- Sep 2024
- Feb 2023
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personal.math.vt.edu personal.math.vt.edu
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My experience is that whatever students learn from that formalism is left by the wayside as soon as they move into a mathematical context of any substance.
I've experienced and seen this myself.
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- Sep 2022
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citeseerx.ist.psu.edu citeseerx.ist.psu.edu
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As this quote implies, formalisms are often difficult for people to use because they need to takemany extra steps (and make additional decisions) to specify anything.
I wonder if the formalisms were more carefully baked into the affordances and interactions if it could feel like less work. Since you're getting something for the formalism rather than doing extra work to embed it.
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Experiences with workflow systems, systems which automatically route documents and workthrough defined procedures, show that systems without the ability to handle exceptions to theformalized procedure cannot support the large number of cases when exceptional procedures arerequired (Ellis, Gibbs, Rein, 1991).
Feels like formalizing organic processes will have this outcome. The goal should be to make an actual formal workflow or build tools that enable users rather than trying to mix them.
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Of course, training and supervisionhelped users learning the general techniques for hypermedia authoring, but they tended to avoid(or lose interest in) the more sophisticated formalisms
What affordances were they given in exchange for the formalisms?
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This level of formalization enablesthe system to apply knowledge-based reasoning techniques to support users by performing taskssuch as automated diagnosis, configuration, or planning.
What I'm getting so far is that the formalization is what gives the users affordances to certain features. I'd imagine sophisticated data mining techniques (such as text-search, classification, etc) can alleviate this partially but is always going to be useful. It would be beneficial to opt into the formalism explicitly for the affordances and maintain bidirectional linking between non-formalized representations. In other words, you want the ability to create a formalized view.
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- Feb 2022
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archive.org archive.org
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Diesen gebrochenen Zahlen, welche zunächst als reine Zeichen auftreten, kann in vielen Fällen eine actuelle Bedeutung beigelegt werden.
A presented meaning can in many cases be attributed to these rational numbers, which at first appear as pure signs,
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Wie wir die Regeln der rein formalen Verknüpfungen, d. h. der mit den mentalen Objecten vorzunehmenden Operationen definiren, steht in unserer Willkühr, nur muss eine Bedingung als wesentlich festgehalten werden: nämlich dass irgend welche logische Widersprüche in den- selben nicht implicirt sein dürfen.
How we define the rules of purely formal operations (Verknüpfungen), i.e., of carrying out operations (Operationen) with mental objects, is our arbitrary choice, except that one essential condition must be adhered to: namely that no logical contradiction may be implied in these same rules.
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man sich zu der gegebenen Reihe von Ob- jecten eine inverse hinzudenkt
one adds an inverse in thought to the given series of objects
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Man sieht aber nicht, wie unter — 3 eine reale Substanz verstanden werden kann, wenn das ursprünglich gesetzte Object eine solche ist, und würde im Rechte sein, wenn man — 3 als eine nicht reelle, imaginäre Zahl als eine „falsche" bezeichnete.
one cannot see how a real substance can be understood by -3... and would be within his rights if he refers to -3 as a non-real, imaginary number, as a "false" one.
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Eine andere Definition des Begriffes der formalen Zahlen kann nicht gegeben werden; jede andere muss aus der Anschauung oder Erfahrung Vorstellungen zu Hilfe nehmen, welche zu dem Begriffe in einer nur zufälligen Beziehung stehen, und deren Beschränktheit einer allgemeinen Untersuchung der Rechnungsoperationen unüber- steigliche Hindemisse in den Weg legt..
A different definition of the concept of the formal numbers cannot be given; every other definition must rely on ideas from intuition or experience, which stand in only an accidental relation to the concept, and the limitations of which place insurmountable obstacles in the way of a general investigation of the arithmetic operations.
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Die Bedingung zur Aufstellung einer allgemeinen Arithmetik ist daher eine von aller Anschauung losgelöste, rein intellectuelle Mathem&tik, eine reine Formenlehre, in welcher nicht Quanta oder ihre Bilder, die Zahlen verknüpft werden, sondern intellectuelle Objecte, Gedankendinge, denen actuelle Objecte oder Relationen solcher entsprechen kön- nen, aber nicht müssen.
The condition for the establishment of a general arithmetic is therefore a purely intellectual mathematics detached from all intuition, a pure theory of form, in which quanta or their images, the numbers, are not combined, but rather intellectual objects, thought-things, to which presented objects or relations of such objects can, but need not, correspond.
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- Nov 2020
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www.roambrain.com www.roambrain.com
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Connected to this are Andy Matuschak’s comments about contextual backlinks bootstrapping new concepts before explicit definitions come into play.
What Joel says here about Contextual Backlinks is that they allow you to "bootstrap" a concept (i.e. start working with it) without explicit definitions coming into play (or as Andy would say, the content is empty).
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Cognitive Overhead (aka Cognitive Load): often the task of specifying formalism is extraneous to the primary task, or is just plain annoying to do.
This is the task that you're required to do when you want to save a note in Evernote or Notion. You need to choose where it goes.
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The basic intuition is described well by the Shipman & Marshall paper: users enter information in a mostly informal fashion, and then formalize only later in the task when appropriate formalisms become clear and also (more) immediately useful.
Incremental formalism
Users enter information in an informal fashion. They only formalize later when the appropriate formalism becomes clear and/or immediately useful.
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