**Links to**: [[Sameness]], [[Identity]], [[Thermodynamics]], [[Difference]], [[Equivalence]], [[Différance]], [[Difference and Repetition]], [[Rifference and Depetition]], [[Homotopy]], [[Homotopy Type Theory]], [[Equal]], [[All men are created equal]]. This note is not yet complete, paste notes here. #todo ### [[Postulate]]: difference and equivalence are universal, inevitable, inseparably relational _first predictive principles_. At least at this point in history. >If order and life counter chaos, they are nevertheless conceived from it.^[Prado Casanova, M. “The Noise in Noise”, p. x. ] **Difference**: exploring, salience, relevance, change, noise that makes a difference, etc. **Equivalence**: exploiting, sameness, forgetting, abstracting, noise that makes no difference, etc. ### “ = ” Identity and difference are traditionally diametrically opposed, conceptually speaking. We prefer equivalence because it is not the _opposite_ of difference but a concept which enables its possibility, and vice versa, see: [[Equivalence]]. Identity is impossible without the notion of _equality_, which precedes it: but equality is a lie. Difference is impossible without the notion of equivalence (explain, #todo). The joke remains the same, and that is, which came first: _difference or equivalence_?^[Thinking about the beginning of the universe as presented by the causal image of the Big Bang and the ensuing symmetry-breaking: equivalence, or pure, _actual_ identity, came first. Symmetry-breaking: difference, came second.] Equivalence is preferred over common counter-notions such as *identity* or *repetition*, because equivalence includes both identity _and_ repetition (and even difference, as we can operationalize equivalence to say that A = B). Eq. and difference can be understood as mutually-defining in everyday parlance but also mutually-operationalizable when it comes to talking about programming things, for example.^[Sameness is used in a different way in this thesis, please see [[Sameness]].] We will talk about equivalence and difference in a few ways, starting with biological copying and diversification as explained by Chris Fields. Why do we take for granted that life makes copies of itself? ... %% Before we start: please take a picture of something, now. Look at it for a few seconds, put it away. Look at it again. (Think about your phone asking “please hold still” in dark situations). ![[chris fields copy and diversify.png]] In [[Homotopy Type Theory]] we define equivalence as…. And identity as… and equality as… #todo %% ### Sensing difference and equivalence Sensing-making patterns, which is what spatiotemporal agents do, is tempered by an interplay between difference and equivalence. Novelty, _change_, is all about difference, while _permanence_ all about equivalence. We observe permanence, laws, patterns, etc. because something remains the same, in spite of background differences. But this observation is only possible _because_ of the differences which temper everything, and salience is, precisely, the memorious reportage of difference. Salience-sensing thus requires permanence (equivalence), but also difference. Reference does not point _to_ things, we create equivalence over patterns in the concept of reference, that’s the model of semantics: permanence over change. Memory is generalistic: forgetful. In dealing with complex patterns like human agents—at least by way of what we currently understand as complexity—what we are mostly interested in is in seeing the pattern that makes up another agent, or worse yet: a group of agents, and the interactions between groups of agents. In bearing witness to this level of complexity it is not difficult to conclude that this is how massive disagreements arise in e.g., the family, politics, and social organizing across all scales. Perceptions vary so drastically, desiring-equivalence is impossible. This results in interesting long-term patterns that exceed the scope of a generation, but in weird, contingent experiences for the living agents at hand. Forgetting about the agent involved in a larger abstraction (passion for a political commitment, desire to nurture family or friends) means a reshuffling of the expected difference-equivalence conditions that gave rise to the agent: in order to choose my neighbor’s life above mine I need to forget that I am a thing worth my time. This is confusing. Everything that kicked off the organism makes the organism become irrelevant to itself, because it understands itself as distributed (distributed across the family, the future, etc.). ### Thinking difference and equivalence Joscha Bach says that, ultimately, we only have math and philosophy. Math is too brittle, philosophy is too soft. Philosophy presupposes agency, as it is platformed by language. Mathematics can augment this: philosophy and mathematics can be technically merged in the project that is AI. Truth is not _stateless_ but **computational**, Bach insists. #todo Unfininished notes: - Gödel’s incompleteness/the Turing halting problem: Hilbert and the gap in mathematics, taking Principia Mathematica as a starting point. - Cantor’s infinite set, in number theory, a number is the cardinality of the set. We then discover that the subsets of a set is larger than the members of a set: combinatorial possibilities are larger than combination-elements (!). The infinite set, with its even **larger** subsets, shows that we have multiple infinities. And that some infinities are larger than others. But what about the _total_ set, set of all sets? It’s also going to contain the combinatorial subsets, which means that the total set has a different number of members than the total set itself: the number of members becomes inconsistent. - What Gödel (and subsequently Church, Turing) showed is that a **complete semantic machine** cannot be built. This destroyed Gödel. Truth is a predicate you assign by constructing an algorithm (remember “bounded variable”). - Chaitin says: a proof is a reduction of a statement to its axioms, and the proof is the algorithm you need to get there, a lossless data compression: the assembly. This can also be related this to computational irreducibility. Classical maths sometimes claims that there are infinite steps, but you cannot claim that you can have a result without knowing the **steps**, computationally. ### Hofstadter, Arc, Melanie Mitchell and analogy - Mitchell explains that we should be wary of the generalizing capacities of AI. But: generally, we generalize too much, and overfitting is sometimes actually better. - One shot learning, one example. 0 shot: answer right away. Look at the assumptions about what [[Question]]s are, what is the **shape** we are after. - I only speak this language because I have to speak it in the way I think you will be able to grasp, somehow. Making myself semi-unintelligible, like James Joyce, is more generative than trying to fit some landscape which I generalize about (making big mistakes and telling people how to live). This, in essence, is the point about acting versus reacting. ### Why is the _shortest_ path important? Linking HoTT, Assembly and assemblage The equal sign in Hofstadter: intro. Following on from thoughts on differenc/tiation: the interplay between environment and objects entails compression (i.e. possibilistic reduction), but not **just** that (the implications of compression and shortest paths will also be discussed below). Things decompose over time because of the second law, because of entropy, and that is what we know to be a “fact” **as of yet**: black holes might be able to reverse entropy: [Second law of quantum complexity](https://www.quantamagazine.org/in-new-paradox-black-holes-appear-to-evade-heat-death-20230606/). But: molecules **remaining** molecules means they need to survive via repetition: the rules (the assembly steps), as constraints, survive over time, and these represent new constraints to the atomic environment. The shortest path to get there, the minimum set of rules, to make something preserve itself in light of entropy, is what will _initially_ survive over the things that take longer to make themselves. This is why we observe, e.g. in Longo diagram below: initially we have the reign of simple organisms, but these give way over time. So: complexity does emerge: sometimes the shortest path is not taken, the assumption made here is that this occurs by mere complexity of the fact that once a difference is marked at the very genesis of things, once the very first atom has differentiated itself from total equivalence, all is pure combinatorial complexity thereafter. Imagine a pool of unicellular organisms: the shortest path will win until, by a certain random bouncing around of things, survival will mean being _drastically_ different from the rest. This is all the law of large numbers and tendencies, with the added factor that at some point change emerges (like an atom decays, like a catastrophe of any kind suddenly occurs: this is also what we know to be a _fact_ of our observation of existence/the universe). ![[longo diagram, complexity, unicellular.png]] <small>Longo & Montévil, Time evolution of mass repartition over anti-entropy . The initial condition is a finite mass at almost 0 anti-entropy, thus having the shape of a pulse.</small> ### Equivalence and entropy Total randomness and high complexity can be understood as indistinguishable. This seems obvious: a highly intricate structure cannot be functionally, historically, etc. distinguishable from pure chaos because we don’t know what it can do (cf. not knowing what a body can do). Also a note on perspectivism here: perspective makes the object, because of [[Vantagepointillism]]). But repeated, thinking about the evolution of organisms via assembly theory: copies of things tells us: **pattern**. Whether that be molecules or human beings. Equivalence holds when there are apparent copies, copies hold when there’s a rule that guarantees initial selection and survival. ### Algebraic topology >Algebraic topology can be roughly defined as the study of techniques for forming algebraic images of topological spaces. Most often these algebraic images are groups, but more elaborate structures such as rings, modules, and algebras also arise. The mechanisms that create these images — the ‘lanterns’ of algebraic topology, one might say — are known formally as functors and have the characteristic feature that they form images not only of spaces but also of maps. Thus, continuous maps between spaces are projected onto homomorphisms between their algebraic images, so topologically related spaces have algebraically related images. (Hatcher 2001). ### Scales of difference and equivalence Zf. maybe irrational numbers exist perfectly but our vantagepointillist scale doesn’t allow us to see that. It looks like a function (something that can be put to work) from our perspective: it looks like difference, but it is a type of equivalence at another level. ### Pattern-sensing-making-testing Raven’s progressive matrices https://en.wikipedia.org/wiki/Raven%27s_Progressive_Matrices — but what about music? Would that even be possible? %% ### Email For a while (maybe since before starting the PhD), my thoughts have been circling around the classic question of whether difference is reducible to something more compact (or different!) than itself, and whether equivalence is--not as repetition, which is impossible, but as a more fundamental type of correspondence--in effect, this reduction I am in search of, or whether it is simply trumped by difference underlying it (the latter has been my feeling/conclusion the whole time until recently).  Ok, so. The Kantian schemata (quantity, quality, relation, modality) are, in my eyes, sustained/underlied by these more foundational schemata of difference and equivalence, right? To count we need a plane of equivalence (numbers) and their differential nature. To qualify something like color we need the categorical equivalence of something (e.g., color) and the way one color is different from another. For relations of causality we need the equivalence of continuity, and the necessary changes cause-effectuated. Finally, modally: we need the equivalence generated by modes and what these imply for (temporal) differentiations of all sorts. Following up on negintelligibility, my idea is to start from this schematism of _difference/equivalence,_ and treat it in a variety of ways. First of all via Deleuze, then via Homotopy Type Theory, and if there is spacetime via Marx. I am now in the process of grasping some things about different/ciation. This [lecture](https://ses.library.usyd.edu.au/bitstream/handle/2123/618/adt-NU20051202.14522707appendices.pdf;jsessionid=AF0B119895FFFA9FAC25C51E6C2FA938?sequence=2 "lecture") by Deleuze, p. 98. explains a lot about it, it is truly full of incredibly precise observations. I was particularly taken aback by the meditation on the schematism and the straight line (in the context of infinitesimal approximations to curved lines). Deleuze being anti-representational and a thinker of difference makes the sketches below quite reductive, but: I was thinking about the differentiation/differenciation differentiation (haha): does it not somehow imply a similar kind of duality that the traditional understanding vs. reason division implies? In the sense that: **Differenciation,** according to Deleuze, an operation of qualification/specification _and_ division/distribution, which deals with the actual (i.e. with what already exists, what is apprehended, what we qualify and discretize e.g. via the schematism); **Differentiation**, acc. Deleuze: the realm of the pre-individuated, the ideal (without being abstract), he talks about the Kantian Idea, containing relations and singularities which are capable of becoming actualized in qualities and parts, but crucially: **without** resembling the qualities and parts. Differentiation then deals with the virtual, with the yet-to-be-combinatorial, with an abductive landscape of affordances. I would still want to term it possibilistic, but he explicitly refers to the virtual as _opposed_ to the possible (pp. 98-9), because the possible is a representational concept: whatever is possible resembles something which already exists, while: "The Idea is an image without resemblance; **the virtual does not actualise itself through resemblance**, but through divergence and differenciation. Differenciation or actualisation is always creative in relation towhat they actualise, whereas realisation is always reproductive or limiting." (p. 99, my emphasis). He also calls the virtual a "quasi-cause". In the essay/lecture linked above: "The question of the “_ens omni modo determinatum_” must thus be posed in this way: a thing in its Ideal form can be completely determined (differentiated), and yet lack the determinations which constitute actual existence (it is undifferenciated)." (p. 98). I wonder to what extent this is perfect answer to the classic analytic problem of reference and non-actualized possibles and beyond (Quine, _On What There Is_): Pegasus is virtually differentiated in that it is completely determined what it is, but it is not actually differenciated (unless we consider it to be _precisely_ what it is in order of mythical, representational, etc. appearance: as a phenomenon which should not make one think of an animal but of a process/product of human imagination, and not, indeed, as a possible creature resembling other earthly creatures). In considering the latter: different notions of equivalence give rise to different actualizations of the virtual. Difference is only the template, what tempers. Side note, but as I already mention in the Semantic Noise paper and other places: in this way, for me, it becomes impossible not to think of all concepts as always operating on a virtual level. But I need some help clarifying the Idea/concept distinction traditionally made, and explicitly used throughout the text linked.  In How to recognize structuralism? He talks about concepts like "language" or "society" as structurally virtual: they actualize themselves but there is no "total" language out there. The other (Bergsonian) example he uses to talk about the virtual and the actual is the eye: the condition (constraint regime!) which includes light and surfaces which reflect it, is _virtual_ to a possible processing/interacting operation (like the eye), _before_ eyes appear on the evolutionary scene. Once we have eyes, these are _actual_ responses to the constraint regime (but the implication is, of course, that these are by no means the only possible response). The interesting thing is that it is always something _else_ that contemplates the actual in light of the virtual (as in: e.g., it is now thought, discourse, technology, etc. that processes the evolution of the eye, opening up new virtual domains). The virtual, in a process of differentiation, is what tempers or differenciates the actual. Phenomena such as degeneracy (where difference and equivalence meet full-frontal), in biology, could have never been actual unless they evolved as a response to the virtual constraint regime operating beyond their actual appearance as traits. Complexity increases because of these differences becoming inevitably dynamic/interactive in what we witness as one equivalent combinatorial domain (i.e., spacetime). In constraint-terms: once eyes appear on the scene these create a whole new series of constraints: the world of sight for creatures that can develop hunting/niche-constructing/mating/etc. strategies all based on the affordances generated by the regime of sight. Once concepts appear on the scene: mamma mia. I am struggling, cognitively, trying to understand how this challenges a linear notion of time. This is an open question for you: because Deleuze wants to go against identity and representation, he _needs_ something like the concept(s) of differenc/tiation in order to say something about the process which reaches beyond established ideas like potential, possibility, etc (what I am talking about as the drive of the negintelligible, in another way). This is why I am linking this to traditional understanding/reason dichotomy: where reason is abductive (in the Peircian sense) and reaches 'beyond' presumed foundations (it is able to challenge and change foundations) and the understanding employs finite, established knowledge. Differenc/tiation is trying to get beyond difference and equivalence (more than Deleuze realizes himself) by simply _stating_ the fact of combinatorial complexity and non-linearity, maybe, or am I barking up the wrong tree? Ultimately, we are trying to think (for example) vast jumps through time (in what we can currently only organize as a kind of linear historicity): how did the universe transition from star formation to the phenomenon of slime mold or the periodic table of elements or the Qur'an or bionic robot creatures? Mysterious, complex, yet: actual! Is the impossibility of equivalence due to perception being out of phase w/ itself ([[12 Negintelligibility]])? Sorry, long email, got out of hand. _______ ### Notes: Use SEP Carnap on explication to deal w/ diff, eq: Carnap adduces a number of obvious desiderata for an explication. He does not offer this list as exhaustive, let alone as anything like necessary and/or sufficient conditions for an explication, but merely for clarification of the kind of thing he has in mind. The first desideratum he mentions is _exactness_, by which Carnap means that the explication should where possible be embedded in some sufficiently clear and exact linguistic framework (see the [main entry (Section 1.2)](https://plato.stanford.edu/entries/carnap/index.html#Fram)). The second is _similarity_ to the explicandum (in the sense that at least many of its intended uses, brought out in the clarification step, are preserved in the explicatum). However, the extensions of _C_ and C∗ are not required to be identical. Indeed, “considerable differences are permitted” (Carnap 1950b: 7) when this serves the purposes of explication. First of all, some of the applications of _C_ that might have been indeterminate originally may be decided in the course of explication (e.g., in the case of Tarski’s explication of truth, it may originally have been unclear how to deal with paradoxical sentences, such as the famous Liar sentence ‘This sentence is not true’, but then after explication the same sentence may, e.g., turn out to be non-well-formed). Secondly, even changes of clear-cut applications are acceptable when this serves desiderata other than similarity. (Carnap 1950b gives the example of re-classifying whales as _non-fish_, which happened to be useful for the purposes of biological research, even though the common-sense concept _fish_ originally applied to whales.) Here we see another crucial desideratum in action, that is, _fruitfulness_: the explicatum should be usefully applicable in scientific or philosophical theorizing and discourse, e.g., in the formulation of lawlike statements, or by creating deductive or inductive links to established theories that are themselves sufficiently clear, exact, and successful. Fruitfulness is itself regarded as a comparative notion or a “matter of degree” (cf. Carnap 1956a: 62); the explicatum ought to be _more_ fruitful than the explicandum. Carnap’s final desideratum for explications is _simplicity_. That is, once all other desiderata have been satisfied, the simplest available explication is to be preferred over more complicated alternatives. ### Generalization as equivalence 2016 Massively cited 6000+ article: https://arxiv.org/abs/1611.03530 [Chiyuan Zhang](https://arxiv.org/search/cs?searchtype=author&query=Zhang,+C), [Samy Bengio](https://arxiv.org/search/cs?searchtype=author&query=Bengio,+S), [Moritz Hardt](https://arxiv.org/search/cs?searchtype=author&query=Hardt,+M), [Benjamin Recht](https://arxiv.org/search/cs?searchtype=author&query=Recht,+B), [Oriol Vinyals](https://arxiv.org/search/cs?searchtype=author&query=Vinyals,+O) Despite their massive size, successful deep artificial neural networks can exhibit a remarkably small difference between training and test performance. Conventional wisdom attributes small generalization error either to properties of the model family, or to the regularization techniques used during training. Through extensive systematic experiments, we show how these traditional approaches fail to explain why large neural networks generalize well in practice. Specifically, our experiments establish that state-of-the-art convolutional networks for image classification trained with stochastic gradient methods easily fit a random labeling of the training data. This phenomenon is qualitatively unaffected by explicit regularization, and occurs even if we replace the true images by completely unstructured random noise. We corroborate these experimental findings with a theoretical construction showing that simple depth two neural networks already have perfect finite sample expressivity as soon as the number of parameters exceeds the number of data points as it usually does in practice. We interpret our experimental findings by comparison with traditional models. But use the second one, _still_ requires: https://dl.acm.org/doi/pdf/10.1145/3446776 Psychology and equivalence/difference Melanie Mitchell: AI a guide for thinking humans. Dartmouth goals she writes illustrious founding AI Researchers wanted to find the ways in which to design ai so that it could "create abreactions and concepts." Mitchell comments on how funny this is, given we haven't been able to get there yet. Concepts, she sustains, are the hardest thing. We agree: which is why concept-generation is key to new insights and understanding. This thesis focuses on concept generation _a lot_ (see neologistics). This is not just a deleuzian inheritance but goes deep into my childhood fascination with the limitations of the alphabet: both in terms of amount of symbols, and of sounds. Tell anecdote about asking Jan about Spanish sounds and tell libro de arena fascination and aliens and black holes. A note on either hallucinations or shortcuts: this is the proof that patterns are of functional interest (see assembly entry). Patterns can be _anything_, what matters is their purpose (tells, see teleosemantics and autosemeiosis) %% ### Footnotes