Spring Semester Seminar Style Reading Group
Hi all.
I want to expand my knowledge by tackling some new and/or particularly thick book. You guys have great reading groups on canonical figures, but they usually frustrate me because everyone is coming in with pre-concieved views and opinions. I want to discover an author or book which is mostly new for everyone involved.
My plan is just to do this systematically by laying out a seminar's worth of weekly reading (i.e. 12 weeks) and putting together lecture notes on the material every week as the basis for a discussion. My plan is to do this even if I'm the only person posting in this thread!
If possible, though, I think we might attach it to the audio of a graduate seminar freely available online. Just to take an example -- Who wouldn't (time permitting) want to go through the full audio of Brandom's 12 week course on Sellars, with weekly discussion on here, 'lecture notes' available each week, and maybe sprinkling in some O'Shea or other secondary literature? I think that would make a lot of us, well, a lot smarter.
Or we could just take on a massive work like Making it Explicit, or A Secular Age. We could get really freaky and cover Deleuze on Foucault or Spinoza with his accompanying lectures (for anyone who speaks French). Or we could take a theme -- "Political Aesthetics", "Virtue Ethics" -- and I'll find an accompanying essay/conference video of value each week.
The idea is that for me and any other members interested in coursework (like Wallows!) we can come away with a vague sense of having accomplished the learning equivalent to a philosophy seminar.
Anyway, if anyone is interested let me know. Please feel free to voice a particular philosopher/book/topic of interest. Like I said, my plan is to do this regardless of interest so don't be too embarrassed for me if you see me playing alone in the corner each week.
Whatever topic is chosen, readings should begin with the Spring semester - viz. Early January.
I want to expand my knowledge by tackling some new and/or particularly thick book. You guys have great reading groups on canonical figures, but they usually frustrate me because everyone is coming in with pre-concieved views and opinions. I want to discover an author or book which is mostly new for everyone involved.
My plan is just to do this systematically by laying out a seminar's worth of weekly reading (i.e. 12 weeks) and putting together lecture notes on the material every week as the basis for a discussion. My plan is to do this even if I'm the only person posting in this thread!
If possible, though, I think we might attach it to the audio of a graduate seminar freely available online. Just to take an example -- Who wouldn't (time permitting) want to go through the full audio of Brandom's 12 week course on Sellars, with weekly discussion on here, 'lecture notes' available each week, and maybe sprinkling in some O'Shea or other secondary literature? I think that would make a lot of us, well, a lot smarter.
Or we could just take on a massive work like Making it Explicit, or A Secular Age. We could get really freaky and cover Deleuze on Foucault or Spinoza with his accompanying lectures (for anyone who speaks French). Or we could take a theme -- "Political Aesthetics", "Virtue Ethics" -- and I'll find an accompanying essay/conference video of value each week.
The idea is that for me and any other members interested in coursework (like Wallows!) we can come away with a vague sense of having accomplished the learning equivalent to a philosophy seminar.
Anyway, if anyone is interested let me know. Please feel free to voice a particular philosopher/book/topic of interest. Like I said, my plan is to do this regardless of interest so don't be too embarrassed for me if you see me playing alone in the corner each week.
Whatever topic is chosen, readings should begin with the Spring semester - viz. Early January.
Comments (146)
What kind of works are you interested in? Phenomenology, linguistics, metaphysics, philosophy of logic?
I'm not sure that what I have suggested (e.g. Brandom on Sellars) is too complex! My feeling is that any topic in philosophy is approachable at any level; if we are doing weekly notes to set the tone (rather than anarchic free-for-all) then I think we can discuss even very complex topics in an approachable way. At least I hope.
Out of the four topics you mentioned I'm most interested in phenomenology.
That will only be moreso the case, accompanied by lots of complaints about the philosopher's writing from a formalist perspectivel, if we choose someone like Deleuze. I'm not very fond of continental writing on a formal level, to say the least. But I'll play along with whoever we choose . . . well, unless it would require having to buy books I don't have and it would mean paying $40-50 or whatever for some Derrida or something. In that case I'd just try to read freely available secondary literature on it or something instead.
Quoting Wallows
What would interest you? For example, covering Phenomenology of Perception or an essay (e.g. Cezanne's Doubt)?
I really appreciate your interest, though it sounds like the purpose and objectives I have in mind -- serious reading aimed at improving our understanding of difficult material -- may be at odds with both your highly critical approach to texts and your uncertainty about committing to a text.
I think starting with an essay would be a good idea to just get a feel for how these things work out in practice. I'm open to whatever you propose.
You decide. I'm unsure yet.
Though PhP happens to be my favorite philosophy text so I'm always happy to read what people say about it.
Ooo, I thought you might like it. Didn't know it was your favorite. Phenomenology of Perception would be a worthwhile text to go over IMO. But, since you're already aquainted with it so well, then I'm not sure it's worth your time to go over.
Maybe, but I still plan on participating. Sometimes here it seems like people want to approach texts almost like a disciple approaching their religious tome of choice--there's that sort of remove in the relationship, with some reverence to it, etc.
I approach texts like it's my buddy talking to me while we're hanging out at a bar. There's no remove--the person is engaging in a conversation with me where I'm an equal, and I can be pretty irreverent, challenging, etc.
Or, to be purely selfish, it would be lovely if people were willing to read either Reza Negarestani's Intelligence and Spirit (on rationalism), Gilles Chetalet's Figuring Space (on math), or Giovanni Maddalena's The Philosophy of Gesture (on pragmatism and Pierce) with me - my planned reading, essentially - but that's probably a long shot.
Edit: A work that might have a nice cross-audience appeal is John Haugeland's Dasein Disclosed - an analytic philosopher's take on Heidegger, reading him in a Sellarsian bent. Might be cool.
Personally interested in that short paper. 15 pages are doable for my feeble mind and would be a good trial run on how to approach abstracta into intelligibility for us.
What would be your approach to a topic like time or infinity? These subjects are so broad. with such varied perspectives.
Alright, looks like we should definitely do Riemann! I would definitely also be interested in reading The Structure of Behaviour because it's been an embarrassingly long time since I've read it carefully.
One thing we might want to do is simply take a vote or a little straw poll everytime we finish a work. We could just do the Riemann essay then see how people feel afterwards and what they want to move onto next (if not SB or the Kant essay).
For example, Street, you read a lot of interesting stuff that would be great to do in a group - judging from your reading lists - but those particular books aren't necessarily my thing. Maybe we'll see where you're at after those books? Joanna Seibt intrigues me but I can't find access to a legal PDF of that work which seems like a priority for this sort of reading group.
I guess I would also be interested in comparisons, either reading side-by-side or successively. E.g. (from the other thread) Wittgenstein and Sellars on the game of giving and asking for reasons and the question of how central it is to our linguistic practices.
Quoting Metaphysician Undercover
Same way any philosophy seminar functions. We look at the variety of important essays which contribute to the topic's history, or a single book with an expansive scope though highly particular pov, or we narrow down on some small aspect of the topic in question and look at the current debates.
I realize that plenty of worthwhile, interesting things are only available via purchasing books for $30-40 or more, but probably a lot of people who might participate wouldn't bother if they need to keep buying books--including me, partially because I've been in the process of trying to streamline at home, and out of "principle" I refuse to pay over, say $20 for a kindle book that's just a "normal" 200-300 whatever page book. I do still buy some "physical" books, but I'm extremely selective about what I buy..
Here is a link to the paper. When would reading start? I plan to finish the last section of Chapter 1's value theory in my Marx thread before going into something else with the same detail, only one section left (commodity fetishism) now, and I'll have done what I planned to do in the thread.
Between now and then I am thinking that I might try to squeeze in my own notes / read a Sellars essay -- “Some Reflections on Language Games” -- for anyone who is interested (mostly as an extension of the Wittgenstein thread), so I'm sure I can stretch that out if you need more time to work on your Marx thread.
I might be able to finish what I set out to do in thread this weekend, at least that's my goal. I'll have a good crack at it on Saturday. It shouldn't interfere.
It's not exploitation if it's TPF. :)
Only if you join my hermetic circle of overly mathematical Marxism.
Shall we start preemptively?
https://www.webdeleuze.com/sommaire
Will read the Riemann essay over the weekend.
You can start, I have other things to do.
I don't know where to begin! I'll wait patiently on when others have had time to address the paper.
A bit off from other suggestions, and definitely not the sort of thing you read alongside -- it's too big -- but I've always been meaning to finish my copy of Being and Nothingness. Thus far I've pretty much just done selections.
Regardless I'd just be happy to participate with whatever you set out.
Great to have you on board, Moliere! Honestly, a lot of reading groups seem to have popped up recently for big books so I'm trying to figure out how best we should navigate what I had in mind for this group. I think maybe the best will be to take a poll and set a 'syllabus' of sorts. I'm definitely deeply interested in Merleau-Ponty and Sartre who have both been brought up in this thread, though I worry interest may wain in the group if we do just another big canonical book since there are like half a dozen threads doing the same thing. So I'm thinking maybe some essays, lectures, and very short books? E.g. Sartre, Merleau-Ponty, Sellars, Deleuze, Riemann, Kant, Marx, Ranciere? (Do you have any interest in Sartre's Critique of Dialectical Reason?)
That's a really good point. You convinced me! :D
I'm down for Critique of Dialectical Reason as it's another one of those I haven't gotten to but have wanted to.
Honestly the schedule you have looks great to me. Kind of a survey in interesting writers that we can explore together. I'll have to think a sec for your weeks 11 and 12 though.
I'd be down for Riemann and Kant and would lead the discussion on Riemann if you'd have this ignorant schoolmaster's musings. I hope @StreetlightX finds the time to lead the discussion on Kant. That would be a cool thread.
That's the best way to do it in my humble opinion! Anyway, let's go for it. We can start whenever you feel ready. I think my position is that it's great if someone wants to use this thread as a place to read an essay so I'm happy to either lead or cede any discussion. :up:
Quoting Moliere
Well, if I had to offer advice I'd suggest either "Philosophy as Rigorous Science" or "Crisis of the European Sciences" as a great starting point.
I still have a few bits left before I conclude my Marx thread for now. Think it's 6 paragraphs. I should be able to finish that by next weekend at the latest.
Cool!
I tried to find some free versions online and so far have failed. I think I have copies of those, but I'll come back with a different suggestion once I find something that everyone can have access to without having to spend money. Also it'd be cool if everyone was excited to read it, I think, so I'll try to find something that "fits" with what's up there so far.
What does everyone feel about that one?
I also sort of think of Rorty as an "in-between" philosopher, from what little familiarity I have with him, between analytic and continental approaches -- and I noticed how there was a kind of mixture between these traditions in what's already been picked.
EDIT: Also, don't feel bad if it just doesn't catch you -- just say so and I'll keep digging around. I'm just sort of throwing ideas out there, and I'm fine with finding other stuff. It's more important to me that people actually would like to read what we choose.
I thought that https://www.marxists.org/archive/lukacs/works/1949/existentialism.htm might make a good companion piece -- since they are both sort of "outsider" criticisms (so it appears from the first couple of paragraphs at least) of certain ways of doing philosophy, but one from (what I take to be) a pragmatist view, and this one from a Marxist perspective. Plus it's short, and Lukacs is another writer I haven't really spent time with.
So my initial goal - to take on some theme or topic and explore it over the course of several months through weekly postings -- hasn't changed. But I also think if there is enough interest for people to use a thread like this to catapult discussions of essays they have been wanting to read, then I have no problem starting a second thread along the lines of what fdrake has done with Marx's Capital.
Though if interest wanes in this thread (as it seems to have) then I may just stick whatever I want to do in here.
So I might say that Lukacs criticism of existentialism comes from a couple of different levels. One is that existentialism does not reflect reality, and another is that existentialism does not present any new method but rather has roots in Kantian enlightenment era thinking -- and since epoch-making philosophies are characterized by a novelty in method, existentialism is not epoch-making but rather a fad which has become popular because of the social conditions of the time in which it was written. Existentialism appeals to the feelings and needs of certain intellectuals and so escapes criticism, according to Lukács.
On a secondary level, I think, Lukács also notes how this philosophy helps bourgeois and reactionary political actors by isolating the individual from their. social relationships. But I really think the core of his argument has more to do with the above -- he isn't just saying that existentialism does not agree with Marxism, and helps the enemy, but rather that it fails on its own from proper philosophical considerations: it fails as a third-way, but is just a rehashing of transcendental idealism for the times it was written in.
There was one point in the essay that made me think I'd like to hear more from Lukács where he said ,"A very specialized philosophical dissertation would be required to show the chains of thought, sometimes quite false, sometimes obviously sophistical, by which Sartre seeks to justify his theory of negative judgment." -- but, hey, then this wouldn't be so brief either :D.
It seems that he focuses mostly on Sartre at the end because he was the philosopher at the time most associated with popularity, one, and I suspect he focuses on Sartre too because he was a communist -- so that one couldn't say "well, even one of your own is an existentialist, so surely this is not a reactionary philosophy"
One take-away that I really liked from the essay was Lukács' observation that absolute responsibility is only a shade away from a total lack of responsibility -- and similarly so for freedom -- so that one could feel that one both is responsible yet act cynically. I thought there was some truth to that.
I'm not sure I totally agree with Lukács criticism, though. While I do think of phenomenology, at least, as a kind of "third way" between idealism and materialism, I don't think it's a way between as much as it passes over such questions as not worth asking -- at least, not with respect to phenomenology. I suppose it would depend on just how serious you took phenomenology when considering questions of ontology, but it seems to me that one could easily be a phenomenologist and a materalist without tension.
Addressing some of your points....
Quoting Moliere
I find the methodological points quite interesting. Nowadays, there's a tendency to use the term "existentialism" to refer to 19th century philosophers who take a specific stand on "life" from the perspective of their individual experience (Schopenhauer, Kierkegaard, Dostoyevsky, Nietzsche) and 20th century non-technical philosophers who do the same (Camus, Jaspers), which we tend to distinguish from phenomenology as a movement starting with Husserl, through Levinas, etc. Then we have the term "existential phenomenology" getting thrown around for some of these thinkers who, following Heidegger, are influenced by the 'non-technical' individual-taking-a-stand-on-existence side of existentialism and the 'technical' side of phenomenology as logical analysis of the nature of experience.
Lukács lumps existentialism and phenomenology together, claiming that they are expressions of a particular sort of methodology. This methodology gains traction, philosophically, in the attempt to overcome the debate between materialism (being) and idealism (consciousness) through the development of new philosophical techniques.
This is in my opinion the most disappointing aspect of the paper. He posits that (a) one needs to take a stand on materialism versus idealism, because this is the inescapable debate at the heart of philosophy; (b) the "third way" attempts one finds in 'existential' or 'phenomenological' thinkers is generated by a breathtakingly bourgeois motive -- to overcome taking a stand on real issues by looking for comfortable solutions which will incorporate obdurate antagonisms into the bourgeois order. Let's call this the bourgeois having your philosophical cake and eating it too.
(This leads him to make some eye-raising claims like Nietzsche is essentially a bourgeois working in service of the status-quo and the increasing focus on the role of embodiment is motivated by bourgeois self-absorption.)
Quoting Moliere
Interesting! So I take him to be saying that the jury is still out on the extent to which "existentialism" (which we must keep in mind is essentially used as a stand-in for "Neo-Kantianism" and "Phenomenology" since all three are expressions of the same underlying way of going about things) will define the twentieth century. I think you're right that he thinks existentialism isn't epoch-making in the philosophical sense -- it doesn't open up the possibility of a new world the way that Kantian and Marxist philosophy did -- but I think he does genuinely worry that it might define the epoch insofar as it could aid decades of bourgeois grandstanding and avoidance of taking a stand on the era's meaningful philosophical and social conflicts.
I thought it was really interesting in our own context to be reading this with the historical knowledge that existentialism, phenomenology and marxism will all become marginalized very quickly by both postmodern thought and an increasingly trenchant capitalist system.
Quoting Moliere
Mostly, I agree. I'm not sure the details matter because this is certainly the weakest aspect of his argument. He essentially hunts for the weakest thinkers then reads their neo-Kantianism into much stronger thinkers like Heidegger.
Quoting Moliere
This is funny! I had the opposite reaction. That is the sort of thing I insert into an essay when I want to attack some big author or theme. I just sort of took that as short hand for "I think Sartre is b.s. but I have to acknowledge that I'm treating his arguments pretty flippantly." It would be interesting to know if he was sincere in thinking Sartre is worthy of a dissertation or not.
One thing I was hoping he might jump onto is the relationship between Nietzsche's interest in "positive judgments" (what we would now probably call "value monism") and Sartre's focus on "negative judgments".
Quoting Moliere
I am pretty sure that he is writing before Sartre's turn to Marxism and essentially encouraging that turn.
Quoting Moliere
Yeah, I think this comes from the larger focus on neo-Kantianism. If you start with isolated individual experience and expand outwards you will inevitably distort the matter in question. The world is for your consciousness. The world is your responsibility. Your freedom determines the world.
Again, a lot of this is not all that complicated or difficult to understand. But the fact that he locates all of these problems in fetishism is extremely interesting and I definitely want to re-read that section and post about that idea specifically.
Quoting Moliere
I agree. In my view, I think there is a "third way" between materialism and idealism, which is best exemplified by Wittgenstein's work. To an extent I sympathize with Lukács's criticisms because I think he's right about the bad Kantianism that pops up time and again in a variety of thinkers in the early/mid twentieth century. (Hubert Dreyfus's attacks on Sartre mostly mimic Lukács here and went a long way to killing off interest in Sartre in the anglophone world.)
Quoting Moliere
Yeah. Or, you know, perhaps we could actually just overcome the stupid materialism/idealism debate by drawing on phenomenology! Lukács merely posits the notion that this debate is necessary without offering much in the way of defense. And he attacks the 'third way' attempts to overcome this debate by dissecting and attacking inferior thinkers who, to my mind, exemplify the most feeble-minded aspects of the phenomenological tradition.
No worries. :) We all have lives outside of books, and it's totally understandable -- I was late too.
Quoting John Doe
This is a bit astray, but something I thought about when reading his comments about Nietzsche was how Kaufmann's interpretation of Nietzsche had yet to appear when he made these comments, and even though Nietzsche as a thinker was not a fascist the German fascists did try to appropriate Nietzsche as a sort of philosophical foundation to their political program (mostly thanks to the efforts of his sister after his death).
That was a thought that occurred to me, at least, in thinking about his comments on Neitzsche -- but it's hypothetical at this point because I'm not really sure the extent to which Lukács would be influenced by Anglophone philosophy's reception of Nietzsche, which is largely what I'm basing this off of.
But I hear you about the categories -- I probably wouldn't lump everything together that Lukács is lumping together.
Quoting John Doe
That's true! I agree with your reading. Otherwise why would he be writing on it if he didn't think that existentialism presented a kind of wrong turn in thinking?
Quoting John Doe
That is something that had not occurred to me, but now that you mention it that is interesting. We could take Lukács criticism of existentialism (as he conceives that term) as evidence of just how widespread the philosophy seemed at the time, even outside of strictly academic circles.
Quoting John Doe
Do you group Sartre in with the weakest thinkers?
I'm just curious. The other authors, to be honest, I have never read or even heard of prior to reading this paper so I have no opinion. But I never really thought of Sartre as a weak thinker, though I also didn't think he was really talking about what Heidegger was (not that we have to get caught up on this point!)
Quoting John Doe
This made me laugh :D. I hadn't thought of that, though you're probably right -- it makes sense of why he wouldn't want to spend the time with Sartre to detail where he fails.
That'd be cool to read, I agree.
Quoting John Doe
Hrm! I wasn't aware of that aspect of Sartre interpretation, or my anachronistic thinking there. Thanks for the heads up
Quoting John Doe
Looking forward to hear more.
Quoting John Doe
That definitely is the appeal I see in phenomenology -- since I don't think that a debate between materalism/idealism is really all that interesting. I think that a commitment to materialism might just be seen as important for a commitment to Marxism, which may be why it seems so important.
Really, my meta-philosophical outlook is pluralistic, too -- so I think there are fourth and fifth and so on ways. :D Materialism/Idealism is just one debate peculiar to early modern philosophy that seems to still be hanging around sometimes.
That's my plan too.
What do you guys make of starting on Riemann in about a couple weeks; say, around 13 January?
This does seem to be a problem of sorts for Lukács. It's the same thing he does with Sartre ("One would need a dissertation..."). It's clear that he has a strong perspective on a variety of topics and thinkers, and he doesn't really want to get bogged down in the sort of meta work required to justify his distaste for certain thinkers and trends. So he tries to pin the tail on the philosophical donkey by finding Kantian tendencies or forms of thought in a few phenomenologists and labeling the whole movement a type of dressed up neo-Kantianism. (To answer your questions, this is why I called Scheler and Sartre "weak" thinkers for him to pick on, because neither represents the existential or phenomenological method in its uniqueness or full strength.)
Unfortunately the consequence is that he tends to take a God's eye view throughout most of the essay and it seems to me that he's not particularly responsive to most of the interesting differences or distinctions among the targets of his critique. It is the philosophical equivalent of labeling anyone to the right of Marxism "right wing" and saying that their differences aren't particularly important because they all support something common (say, capitalist oppression).
So I think we have to just follow his logic here. In many ways the enemy in the background of the whole essay is Kant and the idealism he inspires, which is at bottom (for Lukács) an expression of bourgeois morality. That's why he is at such pains to argue that their is no 'third way' between materialism and idealism, and to argue that existentialism, phenomenology, etc. are consequently idealist philosophies. Their lack of methodological innovation is due to the fact that they are neo-Kantian (philosophically) and bourgeois (politically). Returning to the 'Marx' analogy: anything to the right of materialism is (I think on Lukács's view) in an important sense backwards and retrogressive and consequently cut from the same cloth with respect to the God's eye view he wants to take in this essay.
I don't know, do you buy this reading? If so, what do you make of his attack on 19th and 20th century non-materialist philosophies as merely 'neo-Kantian'?
Fine with me!
I do! It's his broad treatment of thinkers that sort of allows his logic to go through, too. Though I might say that the criticism isn't bourgeois morality as much as it is that neo-Kantian thought inhibits the working class and helps the bourgeois political order to justify itself. A bit of a slight difference, but I think I'd at least try to frame things in terms of political power and not morality (though there is an interesting tension in Marx showing here between his avowed nihilism and the fairly apparent moral impulse that generates the project in the first place -- not un-resolvable, but a tension)
I don't exactly buy the argument that idealism and the bourgeois order are linked. I think even materialism can be framed to justify the bourgeois order. As you note this point is not demonstrated at all -- it's sort of an assumption of Lukács that neo-Kantian methodology justifies idealism justifies the bourgeois order, because the working class's beliefs are turning increasingly materialistic.
I mean, I can get the gist of why someone who is a materialist might be inclined to take their role in the (public, political) world more seriously, and why someone who is an idealist might be inclined to remain focused on the (private, personal) world. So I don't think it's entirely out of left field. For myself it's just a matter of reflection: if I could think of some way to link idealism to an anti-bourgeois, public, and political life then the assumption -- on the philosophical level, at least, though perhaps it might in a general way for how it motivates people -- doesn't hold.
Bit of a mouth full there, but Hegel seems like a good example of a philosophical way of linking idealism to just that. Not that Hegel was by any means a political radical, but the whole taking one's place in history towards a stateless society thing is a direct reflection of Hegel in Marx. And the step from Hegel to Marx doesn't require materialism for that to take placed, so it does seem to me that there is an example that disproves the assumption.
Interesting! My reason for thinking that Lukács is dealing at bottom with what he takes to be a moral concern is that Kant explicitly states (if I read him correctly?) that his metaphysics is ultimately justified on moral grounds (in the First Critique). I guess it's probably impossible to come to any definite conclusion about this underdetermined aspect of Lukács's paper but I would love to pursue the point a bit.
Basically, my reasoning for thinking that Lukács finds an inherent connection between idealism and bourgeois morality (which perpetuates a politically retrogressive ideology) was that he's playing off of this connection that Kant draws between his morality and his metaphysics.
For example, from the closing sections of the first Critique: "Hence theology and morality were the two incentives, or better, the points of reference for all the abstract inquiries of reason to which we have always been devoted" (A853/B881).
So I think you're right that since I'm not a Marxist I may be missing the fact that thinking of metaphysics and morality as in this sense more fundamental than politics and ideology is a distinctly bad way of reading a Marxist thinker.
I like your way of looking at this but wonder if you might expand a tad? Definitely curious about how to work within the Marxist framework here to think through the difference between morality and the political order, as well as whatever personal views you might have about tensions in Marx's nihilism. (Isn't Lukács usually read through the lens of the discovery of Marx's early writings and the debate between 'early' and 'later' Marx?)
Just a quick note here -- I couldn't tell you. I was interested in reading the selection because of my ignorance of Lukács :)
Quoting John Doe
Well, I also want to say that I don't want to offer any kind of definitive way of reading Marx too. These are just my opinions I've developed after reading him, but it's not something like a how-to on thinking through a Marxist framework, and like a lot of thinkers which write interesting things; well, there's more than one way to fry an egg.
Getting that off my chest. . .
I think that morality is usually conceived in Marx as a tool of bourgeois power -- and really of all power structures. Morality is the code of conduct taught to people to ensure the order of things. If you can teach people to not just mind the police but even to police themselves then you entrench your position even more.
I think that highlighting Kant is noteworthy, but I'd use it as a foil. Whereas a liberal capitalist at the time would appeal to moral intuitions, a Marxist would appeal to one's position within the social structure. So these neo-Kantians that Lukács criticizes are sort of on the side of established power by bringing things back to moral consciousness or authenticity or idealism. Where a Kantian would point out the relationship between metaphysics and morality, a Marxist (again, according to my reading) would draw that relationship from metaphysics to the social order -- idealism being one fairly esoteric method of justifying God, morality, and a spiritual world which are all the sorts of stories those in power tell those without it to encourage them to police themselves.
The tension I see is pretty apparent in Capital -- while he is mostly drawing a descriptive picture of the mechanisms of capitalism, he also frequently cites horrid conditions of labor that he sees at the time, including child-labor. That is, rather than just making an appeal to one's social position, their historical mission to overthrow class structure, and the fetters of bourgeois morality he seems to make implicit appeals to moral intuitions. In the manifesto he lampoons bourgeois morality, but he does so by pointing out its hypocrisy with itself. Then the overall thrust of his work seems to be getting at a more humane way to live and be human -- an interpretation of Marx is to say that human essence is found in work and technology, and his appeals for a world owned by workers is based upon this essence.
At the same time, though, Marx seemed to believe that capitalism was a necessary stage in human progression, so his nihilism is also somewhat necessary. Capitalism generated both the wealth which would be required for a communist society, and it generated its own instruments of destruction through the creation of a proletarian class. So while there is a tension between his implicit claims and his outright declarations of nihilism, nihilism is sort of a part of the package too in that the evils which Marx highlights are necessary evils for human progress.
I think that for Lukács materialism is seen as an important metaphysical position for proletarian revolt because it focuses the person on what they see, here and now, and notes that what is real are the things we work upon, and not the ideas or spirits of another world or some kind of grand moral order to which they should conform to think of themselves as good persons -- a moral order which, on the whole, is something taught by churches which tend to encourage submissive behavior to authority figures, which includes the bourgeoisie. Hence the emphasis on politics and power over morality -- morality is a kind of tool, and nothing more, and to focus on it is to give up ground to the enemy.
I'm just inclined to say that this is a sort of hinge. One can make moral appeals, and one can make appeals to one's social position. I'm not sure it's actually important -- both, to me, seem more like a framing device than something with factual content. And how one effects or affects people is more of an empirical question.
Typed that kind of quickly, but those are my first thoughts! I hope I addressed some of what you're interested in.
I'll start going through the Riemann paper on the 13th.
We're going to start with the Riemann paper on the 13th. I'm still open to what people want to do beyond that. It seems to me that we should probably just let that conversation go on as long as it goes. Meanwhile, people should feel free to nominate papers or books for discussion after that.
Hi all.
@fdrake has offered to lead a discussion of Bernhard Riemann's "On the Hypotheses which lie at the Bases of Geometry (1873)". Here is a link to the paper. If for some reason that link does not work for you, just google the paper - it seems to be pretty widely available. We will begin sometime around Sunday. It's a really cool paper so I definitely want to encourage people to read it - if you're not too busy it's sure to be worth the time you put into it.
https://www.youtube.com/watch?v=K3Zx-qcNZf4
Don't worry, it's pretty dense for me too. Or worry more!
One of the reasons why it's so dense is because he's inventing lots of, what are now distinct, mathematical concepts at once. But he's not using the usual words for them (most of the time). EG, n-ply extended magnitudes seem to be n dimensional vector spaces, discrete manifoldness and its elements are like countable sets, continuous manifoldness and its elements are like the real numbers between [0,1], quanta are either elements of countable sets or bounded, connected regions of (possibly higher dimensional) space. Things like (paraphrasing) "mathematicians might unhesitatingly found the theory of discrete magnitudes upon the notion that certain things are to be found equivalent' seem to be ur forms of things like natural numbers being defined as bijection isomorphism classes of finite sets (eg, {a,b,c} and {d,e,f} are just relabelled forms of each other and both could represent the number 3). And this is all just stuff on the first page.
So yes, it's hard going, even for someone with lots of training in math.
The basic idea is this: Riemann is saying that space as we know it - the kind of space in which me move around and live - is but a particular case of a more general notion of space which can be constructed from 'general notions of magnitude'. Or, put the other way, one can construct more kinds of spaces out of 'general notions of magnitude' than only the kind of the space in which we live in.
To this degree, the 'general notions of magnitude' are too general to properly model (our) space: if you begin with such general notions and nothing else, you won't be able to properly model our space with enough specificity. So to such general notions, one needs to add another ingredient: 'experience'. Hence:
"The propositions of geometry cannot be derived from general notions of magnitude, but ... the properties which distinguish space from other conceivably triply extended magnitudes are only to be deduced from experience".
So Riemann here develops a tension between general notions on the one hand and experience on the other, the latter of which includes 'matters of fact' (a kind of rationalism vs. empiricism dichotomy). And each 'side' has its own issues when it comes to space. On the 'side' of the rational, general notions are 'too general' to get at the specificity of space, and on the side of the empirical, 'several systems of matters of fact' can be used to 'determine the measure-relations of space'.
Or to put it differently, if there is a one-to-many relation between general notions of magnitude and our space (the general notion can give rise to many notions of space, of which ours is only one), there is, on the other hand, a many-to-one relation between 'systems of matters of fact' and our space (several systems of matters of fact can give rise to our space). That all said, the main 'system of matters of fact' has so far been Euclidean geometry (there can be others).
So given that there is no one-to-one correspondence between 'matters of fact' and our space, Riemann wants to ask after the 'probability' of such matters of fact - which I understand to be something like 'how probable is it that these matters of fact obtain, and not others?'. And also, given that we can go 'beyond the limits of observation', both at the level of facts and at the level of notions, how 'justified' would we be in doing so? Which I read as something like: 'is there any good reason to 'go beyond the limits of observation' and indulge in what we 'can' do over what 'is'?
I'm a bit iffy on my reading of this last bit, and am keen to see what others make of it. This bit: ""we may therefore investigate their probability,which within the limits of observation is of course very great, and inquire about he justice of their extension beyond the limits of observation, on the side of both the infinitely great and of the infinitely small".
Section 1: Plan of the Investigation.
I think the first sentence is an attempt to characterise the study of geometry. In my mind here I'm imagining he's talking about Euclid's elements as an example. The logical structure Riemann seems to be given to geometry is as a composite of:
(1) intuitions about space "the notion of space"
(2) (mathematical) rules which characterise the intuitions.
and these things are treated as a unit; that we have characterised all our intuitions about space, the space construction/model within mathematics (Euclidean space), through a successful determination of mathematical rules which characterise them.
So when Riemann writes " She gives definitions of them which are merely nominal, while the true determinations appear in the form of axioms.", he's saying that the supposed unity between (1) and (2) is actually just a predisposition of interpretation, and we can do with (2) alone to characterise a notion of space. Thus the fact that:
is ensured by the duality of mathematical rules characterising the mathematical nature of space, but we suppose that these rules, the axioms, give a complete characterisation of the intuitions we may have, (1). What Riemann is doing here is driving a wedge between intuitions of space as studied in (Euclidean) geometry and the necessity of application of those intuitions to all possible mathematical space concepts.
I imagine anyone with a familiarity of the Transcendental Aesthetic in Kant will already find this argument extremely interesting. So I think Street is very right in their emphasis that:
Quoting StreetlightX
and I would add that this follows from our ability to play with axioms to posit new spaces to have intuitions about. Or inversely to refine and add specificity to our intuitions of (maybe Euclidean) space by codifying them in appropriate axioms.
Hoo boy.
So Riemann's saying that the mathematical accounts in history, while possibly providing different conceptions of space (as comparing Euclid to Legendre and later Gauss), did little to remove the void of darkness between mathematical intuitions of space and their axiomatisations; to name the darkness, I think it is characterised by the questions: "What do our intuitions (1) say about the axioms (2)? And what do the (2) axioms say about our intuitions (1)?", characterising the relationship between (1) and (2) from both sides as it were.
Space magnitudes seem to be treated as 0 dimensional points, 1 dimensional lines, 2 dimensional areas and 3 dimensional volumes. Space-time in the Einsteinian sense would also be a 'multiply expanded magnitude' and space concept.
So what's the commonality here? I believe when Riemann is considering a 'multiply extended magnitude', he's thinking of a vector of the appropriate dimension, a 'position' in a space. So a 1 dimensional line becomes
Note at this point we have a sense for the 'size' of 1 dimensional, 2 dimensional and 3 dimensional magnitudes - length, area, volume-, and we also have multiply extended magnitudes being a collection of independently varying 1 dimensional magnitudes (the x and y directions in the plane, say, are both 1 dimensional magnitudes which together form a 2 dimensional magnitude). So Riemann sets himself the task of:
defining/mathematically characterising/axiomatising notions of size (like length, area, volume) for multiply extended magnitudes (like lines, circles, spheres). But it is worthwhile to note that Riemann is explicitly considering notions of space, so we're considering things 'one layer back' from lines, circles, spheres - we're considering ways of linking geometries to sizes. The first example of which in the paper is trying to construct/axiomatise the usual notion of length/area/volume in Euclidean space. So when Riemann says:
he's talking about the fact that completely characterising, say, the relationship between space and volumes mathematically - you can change this relationship in accord with some notion end up with an inequivalent notion of space. This drives a 'hard wedge' as it were between the necessity of the relationship between (1) the space intuition and (2) its complete axiomatisation; there is now more than one space intuition/notion, revealed by the ability to modify axioms/characterisations of space. Shifting vocabularies, there's no unique mathematical 'space intuition' a priori, since we can characterise others - and perhaps, tentatively, this means the reason for the darkness between (1) and (2) is an elision inherent in previous geometric thought generated by the belief that studying space intuitions always meant articulating a single a priori notion of space (eg Euclidean space, that which is described by Euclid in his Elements). Because of this
(because of the plurality of magnitude notions revealed by Riemann's approach) and thus:
we must recognise what space-like notion is appropriate for whatever purposes we may have. Riemann then seeks to find indicators - necessary and sufficient conditions / characterisations - of notions of space (and multiply extended magnitudes more generally) - of which space concept is appropriate for which purpose.
and moreover the existence of multiple n-dimensional magnitude concepts (like the link between 2 dimensional spaces and areas) severs the a priori connection between Euclidean space(s) and space, as n dimensional magnitude, simpliciter.
One illustrative example here is that for distances on the surface of the Earth, if they're short we can use Euclidean geometry to calculate them, but if they're long we can't - the surface of a sphere is not Euclidean, it wraps around itself, it has curvature and so on.
Thus, the a priori necessity of space being Euclidean space, or more generally of space being uniquely characterised, is broken by the plurality of n dimensional magnitudes and measures of their length/area/volume. In essence, Riemann is playing a game of constructing counterexamples to Euclidean space after characterising precisely what it is! Find the boundaries of the concept, find the exceptions, and vice versa.
Because we no longer have the a priori necessity of our calculations about space, destroyed by the non-uniqueness of space concepts, this renders which space concepts nature can be modelled by as a matter of investigation:
and of purely mathematical consequences/assumptions of them ('beyond the limits of observation').
The relationship to infinitely great and infinitely small probably connotes the fact that Riemann will be studying space at multiple scales; the geometry relationship between 2 dimensions and volumes, say, becomes defined with respect to infinitely small variations (like dx and dy in calculus), and it may be that on larger scales different mathematical patterns can hold even within the same notion of space (like my sphere example above).
Edit: note that when I'm using <> to surround something, that's notation that refers to a coordinate system being in play. So EG
That contrast between space-as-concept and space-as-experienced helped me parse out the last paragraph a lot, so thanks for that. There is something I think I'd highlight in the intro, though, from the beginning:
I think that the "darkness" referred to above is the relations between the assumptions geometry has -- I take it he means the 5 postulates of Euclids system as the primary example, though he does allude to the thought that there can be other axioms. As I read him here it seems that Reimann is motivated to understand the possible justification for just these axioms, and wants to understand the relationship they have to one another -- whether they are necessary, whether they are universal, and whether they are even possible.
Nothing to disagree with here, just highlighting something that leapt out. It does seem, as we go on, that he believes that these axioms are not necessary a priori, but have to be justified by reference to experience -- and so the project changes to ask just how far we are justified in trusting just these axioms.
In particular why this leapt out for me was because of the next sentence I sat puzzling over for awhile:
I pretty much just had to take this on faith to keep on going, but I didn't understand why this was even the next step in the line of reasoning - much less why it was doubtless.
This also means, then, that whenever I'll use a more contemporary mathematical example or analogy, it will make an omission through excessive clarity - like reading the idea of a vacuum back into the aether, or treating phlogiston as a rung on the scientific ladder used to climb toward energy.
The modern notions of coordinate systems and vectors weren't well established at the time of Riemann's writing. Moreover, as will become more clear (I think) in the first section, he sees these 'coordinate system' like concepts as just an expression of something more basic about the objects considered. This desire to treat objects, surfaces and so on 'intrinsically', rather than through the 'extrinsic' ideas/representational powers of coordinate geometry is a style of thinking Riemann helped found in this paper - where we can see many germinal forms of now precisely articulated mathematical concepts.
Third post in a row, sorry fellow mods, I got too excited and hit the 'Post Comment' button too quickly.
I think this is about right, wondering how Euclid's system 'fell out of' our experiences and intuitions is probably something driving Riemann's engagement here; and the ability to define what makes a concept of space like the one dealt with in Euclid invites/renders possible mathematical suppositions contrary to it.
I suppose this relates to broader themes in philosophy. Mathematics is usually considered a largely a priori exercise - yet here we are with a certain plurality and non-determination within it, as if there are competing images/accounts to be had. All of which equally good and true in supposition - treating them purely mathematically -, but now the plurality invites questions of which suppositions are right. To borrow Kantian vocabulary, we either sever the pure intuition of space from the objects it phenomenally conditions or admit a multiplicity of pure intuitions of space which may apply in different mathematical or experiential contexts.
Edit: another possibility is that, say, 'the' pure intuition of space isn't necessarily captured by Euclid's work and presuppositions, driving a wedge between the intuitions embodied in Euclid's assumptions and the necessity of their connection with objects. 'The' pure intuition may admit of multiple space concepts consistent with it.
I imagine that even without understanding the math, as I won't at points without lots of work (I have a first principles/pedagogical differential geometry textbook I'm referring to to orient myself, I can definitely not send it to people if they want it), I still think it's possible to get something out of the flow of the argument. Bracketing the specifics of the mathematical reasoning and treating it like a phenomenology of space; or a highlighter of differences and commonalities between the phenomenology of space and possible ways of conceiving it mathematically.
I'm happy to put up maths notes explaining what I can. I'm also having to learn more differential geometry to write the exegesis, since I've never studied it outside a tiny undergrad reading group.
Sounds good to me. I won't comment much henceforth and there really isn't much to comment on either way for the better. I wish we could group work on a PDF with comments on the side of highlighted text to be more precise as this text contains a lot of content despite its brevity.
I forgot to say, I'll move onto the next section once you've had chance to comment. I might be attuned to the math but I don't think I am to the broader philosophical relevance, despite my efforts to immediately link it to Kant.
That probably should have leapt out as obvious from the beginning, since I know Reimann is associated with notions of space itself bending, but I thought I'd put that out there to see if others agree.
I think this is right. A manifold is essentially an object with a coordinate system associated with it. So like a sphere, or the boundary of a circle, or even the entirety of our infinite space in any number of dimensions.
A key insight in the mathematics of manifolds is that 'the' coordinate systems is kind of extraneous to it, in the same sense that a circle is still the same circle no matter whether you express all its points through the usual (x,y) coordinates, or in terms of a distance from its centre r and an angle of rotation t - the shape doesn't change, the circle is the circle, we've just altered its manner of representation.
This provokes defining manifolds with respect to all the coordinate systems they are compatible with, so definitions of manifolds quantify over coordinate systems and ensure that these coordinate systems 'chart' neighbourhoods around every point on the manifold, and moreover the coordinate systems associated with a manifold have to be able to be transformed seamlessly into each other. If this seems abstract, imagine that we rotate the y axis in the x-y plane 45 degrees to the right, the underlying space is the same, and we have a formula to translate positions assigned in the original x-y plane coordinates to the rotated version (a similar construction exists for the circle example, r^2 = x^2 + y^2, the angle t being measured from the right part of the x-axis through trigonometry, tan(t)=y/x).
The highlights in the modern definitions are:
A (smooth) manifold is a set of points considered together with a set of coordinate systems. These points and the coordinate systems must behave in the following way:
(1) that every neighbourhood of a point in a manifold (think of a closed loop drawn around the point = neighbourhood) has to be able to be expressed in some coordinate system associated with it.
(2) that every coordinate system associates neighbourhoods of points to the usual mathematical Euclidean space in a one-to-one fashion.
(3) The coordinate systems associated with a manifold must be able to be smoothy transformed into each other (like a coffee cup into a donut in the usual example).
(1) mirrors the idea of 'localised geometry' or relating to the focus Riemann has on infinitesimal calculations in the paper; the infinitesimal parts might be Euclidean or not, the space might be locally Euclidean but over large (or infinite) regions of space not; hence concerns of 'going beyond the limits of observation' - manifolds both have an infinitesimal aspect (localised geometries) and an infinite one ('really big' regions might not respect the infinitesimal relationships/local geometries derived or ascribed). [Condition (2) ensures, however, that we have an object which is locally Euclidean, but Riemann introduces this conception later.]
(2) mirrors the concerns Riemann has about quartics and quadratics in the distance notions ('square root of a quadric (quadratic, x^2 terms)' vs 'fourth root of a quartic (x^4 terms)', the notions that have 'square root of a quadric' are locally Euclidean in the way our universe and perceptions (allegedly) are.
(3) mirrors the concerns of intrinsic vs extrinsic thinking, nothing about a shape should depend on how we describe the positions of points on it, this would be like saying the map manipulates the territory, or the surface of the Earth is given its shaped by our lines of longitude and latitude.
So the specialisations bit, this is now discussing §1 in the paper:
We can only have magnitudes expressed mathematically if there exists a general concept of size (relating to previous philosophical concerns) which we can tailor to specific contexts. Like we can imagine lengths as represented by numbers with a length dimension, times as numbers with a time dimension, collections of distinct objects with numbers (cardinalities) that give how many distinct objects there are.
The distinction between continuous and discrete manifoldnesses seems to depend on whether we can 'travel' from one element of the manifold to another 'through other elements laying within the manifoldness' - so we can link two points on a bit of paper with a line, making the sheet of paper continuous manifoldness, but we couldn't link the letters A and B with a letter 'between' them, or the numbers 1 and 2 with a natural number 'between' them.
Another way of constructing a discrete manifoldness, say, would be to imagine two non-overlapping spheres in space. There are points between them, but no points between them which are also points on the sphere - and thus no points between them which are part of the coordinate systems giving the points on either sphere, so paths between them are blocked, making a discrete manifoldness.
The 'ability to travel through/on the manifold from one point to another' is related to points (1) and (2) in my previous post discussing the modern notions of manifold ('manifoldness') that developed after Riemann. It is related to point (1), because this allows us to express 'local changes of position' using the coordinate system. This equates to the condition of being locally connected. But the notion Riemann discusses is more broad, I think, because guaranteeing that all the 'specialisations' (points in the continuous case) have a continuous path from one to the other on/through the manifold would make it a connected space. Though I think that the usual Riemannian manifolds - the one the paper inspires - are indeed fully connected in the 'continuous path from every point to every other' sense. Note that coordinate systems are used to express these paths, but the paths themselves are part of the manifold and not tied to the specific coordinate system used to express the path.
(Edit: as an aside, look at this pathological motherfucker, connected but not path connected or locally connected!)
Riemann goes on to say that:
I imagine this commonality as coming from him having countable or finite collections of object in mind. So we can count chairs or coffee cups or insects or items of food. So the aside he makes:
suggests something like the following procedure. We might define the number, 2, say as the equivalence class of sets with 2 elements, so {A,B} and {C,D} could both be regarded as exemplars of the number 2, since both have 2 elements. They're equivalent because we require the same number of discrete labels (A,B for the first, C,D for the second) to attach to all the elements. This line of thought was actually developed, and numbers can be characterised as collections of sets with their elements relabelled (IE that one-to-one correspondence holds between the two, and all other sets with 2 objects in 'em). I also think he has distinct objects in mind that we usually encounter because he contrasts the massive number of cases where we can count stuff or label stuff exhaustively vs cases where we cannot; examples of continuous manifoldness.
IE, the canonical examples of continuous manifoldness being colour spectra and position in space. But the expressive power of math is much greater than this; suppose there is a continuous path between every pair of elements, then we have a continuous manifoldness by definition. We could do this with space-time, we could express the motion of a car with respect to the driver's wheel turning angles and its forward motion, we could aggregate colours together with sound frequencies or perceived colours together with frequency measurements, and these things would be examples of continuous manifoldness (since they strongly resemble 2-D Euclidean space) since mathematics allows us the freedom to consider both together, at once, as a unitary abstraction and study/ascribe relationships to the components. Note that these space-like concepts do not correspond to our usual phenomenological spatial directions, so the notion of 'multiply extended magnitude' is broader. We could even consider, say, mass as a continuous manifoldness (since it's a real number greater than or equal to 0!), and give it a local geometry (flat in this case).
The notion of splitting manifolds up into components, informally independent directions of variation, is the subject of §2, the next paragraph (but I haven't finished writing about §1 yet).
(Ahh, I messed up the last bit of this table: the type of measure for discrete manifolds should be one magnitude as a measure of another, not as part of another. I copy-pasted the wrong text!).
Now, the tricky thing is to understand the two types of measurement for manifolds. I think a visual representation will be helpful:
(1) Superposition of magnitudes:
(2) One magnitude as part of another:
It is this second form of measurement, conducted on continuous manifolds which cannot be superposed, which "forms a general division of the science of magnitude in which magnitudes are regarded not as existing independently of position and not as expressible in terms of a unit, but as regions in a manifoldness." The language in this is a bit archaic, but a 'general division of the science of magnitude' can be translated into something like "this kind of measurement is one kind of measurement in a larger 'science of measurement' which also includes other kinds of measurement".
Note the two conditions of this kind of measurement:
(A) The magnitude cannot be regarded as independent of position (within the manifold).
(B) The magnitude is not expressible in terms of a unit.
Which one can summarise as: the measure of a magnitude of this kind is immanent to the manifold itself, and not extrinsic to it. Gonna include a quote from Manuel DeLanda which I find very helpful in explaining in why this kind of thing is novel in math:
"In the early nineteenth century, when Gauss began to tap into differential [mathematics], a curved two-dimensional surface was studied using the old Cartesian method: the surface was embedded in a three-dimensional space complete with its own fixed set of axes; then, using those axes, coordinates would be assigned to every point of the surface; finally, the geometric links between points determining the form of the surface would be expressed as algebraic relations between the numbers. But Gauss realized that the calculus, focusing as it does on infinitesimal points on the surface itself (that is, operating entirely with local information), allowed the study of the surface without any reference to a global embedding space. Basically, Gauss developed a method to implant the coordinate axes on the surface itself (that is, a method of “coordinatizing” the surface) and, once points had been so translated into numbers, to use differential (not algebraic) equations to characterize their relations. As the mathematician and historian Morris Kline observes, by getting rid of the global embedding space and dealing with the surface through its own local properties 'Gauss advanced the totally new concept that a surface is a space in itself'.
The idea of studying a surface as a space in itself was further developed by Riemann. Gauss had tackled the two-dimensional case, so one would have expected his disciple to treat the next case, three-dimensional curved surfaces. Instead, Riemann went on to successfully attack a much more general problem: that of N-dimensional surfaces or spaces. It is these N-dimensional curved structures, defined exclusively through their intrinsic features, that were originally referred to by the term “manifold”. Riemann’s was a very bold move, one that took him into a realm of abstract spaces with a variable number of dimensions, spaces which could be studied without the need to embed them into a higher-dimensional (N+1) space" (Delanda, Intensive Science and Virtual Philosophy).
--
The end of §1 basically tries to say why this kind of measurement is so important: it allows one to make good on certain mathematical advances by Abel, Lagrage, etc, and it allows for a fuller investigation of multiply extended manifolds.
Plan of the Investigation - Okay, the organization of this section is straightforward (though where he lands on certain philosophical questions certainly is not). He’s introducing the topic; orienting the conversation; telling us what he thinks is wrong with previous approaches; how he’s going to fill in this gap; what his argument will be. Again, like basically any philosophy paper.
These are the basic questions I think we should be asking ourselves in this section:
(1) What is the meaning of the title - i.e. the scope of the work?
(2) What is the problem the author is addressing?
(3) Which previous solutions or ways of thinking about this problem is he attacking? In what way are they wrong or misguided?
(4) What sort of solution is he going to offer us instead? How is he going to go about doing that?
Paragraph 1:
Literature Review: Geometry assumes as given — let’s call this ‘the Given’ of geometry — (a) the notion of space; (b) first principles of constructions in space.
But geometry gives definitions of this ‘Given’ in name only; the actual work of the sort that interests us philosophically (determinate truth) appears in the form of axioms.
Oversight in the Literature: This leads Riemann to a question which will guide the paper: What is the relationship between this taken-for-granted notion of space and constructions in space (the ‘Given’ of geometers) and the type of determinate truth which geometers arrive at via axiomatization?
Problem: I take it that this is a straightforward philosophical puzzle that actually interests most members of this forum: What does mathematics reveal about the nature of space? How does our intuitive understanding of space (derived from us being a particular sort of creature with a special type of bodily experience of space) affect our mathematics? Is it possible to achieve a more disinterested and perfect mathematical understanding of space if we remain steadfastly committed to untangling the role that anthropocentrism has played in various approaches to geometry? Is a wholly disinterested mathematical notion of space ‘more true’ than the sort of space in which we live and move and have our being? (These are all still very contemporary debates.)
Paragraph 2:
Literature Review: Philosophers and Mathematicians (including the most renowned) have failed to address adequately the concerns raised in Para 1. — Why? — Because of a conceptual failure to understand a new concept that I shall be introducing: Multiply Extended Magnitudes. (MEM)
Problem How should we understand a Multiply Extended Magnitude — and consequently answer the problem set out in this paper (see para 1)? Well, we will have to show how:
General Notions of Magnitude —> Multiply Extended Magnitude.
(So if you're having trouble with the math just remember to contextualize it in this way: We have a mathematical problem, GNM --> MEM, and we're going to see how answering this will be of basic interest to the philosophical problem which likely interests all of us.)
Consequences: (a) Triply Extended Magnitude is more general than Space (i.e. ‘the Given’ of previous geometry; i.e. intuitive space). (b) Experience alone distinguishes intuitive space from other conceivable triply extended magnitudes. (TEM)
Problem: If “experience” is the sole means by which we can understand intuitive space — how do we understand what “experience” is with respect to the role that it’s playing in our ability to understand space mathematically (viz. as one among many conceivable TEM)? How does the determination of ‘intuitive’ space work?
To answer the question of previous posters about “simplest matters of fact” I read this — perhaps stupidly — as a matter of the straightforward “relation” problem being dealt with in para. 1; do we get from mathematics to intuition via facts/theory/axioms/deduction, etc.? Or is there a different sort of intelligibility at work? If the latter then we arrive at the “several systems of matters of fact” — namely, that different creatures will so-to-speak “interpret" space differently in their lived experience, but there may be an insoluble gap between how we can understand these different systems mathematically versus in lived experience.
If I may put this in Heidegger terms. If the intelligibility of the ready-to-hand understanding of space is necessary to arrive at the present-at-hand understanding of space then we may not be able to get at the variety of ready-to-hand understandings which the mathematics suggests would be available to different types of embodied creatures.
To put this in terms of the “transcendental aesthetic”: Aliens may very well experience a different sort of ‘intuitive space’, but they would need this experience in order to understand how the TEM describes this space. Just as we need our intuitions in order to get from the TEM to our intuitive space.
To put the same point in yet another way: Euclidean geometry is possible only because we have lived in human space. An alien could not get to Euclidean geometry from within a different lived intuitive-space.
The Title: I’ll quote in full: “These matters of fact [viz. euclidean geometry] are - like all matters of fact - not necessary, but only of empirical certainty; they are hypotheses.”
Sorry this write-up was a little lame; hopefully in a couple days I will be less distracted and promise to do a better job of it. Really love everything I'm reading from you guys!
definite portions could be elements, like A in {A,B} portions of discrete manifoldness, distinguished by a mark (A). Or they could be points, like the circle with radius 1 centred at 0 in the plain. I want to emphasise along with Street (and highlight since it's super important), that treating the circle as a space unto itself, as a thing with its own properties independent of the coordinate system used to describe it, is a really novel way of thinking that Riemann helped create.
I'll take it that counting is straightforward. But note here that Riemann is operationalising the concept of size with the concept of quantity; that is, we may express sizes using numbers. In the discrete case this is just counting, in the continuous case this is measuring.
I think I read this bit a little differently from @StreetlightX, but our two accounts are complementary rather than opposed. I'm imagining something like the norm of a vector in a vector space. So a norm in a vector space (roughly) is a function that takes the vector to number which represents its length. The usual norm we have in Euclidean spaces is Pythagoras' theorem. If we have the point (3,4), 3 along 4 up in the plane, the distance it is from the origin is the hypotenuse of the triangle:
What this sets up is an embedding of position notions in the plane to position notions on the number line. IE, we have 'superposed' positions (magnitudes) in the plane (x=3,y=4) by taking them both as arguments of a function (f(x,y)=sqrt(x^2+y^2) with x=3, y=4) and used this function to map the position in the plane to a position on the line (sqrt(3^2+4^2)=sqrt(9+16)=sqrt(25)=5), which is then treated as a quantity that expresses the size (magnitude) of the position in the plane. In the absence of such an idea, a metric - a means of measurement -, we can only compare relative sizes through subset relations, such as (-----) being shorter than (-----------), which moreover is achieved without the explicit ascription of quantity.
However, note that this metric requires some kind of coordinate system - a means of expressing positions in terms of quantities -, and Riemann emphasises that we should instead consider the notion of coordinate system as conditioned by/associated with the objects (manifoldnesses/manifolds) whose points are commensurable under them. As he puts it:
What this does to the above idea of 'metric' is that it requires that such metrics, systems of measure on manifolds, become localised and indexed to the local topography of the manifold itself. This is a historical antecedent to condition (1) here, emphasising that systems of measure (coordinate systems and 'superpositions' like metrics) need only obtain locally on a manifold.
In terms of Riemann's argument, this is one of those bits of fluff that you'd send in a research grant application. It's just saying that the inquiry Riemann will do has practical consequences for maths. Riemann then provides an orienting, preparatory remark for §2 in which he'll describe multiply extended magnitudes consistent with the concerns in §1.
Which states the research objective of treating manifolds as spaces unto themselves, but nevertheless finding systems of measurement that express their local topography. Moreover, he'll deal with the 'general case' of 'multiple extension', when we have as many directions of extension (dimensions/independent directions of variation) as we require. For example, 2 for the surface of a sphere despite it being in 3-space.
I think the archetypal example here is that of a line segment. I think a specialisation can be harmlessly read as a dimension, or direction of variation. So say we have the following line segment:
(-------)
the specialisations we pass over are the points which constitute it, and together they form the line - the simply extended manifoldness. There are only two directions to travel, forwards and backwards, and this gives the 'true character' of it. This notion is, however, broader than a line, as the boundary of a circle would also be a simply extended manifoldness: we are only travelling 'forwards' through clockwise rotations and 'backwards' through anticlockwise rotations, assuming we lay on the circle's boundary.
Inherent in this is the idea of parametrising a movement with respect to the changes in a single direction of variation, and one common way of doing this is by parametrising the position on a shape by the distance required to travel to it while remaining within the shape, the arclength.
Thinking about it in terms of travel on the manifold (the line above) reveals how many directions of variation are required to express its variations innately - we only need one, the arc length, because it's one dimensional. This is a simply extended manifoldness, one of a single dimension.
Now we make the above blue line 'pass over' another red line, creating a doubly extended manifoldness:
which, note, can have the points on it described by the arclength along the first blue line and the arclength along the second red one (the arrow). This means it is 2 dimensional (despite it being embedded in 3-space).
And Riemann iterates the procedure, giving a recipe for constructing a manifold of n+1 dimensions from a manifold with n dimensions and a manifold of 1 dimension. This procedure is usually called 'sweeping out' space, and usually first appears in undergraduate or school calculus when discussing volumes of revolution, the Wiki link there has another good picture of 'sweeping out' a 2 dimensional surface using a 1 dimensional surface (and a circle/axis of rotation).
Another thing to highlight here is that Riemann is looking at composing higher dimensional shapes out of lower dimensional shapes; the dependence on the coordinate system used to express either is like map to the territory, the shapes are the shapes, the manifolds are the manifolds, regardless of the particular coordinate system used in their description.
This kind of thing should be relatively familiar with most people. Now, ignoring the 1D point (which doesn't have a magnitude or size), a 1D extended magnitude actually corresponds to the 2D line: the line is the most basic 'extended magnitude' along which one can move backward and forward along. Now, add another dimension (the 3D plane) and you have a doubly (2-ply) extended magnitude. Add yet another, and you have a triply (3-ply) extended magnitude and so on, for all dimensions N - hence, n-ply extended magnitudes. 'Adding dimensions' is what Riemann refers to as a manifold 'passing over into another entirely different manifold in a definite way'.
The only thing to add to this is that extended magnitues, unlike the 'dimensions' we're used to speaking about (pictured above) don't have to be straight. They can be bendy. Or to use a more technical vocabulary, they don't have to be rectilinear, they can be curvilinear, like fdrake's illustrations. So if you begin with a bendy 1D extended magnitude (a curvy line), you can 'pass over' into another magnitude by rotating the curve. From the fdrake's Wiki link:
This is a 1-ply extended magnitude (a bendy 2D line) 'passing over' into a 3-ply extended magnitude (a 4D Volume). This 'skips' the 2-ply extended magnitude because we're rotating the curve, rather than just 'stretching it out' along a single dimension, like was done in fdrake's post.
Point P is specified by 3 variables, x, y, and z, corresponding to each of the spatial dimensions of the coordinate system. If, however, point P were to be part of a 2-ply extended manifold (which, remember, corresponds to a 3D plane), one would only need 2 variables, and not 3. And the reason this is so, is that the measure of the magnitude is no longer extrinsic to the surface (like the Cartesian coordinates), but immanent to it. This also has to do with why the measure of magnitude starts with a curve (a line is species of a curve, btw), and not a point. Only a curve can have a magnitude, which is why n-ply magnitudes are always one number 'down' from a dimension.
Quick corrective note: Riemann equates simply extended magnitudes with 1 dimensional objects, but the points which constitute them are 0 dimensional. So the point/line/surface/volume are 0/1/2/3 dimensional respectively. In my diagrams, we have a curve passing over another curve, and because it's 2 curves passed over we have 2 dimensions.
He switches to the dimension vocabulary in §3
this procedure, taking a variability of one dimension (a curve) from a manifoldness of higher dimension n, reduces the remaining dimensions of the manifold left unaccounted for by 1. So we end up with n-1 directions for variation when we've already set up the 'variability of one dimension' - a 'simply extended magnitude', since it 'varies continuously with the point'.
If you want to consider the volume of the vase, then the whole circle is a 2 dimensional object we're 'passing over' with the curve which is a cross section of the vase boundary, adding the dimensions gives us that the resultant manifoldness would be of 3 dimensions. If instead we rotate the curve which is a cross section vase boundary solely along the boundary of the circle (the full extent of the radius), we end up with a 2 dimensional vase-surface, since the circle boundary is 1 dimensional and the vase boundary is too (which is what's actually pictured in the wiki link, but it is using this procedure to suggest the volume itself is formed from the rotation by setting up the right vase-boundary).
'not constant' is a requirement for being a coordinate axis, say if on the usual real line every number between 0 and 1 was normal, it was associated with the correct real number, but every number above 1 was associated with the number 2. This would mean that this direction of variation cannot discriminate between positions which must be represented as quantities greater than 2, 'folding' all of the real line between 2 and infinity into the natural number 2. Riemann describes the procedure as something that looks like this:
(click to zoom) We also could have used distance along the original blue line and the original red arrow as axes, but I wanted to stress that any other pair of independent directions of variation within the manifold would do the same job. This procedure usually works, so long as the dimension of the manifold is finite and that the coordinate system doesn't have singular points. Riemann stresses that infinite dimensional manifolds do exist and are worthy of study:
such as manifolds whose points consist of functions (function spaces) or shapes. This concludes 3 - Riemann's discussion is just describing the above picture and its limitations.
The black point is what you get when you travel the green point along the blue curve and the lilac point along the red curve. This is equivalent to the usual notion that the coordinate (1,2) in the x-y plane is obtained by going forward along the x/horizontal axis by 1 and forward up the y/vertical axis by 2. Only now we're moving over curves rather than straight lines - transitioning from rectilinear to curvilinear coordinates as @StreetlightX highlighted here.
Riemann follows this connection between measurement/metric and coordinate system in the next section.
Does anyone have anything they want to discuss before we move onto the meatier/more technical bits?
The only way I can think that this is possible is if, on every point of the line there is a unique value for that line -- so that you really do only need 1 number to specify your location, the arclength (or whatever), since your position cannot be any other position due to every position being unique.
But otherwise I'm not tracking.
Switching into the language of degrees of freedom can help. A degree of freedom is a unique direction of variation. Straight lines have 1 degree of freedom. If you remember from school straight lines have the equation:
[math]y=mx+c[/math]
this means if you fix [math]x[/math], you determine [math]y[/math] immediately. The same concept holds for, say, the boundary of a circle. The points laying on the boundary of a circle (with radius 1 centred at the origin) satisfy the equation:
[math]x^2+y^2=1[/math]
this means that if you fix an [math]x[/math], you can determine the [math]y[/math] (up to sign). Another way of seeing this dependence on a single variable is that you can uniquely specify a point on the boundary of a circle through an angle of rotation:
So when the usual degrees of freedom for expressing a position in space using an angle and a distance from the origin are... the angle and the distance from the origin... and if we constrain the distance to be constant (1, here), we lose a degree of freedom (dimension) from the unconstrained space (of 2 dimensions) by applying one constraint (the distance from the origin = 1).
When I showed this with my examples of a 2 D shape with lines on it above, each line creates axes across the shape where the distance would be the same. In the second diagram in that post. I've added the 'lines of constant distance' from the original blue curve we used to sweep over the red one, pictured below.
The green curve denote all the possible positions consistent with being the picked distance travelled on the blue curve, the lilac curve denotes all possible positions consistent with being the picked distance travelled on the red curve. You can see they intersect at one and only one point, the previously specified black one - this is why the black one showed up in the place that it it did.
But the line example is more general than that -- it's not a function, per se, it's just a representation of a manifold which happens to have that shape. So if a manifold were shaped like a sphere, for example, and that sphere is already "given" in the sense that we already know how far each point is from the origin (if we so chose to express it in such and such a coordinate system), then we would only need 2 numbers to find our position on said sphere.
If this sounds right, then I'd say your circle example helped a lot -- I wasn't thinking of it as having already been "fixed" and was stuck on trying to figure out how I'd find where I was on a coordinate system.
Does this look right?
I'm still confused about the OD point: does it count as an extended magnitude or not?
That sounds about right Moliere. You extended it correctly (by my reckoning) to the surface of the sphere, so I think we see eye to eye now.
Quoting StreetlightX
Typically it doesn't, and I don't think it does for Riemann either. From the §1 in section one:
The important line is:
removing the parts which aren't talking about 'continuous manifoldness' or 'points' and paraphrasing:
we have that 'points' are a specialisation of which 'manifoldnesses' consist of just when 'there is a continuous path between (every pair of) points'. Now that we've established that specialisations are points, and continuous manifoldnesses consist of points, Riemann develops the notion of a continuous manifoldness in §2:
So we have that simply extended continuous manifoldnesses consist of points. Simply extended manifoldnesses are 1D objects, because 'in it a continuous progress is possible only on two sides' - like the horizontal axis in Cartesian coordinates; we can go right or left, and going right is the same as going 'inverse' left, like up and down are both part of the vertical dimensions, left and right are both part of the horizontal dimension, 'forward and backward' denote the union of the two directions within a single axis.
In answer then, an isolated point is not a 'simply extended manifoldness', or a 1 dimensional object, because for it no 'progress' is possible. Thinking intrinsically, if you placed your point of view on the point, there are no possible movements you can make in any direction to still remain on the point - you have no degrees of freedom for motion - since your position is completely specified. This total lack of degrees of freedom; corresponding to the complete specification of a location; is why a point is 0 dimensional.
Though Riemann has not ascribed a dimension to points in the paper, only to manifoldnesses which consist of points (in the continuous case). The lowest dimension being a simply extended manifoldness or 1-ply extended magnitude (a line or curve with infinitesimal/0 thickness, strictly speaking I think n-ply magnitudes are associated with n dimensional manifoldnesses, rather than being equivalent to them, but this equivocation here helps more than it hinders. I think later n-ply extended magnitude is a coordinate system notion, which can be used to describe positions on an n dimensional manifoldness).
Empty areas in the 1D case because we're just considering the lines. Black areas in the 2D case mean we're dealing with all those shaded points in the enclosed area. Graduated shading areas in the 3D case means we're dealing with all those shaded points in the enclosed volume. The surface of a sphere is another example of a 2D manifold. All the points in a cube and all the points in a sphere are more examples of 3D manifolds.
Edit: imagine yourself on the manifold, with all the directions of movement the manifold allows you. 1D manifolds - you can only step forward and backwards. 2D manifolds - you can step forward and backward; and left and right, like walking on the floor of your house. 3D manifolds - you can go forward and back, left and right, and up and down - aeroplanes move in 3D, so do swimmers. If you can 'immerse yourself' in the manifold, you're in 3 dimensions. If you can 'move about freely on a surface' you're in 2 dimensions. If your movement is forced to move along a pre-determined path, your only choice being to move forward or back, you're in 1 dimensions. If you have nowhere else to travel, and no directions to travel in, without leaving the manifold you're on - you're on a point.
EDITED: Later developments in math have that 1 dimensional manifolds become associated with lengths, 2 dimensional manifolds become associated with areas, 3 dimensional manifolds becomes associated with volumes (measure theory). However the paper develops notions of inter-point distances (metrics - distance measuring functions) that lay within manifolds, rather then just dealing with the embedding space (example will be drawn later).
But yes, the 'ply' corrections below the shapes in your drawing look right. :D
EDIT: references to areas and volumes that were here have been removed seeing as Riemann doesn't actually discuss them in the paper! He just talks about length notions within manifolds.
II. Measure-relations of which a manifoldness of n dimensions is capable on the assumption that lines have a length independent of position, and consequently that every line may be measured by every other.
The title itself is an orienting remark, the first section was:
I. Notion of an n-ply extended magnitude.
and now we've developed notions of n dimensional manifoldness (curves/surfaces/volumes), n-ply extended magnitudes (coordinate systems of 1/2/3 dimensions), we can link manifolds to measures of length through the following chain of associations:
n dimensional manifoldness -> n ply extended magnitude/n dimensional coordinate system -> measures of length
In moving from n-dimensional manifoldness to n-ply extended magnitude, we needed to be able to associate every point in the manifoldness with a collection of quantities in the n-ply extended magnitude, which takes the notion of the position of a point within/on a manifold and maps it to a corresponding quantity (or set of quantities) that locates the position of the point. This is the basic function of an n-ply extended magnitude or coordinate system.
Now that we have a machine which takes manifolds and labels all their points in a consistent manner, we can take the point labels and start to ask questions using them: what's the distance between a pair of points within/on the manifold and how is this related to the quantities we used to measure their position? IE, how can we link a coordinate system to a notion of length?
In §1 in section 1, Riemann stipulated that:
we first superposed a number of independent directions of variation to describe the position of a point on/within a manifold - taking a point in it and mapping it to a coordinate like (x,y), now we're going to superpose all those independent directions of variation in order to ascribe a length to them, like sqrt(x^2+y^2) - mapping coordinates or relative positions to quantities that represent their size.
The mechanism that described the ascription of a coordinate system (n-ply extended magnitude) to a manifold looks like this:
for every point p on a manifold of dimension n, we have the unique ascription
[math]p \rightarrow x_1 (p), x_2 (p), x_3 (p), ... ,x_n (p)[/math]
where we have n numbers that uniquely and completely specify the position p within the manifold.
Setting up this conception was the topic of section 1, especially the passages on how to build a manifold of n+1 dimension out of a manifold of n dimensions and a manifold of 1 dimension (with its simply extended magnitude). Riemann summarises all of the previous developments in the paper as:
Section 2 however introduces something more similar to the modern notion of the relationship between manifold and coordinate system. It looks very similar to the one developed in section 1, but we 'zoom in' to get a weaker condition - a more local description:
for every point p there exists some neighbourhood around it such that we have the unique ascription:
[math]p \rightarrow x_1 (p), x_2 (p), x_3 (p), ... ,x_n (p)[/math] where we have n numbers that uniquely and completely specify the position p within the manifold.
And this manifold is locally flat when all the numbers [math]x_1,...,x_n[/math] are ascribed consistently with the usual Cartesian coordinates when we zoom far enough in- making the coordinate geometry look flat very close up. What we're going to do now is flesh out the second arrow in the flow chart:
n ply extended magnitude/n dimensional coordinate system -> measures of length
and Riemann summarises these intentions in the next part of the first paragraph:
He then provides preparatory remarks for the remainder of the study:
Firstly,
connotes that we will be superposing the coordinates we have ascribed to points on the manifold in order to derive notions of distance using them. Mathematically this resembles taking every coordinate as part of a function that maps to a single number. Like we can take (x,y) and map it to sqrt(x^2+y^2) to get the distance of the point (x,y) from the origin. More formally, Riemann will be constructing a localised version of something like:
[math]x_1,x_2,...,x_n \rightarrow [0,\infty)[/math]
which takes every coordinate of a point, combines them in some way through algebraic (and differential) operations which produces a single quantity - which is a measurement of length. Riemann will make...
which correspond to this locally-flat condition - since the space we live in (at least in a present-at-hand sense @John Doe) looks to obey Euclidean geometry/be flat on small scales, we can only draw things which have this condition even if we can stipulate different notions.
Riemann gives a final head nod to Gauss before diving right into the characterisation of flat space - what is it that makes flat space flat? It will turn out to be a measure relation of the above form.
I'm thinking of making a post that describes the flow of the argument without going into the mathematical detail. Similar to @John Doe's post earlier in the thread.
Riemann is constraining his discussion to metrics, means of measuring distances in continuous manifoldnesses, which ascribe distances independent of the location on the manifoldness. Note that this is a way of assigning a notion of size to a notion of geometry, rather than measuring a specific shape. This notion is what sets up the meaning of length in a geometry, rather than an instance of measuring any particular distance within it. To be sure, objects (sub-manifoldnesses, neighbhourhoods etc) will have their sizes expressible through this notion of size, but the notion of size itself is a characteriser of the geometry rather than of any particular shape.
When you say the length of lines is independent of their position, what this means is that the distance notion applies the same everywhere in the space - there are no partitions acting on the size notion that create regions of distinct size ascriptions. To make this clear, consider two notions of interpoint distances in our usual 1 dimensional Cartesian coordinates, the real line:
[math](A): d(x,y)=\sqrt{(x-y)^2}[/math]
the usual distance notion
and:
[math](B): d_2 (x,y)=\begin{cases} 0 \leftrightarrow x^2+y^2<1 \\ d_2 (x,y)=d(x,y) \leftrightarrow x^2+y^2 \geq 1 \end{cases}[/math]
(A) computes the distance between the number 2 and the number 1, d(2,1) by sqrt (2-1)^2 = sqrt(1)=1, which is the usual distance between the numbers, and behaves exactly the same over the entire real line. (B) computes distances as 0 if x^2+y^2<1, and computes them exactly as in (A) if x^2 + y^2 is greater than or equal to 1. The picture here is that if we pick two numbers x,y that give a coordinate within the unit circle centred at the origin in the plane, the distance between them is 0, if we pick two numbers that give a coordinate outside of the unit circle, the distance between them is the usual distance on the real line. (A) is a metric in which the size of a line is independent of the position, (B) is a metric in which the size of a line is dependent upon the position.[hide=errata](B) strictly speaking isn't a metric in the modern sense, but it suggests the right idea of position dependence of line length[/hide].
However, the distinction between this 'global sense' of the metric is that (A) operates on the entire embedding space whereas what Riemann's after is a localised version. In order to set up this localised version, however, we still need to have a localised coordinate system (n-ply extended magnitude) of appropriate dimension for the manifold (of n dimensions).
The idea here is that if we take a collection of coordinates [math]x_1 (p), x_2 (p), ... , x_n (p)[/math], we determine a line (1 dimensional manifold) on the overall manifoldness by making all the coordinates a function of a single variable - like the arc length example above shows. We can imagine p as an arc-length along a curve, and all the x's are translations of the arc length to the n-dimensional coordinate system used to chart the (localisations of the) manifold. The problem then is to find a localised/differential expression for the arc-length [math]ds[/math] in terms of the infinitesimal changes (localised changes) in the (local) coordinate system. To do this we consider an infinitesimal increment along the curve, which associates the differentials [math]dx_1(p),dx_2(p),..,dx_n(p)[/math] to it - this can be thought of as a tangent to the curve at the point p, and a localised metric will take these infinitesimal changes; the infinitesimal tangent vectors; and relate them to the infinitesimal arc-length [math]ds[/math]. As Riemann puts it:
the task of finding a localised metric (for a continuous space) is solved by finding an appropriate expression of the localised arc-length [math]ds[/math] in terms of the localised changes [math]dx_1,...,dx_n[/math] - IE setting up the arc-length as a function of these infinitesimal changes. Riemann begins this task by noting various constraints on the functions which can count as localised metrics.
(1) The length [math]ds[/math] at p and the length [math]ds'[/math] at the infinitesimally displaced p' only differ by a function of the variables [math]x_1,...,x_n[/math] and differentials [math]dx_1,...,dx_n[/math] which have an infinitesimally vanishing non-linear component above the quadratic terms; this is to say that the curve is locally linear with constant curvature, so scaling the changes proportionally scales the arc length in infinitesimal regions.
(2) [math]ds[/math] should not depend on the sign of the changes, IE if we replaced [math]dx_1[/math] with [math]-dx_1[/math] in whatever function we have, the function should be unchanged. An example here is the function f(x)=x^2, we have that f(-x) = (-x)^2=x^2 (which is the case Riemann will actually use).
Riemann then takes these two conditions and finds the simplest possible set of examples.
If n=3, we have a 2 dimensional manifold which is everywhere equally distant from the origin of the space - the surface of a sphere. If n=2, we have a 1 dimensional manifold with the same condition - the boundary of a circle. We imagine wrapping such a boundary of constant distance around a manifold - and then we increment out infinitesimally from the origin, each increment gives an n-1 dimensional sphere surface (of constant infinitesimal distance from the origin of the curve). If we're going out from the origin in all directions, this means that the increments must all be increasing (getting more positive) or that the increments are decreasing (getting more negative), either way they are getting further away from 0 uniformly. From (2) we have that this is a symmetry of the problem, so Riemann can deal just with the case where all the differentials are increasing.
Since the arc-length increases going away from the origin in both directions, the arc-length must have a minimum at this point, which from basic calculus means the first derivative of the arc-length with respect to the point vanishes. So long as we assume that the differentials are bounded, anyway (like we're not going into a region with infinite curvature).
The vanishing behaviour ensures that the the second order differential of s, the curvature, remains constant when the infinitesimal increment in the arc length remains constant; thus we have constant curvature at a point on the manifold, which ensures that the lengths of lines within this infinitesimal region of constant curvature do not depend on their position! Combining this with (1) ensures that the arc-length, the localised distance measure, has the following properties (restating the first two):
(1) The length [math]ds[/math] at p and the length [math]ds'[/math] at the infinitesimally displaced p' only differ by a function of the variables [math]x_1,...,x_n[/math] and differentials [math]dx_1,...,dx_n[/math] which have an infinitesimally vanishing non-linear component above the quadratic terms [hide=errata]when dividing by the norm of the position vector in the coordinate system[/hide]; this is to say that the curve is locally linear, so scaling the changes proportionally scales the arc length.
(2) [math]ds[/math] should not depend on the sign of the changes, IE if we replaced [math]dx_1[/math] with [math]-dx_1[/math] in whatever function we have, the function should be unchanged. An example here is the function f(x)=x^2, we have that f(-x) = (-x)^2=x^2 (which is the case Riemann will actually use).
(3) if we map [math]x_1(p),x_2 (p),...,x_n (p)[/math] to [math]ax_1(p),ax_2(p),...,ax_n (p)[/math], scaling by a positive constant a, this maps the arc length [math]ds(p)[/math] to [math]ads(p)[/math].
(4) [math]ds\geq 0[/math]
The simplest example of this is the usual distance measure (A), which is characteristic of flat Euclidean space. Riemann restricts his discussion to manifolds which can be locally geometrically represented - namely those whose arc-length element is the square root of a quadratic function of the coordinate system differentials. As he puts it:
The numbers seem like they're coming from mid air, but they actually just come from the combinatoric structure of quadratic equations in n variables. If we have 2 variables x and y, there are 3 possible quadratic terms. x^2, y^2, xy. This is 2*3/2, ie 0.5 n(n+1) with n=2. If we have 3 variables x y z, there are 6 possible quadratic terms, x^2, y^2, z^2, xy, xz, yz, and so on. What this is saying is if we take some set of coordinates:
[math]x_1, x_2, ... , x_n[/math]
and consider the possible quadratic functions possible from this set:
[math]\sum_{i}a_i x_i \sum_{i} b_i x_i ^2 + \sum_{i}x_i \sum_{i\neq j} c_{ij} x_{j}[/math]
where the things besides x's and y's are just numbers. We have more required coefficients than just a simple linear relation would require for a new coordinate system [math]y_1,...,y_n[/math]. In particular, comparing coefficients between the equation of the two quadratics:
[math]\sum_{i}a_i x_i \sum_{i} b_i x_i ^2 + \sum_{i}x_i \sum_{i\neq j} c_{ij} x_{j}=\sum_{i}d_i y_i \sum_{i} e_i x_i ^2 + \sum_{i}x_i \sum_{i\neq j} f_{ij} x_{j}[/math]
the first sum deals with the linear terms, the second two deal with all the quadratic terms. The terms without x or y in are just constants. Comparing coefficients of the raw coordinates here only fixes the linear terms. The remaining 0.5(n+1)n-n=0.5n(n-1) terms, then, must be determined entirely from the local structure of the manifold as given by continuous functions of position. The situation here is analogous to looking at a Taylor expansion of the y coordinates in terms of the x coordinates:
[math]y=f(x_0)+Ax+O(\text{quadratic terms in the x variables})[/math]
this higher dimensionality - needing more coefficients to specify - attained by the curvature is why curvature is associated with a tensor! It needs more information than the linear terms and their associated square matrix to specify. .
the x and y are vectors (the complete coordinate specification for the same point in two different systems), the capital A denotes an invertible matrix, and the quadratic terms determine the curvature. Emphasising this point, the matrix A only codifies the linear relation of the coordinate systems y and x to each other - the first derivatives/tangent vectors -, the remaining parts 'spread out' along local topography of the manifold and encode its curvature [hide=errata]Since we're dealing with infinitely small quantities, we're really considering the limit of this expression as the norm of x and y go to zero[/hide]. When the space is flat everywhere, the quadratic terms vanish - meaning flat spaces have no intrinsic curvature. When the curvature vanishes at a point, the neighbourhood around that point is locally flat. The remaining part of §1 is a preparatory remark to set up for Riemann's study of the more general spaces with constant (nonzero) curvature in §2.
[math]\sum_{i=1}^n l_mx_m + \sum_{i = 1} x_i \sum_{i \neq j} p_{ij}x_j[/math]
From your last post:
That doesn't have any terms of the form:
[math]\sum_{m=1}^n l_mx_m[/math]
But rather has something like:
[math]\sum_{m=1}^n l_mx_m^2[/math]
I think that first term could be dropped and included in the second term if the second term is modified to be:
[math]\sum_{i = 1}^n x_i\sum_{j=1}^i p_{ij}x_j[/math]
This actually gives you the right number of terms, namely 0.5n(n+1) - the [math]p_{ij}[/math] would essentially compose an n x n matrix with the only non-zero terms being in the lower triangular portion. The justification for the lower triangular matrix is that in metrical relations we would always want [math]x_{ij} = x_{ji}[/math]. If we include the full n x n matrix (i.e. include non-zero terms from above the diagonal), I think we will be double counting the distance of the [math]x_{ij}[/math] entries.
Yeah. I screwed up the formulas a few times and have been editing them since. That post's very much a work in progress. It should read something like:
[math]\sum_{i}a_i x_i+ \sum_{i} b_i x_i ^2 + \sum_{i}x_i \sum_{i\neq j} c_{ij} x_{j}[/math]
the reason I was struggling with it was because I wanted to present the overall expression as a sum of linear and quadratic terms, with one sum/sigma-notation for each group. I also glossed over the double counting because if we double count a term its coefficient will be the sum of two others. Really all these issues go away if I gave up on trying to represent it as two sums. I could achieve the same effect by just grouping the sum of sums using brackets!
Edit: I've updated the previous post to use the three sums and provided a comment below its first instance to highlight which parts are which.
[math]ds^2[/math] = x P [math]x^t[/math]
Edit: For some reason I thought we were discussing the matrix A above rather than the curvature, apologies for confusions. I've changed this comment to describe the curvature rather than the translation of the linear bits of the coordinate systems to each other.
The P there generalises the relationship away from flat space, for an arbitrary invertible P this would present a quadratic with all the cross terms like [math]x_1 x_2[/math] thrown in - which I'm thinking of as the two directions 'interacting' in the surface, so that change in the local topography can't be neatly partitioned into independent directions. For the case where P is the identity matrix (the 1 in matrix algebra, multiplying by it changes nothing), we end up with [math]xx^t[/math], which through the usual inner product/norm rules is just [math]\sum x_i ^2[/math], if we treat this as an infinitesimal displacement we end up with Riemann's characterisation of flat space [math]ds^2 = \sum dx_i ^2 [/math].
But yes, I think this gives the correct expression, and you can force the matrix to be either a lower or upper triangle at your whim. Engineer's proof for n=2:
[math](x_1,x_2)\begin{bmatrix}p_{11} & p_{12}\\p_{21} & p_{22}\end{bmatrix}(x_1,x_2)^T[/math]
equals
[math]\begin{bmatrix}p_{11} x_1 + p_{21} x_2 & p_{12} x_1 + p_22 x_2\end{bmatrix}(x_1,x_2)^T[/math]
equals
[math]p_{11} x_1^2 + p_{21} x_2 x_1 + p_{12} x_1 x_2 + p_{22} x_2 ^2[/math]
we can zero out [math]p_{12}[/math] without losing any expressive power, as you said (and as I abused by letting myself double count). Which means we have:
[math]p_{11} x_1^2 + p_{21} x_2 x_1 + p_{22} x_2 ^2[/math]
an arbitrary quadratic in two variables [math]x_1,x_2[/math]
I wanna highlight something in this post because it's cool, and I just grokked a connection. Jon suggested writing the general quadratic equation (without linear terms) as:
[math]\sum_{i} p_{ii} x_i ^2 + \sum_{i}x_i \sum_{i\neq j} p_{ij} x_{j}=xPx^T[/math]
see here for a worked example. The usual way we assign a size, called a norm, to a vector is through Pythagoras' theorem: the distance of the hypotenuse (squared) [math]s^2[/math] is the sum of all the squared components of displacements [math]\sum x_i ^2[/math], so we write
[math]s^2=\sum x_i ^2[/math]
this sum is equal to [math]x x^{T}[/math], and is the squared (Euclidean) norm of [math]x[/math].
now imagine that instead of it just being orthogonal directions and terms involving [math]x_i ^2[/math] alone, we instead replace the expression for [math]s^2[/math] by an arbitrary quadratic in the same variables:
[math]s^2=\sum_{i} p_{ii} x_i ^2 + \sum_{i}x_i \sum_{i\neq j} p_{ij} x_{j}=xPx^T[/math]
this, then, is the condition Riemann plays with when he says:
the Taylor series analogy of transforming coordinates x to coordinates y before links in at this point, now including the quadratic terms, then looks something like:
[math]y=f(x_0)+Ax+xPx^T+O(|x|^3)[/math]
if we can zoom in close enough - shrinking the norm [math]|x|[/math] towards zero, the terms above the quadratic term [math]xPx^T[/math], denoted O(|x|^3) will be the first to go, giving the local approximation of the coordinate transformation as:
[math]y=f(x_0)+Ax+xPx^T[/math]
we can think of the successive terms as 'wrapping' the coordinate system of [math]y[/math] onto the coordinate system [math]x[/math] in the region around [math]x_0[/math] with greater degrees of flexibility, more flexible means more adaptive to the local topography and thus more accurate. The approximation to the transformation around [math]x_0[/math] is, in turn, just the raw function evaluation (where [math]f[/math] is the function mapping [math]x[/math]-space to [math]y[/math]space) at the point, then 'a bit further out' the matrix [math]A[/math] allows correction for linear variations around the point, which we can think of as fitting a tangent plane to the manifold at the point [math]x_0[/math] (this information is encoded in the first derivatives of [math]f[/math] with respect to each coordinate), then after we've got as far as the tangent plane will work, we start seeing the influences of the curvature of the manifold crop up - we need to bend the y coordinate system onto the x one!
Now, what's the connection between this bending and the norm [math]s^2[/math]? Riemann is noting that any localised information about the curvature is precisely given by the behaviour of a quadratic [math]xPx^T[/math] function around the point; so the matrix [math]P[/math] gives precisely how to translate the lengths of lines in the [math]x[/math] coordinate system to those in the [math]y[/math] coordinate system. So the bending of a line (1 direction in a coordinate system/a simply extended manifoldness) can be thought of as its wrapping onto the manifold over a region. Comparing how they both wrap into each other lets us derive information about the local curvature.
So, the quadratic terms are simultaneously 'corrections' in translating one coordinate system to another over a slightly larger, though still infinitesimally small, neighbourhood; and they are also a transformation of distance notions between the two coordinate descriptions localised to a point on the manifold. This is why curvature changes the behaviour of 'straight lines' within the manifold, the curvature of the manifold encoded in [math]P[/math] tells us what 'additional ingredients' it takes to get from the flat space metric [math]xx^T[/math] to the curved space metric [math]xPx^T[/math]. Just as [math]A[/math] translates straight bits to straight bits, [math]P[/math] (locally) translates curved bits to curved bits and thus encodes information about the curvature the coordinate systems both represent.
It's really hard going! Spending hours on paragraphs means you know it's hard.
I'm not giving up or anything. It's just taking time.
I'm finding the same thing, what I'm benefitting most from I think is trying to integrate the imaginative background from the first section with the mechanical mathsy bits in the second. I imagine philosophically the first section and the final section would do, so if others feel like this isn't progressing quick enough to the philosophical juicy bits I could summarise the maths so far and then we could move on to §3.
https://www.youtube.com/watch?v=lX5eCfRSCKY
Alright, I have a question about this. I thought I was following until the end here.
As I understand it what Reimann is saying is that the displacement of a line does not alter the length of the line -- it's not like the coordinates themselves follow some kind of progression where the space between 1 and 2 is smaller than the space between 3 and 4. The space is equidistant.
In the quoted bit it seemed to me that he was considering a manifold which is a straight line (segment?), and he is trying to establish the rate of change of x with respect to s (or vice versa?). But then I get lost when he is using the 2nd derivative, because by my figuring that would be equal to zero since he was considering a straight line?
He uses the example of this function in Space:
[math]ds = \sqrt{ \sum (dx)^2 }[/math]
If I can make an analogy, imagine yourself as a point rolling down a hill. When the hill changes shape, so does your acceleration. A 'straight line' on the surface of the hill isn't a 'straight line' in the embedding space. To be sure, if we have that the manifold is locally flat, 'straight lines' of tiny extent on the manifold will look like straight lines in a Euclidean embedding space. But they don't actually have to be straight (in the sense of the embedding space) because of the possibility of curvature.
Though, I think some of your intuition about the derivative is correct. Imagine if we place two points A, B on the hill really close together and draw a smooth path between them, like a piece of string bound tightly to the surface.
A-----B
Imagine that we parametrise this path so that s=0 at point A and s=k at point B - this uniquely specifies every point on the path. The average change in the function (per unit length on the path) that this path corresponds to would be:
[math]\frac{f(k)-f(0)}{k-0}=\frac{k}{k}=1[/math]
so the second derivative would be 0, since the first would be a constant (except if the points A and B coincided). But what does this calculation actually mean? All the function f does is take the arclength along the curve between A and B and spit it back out. IE f(s)=s, with f(A)=0 and f(B)=k. The rate of change of the arclength with respect to itself is always 1. More generally, the rate of change of any function with respect to itself is always 1.
Another thing to note is that the tangent vector to a manifold at a point - a line that is visualised in the embedding space - agrees with the first derivative of the manifold in its direction, but the second derivative of the tangent vector is 0 - whereas the manifold itself 'curves away' from it, showing the presence of a nonzero second derivative of the manifold (with respect to the coordinate system we're using) - curvature.
The situation that we usually have on the manifold is more like the form:
[math]x(s),y(s)[/math]
where there is more than one coordinate required in the specification of the path, if it's a one dimensional curve we also have that y(s) = f(x(s)). This means that we can consider how the curve bends over the coordinate system x(s), y(s). The curve itself, like the length of wiring, can be straightened out, so the curvature it has isn't intrinsic to its shape. This ability to straighten out something precisely means that there is a smooth transformation from distances within the shape to distances like they behave in a Euclidean (flat) space. You can also visualise this as the curvature of spaces rendering the linear approximation to their surface (like the tangent plane or vector) worse and worse when you go away from the point of approximation.
So it might be that we can bend the wire, but we don't introduce any irremovable/intrinsic curvature. What intrinsic curvature actually measures is how movements of oriented objects constrained within the surface change the orientation of those objects when moving around closed paths - paths with the same start and end point.
This even applies to forming a circle out of it, circle boundaries don't have intrinsic curvature whereas the surfaces of spheres do! A tangent vector to a circle at one point, transported around the circle while remaining tangent, is still in the same direction as it was when you started! You can check this by rotating, say, your phone around the top of a coffee mug. So let's talk about the surface of a sphere.
Another task we might imagine is taking a perfectly taut bit of copper wiring and trying to wrap it onto the surface of a sphere. You can press it onto a point, and it'll touch the point. But, if the sphere is absolutely huge relative to the size of the perfectly taut bit of string, the taut bit of wire might resemble the surface of the sphere very well, it wraps away slowly from the wire with respect to spatial changes (points flowing away from the wire). If the sphere is tiny compared to the bit of string, it just glances off it and the sphere quickly wraps away from it. The acceleration with which the sphere wraps away from the taut bit of string is its curvature (no acceleration = flat!) - in the small sphere case you'd have to bend the wire a lot to fit the surface, in the large sphere case you'd have to bend the wire a tiny amount to fit the surface. The curvature of the surface of a sphere is not something that can be removed by these smooth transformations.
The presence of curvature, like on the surface of a sphere, does something pretty strange to straight lines from the embedding space. If you take the iron wire, press it onto a point on the surface of the sphere, and you hold it onto the sphere, what happens when you take the iron wire, still taut and straight, from around a closed path on the sphere? This requirement that it's still tight equates to that the iron wire must be tangent to the sphere. If you move it along a path, where the start and end points of the path are the same (say carving out a quarter of the sphere). By the time it's back to its start, the direction you're holding the iron wire in actually changes. The presence of these changes signals intrinsic curvature.
You can check this with your fist and your mobile phone. Clench your fist with your thumb toward you, take your phone and press it hard onto the knuckle of your index finger. Push the phone away from your body along the line of your knuckles. When it gets past your final knuckle, still keeping it pressed onto your hand, force the phone towards you from the first finger joint on your pinky to the first finger joint on your index finger. Finally, bring it back from the first joint on your index finger to the base of your thumb, and back up again to the first knuckle. You should see that the phone has inverted. Contrast this to the mug and phone thing from earlier.
Have you read the thread so far? We've discussed things relevant to your questions.
(1) “Let us suppose a variable piece of a manifoldness of one dimension” - I’m not sure what work the word ‘variable’ does here in ‘variable piece of a manifoldness’. Can one take an invariable piece of a manifold? And what would this distinction mean?
(2) “Let us take a continuous function of position within the given manifoldness, which, moreover, is not constant throughout any part of that manifoldness.” - Here, I’m not sure what work ‘not constant’ is doing. Is it the variation of position on the manifoldness we are asked to think of is ‘not constant’?
I don’t think I can work my way through the rest of the paragraph without getting these fundamentals down.
I read that as saying variable piece of a manifoldness might be a connected chunk of a manifoldness. But when we take 'a' variable piece of a manifoldness Riemann intends us to be discussing an arbitrary one. The other bit 'of one dimension', connotes that the manifoldness he's considering is just a 1 dimensional curve.
So you might imagine cutting a cylinder down the middle really finely to produce a circle.
Quoting StreetlightX
This continuous function describes what point you are on on the previously considered curve. If the function was something like f(x)=x for x<1 and f(x)=1 for all x>=1, this makes the entire region [1,infinity) map to 1, so it can't be used to uniquely specify the position. More specifically, in this example, if you know the function is f(x) = x, you can take an output of this function f(x) and directly map it to an input x, allowing you to translate between the position described using the function and the position on the manifold. However, when this x becomes greater than 1, all this function tells you is that it's equal to 1. Which means the input which caused the function to be 1 could be anywhere between 1 and infinity - so we can't invert the function to uniquely specify the point on the curve.
Imagine we're on the surface of a wibbly wobbly sphere, and we pick a point O and call it the origin. We're going to look at the distance from O to nearby points.
From O, we move out in every direction along the shortest possible path. We can imagine this as inscribing a wibbly wobbly circle on the wibbly wobbly sphere and drawing lines on it which hug the surface and are as straight as possible.
We then pick another point on the wibbly wobbly sphere within the wibbly wobbly circle. Since we drew the 'system of shortest lines (geodesics)', and drew all of them, this point will lay on one of the geodesics. Therefore, we can relate the position of this point to its position on the geodesic. In order to do this, we need to look at how the geodesic hugs the sphere - which means we need to look at how the geodesic changes over the wibbly wobbly circle within the wibbly wobbly sphere. That is, we need to relate the new point to the old point using the geodesic line and the coordinate system the wibbly wobbly sphere is in (the embedding space).
Riemann's recipe for this is:
The [math]\text{d}x[/math] quantities 'in' the geodesic are the embedding coordinate system quantities which vary with it - like the angle when moving on the boundary of a circle. The ratios [math]\text{d}x_0[/math] are the rates of change of each coordinate with respect to every other. EG, moving around the boundary of a circle, we never change the distance from the origin-we have a fixed radius, so the rate of change with respect to the distance from the origin is 0, whereas the rate of change with respect to the angle from the horizontal axis is 1. IE we are looking at [math]\text{d}r[/math] and [math]\text{d}\theta[/math], the [math](r,\theta)[/math] being the distance of the point in the embedding space from the origin [math]r[/math] and the angle of rotation from the positive x-axis [math]\theta[/math], rotating clockwise around the boundary of the circle by [math]\text{d}\theta[/math] moves [math]\frac{\text{d}\theta}{\text{d}\theta}=1[/math] with respect to [math]\theta[/math] changes and [math]\frac{\text{d}\theta}{\text{d}r}=0[/math] with respect to [math]r[/math]. So the distance between two points on the boundary of a circle only increases with respect to the angle (sweeping out an interpoint distance of [math]r \text{d} \theta[/math] infinitesimally, and does not increase with respect to the radius since the distance from the origin does not change.
Riemann wants to generalise from this notion, instead of necessarily having two independent coordinates, the system of points going out from the origin might (and in general will) be functions of multiple dimensions from the embedding space - a general interpoint distance on the wibbly wobbly surface depends on changes in all the in the embedding space coordinates. This means instead of just looking at independent sums where [math]\frac{dx_i}{dx_j}=0[/math], he wants it to be [math]\frac{dx_i}{dx_j}=k_{ij}[/math]. What this looks like for a curve [math]f[/math] with inputs [math]x_1,x_2,...,x_n[/math] from the embedding space that outputs a position [math]y_1, y_2, ... , y_m[/math] on the surface is.
[math]\text{D}f = 0+J_x f + \text{higher order terms}[/math]
where [math]J_{x}[/math] is a matrix that stores all the [math]k_{ij}[/math] at the chosen origin. This [math]J_x[/math] interfaces with the y variables through the chain rule:
[math]df(y_i (x_1,...,x_n))=\frac{df}{dy_i}\frac{dy_i}{d(x_1,...,x_n)}[/math] and stores the results in the vector [math]J_x f[/math]. This vector can be thought of as a linear displacement from 0 - (edit: like the gradient operator combined with the chain rule), and Riemann then insists that when the displacement is infinitesimally small - when we increment along the curve [math]f[/math] by [math]Df[/math], this gives us the square of the line element:
[math]ds^2 = \sum dy_i ^2 [/math]
the intuition here is that we need to increment along 'all the quantities in the geodesic', which are the [math]y_i[/math], so that the infinitesimal increment in position becomes:
[math]\sum d y_i[/math]
and the norm of this increment is then
[math]\sum (d y_i )^2[/math]
through Pythagoras (and the flat space stuff from before). Which Riemann states as if it is incredibly obvious:
. It should be noted here that the embedding space coordinates are kind of extraneous so long as we are considering increments confined to the surface - IE, whenever we write a y or a dy it's also secretly an x or a dx, and all this chain rule stuff does is express how the changes on the surface coordinates (system of geodesics) work with respect to the embedding coordinates which we initially used to express them. This local linear approximation of changes on the manifold with respect to the embedding space using the variables on the manifold is what Riemann achieves through the proportion construction
by fixing the 'initial value' to be the sum of the squares of the increment's norm. The increment's norm gives us the local linearity, the remaining discussion refines the approximation of [math]ds^2[/math] to include quadratic terms that express the curvature.
For the sake of actually moving on in the reading group, I'm going to stop trying to get the math down precisely. The picture here gives the broad-strokes steps Riemann uses to define the curvature in 2§2. 2§3 is much more qualitative - expressing the 'donut is a coffee cup' and 'boundary of a cylinder is a plane' notions in topology.
I think life happened to all of we intrepid learners.
Badger here! On Nietzsche binge at the moment, but I’m hoping to have a proper go at Logical Investigations soon ... ish! Meaning within next 6 months!
Nietzschewise should be starting On the Geneaology of Morals in a week or so for the first time. Note: I purposely don’t read the translastors intro on the first read so I’ll be completely new to the text.
Riemann abstracts from the analysis of specific surfaces and their coordinate systems to properties which remain the same after the surface has been deformed.
There are two points here: the first is that if you bend a surface, you can account for the bend through a change in coordinate system - you just need to chart where the points started, the initial shape, and where the points ended up; the function which takes you from start to end is a coordinate transformation. The second is that transforming a shape can be thought of as a way of taking the points on it onto other points, a function which maps points to points again.
Constraining the types of function which bend shapes or transform coordinate systems into each other will allow the the study of invariants to those transformations; which are thereby intrinsic properties of the manifold considered.
The methodological move here is to define what is intrinsic to a shape with reference to what does not change when you transform it. This is the idea where the (long ago now) mentioned difference between intensive and extensive types of analysis is partly inspired by.
For a physical analogy, take two balls made entirely of iron, the sizes can differ but the density is the same. Or when you pour water out of your kettle into a cup, the temperature of the water in the cup is the same as the water in the kettle, whereas there's more heat energy in the larger volume. Density and temperature are then intensive properties, volume and heat energy are extensive properties.
Riemann wants to introduce a similar distinction within the study of manifolds; intensive/intrinsic properties are those that do not depend upon any particular coordinate system or object deformation for their expression, extrinsic/extensive properties conversely depend upon a particular coordinate system or object deformation.
He sets this up in the first paragraph of §3.
The specific constraint he places on deformations to make this distinction between intrinsic and extrinsic properties is that the deformation must preserve the length of lines within the object. In this manner, if you draw two dots on a piece of paper and measure the distance between them with a tape measure, then fold the tape measure and the paper together to make a cylinder, the tape measure will still report the same distance between the two dots after the deformation. The existence of a deformation which can transform one object into another without changing the length of lines within the object; or equivalently the interpoint distances constrained to the object; means that two objects can be treated as equivalent.
Therefore, if you can take a plane and bend it into some other shape, the theorems of plane geometry will continue to apply to it. If you cannot bend a plane into another shape, it will have different intrinsic properties; it will require different geometric laws to model.
This connection between geometry and the intrinsic properties of surfaces is the point in the analysis which suggests that there are a plurality of geometries, rather than simply the geometry of a Euclidean embedding space. Geometries track intrinsic properties of manifolds, and constrain the coordinate systems which can be used to express them; by means of limiting whether an appropriate deformation exists between an object in one configuration and an object in another. Riemann gives the example of the surface of a sphere as a manifold which cannot be appropriately deformed into a plane.
In one dimension, for curves, the radius of curvature at a point is the radius of a circle tangent to that curve which fits the closest to it; the curvature at that point is then the reciprocal of the radius of curvature. In higher dimensions, tangents are replaced with tangent spaces, the spheres to their higher dimensional analogues, and the point they touch the surface must have the radius normal to the surface.
The 'two curvatures' of a surface (in 3 space) are the directions in which the curvature changes least and most at a point. These depend on the deformation of an object; and so are extrinsic properties.
The idea behind this is to wrap a surface onto a circle or sphere (the Gauss map), you take the radius of curvature and the direction it lays in, then shift the line segment representing the radius to have its origin in the centre of a circle or sphere. You do this for every point on the desired surface, making a function from the surface you're studying to the circle or sphere. Intuitively, the rate of change of this function with respect to displacements on the original surface measures the angular changes in normal vectors with respect to movement on the surface; if the normal vector never changes angle, the shape is a plane. Of course, for higher dimensional manifolds, there will be various directions we can choose for the displacements on the original surface, meaning we need to represent changes in all directions in the same mathematical object. This is done by finding a basis for the directions in which a normal vector can change, then expressing the rate of change in each direction. The two curvatures (for a 3 dimensional surface), then, quantify the amount of curvature change in directions of maximal and minimal curvature change.
The spherical excess is just another way to measure this angular change. Riemann then wants to generalise this construction to an intrinsic feature of the surface; and he does this by defining the above directions of variation in terms of geodesics which stem from a point; the geodesic is uniquely determined by the initial direction it moves away from the point, and these directions are the directions of variation for a normal vector to that point. Thereby, the notion of displacement in the previous paragraph is made intrinsic to the surface by relating the directions of normal vector change to the directions geodesics can travel away from a point.
The previous discussion in 2§2 then applies when we have chosen appropriate geodesics to express variation on the surface. Riemann summarises this by saying that the above geodesic construction determines the curvature. Since the construction was based on the length of lines, it will also be invariant to transformations which preserve length of lines.
This follows from the previous discussion; it's enough to know the curvature is zero in a subset of independent surface directions to know it's zero in all of them. Riemann then generalises the discussion from surfaces of 0 curvature to surfaces of constant curvature (0 is a constant).
If we have a rubber sheet and roll it into a cylinder, interpoint distances don't change; this follows from thehe thought experiment with the measuring tape pinned to the surface). If you stretch the rubber sheet, then the measuring tape would have to stretch too to mark the new interpoint distance. This follows through contraposition (P=>Q <=> ~Q => ~P), where 'Q' is 'curvature is constant' and 'P' is 'figures may be viewed in them by stretching'. It's easier to imagine ~Q, where the curvature changes - which means the measuring tape would have to stretch to fit snugly to the object -, which would therefore mean the figures inscribed on the surface would have their interpoint distances changed too; and thus are distorted.
Is the contrapositive argument. Inspired by the relationship between curvature and interpoint distances in the previous discussion, and through this contrapositive argument, Riemann seeks to strengthen the implication to P<=>Q, which says 'figures inscribed on the surface are undistorted' if and only if 'the curvature is constant'. The argument here is somewhat informal, as it is demonstrated by 'the measure relations on the manifoldness are entirely determined by the curvature', IE 'curvature behaviour' characterises 'interpoint distance behaviour'. Though this is of course supported by the prior derivations linking interpoint distances to curvatures.
which is just the formula for the relationship between the metric and the curvature (don't know exactly how he gets to it).
The (curved) surface of a cylinder is just a circle boundary swept upwards. This means that any point on the surface of a cylinder can be given by an angle of rotation around its central axis [math]t[/math], the distance from this central axis [math]r[/math]; which is also the radius of its generating circle; and the distance travelled up from the circle's base [math]z[/math].
For ease, fix [math]r=1[/math] and let's make the circle centred at the origin of 3D space, the point (0,0). The typical point on the rolling surface of the cylinder (with radius 1) can then be expressed as:
[math]\begin{bmatrix}x \\ y \\ z \end{bmatrix} = \begin{bmatrix} \cos (t) \\ \sin (t) \\ z \end{bmatrix} = P(t,z)[/math]
The curvature of the cylinder is expressible by how the rates of change of tangent planes at each point interact, which means we need to differentiate [math]P(t,z)[/math] with respect to each coordinate.
[math]P_t (t,z) = \begin{bmatrix} -\sin (t) \\ \cos (t) \\ 0 \end{bmatrix}[/math]
[math]P_z (t,z) = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}[/math]
The angular changes which track the changes in the tangent plane with respect to variation in coordinates are tracked by the dot product between all pairs of rates of change. IE we have:
[math]P_t (t,z) . P_t (t,z) = \sin^2 (t) + \cos^2 (t) = 1[/math]
[math]P_z (t,z) . P_z (t,z) = 1 \times 1 = 1[/math]
[math]P_z(t,z) . P_t (t,z) = 0 [/math]
What each equation says is that (1) differential changes in angle [math]t[/math] as they contribute to the curvature are constant, [math]1[/math], (2) differential changes in height [math]z[/math] as they contribute to the curvature are constant as well [math]1[/math] and that (3) moving a bit around the circle and moving a bit up the circle at the same time doesn't change the curvature.
The first (1) is associated with the infinitesimal length change along [math]t[/math] alone, [math]dt^2[/math]. The second is associated with infinitesimal length change along [math]z[/math] alone, [math]dz^2[/math], and the final tells you how moving around and up at the same time changes the interpoint distance - not at all.
This means that the arc-length [math]ds^2[/math] between two points on the surface can be measured just by how much one is rotated with respect to each other [math]dt^2[/math] and how much one is above or below the other [math]dz^2[/math], the sum of the rotational amount and the distance up gives the infinitesimal change from one point to another. IE:
[math]ds^2 = dt^2 + dz^2[/math]
which is actually a flat metric. Thus, since a coordinate system exists which renders the arc-length between nearby points an independent function of each position, the surface of a cylinder is actually flat in the intrinsic sense.
Riemann takes his previous discussion and tries to relate it to physical space. Specifically, how does being able to see shapes/manifolds as 'spaces unto themselves' change what conceptions of space are possible? And how does this plurality of space concepts relate to empirical science?
He begins in §1 with a recap of the properties which characterise the geometry of an object.
(1) (Space is flat) if and only if (triangle angles sum together to 180 degrees). This is one way for curvature to be constant - everywhere 0.
(2) (Space has constant curvature, like a sphere surface) if and only if (the sum of angles in one triangle is the sum in all others). Note that the equivalence in (1) is also a case of this equivalence.
(3) Even weaker, if line lengths do not vary with origin point or orientation, how inter-point distances constrained to a manifold vary is always expressible as a function of the variables used in defining the manifold. (1) is a special case here since the 0 function is such a function, so is (2) since the constant function is such a function. (3) expresses the dependence of interpoint distances on the variables tracking the point on the manifold.
A discrete manifoldness is a purely categorical structure - it consists of distinct elements like A,B,C, rather than a continuum of points constrained by a function of position. I think all this is saying is that ways of counting discrete objects can error, but conceptually the procedure is quite simple - irrelevant of what the categorical structure is, counting's the same and 'to count' doesn't change meaning with respect to changes in counted elements. Measure relations of continuous manifoldness do, however, since curvature may change and thus change how measuring tapes glued to the manifold work.
Extent probably means something similar to cardinality, or maybe a 'larger' concept. A continuum of points will have an infinite extent of points in it, since there are infinitely many, but nevertheless it may be bounded. Like the line between 0 and 1 on the number line.
Space is three dimensional, this is confirmed in every experience of it. Moreover, it doesn't 'just stop' at any point, making it unbounded (Riemann does not however seem to say why). However, Riemann portrays the unboundedness of space as 'possessing a greater empirical certainty than any external experience'. This is puzzling, on the one hand we have the unboundedness of space being supported by every experience, on the other we have that because or leading on from this unboundedness in situations of perception we can be more 'empirically certain' of this unboundedness than any mere synthesis of experiences allows.
I suspect I am misinterpreting the distinction between unboundedness and infinite extent, and both terms.