# Graded Algebras and Classical Mechanics

Certain algebras possess an important classifying property known as a grading.  This essentially amounts to being able to express these algebras as a special kind of direct sum that respects multiplication.

Definition[Graded Algebra]:  An algebra $A$ over a field $\kappa$ is said to be graded if, as a vector space over $\kappa$, $A$ can be written as the direct sum of a family of subspaces $(A_{i})_{i\in \mathbb{N}}$ – i.e.

$\displaystyle A = \bigoplus_{i\in \mathbb{N}}A_{i}$

and such that multiplication behaves according to

$A_{i}\cdot A_{j} \subseteq A_{i+j}$, for all $i,j\in \mathbb{N}$

In classical mechanics, one deals with smooth manifolds. To understand what these are we need some definitions.

Definition[Chart]:  Let $M$ be a manifold of dimension $n$.  Then a chart is a pair $(U, \psi)$, where $U\subset M$ is open and $\psi: U\to V\subset \mathbb{R}^{n}$ is a homeomorphism to some open $V$.

The collection of all charts such that each $x\in M$ belongs to some chart is called an atlas.  An atlas characterizes a manifold.  More formally,

Definition[Atlas]:  An atlas on a topological space $T$ is a collection of charts $\{(U_{\alpha},\varphi_{\alpha})\}_{\alpha\in I}$ where the $U_{\alpha}$ are open sets that cover $T$, and for each index $\alpha$

$\varphi_{\alpha}: U_{\alpha}\to \mathbb{R}^n$

is a homeomorphism of $U_{\alpha}$ onto an open subset of $n$-dimensional Euclidean space.

Definition[Smooth Atlas]: An atlas $\mathcal{A}$ is called smooth if for all $\varphi_{i}, \varphi_{j}$ we have that

$\varphi_{j}\circ \varphi_{i}^{-1} : \varphi_{i}(U_{i}\cap U_{j})\to \varphi_{j}(U_{i}\cap U_{j})$

is a diffeomorphism.

Definition[Diffeomorphism]: Given two manifolds $M$ and $N$, a differentiable map $f: M\to N$ is called a diffeomorphism if it is a bijection, and its inverse $f^{-1}: N\to M$ is also differentiable.

Now, a manifold may not be generated by a unique atlas.  To account for this, we seek a preferred atlas known as a maximal atlas.  In defining such a thing, we need to know what it means for atlases to be compatible.

Definition[Compatibility]: Let $\mathcal{A}$ and $\mathcal{A}'$ be two smooth atlases.  Then $\mathcal{A}$ and $\mathcal{A}'$ are called compatible if and only if $\mathcal{A}\cup \mathcal{A}'$ is again a smooth atlas.

This notion of compatibility is actually an equivalence relation.  The union over an equivalence class of atlases gives us our maximal atlas.  If we denote an equivalence class by $[\mathcal{A}]$, then we have

Definition[Maximal Atlas]:  A maximal atlas for a manifold $M$ is the union of all smooth atlases in an equivalence class – i.e.

$\mathcal{A}_{max} = \bigcup\{\mathcal{B} : \mathcal{B}\in [\mathcal{A}]\}$

This maximal atlas is said to be the differentiable structure of the manifold.  We are now ready to understand the idea of a smooth manifold.

Definition[Smooth Manifold]:  A smooth manifold is a pair $(M, \mathcal{A}_{max})$ where $M$ is a manifold and $\mathcal{A}_{max}$ is a differentiable structure of $M$.

Smooth manifolds are important in classical mechanics because they allow us to do calculus on them.  Now, to return to the original topic, we are also interested in the graded algebra of smooth differentiable forms on a smooth manifold $M$ with respect to the wedge product, which is denoted by

$\displaystyle \mathcal{A}^{\bullet}(M) = \bigoplus_{k=0}^{n}\mathcal{A}^{k}(M)$

# – Differential Forms –

For now, let’s restrict our focus to what are called differential $1$-forms, or just $1$-forms.  According to one source, “Informally, a differential form is what can be integrated along a path.”

[$1$-Forms]

Starting in $\mathbb{R}^{2}$, let $U$ be some open subset of $\mathbb{R}^{2}$ and let $F, G$ be two real-valued functions defined on $U$.  Then an expression of the form

$F(x,y)dx+G(x,y)dy$

is called a differential $1$-form on $U$.  A particularly important example of a differential $1$-form is the total differential of a $C^{1}$ real-valued function $f$ defined on some open subset $U$.

Definition[Total Differential]:  Let $f$ by a $C^{1}$ real-valued function defined on an open subset $U$ of $\mathbb{R}^{2}$.  Then its total differential $df$ is defined by

$\displaystyle df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy$

This can also be expressed as a kind of dot product of vectors, namely $\nabla f\cdot (dx,dy)$, where $\nabla f$ is the gradient of $f$.  Note that in this case we have $\displaystyle F(x,y) = \frac{\partial f}{\partial x}$ and $\displaystyle G(x,y) = \frac{\partial f}{\partial y}$.

$1$-forms can be defined similarly for higher dimensions.  For instance, a differential $1$-form on an open subset $U$ of $\mathbb{R}^{3}$ is an expression of the form

$F(x,y,z)dx+G(x,y,z)dy+H(x,y,z)dz$

where $F, G$ and $H$ are real-valued functions on $U$.  Again, if $f$ is a $C^{1}$ function defined on $U$, then the total differential

$\displaystyle df = \frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz$

is an example.

In general, a $1$-form on an open subset of $\mathbb{R}^n$ is an expression of the form

$F_{1}(x_{1},x_{2},...,x_{n})dx_{1}+F_{2}(x_{1},x_{2},...,x_{n})dx_{2}+\ldots + F_{n}(x_{1},x_{2},...,x_{n})dx_{n}$

or simply $\displaystyle \sum_{i = 1}^{n}F_{i}(x_{1},x_{2},...,x_{n})dx_{i}$.  If we let $\boldsymbol{x} = (x_{1}, x_{2},..., x_{n})$, then we can write a $1$-form even more succinctly as

$\displaystyle \varphi = \sum_{i=1}^{n}F_{i}(\boldsymbol{x})dx_{i}$

So, if $f$ is a $C^{1}$ real-valued function defined on $U$, the the general total differential is

$\displaystyle df = \sum_{i=1}^{n}\frac{\partial f}{\partial x_{i}}dx_{i}$

Note that each $dx_i$ is a $1$-form.  These are defined by the property that for each vector $\boldsymbol{v} = (v_{1},v_{2},...,v_{n})\in T_{x}\mathbb{R}^{n}$

$dx_{i}(\boldsymbol{v}) = v_{i}$

Clearly, these form a basis for the $1$-forms on $\mathbb{R}^{n}$.

Definition[Smooth $1$-Form]:  A smooth $1$-form $\varphi$ on $\mathbb{R}^{n}$ is a real-valued function on the set of all tangent vectors to $\mathbb{R}^{n}$, i.e. –

$\varphi : T\mathbb{R}^{n}\to \mathbb{R}$

with the properties that

1. $\varphi$ is linear on the tangent space $T_{x}\mathbb{R}^{n}$ for each $x\in \mathbb{R}^{n}$ – i.e. if $\boldsymbol{v}_{1}, \boldsymbol{v}_{2}\in T_{x}\mathbb{R}^{n}$ and $c_{1},c_{2}\in \mathbb{R}$,then $\varphi(c_{1}\boldsymbol{v}_{1}+c_{2}\boldsymbol{v}_{2}) = c_{1}\varphi(\boldsymbol{v}_{1})+c_{2}\varphi(\boldsymbol{v}_{2})$.
2. For any smooth vector field $v = v(x)$, the function $\varphi(v):\mathbb{R}^{n}\to \mathbb{R}$ is smooth.

A smooth $1$-form on an $n$-dimensional manifold $M$ is defined similarly.

Note:  Another source simply says that a smooth $1$-form on an open subset $U$ of $\mathbb{R}^{n}$ is given by an expression

$\displaystyle \varphi = \sum_{i=1}^{n}f_{i}dx_{i}$

where $f_{i}\in C^{\infty}(U)$.

To better understand the graded algebra of smooth differential forms on $M$ with respect to the wedge product we need to understand that the $1$-forms on $\mathbb{R}^{n}$ are part of an algebra called the algebra of differential forms on $\mathbb{R}^{n}$.  Multiplication in this algebra is known as the wedge product and is denoted by the symbol “$\wedge$“.  It has the property of being skew-symmetric or anti-commutative:

$dx_{i}\wedge dx_{j} = -dx_{j}\wedge dx_{i}$

From this it follows that $dx_{i}\wedge dx_{i} = -dx_{i}\wedge dx_{i}$, which implies that $dx_{i}\wedge dx_{i} = 0$ (provided we are not working over a field with characteristic 2).  This means that the wedge product is alternating.

From here we can build up $k$-forms on $\mathbb{R}^{n}$.  If each summand of a differential form $\varphi$ contains $k$ $dx_{i}$‘s, then $\varphi$ is called a $k$-form.

Note: Functions are considered to be $0$-forms.

The set

$\{dx_{i_{1}}\wedge \ldots \wedge dx_{i_{k}} : 1\leq i_{1} < i_{2} < \ldots < i_{k} \leq n\}$

is a basis for the $k$-forms on $\mathbb{R}^{n}$.  Thus, every $k$-form can be expressed in the form

$\displaystyle \varphi = \sum_{|I| = k}f_{I}dx_{i_{1}}\wedge \ldots \wedge dx_{i_{k}}$

where $I$ ranges over all multi-indecies $I = (i_{1}, i_{2},...,i_{k})$ of length $k$.

## The Exterior Derivative

The exterior derivative is an operation that sends a $k$-form to a $(k+1)$-form.

Definition[Exterior Derivative ($1$-Form)]:  Let $f$ be a $0$-form (function).  Then its exterior derivative $df$ is the $1$-form

$\displaystyle df = \sum_{i=1}^{n}\frac{\partial f}{\partial x_{i}}dx_{i}$

Note that this corresponds to the total differential of $f$.

Definition[Exterior Derivative ($k$-form)]:  Let $\varphi$ be a $k$-form.  Then its exterior derivative $d\varphi$ is the $(k+1)$-form obtained from $\varphi$ by applying $lated d$ to each of the functions involved in $\varphi$.

Definition[Directional Derivative]:  Let $f$ be a differentiable function defined on a region $R$ in $\mathbb{R}^{n}$.  Let $\boldsymbol{v}$ be a vector based at the point $p_{0}\in R$.  Then the derivative of $f$ along $\boldsymbol{v}$, denoted $D_{\boldsymbol{v}}f$, is defined as follows.  Let $F(t) = f(p_{0}+t\boldsymbol{v})$.  Then

$D_{\boldsymbol{v}}f = F'(0)$

Theorem:  Let $f$ be a differentiable function defined on a region $R$ of $\mathbb{R}^{n}$.  Let $\varphi$ be defined on tangent vectors to points of $R$ by

$\varphi(\boldsymbol{v}) = D_{\boldsymbol{v}}f$

Then

$\varphi = df$

Proof:

Let $\boldsymbol{v}$ be a vector based at a point $p$ of $R$ with coordinates $(p_{1}, p_{2},..., p_{n})$. Let $\boldsymbol{v}$ have coordinates $(v_{1}, v_{2}, ..., v_{n})$.  Let $\textbf{u}_{i}$ be the $i$th unit vector.  Then

$\displaystyle df(\boldsymbol{v}) = \left(\sum_{i=1}^{n}\frac{\partial f}{\partial x_{i}}dx_{i}\right)\left(\sum_{i=1}^{n}v_{i}\textbf{u}_{i}\right)$

$=\displaystyle \sum_{i=1}^{n}\frac{\partial}{\partial x_{i}}f(p)v_{i}$

Next, let’s evaluate $\varphi(\boldsymbol{v}) = D_{\boldsymbol{v}}f = F'(0)$ where we recall that $F(t) = f(p+t\boldsymbol{v})$.  So, really, we should write $\displaystyle F'(0) = \frac{\partial}{\partial t}f(0)$.  Now, $F$ is actually a composition of two functions, $F(t) = f(g(t))$, where $g(t) = p+t\boldsymbol{v} = (p_{1}+tv_{1}, p_{2}+tv_{2},...,p_{n}+tv_{n})$.  If we now apply the chain rule, then we get

$\displaystyle \frac{\partial}{\partial t}f(0) = \sum_{i=1}^{n}\frac{\partial}{\partial x_{i}}f(g(0))\frac{d}{dt}x_{i}(0)$

But $x_{i} = p_{i}+tv_{i}$, so $\displaystyle \frac{d}{dt}x_{i} = v_{i}$.  Therefore,

$\displaystyle \frac{\partial}{\partial t}f(0) = \sum_{i=1}^{n}\frac{\partial}{\partial x_{i}}f(p)v_{i}$

This finishes the proof.

The exterior derivative obeys both the Leibniz rule and the chain rule:

• $d(fg) = gdf+fdg$         Leibniz Rule
• $d(h(f)) = h'(f)df$        Chain Rule

If $\phi$ is a $p$-form and $\psi$ is  a $q$-form, then the Leibniz rule takes the form

$d(\phi \wedge \psi) = d\phi \wedge \psi+(-1)^{p}\phi\wedge d\psi$

Theorem:  For any differential form $\phi$,

$d(d\phi) = 0$

(Or more succinctly, $d^2 = 0$).

Proof:

First, let $f$ be a function (i.e. $0$-form).  Then

$\displaystyle d(df) = d\left(\sum_{i=1}^{n}\frac{\partial f}{\partial x_{i}}dx_{i}\right)$

$\displaystyle = \sum_{i=1}^{n}d\left(\frac{\partial f}{\partial x_{i}}\right)dx_{i}$

$\displaystyle = \sum_{i=1}^{n}\left(\sum_{j=1}^{n}\frac{\partial f_{x_{i}}}{\partial x_{j}}dx_{j}\right)dx_{i}$

[where $\displaystyle f_{x_{i}} : = \frac{\partial f}{\partial x_{i}}$]

$\displaystyle =\sum_{i=1}^{n}\left(\sum_{j=1}^{n}\frac{\partial^{2} f}{\partial x_{i}\partial x_{j}}dx_{j}\right)dx_{i}$

$\displaystyle =\sum_{i,j}\frac{\partial^{2} f}{\partial x_{i}\partial x_{j}}dx_{j}\wedge dx_{i}$

$\displaystyle =\sum_{i

since mixed partials commute.  Now, since $dx_{i}$ actually means $d(x_{i})$, where $x_{i}$ is the $i$-th coordinate function, we have that $d(dx_{i}) = 0$.  Let

$\displaystyle \phi = \sum_{|I| = k}f_{I}dx_{i_{1}}\wedge \ldots \wedge dx_{i_{k}}$

Then

$\displaystyle d(d\phi) = \left(\sum_{|I|=k}df_{I}\wedge dx_{i_{1}}\wedge \ldots \wedge dx_{i_{k}}\right)$

$\displaystyle = \sum_{|I|=k}\Big(d(df_{I})\wedge dx_{i_{1}}\wedge \ldots \wedge dx_{i_{k}}-df_{I}\wedge d(dx_{i_{1}})\wedge \ldots \wedge dx_{i_{k}}+\ldots\Big)$

[By the Leibniz Rule]

Per the reasoning above with functions, this is clearly $0$.

Much of the above has been described in terms of differential forms on $\mathbb{R}^{n}$.  Most of it transfers fairly seamlessly to general manifolds (i.e. the concepts are defined similarly).  So, to tie this all back to the graded algebra given above, we should understand $\mathcal{A}^{0}(M)$ to be $C^{\infty}(M)$ (i.e. the $0$-forms on $M$) and $\mathcal{A}^{k}(M)$ to be the algebra of smooth differential $k$-forms on $M$. We’ll look more at graded algebras in the next post.

# Evaluating the Moral Arguments

This post is meant to be an initial assessment of two forms of the moral argument.  I have taken them exactly as formulated by Tyler.  I want to apologize up front if I come across as overly pedantic.  I like to break things down as much as possible.  Also, it’s my blog… so deal with it.

# The Arguments Stated

The Epistemological Argument:

(1) If NE is true, belief in objective moral values and duties cannot be warranted.
(2) But belief in objective moral values and duties can be warranted.
(3) Therefore, NE is false.

Note: “NE” stands for the conjunction of naturalism and evolution.

The Classical Moral Argument:

(1) If God doesn’t exist, objective moral values and duties do not exist.
(2) Objective moral values and duties do exist.
(3) Therefore, God exists.

# The Arguments Evaluated

## – The Epistemological Argument –

Let’s start by putting this in symbolic form.  Technically, the proposition NE is true is a compound proposition, namely Naturalism is true and evolution is true.

Let $n$ be the simple proposition Naturalism is true and $e$ the simple proposition Evolution is true.  Finally, let $b$ represent the simple proposition Belief in objective moral values and duties can be warranted.  Then the argument can be expressed as

(1) $(n\wedge e)\rightarrow \sim b$

(2) $b$

(3) $\sim(n\wedge e)$

This form of argument is valid (modus tollens), so the first order of business will be to address any ambiguities and then the soundness of the argument.

I’ll start by pointing out that some clarification is needed as to what is meant by “naturalism”, since the term has no precise meaning in philosophy$^{1}$.  Presumably this term is intended to be a position excluding the supernatural, but then there is the question as to what counts as being super-natural.  For instance, one might subscribe to naturalism, but hold that parallel universes exist or that Platonism is correct, which could be interpreted as “super-natural” in some sense.  Some care also needs to be taken since naturalism is many times distinguished from materialism.

Putting that aside for the moment, let’s examine premise (1) with just a broad understanding of the terms.  Note that the negation of (1) would be $\sim[(n\wedge e)\rightarrow \sim b]$, which is equivalent to $n\wedge e\wedge b$.  In words, this says that naturalism is true, evolution is true and belief in objective moral values and duties can be warranted.

At this point we need to know more about what it means for something to be warranted.  From what I understand, Tyler uses the term in the same sense as Alvin Plantinga.  It is a technical term given to that which distinguishes mere true belief from knowledge.  In particular, warrant is strongly related to the notion of proper function.  This seems to mean something like our faculties being geared toward forming true beliefs when operating correctly.  Thus, what I’ll need to argue, here, is that our faculties can be geared toward forming true beliefs even if they were not designed by an intentional agent.  However, to prevent excessive length (and because this will be a continued discussion), I shall start by merely giving some general ideas.

Something to note right away is that this argument assumes that morality is objective.  I happen to think that morality can be objectively defined, but I’m not entirely convinced that what counts as moral is objective.  More on this later.

If we grant for the moment that there is such a thing as objective moral values and duties, then I imagine that these moral facts would exist in the same sense that, say, logical facts do.  As far as we know, what allows us to be able to access such facts is our capacity to think, reason, abstract, and in the case of morality, empathize.  So, it seems reasonable to take it that any creature constructed similarly enough to the way humans are, will be able to access logical and moral facts.  The question then shifts to: how did we come to be constructed in this way?

Certainly one possibility is that God purposefully made us (somehow) this way out of nothing.  Now, one thing we seem to know with reasonable certainty is that our world (and arguably any possible world) is governed by or written in the language of mathematics.  I maintain that the mathematics that underlies our reality exists eternally and necessarily.  So, it may be that there is a multiverse in which all mathematically possible worlds simply exist.  In at least one of them, namely ours, the structure will allow for creatures to exist in a way that they can access the laws of logic and moral laws.  Thus, the need for special creation is eliminated and it is possible that naturalism is true, evolution is true, and yet we can be warranted in a belief in objective moral values.

The last thing to point out is that the conclusion of this argument is not as strong as the theist might intend.  What I mean is that $\sim(n\wedge e)$ is logically equivalent to $\sim n \vee \sim e$, which simply says that either naturalism is not true or evolution is not true.  Nothing here requires that both are false.

## -The Classical Moral Argument-

This argument is another example of modus tollens.  Since it is valid, let’s consider the premises.  I’ll be a bit shorter with my analysis of this argument to start.

First let me say that I see no reason to accept (1).  As alluded to above, moral facts may exist in the same way that logical facts do.  Second, I contend that (2) is undecidable.  Morality is of such a nature that we cannot tell if it is truly objective.  It certainly feels this way, but this is largely built on intuition deriving from how we are made up as humans.  At best, I think one could only maintain that morality is what I call “locally objective”.  That is, there is a certain set of moral laws $M$ associated with humans (based on how we operate) such that any creature $c$ that is sufficiently similar to humans will be subject to $M$.

Okay, at this point I don’t want to take much more time, so I’ll pass it over to Tyler.  Upon receiving his critique, I will then expand on my thoughts where needed.

# A Friendly Discussion on the Moral Arguments

I am a mathematician.  But as many of you know, the topic of God’s existence is also of great interest to me.  This is in large part due to my desire to understand the ultimate nature of reality.  Some might reckon that pursuing the question of theism is a waste of time.  It has been debated for millennia with seemingly little progress.  While perhaps true, I tend to be a bit more optimistic.  Even if the question is ultimately undecidable, some very interesting ideas and philosophy have come out of the discussion, which have shaped many areas of our thought.

There are many different types of arguments for the existence of God, and even if they ultimately fail, there is no denying that evaluating them has led to great progress in various philosophical topics.  The notion of morality happens to be one of these.  In fact, the moral argument is one of five or so major types of arguments for God’s existence.  I personally find the topic of morality to be one of the most difficult to analyze and nail down.  Because of this, I find the moral argument to be the weakest of all theistic arguments.  Others, like my friend Tyler Dalton McNabb, assess it as among the stronger arguments.

So, this is what I would like to do: Tyler and I have agreed to have a friendly discussion on the moral argument.  He has presented the basic arguments on his blog.  I will give an initial assessment of these arguments on my blog and he will then address my criticisms back on his blog.  It should prove to be a fruitful exchange, so follow along and enjoy.  Comments are also welcome.

# Could Cosmic Inflation Mean That Time Is Quantized?

Cosmological inflation, as of right now, is the leading theory regarding the very earliest moments after the Big Bang. According to mathematical physicist and cosmologist Max Tegmark of MIT this theory suggests that the universe is actually infinite! If that is hard to imagine, then just wait because things get weirder. If the universe is actually infinite, then there is good reason to think that there are infinitely many “copies” of you in other solar systems spread throughout the infinite expanse. Not only that, but there are infinitely many “near-copies” of you living out every possible variation.

Because of the discrete nature of these alternatives, the number of you’s would be represented by the cardinal number $\aleph_{0}$. In other words, it is a countable infinity.

Now, if the possible variations depends on time, then these possibilities should be in one to one correspondence with the variant you’s. This seems to suggest that time is made up of discrete moments. Either that, or causality is in some sense quantized.

# A Quick Intro to Manifolds

An $n$-dimensional manifold is intuitively described as a topological space that locally resembles $\mathbb{R}^n$.  In other words, if you pick a point on a manifold and “zoom in”, it starts to look more and more like a Euclidean space of the same dimension. According to Penrose a manifold is

… a space that can be thought of as ‘curved’ in various ways, but where, locally (i.e. in a small enough neighborhood of any of its points), it looks like a piece of ordinary Euclidean space.

Some examples will help to illustrate.

Example: If we start with $n = 0$, then $\mathbb{R}^{0} = \{0\}$, which is just a singleton.  This means that a $0$-manifold is one in which every point has a neighborhood that looks like a single point.  Thus, a $0$-manifold just is a collection of points.

Example: If $n=1$, then $\mathbb{R}^{1}$ is the real line.  Thus, a $1$-manifold is one in which each point has a neighborhood that looks like an open line segment.  A circle is a good example of this.  In topology, one ignores “bending” so that a small portion of the circle is treated like a small line segment.  Let’s consider this more carefully by considering the unit circle $x^{2}+y^{2} = 1$ and seeing how it “looks like” a line segment at each point.

Begin with the upper half part of the circle described by $y = \sqrt{1-x^{2}},~ -1.

Note that the $y$-coordinate is always positive on this part of the circle.  Since any point on this portion of the circle can be uniquely described by its $x$-coordinate we can use the projection map $\rho_{top}: S^{(+)}\to (-1,1)$ defined by

$\rho_{top}(x,y) = x$

to map the open upper half-circle $S^{(+)}$ to the open interval $(-1,1)$.

This mapping is both continuous and invertible.  Some example projections are shown in the diagram below.

The pair $(\rho_{top}, S^{(+)})$ is called a chart.  There are similar charts for the open lower half-circle, the open right half-circle and the open left half-circle.  Taken together, these charts cover the whole circle forming an atlas.

Example:  For $n=2$, the space $\mathbb{R}^{2}$ is the plane.  Thus, a 2-manifold is one in which every point has a neighborhood that looks like an open portion of the plane.  The surface of a sphere or a torus are good examples.

The formal definition of a manifold is a bit involved, so let’s start with the condensed version and unpack it from there.

Definition[Manifold]: A topological manifold is a second-countable Hausdorff  space that is locally homeomorphic to Euclidean space.

Let’s first look at what it means for a space to be “second-countable”.

Definition[Second-Countable]: A space is said to be second-countable if its topology has a countable base.

Here a base $B$ for a topological space $X$ with topology $T$ is a collection of open sets in $T$ such that every open set in $T$ can be written as a union of elements from $B$.  The base $B$ is said to generate the topology $T$.

That $B$ is countable simply means that there exists an injective function $f: B\to \mathbb{N}$.  In other words, the elements of $B$ can be put in one-to-one correspondence with a subset of the natural numbers.

Next, let’s consider Hausdorff spaces.

Definition[Hausdorff Space]:  A Hausdorff space $H$ is a topological space such that for all $x,y\in H$ with $x\neq y$, there exist neighborhoods $U, V$ in the topology with $x\in U, y\in V$, and $U\cap V = \varnothing$.

Sometimes Hausdorff spaces are referred to as separated spaces.  This is illustrated below.

As Penrose puts it

A Hausdorff space has the defining property that, for any two distinct points of the space, there are open sets containing each which do not intersect.

The last bit we need is to understand what it means to be locally homeomorphic to Euclidean space.

Definition[Homeomorphism]:  Let $S$ and $T$ be two topological spaces.  A homeomorphism $h$ is a function $h:S\to T$ that satisfies the following properties:

1. $h$ is a bijection.
2. $h$ is continuous.
3. $h^{-1}$ is continuous.

Definition[Local Homeomorphism]:  Let $S$ and $T$ be two topological spaces.  A function $f:S\to T$ is called a local homeomorphism if for every point $x\in S$ there exists an open set $U$ containing $x$, such that the image $f(U)$ is open in $T$, and the restriction $f|_{U}: U\to f(U)$ is a homeomorphism.

So, to be locally homeomorphic to Euclidean space means that every point of an $n$-manifold has a neighborhood that is homeomorphic to an open Euclidean $n$-ball

$B^{n} = \{(x_{1}, x_{2}, ..., x_{n}) \in \mathbb{R}^{n}: x_{1}^{2}+x_{2}^{2}+\ldots +x_{n}^{2}<1\}$

In other words, if $H$ is a second-countable Hausdorff space, then it is locally homeomorphic to Euclidean space iff there exists a function $f: H\to \mathbb{R}^{n}$ such that for each $x\in H$ there is a neighborhood $U$ of $x$ such that $f|_{U}:U\to B^{n}$ is a homeomorphism (where $f(U) = B^{n}$).

References:

Differential Forms: A Complement to Vector Calculus, by Steven H. Weintraub

The Road to Reality, by Roger Penrose

Fundamentals of Topology, by Benjamin T. Sims

http://en.wikipedia.org/wiki/Second_countable

http://en.wikipedia.org/wiki/Manifold

http://en.wikipedia.org/wiki/Base_(topology)

http://en.wikipedia.org/wiki/Hausdorff_space

# Chuck Missler and the Existence of Infinity

Along with my (slow) endeavor of exploring and critiquing the ideas undergirding intelligent design, I want to resume a project I started a while back that involved addressing the claims of Chuck Missler.  As I have previously mentioned, Chuck Missler is a well educated man.  That being said, I also get the strong impression that Missler pretends to know a lot more than he really does.  What annoys me the most is that he seems to present himself as an expert in, well, just about everything.

In his Bible studies, Missler lectures his flock on everything from cosmology to quantum mechanics to information theory and beyond.  Missler has everything figured out it seems, and, while he might deny it, presents himself as having just about everything figured out.  It’s shocking that he hasn’t been awarded a Nobel prize.  Listening to his talks, one cannot help but be impressed by the depth and breadth of his knowledge.  However, once one gets over the dazzle of Missler’s apparent expertise in everything, one begins to pick up on very questionable claims and ideas.  Most of the time Missler alludes to very deep ideas, but glosses over them to spare his audience the details.  OR, maybe Missler really doesn’t know or understand the details.  This is my suspicion, which is motivated by several cracks in his intellectual veneer amounting to questionable claims on his part.

The first issue I’d like to address regards the size of the universe.

Missler is fond of proclaiming there are two central mathematical concepts that we don’t find in nature: Infinity and randomness.  I’ll save randomness for another post.

Does Infinity Exist?

Certainly infinity exists conceptually, but the question is whether it describes anything real.  In particular, Missler focuses on size.  This can go in two directions: (1) small scale infinity, and (2) large scale infinity.

Small Scale ∞

Most people are familiar with number lines.

For our purposes we can focus on the interval $[0,1]$.  We can cut this interval in half and get $[0,\frac{1}{2}]$.  This can be cut in half again to get $[0,\frac{1}{4}]$.  In fact, we could carry out this cutting procedure indefinitely, always obtaining a new interval $[0, \frac{1}{2^{n}}]$.  This means that we can make the interval as small as we like.  Put differently, we can cut the interval infinitely many times in the sense that there will never be a limit to the number of times we can cut the interval in half.

Now suppose that our interval $[0,1]$ models a length in space-time, say an inch.  If the correspondence were true, then it would follow that space could be indefinitely halved.  According to Missler, this is actually not true.  It turns out that there is a limit to smallness in space-time.  In other words, there is a smallest distance.  This distance is known as the Planck length, which is defined by

$\ell_{p} = \sqrt{\frac{\hbar G}{c^{3}}}$

where $\hbar$ is Planck’s constant, $G$ is the gravitational constant, and $c$ is the speed of light.  This length is exceedingly small, a mere $1.6\times 10^{-35}$ meters (approximately).  Missler is fond of saying that if at any point you divide a distance into lengths smaller than $\ell_{p}$, then you lose locality, the thing you are cutting is suddenly everywhere all at once.  What he concludes is that our reality is actually a “digital simulation”, terminology he uses purposely to insinuate that our world is created by God.  Such loaded language seems to be characteristic of Missler.

So, what is the nature of this mysterious length?  Where does it come from and how do we know it is the smallest length?

The Planck length has profound relevance in quantum gravity.  Put differently, it is at this absurdly tiny scale that quantum effects become relevant and the question regards how gravity behaves or should be understood at this scale.  Both general relativity and quantum field theory must be taken into account.  The definition of the Planck length now makes sense since the speed of light $c$ is the natural unit that relates time and space, $G$ is the constant of gravity, and $\hbar$ is the constant of quantum mechanics. So the Planck scale defines the meeting point of gravity, quantum mechanics, time and space.

Theoretically, it is considered problematic to think of time and space as continuous because we don’t appear to be able to meaningfully discuss distances smaller than the Planck length.  Unfortunately, there is no proven physical significance to the Planck length because current technology is incapable of probing this scale.  Nevertheless, current attempts to unify gravity and QM, such as String Theory and Loop Quantum Gravity, yield a minimal length, which is on the order of the Planck distance.  This arises when quantum fluctuations of the gravitational field are taken into account.  In the theory, distances smaller than $\ell_{p}$ are physically meaningless.  Two “points” at distances smaller than $\ell_{p}$ cannot be differentiated.  This seems to suggest that space-time may have a discrete or “foamy” nature rather than a continuous one.  Unfortunately for Missler, however, we just don’t know at this point.  Thus, the declarative nature of his claims are hasty.

Nevertheless, even if this turns out to be the case, the question becomes: what follows from that?  As mentioned above, Missler is fond of saying that the universe is a “digital simulation”.  Certainly the term “digital” would be apropos, but his use of “simulation” seems loaded and dubious.  “Simulation” suggests that this world isn’t the “real” world.  It suggests that our world merely imitates some meta-world.  Of course, Missler purposely uses the term as a way to smuggle in a simulator.  That simulator is God, and the meta-world is the spiritual world.

Large Scale ∞

The more questionable of Missler’s claims regards the size of the universe.  He brazenly declares that we have discovered the universe to be finite.  This is just flat out false, and such carelessness makes me question his credibility.  It is likely that he is simply confusing the observable universe with the universe proper.  There is no doubt that the observable universe is finite.  It is estimated to have a radius of 46 billion light years and due to expansion grows ever larger.  However, this does not necessarily imply that the universe proper is finite in size.  Sir Roger Penrose, one of the most respected mathematical physicists in the world, says, “it may well be that the universe is spatially infinite, like the FLRW models with $K = 0$ or $K<0$.” (see The Road To Reality, p. 731)

Note: FLRW models refers to  Friedmann–Lemaître–Robertson–Walker models and $K$ is a density parameter governing the curvature of the universe.

Even a quick search on Wikipedia reveals that, “The size of the Universe is unknown; it may be infinite. The region visible from Earth (the observable universe) is a sphere with a radius of about 46 billion light years, based on where the expansion of space has taken the most distant objects observed.”

In fact, the possibility of an infinite universe is the stuff of some multiverse models.  Max Tegmark, a mathematical physicist at MIT puts it this way:

If space is infinite and the distribution of matter is sufficiently uniform on large scales, then even the most unlikely events must take place somewhere. In particular, there are infinitely many other inhabited planets, including not just one but infinitely many with people with the same appearance, name and memories as you. Indeed, there are infinitely many other regions the size of our observable universe, where every possible cosmic history is played out. This is the Level I multiverse.

Tegmark goes on

Although the implications may seem crazy and counter-intuitive, this spatially infinite cosmological model is in fact the simplest and most popular one on the market today. It is part of the cosmological concordance model, which agrees with all current observational evidence and is used as the basis for most calculations and simulations presented at cosmology conferences. In contrast, alternatives such as a fractal universe, a closed universe and a multiply connected universe have been seriously challenged by observations. Yet the Level I multiverse idea has been controversial (indeed, an assertion along these lines was one of the heresies for which the Vatican had Giordano Bruno burned at the stake in 1600†), so let us review the status of the two assumptions (infinite space and “sufficiently uniform” distribution). How large is space? Observationally, the lower bound has grown dramatically (Figure 2) with no indication of an upper bound. We all accept the existence of things that we cannot see but could see if we moved or waited, like ships beyond the horizon. Objects beyond cosmic horizon have similar status, since the observable universe grows by a light-year every year as light from further away has time to reach us‡. Since we are all taught about simple Euclidean space in school, it can therefore be difficult to imagine how space could not be infinite — for what would lie beyond the sign saying“SPACE ENDS HERE — MIND THE GAP”? Yet Einstein’s theory of gravity allows space to be finite by being differently connected than Euclidean space, say with the topology of  a four-dimensional sphere or a doughnut so that traveling far in one direction could bring you back from the opposite direction. The cosmic microwave background allows sensitive tests of such finite models, but has so far produced no support for them — flat infinite models fit the data fine and strong limits have been placed on both spatial curvature and multiply connected topologies. In addition, a spatially infinite universe is a generic prediction of the cosmological theory of inflation (Garriga & Vilenkin 2001b). The striking successes of inflation listed below therefore lend further support to the idea that space is after all simple and infinite just as we learned in school.

So, it seems that unless Missler knows something all other physicists don’t, he is being much too hasty and cherry picking possibilities to support what he wants to be true.

# The Thesis of Modal Realism

A few post ago I was talking about formal systems.  As an example I talked a little about Hofstadter’s example of the MIU system and promised to talk about decision procedures.  I would hate to be a liar and so I do plan on returning to this matter.  However, it seems appropriate to me to take a diversion into the philosophical foundations of Tegmark’s view before delving into the more mathematical aspects.  For this reason, and in keeping with the announcement of my previous post, I am going to interact a bit with David Lewis’ work On the Plurality of Worlds.

So, what is Modal Realism?  From what I gather, the name was actually coined by Lewis, though he expresses certain regrets for calling it this based on some ambiguity in the term ‘realism’ and what it has come to mean in academia.  As one who is less familiar with the larger body of literature on realism, I find the name perfectly appropriate and satisfying.

The term ‘modal’ comes from modal logic which analyses the notions of necessity and possibility.  G.E. Hughes and M.J. Cresswell summarize this discipline similarly:

Modal logic can be described briefly as the logic of necessity and possibility, of ‘must be’ and ‘may be’.

‘Realism’ as Lewis intends to use it deals with ontology.  Put together, modal realism is

… the thesis that the world we are part of is but one of a plurality of worlds, and that we who inhabit this world are only a few out of all the inhabitants of all the worlds.

In other words, modal realism takes possible worlds semantics quite seriously as well as literally.  Upon reading this description, one may be tempted to think of modal realism as a multi-verse theory and indeed it is (we’ll see that it corresponds to Tegmark’s Level IV multiverse), but not in the sense often portrayed in popular literature.  The worlds of modal realism are not some distant universes located some vast, albeit finite, distance away.  Nor are they located in some alternate dimension of this reality.  As Lewis puts it,

They are isolated; there are no spatiotemporal relations at all between things that belong to different worlds.  Nor does anything that happens at one world cause anything to happen at another.

One should liken this to the way in which the vector space $\mathbb{R}$ is isolated from the finite field $\mathbb{Z}_{7}$, to sneak in some mathematics.

Lewis goes on to say,

The difference between this and the other worlds is not a categorial difference.  Nor does this world differ from the others in its manner of existing.  I do not have the slightest idea what a difference in manner of existing is supposed to be. [emphasis mine]

Thank you!!!  I expressed this very sentiment in A Philosophical Interlude.  It always feels amazing when one’s own thoughts are legitimized by world renown experts.  Okay, enough tooting my own horn.  The idea, here, is that some things exist, say, on Earth, while other things exist elsewhere and this is a difference in location of existing things (not a difference in any mode of existence).  Similarly, though somewhat more controversial, some things exist in the past, others now and still others in the future.  This is a temporal difference (we will deal with the nature of time in another post).  To extend this further, it seems reasonable that the same could be said for worlds.  Some things exist in this world and other things exist in other worlds and this is still a difference between existing things as opposed to a difference in any mode of existing.

For those not familiar with the language of possible worlds, I should hasten to point out that a world need not be a synonym for a universe (though it might be).  Rather, a world is a complete state of affairs, an entire reality.  Thus, in our world, if there is a God who created the universe, then clearly our world is more than just our universe.  Alvin Plantinga in his book The Nature of Necessity defines a possible world to be a maximal state of affairs and a maximal state of affairs $S$ is one such that for every state of affairs $S'$, $S$ includes $S'$ or precludes $S'$.  Note the similarity of this definition with that of completeness for a mathematical system:

A system is complete if, for every statement $p$, we can find a proof of $p$, or a proof of not-$p$.

Here “proof” refers to a demonstration within the system in question using the axioms and the rules of inference. In other words, a “proof”, here, of a statement $p$ within a system $S$ “consists of a sequence of statements, each of which is either an axiom or a logical consequence of certain preceding statements in the list, such that the last statement in the list is $p$.”

At this point, we needn’t get bogged down in technicalities.  The basic idea is that modality is best understood by actually taking possible worlds semantics as more than just semantics.  According to modal realism, all possible worlds are actual worlds, where actuality is taken to be indexical.