Functions Are Vectors (2023)
Intuition vs Formalism
- One camp argues visualizations of abstract math (infinite dimensions, function spaces) ultimately mislead; better to think “linguistically” in terms of definitions and vector-space axioms, just manipulating symbols.
- Another camp finds axiomatic presentations unusable; they build intuition via physical metaphors (arrows, quantum states, local linearization) and see formal vector-space axioms as nearly irrelevant for understanding.
- Several suggest a synthesis: start from strong geometric/physical intuition, then let formal definitions correct and refine it; intuition alone can go wrong without rigor.
Are Functions Really Vectors? Scope and Rigor
- Many point out that real-valued functions form a vector space under pointwise addition and scalar multiplication, so in the abstract sense they are vectors.
- Critics say the article leans on the everyday “finite list of numbers” notion of vectors to sell a much harder idea: infinite-dimensional function spaces, uncountable index sets, and functional analysis.
- There’s debate over bases:
- Abstractly, any vector space (including ℝ→ℝ) has a basis if you assume the axiom of choice, but such bases are nonconstructive and not useful computationally.
- Practical “bases” like Fourier or eigenfunction families are Hilbert bases requiring infinite sums; they are not bases in the finite-linear-combination sense.
Infinite-Dimensional Structure and Hilbert Spaces
- Discussion of Hilbert spaces, inner products, and Cauchy–Schwarz for functions; note that not all function spaces are Hilbert spaces.
- Clarifications around function spaces: finite- and countable-dimensional cases (polynomials, sequences), bandlimited subspaces, and approximations via finite elements or Galerkin methods.
Applications and Motivation
- Several note the importance of this perspective for signal processing, PDEs, machine learning (kernel methods, boosting in function space, Gaussian processes), and quantum mechanics.
- Some readers wish applications were foregrounded earlier to motivate the abstractions; others defend “math for math’s sake” and say the article rightly assumes intrinsic interest.
Pedagogy, Culture, and Tooling
- Extended comparison of pure-math vs physics pedagogy: proofs-first vs intuition-first, and complaints about both math departments and physics curricula.
- Proof-writing is compared to programming using definitions/theorems as a “standard library”; this links to interactive theorem provers like Lean and Curry–Howard.
- Minor disputes over diagrams (vectors not all drawn from the origin) and terminology (codomain, sequences) reflect differing levels of strictness.
Miscellaneous Threads
- Side explorations on “fuzzy” numbers and low-pass–filtered Fourier analysis, operator SVD analogues, geometric algebra’s view of vectors-as-operators, and sampling/Dirac kernels.
- Numerous readers praise the article as exceptionally clear and appetite-whetting, even if not fully rigorous or comprehensive.