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