation).
An infinitely differentiable function has all derivatives; the sine function is infinitely differentiable.
There are also piecewise- versions of all these (piecewise-continuous, etc.).
You are right, it's a sufficient condition for the failure of the arc-length functional to respect the limiting procedure, not a necessary one. I believe the staircase could be approximated by some differentiable curve (replace the discontinuities with regions of sufficiently high growth, I believe polynomials would work) and cause the same issues. — fdrake
Yes, that's just what I did with the half-circle curve, and I think the sine curve (with proper scaling) would work as well.
Do you know a sufficient and necessary condition that characterises this sort of pathology? Other than stating "the arc-length map of the limit of the approximating series of functions is not necessarily the limit of the arc-length map of the approximating series of functions". — fdrake
Interesting question, but beyond my modest pay grade, I am afraid :) -
The Diagonal or Staircase ParadoxThe half-circle wave is smooth though, i.e. the tangent (first derivative) exists everywhere. I used it for simplicity, but if we want an honest to goodness infinitely differentiable curve, we can use the sine function as in John's other example - just scale it by an additional factor of to make it behave. (I think this will work...)
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The Diagonal or Staircase ParadoxIt's in the second half of .
There must be some neat identity for elliptical functions at work here, because otherwise I wouldn't know how to calculate such a limit.
By the way, and seemed to suggest that the key to the staircase "paradox" is in some pathology of the shape, namely its corners, where the curve is not differentiable. But this is not so. Consider a similar example, where in place of straight lines there are smooth curves. I'll use half-circle arcs for simplicity:
No corners here, the curve is everywhere differentiable (although the second derivative does jump around at the intersections). As with the staircase, the amplitude of the wave tends to zero as the number of crests increases without bound. But as with the staircase, the length of the curve does not approach the length of the diagonal. We don't even have to do the calculation to see that the length of the curve does not depend on the number of crests (this is because the length of each half-circle is proportional to its diameter, and the total length of all diameters is the length of the diagonal). And so the length of the wavy curve is always .
What's more, with a simple modification we can make the length of the wavy curve increase without bound, just as in @jgill's example. Just replace half-circles with ellipses whose major axis is perpendicular to the diagonal and scaled by a factor of . -
Is Bong Joon Ho's Parasite Subversively Conservative?I thought back to Parasite after seeing another film with a somewhat similar theme. To be honest, for all its sleek execution and obvious talent, Parasite turned me off by its heavy-handed messaging (in retrospect, the Oscar win wasn't very surprising). The dream sequence at the close of the film is a good illustration of that. At first I thought that the filmmakers were finally going for a little subtlety by ending the film right there at the happy reunion. Let the viewers work it out for themselves and then congratulate themselves on their perspicacity. But no, they had to drive the message home, lest someone might be confused by the ambiguity, and so they cut back to the half-crazed boy sitting in his squalid banjiha.
The socially conscious film that I would like to recommend as an antidote is Shoplifters (2018) by the Japanese director Hirokazu Kore-eda. And if you like that, see Nobody Knows (2004) and Still Walking (2008), which I think are even better. I hear that Kore-eda's output is uneven, but these three are excellent.
Also, anything by the Dardenne brothers. -
Q. on Fallacy of False DichotomyWell, if someone says something to the effect of (A or B), and it is not the case that (A or B), then a logical fallacy has been committed. How damaging that is to their overall argument depends on the context. Sometimes, as you say, not very damaging, and the argument can be repaired without much difficulty.
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Regulating procreationthe popular myth merely unilateral or blanket statements based on some silly and highly questionable pop cultural myth or axiom accepted or taken for granted on the basis of faith, nonsensical circular reasoning and rote regurgitation outdated 19th century myths and archaisms archaic and highly debatable or questionable or easily disprovable and contradictroy — IvoryBlackBishop
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Thought as a barrier to understandingIf thought were the natural outcome or effect, brought on by confusion, then the more you think, the more confused you will get. — Antidote
That's a nice example of a self-undercutting argument. If the premise is assumed, then everything that follows from it can be dismissed as confused ramblings. No need to go any further. -
Intuitions About TimeThink Heraclitus and Parmenides. — Pneumenon
I was looking for something in the way of critical reflection, but I find only free-floating metaphors here. I've also been thinking about the metaphysics of time lately, but I prefer a more grounded approach. -
Intuitions About TimeTake these two:
1. Reality is fundamentally flux, and permanency is constructed
2. Reality fundamentally is, and change is an illusion — Pneumenon
I don't even know what either of these statements are saying. It seems like they make some substantive claims about reality, but when I try to nail these claims, they just slip out of my hands. -
The Diagonal or Staircase ParadoxI find no error in this. — tim wood
There is no error, hence the apparent paradox.
Here is another take I just thought of. We can define two functionals, one that gives some measure of the distance between the points of the stair functions and the diagonal (e.g. the average distance), and the other that gives the length of those functions. The shape functional steadily converges to zero, but the length functional does not. -
The Diagonal or Staircase ParadoxNice.
he number of "stairs" tells something similar how polygons start resembling a circle: — ssu
The difference is that in the case of polygons approximating a circle, with each successive step the error decreases (don't ask me for a proof - it's pretty messy, from what I remember), whereas in the case of the staircase the error stays the same throughout. -
Can I deal with 'free will' issue like this?People define free-will in different ways. And so they argue about different things. But it really goes back to the concept of "you". You like others, will say you have a body, you have a brain, you have... maybe a spirit or soul... two arms and two legs. Who is "you"? The idea of there being a "you" and the continuation of self is intertwined with all definitions of free-will. — Malice
Nature as in, our exact state. — Malice
Our nature is not identical with our physical state. I am not making some dualistic statement here - I am just agreeing with what you said earlier that I quoted here. What constitutes "you" or "your nature" isn't simply reducible to your physical body (or else your personal identity would be extremely volatile). It is a somewhat vague notion rooted more in psychology and culture than in physics - you know, like "free will." -
Can I deal with 'free will' issue like this?The solution to deal with its demoralizing power. ‘Solution’ sounds confusing, I’ll change that — Rystiya
What does "its" refer to here? You never say. -
Secular moralityWhat do you even mean by "being moral"? — Pfhorrest
Making right moral decisions. What is right is, of course, the very subject of morality.
The criteria for the success of what? A moral science, or generally any system of morality? The criteria for success of those things is to provide a means of answering questions about morality. When someone wonders what is moral, how do they figure it out? When two people disagree about what is moral, how do they resolve those difference? Answering how to do that, how to figure out those answers to questions about morality, is the criteria for the success of a system of morality. — Pfhorrest
So any procedure for answering moral questions will do, as long as it is comprehensive? No other criteria of success are required?
That you think I'm even trying to do that shows you haven't understood a word that I've said so far. — Pfhorrest
If by that you mean that I haven't read your articles on morality, then no, I haven't. That wasn't the subject of this thread.
I predict you'd respond here "aha! So you're starting with a system of morality already, your 'ought' premises, just like I said!" But no, no more than the physical sciences start with some set of unquestionable "is" premises. — Pfhorrest
That sort of Cartesian scheme that you outline doesn't remotely resemble the way science is done. But even if your approach was better at aping science, that still wouldn't make it any better than a cargo cult, because you still aren't thinking about why you do what you do. Why should morality resemble science? -
Do professional philosophers take Tegmark's MUH seriously?It's certainly incompatible with materialism. A mathematical ontology isn't compatible with there being stuff, so I don't see how it's physical. But I guess if we're allowed to redefine the meaning of "physical" to be whatever is consistent with physical models. — Marchesk
There is nothing to redefine here, because there aren't any commonly established definitions for physicalism or materialism. All these arguments over this or that being compatible with physicalism/materialism is pure wankery, in my opinion. Argue substance, not isms. (Same goes for realism, of course. Crispin Wright famously wrote that "a philosopher who asserts that she is a realist about theoretical science, for example, or ethics, has probably, for most philosophical audiences, accomplished little more than to clear her throat.") -
Can I deal with 'free will' issue like this?It seems that it's hard to say whether we have free will or not. — Rystiya
It's hard to say what free will is. It's hard to even get people to think seriously about the question: they would rather argue endlessly about "free will" than think about the question that ought to be addressed before anything else.
The solution is simple — Rystiya
The solution to what? -
Secular moralityBecause while you busy yourself with procedural details of how to reduce morality to a utilitarian optimization, you don't ask what any of that has to do with being moral. What are the criteria of success (other than aping the superficial trappings of science)? How do you jump the is-ought gap?
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Mathematics is 75% Invented, 25% DiscoveredThanks, I got it when I thought about it again. I got into a habit of thinking of numbers as abstract objects that just satisfy certain requirements, like Peano axioms, whose representation and internal structure need not concern us. I lost track of the obvious idea that a number can also be thought of as a collection with that many distinct objects - a cardinal*. When you mentioned von Neumann construction, I thought about Zermelo's construction with its Russian doll structure - that wouldn't work so well. I guess that's one of the advantages of the von Neumann model, right?
* Heh, now that I think about it, that was exactly how numbers were introduced to us in preschool, using little sacks with different amounts of marbles :) -
Mathematics is 75% Invented, 25% DiscoveredBeing correct and being clear is not the same thing. I took ten or so different courses of mathematics in college. It's been a long time and I have forgotten much of it, but I still understand basic combinatorics, thank you very much.
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Secular morality"Methodically" is the key word here. Yes, you have a method, but that's not enough to qualify as "science." If you don't understand why you do what you do, then your method is no more scientific than astrology or divination.
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Mathematics is 75% Invented, 25% DiscoveredThis wasn't very helpful.
It's still not clear to me how the use of this set theoretic representation explains "the number of functions from 2 into 2 is 4". Whatever representation you use, you still have one number (in this case, one set) in the domain and one number (one set) in the codomain. I mean, I can see (after your explanation) what he is doing, but I am not even sure how to formulate that correctly without specifically referring to the internal structure of a von Neumann ordinal. -
Mathematics is 75% Invented, 25% DiscoveredThe number of functions from 2 into 2 is 4. — GrandMinnow
I don't understand.
How is this four functions?
(FWIW in calculus texts the exponential function is usually defined generically as a power series.) -
Secular moralityAll we need for an "ethical science" is that kind of broad agreement. — Pfhorrest
Why do we need an "ethical science"? You never stop to ask yourself this question. This is cargo cult. -
Mathematics is 75% Invented, 25% Discovered2^0 = 1
The number of functions from 0 into 2 is 1. — GrandMinnow
x^y = the cardinality of {f | f is a function & domain(f) = y & range(f) is a subset of x}. — GrandMinnow
OK, but by the same token 2^2 = 1, because the number of functions from 2 into 2 is 1. What am I missing? -
Mathematics is 75% Invented, 25% Discoveredx^y may be defined as the number of functions from y into x. — GrandMinnow
What do you mean by that? -
Does anybody actually agree here?I'm wondering if everybody perceives themselves as all alone with nobody of the same "general color" as them, or if everybody else feels like they're in good company with like-minded people who just have "shades of disagreement". — Pfhorrest
That's an odd wish, from my point of view, to have a camp of people that agree with you on everything. But you strike me as an opinionated fellow, with a definite position on everything, so I can kind of see how you would expect all water to flow to the same level. I like to think of myself as too much of a chameleon to be the same color with anyone (though I am probably deluding myself).
I think it may be related to the Uncanny Valley effect: someone sufficiently different is just an Other, but someone who's a lot like us but slightly off is just... sick somehow, disgusting. — Pfhorrest
Intraspecific competition is the most vicious. -
Secular moralityIt is impossible to do science without agreement on foundational things like empiricism and realism and some form of rationalism (as in rejecting appeals to intuition, authority, etc). Those practicing scientists may not have all made explicit their philosophical assumptions, but the work they did as a community had to take them for granted; those who continued to dispute those principles did not become part of the scientific community, but instead became its opponents, disputing its results on what scientists consider fallacious philosophical grounds. Because those scientists had at least an implicit philosophical framework in common. — Pfhorrest
You are conflating actual practice with its philosophical interpretations. Science is done the way it is done not because scientists have come to an agreement about its philosophical foundations (even the philosophical community is far from such an agreement), but because science is a fairly distinctive enterprise and there is a particular way in which it is practiced, which scientists learn in the course of their training. This is not to say that science is a game with arbitrary rules institutionalized by tradition. I believe that modern science is a product of cultural evolution, the seed of which is just our instinctive way of understanding our environment, one which we practice on an everyday basis. Moreover, the weightiest normative criteria in science - closeness of fit and parsimony - are objective to an extent that few other activities can boast (which partly explains the trust that we put in science). It is because science is constrained between its determinate natural origins and its semi-determinate goals that we think we can retrofit it with determinate philosophical foundations.
Which leads us to the contrast case: morality. On the one hand, morality, like science, has deep evolutionary, cultural, social and psychological roots, which makes it fairly determinate and eminently suitable as an object of study. But the other, normative end does not hold up, because of course morality is itself normative. This immediately short-circuits any question about what ought to be moral - what ought to be moral is what is moral, duh!
There is a clear trend of moral thinking moving toward a more “scientific” methodology based on common experience and critical reasoning, we just haven’t fully developed a consensus on how exactly those principles all fit together yet. — Pfhorrest
Yeah, this is just cargo cult science, I am afraid. There is a science for everything nowadays (or rather since the Enlightenment), so there must be a science of morality! Never mind that it makes no sense - to be intellectually respectable it gotta look like science. -
CoronavirusThat can't be right. Death is a symptom. If you are asymptomatic, you don't die any more than you cough or have a temperature. And if you are asymptomatic, in most cases you don't get tested. — unenlightened
That's what I said. Death rate is estimated on the basis of confirmed infections. Adding a hypothetical number of untested and asymptomatic cases doesn't change anything if what you want to know is how dangerous and disruptive the epidemic will be, or what your chances of falling ill are, or what your chances of dying will be if you develop symptoms.
That is why the quarantined ship makes a good statistical sample - everyone was tested. — unenlightened
Yes, I mentioned that too. When the entire population or subpopulation is tested regardless of symptoms, that is where the true fatality rate becomes relevant. But such testing is done in a small minority of cases.
Bottom line is that statistics should be used with care. -
CoronavirusYou like reasoning and argument, don't you? Here's some reasoning and argument.
https://slate.com/technology/2020/03/coronavirus-mortality-rate-lower-than-we-think.html?fbclid=IwAR2Di6GSNzwF8WJ4RDVmkLjwJr7x8sY5Bnwq-UJY_Uv3WmUu4EmjHP3XRZU — unenlightened
There are reasonable points here, but the article is too focused on massaging the fatality rate. Yes, if you include asymptomatic cases, the fatality rate will go down. But the hypothetical asymptomatic cases that were not counted for fatality rate were also not counted for infection rate, so the net result is zero. All you've shown is that in addition to cases with pronounced symptoms, having fatality rate f, there are also X asymptomatic cases.
Now, if people were tested regardless of their symptoms, then knowing the true fatality rate would be relevant. This is the case with those who are quarantined and tested because of their previous contact with an infected person. But for most people testing is still confined to those with pronounced symptoms. -
Secular moralityWhy do those ones deserve an exception?
The physical sciences we have today began as a branch of philosophy, "natural philosophy", that pretty much solved its foundational questions and then went on to do the business of applying them. — Pfhorrest
Science didn't wait around for its foundational questions to be solved before it could get off the ground - if it did, it would have been waiting to this day. Historical nomenclature aside, what we today recognize as science came together haphazardly as a living practice, rather than as a systematic application of a fully developed philosophical program. If anything, metaphysics and epistemology have for the most part been playing catch-up to science, taking its practice and its findings as a subject of study.
There is no reason to think that moral philosophy cannot do the same thing, solve those foundational questions, and go on to start doing ethical sciences by applying those. — Pfhorrest
Well, since in truth nothing in history has followed this path, then there is nothing for moral philosophy to imitate here. -
Can I say this to divine command theory?Are there something else in our mind makes us know that divine commands are moral? If we have it, we don’t need divine commands, as our hearts know what to do. — Rystiya
No, you don't second-guess God, that's not how divine command works. All you need to know is that you must accept God's authority. This is where your role as a moral agent ends and God's begins. -
Secular moralityNo, it's a meta-ethical question. Just like the foundations of the physical sciences are found in answers to meta-physical questions (broadly, including epistemology in there). — Pfhorrest
The relationship of science and philosophy is a complex one. Philosophy can and does take science as an object of study, just like anything else, but its prescriptive role is very limited. I personally believe that there can be some fruitful cross-pollination between science and philosophy, but it would be the height of hubris to think that science is principally guided by philosophical doctrines, other than the ones that emerged organically in the course of its own development.
The situation with ethics is different though. You can think up metaphysical interpretations and epistemological models, you can package ethics into systems - all as part of the descriptive program of philosophy. But philosophy has no prescriptive role to play with respect to ethics, because at the end of the day, the question that ethics is answering is what one ought to do. Ought questions cannot be decided by anything other than moral judgement. They are like the universal acid: any philosophy that you throw at them will be cut through to the foundation by this stubborn ought: Why ought this be so? -
CoronavirusIf ideological spam is acceptable here, then responding to it with ideological trolling is fair game.
SophistiCat
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