1Introduction

Computer science can give back to philosophy of mathematics, because computers are inherently finite.

The words "finitistic" and "finitistically" are unpleasant to write and read.

The word "finitely" is already taken to mean something else.

The phrase "according to finitism" is somewhat long.

2Set theory

2.1The set of all natural numbers is not finite (does not exist, if we reject the axiom of infinity).

Here I argue that fixed-point theory suggests that \(\Nat\) should not exist.

Let \(f\) be a function.

A fixed point of \(f\) is an \(x\) such that \(x=f(x)\).

It is tautological that if \(x \neq f(x)\) for all applicable \(x\), then \(f\) has no fixed points, that is, there is no \(x\) such that \(x=f(x)\). We readily agree with that if \(x\) is a natural number, and it shouldn't be any different if \(x\) is a set.

Suppose we encode natural numbers as von Neumann ordinals (VNOs). Let \(S\) be the VNO successor function where \(S(X) = \{X\} \cup X\). The idea of the successor of \(X\) is to enlarge \(X\) with something that is surely not already in \(X\). (Can we be sure that \(X \not\in X\)? What does this have to do with Russell's paradox?) Let 0 be represented by \(\emptyset\). Thus 1 is represented by \(S(\emptyset) = \{\emptyset\}\), 2 is represented by \(\{\{\emptyset\}, \emptyset\}\), and so on.

The set \(\Nat\) would have to satisfy the equation \(\Nat = S(\Nat)\).

My argument ends here.

If we embraced infinitism, we could declare that \(\Nat\) is the least fixed point of \(S\).

Now that the set \(\Nat\) doesn't exist, what do we mean by the notation \(n \in \Nat\)? We don't want to invent a new notation, so we invent a new reading: the phrase \( n \in \Nat \) is to be read as "\(n\) is a natural number", not "\(n\) is an element of the set of natural numbers". The meaning of the phrase \(n \in \Nat\) is inductively defined by the base case \(0 \in \Nat\) and inductive case \(n \in \Nat \rightarrow (n+1) \in \Nat\).

We don't need the axiom of infinity if we use predicates directly and don't use sets at all. We can read \(\in\) as a binary predicate, and define \(x \in S\) to read "an \(x\) that satisfies the predicate \(S\) (that is, an \(x\) such that \(S(x)\) is true)". We can define these satisfiers using induction (inductive data types in functional programming).

3Geometry

3.1Other people's ideas

Confusing names: Discrete geometry is not finitist geometry. Discrete geometry is combinatorical geometry. But there is Digital geometry and Digital topology.

TODO: Summarize Jean Paul Van Bendegem's "Finitism in geometry" https://plato.stanford.edu/entries/geometry-finitism/.

"One such possibility for detection [of spacetime anisotropy] concerns the following curious phenomenon […]" (smooth continuous version, but chaotic discrete version)

A finitist geometry must overcome several problems: Weyl's distance function problem, the isotropy problem, Fritz no-go theorem, the identity problem, and what else?

Peter Forrest (1995) formulated a family of discrete spaces \(E_{n,m}\) where \(n\) is dimension and \(m\) is adjacency criterion. A two-dimensional point is modeled by a pair of integers \((i,j)\). Two points \((i,j)\) and \((i',j')\) are adjacent iff \(0 < (i'-i)^2 + (j'-j)^2 \le m\). "Once adjacency has been stipulated, a distance function can be easily derived". As \(m\) grows, the distance function approaches Euclidean distance.

If geometry is discrete, then shouldn't spacetime be anisotropic? How do we test spacetime anisotropy?

Hodons.

Physics, geometry and topology of the Universe, anisotropy of space

  • 1998 "Measuring the topology of the Universe" http://www.pnas.org/content/pnas/95/1/82.full.pdf
    • "Observations of microwave background fluctuations can yield information not only about the geometry of the universe but potentially about the topology of the universe."
    • "There is growing evidence that we live in a negatively curved universe."
  • 2003 "Constraining the Topology of the Universe" https://arxiv.org/pdf/astro-ph/0310233.pdf
    • This interprets data from the Wilkinson Microwave Anisotropy Probe.

3.2What is the essence of geometry? Which concepts are primitive?

Point? Line? Adjacency? Distance?

The Pythagorean theorem \( a^2 + b^2 = c^2 \) is not fundamental. It assumes flat Euclidean spaces. It doesn't work on spherical geometry. However, the spherical Pythagorean theorem does converge to the Pythagorean theorem as the sphere radius grows unbounded.

https://en.wikipedia.org/wiki/Elliptic_geometry "The ph th fails in ell geom"

A more fundamental law is the law of cosines. https://math.stackexchange.com/questions/1374058/why-does-the-pythagorean-theorem-have-its-simple-form-only-in-euclidean-geometry

A more fundamental property of distance is the triangle inequality. The Pythagorean theorem is a special case of the triangle inequality.

https://philpapers.org/rec/VANFIG

https://mathoverflow.net/questions/23113/is-there-any-geometry-where-the-triangle-inquality-fails

3.3Hypothesis: The Universe has no edges? Cosmic-scale Magellan expedition?

"Current astronomical observations support the model of an infinite flat universe." Really? https://www.reddit.com/r/math/comments/1uhj1a/what_do_you_mathematicians_think_about_finitism/

The Universe "wraps around", like a FreeCiv map that wraps around.

"They may wrap at the north-south and east-west directions to form a flat map, a cylinder, or a torus (donut)." http://freeciv.wikia.com/wiki/Map_format

If we fire a photon, it will come back to us?

How do we do a cosmic-scale Magellan expedition?

3.4What I'm somewhat sure of

Every point must have a finite number of neighbors. A strong reason for this is to avoid Zeno's paradox. A weak reason for this is Democritus's atomism. A remote reason for this is that if the Universe is a computer, then it should be finite.

3.5What is an example implementation of a quantum-mechanical potential well?

An electron trap? An electron in an electric field?

3.6Synthetic finite model of geometry via discrete topology, graph theory, and discrete rotation

We assume these global parameters \(n\) (neighborhood granularity) and \(d\) (number of dimensions).

A point has \(n\) neighbors. A two-dimensional point is like the surface of a floor tile, but it doesn't have to have four sides; the geometry's number of dimensions doesn't have to be related with \(n\). Because there are finite directions, angles are quantized.

The distance between a point to its neighbor is 1. For every point \(P\), there are \(n\) points \(Q_1,\ldots,Q_n\) such that \(d(P,Q_k) = 1\).

Angle is distance in projective geometry.

In three-dimensional geometry, is a point a zero-dimensional object or a three-dimensional object? In reality, a point has volume, although tiny.

A rotation is the mapping from \(v_0, \ldots, v_k\) to \(v_1, \ldots, v_k, v_0\). It is the shifting of the sequence of vectors in the same plane.

Don't imagine tiles. Imagine a graph.

There are \(n\) possible directions (unit vectors) from a point to one of its neighbors. For example, we can discretize two-dimensional Euclidean geometry with 100 unit vectors (similar to dividing a circle into 100 sectors). We can discretize three-dimensional Euclidean geometry with 1000 unit vectors (similar to cutting an orange into 1000 same-sized parts).

The vector from point A to point B is the sum of vectors in the shortest path from A to B.

Every direction has a reverse. Thus \(n\) is divisible by \(2d\) where \(d\) is the dimension. If \( X + v = Y \) then \( X = Y + (-v) \). If \(Y\) is the \(v\)-ward neighbor of \(X\), then \(X\) is the reverse-\(v\)-ward neighbor of \(Y\).

A line is a path in the graph.

A straight line is the shortest path between two points.

We want triangle inequality: \(d(A,B) + d(B,C) \ge d(A,C)\).

Euclidean geometry is limit of space-filling curve? https://en.wikipedia.org/wiki/Space-filling_curve

3.7What if volumes/lines, not points, were fundamental?

A point is line that is shortened until it can't go any shorter.

What is a line? What happens if we zoom very close to a line?

3.8Lattice geometry?

Two-dimensional geometry.

Pick one point \(O\) (origin) and two vectors \(\vec{e}_1\) and \(\vec{e}_2\) (unit axis vectors). Assume that those vectors are perpendicular (orthogonal) to each other.

A vector \(\vec{v}\) is \(v_1 \vec{e}_1 + v_2 \vec{e}_2\) where \(v_1\) is an integer and \(v_2\) is an integer.

The result of mirroring the direction \(v_i\) with respect to direction \(v_m\) is \(v_{m+(m-i)}\).

The result of rotating \(v_k\) by one unit angle is \(v_{k+1}\).

Rotation, translation, symmetry, mirroring

Too formal? 2009, "Strict Finitism and the Logic of Mathematical Applications" http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.603.1574&rep=rep1&type=pdf

That article finitizes Hilbert spaces?

3.9Bresenham's line algorithm, contiguousness, neighbors, and distance

Idea: Define the distance between two points as the number of points filled by Bresenham's line algorithm.

This is an exciting way for computer graphics to give back to the philosophy of mathematics! Idea: Finitist geometry is computer graphics with the assumption that the computer can get as big and powerful as we want.

What is "distance" in two-dimensional finitist geometry?

Let \( h = h_x h_y \) be a unit area.

The area of line is \( n h_x h_y \) where \(n\) is the number of pixels in that line.

The finitist Pythagorean theorem:

\( (n_x h_x)^2 + (n_y h_y)^2 = n^2 \cdot (h_x^2 + h_y^2) + ??? \)

A discrete line that "best approximates" the ideal Platonic-infinitist's line?

Bresenham's line algorithm?

https://en.wikipedia.org/wiki/Bresenham%27s_line_algorithm

A line is a "contiguous" set of points.

Two points are "contiguous" iff they are each other's "neighbors".

4-neighbor rule or 8-neighbor rule?

The distance between point A and point B is the square root of the number of pixels in the line from A to B?

How do we measure the units of the Universe? Rounding errors? Quantization artifacts, like JPEG artifacts?

What can finitism say about general relativity and quantum mechanics?

3.9.1Finitism and Newtonian mechanics collisions?

How does calculus finitization affect Newtonian physics?

What does a vector become?

3.10Hypothesis: A photon does not travel slower in medium.

A photon does not travel slower in medium. A photon is scattered. A photon hits atoms. A photon travels farther, not slower, in medium.

4Real analysis

2004, "'real' analysis is a degenerate case of discrete analysis" http://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/real.pdf

Every definition here assumes that the unit. It is a global parameter. It is usually written \(h\).

Assuming unit \(h\), we define the "derivative of \(f : \Real \to \Real\) at \(x\)", written \( (Df)(x) \), as \( [f(x+h)-f(x)]/h \).

Higher-order derivatives are a straightforward matter of substitution:

\[\begin{align*} D^2f = D(Df) &= D\left(x \mapsto \frac{f(x+h)-f(x)}{h}\right) \\ &= \frac{\frac{f(x+h+h)-f(x+h)}{h} - \frac{f(x+h)-f(x)}{h}}{h} \\ &= \frac{f(x+2h) - 2 \cdot f(x+h) + f(x)}{h^2} \end{align*} \]

Pascal's triangle with alternating signs. Similar to polynomial coefficients of \((a-b)^n\).

4.1Solving a differential equation by detouring to its analogous difference equation

Here we use finitism, and arrive at a result consistent with infinitism! This is only a motivating example. This does not prove that finitism subsumes infinitism.

We consider the equation \( y = Dy \).

We expand the equation according to finitism:

\[\begin{align*} y(x) &= (Dy)(x) \\ y(x) &= \frac{y(x+h)-y(x)}{h} \\ h \cdot y(x) &= y(x+h) - y(x) \\ (1 + h) \cdot y(x) &= y(x+h) \\ 1 + h &= \frac{y(x+h)}{y(x)} \end{align*} \]

Now we're going to do some trick with these substitutions: substitute \(x\) with \(hu\), and substitute \(z(u) = y(hu)\).

We transform the above into a recurrence relation:

\[\begin{align*} (1 + h) \cdot y(x) &= y(x+h) \\ (1 + h) \cdot y(hu) &= y(hu+h) \\ (1 + h) \cdot y(hu) &= y(h\cdot(u+1)) \\ (1 + h) \cdot z(u) &= z(u+1) \\ 0 &= z(u + 1) - (1 + h) \cdot z(u) \end{align*} \]

A possible solution is \( z(u) = (1+h)^u \).

Now the second part of the trick: do the same substitutions: substitute back \( x = hu \) and \( z(u) = y(hu) \).

We get:

\[\begin{align*} z(u) &= (1+h)^u \\ y(hu) &= (1+h)^u \\ y(x) &= (1+h)^{x/h} \end{align*} \]

Now this is where finitism and infinitism "coincide": \[ \lim_{h \to 0} (1+h)^{x/h} = e^x \]

This seems promising!

What does the Taylor series become?

What is the relationship between finitism and non-standard analysis?

4.1.1Bias toward the positive side?

A problem is that \( Df \) is biased toward the positive side. We could try a symmetric definition such as \( (Df)(x) = [f(x+h)-f(x-h)]/(2h) \), but this doesn't use \(f(x)\), but why is this a problem?

5Probability theory

5.1Some definitions

A sample space is a set.

An event is a subset of a sample space.

A probability mass function \( p : \Omega \to \Real \) maps each sample to a probability such that the probabilities add up to one: \( \sum_{x \in \Omega} p(x) = 1 \).

The probability function \(P : 2^\Omega \to \Real \) satisfies

\( P(E) = \sum_{e \in E} p(e) \)

\( P(\Omega) = 1 \)

\( P(\emptyset) = 0 \)

Conditional probability: we define the notation \( P(A|B) := P(A \cap B) / P(B) \).

A random variable is a function whose domain is the sample space. An \(R\)-valued random variable is a function \(\Omega \to R\). The codomain depends on your modeling.

A random variable can model a player's profit of an outcome of a gambling round. Example: a game of fair coin toss: \( \Omega = \{ L, W \} \) where \( \{L\} \) represents "lose" and \( \{W\} \) represents "win". \( p(L) = p(R) = 1/2 \). \( X(L) = -1, X(W) = 1 \).

5.2Random variable probability notation

This is widely used but is rarely explained.

Remember that a random variable is a function, not a variable.

Suppose that we are discussing about some random variables.

Let \(\phi\) be a logic formula containing logic variables with the same "name" as some random variables in the context of our discussion. Some example formulas are \(X = 0\) and \(2 X < 5\). These logic variables have the same name as our random variables, but these logic variables look like algebraic variables.

Let \(\phi'(s)\) be \(\phi\) but with every occurrence of every logical variable \(X\) replaced with \(X(s)\) (the application of random variable \(X\) to sample \(s\)). If the letter \(s\) is already used in \(\phi\), use another unused letter.

We define this notation:

\[\begin{align*} P(\phi) := P(\{ s ~|~ s \in \Omega, ~ \phi'(s) \}) \end{align*} \]

Here are some examples of that notation in action:

\[\begin{align*} P(X \in S) &:= P(\{ s ~|~ s \in \Omega, ~ X(s) \in S \}) \\ P(X = s) &:= P(\{ s ~|~ s \in \Omega, ~ X(s) = s \}) \\ P(X < s) &:= P(\{ s ~|~ s \in \Omega, ~ X(s) < s \}) \\ P(X^2 + X + 1 = 0) &:= P(\{ s ~|~ s \in \Omega, ~ [X(s)]^2 + X(s) + 1 = 0 \}) \\ P(f(X) = 0) &:= P(\{ s ~|~ s \in \Omega, ~ f(X(s)) = 0 \}) \\ P(e^X = 1) &:= P(\{ s ~|~ s \in \Omega, ~ e^{X(s)} = 1 \}) \end{align*} \]

The notation also works with many random variables at once:

\[\begin{align*} P(X < Y) &:= P(\{ s ~|~ s \in \Omega, ~ X(s) < Y(s) \}) \\ P(X + Y + Z = 0) &:= P(\{ s ~|~ s \in \Omega, ~ X(s) + Y(s) + Z(s) = 0 \}) \end{align*} \]

Note how the notation makes a random variable look like an algebraic variable; remember that a random variable is a function, not a variable.

5.3Expected value

Expected value of real-valued random variable: \( E(X) = \sum_{s \in \Omega} p(s) \cdot X(s) \).

5.4Random walk

A random process (a stochastic process) is a sequence of random variables. \( Y : \Nat \to (\Omega \to R) \).

A martingale is a random process […]

A random walk […]

Understanding one-dimensional Brownian motion?

"In 1906 Smoluchowski published a one-dimensional model to describe a particle undergoing Brownian motion." https://en.wikipedia.org/wiki/Brownian_motion

Example: one-dimensional Brownian motion: we assume that at every time step, the particle of interest is hit by another particle: sample space \( \Omega = \{ L, R \} \), where \(\{L\}\) represents the event that the particle of interest is hit from the left, probability mass function \( p(L) = p(R) = 1/2 \), which means that a hit from the left and a hit from the right are equally likely; random variable \( X : \Omega \to \{ -1,+1 \} \) where \( X(L) = -1, X(R) = +1 \).

A Wiener process […]

What does stochastic calculus become?

5.5TODO

One dimensional random process Statistical mechanics Gas in a box https://en.wikipedia.org/wiki/Brownian_motion

6What?

"A New constructive axiomatic scheme for the geometry of space-time" http://inspirehep.net/record/353369/