24 Matching Annotations
  1. May 2023
    1. 4.1-6 Zorn's lemma.

      Useful for abstract vector spaces Banach Hahn.

    2. 4.8-3 Lemma (Weak convergence).

      weak convergences and its lemmas.

    3. 4.6-2 Lemma (Canonical mapping).

      The canonical mapping properly defined



  2. Apr 2023
    1. Lemma 2.13

      The cute formula, generalized version.



    1. 2.9-2 Lemma· (Zero vector)
    2. 2.4-1 Lemma (Linear combinations)

      Norm of the linear combinations of vectors are, larger than the sum of absolute value of all the scalar weight, by a strictly positive constant. Only for finite dimensional spaces.

      \( \Vert \alpha_1 + \cdots + \alpha_n\Vert \ge c(|\alpha_1| + \cdots + |\alpha_n|) \)

    3. 2.7-2 Lemma (Norm).

      operator norm is actually a type of norm. Linear operators as homomorphism between the vector spaces are indeed a vector space itself. See here.

      The properties of the operator norm basically interits the properties from the normed vector space. This is direct from the definition. However, these results are only for the bounded linear operators.

    4. 2.6-11 Lemma (Inverse of product)
    5. 2.5-4 F. Riesz's Lemma

      Riesz's Lemma, preparing for a theorem about the norm ball in finite dimensional spaces.

    6. 2.5-2 Lemma (Compactness)



  3. Mar 2023
    1. Lemma 7.12.

      Upper bound for the distance label of the preflow push algorithm

    2. Lemma 7.11.

      dipath connection for all nodes in the graph.

    3. Lemma 7.5

      The shortest augmenting path matains a correct distance label for all the vertices during the iterations. By valid, we mean that there exists some way such that \(d(j) = d(i) + 1\), where \((i, j)\in A\), for all \(i\in N\).

  4. Feb 2023
    1. 5.1-5 Lemma (Continuity).

      This is direct because the characterization contractive mapping resemble the Lipschitz Continuity in real analysis.



  5. Jul 2022
    1. some people will 01:52:34 read nagarjuna as allowing for the existence of true contradictions that something can be both true and false at the same time and uh graham priest is a philosopher who has a 01:52:46 uh reading of nagarjuna as under his uh dialethis logic which allows for certain uh contradictions to be true um [Music] i don't think that actually works in the 01:52:58 case of i think nagarjuna seems to presume the principle of non-principle of non-contradiction in order to run these kinds of reduction reductio absurdum type arguments um by drawing contradictions and incoherencies within 01:53:11 a given concept under analysis and then showing how it leads to contradiction so we should reject that concept um uh yeah do you have any thoughts about uh about 01:53:23 you know quantum physics is is sort of notorious for seeming to violate basic laws of of logic like say the law of non-contradiction or law of excluded middle or uh and so on and 01:53:35 so do you think that um our conventional logic you know it's like say classical logic is uh in if if there is no ultimate reality for madhyamako or for your your 01:53:48 understanding of uh quantum physics slash medium um then should the tools of classical logic what are the tools within conventional discourse broadly speaking as well for um 01:54:01 capturing um what madhyamaka is saying or what quantum physics as you understand it are saying so yeah let me answer specifically um uh 01:54:13 nagarjuna uh main negotiations from one perspective can be viewed as a logician right i mean it's a it's it's a his way of presenting things 01:54:25 uh uh it's it's a characteristic of somebody who's uh who's a legitimation you use logic uh but from from where's the perspective the first first of impact it sounds strange because uh his main tool is the 01:54:38 tetra of course which um somehow uh presents uh the impossibility of four alternative one being a something i don't know time exist uh one being non-a say time does not 01:54:57 exist and the third being um neither a nor not a and the fourth is uh both a and known a so it seems that wait a moment uh we we 01:55:09 we we are talked in logic 101 um that uh uh either a or not a and there is um beginning of logic so it seemed to be a clash here uh my 01:55:23 impression that there's no clash is that the known of non-a is not the same known as uh um as they restotelien known and we can uh we can think of innumerable uh everyday experience in which this whole 01:55:36 possibility it's exactly what we would uh we would consider so the exhaustive thing is the four there's four possibilities i don't want to go technically specifically but so it's not a an alternative logic here it's just a 01:55:48 different way of using known um so i don't see any clash between what we call logic uh in in in it's an interesting articulation but not 01:56:00 any any club it's not a mag logic um the same is true with quantum mechanics uh people been arguing that we can understand quantum mechanics by changing the logic i find it yeah but i find it 01:56:13 it's not really particularly clarifying um it's true i mean the particle doesn't go here normal goes there so if we think of these are two alternative quantum mechanics can be thought of can be 01:56:25 phrased if an alternative logic but all the alternative logic that i found they can be rephrased in terms of logic with different definitions so i don't i don't i don't think that this is the point um that's this is this is the 01:56:38 answer to your your question about logic you know the uh mutha madhyamakar karika his main treatise which we're talking about nagarjuna's text um 01:56:50 it's very short as you mentioned carlos and some of the things that are not there that are not written that are implied and also make it such a difficult text to understand is that he's refuting many different schools 01:57:05 of understanding an essence in reality and so when he does the tetralemma one of the usages is to be complete in terms of all the different you know 01:57:17 traditions or schools that are claiming some essence in reality to refute them and some do say that there's nothing you know not neither alternative and some say things 01:57:29 do exist and do not exist the both so i think he's using that more pedagogically if you will to um to refute all possible understandings 01:57:40 of an intrinsic existence and that's some of the beauty of his work and it's some of the difficulty in understanding it because you know unless you're really well read and really 01:57:53 understand fully all the different positions uh you it's hard to really know what he's doing at any one time um i could comment on this because it could be interesting um 01:58:08 so there is this uh sense in which barry explained that uh somehow answering 12 possible counter arguments at the same time and there's also a very simple way that you can see that this is not really 01:58:20 about a different logic so take the double slit experiment in quantum mechanics what's the point there that you try to explain a certain set of experimental data 01:58:32 by saying where does the particle go does it go through slit a does it go through slit beam let's go through both does it not go to neither and none of these four possibilities explains what you're seeing on the 01:58:45 screen so what do you do there it's not that you've reached the conclusion that everything is wrong is that you uh throw away the presupposition what was the presupposition that the particle 01:58:56 is somewhere so this straightforward use of logic it seems to me that i don't see any [Music] weird logic going on there yeah 01:59:08 you also throw away the the notion of a particle then if particles are that which have to be somewhere no you throw away the doctor there is an intrinsic reality that's what nagarjuna does if you continue doing that then you throw away 01:59:22 everything i i don't agree with uh personally if you ask me i agree that there is no interesting reality um [Music] in the sense that whenever you assume 01:59:37 such a thing you're going to fall into contradictions

      This question regards the use of logic by Nagarjuna in his tetralemma and parallels in quantum mechanics.

      Jay L. Garfield has some interesting and insightful observations about how Nagarjuna's logic works, and it relates to the different types of experiences where such statements could make sense.


    2. it works on on logic essentially and shows look if you if you take this entity as existence you get into a contradiction for this and this reason 00:36:31 and it slowly demolishes all the possible foundations of our thinking not only objects but also causation itself also time itself also the self 00:36:44 itself and so on and so forth one one by one um showing that uh thinking that they are foundational they're they're they have intrinsic existence uh doesn't doesn't hold

      Nagarjuna uses his tetralema to deconstruct logical arguments of thinking, existence of self, causation, time, intrinsic existence using logical arguments.j

      For other viewpoints of Nagarjuna's Tetralemma, visit:

      Judith Ragir discusses parallels between Dogen and Nagarjuna and employs Trungpha Rinpoche's Diamond Sliver diagram https://hyp.is/go?url=http%3A%2F%2Fwww.judithragir.org%2F2017%2F08%2Fdogen-nagarjunas-tetralemma-6%2F&group=world Graham Priest's paper on the Catuskoti / Tetralemma technique https://hyp.is/go?url=https%3A%2F%2Fbafybeifum5ioeus3y3hl4lqdwclgxpd6in4muleocuhsk3jev2rd7j3hpu.ipfs.dweb.link%2FThe-Logic-of-the-Catuskoti-by-G.-Priest.pdf&group=world

      Professor Peter Adamson:of King's College London on his "History of Philosophy without any Gaps" podcast series https://hyp.is/go?url=https%3A%2F%2Fhistoryofphilosophy.net%2Fnagarjuna-tetralemma&group=world Here Peter interviews Nagarjuna expert, Jan Westerhoff of Oxford,in an insightful interview https://historyofphilosophy.net/nagarjuna-westerhoff

    1. Dogen will take all the 4 positions of Nagarjuna’s tetralemma (either, or, both, neither) and present them in one sentence or one paragraph. He might even debate the righteousness of one point of view and then combat it with the other point of view. He breaks up our attachment to our point of views and points his finger at total dynamic functioning.

      Obviously this will appear confusing and frustrating to someone still trying to find "the right perspective" or "the right concept".

    2. After bringing the opposites to our consciousness, we then want to decide; do they happen as “both” (at the top of the chart) or neither (which is at the bottom of the chart).  Is it the truth that everything is annihilated by the word “neither”?

      Is it the truth that both form and emptiness coexist (Top of diagram)? Is that enough? No! THAT is still a position, that both exists. Is it then that neither exists? Is that enough? No! For even if neither exists, REALITY still exists!

      The whole exercise gets us to rise above our thinking and look at the thinking process itself.

  6. Jul 2021
    1. The term 'lemma' comes from the practice in Greco-Roman antiquity of using the word to refer to the headwords of marginal glosses in scholia; for this reason, the Ancient Greek plural form is sometimes used, namely lemmata (Greek λῆμμα, pl. λήμματα).


      No mention here of the use of headwords within the commonplace book tradition.