Loose Ends

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34 Responses to “Loose Ends”

  1. dcorfield Says:

    Let me do so first. Over here, Urs noted

    …there is not much of an exchange of arguments between the “criticism” that is being voiced and the replies that are being given.

    As I wrote in my book,

    …mathematicians, unlike philosophers, have no particular inclination to engage in sustained argumentative activity. I am inclined to believe that this is due to a deficiency in mathematical training, rather than because it is unnecessary. (p. 214)

    Come on, people. Prove me wrong. One of my motivations for starting the Café was to provide an arena for such discussions. And I’m only encouraging you to become more civilized:

    Being civilized means living so far as possible dialectically, that is, in constant endeavour to convert every occasion of non-agreement into an occasion of agreement. A degree of force is inevitable in human life, but being civilized means cutting it down, and becoming more civilized means cutting down still further. (Collingwood, The New Leviathan, 39.15)

    Perhaps the intellectual sphere is not quite what Collingwood had in mind, but when a group believes that another group has taken over an intellectual field for reasons they don’t agree with, it is experienced as being done by force.

  2. Mike Shulman Says:

    How does agreeing to disagree fit into Collingwood’s description of civilization? My feeling is that mathematicians may disagree on particular points, but nevertheless agree that everyone’s time is more productively employed in doing mathematics than in arguing, especially when the arguments seem unlikely to produce any more agreement. More generally, I don’t think it follows from “civilization means cutting down on force” that civilization necessarily means trying to make all disagreements into agreements; aren’t some disagreements likely to prove impossible to rectify? Civilization just means not fighting even when you disagree.

    • bane Says:

      This is probably a naive comment, but I’ve always thought that argument (in the sense of trying to convince someone about something by semi-rigorous logical and “empirically suggestive” points) is pretty much inversely proportional to the ability to do “definitive” experiments (or in mathematics constructs concrete theories and calculations). No one argues much about the validity and domain of applicability of Newtonian mathematical physics because the people who care can do experiments to resolve the issue, whereas the areas of physics where people are primarily arguing are the ones where there’s little experimental accessibility. So it’s surely where there’s no direct “experiment” one can do where one should see arguments (in the above sense). And so, take for example Arnold Neumaier’s posts on FMathML: my (possibly incomplete) understanding, was that everything can be acheived with the “representation” of either side, the debate is about which one has “nicer” properties which are somewhat personal and difficult to define. This is precisely where one would expect to see “argument”.

    • dcorfield Says:

      Mike, yes nobody’s going to believe that all disagreements are resolvable to the satisfaction of all parties. And one shouldn’t underestimate the time and effort required to re-enact the other side’s thought sufficiently well so as to bring two bodies of thinking together to be in a position to be able to understand where the fundamental disagreement lies. It often feels that that time would be better spent developing your own point of view with the hope that ‘truth will out’.

      So it’s a question of a balance with dangers from weighing down either pan. My thought is that the ‘getting to the heart of the difference’ pan is more often neglected than the ‘blast ahead with my own ideas’ pan. It is easy to pick up on discontent from those who feel their own position has been paid insufficient attention. Of course, it may be right to overlook theor work, or they may be partially to blame for failing to explain it, but I dare say we can dig up examples of unfairly neglected work.

      Bane, I agree some of the considerations as to which approach is faring better are subtle and often inconclusive. You can see why the ability to prove longstanding problems is taken to be an important indication, even if it can be argued that the importance of a particular problem has been overrated. But still some people tend to steer mathematics down eventually more fruitful paths than others do.

    • Mike Shulman Says:

      Ok. I agree that in this particular case, there is more “getting to the heart of the difference” that ought to be done.

  3. ursschreiber Says:

    I’d be happy to agree to disagree, but was still trying to find out what the disagreement actually is.

    I was hoping we could have a productive discussion on the disagreement that moves us all forward. But if my attempts at this are regarded as an attempt to pick a fight I’ll better stop.

  4. Thomas Says:

    I would of course be curious about the “platonism in mathematics” theme. E.g. which implications “platonism” would have for teaching mathematics, If “platonism” would be more like a personality (or even anthropological) trait than an ontological issue, which implications would that have?

  5. Nick Gurski Says:

    I tried posting this yesterday, but obviously failed.

    No one on the skeptical side of this discussion seems particularly interested in talking details – probably for the obvious reason that these objections pretty clearly point to dissatisfaction with things that a lot of people here are interested in – so I thought I would throw out some thoughts as a response to Urs. I will preface this by saying that I have nothing against simplicial or homotopical methods in category theory as should be obvious from the research I do, but I am still a category theorist first and I think that means that deep down I have certain preferences. I also want to say that nothing here means I think the work that other people have done is bad or in some way wrong, but that does not stop me from saying that some results are not the ones that I would want to be the final story.

    I want to point out three ways in which I think people could be skeptical of the current trend of doing everything quasicategorically: a philosophical point on things vs models of things, the question of how algebraic something is, and the question of how algebraic the totality of some objects is.

    First, people should realize that saying `an (\infty,1)-category is a simplicial set such that something’ is quite different from saying `there is a model for (\infty,1)-categories given by simplicial sets such that something’ even if the theorems all come out the same. In both cases there is a standard for knowing what it means to be talking about (\infty,1)-categories, but they are very different standards. I think a major source of discomfort with how these issues get discussed is that the first of these statements seems to imply something like `the right definition of (\infty,1)-category is the one that proves the most theorems in the shortest amount of time’ while in fact most people that have been thinking about higher categories for a while probably already have something in mind, and that something is almost certainly not a simplicial definition. The key is that no one is arguing about whether some theorem is true, but instead that some use of language goes against our basic notions about what higher categories are. This is nit-picky and might even seem silly, but is something that I think is lurking underneath the discussion.

    My second point is to respond directly to Urs on the topic of how algebraic these things are. Personally, I would say not very. In particular, I would say that the algebraic Kan complexes of Thomas Nikolaus, while definitely being interesting, do not really feel that algebraic to me. I think the issue there is that for me, algebraic structures have two parts: operations and axioms. On the other hand, the reason that using quasicategories is so successful is that the axioms are pushed off to infinity, and I don’t think that changes by adding specified fillers. I liked Tom’s spectrum of how algebraic things are because it points out something really important – there are different grades of being algebraic, and there are probably a lot of people that will not be happy to call something algebraic until it is about 99.99% algebra. There is not anything wrong with that, it gives people that like both sides of the story something pretty hard to work towards, namely taking non-algebraic structures and making them really, extremely algebraic.

    Third, I would like to make very explicit something that I consider important and that John almost touched upon at the beginning of this discussion – if you take some algebraic objects and then declare them to be the same by reference to some external notion of sameness, then you are quite possibly doing something non-algebraic. For instance, we can declare two categories to be `the same’ if they have homotopy equivalent nerves. This is an extremely useful thing to do for certain purposes, but it is not algebra. That is why I think the Homotopy Hypothesis is a really serious thing: to prove it, you should really show that something completely algebraic is somehow the same as something completely non-algebraic.

  6. Stephen Harris Says:

    This post was prompted by a comment John Baez made in regard to Ulf in the Platonism thread: “Ulf needs a more intelligent spell-checker. He speaks of “rightest political views”, and the “a priory”.” What I mind more than the OED British confusing me on how to spell s_etpic
    are blatant logical fallacies like equivocation.

    http://www.fortunecity.com/emachines/e11/86/mathex4.html

    The Mathematical Experience by Philip J Davis & Reuben Hersh
    “Attempts have been made to analyze mathematical aesthetics into components- alternation of tension and relief, realization of expectations, surprise upon perception of unexpected relationships and unities, sensuous visual pleasure, pleasure at the juxtaposition of the simple and the complex, of freedom and constraint, and, of course, into the elements familiar from the arts, harmony, balance, contrast, etc.
    Further attempts have been made to locate the source of these feelings at a deeper level, in psychophysiology or in the mystical collective unconscious of Jung. While most practitioners feel strongly about the importance of aesthetics and would augment this list with their own aesthetic categories, they would tend to be skeptical about deeper explanations.”

    Let Platonism Live! by Ulf Persson
    “The alternative provided by Hersch would be to posit some kind of Jungian collective unconscious as the basis for mathematics(7). We are now by making those claims stepping from the mere metaphorical to the metaphysical.”
    (7) The choice of the affiliation Jungian is of course provocative, yet when it comes to language, the notion seems quite apposite.

    SH: Ulf reveals himself as master of the awkward sentence construction and attacks on the man of straw. Hersch is skeptically reporting other people’s “deeper” efforts, not endorsing the Jungian collective unconscious (JCU) or offering it as an alternative as misstated by Ulf Persson. …yet of course when it comes to language, the notion [(JCU)] seens quite apposite?!
    I suppose if English isn’t your “native tongue” that entitles one to conclude howsomever.

    David’s article was well-written and as always, introduces some new subject that I haven’t yet discovered/invented. There was one paragraph which I thought had more than one interpretation. I notice that there isn’t a Preview button, which makescopy and paste readable.

    Staying on the theme of games, the mathematician Alexandre Borovik
    once told me he thinks of mathematics as a Massively-Multiplayer Online Role-Playing Game. If so, it would show up very clearly the difference between internal and external viewpoints. …
    But someone observing them from the outside wants to shout: “You’re
    not dealing with anything real. You’ve just got a silly virtual reality helmet on.” External nominalists say the same thing, if more politely, to
    mathematical practitioners. But in an important way the analogy breaks down. Even if the players interact with the game to change its functioning in unforeseen ways, there were the original programmers who set the bounds for what is possible by the choices they made. When they release the next version of the game they will have made changes to allow new things to happen. In the case of mathematics, it’s the players themselves who make these choices. There’s no further layer outside.

    I think Borovik’s metaphor would more plausibly be applied to online virtual communities such as n-Category Cafe in which the rules for the roles are provided by life, partly by our human culture, and more constantly by the physical rules of the universe which evolved us so that we share many of the same perspectives/perceptions of reality.

    • Thomas Says:

      How does Sasha Borovik’s metapher explain that the “role players” play all the same game? Conc. “aesthetics”, whatever that shall rest upon, I find it interesting that if one follows just one’s uninfluenced curiosity and such “aesthetic” impressions, one quickly ends up precisely at what mathematicians estimate as hot topics. Were mathematics just a social construct, natural intuition would not be such a usefull guide for orientation. I doubt that the distinction internal/external “realism” fits to that. An other question is why ever should people be interested to read e.g. “messy” number theory, if not by some “platonoid” mindset.

      • Stephen Harris Says:

        Thomas wrote: “How does Sasha Borovik’s metaphor explain that the “role players” play all the same game?

        SH: I thought the weakness that David pointed out with Boravik’s metaphor, that the programmer’s could put out a new version of the game and change the rules was a valid point, but at the same time Boravik wasn’t far wrong. As you may know there is the Social Constructivist approach to explaining Mathematics without Platonic or as I prefer mathematical realism. For instance the history of this proof,

        “Yuri Matiyasevich utilized an ingenious trick involving Fibonacci numbers in order to show that solutions to Diophantine equations may grow exponentially. Earlier work by Julia Robinson, Martin Davis and Hilary Putnam had shown that this suffices to show that every recursively enumerable set is Diophantine. … Hilbert’s tenth problem asks for a general algorithm deciding the solvability of Diophantine equations. The conjunction of Matiyasevich’s theorem with a result discovered in the 1930s implies that a solution to Hilbert’s tenth problem is impossible.”

        SH: Many people will see this as a decade spanning effort of cultural cooperation among mathematicians. The interactions that go on within a culture can be conceptualized as a game with rules. They will think that the rules used by these various mathematicians were provided by education, textbooks which provided a common foundation for them.
        It is not necessary to assume an extra explanatory step such as all of these mathematicians were in touch with inspirations provided from some eternal abstract realm. I haven’t heard of any method which distinguishes invented mathematical results having a rather natural explanation, such as the conceptual growth of counting, from mathematics which is discovered through contact from a Platonic realm. The simplest explanation which still accounts for all the results is preferred (Okham).
        Another set of rules if generated by natural selection. Our brains and the organization of our minds is shaped by common experiences for all humans which selects for genes which map how our brains are constructed. We share a common heritage. Thus healthy humans mostly have the same set of instincts and similar native survival strategies. A new baby is born with the ability to learn from its environment, it doesn’t have to be taught to learn about reality. Reality itself has a structure which the human mind can model and approximate because that mind has been shaped by that reality. The baby learns spatial relationships which have a mathematical description; that doesn’t mean that reality itself is mathematical any more than reality owns any other anthropomorphic properties, such as a sense of justice. Those are abstractions which have been created by the human mind and projected onto reality. Wigner’s “The Unreasonable Effectiveness of Mathematics in the Natural Sciences” doesn’t require more than a natural explanation, so one isn’t assumed. The Platonic realm is an abstraction which is defined not to exist within physical reality, so it is a non-natural explanation.
        I don’t know much about aesthetics or where its intuitions lead one. Hersh was quoted by me to clarify what he wrote about Jung and his remarks which included aesthetics were incidental to that purpose. Our aesthetics in general, how we can often share and agree upon an appreciation of a beautiful painting or a particularly gifted translation of The Iliad, are due to shared values. We are a social species, belonging more to the herd mentality than the using predator behavior. However, a rule generalizing such choices or predicting membership is going to establish that the exceptions prove the rule. I think in the 1920’s or so, the male ideal of feminine pulchritude was obesity and compare it to the pressure now on women to exercise and be slender. I’ve visited your website and I think you have a more mystical outlook on reality, whereas I think nearly anything which appears mystical has a physical explanation or else is a product of wishful thinking (an addiction).

        • Stephen Harris Says:

          My example of it ain’t over till the fat lady sings was for Italian men.

        • Stephen Harris Says:

          Richard Feynman expressed his yearning for a more effective mathematics:

          “The next great awakening of human intellect may well produce a method of understanding the qualitative content of equations. Today we cannot…Today, we cannot see whether Schrödinger’s equation contains frogs, musical composers, or morality – or whether it does not. We cannot say whether something beyond it like God is needed, or not. And so we can all hold strong opinions either way.”

        • Thomas Says:

          “I think you have a more mystical outlook on reality” How that? I don’t and the other contributers don’t too. If you point to the parts giving such an impression, I could disambiguify them.

          • Stephen Harris Says:

            I’m going to start a new comment (at bottom) because we are running
            out of space.

      • Stephen Harris Says:

        S Y M M E T RY A Journey into the Patterns of Nature
        by Marcus du Sautoy, quoted at the beginning of the first chapter:

        “The universe is built on a plan the profound symmetry of which is somehow present in the inner structure of our intellect.” Paul Valery

  7. ursschreiber Says:

    Nick Gurski wrote:

    would throw out some thoughts as a response to Urs.

    Thanks, Nick! Very nice, I appreciate it a lot.

    I’ll reply briefly point-by-point:

    Re “quasicategories are just one model”.

    Yes, certainly! In fact it looks like it is not even going to be the preferred model among those who you might think prefer it. Most recent development revolve around the model given by complete Segal space. This is a pretty immediate model of the idea “category weakly enriched in oo-groupoids”.

    But quasicategories are a model that accomplish one important aspect: the proof of existence of (oo,1)-category theory. from the results of Joyal and Lurie it seems clear that whatever other definition of (oo,1)-category you come up with, it should have a simplicial nerve that identifies it with a quasicategory.

    I think we should take category theory seriously: the way we think of our objects (here: (oo,1)-categories) as being presented is irrelevant . What matters is the interrelation between all possible such objects: the category they form.

    Would you agree with that?

    Apart from this I can’t quite see what is so ugly about Kan complexes and quasi-categories. If you just hold two pages next to each other, one carrying the definition of quasi-category, the other carrying the definition of a more algebraic, say operadic, definition, the second one will look way more involved. For one, it will likely not fit on that one page. So I am not sure I see the intuition for why that is somehow intrinsicaly better. But also, by the above remark, I don’t care too much, what matters is not the presentation of the objects, but their interrelation.

    Re: algebraicity of Nikolaus’ algebraic quasi-catgeories:

    Maybe it serves to unwind a bit what it actually is that Thomas has there: I think it is really exactly the typical description of bicategory, tricategory, tetracategory, etc, fully systematized for the (n,1)-case.

    Here is what I mean: what is called a “choice of fillers” is really: “an algebraic specification of composition with its coherences”.

    Let’s look at it starting in low dimensions: First for each pair of composable morphisms, you are to choose a 2-morphism with these as source. The target of that 2-morphism is the chosen composite. The collection of all these chosen 2-morphisms is the composition operation.

    Then for every triple of composable morphisms, you are to choose a 3-morphism going between the corresponding four composition 2-morphisms. The choice of that 3-morphism is a choice of associator.

    Nect, for sequences of five composable morphisms, you are to chose a 4-morphism between the corresponding five associators. That’s the corresponding pentagonator.

    And so on. If you decide to look at an object where this is truncated at some point, this means in the next step you assume to be able to choose these fillers to be identity n-morphisms. The fact that these exist is then the coherence law for all the structure that you have chosen before.

    So one “algebraic quasi-category” in Thomas Nikolaus’s sense is really essentially exactly a choice of weak (n,1)-category in the style of bicategory, tricategory, etc.

    Do you see what I mean?

    Re point 3:

    I am maybe still not sure if I see what these arguments (those you give and those you refer to) are supposed to say. If we are talking about pure catgeory theoretic higher category theory, we shouldn’t care much about topological spaces. I don’t care much about them.

    What I care about is that Kan complexes and quasi-categories are sensible in themselves have been proven to be models for (oo,0)-category theory and (oo,1)-category theory in every conceivable sense of what this should mean. So for me the important point here seems to be that whatever other model for (n,1)-category that one comes up with, it will be very helpful to check if it matches the behaviour of quasi-catgeories under its notion of simplicial nerve. For if it does, it immediately proves that good (oo,1)-category theory will also exist for your model. If it doess not, then, conversely, this is a bit worrisome, since it is unlikely that there will be two inequivalent good “(oo,1)-category theories”.

    Concenring algebraic definition of equivalence. if you look at what equivakence of Nikolaus-quasi-categories means, you will see that it means precisely the kind of equivalence that we are used to from bicategories, tricategories, etc.: you map all the k-moprhism over such that all composition is respected up to corresponding images of corresponding compositors.

    Or so I think. Let me know what you think.

  8. Mike Shulman Says:

    I’ll respond more later, but briefly to one of Nick’s points, it seems to me that the idea of “pushing axioms off to infinity” is not special to quasicategories, but is a general feature of all kinds of weak omega-categories. Why are quasicategories special?

  9. Tim Porter Says:

    When I started to work with Cordier on the Homotopy Coherent Nerve (and hence on quasi-categories) and to think of Kan complexes as one of a possible set of models for weak infinity groupoids, I was concerned by exactly the sort of points that Nick is making. Ronnie Brown and his students had been introducing T-complexes and it was clear that giving a specification of a filler for each horn, satisfying certain axioms (just three which was marvellous), was completely algebraic. It was clear also that the very pretty T-complex structure on simplicial groups (satisfying one condition on the Moore complex) was encoding a lot of structure in a very compressed form. Laxifying the algebra, relaxing the conditions hence allowing a set of fillers but only in certain dimensions, and adjusting the axioms accordingly, is a very fruitful route to follow. Note that in a simplicial group or groupoid the filling algorithm (given in May for instance) is explicit and gives very nicely structured fillers. (Some work by Ali Mutlu and myself may be useful if you want to see what I mean here. I think this should be considered algebraic, but there is a combinatorial side that adds in something as well. )

    If you have a Kan complex with extra axioms and or structure in this way you can algebraicise it by passing to the Wbar of the loop groupoid, and can see to what extent this gives you new filler information. (For instance if $G$ is a simplicial group satisfying the Moore complex condition mentioned above then Wbar(G) is a T-complex in a natural way and you can manipulate things explicitly. (The working can be tedious and is fraught with problems of a human nature … slips are extremely easy to make and difficult to find… but it can be done.) SImplicial groups also have explicit formulae for Whitehead products and other operations, and these can be interpreted both algeraically and geometrically.

    If one relaxes the T-complex axioms you can still get somewhere algebraically. For instance asking for unique T fillers above a certain dimension but not worrying below that. (This needs careful handling and is easiest to see in the group T-complex case with the Moore complex condition.)

    Finally returning to Kan complexes, and a side comment, it always seemed to me that they could be very useful for modelling things outside our normal area. A Kan complex has fillers,…. but are they all equally probable!!!!! Suppose one laxifies things from T-complex by considering a probability distribution on each set of fillers, ‘the composite of two composable arrows is probably this one ….., with probability whatever.’ There do seem to be situations (e.g. in Tom’s area of interest), where this may be worth looking into.

  10. Urs Schreiber Says:

    >introducing T-complexes and it was clear that giving a specification of a filler for each horn, satisfying certain axioms (just three which was marvellous)

    Can you say why we need any extra axioms at all? Just picking fillers seems to do the job.

  11. Tim Porter Says:

    The axioms gave the algebraic nature of things. For instance the third T-complex axiom implies associativity of the composition given by the fillers. This gave strict structures.

    I am just saying that the `strict’ algebraic case corresponded to those three axioms. If you just have fillers then it is a bit like in universal algebra, fine for theory but without the equations governing the objects the calculation is more or less empty of meaning. It gives the operations only.
    If you take a free resolution of a group, it presents the group exactly BUT with a lot of redundancy.
    Pragmatically the free fillers added are going to give objects that will need a new set of tools for developing their calculation.

  12. Stephen Harris Says:

    SH wrote: “I think you have a more mystical outlook on reality”

    Thomas challenged: “How that? I don’t and the other contributors don’t too. If you point to the parts giving such an impression, I could disambiguify them.”

    SH: I told you that I read your website. The whole theme is mystical. Mabye you have a different idea of what mystical means. You wrote,

    “Like the extases caused by visiting spirits in shamans, the early philosopher’s job was not a voluntarily chosen one, but a vocation. As the separation of the personal ego from the egofree part of the mind connected with “platonism”, this needs not to be an enjoyable experience.”

    SH: Shamans communicating with spirit guides, that is called a mystical experience. Cultivating communion with the Higher/True Self is a mystical path. It’s considered mystical because it is supernatural. The scientific method applies to the physical universe, which is natural. The Platonic Realm is not described as existing within the physical universe, so it doesn’t fall under the auspices of the natural physical universe.

    I used the word mystical because “New Age” is too modern and you spoke of ancient roots, and mystical is an older term. As for the other contributors, none that I know of qualify for the word mystical, not even the Buddhist members. And if they looked at your website they would concur that the viewpoint expressed is mystical even if they agreed that what you wrote fairly truly depicted reality. There is no physical evidence for anything spooky: ghosts, guides, higher realms or planes of existence that Magickal orders presume and so on. You probably don’t know that, haven’t decided that you must have proof of the esoteric.

    I surely didn’t say that you were not very bright. Very bright people tend to be the most gullible. I know about this stuff because I spent twenty years believing, and looking for confirming evidence. Also, you are well-read, but that said, Gromov would not impress you as much as he does now, if you had come across his writing after you had spent thirty more polymath years yourself. Also I have the highest regard for Yuri Manin and Roger Penrose who I presume to both be Platonists. You left this part out when you quoted Manin:

    Manin: “Here I take a position that sets me apart from many good colleagues. I’ve heard many arguments against me on this subject. I must explain to you how I imagine mathematics. I am an emotional
    Platonist (not a rational one: there are no rational arguments in favor of Platonism). Somehow or other, for me mathematical research is a discovery, not an invention.”

    Penrose tried to use Godelian Incompleteness to attack strong AI. He felt that the mathematician could obtain insights that no computer would be capable of. It seemed like he linked his argument to the brain collapsing quantum information, with macrotubules and OR. Penrose realized he needed to demonstrate a physical mechanism for the mathematician to have insights that a computer AI could not. Most people judge that he failed like Lucas before him.

    So when I said I think you have a more mystical outlook on reality, I meant that it was more mystical than mine. Would you have preferred that I wrote that I have a more rational outlook on reality because Manin has said “there are no rational arguments in favor of Platonism.” and you were attempting a rational elucidation of Platonism on your website? I was trying to be polite; I used *mystical* which means “A belief in the existence of realities beyond perceptual or intellectual apprehension that are central to being and directly accessible by subjective experience.” because your website describes this practice.

  13. Thomas Says:

    Conc. “you were attempting a rational elucidation of Platonism”: You missed the statements in the introduction, that I write about the history-of-ideas background of “platonism” and that, independent from that, I regard “platonism” as a special mindset, what could led to other questions. I don’t see any sense in ontological or other such discussions connected with a mentality, and find it quite interesting that history-of-ideas tells pretty much the same. Perhaps some themes usually associated with “platonism” are in fact only characteristica of degenerated variants? Conc.: “Gromov would not impress”: Even the tiny fraction of his ideas I know (e.g. pseudoholomorphic curves) appear to me very astonishing and forcefull.

    • Stephen Harris Says:

      Here is an example of where your belief in the possibility of Platonism steps out from behind the largely historical scenes.

      Thomas website:
      “That may sound rather detached from immediate “reality” (whatever that is) and about something like a dodecahedral cloud floating in the air. However, the idea of a screwdriver illustrates the meaning of a “Platonic idea” probably much better. Even a superficial browsing of contemporary mathematics shows that “platonic” programs and concepts are among the leading forces for actual research. E.g. there was hardly ever a more “platonic” insight than Grothendieck’s quest for the “mysterious functor” relating different ways to do geometry (=”cohomology theories”) in number theory, which became one of the most usefull tools in the applied business of cryptography.”

      Compare what you wrote with what Yuri Manin, an avowed Platonist, wrote where he does communicate a belief in an abstract Platonic realm = “Platonian reality”. Your term is “platonic insight”.

      mathcult.ps Yuri Manin
      “Some of the most in Research Programs of the last decades were
      expressions of insights into the complex structure of Platonian reality. A. Weil guessed the existence of cohomology theories for algebraic manifolds in finite characteristics. Grothendieck constructed them, thus forever changing our understanding of the relationships between continuous and discrete.”

      What is a “platonic insight” if not a subjective access to the alleged Platonic Realm? Thomas wrote:
      An other question is why ever should people be interested to read e.g. “messy” number theory, if not by some “platonoid” mindset.

      I think mathematicians pursue what they find interesting. They don’t need an opinion one way or another on whether mathematics is discovered or invented, in order to be stimulated by deeper questions of number theory. The Platonoid mindset has no monopoly on Pure mathematics. I like Manin’s choice of word, emerge.
      Yuri Manin: “Pure mathematics is an immense organism built entirely and exclusively of ideas that emerge in the minds of mathematicians and live within these minds.”
      Suppose you are lost in a forest. Usually by trial and error one discovers a way out, thus emerging from the forest. Another way is to have a method. I think Yuri Matiyasevich had a plan to solve Hilbert’s Tenth problem built on already known structure. I would say he invented the solution. That surely was an interesting result that didn’t require a platonoid mindset. Matiyasevich didn’t have to get up in the morning to start work on his project and wonder or worry whether his mathematical muse would be tuned in that day.

      I will admit that your support of the Platonic realm was a good deal more obvious in the thread, The Sacred and the Profane.

      Jonathan Vos Post in the thread, The Sacred and the Profane
      “Eliade’s The Sacred and the Profane are three categories: the Sacred (which is a transcendent referent such as the gods, God, or Nirvana), hierophany (which is the breakthrough of the sacred into human experience, i.e. a revelation), and homo religiosus (the being par excellence prepared to appreciate such a breakthrough).”

      Thomas in the thread The Sacred and the Profane
      “Sacred” in mathematics obviously refers to “platonic” in contrast to other parts of mathematics. The reference to Eliade fits very well, because “platonism” finally goes back to “shamanism” about which Eliade wrote.”

      Eliade defines a shaman as follows:
      “he is believed to cure, like all doctors, and to perform miracles of the fakir type, like all magicians [...] But beyond this, he is a psychopomp, and he may also be a priest, *mystic*, and poet”

      SH: You see that word mystic. People who believe in hierophany, whether in shamanic nature visions or inspiration from a Platonic realm, a supposed abstract entity existing outside of the physical universe, are considered to have a belief in mysticism = hierophany.

      I think your writing clearly indicates that you do entertain a belief in the possibility of a reality which includes Shamanism or inspiration from the Platonic realm.

      I do not believe such things are possible because they require a belief in the non-physical universe. In comparison to me, you have a more mystical (which is the accepted word to classify Shamanism/Platonism) because you believe these notion might be possible or plausible.

      SH wrote: “I think you have a more mystical outlook on reality”

      Thomas website wrote: “Mentalities can be learned by practice, not by just reading dialogues. And this was his [Plato] reason for advertizing the yoga of mathematics.

      SH: You describe this as if the aspirant is cultivating enlightenment, or opening communication between her atman and the universal Atman, and calling it learning a mentality which has a mindset.

      • Stephen Harris Says:

        I think you are right, this topic is too ambiguous to pursue profitably. Cheers

        • Thomas Says:

          It’s very simple: a history-of-ideas is not a description of anyone’s opinions, “concept A developing out of concept B” does not imply “A=B”, a mentality or mindset is not an ontological proposition. Aside that, many texts on the mentioned concepts don’t meet their real meaning (whose frequent occurance made me write the text). The sketch of the “platonic” mindset seems to fit esp. that of number theorists. An interesting suggestion I received is about Wilfried Owen’s poem “The naming of the parts” as illustration of emotions related to “naming” and the short stories of Tim OBrien “The things they carried” on that and a complex concept of “reality” similar to those of the ancient greeks. An other interesting reaction was about the power of “platonic” ideas even when they seem not to work. There I wonder if the story of p-adic Birch Swinnetron-Dyer, where the p-adification of a classical theory run into unexpected troubles, were an example.

          • Stephen Harris Says:

            http://www.jstor.org/stable/428160
            “According to Jung, poetry is the intelligible statement of submerged truths, the communication of archetypal patterns which reside within the collective unconscious. A poem is primarily an individual’s expression of these commonly-shared ideas which well up from the primordial region of the mind; and the first task of the poet, in the terms of another critic, is to light upon objective correlatives in which he may embody these formless, almost ineffable ideas. To quote a Shakespearean passage often cited by Coleridge and Wordsworth, the poet’s mind, “in a fine frenzy of” Platonic afflatus, “gives to airy nothing a local habitation and a name”; “imagination bodies forth the forms of things unknown.” The symbols used by the poet bring into our ephemeral consciousness those timeless psychic experiences shared with our ancestors. Coleridge frequently recognized the intrusion of archetypal patterns:
            Oft of some Unknown Past such fancies roll
            Swift o’er my brain, as make the Present seem,
            For a brief moment, like a most strange Dream.

            Again in his notebooks Coleridge acknowledged the search for physical images by which to identify his primordial ideas:”

  14. Normal Service is Resumed! « The n-Category Café Says:

    [...] out. (And if you just can’t wait to get that comment into cyberspace, you can put it at the loose ends entry at this [...]

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  16. Tom Ellis Says:

    I’m interested in the discussion of circuits in This Week’s Finds 296, so I hope someone is reading here whilst comments on that blog are down.

    The reason that Q = (x + y – 2z)^2 (or in fact more simply Q = (x + y)^2) is not the power form of a circuit is that the power must not increase when the potentials are brought closer together.

    Rigourously, define T(x) = min(1, max(0,x)). Then a Dirichlet form must satisfy QT <= Q, i.e. apply T to each coordinate separately.

    All Q that come from a circuit must be Dirichlet, but I don't know if the converse is true. I suspect so.

    Dirichlet forms are important in probability theory. The link between stochastic processes and circuits is, as usual, the Laplacian.

    I'm interested in this categorization of circuits, because circuits are related to interacting particle models, such as the Ising model, and percolation.

    • Tom Ellis Says:

      This is an interesting paper on the topic of circuits:

      http://www-an.acs.i.kyoto-u.ac.jp/~kigami/AOF.pdf

      In it, the author shows that Dirichlet forms that I defined above correspond exactly with “Laplacians” which are the linear maps f from potentials to currents that you wrote about in week296, John.

      The condition on Laplacians which corresponds to the Markov condition (QT <= Q) of Dirichlet forms is exactly Ben Tilly's intuition that the off-diagonal elements must be non-positive (he has the inverse sense so he writes non-negative).

      He also demonstrates that every Dirichlet form, i.e. every Laplacian, comes from an electrical network.

    • John Baez Says:

      Thanks very much for offering a solution to my problem, Tom! This sounds wonderful!

      Alan Weinstein pointed me to this article, which you may enjoy too:

      Christophe Sabot, Electrical networks, symplectic reductions, and application to the renormalization map of self-similar lattices, available as arXiv:math-ph/0304015.

      It proposes a class of quadratic forms which it uses as a way to formalize electrical circuits made of resistors. It does a lot of interesting things with quadratic forms of this type. However, I didn’t see a proof that every circuit made of resistors gives a quadratic form of this type, or conversely. I should reread it and compare it to the reference you provide.

  17. Tom Ellis Says:

    John, that paper contains a reference to this one:

    Existence and uniqueness of diffusions on finitely ramified self-similar fractals also by C. Sabot.

    http://linkinghub.elsevier.com/retrieve/pii/S001295939789934X

    In particular the latter contains (p609, i.e. p5 in the PDF):

    “With each bond conductivity matrix J [circuit] we associate a positive bilinear form A on E x E by [the energy form]

    The following result is well-known:

    PROPOSITION 1.7. – The quadratic form A is a Dirichlet form on F …
    The map J |-> A is bijective from the set of bond conductivity matrices [circuits] to the set of Dirichlet forms.”

    I rather dislike these “This result is well known” appeals to folk theorems, but oh well, at least you’ve got another reference for your result!

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