Here is a detailed map of major neuroscience subfields: Neuroscience ├── Cellular and Molecular Neuroscience │ ├── Neurogenetics │ ├── Neuronal cell biology │ ├── Neurochemistry │ ├── Molecular neurobiology │ └── Neuropharmacology ├── Developmental Neuroscience │ ├── Neural stem cells and neurogenesis │ ├── Axon guidance and synaptogenesis │ ├── Neuronal migration and differentiation │ └── Neurodevelopmental disorders ├── Systems Neuroscience │ ├── Sensory systems │ │ ├── Visual neuroscience │ │ ├── Auditory neuroscience │ │ ├── Somatosensory neuroscience │ │ ├── Olfactory neuroscience │ │ └── Gustatory neuroscience │ ├── Motor systems │ │ ├── Motor control │ │ ├── Basal ganglia │ │ └── Cerebellum │ ├── Autonomic nervous system │ ├── Sleep neuroscience │ ├── Circadian rhythms │ └── Neuroendocrinology ├── Cognitive Neuroscience │ ├── Attention │ ├── Perception │ ├── Learning and memory │ ├── Language │ ├── Decision-making │ ├── Emotion │ ├── Social neuroscience │ └── Consciousness ├── Behavioral Neuroscience │ ├── Motivation and reward │ ├── Addiction neuroscience │ ├── Fear and anxiety │ ├── Aggression │ └── Sexual behavior ├── Computational Neuroscience │ ├── Neural networks │ ├── Neural coding and decoding │ ├── Neuroinformatics │ ├── Brain-machine interfaces │ └── Modeling and simulation ├── Translational Neuroscience │ ├── Neurology │ │ ├── Neurodegenerative diseases │ │ ├── Epilepsy │ │ ├── Stroke │ │ ├── Multiple sclerosis │ │ ├── Neuromuscular disorders │ │ └── Sleep disorders │ ├── Psychiatry │ │ ├── Mood disorders │ │ ├── Anxiety disorders │ │ ├── Schizophrenia │ │ ├── Autism spectrum disorders │ │ ├── Attention deficit hyperactivity disorder (ADHD) │ │ └── Substance use disorders │ ├── Neurorehabilitation │ └── Neuroengineering └── Methods in Neuroscience ├── Neuroimaging │ ├── Functional magnetic resonance imaging (fMRI) │ ├── Positron emission tomography (PET) │ ├── Electroencephalography (EEG) │ ├── Magnetoencephalography (MEG) │ └── Optical imaging ├── Electrophysiology │ ├── Patch-clamp recording │ ├── Extracellular recording │ ├── Optogenetics │ └── Chemogenetics ├── Neuroanatomy │ ├── Histology │ ├── Tract tracing │ └── Connectomics ├── Neuropharmacology │ ├── Receptor pharmacology │ ├── Drug discovery and development │ └── Toxicology └── Bioinformatics and data analysis ├── Genomics ├── Proteomics ├── Transcriptomics └── Neuroinformatics This map covers many of the major subfields and areas of research within neuroscience, but it is not exhaustive. Neuroscience is a highly interdisciplinary field that continues to grow and evolve, with new subfields and techniques emerging regularly. Generalizing mathematics and physics "to oblivion" can be an abstract and profound exercise. It involves taking fundamental concepts and principles and extending them to the most generalized forms possible. Here’s an attempt to do that: ### 1. **Mathematics: Generalized Abstractions** #### **Set Theory** - **Generalization**: The concept of a set can be extended to include all kinds of collections, including those that are not strictly defined by membership but by some other relation or structure. - **Example**: Instead of just considering sets of elements, consider classes, categories, and topoi, where the focus is on morphisms and relationships rather than just membership. #### **Algebra** - **Generalization**: Algebraic structures can be generalized from groups to groupoids, categories, and beyond. - **Example**: In category theory, objects and morphisms generalize algebraic structures and functions, leading to the notion of functors and natural transformations which generalize functions between sets and homomorphisms between algebraic structures. #### **Topology** - **Generalization**: Topological spaces can be generalized to locales, which focus on the lattice of open sets rather than points. - **Example**: Abstracting from the notion of a point-based topology to a frame-based approach where the emphasis is on the relationships between open sets. #### **Analysis** - **Generalization**: Measures and integrals can be generalized from Lebesgue measures to more abstract notions like Radon measures or even more general integration theories. - **Example**: Moving from real-valued functions to functions in abstract spaces, leading to the study of Banach spaces, Hilbert spaces, and more general normed vector spaces. ### 2. **Physics: Generalized Abstractions** #### **Classical Mechanics** - **Generalization**: From Newtonian mechanics to Lagrangian and Hamiltonian mechanics, which are formulated in terms of configuration spaces and phase spaces. - **Example**: Generalizing to symplectic manifolds and Hamiltonian systems, where the focus is on the geometry of the phase space and conservation laws. #### **Relativity** - **Generalization**: From special relativity to general relativity, where spacetime is modeled as a four-dimensional pseudo-Riemannian manifold. - **Example**: Extending this to include more general geometries and topologies of spacetime, such as those considered in theories of quantum gravity. #### **Quantum Mechanics** - **Generalization**: From non-relativistic quantum mechanics described by Hilbert spaces to quantum field theory and beyond. - **Example**: Generalizing to algebraic quantum field theory, where fields are operator-valued distributions, and to quantum information theory, which abstracts the principles of quantum mechanics to information processing. #### **Thermodynamics and Statistical Mechanics** - **Generalization**: From classical thermodynamics to statistical mechanics, which explains thermodynamic behavior in terms of statistical properties of microscopic states. - **Example**: Generalizing further to non-equilibrium statistical mechanics and complex systems theory, which deals with systems far from equilibrium and the emergence of complex behavior. ### 3. **General Principles** #### **Dualities** - **Generalization**: The concept of duality (like wave-particle duality, Fourier duality, etc.) can be extended to more abstract forms like category-theoretic dualities, which reveal deep symmetries and correspondences between seemingly different mathematical structures. - **Example**: Pontryagin duality, which relates topological groups to their dual groups, or T-duality in string theory, which relates different string theories. #### **Symmetries and Invariances** - **Generalization**: From specific symmetries (rotational, translational, etc.) to the abstract notion of symmetry groups and group actions. - **Example**: Noether's theorem, which generalizes the relationship between symmetries and conservation laws in physics, can be extended to more abstract settings like gauge theories and fiber bundles. #### **Dimensional Analysis** - **Generalization**: From the specific dimensions (length, time, mass, etc.) to abstract dimensional analysis in various contexts, including fractals and scaling laws. - **Example**: Extending the concept of dimensions to infinite-dimensional spaces, such as function spaces in functional analysis, or to fractal dimensions in chaos theory. In summary, generalizing everything to oblivion means abstracting concepts to their most fundamental forms, often leading to higher levels of abstraction in both mathematics and physics. This process reveals deeper connections and unifying principles underlying various disciplines. Here is my attempt at taking the math and physics and generalizing it to the extreme: At the most fundamental level, the universe can be described by abstract mathematical structures and principles. All of physics emerges from a basic set of equations and symmetries. The core pillars are: - Geometry - The shape and structure of spacetime itself, described by Riemannian geometry and topology. This generalizes to higher dimensional spaces and exotic geometric objects. - Algebra - The rules for abstract mathematical operations, generalizing from real and complex numbers to quaternions, octonions, and beyond to arbitrary algebraic structures. - Analysis - The mathematics of change, limits, derivatives, and integrals, extending from real functions to complex analysis, functional analysis, and the abstract realms of measure theory and nonstandard analysis. - Probability - The science of randomness and uncertainty, generalizing from discrete probability to continuous distributions, stochastic processes, and the abstract foundations of measure-theoretic probability. All of physics can be formulated in terms of these core branches of pure mathematics: - Classical mechanics emerges from variational principles on configuration spaces. - Quantum mechanics arises from linear algebra and functional analysis on abstract Hilbert spaces. - Relativity theory comes from differential geometry and tensor analysis on Lorentzian manifolds. - Statistical mechanics and thermodynamics arise from probability theory and the ergodic hypothesis. - Quantum field theory unifies quantum mechanics, special relativity, and the gauge principle into an abstract algebraic and categorical framework. Even speculative theories aiming to unify physics, like string theory, loop quantum gravity, and causal sets, rely on generalized notions like higher-dimensional Calabi-Yau manifolds, spin foams, and causal order. Abstracting further, we arrive at mathematical universes described by formal systems, type theory, category theory, and mathematical logic. The universe is just an instance of an abstract mathematical structure satisfying certain axioms and equations. Even the notion of a "theory of everything" is generalized to the ultimate ensemble of all mathematical possibility. In this ultimate Platonic paradigm, physics has been reduced to its most general and abstract essence - pure mathematics itself, unfettered by the limitations of manifest reality. Everything is subsumed into an all-encompassing Mathematical Universe. Existence is mathematics, mathematics is existence. Of course, this is taking the generalization to a rather extreme speculative level! But it illustrates how the mathematical foundations of physics can be abstracted and generalized very far from the concrete theories that describe our observable universe. The true nature of reality very well may be far more abstract and mathematical than our current understanding.