Bradley Saul https://bradleysaul.us/bcs-0000/ bcs-0000 /bcs-0000/ Bradley Saul This is a collection of observations, ideas, and other miscellany. Written using Forester. Bradley Saul 2026 1 6 https://bradleysaul.us/bcs-0008/ bcs-0008 /bcs-0008/ About I study where programming meets mathematics and mathematics meets computation. I work in a constructive tradition, influenced by Per Martin-Löf’s type theory, that emphasizes a computational interpretation of mathematics. I am interested in how foundational choices, such as privileging expectation over probability or distinction over equivalence, shape reasoning about approximation, correctness, and failure. Probability, statistics, and numerical algorithms serve as important testing grounds for this work. My background is in statistics, including applied work in causal inference, and I continue to draw on applied domains to ground and stress-test foundational ideas. Software such as the R package geex reflects an early instance of this alignment between mathematical structure and program structure. I explore these ideas using dependent type theory, abstract algebra and functional programming, with the goal of building applications in which correctness is intrinsic. Bradley Saul 2024 1 8 https://bradleysaul.us/bcs-0009/ bcs-0009 /bcs-0009/ Research Interests Causal inference Ecology and Conservation Research software design and engineering Applied Category Theory Bradley Saul https://bradleysaul.us/bcs-0003/ bcs-0003 /bcs-0003/ Papers Bradley Saul 2024 1 8 https://bradleysaul.us/bcs-004P/ bcs-004P /bcs-004P/ Works in progress/Preprints Bradley Saul https://bradleysaul.us/bcs-004O/ bcs-004O /bcs-004O/ Great expectations: Unifying Statistical Theory and Programming Beginning in the 1970s, statistician-cum-logician Per Martin-Löf wrote a series of papers developing what became Martin-Lof type theory, realizing a system where the distinction between mathematics and programming disappears. Inspired by this vision, this paper introduces dependent type theory (of which Martin-Löf type theory is an example) to a statistical audience. Examples from statistics and probability theory demonstrate how dependent type theory and an algebraic perspective can unify the theoretical and computational concerns of statistics, ensuring rigorous, machine-checked proofs and executable software. arxiv repo Bradley Saul https://bradleysaul.us/bcs-004Q/ bcs-004Q /bcs-004Q/ Published Bradley Saul Michael Hudgens 2020 https://bradleysaul.us/saul2020/ saul2020 /saul2020/ The calculus of M-estimation in R with geex reference 10.18637/jss.v092.i02 M-estimation, or estimating equation, methods are widely applicable for point estimation and asymptotic inference. In this paper, we present an R package that can find roots and compute the empirical sandwich variance estimator for any set of user-specified, unbiased estimating equations. Examples from the M-estimation primer (The Calculus of M-estimation) demonstrate use of the software. The package also includes a framework for finite sample, heteroscedastic, and autocorrelation variance corrections, and a website with an extensive collection of tutorials. Bradley Saul Michael Hudgens Michael Mallin 2019 https://bradleysaul.us/saul2019downstream/ saul2019downstream /saul2019downstream/ Downstream effects of upstream causes reference saul2019downstream.bib 10.1080/01621459.2019.1574226 The United States Environmental Protection Agency considers nutrient pollution in stream ecosystems one of the United States’ most pressing environmental challenges. But limited independent replicates, lack of experimental randomization, and space- and time-varying confounding handicap causal inference on effects of nutrient pollution. In this article, the causal g-methods are extended to allow for exposures to vary in time and space in order to assess the effects of nutrient pollution on chlorophyll a—a proxy for algal production. Publicly available data from North Carolina’s Cape Fear River and a simulation study are used to show how causal effects of upstream nutrient concentrations on downstream chlorophyll a levels may be estimated from typical water quality monitoring data. Estimates obtained from the parametric g-formula, a marginal structural model, and a structural nested model indicate that chlorophyll a concentrations at Lock and Dam 1 were influenced by nitrate concentrations measured 86 to 109 km upstream, an area where four major industrial and municipal point sources discharge wastewater. Bradley Saul Michael Hudgens 2017 https://bradleysaul.us/saul2017/ saul2017 /saul2017/ A recipe for inferference: Start with causal inference. Add interference. Mix well with R reference 10.18637/jss.v082.i02 In causal inference, interference occurs when the treatment of one subject affects the outcome of other subjects. Interference can distort research conclusions about causal effects when not accounted for properly. In the absence of interference, inverse probability weighted (IPW) estimators are commonly used to estimate causal effects from observational data. Recently, IPW estimators have been extended to handle interference. Tchetgen Tchetgen and VanderWeele (2012) proposed IPW methods to estimate direct and indirect (or spillover) effects that allow for interference between individuals within groups. In this paper, we present inferference, an R package that computes these IPW causal effect estimates when interference may be present within groups. We illustrate use of the package with examples from political science and infectious disease. Bradley Saul 2024 1 8 https://bradleysaul.us/bcs-0004/ bcs-0004 /bcs-0004/ Software Projects This page lists software projects for which I am/was the primary developer or a significant contributor that are one of the following: in use in a production system (and publicly available), have an associated publication, or I'm interested in publishing at some point in the future. Additional software can be found at: GitHub GitLab Bradley Saul 2024 https://bradleysaul.us/bcs-0041/ bcs-0041 /bcs-0041/ Eudoxus software Formalizing Eudoxus real numbers in Agda. Available on: sourcehut Bradley Saul 2024 https://bradleysaul.us/bcs-000G/ bcs-000G /bcs-000G/ Mini Bird Breeding Survey Website software Bradley Saul 2024 1 9 https://bradleysaul.us/bcs-000H/ bcs-000H /bcs-000H/ Mini Bird Breeding Survey A Mini Breeding Bird Survey (MBBS) is a small-scale version of the full Breeding Bird Survey organized each year by the U.S. Fish and Wildlife Service throughout North America. An MBBS focuses on one county. Volunteers count birds seen or heard along randomly chosen routes on secondary roads. The objective is to determine the distribution of breeding birds in the county and to assess any year-to-year changes in their numbers. Since 2021, Allen Hurlbert and I have managed the survey. I created the project's website using Elm and Vega-lite. Available on: Minibbs.us GitHub Bradley Saul 2024 https://bradleysaul.us/bcs-000E/ bcs-000E /bcs-000E/ fagus software A theory of time based on the temporal logic axiomatized in Moments and points in an Interval-based Temporal Logic, formalized in Agda. After my experience writing the interval algebra package in Haskell, wherein we used QuickCheck to test that our implmentation satisfied the axioms, it was fun to write down the axioms formally in Agda. Available on: GitLab Bradley Saul 2022 https://bradleysaul.us/bcs-0007/ bcs-0007 /bcs-0007/ asclepias software asclepias is a domain specific language written in Haskell for specifying and building epidemiologic cohorts that I developed with my statistical software development team (Leah Jackman, Brendan Brown, David Pritchard, and others) while I was at NoviSci/TargetRWE. To my knowledge, TargetRWE is still using the software. Available on: GitHub Bradley Saul 2022 https://bradleysaul.us/bcs-0002/ bcs-0002 /bcs-0002/ geex software geex provides an extensible API for estimating parameters and their covariance from a set of estimating functions (M-estimation). M-estimation theory has a long history. For an excellent introduction, see The Calculus of M-estimation. Associated paper: The calculus of M-estimation in R with geex Available on: GitHub CRAN Bradley Saul 2021 https://bradleysaul.us/bcs-000D/ bcs-000D /bcs-000D/ Interval Algebra software Interval algebra is a Haskell library that implements the Interval Algebra developed by James F. Allen. It is a key part of asclepias. I wrote and maintained the first several versions, and then Brendan Brown improved it greatly. Available on: GitHub Hackage Bradley Saul 2017 https://bradleysaul.us/bcs-0006/ bcs-0006 /bcs-0006/ inferference software An R package that computes IPW causal effect estimates when interference may be present within groups. Associated paper: A recipe for inferference: Start with causal inference. Add interference. Mix well with R Available on: GitHub CRAN Bradley Saul 2024 1 8 https://bradleysaul.us/bcs-0005/ bcs-0005 /bcs-0005/ Observations Bradley Saul 2025 10 16 https://bradleysaul.us/bcs-004S/ bcs-004S /bcs-004S/ Recursion observation As part of project I'm noodling on, I've been thinking about transfinite recursion. I started collecting a bibliography here: TaylorPaul 2025 https://bradleysaul.us/taylor2025well/ taylor2025well /taylor2025well/ Well founded coalgebras and recursion reference taylor2025well.bib WissmannThorsten MiliusStefan 2024 https://bradleysaul.us/wissmann2024initial/ wissmann2024initial /wissmann2024initial/ Initial Algebras Unchained-A Novel Initial Algebra Construction Formalized in Agda reference 10.1145/3661814 wissmann2024initial.bib LeroyXavier 2024 https://bradleysaul.us/leroy2024well/ leroy2024well /leroy2024well/ Well-founded recursion done right reference leroy2024well.bib BoveAna CaprettaVenanzio 2005 https://bradleysaul.us/bove2005modelling/ bove2005modelling /bove2005modelling/ Modelling general recursion in type theory reference 10.1017/S0960129505004822 bove2005modelling.bib Bradley Saul 2025 10 10 https://bradleysaul.us/bcs-004R/ bcs-004R /bcs-004R/ Great Expectations now on arxiv observation Over the past few months, I've been working on Great expectations: Unifying Statistical Theory and Programming, and I finally put a preprint up on arxiv. Bradley Saul 2025 7 1 https://bradleysaul.us/bcs-004N/ bcs-004N /bcs-004N/ Starting with Apartness observation As I've been working with Heyting Fields from Agda's stdlib in the Great expectations: Unifying Statistical Theory and Programming projects, I began to wonder whether it would be advantageous to push the introduction of apartness further up the hierarchy of algebraic structures. I found this quote in We could demand that every set come with an inequality, which suggests this is a bad idea: MinesRay RichmanFred RuitenburgWim 2025 7 1 https://bradleysaul.us/bcs-004M/ bcs-004M /bcs-004M/ We could demand that every set come with an inequality quote We could demand that every set come with an inequality, putting inequality on the same footing as equality; it would then be natural to demand that all functions be strongly extenstional. With such an approach, whenever we construct a set we must put an inequality on it, and we must check that our functions are strongly extenstional. This is cumbersome and easily forgetten, resulting in incomplete and incorrect proofs. from A course in constructive algebra I'm not convinced by that argument. Yes, it may be easy to forget things when working with pen and paper, but a theorem prover like Agda won't let us forget. In Constructive Semigroups with Apartness: A State of the Art, they do explore semigroups equipped with apartness, and this looks promising to me. Bradley Saul 2024 11 7 https://bradleysaul.us/bcs-0040/ bcs-0040 /bcs-0040/ Eudoxus Reals in Agda observation I started a project (Eudoxus) to formalize the Eudoxus Reals in Agda. Still very experimental at this stage. Bradley Saul 2024 11 5 https://bradleysaul.us/bcs-003Y/ bcs-003Y /bcs-003Y/ Eudoxus Real Numbers KumarAJ LongReese TungAndrew WongIvan 2023 https://bradleysaul.us/kumar2023eudoxus/ kumar2023eudoxus /kumar2023eudoxus/ The Eudoxus Reals reference kumar2023eudoxus.bib A’CampoNorbert 2021 https://bradleysaul.us/acampo2021natural/ acampo2021natural /acampo2021natural/ A natural construction for the real numbers reference acampo2021natural.bib FokmaSJ 2021 https://bradleysaul.us/fokma2021eudoxus/ fokma2021eudoxus /fokma2021eudoxus/ The Eudoxus reals constructed in homotopy type theory reference fokma2021eudoxus.bib MizrahiLeila 2015 https://bradleysaul.us/mizrahi2015thoroughly/ mizrahi2015thoroughly /mizrahi2015thoroughly/ Thoroughly formalizing an uncommon construction of the real numbers reference mizrahi2015thoroughly.bib HarrisonJohn 2012 https://bradleysaul.us/harrison2012theorem/ harrison2012theorem /harrison2012theorem/ Theorem proving with the real numbers reference harrison2012theorem.bib GeuversHerman 2000 https://bradleysaul.us/geuvers2000constructive/ geuvers2000constructive /geuvers2000constructive/ Constructive reals in Coq: Axioms and categoricity reference 10.1007/3-540-45842-5_6 geuvers2000constructive.bib \authoer{NiquiMilad} HarrisonJohn 1994 https://bradleysaul.us/harrison1994constructing/ harrison1994constructing /harrison1994constructing/ Constructing the real numbers in HOL reference 10.1007/BF01384233 harrison1994constructing.bib Related links: Constructing the Real Numbers as Nearly Multiplicative Sequences Existing formalizations in theorem provers: Lean HOL Bradley Saul 2024 10 18 https://bradleysaul.us/bcs-003V/ bcs-003V /bcs-003V/ Notes on On the application of probability theory to agricultural experiments observation ... since we can only repeat the measurement of a particular observed yield, and this measurement can be made with high accuracy, ... How can a measurement of a "particular observed yield" be repeated? ... we have to suppose that the observed yield is essentially equal to $U_i$. $U_i$ is the true yield of a particular variety in plot $i$. This sounds like a causal consistency assumption. But I'm curious about the weasel word: "essentially". What does that mean? To compare $v$ varieties, we will consider that many sequences of numbers, each of them having two indices. One index corresponds to the variety and one to the plot. In other words, $U_{ik}$ is a function from variety (a finite set with $v$ elements) and plot (a finite set with $m$ elements) to some type of (addable) number representing the yield. In other words, we have a matrix where the rows are urns representing varieties and columns are balls representing plots. I'll call the number type R, so we have: Variety = Fin v Plot = Fin m U : Type U = Variety → Plot → R Further suppose that our urns have the property that if one ball is taken from one of them, then balls having the same (plot) label disappear from all the other urns. The phrase "disappear from all the other urns" is weird to me, but it's essentially sampling without replacement. AND this is a statement of the fundamental problem of causal inference: for each unit you only get to observe one exposure (variety in this case). The idea is simple: say you have two plots and two varieties (say wheat and corn). If you observe the yield of wheat in plot 1, then you cannot have (simultaneously) observed the yield of corn in plot 1. Again, the study design is one plot is planted with a single variety. How can this property on the function $U$ be expressed? In modern terminology, lower case $x$, denotes a random variable, and upper case $X$ corresponding realized values. When did this change? Bradley Saul 2024 2 22 https://bradleysaul.us/bcs-003N/ bcs-003N /bcs-003N/ Graphs and linear maps observation Recently, I noticed that (at least some) graphs can be composed from the operations of linear maps, or said more prosaically, it appears that (at least some) graphs can be specificied using the vocabulary of bi-product categories. Here's a few sketches using the operations I've defined in carya. A graph with 3 vertices and no edges: {- 0 2 1 -} graph₀ : 𝟘 ⊸ ((A ⊕ₗ B) ⊕ₗ C) graph₀ = (¡ {𝓜ᴬ = A} ▵ ¡ {𝓜ᴬ = B}) ▵ ¡ {𝓜ᴬ = C} A graph with 3 vertices and multiple "outputs": {- 0 ----------> 2 ↘ ↘ ↘ 1 -} graph₂ : A ⊸ B → A ⊸ C → A ⊸ (B ⊕ₗ C) graph₂ T₁ T₂ = T₁ ▵ T₂ A "confounder" DAG: {- 0 ----------> 2 ↘ ↗ ↘ ↗ ↘ ↗ 1 -} graph₁ : A ⊸ B → B ⊸ C → B ⊸ D → C ⊸ D → A ⊸ D graph₁ T₁ T₂ T₃ T₄ = T₁ · id ▵ T₂ · T₃ ▿ T₄ graph₁' : A ⊸ B → B ⊸ C → (B ⊕ₗ C) ⊸ D → A ⊸ D graph₁' T₁ T₂ T₃ = T₁ · id ▵ T₂ · T₃ A graph with a cycle: {- <---------- 0 ----------> 2 ↘ ↘ ↘ 1 -} graph₃ : A ⊸ B → A ⊸ C → C ⊸ A → (A ⊕ₗ C) ⊸ ((B ⊕ₗ C) ⊕ₗ A) graph₃ T₁ T₂ T₃ = (T₁ ▵ T₂) ⊕ T₃ A disconnected graph: {- <---------- 0 ----------> 2 1 -} graph₄ : A ⊸ C → C ⊸ A → ((A ⊕ₗ C) ⊕ₗ 𝟘 ) ⊸ ((C ⊕ₗ A) ⊕ₗ B) graph₄ T₁ T₂ = (T₁ ⊕ T₂) ⊕ ¡ {𝓜ᴬ = B} How far does this correspondence go? Bradley Saul 2024 2 22 https://bradleysaul.us/bcs-003O/ bcs-003O /bcs-003O/ Relations and bi-product categories observation In my last observation, I noticed that it appears that graphs and linear maps share a vocabulary. Yesterday, I showed that like linear maps relations can also form a bi-product category. This is interesting because graphs, it seems to me, are semantically about (binary) relations. Bradley Saul 2024 1 24 https://bradleysaul.us/bcs-0034/ bcs-0034 /bcs-0034/ Notes while reading Probabilities as Shapes observation Mathematical representations of logical concepts attain meaning not through what they are (i.e., frequencies, sizes, prices) but in how they are used (e.g., for prediction, inference, betting). Reminds me of differences in perspective between set/category theory, functional/object oriented programming, etc. Intuition relates past experiences to present circumstances by detecting similarities between distinct experiences, and using those similarities to guide judgments, actions, and decisions in the present. ... Intuition is therefore self-correcting and yet circular. Common Sense is collective intuition. It is the structure that emerges when the intuitive judgments of a collective of people, all with different experiences, perspectives and intuitions, lead to a common decision. Need common sense only be defined as a feature of human intelligence? Might other intelligences possess common sense? Rather than conceiving of the mind at birth as a blank slate—an empty void— we instead conceive of an initial (null) structure, from which our worldview develops and evolves. Sounds very middle way. On the desiderata of a formalization for Intuition: Judgments about what makes sense must have first-class logical status. on worldview: the framework must express judgments relative to their context and allow judgments to be revised based on new experiences. must be compatible with how humans actually think Per Harry Crane, the above desiderata rule out orthodox logic and probability. On the last desideratum, is the circularity of reason highlighted yet again: how do we judge what "how humans actually think" means? Section 3 (among other things) solidifies an intuition I had when I first started learning constructive logic (there's probably a more clear way to state this): the fact that excluding the law of the excluded middle immediately gets you a notion of uncertainty (we don't know whether something is provable until we prove it). Said more clearly: This isn’t the case [that claims are either true or false] for intuitive reasoning, for which the fundamental judgment isn’t whether a claim is true or false, but whether it make (sic) sense or doesn’t. Unlike true and false, claims do not ‘make sense’ of their own accord. ... In contrast to the 'truth', which is an objective, intrinsic property of a claim, 'making sense' is a subjective, contextual judgment. Therefore, by contrast to classical logic, intuitive reasoning is intensional, constructive and proof relevant. As a programmer, I like the proof relevant part. we envision claims as spatially oriented shapes. Note that the shape of an object is how the object is perceived from the outside, without regard for what it contains on the inside. Moreover, the shape of an object can be perceived differently depending on the perspective from which it is viewed. I like the shape analogy, though I suspect this might be difficult for those entrenched in orthodoxy to wrap their heads around. I don't know HoTT that well, but based on the little know, I can see where this shape idea is going. To conceive of a shape (as a body of evidence) does not require one to identify any specific point on that shape. However, in conceiving the shape, one identifies everything which, if observed, would serve as validation for the corresponding claim. Sounds very much like the distinction between a specification and implementation that Conal Elliott talk about frequently. [Homotopy type theory] fits with a structure-oriented understanding of how probabilistic judgments are made, by consideration of the context, shape, and structure of how relevant facts, outcomes, and experiences fit together. On a personal note, I came across Harry Crane's work, after reading about cubical Agda's interval type, which the docs say "intuitively corresponds to a point in the real unit interval." Huh, I thought, the real unit interval is central to probability, would this be useful in developing a framework for probability? A google search later, I found Harry Crane's work. Intuition and Common Sense defy traditional logical explanation because they are grounded in the natural human instinct to find coherence among personal experiences, rather than in a formal search for 'truth'. The quote above has echoes of Łukasiewicz: "The human mind does not work creatively for the sake of truth." I wonder: if we took Łukasiewicz's ideas of probability (e.g.) and updated them with all the wonderful work in logical the 20th and early 21st centuries, would we end up with Probabilities as Shapes? When our image of a shape is blurry, we cannot clearly discern whether a point near the edge of the shape is on the shape or just outside. With this mental image, if deductive inference is seeing clearly defined relations between images, then inductive inference is squinting, imagining how the blurry images fit together into a coherent scene. This quote made me grab Metaphors We Live By off my shelf. The book argues that: "the human conceptual system is metaphorically structured and defined." It seems to me that metaphor and probability/uncertainty are fundamentally linked in some way. Not judging a proposition as probable is not the same as judging the proposition as not probable. Orthodox probability theory, in the vein of Kolmogorov and de Finetti, doesn’t allow this distinction. This has echoes of the maxim: "absence of evidence is not evidence of absence." For any positive integer and claim , there is an associated claim meaning 'A is probable to degree '. We interpret larger degrees of probability to mean 'more probable' (as in 'makes a greater deal of sense'). If we have the extended natural numbers, is there a sense in which ? Before going further, consider: Why are these the three tenets of inductive logic? See p.25 for the three tenets. We might ask a similar question for the desiderata of plausible reasoning that Probability theory: The logic of science puts forth. For example, why should "Degrees of plausibility [be] represented by real numbers"? Moreover, why is Jayne's robot a good metaphor for reasoning? Additional thoughts/questions Is there any connection between what is proposed in this paper and the idea of plausibility measures described in Halpern (2017) Bradley Saul 2024 1 19 https://bradleysaul.us/bcs-002I/ bcs-002I /bcs-002I/ WIP: Defining Causal Consistency with Types observation WARNING: WORK IN PROGRESS In many treatments of causality, so-called consistency is a core definition/assumption/axiom/theorem of the logic. In this brief note, I examine consistency from a type-theoretic view. I use the Agda programming language as my substrate, hence Martin-Löf type theory specifically. Consistency, as typically stated in terms of potential outcomes, goes something like the following. I suppress any indexing of random variables or potential outcomes, Let be the potential outcome that depends (causally) on the value of the (perhaps random) variable , and let be the outcome data we observe. Consistency as stated by Cole and Frangakis (2009) is: Cole and Frangakis (2009) translate this statement into the natural language sentence: "the observed outcome ... is the potential outcome, as a function of intervention, when the intervention is set to the observed exposure." To simplify, I've removed any mention/notation for the index of experimental units. The essence of the consistency concept doesn't depend on having multiple subjects. I propose that consistency can be stated as a relation between the causal model and observable data. I try make this concrete with an example written in Agda. Suppose we have the simple case where a reasonable model of our study system is a boolean treatment assignment and a boolean outcome. In Agda, we can define the treatment and outcome as simple types as follows: data Assignment : Set where Treated Untreated : Assignment data Outcome : Set where Dead Alive : Outcome I could use Bool or Fin 2 to represent the treatment and outcome, but I hope using types with semantic labels helps the uninitiated understand the code (and prevent boolean blindness). Now we need a type for the causal model for our hypothetical experiment: CausalModel : Set CausalModel = Assignment → Outcome The CausalModel type is the type of functions that map an Assignment to an Outcome. The number of possible functions (i.e. the size of the function set) is , or 4 here. The four possible functions will be familiar to epidemiologists: doomed : CausalModel doomed Untreated = Dead doomed Treated = Dead cured : CausalModel cured Untreated = Dead cured Treated = Alive harmed : CausalModel harmed Untreated = Alive harmed Treated = Dead immune : CausalModel immune Untreated = Alive immune Treated = Alive The fundamental problem of causal inference is usually stated a missing data problem: for a given experimental unit, we only observe one outcome. Said using a different analogy: nature is a black box. We can only observe and manipulate (sometimes only hypothetically) the inputs and outputs of this black box. Unlike the toy situation here where I've written down the CausalModel type and I can manipulate terms of that type, when we actually do science, the CausalModel term for a given experimental unit is unknown. What we do have is a type and values of observable information, which for the scenario herein can be written as: Observation : Set Observation = Assignment × Outcome where the _×_ operator defines the non-dependent (cartesian) product type (the _,_ operator is the data constructor for a term of that type). Like the CausalModel type, I can ennumerate all the possible Observations: o₁ : Observation o₁ = Untreated , Dead o₂ : Observation o₂ = Untreated , Alive o₃ : Observation o₃ = Treated , Dead o₄ : Observation o₄ = Treated , Alive Now that I've defined CausalModel and Observable, I'm in a position to define a relation between them; in particular, I define what it means for Observable to be Consistent with CausalModel: Consistent : CausalModel → Observation → Set Consistent f (a , y) = f a ≡ y Or to align notation with the statement of consistency from Cole and Frangakis (2009): Consistent' : CausalModel → Observation → Set Consistent' Y (x , Yᵒᵇˢ) = Y x ≡ Yᵒᵇˢ The Consistent relation states that the observed outcome is the same outcome you would have gotten had you run the observed exposure through a causal model. For example, the following are proofs that o₁ and o₃ are Consistent with doomed: _ : Consistent doomed o₁ _ = refl _ : Consistent doomed o₃ _ = refl But o₂ and o₄ are not Consistent with doomed: _ : ¬ Consistent doomed o₂ _ = λ () _ : ¬ Consistent doomed o₄ _ = λ () When we step out of toy situtations like here and do the real work of science, we can't actually prove Consistent! We take consistency as a postulate How would we? For a given subject in our experiment, she is one of doomed, cured, harmed, or immune, but we don't and can't know which one. Proving consistency is out of this world. What I've written here is a start that begs for further exposition and probing: I've largely ignored the antecedent () in the statement of causal consistency given at the top of the post. This is on purpose. For one, the notation that equates the random variable with a value is confusing. This abuse of notation is common and goes back a long way, but to unpack a statement like (as innocent as it seems) is beyond the scope of this post. Even the symbol needs to be explained (see below). I set aside issues of randomness and probability. It's not clear to me that a definition of consistency needs to incorporate these concepts. The CausalModel type above should be generalized to any causal model. Moreover, can we define a type (or predicate) for what a causal model is? This deserves more clarity. I defined Consistent using propositional equality. Is that a reasonable notion of equality here? Or is it too restrictive? What are the consequences of inconsistency? Bradley Saul 2024 1 15 https://bradleysaul.us/bcs-0022/ bcs-0022 /bcs-0022/ A "nontechnical" definition of random variable observation On statexchange someone asked: "What do they mean when they say 'random variable'?" This response is interesting. It starts out: In thinking over a recent comment, I notice that all replies so far suffer from the use of undefined terms like "variable" and vague terms like "unknown," or appeal to technical mathematical concepts like "function" and "probability space." What should we say to the non-mathematical person who would like a plain, intuitive, yet accurate definition of "random variable"? After some preliminaries describing a simple model of random phenomena, I provide such a definition that is short enough to fit on one line. Because it might not fully satisfy the cognoscenti, an afterward explains how to extend this to the usual technical definition. Bradley Saul 2024 1 8 https://bradleysaul.us/bcs-0001/ bcs-0001 /bcs-0001/ Diversity and Causal Inference observation Over the weekend, I watched Tom Leinster's Topos Institute seminar on Entropy and Diversity. I've been familiar with Tom's work for a couple of years. His post at the n-category cafe on effective sample size made me wonder if there's a connection between magnitude and the notion of balancing covariates in field of causal inference. I suspect there is. Here's my intuition for a possible analogy. When running an experiment, we ideally want experimental units to be identical in all ways except for the experimental exposure. Said differently, we want experimental units to be similar; that is, lack diversity (here I mean diversity as in Hill numbers). I suspect this may be related to Tom's slogan (discussed roughly this point): magnitude is the effective number of points. Bradley Saul 2024 1 5 https://bradleysaul.us/bcs-000O/ bcs-000O /bcs-000O/ Similarities between basic number systems and trivial module extension observation The operations on the expectation semiring (which I think is a trivial extension of a semimodule) shares similarities with the semiring operations on common number systems (integers, reals, complex) when the type of the number is viewed as pair of other numbers. The first row of the following table shows the operations on the expectation semiring. Here, the type of is the Semiring carrier underlying the semimodule . The shows the differences between the operations. Are the similarities just coincidence? Bradley Saul 2023 7 7 https://bradleysaul.us/bcs-002Y/ bcs-002Y /bcs-002Y/ Idea for Structural Nested Model for coinfection observation Let: : pair of indicators for infection by pathogen and at time : indicators of infection by pathogen and of related units at time Suppose: Example interpretations: : effect of (some function of) infection statuses of neighboring plants (e.g. proportion of neighbors infected) Questions: Target what contrast function? (diff/ratio?) Extend to survival outcomes (time to infection)? Need to account for censoring/competing risks (e.g. death)? Is there some bipartite structure too here? Namely, exposure from plants outside the experimental system? Bradley Saul 2023 2 28 https://bradleysaul.us/bcs-0023/ bcs-0023 /bcs-0023/ Build Systems as collections of knowledge artifacts observation In this episode of the Haskell interlude Andrey Mokhov referred to build systems as something like collections of knowledge artifacts. Would David Spivak's ologs be useful? Bradley Saul 2023 1 16 https://bradleysaul.us/bcs-001V/ bcs-001V /bcs-001V/ Pain points while learning Agda observation Things I've found myself wanting and/or annoyed with in learning/using Agda: I'm not an Emacs user, and while other plugins exist, the docs are really emacs focused discoverability of stdlib is painful. It's big. How do I find what I'm looking for? a hoogle-like search feature would be great. I'd like more examples in the stdlib docs Unicode can be nice, but also can be hard to distinguish symbols even in same module I miss code formatting/linting tools from Haskell ecosystem. I want IDE features like hover over to show type of expression. Bradley Saul 2022 12 14 https://bradleysaul.us/bcs-002P/ bcs-002P /bcs-002P/ Main difficulty of -algebras in classical probability observation On giving computational meaning to concepts in classical probability, Collins (2020) writes: The main difficulty lies in the use of -algebras in classical probability, which have poor computability properties due to the presence of countable unions and complementation. Why is this? Is it a problem of decidability? From nlab: This is why constructivism has few consequences for basic combinatorics and algebra (although it does have important consequences for more advanced topics in those fields). In analysis, in contrast, constructivism matters right away, because the set of real numbers may have very few decidable subsets. Bradley Saul 2022 11 19 https://bradleysaul.us/bcs-001P/ bcs-001P /bcs-001P/ What is the connection between currying and factoring a probability distribution observation Both have a similar shape. Factorization: Currying: Bradley Saul 2022 11 8 https://bradleysaul.us/bcs-0027/ bcs-0027 /bcs-0027/ Any connection between causal "theories" and Lawvere theories observation Is there any connection between "theorems" that Judea Pearl discusses here and Lawvere theories? At any rate, "universal properties" sounds very categorical. Bradley Saul 2022 11 8 https://bradleysaul.us/bcs-001F/ bcs-001F /bcs-001F/ SUTVA and consistency observation I usually seen the no hidden version of treatment assumption embedded in SUTVA. How does this fit with Concerning the Consistency Assumption in Causal Inference tangling this assumption with consistency? Bradley Saul 2012 6 12 https://bradleysaul.us/bcs-003H/ bcs-003H /bcs-003H/ What Poetry to Read observation Whose advice do you take? HoaglandTony quoted in this interview: To me it seems that my students read too exclusively within their own generation, and that this may be ultimately stunting to their growth and vision of poetry, but it’s hard to say. Theordore Roethke: I recommend that you go, on your own, and immediately, to poets closer to your own age. Some may reflect your own confusions — let them be nameless — read them passionately and critically. At this point in my life, it’s most important that I read any poetry I can get my hands on. I’ve recently set to memory a Dickey poem, whose rhythm and force is medicine for my troubled mind. I also find reading Prufock aloud therapeutic. James Dickey nor T.S. Eliot are not of my generation, and I’m generally out of touch with poets of my age. One contemporary poet I’ve enjoyed is Michael McGriff. I read "Catfish and the Sequence of the Night" in American Poetry Review I will take recommendation for other poets my age that I should read.  Bradley Saul 2024 1 9 https://bradleysaul.us/bcs-000K/ bcs-000K /bcs-000K/ Quotes Olga Tokarczuk 2026 2 5 https://bradleysaul.us/bcs-004Y/ bcs-004Y /bcs-004Y/ But we still don't know how it all works quote I want to know, and not give into logic. What do I care about a proof from the outside, framed as a geometric argument? It provides merely a semblance of logical consequence and of an order pleasing to the mind. There's A, and after A comes B, first definitios, then axioms and numbered theorems, some supplementary conclusions -- and you might have the impression that such command is reminiscent of a wonderfully sketched etching in an atlas, where with letters particular sections are marked, where everything seems so clear and transparent. But we still don't know how it all works. from the novel Flights by Olga Tokarczuk Richard Dedekind 2026 1 2 https://bradleysaul.us/bcs-004W/ bcs-004W /bcs-004W/ Without proof quote Nothing capable of proof ought to be accepted without proof quoted in The Great Math War Alexander Grothendieck 2025 11 28 https://bradleysaul.us/bcs-004V/ bcs-004V /bcs-004V/ Understanding was all that mattered quote The question you raise "how can such a formulation lead to computations" doesn't bother me in the least! Throughout my whole life as a mathematician, the possibility of making explicit, elegant computations has always come out by itself, as a byproduct of a thorough conceptual understanding of what was going on. Thus I never bothered about whether what would come out would be suitable for this or that, but just tried to understand -- and it always turned out that understanding was all that mattered. - (in a letter to Ronnie Brown, December 4, 1983) relayed to me by Conal Elliott four thousand weeks: Time Management for Mortals 2025 11 21 https://bradleysaul.us/bcs-004T/ bcs-004T /bcs-004T/ Something essentially human quote But the other reason we might not realize some everyday process is broken is that it isn't broken to begin with -- and that the incovenience involved, which might look like brokeness from the outside, in fact embodies something essentially human. from Four thousand weeks: Time management for mortals MinesRay RichmanFred RuitenburgWim 2025 7 1 https://bradleysaul.us/bcs-004M/ bcs-004M /bcs-004M/ We could demand that every set come with an inequality quote We could demand that every set come with an inequality, putting inequality on the same footing as equality; it would then be natural to demand that all functions be strongly extenstional. With such an approach, whenever we construct a set we must put an inequality on it, and we must check that our functions are strongly extenstional. This is cumbersome and easily forgetten, resulting in incomplete and incorrect proofs. from A course in constructive algebra YoungMolly 2025 5 9 https://bradleysaul.us/bcs-004L/ bcs-004L /bcs-004L/ The Finnish version is a request to acknowledge the existence of other people quote At home in Brooklyn, the library is papered with reminders to "Please keep your voice down." In contradistinction, the signs at Oodi said, "Please let others work in peace!" The two commands are almost -- but meaningfully not -- synonymous. The Brooklyn version is a plea for self-control. The Finnish version is a request to acknowledge the existence of other people. You see the difference. Quote from Molly Young's article in the NYTimes Magazine "My Miserable Week in the ‘Happiest Country on Earth’" Carlo Rovelli 2025 5 9 https://bradleysaul.us/bcs-004K/ bcs-004K /bcs-004K/ We do not think quote We do not think: thoughts pass through us. Asking how we manage to think a given thought may be analogous to asking how a stone in a river manages to raise a wave above it on the water surface. "Consciousness," "free will," "spirituality," "divinity": these words may be nothing more than ways of indicating processes within ourselves whose complexity escapes us. I believe that this simple realization, which goes back to Baruch de Spinoza, is the most trustworthy compass for making our way through the dark forest of our own thoughts. from Anaximander: And the birth of science Carlo Rovelli 2025 4 9 https://bradleysaul.us/bcs-004J/ bcs-004J /bcs-004J/ Sing of our own stupidity quote Each time that we -- as a nation, a group, a continent, a religion -- look inward in celebration of our specific identity, we do nothing but lionize our own limits and sing of our own stupidity. Each time that we open ourselves to diversity and ponder that which is different from us, we enlarge the richness and intelligence of the human race. A Ministry of National Identity, like those established of large in some Western countries, is nothing more than a ministry of national obtuseness. from Anaximander: And the birth of science MosesRobert CobbCharles 2025 3 25 https://bradleysaul.us/bcs-004I/ bcs-004I /bcs-004I/ Work with people and leadership will emerge quote ...work with people and leadership will emerge. You don't have to know in advance who that leadership will be, but it will emerge from the movement that emerges. from Radical equations: Civil rights from Mississippi to the Algebra Project Conor McBride 2025 3 6 https://bradleysaul.us/bcs-004H/ bcs-004H /bcs-004H/ See types as pushing requirements inwards quote we must overcome our type inference training and learn to see types as pushing requirements inwards, as well as pulling guarantees out." from bcs-004F Conor McBride 2025 3 6 https://bradleysaul.us/bcs-004G/ bcs-004G /bcs-004G/ Time we made programs make their own sense.s quote We are used to making sense of programs, but it is we who make the sense, not the programs. It is time we made programs make their own sense. from bcs-004F Henry Petroski 2025 2 27 https://bradleysaul.us/bcs-004E/ bcs-004E /bcs-004E/ Emulating success risks failure quote Things that succeed teach us little beyond the fact that they have been successful; things that fail provide incontrovertible evidence that the limits of design have been exceeded. Emulating success risks failure; studying failure increases our chances of success. The simple principle that is seldom explicitly stated is that the most successful designs are based on the best and most complete assumptions about failure. from Success through failure: The paradox of design Alfred North Whitehead 2025 2 24 https://bradleysaul.us/bcs-004D/ bcs-004D /bcs-004D/ The growth of consciousness is the uprise of abstraction. quote The growth of consciousness is the uprise of abstraction. It is the growth of emphasis. The totality is characterized by a selection from its details. That selection claims attention, enjoyment, action, and purpose, all relative to itself. This concentration evokes an energy of self-realization. It is a step towards unification with that drive towards realization which discloses unity of aim in the historic process. But this enhancement of energy presupposes that the abstraction is preserved with its adequate relevance to the concrete sense of value attainment from which it is derived. In this way, the effect of the abstraction stimulates the vividness and depth of the whole of experience. It stirs the depths. Thus a fortunate use of abstraction is of the essence of upward evolution. But there is no necessity of such good use. Abstractions may function in experience so as to separate them from their relevance to the totality. In that case, the abstractive experience is a flicker of interest which is destroying its own massive basis for survival. It is interesting to note that in the entertainment of abstractions there is always present a preservative instinct aiming at the renewal of connection, which is the reverse of abstraction. This reverse process, partly instinctive and partly conscious, is wisdom of that higher life made possible by abstraction. The above was quoted from Alfred North Whitehead's *Modes of Thought* by Rémy Tuyéra on the ACT Zulip SorensenMorten UrzyczynPawel 2025 2 20 https://bradleysaul.us/bcs-004B/ bcs-004B /bcs-004B/ Curry-Howard not quite isomorphism quote We have just demonstrated that the formulas-as-types analogy can indeed be useful. However, the reader may find Proposition 4.1.1 [the Curry-Horward isomorphism] a little unsatisfactory. If we talk about an "isomorphism" then perhaps the statement of the proposition should have the form of an equivalence? The concluding sentence is indeed of this form, but it only holds on a fairly high level: We must abstract from the proofs and only ask about conclusions. (Or, equivalently, we abstract from terms and only ask which types are non-empty.) To support the idea of an "isomorphism," we would certainly prefer an exact, bijective correspondence between proofs and terms. Unfortunately, we cannot improve Proposition 4.1.1 in this respect, at least not for free. While it is correct to say that lambda-terms are essentially annotated proofs, the problem is that some proofs can be annotated in more than one way. from Lectures on the Curry-Howard isomorphism Henry Petroski 2025 2 19 https://bradleysaul.us/bcs-0049/ bcs-0049 /bcs-0049/ Failure is an unacceptable difference between expected and observed quote "Failure is an unacceptable difference between expected and observed performance," according to the comprehensive defintion used by the Technical Council on Forensice Engineering of the American Society of Civil Engineers. Good design is thus proactive failure analysis, something that both a designer and chooser among designs ought to practice. Anticipating and identifying how a design can fail -- or even just perceived to fail -- is the first step in making it a success. from Success through failure: The paradox of design Colin McGinn 2025 2 18 https://bradleysaul.us/bcs-0046/ bcs-0046 /bcs-0046/ Existence is not a topic that science could expect to deal with quote And existence is not a topic that science could expect to deal with. It is a purely philosophical question, simple but surprisingly confusing. Thinking about it makes you realize that even out most basic concepts are not clear to us; we use them smoothly enough, but we have not articulate understanding of what they involve. And this shows it is wrong to think that all genuine questions are scientific or empirical. Indeed, science itself raises philosophical questions. from The Making of a Philosopher Henry Petroski 2025 2 18 https://bradleysaul.us/bcs-0048/ bcs-0048 /bcs-0048/ This is why failure is the key to design quote Investors, engineers, and other professional designers are constantly criticizing the world of things, which is what leads to new designs for new things. The successful new thing is the one that does not fail in the way that it is intended to supercede did. This is why failure is the key to design. Understanding how things fail -- and might fail -- provides insight into how to redesign them successfullly. from Success through failure: The paradox of design Henry Petroski 2025 2 17 https://bradleysaul.us/bcs-0044/ bcs-0044 /bcs-0044/ Desire is the mother of invention quote Desire, not necessity, is the mother of invention. New things and the ideas for things come from our dissatisfaction with what is there and from the want of a satisfactory things for doing what we want done. More precisely, the development of new artifacts and new technologies follows from the failure of existing ones to perform as promised or as well as can be hoped for or imagined. Frustration and disappoinment associated with the use of a tool or the performance of a system puts a challenge on the table: Improve the thing. from Success through failure: The paradox of design Henry Petroski 2025 2 17 https://bradleysaul.us/bcs-0045/ bcs-0045 /bcs-0045/ Success and failure in design quote Success and failure in design are intertwined. Though a focus on failure can lead to success, too great a reliance on successful precedents can lead to failure. Success is not simply the absence of failure; it also masks potential modes of failure. Emulating success may be efficacious in the short term, but such behavior invariably and surprisingly leads to failure itself. from Success through failure: The paradox of design Douglas Bridges VîţăLuminiţa 2025 1 30 https://bradleysaul.us/bcs-0043/ bcs-0043 /bcs-0043/ Why do we describe the notion of apartness as "computationally more informative than that of proximity"? quote Why do we describe the notion of apartness as "computationally more informative than that of proximity"? Consider what it means, constructively and informally, for two real numbers and to be distinct: if and only if we can compute a rational number that we can place strictly between and . On the other hand, and are equal as real numbers if and only if there is no such rational number . Thus apartness deals with the existence of certain objects, but proximity deals with the non-existence of such objects. Apartness and Uniformity CarterJimmy 2025 1 24 https://bradleysaul.us/bcs-0042/ bcs-0042 /bcs-0042/ What are the goals of a person quote What are the goals of a person or a denomination or a country? They are remarkably the same: a desire for peace; a need for humility, for examining one’s faults and turning away from them a commitment to human rights in the broadest sense of the words, based on a moral society concerned with the alleviation of suffering because of deprivation or hatred or hunger or physical affliction; and a willingness, even an eagerness, to share one’s ideals, one’s faith, with others, to translate love in a person to justice. Per Martin-Löf 2024 11 7 https://bradleysaul.us/bcs-003Z/ bcs-003Z /bcs-003Z/ Distinction between Programming and disappears quote Now, it is the contention of the intuitionists (or constructivists, I shall use these terms synonymously) that the basic mathematical notions, above all the notion of function, ought to be interpreted in such a way that the cleavage between mathematics, classical mathematics, that is, and programming that we are witnessing at present disappears. Found quoted in this article. KlotzLeidy 2024 10 29 https://bradleysaul.us/bcs-003X/ bcs-003X /bcs-003X/ To generate novelty: analogies quote To generate novelty, to go from the known to the unknown, [Nancy] Nersessian finds that revolutionary scientists combine the highest levels of modern scientific practices with a humble and age-old device: analogies. from Subtract: The untapped science of less Andrej Bauer 2024 10 21 https://bradleysaul.us/bcs-003W/ bcs-003W /bcs-003W/ Why proof assistants use type theory quote The working mathematician is familiar with the set-theoretic foundation of mathematics. However, Lean and many other proof assistants use a different formalism known as type theory. This is because type theory is quite close to how mathematics is actually done, and because type theory can be used simultaneously as a programming language and a foundation of mathematics. From Andrej Bauer's course on Formalized mathematics Charles Sanders Peirce 2024 9 19 https://bradleysaul.us/bcs-003U/ bcs-003U /bcs-003U/ Inability to conceive quote Inability to conceive is only a stage through which every man must pass in regard to a number of beliefs, unless endowed with extraordinary obstinacy and obtuseness. His understanding is enslaved to some blind compulsion which a vigorous mind is pretty sure soon to cast off. from The Doctrine of Necessity Examined Charles Sanders Peirce 2024 9 19 https://bradleysaul.us/bcs-003S/ bcs-003S /bcs-003S/ To postulate a proposition quote To "postulate" a proposition is no more than to hope it is true. A bit later... But the whole notion of a postulate being involved in reasoning appertains to a by-gone and false conception of logic. from The Doctrine of Necessity Examined Charles Sanders Peirce 2024 9 19 https://bradleysaul.us/bcs-003T/ bcs-003T /bcs-003T/ What is a postulate? quote It is the formulation of a material fact which we are not entitled to assume as a premise, but the truth of which is requisite to the validity of an inference. from The Doctrine of Necessity Examined Jiddu Krishnamurti 2024 8 26 https://bradleysaul.us/bcs-003R/ bcs-003R /bcs-003R/ Because we play with causes and effects quote Because we play with causes and effects and never go beyond them, except verbally, our lives are empty, without much significance. from Commentaries On Living: First Series in the essay "Politics" Carlo Rovelli 2024 2 25 https://bradleysaul.us/bcs-003P/ bcs-003P /bcs-003P/ Science cannot solve all problems quote But never has it been so clear that science cannot solve all problems. Our splendid intelligence surrenders to a small virus that is little more than a speck of dust. Science is the best tool we have found, let's hold it dear; but we remain fragile when faced with a powerful and indifferent nature. from the essay "This short life feels beautiful to us, now more than ever" in There Are Places in the World Where Rules Are Less Important Than Kindness HarperRobert 2024 2 17 https://bradleysaul.us/bcs-003M/ bcs-003M /bcs-003M/ Constructive logic may be described as logic as if people matter quote Constructive logic may be described as logic as if people matter, as distinct from classical logic, which may be described as the logic of the mind of god. from Harper (2016) ChamberlinThomasC 2024 2 8 https://bradleysaul.us/bcs-003L/ bcs-003L /bcs-003L/ No beguilement more insidious and dangerous quote There is, perhaps, no beguilement more insidious and dangerous than an elaborate and elegant mathematical process built upon unfortified premises. From Chamberlin (1899). Quoted in Armstrong and Green 2022. Carlo Rovelli 2024 1 19 https://bradleysaul.us/bcs-002J/ bcs-002J /bcs-002J/ The man able to convince the pope and Einstein was uninterested in his own ego quote The man who first saw the Big Bang, the man who knew how to convince both the pope and Einstein, was curious about nature, uninterested in his own ego. His message seems to me the deepest and clearest that science has managed to articulate: Don't take yourselves too seriously; stay humble. Even if your name is Einstein, even if you are the pope himself. Even if you are "The Master." From an essay by Carlo Rovelli ("The Master") about George Lemaitre found in There Are Places in the World Where Rules Are Less Important Than Kindness. John Backus 2024 1 15 https://bradleysaul.us/bcs-0024/ bcs-0024 /bcs-0024/ Discussions about programming languages often resemble medieval debates quote Discussions about programming languages often resemble medieval debates about the number of angels that can dance on the head of a pin instead of exciting contests between fundamentally differing concepts. Charles Stein 2024 1 15 https://bradleysaul.us/bcs-0025/ bcs-0025 /bcs-0025/ Statistics gives you a way of thinking about things quote In The Invariant, the Direct and the "Pretentious" the interviewer asks: But statistics takes the guess work out of solving problems. In the old days, you did not know what is going on and you did it by trial and error. Now statistics gives you a way of doing things. To which Charles Stein responds It gives you a way of thinking about things, but you may not come out with the correct conclusions. Edsger Dijkstra 2024 1 12 https://bradleysaul.us/bcs-000Z/ bcs-000Z /bcs-000Z/ The Purpose of Abstraction quote The purpose of abstraction is not to be vague, but to create a new semantic level in which one can be absolutely precise. Aldo Leopold 2024 1 11 https://bradleysaul.us/bcs-000Y/ bcs-000Y /bcs-000Y/ First step in intelligent tinkering quote To keep every cog and wheel is the first precaution of intelligent tinkering. StanleyKenneth LehmanJoel 2024 1 11 https://bradleysaul.us/bcs-000X/ bcs-000X /bcs-000X/ Stop demanding what greatness should be quote Sometimes the best way to achieve something great is to stop trying to achieve a particular great thing. In other words, greatness is possible if you are willing to stop demanding what that greatness should be. from Why greatness cannot be planned: The myth of the objective Cormac McCarthy 2024 1 11 https://bradleysaul.us/bcs-000W/ bcs-000W /bcs-000W/ The probability of the actual is absolute quote The probability of the actual is absolute. That we have no power to guess it out beforehand makes it no less certain. That we may imagine alternative histories means nothing all. from Cities of the Plain p. 286 Cormac McCarthy 2024 1 11 https://bradleysaul.us/bcs-000V/ bcs-000V /bcs-000V/ Where all is known quote Where all is known no narrative is possible. from Cities of the Plain p. 271 G.K. Chesterton 2024 1 10 https://bradleysaul.us/bcs-000T/ bcs-000T /bcs-000T/ Function of the Imagination quote The function of the imagination is not to make strange things settled, so much as to make settled things strange. James Gleick 2024 1 10 https://bradleysaul.us/bcs-000Q/ bcs-000Q /bcs-000Q/ Maxwell's Demon quote It is still puzzling, though, to hang so much of physics on a matter of mere probability. Can it be right to say that nothing in physics is stopping a gas from dividing itself into hot and cold -- that it is only a matter of chance and statistics? Maxwell illustrated this conundrum with a thought experiment. Imagine, he suggested, a "finite being" who stands watch over a tiny hole in the diaphragm dividing the box of gas. This creature can see molecules coming, can tell whether they are fast or slow, and can choose whether or not to let them pass. Tnus he could tilt the odds. By sorting fast from slow, he could make side A hotter and side B colder -- "and yet no work has been done, only the intelligence of a very observant and neat-fingered being has been employed." The being defies ordinary probabilities. The chances are, things get mixed together. To sort them out requires information. from The Information: A history, a theory, a flood Joseph Y. Halpern 2024 1 10 https://bradleysaul.us/bcs-000S/ bcs-000S /bcs-000S/ No one "true" definition of causality quote I do not believe that there is one “true” definition of causality. We use the word in many different, but related, ways. It is unreasonable to expect one definition to capture them all. Moreover, there are a number of closely related notions—causality, blame, responsibility, intention—that clearly are often confounded. Although we can try to disentangle them at a theoretical level, people clearly do not always do so. from Actual Causality James Gleick 2024 1 10 https://bradleysaul.us/bcs-000R/ bcs-000R /bcs-000R/ Second law as entropy quote ...equivalent of probability: the entropy of a given macrostate is the logarithm of the number of possible microstates. The second law, then, is the tendency of the universe to flow from less likely (orderly) to more likely (disorderly) macrostates. from The Information: A history, a theory, a flood James Gleick 2024 1 10 https://bradleysaul.us/bcs-000U/ bcs-000U /bcs-000U/ What theory is quote Later he (Claude Shannon) worked with the mathematician and logician Hermann Weyl, who taught him what theory was: "Theories permit consciousness to 'jump over its own shadow', to leave behind the given, to represent the transcendent, yet, as is self-evident, only in symbols." from The Information: A history, a theory, a flood Jan Łukasiewicz 2024 1 9 https://bradleysaul.us/bcs-000J/ bcs-000J /bcs-000J/ Deductive reasoning can be either inference or verification quote Deductive reasoning can be either inference or verification, and reductive reasoning can be either explanation or proof. If from given reliable judgements we deduce a consequence, we infer; if we look for reasons for given reliable judgements, we explain. If we look for reliable judgements which are consequences of given unreliable judgements, we verify; if we look for reliable judgements of which given unreliable judgements are consequences, we prove. From "Creative Elements in Science" found in Selected Works. Jan Łukasiewicz 2024 1 9 https://bradleysaul.us/bcs-000L/ bcs-000L /bcs-000L/ Logic is the net of science quote Logic, with mathematics, might be compared to a fine net which is cast into the immense abyss of phenomena in order to catch the pearls that are scientific syntheses. It is a powerful instrument of research, but an instrument only. Logical and mathematical judgements are truths only in the world of ideal entities. We shall probably never know whether these entities have counterparts in any real objects. From "Creative Elements in Science" found in Selected Works. Jan Łukasiewicz 2024 1 9 https://bradleysaul.us/bcs-000N/ bcs-000N /bcs-000N/ The essence of probability quote The interpretation of the essence of probability presented here might be called the logical theory of probability. According to this viewpoint probability is only a property of propositions, i.e., of logical entities, and its explanation requires neither psychic processes nor the assumption of objective possibility. Probability, as a purely logical concept, is a creative construction of the human mind, an instrument invented for the purpose of mastermg those facts which cannot be interpreted by universally true judgements (laws of nature). From "Logical Foundations of Probability Theory" found in Selected Works. Jan Łukasiewicz 2024 1 9 https://bradleysaul.us/bcs-000M/ bcs-000M /bcs-000M/ Truth is not the goal of science quote Two kinds of judgements must be distinguished in science: some are supposed to reproduce facts given in experience, the others are produced by the human mind. The judgements of the first category are true, because truth consists in agreement between thought and existence. Are the judgements of the second category true as well? We cannot state categorically that they are false. That which the human mind has produced need not necessarily be a fantasy. But neither are we entitled to consider them as true, for we usually do not know whether they have counterparts in real existence. Nevertheless we include them in science if they are linked by relations of consequence with judgements of the first category and if they do not lead to consquences that are at variance with the facts. Hence it is erroneous to think that truth is the goal of science. The human mind does not work creatively for the sake of truth. The goal of science is to construct syntheses that satisfy the intellectual needs common to humanity. From "Creative Elements in Science" found in Selected Works. Ada Lovelace 2023 11 8 https://bradleysaul.us/bcs-002L/ bcs-002L /bcs-002L/ Lovelace on her singular combination of qualities quote I believe myself to possess a most singular combination of qualities exactly fitted to make me *pre-eminently* a discoverer of the *hidden qualities* of nature ... The belief has been *forced* upon me, and most slow have I been to admit it even. quoted in The Information: A history, a theory, a flood WulfAndrea 2023 10 22 https://bradleysaul.us/bcs-002M/ bcs-002M /bcs-002M/ Ernst Haeckel on being a scientist quote How was he (Ernst Haeckel) supposed to be a scientist in a discipline that felt claustrophobically cramped when nature laid out its tantalizing wares as if in an oriental bazaar? It was so bad, Haeckel wrote to Anna that he could hear 'Mephistopheles' scornful laughter'. from The invention of nature: Alexander von Humboldt's new world Deborah Mayo 2023 10 10 https://bradleysaul.us/bcs-002S/ bcs-002S /bcs-002S/ Three distinct requirements for an objective procedure of inquiry quote There are three distinct requirements for an objective procedure of inquiry: Relevance: It should be relevant to learning about what is being measured; having an uncontroversial way to measure something is not enough to make it relevant to solving a knowledge-based problem of inquiry. Reliably capable: It should not routinely declare the problem solved when it is not (or solved incorrectly); it should be capable of controlling reports of erroneous solutions to problems with reliability. Able to learn from error: If the problem is not solved (or poorly solved) at a given point, the method should set the stage for pinpointing why. from Statistical inference as severe testing TverskyAmos 2023 8 23 https://bradleysaul.us/bcs-002K/ bcs-002K /bcs-002K/ Secret to doing good research quote The secret to doing good research is always to be a little underemployed. You waste years by not being able to waste hours. George Ellis 2023 8 9 https://bradleysaul.us/bcs-002N/ bcs-002N /bcs-002N/ An infinite number of occurrences of everything possible happening quote The other principle George [Ellis] recommends is to not work with infinities. ... "The point I raised [regarding infinities] was that DNA is a finite code, and so if the probability of life is nonzero, then in a large enough volume of spaces you will eventually have used every possible combination of genetic codes and eventually you get an infinite number of genetically identical twins. You see, if you have an infinite universe, as soon as a probability is not zero it gives you an infinite number of occurrences of everything possible happening." from Lost in math: How beauty leads physics astra FischerHans 2023 8 9 https://bradleysaul.us/bcs-001Y/ bcs-001Y /bcs-001Y/ The notion of random variable quote The notion of "random variable" as it is employed in modern probability theory was introduced by Kolmogorov in the 1930s, but this term can still be used largely intuitively. Laplace [1781] himself devised a formula for those probabilities that a sum of "quantités variables" can assume. In 1829, Poisson developed approximations to probabilities that the sum of the "values" ("valeurs") that a (!) "thing" ("chose") receives in various independent experiments remains between certain limits. Hauber [1830], likely motivated by Poisson, emphasized the difference between "undetermined quantities" ("unbestimmte Größen") themselves and the "values that they each can receive with a particular probability". Chebyshev [1867; 1887/90] clearly differentiated between "quantités" and the different "values" they can take, but in his notation he usually made no distinction between these "random variables" themselves and their concrete values. from A History of the Central Limit Theorem Deborah Mayo 2023 7 13 https://bradleysaul.us/bcs-0033/ bcs-0033 /bcs-0033/ On the testing metaphor quote The goal highly well-tested claims differs sufficiently from highly probable ones that you can have your cake and eat it too: retaining both for different contexts. ... The testing metaphor grows out of the idea that before we have evidence for a claim, it must have passed an analysis that could have found it flawed. The probability that a method commits an erronenous interpretation of data is an *error probability*. The value of error probabilities, I argue, is not merely to control error in the long run, but because of what they teach us about the source of the data in front of us. The concept of severe testing is sufficiently general to apply to any of the methods now in use, whether or exploration, estimation, or prediction. from Mayo (2018) Robert Kass 2023 7 13 https://bradleysaul.us/bcs-002O/ bcs-002O /bcs-002O/ Why we care about the philosophy of statistics quote We care about the philosophy of statistics, first and foremost, because statistical inference sheds light on an important part of human existence, inductive reasoning, and we want to understand it. quoted in Statistical inference as severe testing StanleyKenneth LehmanJoel 2023 6 13 https://bradleysaul.us/bcs-0032/ bcs-0032 /bcs-0032/ Campbell’s law on measurement quote Campbell’s law, which is well known in the social sciences: "The more any quantitative social indicator is used for social decision-making, the more subject it will be to corruption pressures and the more apt it will be to distort and corrupt the social processes it is intended to monitor." from Why greatness cannot be planned: The myth of the objective StanleyKenneth LehmanJoel 2023 6 8 https://bradleysaul.us/bcs-002R/ bcs-002R /bcs-002R/ Novelty search is a kind of information accumulator quote Because eventually you have to acquire some kind of knowledge to continue to produce novelty, it means that novelty search is a kind of information accumulator about the world in which it takes place. from Why greatness cannot be planned: The myth of the objective Sandy Maguire 2023 5 1 https://bradleysaul.us/bcs-002T/ bcs-002T /bcs-002T/ Agda isn't just Haskell but with more types quote Agda isn't just Haskell but with more types, but it changes the work I should be doing in the first place. Conor McBride 2023 4 21 https://bradleysaul.us/bcs-002Q/ bcs-002Q /bcs-002Q/ Types warp gravity so correct programs are downhill quote Types warp gravity so correct programs are downhill. shared with me by Conal Elliott who attributes this to Conor McBride. Sofya Kovalevskaya 2023 4 17 https://bradleysaul.us/bcs-001T/ bcs-001T /bcs-001T/ It is impossible to be a mathematician without being a poet in soul quote The NY Times op-ed "The Wondrous Connections Between Mathematics and Literature" quotes Sofia Kovalevskaya: It is impossible to be a mathematician without being a poet in soul ... the poet must see what others do not see, must see more deeply .... And the mathematician must do the same. Deborah Mayo 2023 3 5 https://bradleysaul.us/bcs-0031/ bcs-0031 /bcs-0031/ Meaning of error probability quote We need to be clear on the meaning of error probability. A method of statistical inference moves from data to some inference about the source of the data as modeled. Associated error probabilities refer to the probability the method outputs an erroneous interpretation of the data. Choice of test rule pins down the particular error... from Mayo (2018) Alan Perlis 2023 3 2 https://bradleysaul.us/bcs-002H/ bcs-002H /bcs-002H/ Informal to the formal by formal means quote One can't proceed from the informal to the formal by formal means. Stella Cunliffe 2023 2 5 https://bradleysaul.us/bcs-001X/ bcs-001X /bcs-001X/ The Delight in elegance at the expense of practicality quote The delight in elegance, often at the expense of practicality, appears to me, if I dare say so, to be a rather male attribute ... . quoted in The lady tasting tea: How statistics revolutionized science in the twentieth century Leslie Lamport 2023 1 31 https://bradleysaul.us/bcs-001W/ bcs-001W /bcs-001W/ To think clearly, you need to be able to write quote If you think you understand something, and don't write down your ideas, you only think you're thinking. To think clearly, you need to be able to write down your ideas clearly, which requires being able to write well. Deborah Mayo 2023 1 14 https://bradleysaul.us/bcs-0030/ bcs-0030 /bcs-0030/ Severity requirement (strong) quote If passes a test that was highly capable of finding flaws or discrepancies with , and yet none or few are found, then the passing result, , is an indication of, or evidence for, . from Mayo (2018) Deborah Mayo 2023 1 14 https://bradleysaul.us/bcs-002Z/ bcs-002Z /bcs-002Z/ Severity requirement (weak) quote If data agree with a claim but the method was practically incapable of finding flaws for even if they exist, then is poor evidence for . from Mayo (2018) Robin Wall Kimmerer 2022 12 23 https://bradleysaul.us/bcs-0017/ bcs-0017 /bcs-0017/ Despair is paralysis quote Despair is paralysis. It robs us of agency. It blinds us to our own power and the power of the earth. from Braiding sweetgrass: Indigenous wisdom, scientific knowledge and the teachings of plants Robin Wall Kimmerer 2022 12 23 https://bradleysaul.us/bcs-0016/ bcs-0016 /bcs-0016/ No such thing as random quote One thing I've learned in the woods is that there is no such thing as random. Everything is steeped in meaning, colored by relationships, one thing with another. from Braiding sweetgrass: Indigenous wisdom, scientific knowledge and the teachings of plants Robin Wall Kimmerer 2022 12 23 https://bradleysaul.us/bcs-002G/ bcs-002G /bcs-002G/ The making and unmaking of the world quote The twin grandsons of Skywoman had long struggled over the making and unmaking of the world. Now their struggle came down to this one game. If all the pits came up black, then all the life that had been created would be destroyed. If all the pits were white, then the beautiful earth would remain. from Braiding sweetgrass: Indigenous wisdom, scientific knowledge and the teachings of plants Robin Wall Kimmerer 2022 12 18 https://bradleysaul.us/bcs-0013/ bcs-0013 /bcs-0013/ Ceremonies about the land quote Many Indigenous traditions still recognize the place of ceremony and often focus their celebrations on other species and events in the cycle of seasons. In a colonist society the ceremonies that endure are not about land; they're about family and culture, values that are transportable from the old country. Ceremonies about the land have no doubt existed there, but it seems they did not survive emigration in any substantial way. I think there is wisdom in regenerating them here, as a means to form bonds with this land. from Braiding sweetgrass: Indigenous wisdom, scientific knowledge and the teachings of plants Robin Wall Kimmerer 2022 12 18 https://bradleysaul.us/bcs-0015/ bcs-0015 /bcs-0015/ How it works and nothing about what it meant quote How will people ever care for the fate of moss spiders if we don't teach students to recognize and respond to the world as a gift? I'd told them all about how it works and nothing about what it meant. from Braiding sweetgrass: Indigenous wisdom, scientific knowledge and the teachings of plants Robin Wall Kimmerer 2022 12 18 https://bradleysaul.us/bcs-0014/ bcs-0014 /bcs-0014/ The land is the real teacher quote As an enthusiastic young PhD, colonized by the arrogance of science, I had been fooling myself that I was the only teacher. The land is the real teacher. All we need as students is mindfulness. Paying attention is a form of reciprocity with the living world, receiving the gifts with the open eyes and open heart. from Braiding sweetgrass: Indigenous wisdom, scientific knowledge and the teachings of plants Robin Wall Kimmerer 2022 12 16 https://bradleysaul.us/bcs-001U/ bcs-001U /bcs-001U/ How can we distinguish between that which is given by the earth and that which is not quote How can we distinguish between that which is given by the earth and that which is not? When does taking become outright theft? I think my elders would counsel that there is no one path, that each of us must find our own way. In my wandering with this question, I've found dead ends and clear openings. Discerning all that it might mean is like bushwacking through dense undergrowth. Sometimes I get faint glimpses of a deer trail. from Braiding sweetgrass: Indigenous wisdom, scientific knowledge and the teachings of plants Robin Wall Kimmerer 2022 12 16 https://bradleysaul.us/bcs-001S/ bcs-001S /bcs-001S/ The Honorable Harvest asks us to give back quote The Honorable Harvest asks us to give back, in reciprocity, for what we have been given. Reciprocity helps resolve the moral tension of the taking a life by giving in return something of value that sustains the ones we sustain us. from Braiding sweetgrass: Indigenous wisdom, scientific knowledge and the teachings of plants Alan Watts 2022 12 12 https://bradleysaul.us/bcs-001R/ bcs-001R /bcs-001R/ Faith is an essential virtue of science quote We must here make a clear distinction between belief and faith, because, in general practice, belief has come to mean a state of mind which is almost the opposite of faith. Belief, as I use the word here, is the insistence that the truth is what one would “lief” or wish it to be. The believer will open his mind to the truth on the condition that it fits in with his preconceived ideas and wishes. Faith, on the other hand, is an unreserved opening of the mind to the truth, whatever it may turn out to be. Faith has no preconceptions; it is a plunge into the unknown. Belief clings, but faith lets go. In this sense of the word, faith is the essential virtue of science, and likewise of any religion that is not self-deception. from The Wisdom of Insecurity Robin Wall Kimmerer 2022 12 2 https://bradleysaul.us/bcs-002F/ bcs-002F /bcs-002F/ In the gift economy, gifts are not free quote From the point of view of a private property economy, the "gift" is deemed to be "free" because we obtain it free of charge, at no cost. But in the gift economy, gifts are not free. The essense of a gift is that it creates a set of relationships. The currency of a gift economy is, at its root, reciprocity. In Western thinking, private land is understood to be a "bundle of rights," whereas in a gift economy property has a "bundle of responsibilities" attached. from Braiding sweetgrass: Indigenous wisdom, scientific knowledge and the teachings of plants Robin Wall Kimmerer 2022 12 2 https://bradleysaul.us/bcs-002E/ bcs-002E /bcs-002E/ Song of plants quote A plant scientist, armed with his notebooks and equipment, is exploring the rainforests for new botanical discoveries, and he has hired an Indigenous guide to lead him. Knowing the scientist's interests, the young guide takes care to point out the interesting species. The botanist looks at him appraisingly, surprised by his capacity. "Well, well, young man, you certainly know the names of a lot of these plants." The guide nods and replies with downcast eyes. "Yes, I have learned the names of the all the bushes, but I have yet to learn their songs." from Braiding sweetgrass: Indigenous wisdom, scientific knowledge and the teachings of plants Conal Elliott 2022 11 18 https://bradleysaul.us/bcs-001Q/ bcs-001Q /bcs-001Q/ The essence of tensor programming is Naperian functors quote From Conal Elliott's twitter: .@jer_gib drew an insightful lesson from my Oxford tensor computation talk on Friday: the essence of tensor programming is Naperian functors (generalized tries). Much of the power of what I shared comes from composing these functors/tries from binary product and composition and their identities. The logarithms/indices of these structures are binary sums and products and their identities. These structures are isomorphic to arrays/"tensors" (Fortran-style data), but (unlike arrays) they lead to elegant and powerful deconstruction of parallel algorithms and reconstruction into infinite families of correct variations. The 1D vectors and 2D matrices from linear algebra naturally and elegantly generalize to these trie structures, also yielding infinitely families of correct parallel algorithms for linear algebra. As I mentioned toward the talk's end, even computer memory is really implemented not as array but rather as natural, compositional tries --- specifically, perfect binary leaf trees with their usual logarithm/index type. KahnemanDaniel 2022 11 17 https://bradleysaul.us/bcs-0029/ bcs-0029 /bcs-0029/ The expectation of intelligent gossip is a powerful motive quote The expectation of intelligent gossip is a powerful motive for serious self criticism. from Thinking, fast and slow RehkopfDavid GlymourMarie OsypukTheresa 2022 11 8 https://bradleysaul.us/bcs-001L/ bcs-001L /bcs-001L/ A key distinction in consistency assumption quote The key distinction is whether the attribute violating the consistency assumption is more clearly thought of as a separate factor or a characteristic of the exposure itself. from The consistency assumption for causal inference in social epidemiology: when a rose is not a rose Tyler Vanderweele 2022 11 8 https://bradleysaul.us/bcs-0019/ bcs-0019 /bcs-0019/ Assumptions are at best approximations quote The treatment-variation irrelevance assumption and the consistency assumption, like almost all assumptions, are at best approximations; ... Perhaps one further extension will help make these assumptions more reasonable in some contexts. ... The assumption of treatment-variation irrelevance and the consistency assumption could still be formulated as above but the equalities would be equalities in distribution rather than for single values. from Concerning the Consistency Assumption in Causal Inference Naftali Weinberger Seamus Bradley 2022 11 8 https://bradleysaul.us/bcs-001K/ bcs-001K /bcs-001K/ Consistency assumption links counterfactual claims to what is observed quote The consistency assumption links counterfactual claims to what is in fact observed: is understood as a counterfactual claim saying "if were to receive treatment then the outcome would be ", and consistency then says that if the antecedent is true in the actual world, then the consequent must be true as well. from Making sense of non-factual disagreement in science RehkopfDavid GlymourMarie OsypukTheresa 2022 11 8 https://bradleysaul.us/bcs-001H/ bcs-001H /bcs-001H/ Consistency implies that the exposure must have enough precision quote consistency critically implies that the exposure specified in the analysis must have enough precision that any variation within the exposure specification would not result in a different outcome. from The consistency assumption for causal inference in social epidemiology: when a rose is not a rose Naftali Weinberger Seamus Bradley 2022 11 8 https://bradleysaul.us/bcs-001J/ bcs-001J /bcs-001J/ Debate on consistency looks like operationalism vs realism quote The position of the consistency-is-an-assumption advocates, if taken to the extreme is saying that we should not represent anything going beyond the content of our experiments, begins to look like an untenable form of operationalism. Pearl's position, when presented in a way that treats thorny and difficult empirical questions as secondary issues, begins to look like a realism achieved without the honest toil required to ground it empirically. from Making sense of non-factual disagreement in science Judea Pearl 2022 11 8 https://bradleysaul.us/bcs-001M/ bcs-001M /bcs-001M/ Distinction between an assumption and a theorem quote ...the distinction between an 'assumption' and a 'theorem' is not just a matter of semantics, but rather carries profound implications in research, communication and education, not unlike the implications of labeling the Pythagorean Theorem as a 'theorem,' not an 'assumption.' from On the consistency rule in causal inference: Axiom, definition, assumption, or theorem? Philip Dawid 2022 11 8 https://bradleysaul.us/bcs-0028/ bcs-0028 /bcs-0028/ Distributional consistency is the fundamental property quote where denotes 'has the same distribution as'. Distributional consistency is the fundamental property linking the observational and interventional regimes. It is our, weaker, version of the (functional) consistency property usually invoked in the potential outcome approach to causality. from Decision-theoretic foundations for statistical causality GibbardAllan HarperWilliam 2022 11 8 https://bradleysaul.us/bcs-002X/ bcs-002X /bcs-002X/ Gibbard (1976) statement of consistency quote Counterfactuals and Two Kinds of Expected Utility state consistency as: if I actually do , then the -world which, at is most like the actual world will be the actual world itself. I need to wrap my head around axiom 2. I think it says since counterfactual is true iff holds in world and also the counterfactual is true iff holds in world , then and are not the same. At any rate, they go on: We shall invoke these axioms, not because they follow from the rough theory we have given, and not because we regard them as intuitively self-evident, but because they simplify matters. The rationales just given for these two axioms are shaky. The rationale we gave for Axiom 1 fails if the world is indeterministic, so that a physically possible world exactly like the actual world at time could later diverge from the actual world. The rationale we gave for Axiom 2 fails if there is not unique -world which is closest at to the actual world. (emphasis mine) Tyler Vanderweele 2022 11 8 https://bradleysaul.us/bcs-001O/ bcs-001O /bcs-001O/ Indexing treatment variations quote One issue that is not addressed in their commentary is that the range of $k$ will generally vary with and we will in general need to index treatment variations by , which may be different for each treatment option . Concerning the Consistency Assumption in Causal Inference discussing The Consistency Statement in Causal Inference Judea Pearl 2022 11 8 https://bradleysaul.us/bcs-0026/ bcs-0026 /bcs-0026/ Lewis's 'closest-world' interpretation of counterfactuals entails certain universal properties quote Lewis's 'closest-world' interpretation of counterfactuals entails certain universal properties, called 'theorems', that hold true regardless of similarity measure used in ranking worlds. An example theorem is the consistency rule stated in Counterfactuals and Two Kinds of Expected Utility: "It reads as follows: For all and , if is true, then if would have prevailed (counterfactually) had been true, it must be true already." from On the consistency rule in causal inference: Axiom, definition, assumption, or theorem? Thomas Richardson Jamie Robins 2022 11 8 https://bradleysaul.us/bcs-001G/ bcs-001G /bcs-001G/ Modularity follows from consistency quote Given the factorization, the modularity property follows directly from the consistency condition: . For example, from richardson2013single-world Naftali Weinberger Seamus Bradley 2022 11 8 https://bradleysaul.us/bcs-001I/ bcs-001I /bcs-001I/ The dispute about consistency quote the dispute results from differing perspectives on the relationship between models and experiments. The tradition of treating potential outcomes as primitives is motivated by the idea that we can treat the relationship between a treatment and outcome in a particular experiment as an observed random variable. from Making sense of non-factual disagreement in science Tyler Vanderweele 2022 11 8 https://bradleysaul.us/bcs-001N/ bcs-001N /bcs-001N/ Vanderweele's two axioms of consistency quote C1 is a protocol fudge-factor property: "The condition requires that for each that the potential outcomes take the same value irrespective of what means $k_x$ is used to set to so long as in ." C2 is an existential quantifier: C2 "then requires that for some the potential outcome is equal to the observed outcome when . When both C1 and C2 are true they collapse to the standard statement of consistency. from Concerning the Consistency Assumption in Causal Inference Stephen R. Cole Constantine Frangakis 2022 11 8 https://bradleysaul.us/bcs-0018/ bcs-0018 /bcs-0018/ formalizing the assumption of consistency quote ...formalizing the assumption of consistency may help elevate discussions about exposure specifications. In the absence of a formal foundation, such discussions may devolve into what may seem to be subjective preferences. The exercise of reducing the number of components in $k$ by better specifying the exposure $x$ is crucial... from The Consistency Statement in Causal Inference Alan Chalmers 2022 10 5 https://bradleysaul.us/bcs-0010/ bcs-0010 /bcs-0010/ Typical History of a Concept quote A case could be made to the effect that the typical history of a concept, whether it be 'chemical element', 'atom', 'the unconscious' or whatever, involves the initial emergence of the concept as a vague idea, followed by its gradual clarification as the theory in which it plays a part takes a more precise and coherent form. from What is This Thing Called Science HoaglandTony 2012 6 26 https://bradleysaul.us/bcs-003I/ bcs-003I /bcs-003I/ Hello, my fellow sufferer quote HoaglandTony quoted in this interview: If I were honest and compassionate, I would greet every other person I meet with the phrase, 'Hello, my fellow sufferer,' or 'Greetings, you hopelessly foolish soul.' James Dickey 2012 6 14 https://bradleysaul.us/bcs-003J/ bcs-003J /bcs-003J/ The corrrect poetic attitude quote James Dickey on the poetry of MooreMarianne in From Babel to Byzantium: She persuades us that the human mind is nothing more or less than an organ for loving things in both complicated and blindingly simple ways, and is organized so as to be able to love in an unlimited number of fashions and for an unlimited number of reasons. This seems to me to constitute the correct poetic attitude, which is essentially a life-attitude, for it stands forever against the notion that the earth is an apathetic limbo lost in space. SmithRandall 2012 5 11 https://bradleysaul.us/bcs-003G/ bcs-003G /bcs-003G/ Dickey scared the shit out of us quote I do not think I exaggerate to say that Dickey stood wordless by that window for two or three full minutes. Some brave students risk little smart-ass smiles at one another; however, one girl across the table from me looks sick, or like she might have to drop out of graduate school altogether. Finally, Dickey returns to the head of the table, takes his seat, clears a path between the books, and does what even to this day makes me nervous when I remember it — he stares at each of us, without speaking and without smiling, for fifteen or twenty seconds apiece. By this time, the silence in the room is palpable; I have begun to wonder if I can control my own bowels; I can tell by the faces of others that I am not alone. Finally — and I mean finally — Dickey breaks his silent stranglehold by singling out one poor guy at the other end of the table and confronting him, "Son, why are you in this class?” In turn, without comment from himself or facial response, Dickey poses that question to each of us, and we each answer, wondering if this is some test for remaining in the class or even at the University. Afterwards, Dickey looked at us all and said (paraphrasing a quotation from Auden I think), "The only reason for being in here is that you like to fool around with words." That day — and I wonder if it happened literally to someone in that room — Dickey scared the shit out of us. I copied this quote down years from an essay by (I think) Randall Smith called (I think) "Writer, Reader, Student: Into the Maw of the Monster" about taking a class with James Dickey. Sadly, I can no longer find the original essay. DelilloDon 2012 2 23 https://bradleysaul.us/bcs-003E/ bcs-003E /bcs-003E/ The nature of modern death quote This is the nature of modern death. It has a life independent of us. It is growing in prestige and dimension. It has a sweep it never had before. We study it objectively. We can predicts its appearance, trace its path in the body. We can take cross-section pictures of it, tapes its tremors and waves. We’ve never been so close to it, so familiar with its habits and attitudes. We know it intimately. But it continues to grow, to acquire breadth and scope, new outlets, new passages and means. The more we learn, the more it grows. Is this some law of physics? Every advance in knowledge and technique is matched by a new kind of death, a new strain. Death adapts, like a viral agent. Is it a law of nature? Or some private superstition of mine? I sense that the dead are closer to us than ever. I sense that we inhabit the same air as the dead. Remember Lao Tse. 'There is no difference between the quick and the dead. They are one channel of vitality.' He said this six hundred years before Christ. It is true once again, perhaps more true than ever. from White Noise TooleJohnKennedy 2012 https://bradleysaul.us/bcs-002V/ bcs-002V /bcs-002V/ Any connection between American art and American nature is purely coincidental quote Any connection between American art and American nature is purely coincidental, but this only because the nation as a whole has no contact with reality. That is only one of the reasons why I have always been forced to exist on the fringes of society, consigned to the Limbo reserved for those who do know reality when they see it. from A Confederacy of Dunces DelilloDon 2012 https://bradleysaul.us/bcs-002W/ bcs-002W /bcs-002W/ Tibetans try to see death for what it is quote Tibetans try to see death for what it is. It is the end of attachment to things. The simple truth is hard to fathom. But once we stop denying death, we can proceed calmly to die and then go on to experience uterine rebirth or Judeo-Christian afterlife or out-of-body experience or a trip on a UFO or whatever we wish to call it. We can do so with clear vision, without awe or teror. We don't have to cling to life artificially, or to death for that matter. We simply walk toward the sliding doors. Waves and radiation. Look how well-lighted everything is. The place is sealed off, self-contained. It is timeless. Another reason why I think of Tibet. Dying is an art in Tibet. A priest walks in, sits down, tells the weeping relatives to get out and has the room sealed. Doors, windows sealed. He has serious business to see to. Chants, numerology, horoscopes, recitation. Here we don't die, we shop. But the difference is less marked than you think. from White Noise DelilloDon 2012 1 1 https://bradleysaul.us/bcs-002U/ bcs-002U /bcs-002U/ When he switched from English to German, it was as though a cord had been twisted in his larynx quote When he switched from English to German, it was as though a cord had been twisted in his larynx. An abrupt emotion entered his voice, a scrape and gargle that sounded like the stirring of some beast's ambition. He gaped at me and gestured, he croaked, he verged on strangulation. Sounds came spewing from the base of his tongue, harsh noises damp with passion. He was only demonstrating certain basic pronunciation patterns but the transformation in his face and voice made me think he was making a passage between levels of being. from White Noise WallaceDavidFoster 2011 12 20 https://bradleysaul.us/bcs-003C/ bcs-003C /bcs-003C/ Cantorian tennis quote Were he still now among the living, Dr. Incandenza would describe tennis in the paradoxical terms of what’s now called 'Extra-Linear Dynamics'. And Schitt, whose knowledge of formal math is probably about equivalent to that of a Taiwanese kindergartner, nevertheless seemed to know what Hopman and van der Meer and Bollettieri seemed not to know: that locating beauty and art and magic and improvement and keys to excellence and victory in the prolix flux of match play is not a fractal matter of reducing chaos to pattern. Seemed intuitively to sense that it was a matter not of reduction at all, but perversely — of expansion, the aleatory flutter of uncontrolled, metastatic growth — each well-shot ball admitting of n possible responses, 2^n possible responses to those responses, and on into what Incandenza would articulate to anyone who shared both his backgrounds as a Cantorian continuum of infinities of possible move and response, Cantorian and beautiful because infoliating, contained, and diagnate infinity of infinities of choice and execution, mathematically uncontrolled but humanly contained, bounded by the talent and imagination of self and opponent, bent in on itself by the containing boundaries of skill and imagination that brought one player finally down, that kept both from winning, that made it, finally, a game, these boundaries of self. from Infinite Jest HolmesRichard 2011 6 22 https://bradleysaul.us/bcs-003A/ bcs-003A /bcs-003A/ ColeridgeSamuelTaylor's thoughts on science quote [Samuel Taylor Coleridge] thought that science, a human activity, 'being necessarily performed with the passion of Hope, it was poetical'. Science, like poetry, was not merely 'progressive'. It directed a particular kind of moral energy and imaginative longing into the future. It enshrined the implicit belief that mankind could achieve a better, happier world. from The age of wonder NorvigPeter 2011 5 30 https://bradleysaul.us/bcs-0038/ bcs-0038 /bcs-0038/ [NorvigPeter] on ChomskyNoam's ideas on statistical language models quote From On Chomsky and the Two Cultures of Statistical Learning Chomsky has a philosophy based on the idea that we should focus on the deep whys and that mere explanations of reality don’t matter. In this, Chomsky is in complete agreement with O’Reilly. (I recognize that the previous sentence would have an extremely low probability in a probabilistic model trained on a newspaper or TV corpus.) Chomsky believes a theory of language should be simple and understandable, like a linear regression model where we know the underlying process is a straight line, and all we have to do is estimate the slope and intercept. PaulosJohnAllen 2011 4 25 https://bradleysaul.us/bcs-0035/ bcs-0035 /bcs-0035/ Suspend belief when evaluating statistics quote In listening to stories we tend to suspend disbelief in order to be entertained, whereas in evaluating statistics we generally have an opposite inclination to suspend belief in order not to be beguiled. Seen on NYTimes.com EdmondsJeff 2011 4 5 https://bradleysaul.us/bcs-0036/ bcs-0036 /bcs-0036/ On the paradox of winning quote ...although no society in human history has ever extolled to such a high extent the value of the competitive spirit and individual effort, no society has ever been so filled with human beings that seem unable to give any effort at all. The complete post on The Logic of Long Distance is worth a read. EwartTheodore 2010 6 23 https://bradleysaul.us/bcs-002C/ bcs-002C /bcs-002C/ Cycling for the insane quote For most of us the exquisite loveliness and delight of a fine summer’s day have a special charm. The very life is luxury. The air is full of sound and sunshine, of the song of birds, and the murmur of insects; the meadows gleam with golden buttercups, we almost fancy we can see the grass grow and the buds open; the bees hum for very joy; there are a thou sand scents, above all, perhaps, that of new-mown hay. There are doubtless many patients before whom "all the glories of heaven and earth may pass in daily succession without touching their hearts or elevating their minds," but, in time, it is possible even these would, by means of cycling, have their love of Nature, which had been frozen or crushed out, restored. Thus all Nature, which is full of beauties, would not only be a never-failing source of pleasure and interest, but lift them above the petty troubles and sorrows of their daily life. from Cycling for the Insane Bradley Saul 2024 1 9 https://bradleysaul.us/bcs-0012/ bcs-0012 /bcs-0012/ Writing Bradley Saul 2012 7 25 https://bradleysaul.us/bcs-003K/ bcs-003K /bcs-003K/ Upon These Dead Roads poem I climb Harrison Grade for the last time, seeing the illegal roadside dump, the steep embankment, topography’s spin cycle for economy’s toughest stains. I have a laundry list, wet and folded, in my jersey pocket. Things to do today are tomorrow’s tire tracks, as you can ruin a landscape with nostalgia. At the intersection of Morelli, I stray from my lucid route. Down a forgotten road, across East Austin Creek, past where men once prayed to get out now they pray to go in. Hold on for prayer on the fall to Highway One to the beach at last. Feet numb, I dive into the nuclear ocean trailing the dreams of centuries of young men: Go West. I rise to the surface in the town of Graton. Pass the house where I met my wife. The roads around here are dead, full of potholes. Finally east on Occidental, the sun at my back, faded paint on the pavement: So much upon these dead roads I have written. So much alive these roads have written me. Bradley Saul 2012 3 23 https://bradleysaul.us/bcs-003F/ bcs-003F /bcs-003F/ Thursday at Eno River poem A poem roots, and I look for a rock to rest my head on something difficult, a trunk for flight-weary legs. I sit finally on the bank, with a sense of finality, making a lasting impression on the wet earth whose fecundity paints violet trail blazes. The violence of the river to my left I find comforting. Who am I to set a leaf adrift? I dare not look this upturned tree in the shimmering mirror. Rotors beat the sky hard. I can’t help to turn away and see the process — rock, roots, death and beetles — wait comfortably for the next flood. Bradley Saul 2012 1 21 https://bradleysaul.us/bcs-003D/ bcs-003D /bcs-003D/ Instinct poem I died surviving wasted on needles and berries, sap-filled cavities. My bow unbroken, yet untrue. Its arrows lost to the sky. The dented shadows waited still. This instinct to kill awoke me from the woman’s embrace, forced the pen in my hand, so I began to write, and so undoing, learned to live. Bradley Saul 2011 8 11 https://bradleysaul.us/bcs-003B/ bcs-003B /bcs-003B/ Briarpatch poem I imagine there are two types of people in the world. There are those who, on going blackberry picking for the first time, decide that they will never buy another blackberry. Then there are those who commit to paying whatever price necessary to feed their fresh berry habit without getting their legs mangled. I think poets, accustomed to having their hands dirty with juicy ink and simultaneously tasting tart words and sweet idioms pick the former group. Seamus wrote of blackberry picking. Plath called it of blackberrying. Hass goes so far as to say, "there is in this world no one thing / to which the bramble of blackberry corresponds, / a word is elegy to what it signifies." I wouldn’t mind looking at the blackberries and other fruits that Langston Hughes describes. Lorca, you Casanova you, you would never lay a lady down in a blackberry patch, only next to one. I count myself in good company, then, in writing a poem about blackberries. Two kittens scurry off trail into a blackberry bush. Stuck. in. the. present. moment. Feral now. Lynx posed to catch a rabbit in a diorama. Wildness past and preserved. Rabid for knowledge the student went to the master everyday. He studied. He learned. He saw two kittens scurry off trail into a blackberry bush. He became enlightened. Bradley Saul 2011 6 26 https://bradleysaul.us/bcs-0039/ bcs-0039 /bcs-0039/ Conversation Between a Grasshopper and a Grasshopper Sparrow poem You flathead on the fence trilling such prattle. The last note of your tune has too much rattle. Young grasshopper, be more circumspect. I’m a bird, not a mere insect. A bird! said the grasshopper as he scurried away, Life is too melodic and sweet to be prey! Oy vey, thought the sparrow, these hoppers are mad. I’m called a ‘life bird’, he should be glad. I wrote this after SwallowBill and his wife graciously took me birding when I attended SIBS. Bradley Saul 2011 6 11 https://bradleysaul.us/bcs-0037/ bcs-0037 /bcs-0037/ God’s death was the big bang poem God’s death was the big bang, Stars the embers of the pyre. Life has been mourning ever since. Rumi’s caravan spread across eons of time,  Picking up shreds of evidence of God’s existence. I listened in a room full of believers, ones who Tell us to revel in life. I want to tell them they are wrong, wrong, Wrong. The more we look, the smaller  We get. And we should be like this And cheer? If I’m nothing but fading warmth In dying ash, I at least want the strength To touch the next star and whisper something beautiful. Bradley Saul 2011 3 11 https://bradleysaul.us/bcs-002D/ bcs-002D /bcs-002D/ My health care plan for America poem Have the poets become doctors. Those Bards will know what to do with a diaeresis or epanalepsis.  They’ll alliterate the appendix with the rondelet, prescribe tropes and tropes of chthonic for a nasty limerick. They’ll scan meter  and brain matter, listening for  iambic pentameter through a stethoscope. O apostrophe,  they’ll say, you’ve had your odes, now is the time for surgery on your sonnets. They’ll ban  the cruel practice of vivisecting villanelles and no one will suffer of enjambment again! They’re cheap – anapaests  can be removed for a couplet  of bucks. The vaccine for Haiku  flu has no side effects and save for  an epic case, a poem is much  less paperwork. Irony can  finally be eradicated, though lord  save us if there’s an outbreak  of anacrusis. Call them quacks, call them ryhmesters, but the public loves the option of a heart crushing ballad or bone setting verse. Bradley Saul 2009 12 29 https://bradleysaul.us/bcs-002B/ bcs-002B /bcs-002B/ A sample of my universe poem I followed the recipe exactly: about 10 pounds of matter in a small space and gravity. The big bang was not as climatic as the books tell us. You cannot sense my universe. See it. Hear it. Even feel it brushing your skin. You can measure its ever expanding creation with highly sensitive instruments. Tools that study stars. After your hands clap, or watching all the leaves fall from the bristlecone tree, my waves merge with your waves Ones being one being one being one. Bradley Saul 2009 12 26 https://bradleysaul.us/bcs-002A/ bcs-002A /bcs-002A/ America was made before I was born poem I planned to write a poem about Carl Sandburg, Jesse Helms, being American. They may have made a National Park from a socialist’s home, and I imagine myself walking amongst the goat herd of Connemara, not drinking a drop of milk. Bradley Saul 2009 11 27 https://bradleysaul.us/bcs-0011/ bcs-0011 /bcs-0011/ People on Bicycles fiction People on bicycles usually pass by this time of day. The clicking sprockets are unusually quiet today. I peek up to the cloudy sky. The darkening clouds to the north say rain all day. One bicyclist wurrs by. Her rain jacket flapping loudly. The roads are dry. A cruel March frost sacrificed the daffodils, as the warm sun of February tempted them out early. I look at them drooping over the edge of my white window box. The paint flaking in large chunks. I think to myself, "I should take better care of this place." I chose this spot to watch the bicycles go by and to write. I thought moving to a small town would help me write. I've been here for five years. I haven't seen or heard from my landlord in 4 years. I wonder if I should keep writing checks. References Context Backlinks Related Contributions