55 Matching Annotations
  1. Last 7 days
    1. for settling in a finite number of steps, whether a relevant object hasproperty P.Relatedly, the answer to a question Q is effectively decidable ifand only if there is an algorithm which gives the answer, again by adeterministic computation, in a finite number of steps.

      Missing highlight from preceding page:

      A property \( P \) is effectively decidible if and only if there is an algorithm (a finite set of instructions for a deterministic computation) ...

      Isn't this related to the idea of left & right adjoints in category theory? iirc, there was something about the "canonical construction" of something X being the best solution to a particular problem Y (which had another framing like, "Problem Y is the most difficult problem for which X is a solution")

      Different thought: the Curry-Howard-Lambek correspondance connects intuitionistic logic, typed lambda calculus, and cartesian closed categories.

  2. Nov 2022
    1. okay so remind you what is a sheath so a sheep is something that allows me to 00:05:37 translate between physical sources or physical realms of data and physical regions so these are various 00:05:49 open sets or translation between them by taking a look at restrictions overlaps 00:06:02 and then inferring

      Fixed typos in transcript:

      Just generally speaking, what can I do with this sheaf-theoretic data structure that I've got? Okay, [I'll] remind you what is a sheaf. A sheaf is something that allows me to translate between physical sources or physical realms of data [in the left diagram] and the data that are associated with those physical regions [in the right diagram]

      So these [on the left] are various open sets [an example being] simplices in a [simplicial complex which is an example of a] topological space.

      And these [on the right] are the data spaces and I'm able to make some translation between [the left and the right diagrams] by taking a look at restrictions of overlaps [a on the left] and inferring back to the union.

      So that's what a sheaf is [regarding data structures]. It's something that allows me to make an inference, an inferential machine.

    1. i think so like in social terms the conservatives would say well i like that it benefits from the wisdom of math already invented you're not 00:36:39 throwing anything away you're not you're not throwing it all away and starting over you're taking what we already have and you're you're using it that's great and a libertarian might say i really like that you're free to create as you see fit you can make anything you 00:36:52 want and you're working within this background framework that's minimally invasive it doesn't make a lot of rules for you but it is highly functional i like that it kind of keeps everyone in line while 00:37:03 like satisfying some formal contracts or something while still being uh i'm still free to create and a progressive might say i like about category that theory that everyone can contribute to 00:37:15 making their own world making it more rich adding new ideas uh making it more meaningful understanding connections between things a modern viewpoint would say i like that 00:37:26 it's completely rigorous that it's been used in proving well-known conjectures that people thought were important to prove but also that it's interesting it's useful in science and technology and a postmodern person might say i like 00:37:40 that um that no perspective is right that that there's just all sorts of different categories but that navigating between these perspectives lets you look at problems from all sides or a hippie might say i like that it's 00:37:53 all about relationship and connection or irrelevant i don't know what that means maybe a practical person might say that i like that it's that we can actually use it to organize and learn from big data in 00:38:06 today's world or to manage complexity of software projects that are that are very large and changing all the time i like that you can think about ai and other complex systems with this stuff i think it's relevant and 00:38:19 practical for right now so that's that's my uh tutorial or that's the the part i'm going to record and now i'm going to open it up for questions

      David Spivak discusses how category theory may appeal to different political ideologies for a variety of reasons.

  3. Oct 2022
    1. https://www.loom.com/share/a05f636661cb41628b9cb7061bd749ae

      Synopsis: Maggie Delano looks at some of the affordances supplied by Tana (compared to Roam Research) in terms of providing better block-based user interface for note type creation, search, and filtering.


      These sorts of tools and programmable note implementations remind me of Beatrice Webb's idea of scientific note taking or using her note cards like a database to sort and search for data to analyze it and create new results and insight.

      It would seem that many of these note taking tools like Roam and Tana are using blocks and sub blocks as a means of defining atomic notes or database-like data in a way in which sub-blocks are linked to or "filed underneath" their parent blocks. In reality it would seem that they're still using a broadly defined index card type system as used in the late 1800s/early 1900s to implement a set up that otherwise would be a traditional database in the Microsoft Excel or MySQL sort of fashion, the major difference being that the user interface is cognitively easier to understand for most people.

      These allow people to take a form of structured textual notes to which might be attached other smaller data or meta data chunks that can be easily searched, sorted, and filtered to allow for quicker or easier use.

      Ostensibly from a mathematical (or set theoretic and even topological) point of view there should be a variety of one-to-one and onto relationships (some might even extend these to "links") between these sorts of notes and database representations such that one should be able to implement their note taking system in Excel or MySQL and do all of these sorts of things.

      Cascading Idea Sheets or Cascading Idea Relationships

      One might analogize these sorts of note taking interfaces to Cascading Style Sheets (CSS). While there is the perennial question about whether or not CSS is a programming language, if we presume that it is (and it is), then we can apply the same sorts of class, id, and inheritance structures to our notes and their meta data. Thus one could have an incredibly atomic word, phrase, or even number(s) which inherits a set of semantic relationships to those ideas which it sits below. These links and relationships then more clearly define and contextualize them with respect to other similar ideas that may be situated outside of or adjacent to them. Once one has done this then there is a variety of Boolean operations which might be applied to various similar sets and classes of ideas.

      If one wanted to go an additional level of abstraction further, then one could apply the ideas of category theory to one's notes to generate new ideas and structures. This may allow using abstractions in one field of academic research to others much further afield.

      The user interface then becomes the key differentiator when bringing these ideas to the masses. Developers and designers should be endeavoring to allow the power of complex searches, sorts, and filtering while minimizing the sorts of advanced search queries that an average person would be expected to execute for themselves while also allowing some reasonable flexibility in the sorts of ways that users might (most easily for them) add data and meta data to their ideas.


      Jupyter programmable notebooks are of this sort, but do they have the same sort of hierarchical "card" type (or atomic note type) implementation?

  4. Aug 2022
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  6. Jan 2022
    1. https://www.youtube.com/watch?v=z3Tvjf0buc8

      graph thinking

      • intuitive
      • speed, agility
      • adaptability

      ; graph thinking : focuses on relationships to turn data into information and uses patterns to find meaning

      property graph data model

      • relationships (connectors with verbs which can have properties)
      • nodes (have names and can have properties)

      Examples:

      • Purchase recommendations for products in real time
      • Fraud detection

      Use for dependency analysis

  7. Sep 2021
    1. The CommunitySensor community network ontology can be positioned somewhere in the middle of this spectrum: community network representatives are totally free to come up with their own terms for element and connection types in their own ontologies. However, these terms are organized in a deep structure with community-specific element and connection types being classified by higher-order element and connection type (sub)categories described in the CommunitySensor community network conceptual model.
  8. Aug 2021
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  11. Feb 2021
    1. Though rarer in computer science, one can use category theory directly, which defines a monad as a functor with two additional natural transformations. So to begin, a structure requires a higher-order function (or "functional") named map to qualify as a functor:

      rare in computer science using category theory directly in computer science What other areas of math can be used / are rare to use directly in computer science?

    1. It's hard to say why people think so because you certainly don't need to know category theory for using them, just like you don't need it for, say, using functions.
  12. Oct 2020
    1. In some sense, by studying one model deeply enough, we can study them all.

      This may be where math like category theory is particularly powerful as a map between these different areas which are really the same (isomorphic).

  13. Apr 2020
  14. Jan 2020
  15. Oct 2019
    1. categorical formalism should provide a much needed high level language for theory of computation, flexible enough to allow abstracting away the low level implementation details when they are irrelevant, or taking them into account when they are genuinely needed. A salient feature of the approach through monoidal categories is the formal graphical language of string diagrams, which supports visual reasoning about programs and computations. In the present paper, we provide a coalgebraic characterization of monoidal computer. It turns out that the availability of interpreters and specializers, that make a monoidal category into a monoidal computer, is equivalent with the existence of a *universal state space*, that carries a weakly final state machine for any pair of input and output types. Being able to program state machines in monoidal computers allows us to represent Turing machines, to capture their execution, count their steps, as well as, e.g., the memory cells that they use. The coalgebraic view of monoidal computer thus provides a convenient diagrammatic language for studying computability and complexity.

      monoidal (category -> computer)

  16. Sep 2019
    1. from falsehood you can derive everything ** false \leq truerestrict: don't talk about elements -> you have to talk about arrows (relations) .... interview the friends *product types: [pairs, tuples, records,...]

  17. Aug 2019
    1. But there is an alternative. It’s called denotational semantics and it’s based on math. In denotational semantics every programing construct is given its mathematical interpretation. With that, if you want to prove a property of a program, you just prove a mathematical theorem
    1. hierarchy of questions: "What about the relationships between the relationships between the relationships between the...?" This leads to infinity categories. [And a possible brain freeze.] For more, see here.)  As pie-in-the-sky as this may sound, these ideas---categories, functors, and natural transformations---lead to a treasure trove of theory that shows up almost everywhere.

      Turtles all the way up

  18. Jul 2018
    1. “pulls it back”

      minor quibble, maybe this should be surrounded by parantheses

  19. Jun 2018
    1. Exercise1.75.Doesbù3chave a right adjointR:N!N? If not, why? If so, does itsright adjoint have a right adjoint?
    2. Remark1.73.IfPandQare total orders andf:P!Qand1:Q!Pare drawn witharrows bending as in Exercise 1.72, we believe thatfis left adjoint to1iff the arrows donot cross. But we have not proved this, mainly because it is difficult to state precisely,and the total order case is not particularly general
    3. The preservation of meets and joins, and hence whether a monotone map sustainsgenerative effects, is tightly related to the concept of a Galois connection, or moregenerally an adjunction.
    4. Galois connections between posets were first considered by Évariste Galois—whodidn’t call them by that name—in the context of a connection he found between “fieldextensions” and “automorphism groups”. We will not discuss this further,
    5. In his work on generative effects, Adam restricts his attention to maps that preservemeets, even while they do not preserve joins. The preservation of meets implies that themapbehaves well when restricting to a subsystem, even if it can throw up surpriseswhen joining systems
    6. n [Ada17], Adam thinks of monotone maps as observations. A monotone map:P!Qis a phenomenon ofPas observed byQ. He defines generative effects of such a mapto be its failure to preserve joins (or more generally, for categories, its failure topreserve colimits)
    7. Example1.61.Consider the two-element setPfp;q;rgwith the discrete ordering.The setAfp;qgdoes not have a join inPbecause ifxwas a join, we would needpxandqx, and there is no such elementx.Example1.62.In any posetP, we havep_pp^pp.Example1.63.In a power set, the meet of a collection of subsets is their intersection,while the join is their union. This justifies the terminology.Example1.64.In a total order, the meet of a set is its infimum, while the join of a set isits supremum.Exercise1.65.Recall the division ordering onNfrom Example 1.29: we say thatnmifndivides perfectly intom. What is the meet of two numbers in this poset? Whatabout the join?

      These are all great examples. I htink 1.65 is gcd and lcm.

    8. These notions will have correlates in category theory, called limits and colimits,which we will discuss in the Chapter 3. For now, we want to make the definition ofgreatest lower bounds and least upper bounds, called meets and joins, precise.
    9. Ifxyandyx, we writexyand sayxandyareequivalent. We call a set with a preorder aposet.
    10. Example1.49.Recall from Example 1.36 that given a setXwe defineEXto be theset of partitions onX, and that a partition may be defined using a surjective functions:XPfor some setP.Any surjective functionf:X!Yinduces a monotone mapf:EY! EX, going“backwards”. It is defined by sending a partitions:YPto the compositef:s:XP
    11. Example1.42 (Opposite poset).Given a posetπP;∫, we may define the opposite posetπP;op∫to have the same set of elements, but withpopqif and only ifqp.
    12. Example1.40 (Product poset).Given posetsπP;∫andπQ;∫, we may define a posetstructure on the product setPQby settingπp;q∫  πp0;q0∫if and only ifpp0andqq0. We call this theproduct poset. This is a basic example of a more generalconstruction known as the product of categories
    13. Contrary to the definition we’ve chosen, the term poset frequently is used to meanpartiallyordered set, rather than preordered set. In category theory terminology, therequirement thatxyimpliesxyis known asskeletality. We thus call partiallyordered setsskeletal posets
  20. Dec 2015
    1. Lemma 2.5.1.14

      Invoke universal property of products

    2. Since ducks can both swim and fly, each duck is found twice inC, once labeled as aflyer and once labeled as a swimmer. The typesAandBare kept disjoint inC, whichjustifies the name “disjoint union.”

      The disjoint union reminds me of algebraic datatypes in functional programming languages, whereas a set-theoretic union is more like a union in CS: the union has no label associated with it, so additional computation (or errors) may arise due to a lack of ready information about elements in the union.

    3. facts, which are simply “path equivalences” in an olog. It isthe notion of path equivalences that make category theory so powerful.Apathin an olog is a head-to-tail sequence of arrows
    4. Consider the aspectpan objectqhas››››—pa weightq. At some point in history, thiswould have been considered a valid function. Now we know that the same objectwould have a different weight on the moon than it has on earth. Thus as world-views change, we often need to add more information to our olog. Even the validityofpan object on earthqhas››››—pa weightqis questionable. However to build a modelwe need to choose a level of granularity and try to stay within it, or the whole modelevaporates into the nothingness of truth!
    5. An aspect of a thingxis a way of viewing it, a particular way in whichxcan be regardedor measured. For example, a woman can be regarded as a person; hence “being a person”is an aspect of a woman. A molecule has a molecular mass (say in daltons), so “havinga molecular mass” is an aspect of a molecule. In other words, byaspectwe simply meana function. The domainAof the functionf:A—Bis the thing we are measuring, andthe codomain is the set of possible “answers” or results of the measurement.

      Naïvely (since my understanding of type theory is naïve), this seems to mesh with the concepts of inheritance for the "is" relationships, and also with type-theory more generally for "has" relationships, since I believe we can view any object or "compound type", as defined here, as being a subtype of another type 'o' if one of its elements is of type 'o'. Though we have to be careful for functional mapping when thinking of aspects: we can't just say Int is an aspect of Pair(Int, Int), since this is ambiguous (there are two ints) --- we must denote which Int we mean.

    6. We represent eachtype as a box containing asingular indefinite noun phrase.
    7. Data gathering is ubiquitous in science. Giant databases are currently being minedfor unknown patterns, but in fact there are many (many) known patterns that simplyhave not been catalogued. Consider the well-known case of medical records. A patient’smedical history is often known by various individual doctor-offices but quite inadequatelyshared between them. Sharing medical records often means faxing a hand-written noteor a filled-in house-created form between offices.
    8. As mentioned above category theory has branched out into certain areas of scienceas well. Baez and Dolan have shown its value in making sense of quantum physics, itis well established in computer science, and it has found proponents in several otherfields as well. But to my mind, we are the very beginning of its venture into scientificmethodology. Category theory was invented as a bridge and it will continue to serve inthat role.
    9. All this time, however, category theory was consistently seen by much of the mathe-matical community as ridiculously abstract. But in the 21st century it has finally cometo find healthy respect within the larger community of pure mathematics. It is the lan-guage of choice for graduate-level algebra and topology courses, and in my opinion willcontinue to establish itself as the basic framework in which mathematics is done
    10. In 1980 Joachim Lambek showed that the types and programs used in computerscience form a specific kind of category. This provided a new semantics for talking aboutprograms, allowing people to investigate how programs combine and compose to createother programs, without caring about the specifics of implementation. Eugenio Moggibrought the category theoretic notion of monads into computer science to encapsulateideas that up to that point were considered outside the realm of such theory.
    11. Bill Lawvere saw category theory as a new foundation for all mathematical thought.Mathematicians had been searching for foundations in the 19th century and were reason-ably satisfied with set theory asthe foundation. But Lawvere showed that the categoryof sets is simply a category with certain nice properties, not necessarily the center ofthe mathematical universe. He explained how whole algebraic theories can be viewedas examples of a single system. He and others went on to show that higher order logicwas beautifully captured in the setting of category theory (more specifically toposes).It is here also that Grothendieck and his school worked out major results in algebraicgeometry.

      I haven't studied toposes, but I can at least see how introductory algebraic geometry, i.e. the study of Groebner bases, relates to propositional logic.

    12. The paradigm shift brought on by Einstein’s theory of relativity brought on the real-ization that there is no single perspective from which to view the world. There is nobackground framework that we need to find; there are infinitely many different frame-works and perspectives, and the real power lies in being able to translate between them.It is in this historical context that category theory got its start.
    13. These theorems have not made theirway out into the world of science, but they are directly applicable there. Hierarchies arepartial orders, symmetries are group elements, data models are categories, agent actionsare monoid actions, local-to-global principles are sheaves, self-similarity is modeled byoperads, context can be modeled by monads.
    14. No one would dispute that vector spaces are ubiquitous.But so are hierarchies, symmetries, actions of agents on objects, data models, globalbehavior emerging as the aggregate of local behavior, self-similarity, and the effect ofmethodological context.
    15. I will use a mathematical tool calledologs, or ontology logs, to givesome structure to the kinds of ideas that are often communicated in pictures like theone on the cover. Each olog inherently offers a framework in which to record data aboutthe subject. More precisely it encompasses adatabase schema, which means a system ofinterconnected tables that are initially empty but into which data can be entered.
    16. Agreementis the good stuff in science; it’s the high fives.But it is easy to think we’re in agreement, when really we’re not. Modeling ourthoughts on heuristics and pictures may be convenient for quick travel down the road,but we’re liable to miss our turnoff at the first mile. The danger is in mistaking ourconvenient conceptualizations for what’s actually there. It is imperative that we havethe ability at any time to ground out in reality.