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Welcome to the Objective Mathematics wiki.

Objective Mathematics is an ongoing project which aims to make mathematics more objective by connecting it to reality. This is being done by painstakingly going through each math concept, and thinking about the concrete, perceptual data to which it ultimately refers. If a math concept is not reducible to perceptual concretes, it is invalid; it is a floating abstraction. Objective Mathematics rejects Platonism: math concepts do not refer to Platonic forms inhabiting an otherworldly realm, but rather they refer directly to physical, perceivable things. Objective Mathematics rejects Intuitionism: math is not a process of construction and intuition, but rather a process of identification and measurement. Objective Mathematics rejects Formalism and Logicism: math is not a meaningless game of symbol manipulation, but rather its statements have semantic content (just like propositions about dogs, tea, beeswax, or anything else).

According to Objective Mathematics, mathematics is the science of measurement. [TODO expand]

Besides mathematics, many of the pages on Objective Mathematics wiki are about foundational concepts of physics, computer science, and philosophy. Although I accept the distinction between these four subjects, I think that it is more blurry than is sometimes supposed. The unifying theme of these subjects, and the reason why they all appear on Objective Mathematics wiki, is that each subject is foundational, meaning that it can be contemplated on its own (in the case of philosophy), or that it can be contemplated without taking anything beyond basic metaphysics and epistemology for granted (in the case of the others). For example, what I deem to be the fundamental concepts of computer science, despite what the name of the subject may suggest, could in principle be formed by someone with no knowledge whatsoever about computing machines (by say, an Ancient Greek philosopher).

[TODO I should be very careful in dismissing these ideas; they contain significant elements of the truth] In the context of math, it is common for people to say things like "this [mathematical concept] is an idealization of that [real-world thing]," or "this [collection of mathematical ideas] is a model of that." Such ideas are often very wrong. A mathematical sphere is not an idealization of a real world sphere, it is a real world sphere. And what does it mean for one thing A to be a model of another thing B? It means that A is some sort of object, which shares some essential properties with B, which represents B in some way, but is easier to understand or work with than B itself. What sort of object is A supposed to be, when A is a mathematical model? If A is supposed to be a Platonic form, that's mysticism. And if A is supposed to be the mathematical concepts themselves, that's also wrong: It is totally inappropriate to think about concepts as models, because there can be no means of understanding or working with objects other than by using man's distinctive mode of cognition---i.e. by using concepts. When people call man's understanding of existence man's "model" of existence, they have some absurd picture like this [TODO insert HB's picture of the homunculus perceiving the man's perception of the tree] in mind.

Ayn Rand [TODO cite letter to Boris Spasky] says, about chess, that it is

an escape—an escape from reality. It is an “out,” a kind of “make-work” for a man of higher than average intelligence who was afraid to live, but could not leave his mind unemployed and devoted it to a placebo—thus surrendering to others the living world he had rejected as too hard to understand.

I have the exact same opinion about standard mathematics.

Recommended reading order

[TODO]

All Pages

Math pages:

• Category theory
• Coordinates
• Derivative
• Sequence
• Set
• Continuity
• Number
• Multiplication
• Natural number (obsolete. See multitude instead)
• Multitude
• Magnitude
• Integer
• Fraction (obsolete. See magnitude instead)
• Radical
• Real number
• Imaginary number
• Vector space
• Triangulation
• Function
• Polynomial
• Circle
• Line
• Point
• Curve
• Surface
• Solid
• π
• e
• Limit
• Nill
• Induction
• Ordering
• Symmetry
• Probability
• Group

Physics pages:

• Entropy
• Newton's laws
• Uncertainty
• Perturbation theory
• Velocity
• Notes on "On the Electrodynamics of Moving Bodies" by Albert Einstein
• Notes on "On the Law of Distribution of Energy in the Normal Spectrum" by Max Planck
• Notes on "On a Heuristic Viewpoint Concerning the Production and Transformation of Light" by Albert Einstein

Computer Science pages:

• Algorithm
• Type
• Information
• Computation
• P

Philosophy pages:

• Logic
• Against models (essay)
• Category theory: abstracting mathematical construction (essay)
• Existent
• Concept
• Entity
• Unit
• Identity
• Definition
• Platonism
• Intuitionism
• Formalism
• Induction
• Objective Mathematics
• Zeno's Paradox
• Counterfactuals
• Occam's razor
• Perspective Theory

Essays for Harry Binswanger's Philosophy of Mathematics course:

• Limits (essay) [TODO this is old, replace]
• The Limits of Limits

Other essays:

• Coordinate invariance: a manifesto

Notation

Instead of having set inclusion as one of its fundamental concepts, Objective Mathematics has conceptual identification as one of its fundamental concepts. For conceptual identification, it uses the notation of Type Theory. It is easiest to demonstrate what is meant by this through examples:

• $q={\frac {2}{3}}$ is a fraction, and I denote this fact---this identification---by writing $q:\mathbb {Q}$.
• Any integer is a fraction, and I denote this fact by writing $\mathbb {Z} :\mathbb {Q}$.

Donate

If you would like to help pay to keep the Objective Mathematics wiki afloat, consider donating [TODO].

Contact

If you are interested in Objective Mathematics and would like to discuss it, please email me. You already know my email if you are reading this and you are not a bot. [TODO]

Legal

All writing on this website is (c) Liam M. Fox.

Some images on this website are public domain, some are (c) Liam M. Fox. Check the image descriptions to see which [TODO].