Before you can build aby scientific mathematical models, we have to pick a mathematical foundation ==Mathematics -> Foundations -> Logic== Logic studies potential consistent rules. ==Mathematics -> Foundations -> Logic -> Classical logic== [[Classical logic - Wikipedia](https://en.wikipedia.org/wiki/Classical_logic) Classical logic - Wikipedia] ==Mathematics -> Foundations -> Logic -> Constructivist intuistionic homotopy type theory== From pure mathematics perspective: [[Modern Foundations of Mathematics - YouTube](https://youtube.com/playlist?list=PLCTMeyjMKRkqTM2-9HXH81tvpdROs-nz3&si=yIoxuDW8wJHqAaUs) Modern Foundations of Mathematics Lectures by Richard Southwell on Youtube] Animated 7 minute introduction: [[Homotopy Type Theory - An Introduction to Topology, Formal Logic, and HoTT #SoME3 - YouTube](https://youtu.be/wDxDtWYjmgA?si=J3e49ATrmpvynsOm) Homotopy Type Theory - An Introduction to Topology, Formal Logic, and HoTT #SoME3 - YouTube] 1lab is an online reference maintained by Amélia Liao primarily for category theory done in univalent cubical type theory implemented in cubical Agda: [[1lab in nLab](https://ncatlab.org/nlab/show/1lab) 1lab in nLab] ==Mathematics -> Foundations -> Set theory== One of the first succesfull attempts at rigorou foundations of mathematics was naive set theory, but it included paradoxes like the Russel's paradox: [[Naive set theory - Wikipedia](https://en.wikipedia.org/wiki/Naive_set_theory) Naive set theory Wikipedia] [[Russell's paradox - Wikipedia](https://en.wikipedia.org/wiki/Russell%27s_paradox) Russell's paradox Wikipedia] This was fixed by introducing axiomatic restrictions that disallowed such paradoxical constructs: [[Zermelo–Fraenkel set theory - Wikipedia]([Zermelo–Fraenkel set theory - Wikipedia]([Zermelo–Fraenkel set theory - Wikipedia](https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory)))?wprov=sfla1 Zermelo–Fraenkel set theory - Wikipedia] ==Mathematics -> Foundations -> Category theory== Category theory defines mathematical objects by their relationships to all other objects, while set theory defines their internal structure instead. [[Category Theory For Beginners - YouTube](https://youtube.com/playlist?list=PLCTMeyjMKRkoS699U0OJ3ymr3r01sI08l&si=WSGxG8BvMTr8zRWY) Category Theory For Beginners by Richard Southwell on Youtube] [[∞-Category Theory for Undergraduates - YouTube](https://youtu.be/A6hXn6QCu0k?si=URkY8G3W2_3obeWL) ∞-Category Theory for Undergraduates - YouTube] [[Category Theory 1.1: Motivation and Philosophy - YouTube](https://www.youtube.com/watch?v=I8LbkfSSR58) Category Theory for programmers by Bartosz Milewski on Youtube] ==Mathematics -> Paradoxes== Godel's incompleteness theorems, cantor's theorem, russel's paradox, halting problem, Y combinator, liar paradox expressed as general diagonal argument in category theory: [[What A General Diagonal Argument Looks Like (Category Theory) - YouTube](https://www.youtube.com/watch?v=dwNxVpbEVcc) What A General Diagonal Argument Looks Like (Category Theory)] ==Mathematics -> Fields== [[Mathematics of theoretical physics - Wikiversity](https://en.wikiversity.org/wiki/Mathematics_of_theoretical_physics) Mathematics of theoretical physics - Wikiversity] ==Mathematics -> Fields -> Algebra== [[Algebra - Wikipedia]([Algebra - Wikipedia](https://en.wikipedia.org/wiki/Algebra)) Algebra wikipedia] ==Mathematics -> Fields -> Geometry== ==Mathematics -> Fields -> Analysis== ==Mathematics -> Fields -> Number theory == ==Science== Philosophy can create arbitrary linguistic constructs. Mathematics constrains it by formalScience studies reality. ==Science -> Physics== Physics is the natural science of matter, involving the study of matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. Physics studies the dynamics of things Highlevel nonmathematical intuitive overview of physics subdisciplines in 10 minute video: [[The Map of Physics - YouTube]([The Map of Physics - YouTube](https://www.youtube.com/watch?v=ZihywtixUYo)) The Map of Physics by Domain of Science on Youtube] Wikipedia's neverending physics tree rabbithole: [[Outline of physics - Wikipedia]([Outline of physics - Wikipedia]([Outline of physics - Wikipedia](https://en.wikipedia.org/wiki/Outline_of_physics))) Outline of physics - Wikipedia] The best lectures on all of basic physics from the creator and leader of String theory Leonard Susskind. You don't have to be fan of string theory, the way he teaches everything else in physics holistically is magical!: [[Course Catalogue | The Theoretical Minimum]([Course Catalogue | The Theoretical Minimum]([Course Catalogue | The Theoretical Minimum]([Course Catalogue | The Theoretical Minimum](https://theoreticalminimum.com/courses)))) The Theoretical Minimum: What You Need to Know to Start Doing Physics Book by Leonard Susskind] [[Course Catalogue | The Theoretical Minimum]([Course Catalogue | The Theoretical Minimum]([Course Catalogue | The Theoretical Minimum]([Course Catalogue | The Theoretical Minimum](https://theoreticalminimum.com/courses)))) The Theoretical Minimum Lectures by Leonard Susskind] [[Find Courses by Topic | MIT OpenCourseWare | Free Online Course Materials](https://ocw-origin.odl.mit.edu/courses/find-by-topic/#cat=science&subcat=physics) Physics lectures at MITOpenCourseware] [[Imgur: The magic of the Internet](https://imgur.com/OMqtBWR) List of physics fields, their subfields, major theories and concepts] [[Phenomenology (physics) - Wikipedia]([Phenomenology (physics) - Wikipedia](https://en.wikipedia.org/wiki/Phenomenology_(physics))) Phenomenology (physics) - Wikipedia] [[physics in nLab](https://ncatlab.org/nlab/show/physics) physics in nLab] [[higher category theory and physics in nLab](https://ncatlab.org/nlab/show/higher+category+theory+and+physics) higher category theory and physics in nLab] [[Mathematical physics - Wikipedia]([Mathematical physics - Wikipedia]([Mathematical physics - Wikipedia](https://en.wikipedia.org/wiki/Mathematical_physics))) Mathematical physics - Wikipedia] [[List of unsolved problems in physics - Wikipedia]([List of unsolved problems in physics - Wikipedia](https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_physics))?wprov=sfla1 List of unsolved problems in physics Wikipedia] Best map of how various theories of veerything solve various problems in physics: [[Quanta Magazine]([Quanta Magazine]([Quanta Magazine]([Quanta Magazine]([Quanta Magazine]([Quanta Magazine]([Quanta Magazine]([Quanta Magazine](https://www.quantamagazine.org/theories-of-everything-mapped-20150803/)))))))) Quanta Magazine] [[Theoretical physics - Wikipedia]([Theoretical physics - Wikipedia]([Theoretical physics - Wikipedia]([Theoretical physics - Wikipedia](https://en.wikipedia.org/wiki/Theoretical_physics)))) Theoretical physics - Wikipedia] [[Encyclopedia:Physics - Scholarpedia](http://www.scholarpedia.org/article/Encyclopedia:Physics) Encyclopedia:Physics - Scholarpedia] [[Episode 45: Leonard Susskind on Quantum Information, Quantum Gravity, and Holography - YouTube]([Episode 45: Leonard Susskind on Quantum Information, Quantum Gravity, and Holography - YouTube]([Episode 45: Leonard Susskind on Quantum Information, Quantum Gravity, and Holography - YouTube]([Episode 45: Leonard Susskind on Quantum Information, Quantum Gravity, and Holography - YouTube]([Episode 45: Leonard Susskind on Quantum Information, Quantum Gravity, and Holography - YouTube](https://youtu.be/XTSdPSOcdjI))))) Episode 45: Leonard Susskind on Quantum Information, Quantum Gravity, and Holography - YouTube] [[Sabine Hossenfelder: Superdeterminism & Geometric Unity - YouTube]([Sabine Hossenfelder: Superdeterminism & Geometric Unity - YouTube](https://www.youtube.com/watch?v=walaNM7KiYA)) Sabine Hossenfelder: Superdeterminism & Geometric Unity - YouTube] Brilliant intuitive visualizations: [[General Physics - YouTube](https://www.youtube.com/playlist?list=PLu7cY2CPiRjVSC3dnb4EcPLZoeInpvTCb) General Physics by ScienceClic English on Youtube] [[Quantum World - YouTube](https://www.youtube.com/playlist?list=PLu7cY2CPiRjUa8ykLHubgucgm5EYonVAe) Quantum Physics by ScienceClic English on Youtube] [[Bevor Sie zur Google Suche weitergehen](https://www.google.com/search?q=best+resources+for+physics+site:www.reddit.com&sca_esv=569007408&sxsrf=AM9HkKmQ50BjANGGtxXCmjPE-Oci-sKzIA:1695864004598&sa=X&ved=2ahUKEwiNgdPckcyBAxWYSvEDHZlbCwEQrQIoBHoECBIQBQ&biw=1745&bih=846&dpr=1.1#ip=1) Reddit asking for best physics resources] [Theoretical physics - Wikipedia]([Theoretical physics - Wikipedia]([Theoretical physics - Wikipedia]([Theoretical physics - Wikipedia](https://en.wikipedia.org/wiki/Theoretical_physics)))) [Outline of physics - Wikipedia]([Outline of physics - Wikipedia]([Outline of physics - Wikipedia](https://en.wikipedia.org/wiki/Outline_of_physics))) ==Science -> Physics -> Mechanics== Mechanics is concerned with the relationships between force, matter, and motion among physical objects. [Mechanics - Wikipedia](https://en.wikipedia.org/wiki/Mechanics)#Sub-disciplines ==Science -> Physics -> Classical -> Classical mechanics== Classical mechanics is a physical theory describing the motion of macroscopic objects, from apples falling down to projectiles to parts of machinery and astronomical objects, such as spacecraft, planets, stars, and galaxies. The earliest formulation of classical mechanics is often referred to as Newtonian mechanics with its equations of motion. [[Course Catalogue | The Theoretical Minimum]([Course Catalogue | The Theoretical Minimum]([Course Catalogue | The Theoretical Minimum]([Course Catalogue | The Theoretical Minimum](https://theoreticalminimum.com/courses))))/classical-mechanics/2011/fall Classical Mechanics (Fall, 2011) | The Theoretical Minimum] [[Course Catalogue | The Theoretical Minimum]([Course Catalogue | The Theoretical Minimum]([Course Catalogue | The Theoretical Minimum]([Course Catalogue | The Theoretical Minimum](https://theoreticalminimum.com/courses))))/special-relativity-and-electrodynamics/2012/spring Special Relativity and Electrodynamics (Spring, 2012) | The Theoretical Minimum] [[classical mechanics in nLab](https://ncatlab.org/nlab/show/classical+mechanics) classical mechanics in nLab] [[Classical mechanics - Wikipedia](https://en.wikipedia.org/wiki/Classical_mechanics) Classical mechanics - Wikipedia] ==Science -> Physics -> Classical -> Classical mechanics -> Intuitive description== ==Science -> Physics -> Classical -> Classical mechanics -> Mathematical description== But instead I'll be using Zermelo–Fraenkel set theory foundations. [Zermelo–Fraenkel set theory - Wikipedia]([Zermelo–Fraenkel set theory - Wikipedia]([Zermelo–Fraenkel set theory - Wikipedia](https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory))) [Analytic geometry - Wikipedia](https://en.wikipedia.org/wiki/Analytic_geometry)#Cartesian_coordinates_(in_a_plane_or_space) [Cartesian coordinate system - Wikipedia](https://en.wikipedia.org/wiki/Cartesian_coordinate_system) [Principle of Least Action — an Introduction | by Yash | Quantaphy | Medium](https://medium.com/quantaphy/the-principle-of-least-action-an-exposition-7d60fe554422) The average kinetic energy less the average potential energy is as little as possible for the path of an object going from one point to another ==Science -> Physics -> Classical -> Classical mechanics -> Formulations -> Newtonian mechanics== ==Science -> Physics -> Classical -> Classical mechanics -> Formulations -> Newtonian mechanics -> Ḱinematics== ==Science -> Physics -> Classical -> Classical mechanics -> Formulations -> Analytical mechanics== [Analytical mechanics - Wikipedia](https://en.wikipedia.org/wiki/Analytical_mechanics) ==Science -> Physics -> Classical -> Classical mechanics -> Formulations -> Analytical mechanics -> Lagrangian mechanics== ==Science -> Physics -> Classical -> Classical mechanics -> Formulations -> Analytical mechanics -> Hamiltonian mechanics== ==Science -> Physics -> Classical -> Classical mechanics -> Subdisciplines -> Continuum mechanics== Branch of mechanics that deals with the deformation of and transmission of forces through materials modeled as a continuous medium (also called a continuum) rather than as discrete particles. [[Continuum mechanics - Wikipedia](https://en.wikipedia.org/wiki/Continuum_mechanics) Continuum mechanics wiki] ==Science -> Physics -> Classical -> Classical mechanics -> Subdisciplines -> Continuum mechanics -> Fluid mechanics== [[Fluid mechanics - Wikipedia](https://en.wikipedia.org/wiki/Fluid_mechanics) Fluid mechanics wiki] ==Science -> Physics -> Classical -> Classical mechanics -> Subdisciplines -> Continuum mechanics -> Newtonian fluids== [[Newtonian fluid - Wikipedia](https://en.wikipedia.org/wiki/Newtonian_fluid) Newtonian fluid wiki] ==Science -> Physics -> Classical -> Classical mechanics -> Subdisciplines -> Continuum mechanics -> Non-newtonian fluids== Do not undergo strain rates proportional to the applied shear stress. [[Non-Newtonian fluid - Wikipedia](https://en.wikipedia.org/wiki/Non-Newtonian_fluid) Non-Newtonian fluid wiki] ==Science -> Physics -> Classical -> Classical mechanics -> Subdisciplines -> Harmonic analysis== [[Harmonic analysis - Wikipedia]([Harmonic analysis - Wikipedia](https://en.wikipedia.org/wiki/Harmonic_analysis)) Harmonic analysis - Wikipedia] [[But what is the Fourier Transform? A visual introduction. - YouTube](https://www.youtube.com/watch?v=spUNpyF58BY) But what is the Fourier Transform? A visual introduction. - YouTube] ==Science -> Physics -> Classical -> Statistical mechanics -> Tools -> Annealing== Useful mathematical tool to model annealing that we will use later in neurophenomenology and social sciences. [[Annealing (materials science) - Wikipedia](https://en.wikipedia.org/wiki/Annealing_(materials_science)) Annealing (materials science) - Wikipedia] [[Simulated annealing - Wikipedia]([Simulated annealing - Wikipedia]([Simulated annealing - Wikipedia]([Simulated annealing - Wikipedia]([Simulated annealing - Wikipedia](https://en.wikipedia.org/wiki/Simulated_annealing))))) Simulated annealing - Wikipedia] ==Science -> Physics -> Classical -> Statistical mechanics== Statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Its applications include many problems in the fields of physics, biology, chemistry, and neuroscience. Its main purpose is to clarify the properties of matter in aggregate, in terms of physical laws governing atomic motion. [[Course Catalogue | The Theoretical Minimum]([Course Catalogue | The Theoretical Minimum]([Course Catalogue | The Theoretical Minimum]([Course Catalogue | The Theoretical Minimum](https://theoreticalminimum.com/courses))))/statistical-mechanics/2013/spring Statistical Mechanics (Spring, 2013) | The Theoretical Minimum] [[statistical mechanics in nLab](https://ncatlab.org/nlab/show/statistical+mechanics) statistical mechanics in nLab] [[Statistical mechanics - Wikipedia](https://en.wikipedia.org/wiki/Statistical_mechanics) Statistical mechanics - Wikipedia] ==Science -> Physics -> Classical -> Statistical mechanics -> Thermodynamics== Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. There is second law of thermodynamics, saying that any closed system will go towards entropy. It connects entropy with information. [[Thermodynamics - Wikipedia](https://en.wikipedia.org/wiki/Thermodynamics) Thermodynamics - Wikipedia] [[thermodynamics in nLab](https://ncatlab.org/nlab/show/thermodynamics) thermodynamics in nLab] ==Science -> Physics -> Classical -> Statistical mechanics -> Thermodynamics -> Nonequilibrium thermodynamics== Non-equilibrium thermodynamics is a branch of thermodynamics that deals with physical systems that are not in thermodynamic equilibrium and resisting the tendency towards entropy, such as biological organisms. [[Non-equilibrium thermodynamics - Wikipedia](https://en.wikipedia.org/wiki/Non-equilibrium_thermodynamics) Non-equilibrium thermodynamics - Wikipedia] ==Science -> Physics -> Classical -> Statistical mechanics -> Ising model == [[Ising model in nLab](https://ncatlab.org/nlab/show/Ising+model) Ising model in nLab] [[Ising model - Wikipedia](https://en.wikipedia.org/wiki/Ising_model) Ising model - Wikipedia] [[The hardest sum aka the Ising model #SoME3 - YouTube]([The hardest sum aka the Ising model #SoME3 - YouTube](https://www.youtube.com/watch?v=1WB4ovaNepg)) The hardest sum aka the Ising model #SoME3 - YouTube] ==Science -> Physics -> Classical -> Information theory== [[information theory in nLab](https://ncatlab.org/nlab/show/information+theory) information theory in nLab] [[Information theory - Wikipedia]([Information theory - Wikipedia](https://en.wikipedia.org/wiki/Information_theory)) Information theory - Wikipedia] ==Science -> Physics -> Classical -> Unity== ==Science -> Physics -> Classical -> Theory of relativity== Special relativity is a physic theory of the relationship between space and time, giving us time dilation and length contraction. General theory of relativity is the geometric theory of gravitation, where gravity is the curvature of spacetime. [[Course Catalogue | The Theoretical Minimum]([Course Catalogue | The Theoretical Minimum]([Course Catalogue | The Theoretical Minimum]([Course Catalogue | The Theoretical Minimum](https://theoreticalminimum.com/courses))))/special-relativity-and-electrodynamics/2012/spring Special Relativity and Electrodynamics (Spring, 2012) | The Theoretical Minimum] [[Course Catalogue | The Theoretical Minimum]([Course Catalogue | The Theoretical Minimum]([Course Catalogue | The Theoretical Minimum]([Course Catalogue | The Theoretical Minimum](https://theoreticalminimum.com/courses))))/general-relativity/2012/fall General Relativity (Fall, 2012) | The Theoretical Minimum] [[Course Catalogue | The Theoretical Minimum]([Course Catalogue | The Theoretical Minimum]([Course Catalogue | The Theoretical Minimum]([Course Catalogue | The Theoretical Minimum](https://theoreticalminimum.com/courses))))/relativity/2007/spring Relativity (Spring, 2007) | The Theoretical Minimum] [[special relativity in nLab](https://ncatlab.org/nlab/show/special+relativity) special relativity in nLab] [[general relativity in nLab](https://ncatlab.org/nlab/show/general+relativity) general relativity in nLab] [[Deriving Einstein's most famous equation: Why does energy = mass x speed of light squared? - YouTube]([Deriving Einstein's most famous equation: Why does energy = mass x speed of light squared? - YouTube]([Deriving Einstein's most famous equation: Why does energy = mass x speed of light squared? - YouTube](https://www.youtube.com/watch?v=KZ8G4VKoSpQ))) Deriving Einstein's most famous equation: Why does energy = mass x speed of light squared? - YouTube] [[The Maths of General Relativity - YouTube]([The Maths of General Relativity - YouTube]([The Maths of General Relativity - YouTube](https://www.youtube.com/playlist?list=PLu7cY2CPiRjVY-VaUZ69bXHZr5QslKbzo))) Bevor Sie zu YouTube weitergehen] ==Science -> Physics -> Classical -> Cosmology== The science of the origin and development of the universe. [[Course Catalogue | The Theoretical Minimum]([Course Catalogue | The Theoretical Minimum]([Course Catalogue | The Theoretical Minimum]([Course Catalogue | The Theoretical Minimum](https://theoreticalminimum.com/courses))))/cosmology/2013/winter Cosmology (Winter, 2013) | The Theoretical Minimum] [[Astrophysics and Cosmos - YouTube](https://www.youtube.com/playlist?list=PLu7cY2CPiRjXasFhJrMnjLNJl0PeNSvPU) Bevor Sie zu YouTube weitergehen] ==Science -> Physics -> Quantum==