The Dirac Delta function (0 everywhere except at x=0, there infinite) can be thought of in terms of infinitesimals — jgill

It's funny how ubiquitous the delta function still is in physical and engineering mathematics, and yet it is completely non-kosher from the point of view of standard analysis. It was so useful that it survived Weierstrass's reforms, which did away with non-rigorous infinitesimals of the early calculus.

The situation with surreals vs. reals is a little different than that with reals vs. rationals. Though it may seem as if rationals completely fill their number line, being as you can fit arbitrarily many points between any two points (i.e. they are densely ordered), rationals are actually full of holes, in a sense. You can construct a sequence of rational numbers that definitely converges to... something. But that something is not a rational number. There are lots and lots of such holes between rationals - in a way, the rational number line almost entirely consists of holes. And that is where real numbers come into the picture: they fill those holes.

With real numbers the situation is different: they are complete. Any converging sequence of real numbers most definitely converges to a real number. There are no holes to fill - at least not in that sense. You have to work a little harder to find what those smug bastards are lacking: you have to violate the Archimedean property.

At cosmological scales time translation symmetry breaks down and, as a consequence, so does energy conservation. — InPitzotl

It breaks down in the sense that in a dynamic, curved and possibly infinite spacetime there is no uniquely correct way of calculating and keeping track of the total energy. Physicists don't seem to be much bothered by that though, because energy is mostly useful as a budget in local transactions.

True story. Back in the 60s - that's the 1960s, not the 1860s - I was an under-grad Physics major. Thermodynamics was not on the undergrad curriculum. — EricH

Wha...? What was on the curriculum?

At the end, during the Q&A period I asked how it was that the universe had such a low entropy value. The professor's response??

"When God created the universe he created the Second Law of Thermodynamics" — EricH

Well, that's a crap answer and not even a good joke. The past hypothesis, as the (supposedly) low initial entropy of the universe is known (after David Albert), is an interesting and contentious issue.

The double slit and various related experiments do come close to suggesting the universe likes paradox. But probably we just don't understand what's going on. — Marchesk

No, it doesn't, and yes, we do. This is a typical situation where informal or sloppy language can result in an apparent paradox. IOW is right.

Lancet just published a large observational study of chloroquine-based treatmenets of COVID patients. It found that all treatments that are widely used in hospitals increase the overall mortality. Moreover, it appears that the already known heart complications are not the whole story - the drugs may actually worsen COVID symptoms.

In more cheerful news: a (non-peer-reviewed, preclinical) Canadian study shows potential for medical cannabis to treat COVID-19.

Why didn't influenza stick around? Did it kill too many people back in 1918/19? — Marchesk

The Spanish Flu most likely was never eradicated, in the sense of going completely extinct - more likely, it mutated into less dangerous forms and may still be circulating. Of course, Covid-19 is very different from a flu virus, so direct comparisons are not apt. Nevertheless, the possibility exists that over time it will similarly evolve into something less lethal, as pathogens in general tend to do.

As mentioned earlier, this strategy was successful with the Ebola epidemic in 2014. — Andrew M

Ebola wasn't eradicated though, it is endemic and is certain to reemerge from time to time (in fact there were confirmed cases in April).

Those symbols are just Unicode characters that you can copy/paste from anywhere (e.g. the first Google hit for "set symbols"). But this site also supports Latex.

It makes sense because most masks were not manufactured for the purpose of blocking viruses or very small droplets of virus-carrying moisture. They were designed to reduce inhalation of hazardous dusts and pollution (smoke, for instance). — Bitter Crank

Oh so that's what surgical masks are for? To make those notoriously smoke-filled, asbestos-lined hospitals more safe for medical personnel? Who would've thunk.

There has been largely consistent randomized controlled trial (RCT) evidence in health care workers that wearing surgical masks and N95 respirators can reduce the risks of respiratory illnesses [including severe acute respiratory syndrome (SARS)] by 40–60%, after accounting for key confounders such as other protective equipment or hygiene measures.8,11 However, uncertainty remains as to whether surgical masks are inferior to N95 respirators in preventing infection. A recent meta-analysis shows that, compared with surgical mask use, use of N95 respirators is associated with a >50% reduced risk of overall clinical respiratory illness but has no apparent superiority in preventing viral infection,11 which is supported by a more recent large-scale RCT in an outpatient setting.8 Despite the potential superiority of N95 respirators over surgical masks, the evidence in health care workers defies a common claim that surgical masks are ineffective for prevention because some coronaviruses (e.g. SARS-CoV-2) may be airborne in specific scenarios (e.g. during aerosol generating procedures) and/or can infect people through the mucous membranes of the eyes.

Trial evidence in the general population is, however, more limited, because it is practically challenging to carry out and there is high risk of non-compliance and cross-contamination.15–17 Nonetheless, several case-control studies conducted in the general population in Hong Kong and Beijing during the 2003 SARS-CoV-1 outbreak found that frequent use of facemasks (predominantly surgical masks in both studies) in public spaces was associated with a >60% lower odds of contracting SARS compared with infrequent use, after accounting for key confounders.18,19 Although the effectiveness could be overestimated in observational studies (as seen in studies among health care workers11) the lack of conclusive evidence does not substantiate claims that surgical masks are ineffective for the public, but calls for further research, particularly on the reason behind the failure of transferring the effectiveness observed in health care workers to the general population, and the strategies needed to boost the effectiveness. — COVID-19 epidemic: disentangling the re-emerging controversy about medical facemasks from an epidemiological perspective (Int J Epidemiol. 2020)

The great thing about 'virtue signaling' is that people identify the signaler as virtuous, without the signaler having to actually go to the considerable inconvenience of being virtuous. — Bitter Crank

On the flip side, "covidiot" shaming is also a thing now.

I'll go with the quotational approach - that's the opposite of disquotational, as in the disquotational theory of truth.

The answer to this question is "The answer to this question."

The scientific consensus seems to be that unless one is wearing an N95 mask, and wearing it properly, one is probably not limiting the distribution of corona virus much. — Bitter Crank

There is no such scientific consensus. The evidence is mixed, but the consensus, if anything, is that masks are somewhat effective, some more than others. Don't fall victim to all-or-nothing thinking: even a 20% reduction of the probability of transmission is better than nothing.

Let 'All sets that do not contain themselves as members' be

a = {x}
b = {y}
c = {z}
d = ... and these sets go on for as long as is necessary, e, f, g, h,... — EnPassant

Why are they all singletons?

Yes, but X is a set of sets so X = {{a}, {b}, {c},...} but {a, b, c, ...} might be correct too as long as the logic of what I'm saying holds up. — EnPassant

I don't see what logic could imply that {{a}, {b}, {c},...} is the same as {a, b, c, ...}

You keep making the same mistake over and over again:

The paradox asks if {X} is a member of X — EnPassant

No!

The paradox asks if X is a member of X.

X ≠ {X}

{X} is a set with one member: X

Set X = {{x}, {y}, {z}}

If X is included

X = {{x}, {y}, {z}, {{x}, {y}, {z}}} — EnPassant

No, that's not how it works.

X = {{x}, {y}, {z}}

X' = {{x}, {y}, {z}, {{x}, {y}, {z}}}

X ≠ X'

X ∈ X'

X' ∉ X'

How much of what has been written about philosophy is indubitable? — A Seagull

What would be the point of writing down what is indubitable, except as a jumping off point (like Descartes' cogito)?

No, I am saying there are infinite collections of things that are not a set.
See this link https://math.stackexchange.com/questions/24507/why-did-mathematicians-take-russells-paradox-seriously — EnPassant

That's an interesting discussion there. Most of us here are non-mathematicians, and among mathematicians only a small fraction are working in or at least interested in foundations.

The paradox asks the question "Is X a member of itself?"

Let's say Set X = {{a}, {b}, {c},....}

If {X} is a member of X then

Set X = {{a}, {b}, {c},....{X}} — EnPassant

Your notation is confusing. If you want to say that a is a member of X (a ∈ X), you would write that as

X = {a, ...}

which is not the same as

X = {{a}, ...}

{a} is a singleton set with a as the sole member.

Just discovered this early piece by György Ligeti, a beautiful cello sonata:



Bartok's influence, of course, but also very Bach-inspired. Somehow reminds me of
Ysaÿe violin sonatas.

Set A = {a, w}
Set B = {a, x}
Set C = {a, y}

Set X = the set of sets that have {a} as an subset.

Set X = {A, B, C,...}

{a} is in X (because {a} is in A, B, C,...)

therefore X contains X — EnPassant

You need to understand the difference between being a member of and being a subset of.

Set X = {A, B, C} = { {a, w}, {a, x}, {a, y} }

a is a member of A, B and C, but not a member of X. {a} is a subset of A, B and C, but not a subset of X.

I think Russel's Paradox is superficial and I never believed it "undermines mathematics" which strikes me as an unjustifiably dramatic statement. — EnPassant

Who ever said that Russel's Paradox "undermines mathematics"? It undermines what is now known as "naive set theory" (an early attempt at an axiomatic set theory).

In fact it is a trick question because of the way it is stated: "The set of all sets that do not contain themselves as subsets." Why are they calling it a set? — EnPassant

Because of the axiom of unrestricted comprehension, which is what had to be ditched in the wake of Russel's paradox:

There exists a set B whose members are precisely those objects that satisfy the predicate φ.

At the White House, the press secretary, Kayleigh McEnany, told CBS News Trump was taking hydroxychloroquine.

“I can absolutely confirm that,” she said.

“The president said himself he’s taking it. That’s a given fact. He said it. The president should be taken at his word.” — The Graundian

That's a good one.

It is odd. The normal Trump strategy would be to switch to some other miracle drug once discouraging evidence becomes undeniable. And then of course lie about ever being in favour of it in the first place.

Though it cannot be discounted that Trump drank his own Kool-Aid and is actually personally convinced it's a miracle drug. — Echarmion

The normal Trump strategy is to double down on the bullshit when he is called out on it. And he is perfectly capable of believing his own bullshit after repeating it enough times. That some nasty people are telling him that he is wrong only serves to reassure him.

he idea of any act being inherently good or bad is unfounded in any context, even legal or religious. Ultimately everything is conequentialist as everything is good or bad because you are working for or against what someone or something commanded. Murder is not inherently bad, it is bad because it goes against the law, or against what god said or against what you yourself said. — Duckweed Jones

I am not sure why you think this fits into consequentialism. You don't offer much of an argument, but your last sentence hints at a traditional deontological duty theory, as it contains a paraphrase of the well-known categorization of ethical duties: duties to God, duties to oneself, and duties to others.

Can something similar not be done here ? — Amity

I think the inclusion of essays as an 'ideal introduction' for beginners would be welcome on TPF. — Amity

Why? If you like the 1000wordphilosophy project, why not just put a link in Resources section? What would be the point of attempting to reproduce the same thing here?

Besides, I think you vastly overestimate our resources. 1000wordphilosophy apparently solicits their articles from professional philosophers. There are hardly any professionals participating on this forum.

I don't see the asymmetry. — bert1

This banana is the only banana in the world that is this and not any other. What's so hard about this?

Yes, I know that you want to work the self-selection of the first-person perspective into the problem, but the original "problem" as stated does not display this feature. Indeed, the OP rejects it at the outset.

But unless the banana is conscious, there is no asymmetry (that is relevant to this issue anyway) between one banana and another, and this banana can happily be self-identical without raising any philosophical issues. If a banana is conscious however, then there is an asymmetry, and it would make sense for the banana to ask of itself, why am I this banana, and not my yellow friend over there. — bert1

The asymmetry arises as soon as the banana becomes this banana. Consciousness has nothing to do with it.

This is not a trivial question, I am not asking why a banana is a banana. — bizso09

Rather, you are asking: why this banana is this banana. This means that this banana is somehow special, compared to the others, because it has the property of "this".

This broke my heart... again...

Natural language draws no such distinction, AFAIK. So you're already inventing technical language precisely suited to denying the identity of indiscernibles. — Snakes Alive

I disagree, I think there is a difference in usage between "the same" and "alike." In any case, technical or not, this is more than a stripped-down language of basic logical systems.

Sure, but you can just say 'call one A, the other B.' Problem solved. — Snakes Alive

Before you can call one A, the other B, you have to indicate which is A and which is B - otherwise A and B will be interchangeable and you are back where you started. So it's haecceity or bust, it seems.

I want to home in on the problem here: if you have no way to refer to them separately, you can't even coherently frame the scenario. If you do, then you have a way to distinguish their properties. You cannot have it both ways, where you have the vocab to frame the scenario, but not to distinguish between the two. — Snakes Alive

I agree with the spirit of what you are saying, but I still maintain that individual properties are not always the way to go. They are very convenient, but only up to a point. When we start pushing against edge cases, we should be prepared to give them up. We don't always need to refer to individual parts by their properties in order to know that they are there - we can infer their existence from the system as a whole.

You can describe the system as a whole so as to imply that there are two distinct objects in it. That should satisfy your reasonable demand that we should be account for what we know. But further steps, such as introducing pseudo-properties, are superfluous.

If this is what property is taken to mean, that they are intrinsic to objects and don't require external context, then I would do away with the notion of properties entirely. This is where my structuralist perspective comes in, because I don't believe objects have intrinsic qualities, rather that they are defined by their places in some structure, where the structure is just the composite of relations between things. However, property could still reasonably be defined of a thing as any relationship it has derived from the structure. This is why PII seems so important to me: if it were the case that two things related to everything else in exactly the same way, and those things were not actually the same thing, it would just shatter my worldview. — QuixoticAgnostic

Then you are in trouble, because in a symmetric structure, such as the one with the two spheres, you still can't individuate an isolated part, even using relational properties, because symmetrical parts will have identical relational properties. But I don't think this undermines the structuralist view. The structuralist view is holistic, and when describing the structure of the system as a whole you don't run into any such difficulties. That's because you are not obligated to produce an exhaustive specification of an isolated part of the system; you always have the entire system as a background, so there is no danger of collapsing distinct objects into one, simply because you can't individuate them separately from the rest of the system. There is a difference between a structure with two spheres and a structure with one - so what does it matter that you cannot identify one of these spheres by its properties?

Traditional ontological ideas come under even more pressure in modern physics, where the notion of a bounded material object does not sit well with such "things" as fields. And event if we grudgingly grant objecthood to subatomic particles, we then have to deal with bosons (integer-spin particles, such as photons) that can share all of their properties, including their position. So that while we may know that there is more than one particle, it is not possible, even in principle, and even with the loosest definition of a property, to individuate any of these particles. All we can say is that there are n particles in the system sharing the same state (and even that n may not be precisely known, being subject to Heisenberg uncertainty).

I actually don't think this is right. You can't block haecceity in natural language, either, which is why you need to come up with an artificial language that blocks it.

In English, for example, you'd have to say: "suppose there are two distinct spheres, but they're the same in every way." — Snakes Alive

You might instead want to say "they are alike in every way," making it clear that you mean "internal", or "qualitative" properties.

The response is: what do you mean? You just said they're distinct. Surely the one is not the other, then – but I've just predicated, in the natural language, a property of one that the other doesn't have, viz. the property of being the one as opposed to the other. — Snakes Alive

Ah, but unless you point at one or the other sphere as you say "one" and "the other," this will not get you out of the bind. Because without ostention, the sentence "the property of being the one as opposed to the other" is equally applicable to both spheres; the "property" therefore is exactly the same. It's frustrating, but there you are. If you are forbidden to point, you may find that you lack the words to distinguish between the two spheres in the absence of some additional structure in the world, such as an asymmetrically positioned observer. But this only says something about the limitations of expression.

Did you mean numerically identical? Because if they're already numerically distinct, it wouldn't matter if you predicated another distinction. Although, if you did mean identical, I don't think it would still make sense because if we take PII to be true, it would be impossible for two things to be numerically identical in the first place, so we couldn't predicate anything to make a distinction. — QuixoticAgnostic

No, I meant numerically distinct, as in the example of two spheres. If you are allowed to predicate anything by way of specifying a property, up to and including haecceity, then you will find a way to distinguish two apparently distinct objects, no matter how qualitatively alike they are.

My point is similar to that of @Snakes Alive, except that he too readily trivializes the discussion by reinterpreting it in the formal logic context. Informal language is richer than simple logics in that you can block moves like counting haecceity as a property, without violating any rules. But then again, even if you find yourself in a situation in which, playing by the rules of the game, you cannot express something with words that is otherwise apparent, all that this signifies is that the game has its limitations.

If there is a wider lesson here, it is that the traditional discourse of properties with its atomistic character, in which objects subsist without any external context, is inadequate.

Why not read what Mackie had to say and find out?

It all hinges on what is understood by 'property'. The only sense that makes PII true is the most inclusive (and deflationary): a property of a thing is anything that can be predicated of the thing, i.e. any true proposition with the thing as its subject. But that trivializes PII, because if two objects actually present themselves as numerically distinct, then you can always predicate something that would imply a distinction, if only by ostention ("A is this and B is that"). If this seems question-begging, I think that's because the PII doesn't really say anything metaphysically substantive - it is basically an explication of the concept of numerical identity.

This definitely warrants discussion too, but my main point is that many philosophers have been pretty clear that their goal isn't to be "concise and simple", some wrote for themselves, some a select few, others embraced different degrees of obscurantism, mysticism and "make you think" provocation. — boethius

That may be so, but I don't see the relevance of bringing up the fact that some philosophers did not write well as an objection to an admonition for philosophers to write well. Also, the advice is, obviously, not to be simple at the expense of depth or concise at the expense of scope. Rather, it is "as simple as possible, but no simpler," etc.

If we interpret "simple and concise" to mean "not challenging", then we may not only fail to rouse the curiosity of the reader but also fail to convey the argument. If an argument is not completely clear (due to complicated sentences, qualifications and diction), it requires serious thinking to "get it", and that experience is richer and more memorable than a "pre-chewed" version of the same thing. — boethius

Yeah, no, I have zero respect for this snobbery.

"Many famous philosophers were miserable communicators" is not an argument against good writing tips (unless you want to argue that they were great because they were miserable communicators).

Whereas the explanation of the sphere effortlessly and beautifully explained the problem even before it was shown to be true and the flat earth theory was disproven. — TheArchitectOfTheGods

Ancient cosmogonies were not overly constrained by empirical observations, the way (we like to think) our modern cosmology is; ancient people gave a lot of leeway to their metaphysical and religious imaginations. If people back then were really more comfortable with the concept of a spherical earth, what would have stopped them from conceiving it that way? And yet all the ancient cosmogonies that we know about posited a more-or-less flat earth, sometimes surrounded by water (not all ancient cultures thought this out all the way through, and some didn't bother with cosmogony at all). Spherical earth theory, at least as it developed in ancient Greece, was on the contrary prompted by empirical observations that did not sit well with a flat earth, such as the phenomenon of the receding horizon.

But this is not to say that present day cosmology is guided strictly by empirical considerations. Even setting aside the fact that what we think of as modern scientific empiricism is a philosophy in itself, science edges beyond empiricism when it comes to theory selection at its more speculative reaches. The principle of "naturalness" in particle physics and cosmology is a prime example of that.

I am not an expert, but I don't have the impression that the shape of the universe is a particularly common issue on which to make a philosophical stand. Check out Stoeger, Ellis and Kirchner's Multiverses and Cosmology: Philosophical Issues though, where they do just that, arguing (unpersuasively, IMHO) for a finite universe.

By the way, to confound things even more, the universe doesn't even have to be a hypersphere to have a closed topology. It can have zero intrinsic curvature everywhere, just like a flat sheet, and yet have a closed, finite topology of a hypertorus, or something even more exotic.

Thanks. But aside from quotations, what does this mean? — Zophie

I am not sure what you are asking. What do the designations mean? You can read the links and follow the references inside, but I think you already know something about this area.

Existing research understandably focuses on fairly modest cognitive functions that can be tested experimentally, such as color discrimination (thanks for the video, by the way); the more ambitious the hypothesis, the more speculative it is likely to be.

Maybe the relativity is located deeper than language and culture, but is actually a relativity within the individual self, which might explain the conflicting results, with schematic thinking induced in only some investigative situations. — Enrique

I don't know much about this, but my take, for what it's worth, is that cognitive activity, including but not limited to abstract thought, is thoroughly entangled with language, so that one should expect some causal entanglement as well. But I have a feeling that in most cases, causal factors going between language and thought are not clearly separable from other causal factors and contingencies. We'll see. When you read about this psychological research, you can't help but admire the ingenuity with which researchers find ways to tease out causal links.

The terms of art for this topic are Linguistic Relativity and Linguistic Determinism, the latter being a stronger form of the former.

Among the strongest statements of this position are those by Benjamin Lee Whorf and his teacher, Edward Sapir, in the first half of this century—hence the label, 'The Sapir-Whorf Hypothesis', for the theory of linguistic relativity and determinism. Whorf proposed: 'We cut nature up, organize it into concepts, and ascribe significances as we do, largely because we are parties to an agreement to organize it in this way—an agreement that holds throughout our speech community and is codified in the patterns of our language' (Whorf, 1940; in Carroll, 1956, pp. 213-4). And, in the words of Sapir: 'Human beings...are very much at the mercy of the particular language which has become the medium of expression for their society. ...The fact of the matter is that the "real world" is to a large extent unconsciously built up on the language habits of the group' (Sapir, 1929; in Manlbaum, 1958, p. 162). — Language and Thought

These theses are understandably controversial. There is plenty of evidence indicating that language and thought are intertwined, but identifying causality is not so straightforward. Does thought drive language or the other way around? Or do other factors like ecology drive both thought and language?

Here is one recent review:

The central question in research on linguistic relativity, or the Whorfian hypothesis, is whether people who speak different languages think differently. The recent resurgence of research on this question can be attributed, in part, to new insights about the ways in which language might impact thought. We identify seven categories of hypotheses about the possible effects of language on thought across a wide range of domains, including motion, color, spatial relations, number, and false belief understanding. While we do not find support for the idea that language determines the basic categories of thought or that it overwrites preexisting conceptual distinctions, we do find support for the proposal that language can make some distinctions difficult to avoid, as well as for the proposal that language can augment certain types of thinking. Further, we highlight recent evidence suggesting that language may induce a relatively schematic mode of thinking. Although the literature on linguistic relativity remains contentious, there is growing support for the view that language has a profound effect on thought. — Phillip Wolff and Kevin J. Holmes, Linguistic relativity

However, you will also find both stronger and more skeptical claims, more-or-less supported by research.

I don't think that our esthetic preferences, habits of thought or limits of imagination should dictate what we believe about the world. On the other hand, evidence for a closed universe may be easier to discover, and if so then we should certainly go for the low-hanging fruit first.

Most of the time, when weighing competing cosmological theories against each other, we have to settle for the one that accounts for the available evidence as well as any other and has the practical advantage of being simpler than the rest - which is why, when pressed to make a choice, cosmologists pick the flat, infinite universe as their default.

No doubt, finding the cosmic equivalent of an image of the back of your head out in the distance would be a smoking-gun evidence that we can usually only dream of. The evidence in favor of a closed universe that has been put forward so far is much more subtle and controversial.

It is generally assumed that it is not meaningful to talk about the center of the universe because all locations could equally claim to be the center, like any location on the surface of a sphere. — TheArchitectOfTheGods

The reason for saying that there is no center of the universe is the observation/assumption of symmetry, or homogeneity of cosmos. If, on a large scale, the universe everywhere has the same properties, then there is no reason to single out any place as the "center." For this to be the case the topology does not need to be spherical. Think of a blank sheet, for instance, extended infinitely in all directions. It is featureless, and therefore does not have an obvious center.

As for a "hall of mirrors" universe, that is indeed a live hypothesis, which astronomers are trying to test, but so far there is no definitive evidence either way. This has nothing to do with the Galilean principle, but rather with the fact that we simply don't know what the topology of space is, other than that it is pretty flat around where we are. So we try to find whatever clues we can.

Does language channel our focus in such a way that it affects what we observe even at the level of basic percepts? — Enrique

Yes, though it doesn't pay to run too far with that idea (as was the fashion at one time). Still, there's a lot of fun, unexpected things that we have learned about this. See for instance Lera Boroditsky's publications, such as How language shapes thought.