The Moral Arguments: A Rejoinder

Recently I began a delightful discussion/debate concerning two versions of the moral argument for the existence of God with my friend Tyler Dalton McNabb. His opening post can be read here. My first response can be read here and his followup response here.

In response to Tyler’s latest post, I’d like to start by pointing out that there are two central issues at play in this discussion. One regards the nature of morality itself and the other regards the “proper function” of our faculties and our ability to access truth.

As of right now, I hold firm to the position that the true nature of morality is undecidable. I contend that this is enough to neutralize the classical moral argument, so I’ll focus more on the epistemological version.

Now, while I hesitate to agree that belief in objective moral values and duties is warranted (since “warranted” is a technical term that still isn’t adequately understood), I will admit that belief in objective moral values and duties can be reasonable. Where I think the epistemological argument truly fails is premise (1):

If NE is true, belief in objective moral values and duties cannot be warranted.

Tyler conceded that premise (1), so stated, is false, since it is possible on NE that our faculties be such that they can access truth and form corresponding true beliefs. What Tyler seems to resort to is something like the following:

Let \alpha represent the actual world (at least with respect to our perspective). Furthermore, let W be the event that belief in objective moral values and duties is warranted on NE in \alpha. Then

P(W) < \frac{1}{2}

which says that the probability of W being the case is less than 50%. If this is the case, then it is more reasonable to think that one is not warranted in believing in objective moral values and duties in \alpha. But is this really the case?

First, I’m not sure that there is a well-defined probability measure here. For instance, if there are infinitely many universes, then infinitely many of them have beings with the right cognitive equipment to access truth.

Second, Tyler says that he sees no reason to think that our cognitive faculties are reliable. This is strange to me, since I see every (well maybe not every) reason to think that our faculties are reliable. Why?

(a) This may be a necessary part of consciousness. That is, being conscious may very well entail the ability to form true beliefs and to check them on some level.

(b) We can test our faculties against real life. The fact that we don’t die and the fact that we can successfully navigate our world environment attest to the fact that our brains produce true beliefs.

For these reasons, I think the epistemological argument fails to go through. It is entirely possible, even likely, that our cognitive faculties are reliable even on NE. At this point, I’ll now pass the ball back to Tyler.

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Bill Nye vs Ken Ham: A Break Down of the Debate – Part 1

Lately the interweb has been abuzz with anticipation and now reflection upon the debate between Bill Nye and creationist Ken Ham of Answers in Genesis. As expected, the debate was more polarizing than productive. As far as the actual debate, however, Nye won by a landslide. But that’s easy to say (or type), so let me tell you why Nye was the clear winner.

I’ll analyze each interlocutor in turn.

Ken Ham: Opening Statement

Despite my distaste for Ham, I’ll admit that he started out in a promising way.  He clearly laid out the debate topic, which was

Is creation a viable model of origins in today’s modern scientific era?

From here one would hope, if not expect, that one arguing in the affirmative would proceed to marshal evidence and justification for creation being a viable model of origins. Unfortunately, as soon as Ham sets up an appropriate trajectory, he deviates from it to address a completely different matter, namely, whether creationists can be scientists. He laments that secularists have hijacked the term “science” and then proceeds to introduce us to several scientists who happen to be Biblical creationists. While I can understand that Ham is trying to dispel a misconception, it seems that he is bordering on an argument from authority. The salient issue, however, is not whether creationists can be scientists, but whether the idea of creation itself is (a) scientific in any relevant sense and (b) whether it holds up under scrutiny (of any kind).

Ham seems to get back on track (briefly) by rightly pointing out the need to define terms correctly. This is always important for fruitful communication. Beginning with “science”, Ham distinguishes between observational science and what he calls historical science. Observational science is what produces technology, whereas historical science obviously deals with past events that cannot be directly observed.  Under such a distinction, origins would clearly belong to the latter category. But is there really such a fine distinction? It is true that we cannot directly observe past events, but that does not mean that observational science is not involved in studying the past. What we observe are the clues of the present. Past events get encoded (more or less) into the future. We observe this and then attempt to work the “encoding path” backwards to the event in question.  This is how crime scene investigation works. Experts in arson can tell a lot about how a fire started, how fast it burned, where it started, and many other things by examining the remains of burnt houses and buildings. The event is in the past, but many important bits of information are encoded that can be deciphered.  Yes, Ham is right, no one was there to witness our origins, which is why we are trying to determine the best explanation and model.

Ham seems to think that this gives him an out. No one was there to witness these things, therefore it is legitimate and viable to simply base one’s views on (a very particular interpretation of) the Bible. While I will agree to the importance of world views and starting assumptions, Ham has not demonstrated the legitimacy of using an ancient religious text as a foundational starting point.

After rambling on about there being a difference in philosophical world views, Ham ends with a doozy.  He literally concludes that

Creation is the only viable model of historical science confirmed by observational science in today’s modern scientific era (emphasis mine).

Ummmm…. what?  At no point in his presentation did he present anything remotely close to evidence for this audacious conclusion.  He spent the entire time complaining about secularist hijackers and concluded that creation is the only viable model of origins.

So, Ham had a promising start, but went down in a ball of flames. In the next post, I’ll critique Nye’s opening presentation.

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 ith 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<j}\left(\frac{\partial^{2}f}{\partial x_{j}\partial x_{i}} - \frac{\partial^{2}f}{\partial x_{i}\partial x_{j}}\right)dx_{i}\wedge dx_{j} = 0

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.

1. http://plato.stanford.edu/entries/naturalism/

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.

LUFTOCirc

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

LUFTOUpperHalfCirc

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.

LUFTOUpperHalfProj

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.

LUFTOHausdorff

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