I really think it is useful to consider prior work here: Essentially there are two major schools of thought on what "abstract" concepts like "time", "space", "collections", etc. are in relation to "consciousness", although the traditional name for this part of our mind really is reason (and "consciousness" is more the ability to have qualia, subjective experiences)
There's an "a priori" school, whose modern incarnation starts with Kant, of people who believe that our ability to e.g distinguish individual objects from a single "whole" and to divide a whole into a collection of parts etc. (the categories of quantity in Kantian) is an innate part of reason: This is not a learned ability, not based in experience, it is rather a precondition to process experience, an intrinsic part of reason
And there's an "a posteriori" school, whose modern incarnation starts perhaps with Hume (to whom Kant's philosophy is effectively a response) who think that ideas are formed by recognizing patterns in our experiences, like cause and effect, etc. The buzzword here is Locke's "tabula rasa" - for this school, a mind without any experience at all is incapable of thought, it is a blank slate, i.e. no ideas at all are innate to us, they are all formed from experience
this mirrors the "nature-nuture" debate we have for many human qualities and is similarily hard to definitively resolve one way or the other in practice
I mostly agree with Kant' school here, except I think sentient existence may be strictly greater than the notion of collections. Some very primitive existence like a fetus which cant comprehend collections. About the other school, what are we experiencing according to them? Do collections exist out there in the Plantonic sense according to that school?
And we are experiencing these collections and learning from them
Is the second school closely related to Platonism? @ACuriousMind
A crucial part of Hume's skepticism and the problem of induction is that Hume crucially does not believe all the ideas we generate from experience "exist": Just because everything we have observed so far has obeyed the pattern of cause and effect, that doesn't mean cause and effect is a universal law.
Similarly, I imagine he would argue that just because thinking about the world as being composed of discrete individual object is a useful way to organize our perception does not mean the world "really" consists of such objects.
I mean Kant is saying collections are intrinsic to consciousness. So Kant must be against collections existing in the Plantonic sense. Hume is explicitly against that.
the question I'm talking about right now - and the question you were talking about - is where, exactly, notions such as "one object, two objects, three objects" arise in the mind, whether they are innate or learned
this, I think, is a useful question, as it has in principle - if not in practice - implications on how our mind works
Platonism and the rest is more about metaphysics, do concepts "exist" etc.
Oh, so both schools are only talking about the mind here. They only differ in whether collections r acquired slowly like a fetus growing, or whether it has to be there from the beginning of mind
I first thought Hume's school was talking about aquiring collections from the outside existing world. But he is more of the opinion that mind is strictly greater than the notion of collections. Mind can exist without the latter
@RyderRude yes, my point is that your talk about how consciousness relates to ideas is more akin to the questions Kant and Hume are trying to answer here than to the sort of metaphysics associated with the "existence" of ideas
The Humean mind cannot exist without perceptions, but the ideas the mind forms are not necessarily "part" of the perceptions. Because the mind develops the idea of cause and effect does not mean the problem of induction goes away and so the world has to obey the principle of cause and effect.
The Kantian mind does exist without perceptions, and generates the categories of quantity and causation and whatnot because they are preconditions for any sort of logic and thought. These categories are a priori, they relate to the perceived external world only in so far that any thought about perceptions necessarily needs to employ them
@ACuriousMind So does Hume say or not say that these perceptions come from the outside world which has to objectively obey cause and effect? Does he care about the question of where perceptions come from?
@RyderRude the very question that the perceptions "come from" anywhere assumes some notion of causation, i.e. the the perceptions are caused by some "outside world"
but we cannot ever show that causation, in fact, holds always, so the question is ill-formed
Okay. I'm finding myself with Hume on this one then. Fetuses, at some point in their development, are sentient and have perceptions but cant comprehend collections
@RyderRude yes but arguing about babies supposes already a lot of scientific knowledge, some notion of an external world in which other minds not only exist but are borna nd develop etc.
but e.g. Kant's whole theory of ethics is based on the universalizable aspects of reason: Because Kant's reason has a priori features, these features are shared by all minds, and hence he wants to build universal ethics based on these necessary features of all minds!
Hume is perhaps a bit more depressing that department but essentially I think the question of other minds' "existence" doesn't really matter to him that much: When we feel e.g. compassion for another being, it doesn't matter whether that being "exists"
honestly most epistemologies end up doing ethics somewhere down the road: Very few people are interested in "what we can know" for its own sake, it is usually followed by the question "so what do we do about it?" :P
Is the answer to my foliation question clear-cut? I am very inclined to think it's highly non-obvious whether such a foliation exists. Am i missing something?
I dont find the ethics philosophy that interesting. This is the only conclusions Ive arrived at in that department : "Ethics is what people decide to be ethical"
One thing I find strange about axiology (the study of values) is that it seems very interested about ethics and art but not so much about what makes a good soup
I think people often have this impression that because the ethicists usually are very strongly convinced of their own ethics it's not worth bothering with them - they're all moral absolutists and if you're a relativist there's nothing worthwhile there. But I think there's plenty of value in understand why some people might consider certain actions immoral even if you don't share their exact arguments
@RyderRude well...let's say his political theory is more authoritarian than most modern humans would feel comfortable with
states with all-powerful sovereigns you aren't morally allowed to resist is perhaps not the end state you imagine when you hear "people just want to survive" but it's where Hobbes ends up
but I think this is the value of engaging with ethical philosophy: If I hold some belief, and I find someone who shares it but ends up in a place I don't like, am I wrong or are they? Can I actually point out where we're different? Does my personal stance end up being inconsistent?
hello -- i have a question about the formulation of the momentum operator as the generator of translations in matrix form. overall, what i am trying to do is understand how all elements of the poincare group take on the same form (which i believe should be a rotation matrix). [...]
[..] for the 6 elements that are lorentz transformations, this is more simple, but for the remaining four that are translations, i struggle to see how they can take on a matrix form. [...]
[...] initially, i thought i could use the idea that momentum is the generator of translations to construct this type of operator, but i am stuck at how to convert this exponential operator into the form of a rotation matrix and i wondered how i can do this? i tried to look into it, but i wasnt able to find the translations represented in this way
this is not a homework problem by the way -- i am just generally confused how these 10 elements form a group because i think this means that all elements take on the same form
@Relativisticcucumber what size of matrix are you trying to represent them as?
because translations in $\mathbb{R}^4$ do not have a representation as 4-by-4 matrices - they are not linear operations since $x\mapsto x + c$ maps the zero vector to a non-zero vector but one part of the definition of linear maps is that they must map zero to zero
@Slereah yeah you may be right about that, it definitely makes sense that you can foliate 3 dim. minkowski space minus a line with the class of surfaces $S=\Bbb R^{1,1}$ minus a point in a non-singular manner. I'm just confused and second guessing myself because a geometric topologist said he didn't know how construct an example of this in the Riemannian setting
Take $X=(0,1)^3.$ Fix points $p,q$ s.t. $\text{dist}_3(p,q)=\sqrt{3}.$ Construct a smooth regular foliation of $X$ with $(3-1)-$dim. leaves which are topologically $(0,\sqrt{3})\times S^{3-2} $ accumulating to $p,q.$ This problem was reduced by noting that the leaves are cylinders if you enlarge the caps around $p,q.$ Then you reduce the cylinders to annuli, then reduce the annuli to punctured planes. Therefore the equivalent problem
at least would seem to be a smooth foliation of $\Bbb R^3$ by punctured planes. (cylinder is diffeomorphic to annulus and annulus is diffeomorphic to punctured plane)
well i guess i never thoughts about this because i only know a group is a set and the elements have to obey the group axioms hm i guess i never considered the elements can achieve this by being different entities
a linear representation of a group on a vector space has to have them all be matrices on a vector space but in general there is no requirement that groups should be matrices (and in fact there are groups that cannot be faithfully represented as matrices)
since translations are affine though we don't have to go full abstraction, you can just do the 5-by-5 matrix representation from my answer, also called augmented matrices
@ACuriousMind and why dont we have the issue that, as you said, matrices are linear transformations, so we cannot represent translations, which are nonlinear, by matrices? why does them being 5x5 solve the problem?
@ACuriousMind ok i took a look at this and i think it makes sense, but one thing im confused about is in your answer, you have that the elements can be modeled as a subgroup of GL($\mathbb{R}^5$), right? and i see that this gives us the lorentz transformations in a 5x5 form, but im still not seeing how this accounts for the translations?
I mean it should be a theorem that if $(0,1)^3$ is diffeomorphic to $\mathbb{R}^3$ and $D$ is diffeomorphic to a line in $\mathbb{R}^3$ then their differences are also diffeomorphic. But I made that theorem up now I don't know if it's provable :)
hello -- i have a question about this excerpt below -- i dont understand why, if we look at the second counterexample where a point is removed from the manifold and the curve ended arbitrarily close to this point, we cannot just have an endpoint that is arbitrarily close to P. thus, since this endpoint exists, the curve is still extendable?
@Relativisticcucumber Take some other candidate endpoint $P'$. Hausdorffness means there are neighbourhoods $U$ of $P$ and $U'$ of $P'$ that are disjoint. Since $P$ is an endpoint, there is some $s_0$ such that $x(s)$ lies in $U$ for all $s>s_0$. But since $U$ and $U'$ are disjoint, no $x(s)$ with $s>s_0$ lies in $U'$. So $P'$ is not an endpoint.
@Relativisticcucumber manifolds are usually assumed Hausdorff, yes
if you think the Hausdorff property is weird, it's not: It's just saying that two points can be "separated" in the sense that I can always find two neighbourhoods that don't intersect
in $\mathbb{R}^n$ this is very simple: for any $x$ and $y$ just take two open balls around $x$ and $y$ with radius smaller than half the Euclidean distance between them
there are even weirder surfaces than a once punctured plane which do foliate $(0,1)^3$ without any singularities whatsoever lol. I wonder if this holds when passing to semi-riemannian landscape..
one example is the class $S = \mathbb R^2 - (C \times \{0\})$ where $C$ is the Cantor set
In English General Relativity=GR, while in Italian Relatività Generale=RG. On the other hand Renormalization group=RG, so I'm having a hard time in spoken conversations
@SillyGoose I don't really mean that you just should do calculations and not think about them, I rather mean that we don't really need to think about quantum mechanics as a theory about the entire universe or as applying to every weird thought experiment you can come up with in order to use it
for essentially every practical use case of QM, your interpretation doesn't matter
it only matters when you start waffling about fundamental theories of nature and whether or not the wavefunction is "real" etc., but in the end I think all the paradoxes like Wigner's friend are like all these terrible "paradoxes" in special relativity: It's just stuff that's weird if you insist on classical view of the world
but then again I seem to have a far larger tolerance for not knowing the absolute truth about the workings of the universe than most other physicists :P
I find it fascinating how polarising QM interpretations are. "My interpretation is mostly satisfactory, it just has 1 or 2 problem areas. But your interpretation is patently absurd!" ;)
lol, if you mean that philosophy necessarily becomes worse when you have better data I can see why... or do you mean they just aren't really "trained philosophers" as the various sages of old were?
@SillyGoose Posting stuff like that in comments bypasses the voting mechanism. Sure, we have comment upvotes, but they're just a "me too" mechanism to reduce comment duplication.
@Amit I mean that a lot of discussions by physicists are very much hampered by the fact that they tend to a) look down on philosophy as useless and b) don't actually know a lot about philosophy
though honestly, i do think he kinda has a point inasmuch as - we're literally undergrads. we do not know enough quantum to theorize about its interpretation lol
though on the other hand, i think there is probably a better middle ground position somewhere
Philosophers who are properly trained in physics are pretty rare, and likewise, practicing physicists have rarely invested much effort in studying philosophy in much depth. But there are exceptions, eg John Norton. sites.pitt.edu/~jdnorton/jdnorton.html
