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nLab > nLab General Discussions: New Member - a General Question and some musings.

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  1.  
    • CommentRowNumber1.
    • CommentAuthorChip Cooper
    • CommentTimeJun 29th 2018
    First, Hello! I was first directed to this site through a comment made in a Quantum Magazine article "There Are No Laws of Physics. There’s Only the Landscape." which led me to the page: 'StringMath2017' [https://ncatlab.org/schreiber/show/StringMath2017].

    I am 64yrs old and have been studying physics for most of my life. On visiting the site I was encouraged when discovering the article was replete with links to pertinent information. Somewhere along the path I discovered that it is a guiding principle of the site, for which I can only say Wonderful! As often occurs when I research new concepts, I found myself buried several layers deep. I am familiar with this, and it is welcome when it guides me into the most fundamental principles for clarity of the topic. My first problem is one with which I am also too familiar, I reach the end of the ability to link deeper. This often happens because I have reached concepts that are well known by the author, or there are problems in making links to the needed information; or, in this case, Symbolism for which no reference is given but is a required and necessary requisite for understanding. Specifically here: I have encountered many times the symbols of set logic, and over time many are familiar but not all. This is problematic for me.. here (I hope) I can provide an example:
    A≠B⇔∃x((x∈A∧x∉B)∨(x∈B∧x∉A)) Wow, which to my surprise actually worked... copy/paste! lol
    Simply put, I understand some of these; but not all. What I'm looking for is a definitive source for their meanings as this 'definition' is intrinsic to understanding the page. [https://ncatlab.org/nlab/show/inhabited+set]
    In this case, the good news is I feel confident I completely understand the meaning with no further need for references: however to confirm this is true in this context, I would like to conceptually confirm this internally as it relates to the subject.

    I believe what I seek may exist in my clarifying concepts of perturbative and non-perturbative

    The lack of a source for the symbols used in mathematical, or logical, or any type formula or expression has long been a bane and stumbling block to my understanding concepts expressed by "the community". In physics, understanding what the variable represents is the foundation of its meaning. In the case of set logic, I believe I understand the variable as being a set or 'sub-set'(sic), where the content thereof is being defined by the logic that follows; but, the symbols have definitions. It is those definitions I lack.

    I have long held an interest in all subjects related to physics, as I mentioned above. My desire to learn is driven by an innate curiosity (clinically defined as Asperger's syndrome), a lifetime of study, and what I can only describe as an intrinsic inherently cognitive philosophical event (in my 30's) in which I experienced a cosmic connected nature of everything that IS. My search is for a symbolic set that can express an observation of the inherent nature of physical existence which I have observed but have no means of communicating outside of myself. When I run across information that resonates in some cognitive way to this understanding, I am driven to understand the fullness of the concept 'found'(sic) to determine if there is anything useful to my goal that is contained within it, even to the point of considering how the concepts of intuition are accounted for in mathematics and geometry.

    I was 'not' surprised to see that philosophy is included in the forum. For my personal approach to philosophy I have adopted Peter Suber's logical approach described in his paper "Logical Rudeness". While I feel aspects of my knowledge appear to be on the verge of a 'belief', that is not my goal. Therefore when it comes to belief systems, I respect all beliefs, and consider faith based disagreements to be a good faith bona fide request for dismissal. One irony for me is that the concepts I seek to express are more easily understood than defined, somewhat akin to the concept of negation; related concepts of reality and existence; construction & deconstruction vs destruction and perhaps the superpoint [https://ncatlab.org/nlab/show/superpoint] all concepts I'm trying to validate and discover whether my understanding is is equivalent to what is defined here, which I imagine to be something between the deconstructed zeropoint (π-π) and two superpoints (π,-π) perhaps... between perturbative & non-perturbative, harmonic and dissonant (2π) manifold. In this case π refers to a geometric concept of a circle, or wave function and π being 1/2 of that mathematical or geometric definition which is unitless, and relating to concepts in harmonics or fractions of wave functions and a possibly inherent dissonance in π as a concept related to integers, it being transcendental, which just happens to be an interesting philosophical concept, wherein I conceptualize that purterbation and non-perterbation reside geometrically (space-time) & energy. Which perhaps I try too soon to represent with words as symbols, as it has an appearance of islands & bodies of water found in fractal sets, like space and particles. Yeah (sigh) too vague.

    But, my question basically is where does one find references to the symbolism used in the set theory?
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJun 30th 2018

    where does one find references to the symbolism used in the set theory?

    We could easily create nnLab pages, where missing, that explain whichever symbol needs explaining.

    You mention:

    A≠B⇔∃x((x∈A∧x∉B)∨(x∈B∧x∉A)

    Which of these symbols are unclear? All? Some? Let’s start with one of them.

    • CommentRowNumber3.
    • CommentAuthorChip Cooper
    • CommentTimeJul 2nd 2018
    Thanks for replying to me...
    These symbols were looking more familiar to me today. I just looked up one symbol I 'thought' I knew for sure, which led me to find I was wrong; but, did end with some clarity perhaps from a site which defined many symbols.

    These appear to be both set theory and math logic symbols? I 'assume' yes as with it I understand the phrase. it's defining an inequality between two sets as a member inclusion and exclusion. I wasn't thinking the symbol set as a mixture... my bad.

    However, I looked for, but couldn't find any nLab pages where these symbols were defined. Perhaps you could direct me to a parent topic where I would find those that are here. I presume to understand that these symbols would be difficult to link because they're (La)Tex. Perhaps this now has become more a meta discussion on a way to easily find particular symbols when I encounter them. (it's a big problem on the internet) I imagine constantly posting to request clarification could and would become tedious. I'm also thinking if I become aware of 'the' system you already have in place, I could perhaps find these symbols that exist, "where NOT missing", without much ado. Then only when necessary, bring up what is missing from an nLab page, or by me misunderstood or completely unknown. Plus while reading and digging into these works; it sure would be nice to have a ready reference; I would help however I could.


    Musings... rhetorical plea for help?

    To say I'm lost, is in many ways correct. But, not in the customary sense. I need help; and, feel that this site is replete with the type of help I undoubtedly need. Where I really need help is in correlating previously visualized concepts to words and their definitions. In other words concepts hitherto known, but never symbolically converted into words. Admittedly some are unknown; For example: In spite of many attempts, I have no visual nor mathematical conceptual understanding of a Lorenzian Manifold (I don't seem to be able to see the relationships between the variables and when shown an image, it looks like nothing to me).

    That said, I was born with an innate ability for 3D visualization), later extending that into a 4th dimension (i) was intuitive, and even later to 6 and more.... seen but requiring deeper meditations to visualize. However these are as I said, for me, visualized. Other than the customary basic @90° to one another, I lack words. Ultimately, I found myself understanding but questioning the simple concepts in Heisenberg's uncertainty principle, comprehending fully as it's understood by the community; but lacking in acceptance as Copenhagen's definition of a 'wave function' as the "the basic fundamental entity" I find visually a conceptual manifold at length of 1/2 Planck (finding words or math for this visual is one goal) [Ironically, in nLab I found a concept of superpoint which is intriguing].

    Bottom line for me is where do I start to understand these definitions and links in nlab? [I dive deep into the links, until I get lost, never resolving, or circling circling. Ghostly concepts, at times, rapid heartbeats of excitement; then, wondering where it all went, and I'm lost. The basics are too simple; some definitions are known, others hauntingly familiar in a visual way briefly... but remembering that *that* particular almost foreign word defines and means what I understand and visualize intrinsically... difficult to hold on to... perhaps it's age, perhaps I've missed too much of a subject, I know not what subject.

    I've always been a visual person, even with math. Linear Algebra, multi-dimensional matrix identities and such appear visually to me much like calculus proofs, intuitively we know that... here now, then gone only to be seen again if needed and sought. Visual memories and cognition understood at a level beyond the need for their roots and words. And yet, to massage and manipulate, to reformulate the formula approaches unavailability for me. I need a systematic approach. A study. Something that will address a missing foundation; but, will not be so familiar as to be replete with boredom. (I dove deep into geometry, but without a teacher) Perhaps I'm just some unfortunate visionary that realized too late that while seeing may be believing; it's not enough for convincing, ultimately to create something useful from it.

    So, I look for help, without knowing exactly what I'm looking for. So, this site is great! But, I end up too deep into the links... (too many open pages) often circling around to cross over subjects visited. I'll admit; I'd be one of the first people to extol the virtues of the forest vs trees, expounding that the relationship of one is undefined without the other. But, what have I found here? Something that I both recognize and do not recognize as a lot of familiarity. The argument, in my opinion, to be sure is one of ontology vs epistemology, I'm I glimpsing a serious discussion hidden in the nLab pages? How DOES one prove the unobservable? Is the indirectly observable connected to the unobservable? Or, is the unobservable always been here; but, in it's observation unrecognized due to symmetry?

    Sincerely, I hope I do not ramble so much that my desire to learn and gain clarity is lost in my musings? I don't know if what here is pertinent to what I need; nor do I know where to begin. The answer seems to be 'anyplace one desires', and perhaps all that I really need is perseverance, and with hope the trees will begin to resolve. I already know the forest, I'm willing to invest in knowing the trees to prove it. Symbols are a good start. Without rambling and musing, let me address it as I continue to dig in, and dig back out.
    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 2nd 2018
    • (edited Jul 2nd 2018)

    Maybe start with https://en.m.wikipedia.org/wiki/Set_theory#Basic_concepts_and_notation and/or perhaps check out some books on elementary set theory. The website mathematics.stackexchange is a great place to ask focussed questions on specific mathematical concepts or problems you may encounter. I strongly suggest browsing the site to see what style is the norm there, and what leads to fruitful answers. All the best.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJul 3rd 2018
    • (edited Jul 3rd 2018)

    David, why that urge to move discussion away from the nnLab, here and elsewhere?

    I have started logic symbols – table. People should be invited to expand on it.

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 3rd 2018

    It’s makes for quite an exercise to see what it would take to understand Chip’s A≠B⇔∃x((x∈A∧x∉B)∨(x∈B∧x∉A) solely through nLab pages.

    He needs at least to know that material set theory is generally presented as a first-order theory with a binary relation ∈, hence the mix of logic and set symbols (#3), and that this theory expresses set identity through extension.

    How would you find this out? If by chance you reached material set theory, until 5 minutes ago you wouldn’t have had a link to the axiom of extensionality. Now you do, you find that ∈ is an extensional relation. So now ∈ is to be compared to ≺\prec at the latter page. But how many beginners will see that the ’set’ on which ≺\prec is a relation is a set of sets for ∈?

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJul 3rd 2018
    • (edited Jul 3rd 2018)

    Let’s not make it more complicated than it is. He was asking for the meaning of the basic symbols.

    Sometimes it is good to remember that there are people out there who have yet learn everything from scratch. We can easily provide a little guidance, long before it gets to subtleties as the distinction between material and structural set theory.

    I just talked to a lot of students at Nesin Math Village in Turkey. There, Wikipedia is blocked, and I learned that the nLab is a common source of information. This is young human beings who want to get basic information. And it’s not hard to provide, I just did it: logic symbols – table.

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 3rd 2018
    • (edited Jul 3rd 2018)

    Of course, I share your wish that people have access to basic information, and it’s great to give whatever we can. Better to have that table than not have that table.

    My point was rather to think about how difficult it would be to find one’s way about the nLab if you wanted to understand the meaning of A≠B⇔∃x((x∈A∧x∉B)∨(x∈B∧x∉A) and were surprised to find set theoretic and logical notation combined. Perhaps Chip could tell us whether the logic symbols – table suffices. I’ve just added in ∉.

    Actually, things are even trickier. Looking back, Chip (#1) was struggling with the page inhabited set. Imagine you were present watching a student who had somehow navigated to this page. Wouldn’t you want to warn them that there are subtle issues regarding constructive thinking going on there that could be left to another day?

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeJul 3rd 2018

    I just talked to a lot of students at Nesin Math Village in Turkey. There, Wikipedia is blocked, and I learned that the nLab is a common source of information.

    Wow. That’s just… wow. Maybe I’ll mention that in the Broader Impacts section of my grant proposal. (What level were these students? College? High school? What pages were they looking at?)

    I agree that it’s great to add basic information if we are able to spend the time. I think David R’s point is that here at the nLab/nForum there are a very limited number of people, and our primary focus is not on helping people with very basic questions, whereas MSE (for instance) has a huge number of people and is explicitly for the purpose of helping people with even very basic questions. So someone with very basic questions may be better served by asking them at MSE rather than here.

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeJul 3rd 2018

    Did they mention whether stackexchange is also blocked at Nesin? What about the arxiv?

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJul 3rd 2018
    • (edited Jul 3rd 2018)

    Hi Mike,

    the participants in our workshop were master students and PhD students. They told me that they use the nnLab regularly, but find it often frustrating that the entries are too advanced. I think if we could, slowly but surely, have more textbook-style basic material, it could eventually have a big impact.

    It is only Wikipedia that is blocked in Turkey. Sites such as MathOverflow and the arXiv (and the nnLab) work just fine. I heard that the situation in China is similar, and presumably for similar reasons: There is no politics on the nnLab.

    The main business of Nesin Math Village is to prepare highschool students for their final exam. At any time, there is a swarm of them around. My understanding is that tuition fees of these students is what keeps the site running. But cross-financed by that, there is advanced and research activity for a select few. It feels a bit like a combination of Oberwolfach and a mediterranean vacation resort. The scenery is dramatic, the housing is rustic (for advisors, that is, the students sleep in tents), all buildings are completely in natural stone. The bread is baked on-site in a wood fire oven.

    It felt at times too good to be true that such a place can exist. I have to admit that I didn’t know about it before I went there. But I learned that in Turkey the Math Village is famous and its founder, Ali Nesin, is a national celebrity, as was his father. There are regularly tourists passing through the Village. It happened that I was looking up from my computer while in the commons area, only to see that some family was taking pictures of us working.

    When wondering about this, it was highlighted to me that it is not for nothing that Turkey is one of the few (the only?) country on Earth that has, with Arf, a contemporary mathematician portrayed on their banknotes. There used to be Gauss and his bell curve portrayed on the 5 Deutsche Mark note before the Euro was introduced, but that seems no match to having the expression for the Arf-invariant on the 10 Lira bill!

    My lecturer colleague Murad Özaydin, who did his B.S. with Arf, joked, after paying for dinner: "I just gave them a piece of paper with the picture of my advisor, and they were happy."

    • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeJul 3rd 2018

    Thanks for the information! I definitely support the gradual inclusion of more textbook-style basic material.

    • CommentRowNumber13.
    • CommentAuthorTim_Porter
    • CommentTimeJul 4th 2018

    How ’basic’?

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJul 4th 2018

    How ’basic’?

    Usual math style. Introduce the concepts before using them.

    • CommentRowNumber15.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 4th 2018

    I’m still thinking about someone quite early in their education reaching HomePage and the next steps they might take. We don’t really say in the ’Purpose’ section that some pages are expositions of introductory material. Since we have a collection of more introductory articles, how do people feel about a link from there to a page which lists designated introductory material?

    It may not always be possible to separate out the introductory material, but we could point to pages which are pointedly such. In particular these are Urs’s pages of his courses

    and other pages

    Logic isn’t so well served, contrast homotopy type theory FAQ with string theory FAQ. logic and type theory could be included. We could do with someone teaching an nLab-based course on these areas.

    • CommentRowNumber16.
    • CommentAuthorMike Shulman
    • CommentTimeJul 6th 2018

    I agree our pages on logic could use some love.

    • CommentRowNumber17.
    • CommentAuthorTim_Porter
    • CommentTimeJul 6th 2018

    I certainly think that idea if a list of introductory material would be useful. I did start some pages on modal logic, but (i) they probably were not introductory enough, and (ii) the importance of the general area has exploded so that the introductory aim of those pages has been superceded to some extent.

    • CommentRowNumber18.
    • CommentAuthorChip Cooper
    • CommentTimeJul 11th 2018
    • (edited Jul 11th 2018)
    I'm still reading through all these comments; however, starting with Urs in #4, #5 and multiple others there is a reference which in red says [Math Processing Error]. I presume this is due to some formula which isn't converting for my viewing; but, otherwise is unknown. I can look at it in 'Original MathML" in which I see it's a link. Is this the method for passing links? Or is there something else here I should be seeing?

    Since I'm too much aware that passing an unknown symbol or word leads to misunderstanding of a sentence, and a misunderstood sentence to a paragraph, ad nauseam; I'll come back to read more.. (wife's on vacation) and make a response... in the meantime. When possible, I'll be digging and reading. Thanks to everyone for what's here so far...

    I would be interested in seeing the "contrasted homotopy type theory FAQ with string theory FAQ." I know that's a primary focus for me here based on my entry point; though I might get just as deeply or circularly lost as I did just reading 'homotopy type theory FAQ'.

    #17 Tim_Porter, not following per-se all you said, but I would look at this 'introductory material'. I, having discovered by having a fairly large vocabulary, find that American-English if not humans and all language in general tend to use related words with related meanings when coining words for new concepts in (what are or once were) new subjects (dictionaries do not always include those sub meanings); yet one can observe there often an undeniable base conceptual context which applies to the newly (are or once were) coined terms which is sufficient to grasp a recognizable gestalt of the subject in question, unless of course the newly encountered word is not coined, and is 100% original or without recognizable (by the student) root origins.

    Musing: Like a personal concept I explore and observe where it seems at times that nature is capable of behaviors(?) for which mathematics falls short? This concept for which I also have no words, as when a line approaches a point asymptotically from either (or both) sides of the line and tends toward +infinity on one side, and -infinity on the other. I have concepts where I find nature doesn't have an issue with that. Falsely similar to Zeno's paradox, to give an extremely poor and invalid example. But, more closely resembling something in major (and minor?) zero's and de-constructive (sometimes misnamed destructive) interference.
    • CommentRowNumber19.
    • CommentAuthorMike Shulman
    • CommentTimeJul 11th 2018

    Chip, what browser and OS are you using?

    • CommentRowNumber20.
    • CommentAuthorChip Cooper
    • CommentTimeJul 11th 2018
    Mike, I'm using Chrome and Windows 10 Professional; and for whatever reason, those links are now showing properly :-) Thank you
    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeJul 11th 2018

    I might get just as deeply or circularly lost as I did just reading ’homotopy type theory FAQ’.

    That page is left in skeletal state and not meant for public consumption. Maybe it should be deleted.

    I had started writing this, ever optimistically. But I gave up after we got bogged down in discussions of some basic point right away.

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeJul 11th 2018
    • (edited Jul 11th 2018)

    Chip, regarding “musings” as in the last part of your #18 and similar musings in previous messages of yours: This is not the kind of exchange suitable for this forum. You should stick/get back to actual maths questions, if you have any.

    You had initially asked for the meaning of various logical symbols, and we had given you a first pointer here. I doubt that the entries linked to there are expository enough to actually serve as decent explanation of these symbols. So you should ask a followup question and thus make somebody improve the respective entries!

    • CommentRowNumber23.
    • CommentAuthorChip Cooper
    • CommentTimeJul 11th 2018
    Re: Mike #8 &

    "the logic symbols – table": Yes, something like that would be great. However, even as I was digging into the references for the table, I ran into other symbols which are not in that table... but, I reiterate this lack of symbolic references is not exclusive to nLab. In fact one of the references to Wikipedia for basic information also runs into some of the same problems with symbols. see DavidRoberts #4.

    Mike said, "Actually, things are even trickier. Looking back, Chip (#1) was struggling with the page inhabited set. Imagine you were present watching a student who had somehow navigated to this page. Wouldn’t you want to warn them that there are subtle issues regarding constructive thinking going on there that could be left to another day?"

    Me: I concur and thanks for the link, I lost that page. Going back to it, I see the subtleties you mention. I do want it to be kept in mind, so far, most of the concepts I'm encountering aren't completely foreign (so for , the difficulty is in recognizing the concept in written form, for example seeing mention of the intuitive sense in concepts of negation, and then understanding the subtleties as in De Morgan duality; many of which we encounter in life (an innate,if not incorrect understanding) if we've lived long enough, others an intuitive relationship or dual intuitionistic understanding as seen in the ⇒ (conditional) and its negation ∖ (without?) where a link and definition are elusive and not included except by 'intuitively we understand... subtleties of exceptions to negations and a dual nature. Which (I digress, reminds me of a koan "Dwell not within the inner void, seek not the outer entanglements, be content with the oneness of things and duality vanishes by itself." where on can only indirectly, and for some vaguely, point to a concept that is both intuitionistic and dual-intuitionistic paraconsistent logic but only after it's experienced. :-)

    David #6 says "It’s makes for quite an exercise to see what it would take to understand Chip’s A≠B⇔∃x((x∈A∧x∉B)∨(x∈B∧x∉A) solely through nLab pages. He needs at least to know that ... But how many beginners will see that the ’set’ on which ≺ is a relation is a set of sets for ∈?
    Me: Indeed, impressive as a concept on first blush; but I've yet to dive into that which you say; but I will keep this in mind when I review the material. :-) At least I now know ≺, as a symbol exists and as a set of sets for ∈ and I'm beginning to realize how succinctly those sets for ∈ are defined. In many ways I see that this site exists as the foundation of proofs leading to my entry point. StringMath2017' and in the timely discussion of the ontology or epistemology of wave functions fundamental to strings and branes. Axioms are easily understood foundations... it goes without saying.. it's true... it true. Thanks for the help... I continue to read... but for now... a break.
    • CommentRowNumber24.
    • CommentAuthorChip Cooper
    • CommentTimeJul 11th 2018
    • (edited Jul 11th 2018)
    For what it's worth, as long as words specific to nLab are defined, as they seem to be and/or correlate to well known definitions, and ultimately I find definitions or at least how to say the symbols, I'm not having any issues understanding the material here. It's only, or so appears, that when I run across unknown symbols I hit a wall. My keystone to understanding always takes me back to my entry point, the reason I came here. I am making progress; and for that I thank everyone for their input.

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