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the #math book I was waiting 4 & how concept of curvature should be introduced quickly, bought it 4 ~20 GBP @ @heffersbookshop, will study it pseudo-religiously on a daily basis or as a humanist scholar but will benefit from it as the scientist/biologist I am :) thx @wtgowers&co
@heffersbookshop @wtgowers already had exposure to some articles in this book, a true companion, when using Safari Books Online as an @emblebi-er
@heffersbookshop @wtgowers @emblebi Day #2 studying the Princeton Companion to Mathematics (PCM from now on) Dynamical Systems and Chaos entry made me think about systems w/continuous and discrete consecutive phases/descriptions mixed & read Leibniz article, L constructed 1st four-function calculating machine
@heffersbookshop @wtgowers @emblebi Day #3 studying PCM VII.2 Mathematical Biology/Correlation & Causality 'It is these networks, in which proteins control genes & gene control proteins, that carry out & control specific cellular functions' why do we still call these 'gene networks' then? DNA is not the director
@heffersbookshop @wtgowers @emblebi Day #3 studying PCM p1004 Bela Bollobas' tautological hint: 'you should choose problems you find it difficult not to think about' to me THE problem couldn't help but think a lot on are cell lineage trees & aging, peerj.com/preprints/2782… discussed it w/Bollobas @ @TrinCollCam :)
@heffersbookshop @wtgowers @emblebi @TrinCollCam Day #4 studying PCM p215 Graphs: Ramsey Theory type question on existence of n vertices 2 always contain k vertices all joined or disjoint #math my question was about 'and' not 'or' & check tree with 6 vertices, looks harder to find existence of such n
@heffersbookshop @wtgowers @emblebi @TrinCollCam Day #4 studying PCM p181 started III.17 Dimension, 5 different dimension concepts discussed, asked my 9 yo who wandered around in room what dimension is she thinks, said a dimension is a separate world, then we checked dimensionality of this object, got diff numbers back #math
Getting ready for my evening study, also prepping 4 2 upcoming talks, aperitif, main course, side dishes, dessert, quantitative biology, math, causal models, political philosophy & continental light reading
Day #5 studying PCM 1.2 p9-13 Four basic concepts: sets, functions, relations, binary operators, technically latter are functions, sets are static, functions are dynamic, loved 2 alternative definitions of equivalence relations 3 well-known properties R/S/T being just DEF2 #math
Day #5 studying PCM p74-5 2 types of definitions: 1. shorthands & 2. knowledge extensions eg. derivative of a function helps you find smallest value of f(x), liked distinction definition- and theorem-oriented mathematicians, finding a good def is as dynamic as proving a theorem
Day #6 studying PCM main branches of #math p3-8, 3rd view of combinatorics concerned with mathematical structures that have 'few constraints', curious to see later how combinatorics can be combined with analysis, after all 'few constraints' might still need 'limiting processes'
Day #7 studying PCM p15 handling of free & bound variables left me bit unsatisfied esp. latter. Tension between the term 'free' and a free variable denoting a targeted/stated/examined special object and tension between 'bound' and a bound variable not denoting a specific object
Day #7 studying PCM p16 Levels of formality. fun to realise deep connection between induction principle and ordering, so deep there's equivalence (each can be used to prove the other) between mathematical induction & well-ordering principle, reduced # of placeholders in my head
Day #8 studying PCM p841-2 Geometry & Topology of Macromolecules 1. p841 molecular dynamics, conceptual, not technical-looking problem is that the 'choice of energy terms is speculative' 2. p842 cool quick intro into modelling geometry of binding regions w/Voronoi surfaces #math
Day #9 studying PCM p18 Complex numbers: had problem when i is treated as a 'variable' in complex addition/multiplication & i2 turns -1, what kind of variable is that? 'free' but it can take only 1 value, 'bound' but it does denote specific object? Also no diagrams?
Day #9 studying PCM p19-20 Groups (see pic in previous tweet) did all the group example symmetry transformations through, abstract algebra is a non-intuitive mixture of statics and movement, probably will get used to it by the end of the book
Day 10 studying THE book, PCM, the most expected intellectual treasure of the day, this time in the bath
Day 10, studying PCM p20-2 Fields, vector spaces, p30-1 Limits, it’s the ultimate joke of mathematical analysis that this ‘loose’ idea of specifying mathematical objects indirectly w/better&better approximations, w/limit processes turned into most rigorous & hard to grasp field
Day #11 studying PCM Continuity p32-3, the most beautiful concept & definition of the book so far and the way continuous functions are generalised through replacing differences by distances and then by limits of convergent sequences ...
or the way non-functional concepts of sequence and limits turn into functional concepts by applying same 'getting close, staying close' rule to both values in domain and co-domain of a function, double indemnity Day #11 #studyingPCM Continuity p32-3
Day #12 #studyingPCM p725-6 Robertson-Seymour theorem and graphs well-quasi-ordered by their minors & 'nearly embeddable' into fixed surfaces, deep connection also between minors and chromatic numbers, see Hadwiger-conjecture en.wikipedia.org/wiki/Hadwiger_…
Day #13 #studyingPCM p22-3 rings, creating new structures out of old ones & substructures, fascinating how a substructure can be more interesting than a new one built from scratch, must have relevant closure properties if it’s an algebraic one
Day #14 #studyingPCM p23-24 products of algebraic structures: must admit some examples of simpler & complicated products of groups/fields/vector spaces r a bit too free floating now, p77-8 Numbers in early math, learned about Mesopotamian sexagesimal sys from where 60 mins came
Day #15 #studyingPCM p24-6 Quotients, left me w/lot of questions, structures defined w/help of very generous equivalence relations, arbitrary looking
analogies eg simple groups as prime numbers of algebraic structures but aids in defining topological/geometrical objects
Day #17 #studyingPCM p27-8 Homo/iso/automorphisms 1. philosophy: isomorphic: structure everything, participating objects nothing 2. automorphisms of a structure X form a group 3. no algebraic diff between +/- square roots 4. useful, but 0 ambition to become an abstract algebraist
Day #18 #studyingPCM p28-32 Linear Maps & Matrices, integration example w/3 levels of complexity 1. real numbers 2. functions from real-to-real 3 higher order function & liked 'very close analogy' between 2 variable u(x,y) & matrix, which can be thought as double integer function
Day #19 #studyingPCM p30 Eigenvalues and Eigenvectors or everything what 1 wants to know about linear maps or pagerank, no, not really
Day #20 #studyingPCM p928-30, 934-5 Analysis, Mathematical & Philosophical, hilariously biased & funny piece by John P. Burgess of
@Princeton, see starting sentence's little metaphilosophy also the term 'continentalising philospohasters', philo analysis as data prep stage
@Princeton Day #20 #studyingPCM p930 Burgess mentions on how solving Millenium problems wouldn't have impact on analytical philosophers, in this context could have mentioned Hilary Putnam's role in solving Hilbert's 10th problem referenced in PCM p50 cc @wtgowers en.wikipedia.org/wiki/Hilary_Pu…
@Princeton @wtgowers Day #21 #studyingPCM 33-4 Differentiation, please see 2 versions of limits defined w/formulas @ the bottom, the well-known delta & a sequence vs a fraction of epsilon @ diff/diff & book says latter is slightly more general notion of limit than former, why? Simpler, less symbols?
@Princeton @wtgowers Day #22 #studyingPCM p34-5 Partial Differential Equations heat equation example shows how lucky are physicists with linear approximations turning out to be arithmetic means and second derivatives offering comparisons between a value at x and average values in the neighbourhood
@Princeton @wtgowers Day #23 #studyingPCM p35-7 Integration demoing how integration is not simple antidifferentiation, eg approximating rock mass from density w/cubioid via Riemann is cool, diff good 4 critical values, integration 2 give typical from critical, what gives critical from typical values?
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