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<x<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}).


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






Christian “Trump Cards” – Part 2

Recently I began compiling something of a list of, what appear to be, commonly appealed to “trump cards” by certain Christians during attempts at rational dialogue.  While I certainly don’t intend to implicate all Christians (or even only Christians), I have noticed a regular occurrence of these “moves” in a multitude of discussions.  In Part 1 I briefly discussed the appeal to faith.  From my own experience discussions seem to either begin or end with this “trump card”.  For part 2 I would like to analyze what seems to be a background assumption of many Christians.  This “trump card” isn’t necessarily appealed to directly; it most often operates behind the scenes, but has noticeable effects.  Here it is:

A: If one does not believe that Christianity is true, then that person hates God and rejects “His” gift of salvation.

The difficulty with this background assumption is that it goes to the heart of the Christian message.  Humans are the creation of God, but are separated from “Him” because of sin.  Ultimately, this means that humans are headed for one of two fates: eternal heaven or eternal hell.  Those who repent and accept Christ are granted eternal heaven, while those who do not get eternal hell.

Now, eternal hell (which is generally taken as never-ending punishment) is rather harsh.  So, to alleviate the uncomfortable dissonance, it is my opinion that many Christians are driven to hold A.  It is much easier to believe that unrepentant God-haters who despise “His” sacrifice deserve never-ending punishment than it is to accept that honest seekers and skeptics could somehow “miss the boat” and end up in eternal torment.

The problem, of course, is that A is a completely unwarranted assumption.  More than that, it is unfalsifiable in the sense that all counter-examples can be dismissed.  Since no one can expose his or her first person perspective to direct analysis, we must always go off of what people report about themselves.  The person who accepts A, however, can simply maintain that the non-believer is deceiving himself or herself by suppressing the truth (more on that at a later time).

Nevertheless, let’s take a look at A itself and see if it makes better sense to adopt its negation.  Note that A is a conditional statement.  Let  stand for the simple proposition A person P believes that Christianity is true.  Let H represent the proposition P hates God and rejects “His” gift of salvation.  Then the assumption A can be expressed as

\lnot C\rightarrow H

The negation of this is therefore

\lnot(\lnot C\rightarrow H)\equiv \lnot C \wedge \lnot H

In English this would read

\lnot A: Person P does not believe that Christianity is true, but does not hate God or reject “His” gift of salvation.

Let’s see why \lnot A is more likely true than A.  If we look again at the component propositions \lnot C and H we run into an immediate problem.  Notice that \lnot C is a proposition that refers to the noetic status of a person.  It makes a claim about what a person believes.  In other words, it reports that some person takes the Christian system of belief to be false or not to correspond to reality.  The proposition H, by contrast, refers to a directed feeling or emotion of a person.  So, the assumption A claims that a certain state of unbelief with respect to some propositions implies an associated emotion with respect to the content of those propositions.  This is a very queer claim indeed.  Generally, the status of one’s belief has no connection to how one feels about the content of the belief.  For instance, I don’t believe in fire-breathing dragons.  In other words, I take it that they do not exist.  But this says nothing of how I feel about fire-breathing dragons.  In fact, I think fire-breathing dragons, while scary, are pretty awesome.  So, if this connection fails to hold in general, why should we believe it holds in the specific case of Christianity?  It is true that some people hate the idea of the Christian God, but there is no evidence to suggest that every non-believer does.

The underlying issue here is that many Christians conflate two types of “rejection”.  Let’s call the first type of rejection relational rejection and the second type propositional rejection.  Relational rejection involves rejecting a person or something a person is offering.  It involves a negative feeling toward the person or thing offered by the person.  For instance, if a boy asks out a girl and she says “no”, then she has relationally rejected the poor boy.  She is saying that she does not like the idea of having a particular type of relationship with him.  Propositional rejection, by contrast, is simply to not accept a proposition as being true.  For example, suppose a girl is asked whether she thinks that some boy will ask her out.  Suppose she says “no”.  Then in this case she is engaging in propositional rejection.  That is, she is rejecting the proposition that some boy is going to ask her out.  Notice that this rejection says absolutely nothing about whether she wants the boy to ask her or not.

The confusion arises from a failure to see the distinction between the proposition that one thinks is either true or false, and the content of the proposition that one may or may not have a feeling about.  For the Christian, the content of the belief is so central and personal that disbelief is automatically taken to be personal.  But there is no reason to think that disbelief is always (or even often) of a personal nature.  Thus, the Christian who holds A is going to have to summon some really compelling evidence.  Next time I’ll address one such argument claiming that truth is a person and hence rejection of the “truth” amounts to rejecting the person.

Some Common Christian “Trump Cards” – Part 1

Talking with Christians (and religious people in general) is a mixed bag.  Sometimes you get awesome intelligent people along with a really great discussion (even if you end up disagreeing).  Most of the time, however, the discussion ends up in frustration.  Having quite a bit of experience, I have put together a list of some of the most commonly used discussion killing “trump cards” played by many Christians.  I’d like to address each one in a separate post.  In analyzing each, I hope this will be useful to both skeptics and believers to promote more fruitful dialogue.  But before I get to the list, I’d like to give an explanation as to the root of these “trump cards”.

Controlling Assumptions

While there is no doubt that humans are rational beings, there is also no doubt that humans are emotional beings.  Unfortunately, at the conscious level, we are much more influenced by the latter than the former.  Emotions and other psychological effects are so powerful, that humans have to work quite hard to be rational in the midst of them.  This is not completely bad, since our drive, hope, and motivation are essential aspects our success.  The problem is striking a correct balance, which is obviously fairly difficult.

Here lies one of the difficulties with Christianity.  Not only is Christianity a system of belief, it is a system of belief saturated in emotion.  It is a system designed to address the core of human longings: purpose, meaning, forgiveness, justice, belonging, love, hope, peace, etc.  It provides a complete mental framework from which to operate.   This, of course, is not a bad thing, but the practical effects of adopting it can make it rather difficult to maintain objectivity.  One is easily blinded to the cold matters of truth when it just feels right.

This is an example of something I call a controlling assumption.

Definition [Controlling Assumption]A controlling assumption is an assumption that once adopted sets a mental framework that interprets all data to be consistent with the assumption, even data contrary to the assumption itself.

One might describe a controlling assumption as a self-preserving assumption.  It is intimately related to the idea of confirmation bias.  For example, consider the famous psychological experiment where several researchers checked themselves into a mental hospital.  Once admitted, they acted completely normal.  One would hope that the doctors of the hospital would be able to recognize that these “impostors” were completely sane and free of mental illness.  In other words, the healthy should be distinguishable from the sick.  The problem, however, is that the doctors were under the influence of a controlling assumption, namely People who enter the hospital as patients have a mental illness.  Because of this, the doctors interpreted the researcher’s normal behavior as symptomatic of their supposed “neuroses/psychoses”.  Even as the researchers documented these things with meticulous notes, the doctors recorded in their charts that “Patients engage in note taking behavior” as if it was pathological.  Ironically, those who were actually mentally ill caught on to the researchers almost immediately.

Although not always the case, the Christian belief structure can operate much like a controlling assumption, and part of the self-preserving nature of the belief structure manifests itself in various “trump cards” when being challenged.  I address the first of these below.

“Trump Card” 1  -[You just have to have faith]

Generally, the very first response to any rational challenge is an appeal to faith.  How “faith” is being used, however, is not generally very clear.  When pressed for a definition, most respond by quoting Hebrew 11:1, which says, “Now faith is the assurance of things hoped for, the conviction of things not seen.” (NAS)  Using this as a definition, what the Christian is saying is that one just needs to have assurance.  My initial thought is, “Oh, is that all I need?”.  Of course, even if Christianity is something I hope for, the problem is that I don’t have assurance.  This is what I am seeking, but not finding.  It strikes me a bit like saying to an addict who is asking how to stop, “Well, you just need to stop.”  That won’t be received well, because that is the very problem that needs addressing.

The go to response from here is generally to deny that faith has anything to do with the intellect, but is a matter of the heart.  Again, it isn’t at all clear what this is supposed to mean, nor is it clear where such an idea is expressed in the Bible.  In fact, the Greek word for “heart” in the New Testament is καρδια or kardia, which was taken to include the whole self, including the faculty and seat of the intelligence.  Thus, to say that faith is a matter of the heart and not the mind is an artificial and incorrect distinction even measured against the Bible.

Finally, telling someone that faith is required only pushes the problem back a step.  Why is faith required?  How does one know this?  And what guides where I place my faith?  After all, many religions appeal to the very same requirement.  So, which does one choose?  I think the issue clearly reveals that faith is not an epistemological tool that yields knowledge.  This can be more rigorously demonstrated as follows.

(1) Suppose that faith provides a means of knowing something.

(2) The Christian has faith and so knows Christianity to be true.

(3) Therefore, from the definition of “know” it follows that Christianity is true.

(4)  The Mormon has the same sort of faith and so knows Mormonism to be true.

(5)  Therefore, Mormonism is true.

(6)  Christianity and Mormonism are incompatible systems of belief.

(7)  Therefore, either Christianity or Mormonism (or both) is false.

(8)  If Christianity is false, then we get a contradiction with (3).

(9) Thus, Christianity must be true and Mormonism false.

(10)  If Mormonism is false, then we get a contradiction with (5).

(11)  Thus, Mormonism is also true, which contradicts (7).

(12)  Therefore, since our initial assumption leads to a contradiction, it must be the case that (1) is false.

Of course, one could deny that Christians and Mormons have the same faith, but then one would have to wonder how we could possibly distinguish between “real” faith and “fake” faith.  One would then have to appeal to something other than faith anyway.