Mathematics FAQ

During a discussion on how our universe may have been created, a Christian Apologist is introduced to the concept of mathematical modelling and its role in helping humans to understand reality, in this case, within the field of cosmology.  His reaction is to attempt to discredit the field of study of mathematics! But all this does is reveal a lack of understanding of the nature of mathematics.

So here is a mathematics FAQ for Apologists based on questions that I have actually been asked...

1 Why is M-Theory a good explanation for the creation of the universe?
Mainly because it is consistent with other hypotheses, and rationalises what used to be apparent contradictions in other hypotheses, such as the various manifestations of string theory. It requires no pre-existing intelligent creator being(s) which makes it self-sufficient.  It also has a basis in mathematics, it is consistent within itself, and it is testable. 

2 But how can mathematics be reliable when it has "built in glitches"  like the ones listed here?

2.1 The list is of unsolved problems in mathematics from Wikipedia, but they are not "glitches".  A glitch is a malfunction or a fault. An unsolved problem is not a malfunction, it is a problem waiting to be solved.  In the context of a  mathematical solution, a glitch would be a mistake in reasoning caused by human error, resulting in an incorrect solution which would be self evident by peer review or double-checking. 

3 Is division by zero a fundamental glitch in mathematics?

3.1 No, because it is not a malfunction or a fault. Division by zero is a well known example of an arithmetic operation with an undefined result.  Other examples are...

0/0

∞/∞

0 × ∞

∞ − ∞

1^∞

∞^0

0^0

3.2 There is a famous arithmetic "proof" that 1=2 which includes an erroneous step where a divide by zero is assumed to have a defined result, but many people fail to spot it. That's not a glitch in mathematics, rather the person who wrote the "proof" broke a mathematical rule. In philosophical terms they broke the rules of reality and that's how they arrived at an unreal result! 

3.3 In the philosophical context of mathematics as a means of modelling reality, perhaps expressions which lead to undefined results are revealing an important aspect of reality. Could they indicate evidence of God?

3.4 Divide by zero can be used to illustrate how mathematics models reality. For example:

3.4.1 Imagine you have to share 12 apples among 4 people. Each person gets 3 apples. If there are 2 people, they get 6 apples each. If there's one person, they get 12 apples. But what if you had to share 12 apples among no people? How many apples does each person get? The answer is undefined because you can't give something to no one. This isn't a glitch in mathematics - it's mathematics describing reality.

3.5 If, when solving a problem mathematically one  encounters a divide by zero, is that a dead end?
3.5.1 No! Mathematics can be used to analyse the reality of this situation by considering limits. It's a very simple idea - if we can't divide something by zero, then let's divide it by numbers which get smaller and smaller so that they approach zero but never get there. 

Consider the expression, 1/x  We can see this will be undefined if x =0. But if we try different values for x which approach zero we see that 1/x tends to infinity as x tends to zero.  In fact the concept of limits is one of the most powerful in mathematics. 

4 Divide by zero causes computer programmes to crash - doesn't this demonstrate a glitch in mathematics?
4.1 No.  Any indeterminate form can result in malfunctions if they are assumed to have results that are not undefined, but that is a mistake by the programmer.  it is not a glitch in the mathematics, it is a bug in the code.  It's human error.  The code does not reflect reality.  Consider a program which calculates how to share a number of apples (a) among a number of people (p) but hasn't been coded to deal with the situation when there are no people and p=0  (see 3.4.1) 

4.2 Let's also consider Javascript (among other computer languages) which does not crash when there is a divide by zero. Does this mean Netscape have resolved  a "glitch" in mathematics? Of course not! They've developed a programming language which can model reality better than other computer languages.

4.3 When division by zero pops up in algorithms and computer processes, doesn't this create a glitch?
4.4 It only creates a glitch (more commonly known as a "bug") if the algorithm or computer process isn't written correctly in order to deal with indeterminate results.  See 4.1

5 Is mathematics a human construct?

5.1 There is a philosophical debate as to whether mathematics is discovered (as described by Plato) or invented. There is no consensus, but it is a fact that on many occasions over thousands of years, mathematical models which appeared to be "unreal", perhaps philosophical curiosities, have turned out to be models with applications in physics, science and technology.  Perhaps the best answer is to say it doesn't matter because calculations provide the same results regardless of what we believe.

5.2 If divide by zero is a "glitch" which results from mathematics being a "human construct" then this implies an intelligence greater than human intelligence (or perhaps God) could share 12 apples among zero people. Mathematics tells us it's impossible. Could God, or non-humans with superior intelligence, perform this impossible feat? 

6 Does the use of different number bases (such as base ten and binary) illustrate that mathematics is a human construct?

6.1 No. Number bases are an example of using different languages to describe the same thing. The reality is not changed by the language we use.  

6.2 For example, a person who weighs 140 lbs using base ten, would weigh 10001100 lbs in binary, 214 lbs in base 8 and 8C lbs in base 16. But of course, their mass is identical in every example, regardless of which number base is used. 

6.3 So the language and symbols of mathematics are invented by humans, but the reality that is modelled by mathematics is not affected by the language or symbols used.  Perhaps if human beings had 8 fingers and toes instead of 10, we'd all use base 8 intuitively instead of base 10. 


Interesting stuff for future discussion...

The best book to read in order to understand mathematics and reality is "The Road to Reality" by Penrose which is here

Principia Mathematica - an attempt to describe a set of axioms and inference rules in symbolic logic from which all mathematical truths could in principle be proven. It takes over 360 pages to prove that 1+1=2 - You can find it here

Benford's Law
http://en.wikipedia.org/wiki/Benford's_law

The History of complex numbers
http://en.wikipedia.org/wiki/Complex_number#History

The history of negative numbers
http://en.wikipedia.org/wiki/Negative_number#History

The history of zero
http://en.wikipedia.org/wiki/0_(number)#History

Types of infinity
http://vihart.com/how-many-kinds-of-infinity-are-there/

Does Infinity exist?
http://www.livescience.com/37077-infinity-existence-debate.html

How To Evangelise

Before the "how" it's useful to know the "why". Three reasons :

A) Jesus commands it
B) Everyone’s greatest need is eternal salvation
C) To see God glorified

Explained here.

So... How to evangelise? Here are some handy tips:

For example: 

Lead with a probing question. Ask a question that will lower the person's guard and get them thinking about larger existential issues, making them receptive to an exchange of ideas. A question like, "What do you think happens when you die?" or "Do you believe in an afterlife?" can be effective at transitioning the conversation into your territory.

A most effective evangelistic tool that you can use is a survey. You can ask four questions about an individual's life, and after you know the needs and beliefs, witness to each based on the point of view.

Perspectives

A Christian Apologist becomes fed up defending his position...

You challenge the Christian perspective, you like to ask all of the questions yet you provide few answers regarding the basis for your own perspective and their underlying assumptions. Perhaps you have well thought-out positions on the matter, but I have yet to see them. It seems you have no logical basis for justifying your own perspective, and all you offer are challenges to mask that fact.

I can't help thinking... "The lady doth protest too much, methinks."  

In fact I enjoy providing detailed, logical and reasoned responses to the many questions the Apologist asks me (and this blog wouldn't exist if it wasn't for questions from Apologists!)  Anyway, here are some of the main topics...


God

Gods compared 
Analysis of arguments for God
The existence of God
Models of reality - is God necessary?
Quantum Physics and God

Can God be defined? 
Do our values come from God
Where does the idea of God come from?
God's Atrocities in the OT

God from the point of view of Holocaust Survivors Part 1
God from the point of view of Holocaust Survivors Part 2
Can there be evidence that God does not exist?
Is belief in God a function of the brain?
Is God a conspiracy theory?
Is there evidence for God?
Which gods are the least improbable?

The Origin of Our Universe

Explanations of how our universe came to be
Explanations compared

The Nature of Reality

Models of Reality

Philosophy

Verification 
Needs
Burden of Proof.
Free Will is an Illusion
Determinism

Morality

Morality: "atheistic regimes"[1]
Morality: "atheistic regimes" [2]
Morality: "The Bible provides an absolute moral reference"
Morality: "Reigns of Terror" 
Morality: "Moral Relativism"

The Watchmaker Variation

Introduction
A Christian apologist comes up with a version of Paley's Watchmaker argument during a discussion on M-theory and the creation of our universe. He uses a Rolls Royce instead of a watch. He states it thus...

"A Rolls Royce with all of the options is unlikely to occur though mindless mechanistic forces minus the input and impact of applied intelligence. If intelligent forces can alter dynamics, and non-intelligent forces can alter dynamics, then the impact of both ought to be considered in a comprehensive perspective."

If we translate this into plain English, the first part of the argument appears to be that a Rolls Royce does not occur though natural processes and is the result of design and conscious actions by designers and manufacturers.  This is almost true, the only problem being the word "unlikely" when a better word would be "impossible". It is actually impossible for a Rolls Royce to occur by natural processes because it's not a product of nature. It doesn't display any of the attributes that are seen in the structure of nature. 

The apologist then extends the analogy to the creation of the universe. The Apologist is saying that our universe could have been designed and created by a being who is more advanced than us, and that science should consider this possibility. 

This is also true, and science does consider this possibility. It seems a universe can indeed be manufactured, using advanced technology, perhaps using particle accelerators or as simulations.  (More information here). This is rather like human beings creating diamonds or snowflakes. Perhaps one day we will be able to examine the fabric of the universe and tell if it is natural or artificial in the same way that we do with diamonds. But at the moment there's no way of knowing. But there is with a Rolls Royce (or a pocket watch).

1 Car maker / Watchmaker
Applying this Rolls Royce analogy to naturally occurring universes, such as explained by M-Theory, Cyclic Models, Quantum Fluctuations etc. is a restatement of Paley's Watchmaker argument which argues for a designed and created universe (which then becomes an axiom for the existence of God). 

The common objections to this analogy (it's actually a disanalogy - in fact it's a collection of disanalogies) are well known:

1a) It is a category mistake to make an assumption based on the processes that produce a manufactured artefact and apply that assumption to the products of natural processes. 

1b) The designers of a Rolls Royce have foresight and a future purpose in mind. The processes described in natural processes, have no purpose or foresight and therefore are incapable of design and have no requirement for a designer.  

1c) Assuming that the universe is designed because an item that we know to be designed, was designed, is just begging the question. 

1d) It's possible God created the Rolls Royce but for some reason, Religious Apologists never claim that man made objects were created by God. If we ask the Apologist to explain how we know the Rolls Royce was not created by God, the argument collapses.

1e) The disanalogy between a car maker and a universe-maker is significant. Not only is the term "universe-maker," beyond the bounds of possible experience, but also the hundreds of people involved in the construction of a Rolls Royce - from the miners of the metals, the farmers of the leather, to the draftsmen, craftsmen, factory workers, and distributors - would suggest many gods are involved in universe-making. 

1f) Another disanalogy exists with respect to the Rolls Royce designers - they were created by their parents but the universe-maker is described as having no parents

1g) Some parts of the Rolls Royce do not work perfectly and/or require maintenance yet theologians claim Creation is perfect. So again the Rolls Royce is a disanology when compared to Creation. 

1h) If the design of the universe is assumed to be imperfect (as with man made artefacts) then this implies the Creator of our universe is neither all good or all-powerful which contradicts the alleged attributes of God.

1i) The purpose of a Rolls Royce is evident by observation, but the purpose of the universe is not.

Time, Quantum Mechanics and Quantum Fields

Why is time not an operator in Quantum Mechanics even though it is observeable?





One should be careful to distinguish between non relativistic quantum mechanics (Schrödinger equation, etc.) and quantum field theory.

For nonrelativistic quantum mechanics, it is not so surprising that time and space are treated differently, with position being an operator and not time. After all, this is also what happens in Newtonian mechanics: time is absolute, and part of the background, and all other observables are functions of time. This paradigm underlies the formulation of the fundamental problem of Newtonian physics: to determine how a system evolves in time. Time cannot be an observable because an observable is a function of what we consider the system's "state", but the state is considered a function of time in the first place (so time is the independent variable).

Quantum field theory is fully compatible with special relativity, and therefore must treat space and time on equal footing. In nonrelativistic quantum mechanics, position is an observable whereas time is a parameter. That is, position is a function of the state, whereas time is used to label states. In formulating quantum field theory, we therefore have a choice between making spatial coordinates into parameters, or making time into an observable. This choice is discussed by Srednicki:

We can solve our problem, but we must put space and time on an equal footing at the outset. There are two ways to do this. One is to demote position from its stat us as an operator, and render it as an extra label, like time. The other is to promote time to an operator.

Let us discuss the second option first. If time becomes an operator, what do we use as the time parameter in the Schrodinger equation? Happily, in relativistic theories, there is more than one notion of time. We can use the proper time τ$\tau$ of the particle (the time measured by a clock that moves with it) as the time parameter. The coordinate time T$T$ (the time measured by a stationary clock in an inertial frame) is then promoted to an operator. In the Heisenberg picture (where the state of the system is fixed, but the operators are functions of time that obey the classical equations of motion), we would have operators Xμ(τ)$X^{\mu }(\tau )$, where X0=T$X^0=T$. Relativistic quantum mechanics can indeed be developed along these lines, but it is surprisingly complicated to do so. (The many times are the problem; any monotonic function of τ$\tau$ is just as good a candidate as τ$\tau$ itself for the proper time, and this infinite redundancy of descriptions must be understood and accounted for.)

One of the advantages of considering different formalisms is that they may suggest different directions for generalizations. For example, once we have Xμ(τ)$X^{\mu }(\tau )$, why not consider adding some more parameters? Then we would have, for example, Xμ(σ,τ)$X^{\mu }(\sigma ,\tau )$. Classically, this would give us a continuous family of worldlines, what we might call a worldsheet, and so Xμ(σ,τ)$X^{\mu }(\sigma ,\tau )$ would describe a propagating string. This is indeed the starting point for string theory.

Thus, promoting time to an operator is a viable option, but is complicated in practice. Let us then turn to the other option, demoting position to a label. The first question is, label on what? The answer is, on operators. Thus, consider assigning an operator to each point x$\mathbf{x}$ in space; call these operators ϕ(x)$\varphi (\mathbf{x})$. A set of operators like this is called a quantum field. In the Heisenberg picture, the operators are also time dependent:

ϕ(x,t)=eiHt/ℏϕ(x,0)e−iHt/ℏ$\varphi (\mathbf{x},t)=e^{iHt/\hslash }\varphi (\mathbf{x},0)e^{-iHt/\hslash }$.

Thus, both position and (in the Heisenberg picture) time are now labels on operators; neither is itself the eigenvalue of an operator.

So, now we have two different approaches to relativistic quantum theory, approaches that might, in principle, yield different results. This, however, is not the case: it turns out that any relativistic quantum physics that can be treated in one formalism can also be treated in the other. Which we use is a matter of convenience and taste. And, quantum field theory, the formalism in which position and time are both labels on operators, is much more convenient and efficient for most problems.