*My interest in this piece is the fact that I am gathering information from various sources in order to validate the work of Cornelius Van Til. Van Til is recognized as formulating a defence of the Christian faith variously known as the Transcendental Argument or (more broadly) Presuppositional Apologetics.*

Presuppositional Apologetics recognizes that people come to various truth claims with a readymade worldview through which all truth claims are interpreted. Therefore a persons epistemological background is seen to play a crucial part in what is recognized as true or not true (or whether "truth" is even a valid category!). It is therefore the presuppositions a person holds that determine how those claims are judged. The high claim of Presuppositional Apologetics is that unless one presupposes the existence of God, no other presupposition can logically account for anything. This claim is so profound that

In this post we look at Mathematics. And we look at it through the eyes of research mathmetician Leland McInnes Phd. The point of interest in his blog is not only the inexplicable fact of maths, (from a presumed naturalistic worldview) but the extraordinary ability of mathematics to correspond to the problems of physics sometimes even finding a mathematical solution to a "real-world" problem before the problem is even raised in a maths-related context.

Presuppositional Apologetics recognizes that people come to various truth claims with a readymade worldview through which all truth claims are interpreted. Therefore a persons epistemological background is seen to play a crucial part in what is recognized as true or not true (or whether "truth" is even a valid category!). It is therefore the presuppositions a person holds that determine how those claims are judged. The high claim of Presuppositional Apologetics is that unless one presupposes the existence of God, no other presupposition can logically account for anything. This claim is so profound that

**any**other presupposition will, when pushed to ultimate conclusions lead to irrationality. This post and others like it is concerned to point out the validity of this extraordinary claim by pointing to other writers who have seen and logically laid claim to a) either show non-theistic accounts to be using faulty logic or b) where an expert in a particular field has come to admit that from their particular world-view a certain phenomena is inexplicable.In this post we look at Mathematics. And we look at it through the eyes of research mathmetician Leland McInnes Phd. The point of interest in his blog is not only the inexplicable fact of maths, (from a presumed naturalistic worldview) but the extraordinary ability of mathematics to correspond to the problems of physics sometimes even finding a mathematical solution to a "real-world" problem before the problem is even raised in a maths-related context.

I was browsing the Internet the other day looking for materials to teach math to our twelve year old when I stumbled across this blog:

**The Narrow Road**by Leland McInnes.

Leland McInnes is a mathematician who recently completed a Ph.D. in topological algebra at the University of Western Ontario. He lives in Ottawa.

http://en.wikipedia.org/wiki/User:Leland_McInnes

Now if math for you was anything like it was for me when I went to school you probably wouldn’t have given the blog another look, but for reasons unknown to me I stayed awhile and wandered down the paths that the author obviously found so intriguing and beautiful.

All that is in

*blue Italics*comes from

**“The Narrow Road”**by Leland McInnes or

*green italics*from the book

**“Orthodoxy”**by G.K. Chesterton. Anything in

**Bold**I have added to make a point or emphasize. If you can wade through the lengthy logic of 1+1=2 given below you will pan out nuggets further on but otherwise don't be to daunted by it all! (Actually the length of the explanation serves well even if you don’t understand it, it is merely there to give weight to the point that maths actually makes complex logical operations palpable to elementary school kids.)

One of the first things I learned from McInnes is that what is seemingly basic and obvious is not necessarily so:

*Natural numbers (also known as counting numbers) are one of those remarkable abstractions, like objects falling to the ground, that we take for granted. Natural numbers are, however, a very abstract concept -*

**they simply don’t exist in the external world outside your own head.**If you think otherwise, I challenge you to show me where the number 3 exists. You can point to some collection of 3 things, but that is only ever a particular instance that possesses the property common to all the particular instances from which the number 3 is generalised. Because we take natural numbers for granted**we tend to make assumptions**about how they work without thinking through the details. For instance we all know that 1 + 1 = 2 - but does it? Consider a raindrop running down a windowpane. Another raindrop can run down to meet it and the separate raindrops will merge into a single raindrop. One raindrop, plus another raindrop, results in one raindrop: 1 + 1 = 1. The correct response to this challenge to common sense is to say “but that’s not what I mean by 1 + 1 = 2″ and going on to explain why this particular example doesn’t qualify. This, however, raises the question of what exactly we do mean when we say that 1 + 1 = 2. To properly specify what we mean, and rule out examples like putting 1 rabbit plus another rabbit in a box and (eventually) ending up with more than 2 rabbits, is rather harder than you might think, and requires a lot of pedantry. I’ll quote the always lucid Bertrand Russell** to explain exactly what we mean when we say that 1 + 1 = 2:The reader who is not especially interested in mathematics per se is cordially invited to skip the lengthy explanation given below,

**but note the paper saving properties of maths when we realize just how lengthy the proposition (1+1=2) is when we use language instead of numbers!**It may be helpful to take up the comments beginning with the words:

**"**

**As you can see..."**

Omitting some niceties, the proposition ‘1 + 1 = 2′ can be interpreted as follows.

We shall say that φ is a unit property if it has the two following properties:

1. there is an object a having the property φ;

2. whatever property f may be, and whatever object x may be, if a has property f and x does not, then x does not have the property φ.

We shall say that χ is a dual property if there is an object c such that there is an object d such that:

1. there is a property F belonging to c but not to d;

2. c has the property χ and d has the property χ;

3. whatever properties f and g may be, and whatever object x may be, if c has the property f and d has the property g and x has neither, then x does not have property χ.

We can now enunciate ‘1 + 1 = 2′ as follows: If φ and ψ are unit properties, and there is an object which has property φ but not the property ψ, then “φ or ψ” is a dual property.

It is a tribute to the giant intellects of school children that they grasp this great truth so readily.

We define ‘1′ as being the property of being a unit property and ‘2′ as being the property of being a dual property.

The point of this rigmarole is to show that ‘1 + 1 = 2′ can be enunciated without mention of either ‘1′ or ‘2′. The point may become clearer if we take an illustration. Suppose Mr A has one son and one daughter. It is required to prove that he has two children. We intend to state the premise and the conclusion in a way not involving the words ‘one’ or ‘two’.

We translate the above general statement by putting:

φx. = .x is a son of Mr A,

ψx. = .x is a daughter of Mr A.

Then there is an object having the property φ, namely Mr A junior; whatever x may be, if it has some property that Mr A junior does not have, it is not Mr A junior, and therefore not a son of Mr A senior. This is what we mean by saying that ‘being a son of Mr A’ is a unit property. Similarly ‘being a daughter of Mr A’ is a unit property. Now consider the property ‘being a son or daughter of Mr A’, which we will call χ. There are objects, the son and daughter, of which (1) the son has the property of being male, which the daughter has not; (2) the son has the property χ and the daughter has the property χ; (3) if x is an object which lacks some property possessed by the son and also some property possessed by the daughter, then x is not a son or a daughter of Mr A. It follows that χ is a dual property. In short … a man who has one son and one daughter has two children.

Omitting some niceties, the proposition ‘1 + 1 = 2′ can be interpreted as follows.

We shall say that φ is a unit property if it has the two following properties:

1. there is an object a having the property φ;

2. whatever property f may be, and whatever object x may be, if a has property f and x does not, then x does not have the property φ.

We shall say that χ is a dual property if there is an object c such that there is an object d such that:

1. there is a property F belonging to c but not to d;

2. c has the property χ and d has the property χ;

3. whatever properties f and g may be, and whatever object x may be, if c has the property f and d has the property g and x has neither, then x does not have property χ.

We can now enunciate ‘1 + 1 = 2′ as follows: If φ and ψ are unit properties, and there is an object which has property φ but not the property ψ, then “φ or ψ” is a dual property.

It is a tribute to the giant intellects of school children that they grasp this great truth so readily.

We define ‘1′ as being the property of being a unit property and ‘2′ as being the property of being a dual property.

The point of this rigmarole is to show that ‘1 + 1 = 2′ can be enunciated without mention of either ‘1′ or ‘2′. The point may become clearer if we take an illustration. Suppose Mr A has one son and one daughter. It is required to prove that he has two children. We intend to state the premise and the conclusion in a way not involving the words ‘one’ or ‘two’.

We translate the above general statement by putting:

φx. = .x is a son of Mr A,

ψx. = .x is a daughter of Mr A.

Then there is an object having the property φ, namely Mr A junior; whatever x may be, if it has some property that Mr A junior does not have, it is not Mr A junior, and therefore not a son of Mr A senior. This is what we mean by saying that ‘being a son of Mr A’ is a unit property. Similarly ‘being a daughter of Mr A’ is a unit property. Now consider the property ‘being a son or daughter of Mr A’, which we will call χ. There are objects, the son and daughter, of which (1) the son has the property of being male, which the daughter has not; (2) the son has the property χ and the daughter has the property χ; (3) if x is an object which lacks some property possessed by the son and also some property possessed by the daughter, then x is not a son or a daughter of Mr A. It follows that χ is a dual property. In short … a man who has one son and one daughter has two children.

**As you can see**, once we try to be specific about exactly what we mean, even simple facts that we assume to be self-evident become mired in technical detail. For the most part people have a sufficiently solid intuitive grasp of concepts like “number” and “addition” that they can see that 1 + 1 = 2 without having to worry about the technical pedantry. The more abstractions we pile atop one another, however, the less intuition people have about the concepts involved and it becomes increasingly important to spell things out explicitly.

*We can see from the above that math is about logic or reasoning that otherwise would be very involved if it weren’t for the “shorthand” system of numbers. It was when I understood the idea that logic was endemic to math that my ears pricked up. Also when he talks about the “beauty of math”. I had not long ago written about beauty in the eye of the beholder (see*

**"The eternal mystery of the world is its comprehensibility"**

*One of my pet subjects is the veneration of science in our day so when he started speaking of maths as being the foundation of modern science this also interested me.*

**The discovery and development of mathematics is one of the great achievements of mankind — it provides the foundation upon which almost of all modern science and technology rests.**This is because mathematics, as*the art of abstraction*

*, provides us the with ability to make simple statements that have incredibly broad application. For example, the reason that numbers and arithmetic are so unreasonably effective is that they describe a single simple property that every possible collection possesses, and a set of rules that are unchanged regardless of the specific nature of the collections involved. No matter what collection you consider, abstract or concrete, it has a number that describes its size; no matter what type of objects your collections are made up of, the results of arithmetic operations will always describe the resulting collection accurately. Thus the simple statement that 2 + 3 = 5 is a statement that describes the behaviour of every possible collection of 2 objects, and every possible collection of 3 objects.*

In the above we see something of the universality of mathematics and the unity in diversity that philosophers have for ages attempted to comprehend, we can understand a little of McInnes’s (and our own?) fascination and awe of it all.

And again when he started talking about logical paradoxes and the philosophical considerations and arguments currently taking place amongst the mathematician elite

**I began to see a familiar pattern**somewhat akin to the conspiracy of facts that G.K. Chesterton speaks of in his book Orthodoxy:

*The Paradoxes of Christianity.*

The real trouble with this world of ours is not that it is an unreasonable world, nor even that it is a reasonable one. The commonest kind of trouble is that it is nearly reasonable, but not quite.

The real trouble with this world of ours is not that it is an unreasonable world, nor even that it is a reasonable one. The commonest kind of trouble is that it is nearly reasonable, but not quite.

**Life is not an illogicality; yet it is a trap for logicians**. It looks just a little more mathematical and regular than it is; its exactitude is obvious, but its inexactitude is hidden; its wildness lies in wait. I give one coarse instance of what I mean. Suppose some mathematical creature from the moon were to reckon up the human body; he would at once see that the essential thing about it was that it was duplicate. A man is two men, he on the right exactly resembling him on the left. Having noted that there was an arm on the right and one on the left, he might go further and still find on each side the same number of fingers, the same number of toes, twin eyes, twin ears, twin nostrils, and even twin lobes of the brain. At last he would take it as a law; and then, where he found a heart on one side, would deduce that there was another heart on the other. And just then, where he most felt he was right, he would be wrong. It is this silent swerving from accuracy by an inch that is the uncanny element in everything. It seems a sort of secret treason in the universe. An apple or an orange is round enough to get itself called round, and yet is not round after all. The earth itself is shaped like an orange in order to lure some simple astronomer into calling it a globe. A blade of grass is called after the blade of a sword, because it comes to a point; but it doesn't. Everywhere in things there is this element of the quiet and incalculable. It escapes the rationalists, but it never escapes till the last moment. From the grand curve of our earth it could easily be inferred that every inch of it was thus curved. It would seem rational that as a man has a brain on both sides, he should have a heart on both sides. Yet scientific men are still organizing expeditions to find the North Pole, because they still organizing expeditions to find a man's heart; and when they try to find it, they generally get on the wrong side of him.**Now, actual insight or inspiration is best tested by whether it guesses these hidden malformations or surprises**. If our mathematician from the moon saw the two arms and the two ears, he might deduce the two shoulder-blades and the two halves of the brain. But if he guessed that the man's heart was in the right place, then I should call him something more than a mathematician. Now, this is exactly the claim which I have since come to propound for Christianity. Not merely that it deduces logical truths, but that when it suddenly becomes illogical, it has found, so to speak, an illogical truth. It not only goes right about things, but it goes wrong (if one may say so) exactly where the things go wrong. Its plan suits the secret irregularities, and expects the unexpected. It is simple about the simple truth; but it is stubborn about the subtle truth. It will admit that a man has two hands; it will not admit (though all the Modernists wail to it) the obvious deduction that he has two hearts. It is my only purpose in this chapter to point this out; to show that whenever we feel there is something odd in Christian theology, we shall generally find that there is something odd in the truth.McInnes pays tacit approval to Chestertons appraisal of reality with the following lines, in effect saying just when you think your math is able to explain reality down to the last dotted “i” and crossed “t” it all goes to custard and challenges the very roots of logic.

As an example he waxes lyrical about the impossibility of expressing the square root of 2 as a fraction.(See http://jedidiah.stuff.gen.nz/wp/?p=14 )

*Paradoxes of the Continuum, Part II*

*Mathematical arguments can be very persuasive. They lead inexorably toward their conclusion; barring any mistakes in the argument, to argue is to argue with the foundations of logic itself. That is why it is particularly disconcerting when a mathematical argument leads you down an unexpected path and leaves you face to face with a bewildering conclusion. Naturally you run back and retrace the path, looking, often in vain, for the wrong turn where things went off track.*

As we noted at the start of this post, when a mathematical argument leads you somewhere you don’t wish to go you are left having to challenge the very foundations of logic itself. Surprisingly, that turns out to be the resolution: smooth infinitesimal analysis rejects

**People often don’t deal well with challenges to their world-view.**When a winding mountain path leads around a corner to present a view of a new and strange landscape, you realise that the world may be much larger, and much stranger, than you had ever imagined. When faced with such a realisation, some people flee in horror and pretend that such a place doesn’t exist; the true challenge is to accept it, and try to understand the vast new world. It is time for us to round a corner and glimpse new and strange landscapes; I invite you to follow me down, in the coming entries, and explore the strange hidden valley.As we noted at the start of this post, when a mathematical argument leads you somewhere you don’t wish to go you are left having to challenge the very foundations of logic itself. Surprisingly, that turns out to be the resolution: smooth infinitesimal analysis rejects

*the law of the excluded middle*

*.*

So that while celebrating- if you like- the exquisite logical nature of maths at the same time when we come across these paradoxes we have to embrace either our inadequate grasp of higher logic or “challenge the very foundations of logic itself”

This reminds me of the Christian paradoxes like the Trinity, which is, after all a mathematical expression.

Now thus far you may be piqued at my attempts to elucidate a conspiracy of facts which, to my mind point to not a little evidence that all of these things are in fact the orchestration of an extraordinary, dare I say supernatural mind. But here it gets even more interesting. Given that McInnes is just an ordinary if somewhat brilliant mathematician not having any particular interest or view on metaphysics, or perhaps being even unabashedly secular in his thinking, the following observation of his becomes even more astonishing.

*So far we’ve been discussing the completely abstract.*

**We’ve climbed up a path of increasingly theoretical constructions built one atop the other to reach a point of potentially perilous irrelevance**. Now it’s time to step into the concrete world. Our universe can be thought of as a space with some structure - the laws of physics. We can ask what functions or transforms of space and time will preserve certain structures (like, say, Maxwell’s equations). This leads us to a group of transformations of space-time that preserve desired properties - as I said, groups, once you know to look for them, crop up everywhere. Now something intriguing happens: the internal structures of the group correspond in a natural way with physical conservation laws, such as the law of conservation of energy, or the law of conservation of momentum. Things get really interesting, however, when we realise that, given the group, each irreducible representation of the group very naturally corresponds to a different fundamental particle, with force-carrying and non-force-carrying particles neatly delineated from one another according to the degree of the irreducible representation!

*Now I do not lay claim to completely understand what he’s talking about here but you will no-doubt get the gist of it. Especially from what follows:*

**All of a sudden what had seemed like a little axiomatic game that mathematicians were off playing by themselves is having very real impacts on the world in which we live. All of a sudden we’ve come crashing back down into relevance. Which brings us to the start of the truly hard questions: why does it work?!**

*I like to compare his expression here with that of a person doodling with pen on paper. All sorts of lines, shapes and concepts are produced, which although interesting, were thought to bear no relation at all to reality other than the paper and ink they are created on. Suddenly and inexplicably (at least in the naturalistic world-view) someone has made the connection realizing that his or her creations are in fact a reinterpretation of the real world. That is what mathematics is all about.*

*Mathematics works. That much is obvious to anybody - mathematics is essentially the language of much of science, and is seemingly unendingly applicable to the real world. The problem is that it is not at all obvious why this should actually be the case. Philosophers have struggled with this question since Pythagoras, all to no avail.*

**It is entirely all too common for mathematicians to embark on a purely theoretical exploration of the truly abstract and abstruse as a mental game, only to have, decades or centuries later, their work prove stunningly applicable to some very real world problem.**The mathematics that laid the foundations for General Relativity began with mathematicians wondering, purely hypothetically, what would happen if they ignored Euclid’s fifth postulate; much of the work of G.H. Hardy, who famously claimed no discovery of his would ever make a practical difference in the world, has found considerable application in cryptology. To explain its effectiveness it seems we need first to explain what mathematics actually is.*So why does it work? If your worldview precludes the existence of anything greater than the human mind then it would seem your explanation would inevitably fall short!*

**If mathematics is simply the abstraction of our intuitions about the physical world, then why is it so universal?**Surely each mathematician would then develop a personal mathematics from their own intuitions and claims of mathematical objectivity would become muddied. Equally one can ask why it is that a rejection of intuition on purely logical grounds, such as the rejection of Euclid’s fifth postulate, should lead to mathematics that is surprisingly more applicable rather than less.
The same problem has been touched on in the issue of the universal human apreciation of beauty, and also exists in the universality of moral order. As McInnes so rightly points out if it was merely an arbitrary human construct why should a Japanese mathematician find just the same applicable way of doing math as someone from Lithuania? And again why should it then "lead to mathematics that is surprisingly more applicable rather than less" ? These "coincidences stretch the credibility of the naturalistic mindset to breaking point- who then has more "faith" ?

The following comes from Wikipedia articles:

Eugene Paul "E. P." Wigner, November 17, 1902 – January 1, 1995), was a Hungarian American theoretical physicist and mathematician.

He received a share of the Nobel Prize in Physics in 1963 "for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and application of fundamental symmetry principles". Wigner is important for having laid the foundation for the theory of symmetries in quantum mechanics as well as for his research into the structure of the atomic nucleus.

In 1960, Wigner published a now classic article on the philosophy of mathematics and of physics, which has become his best-known work outside of technical mathematics and physics,

*The Unreasonable Effectiveness of Mathematics in the Natural Sciences*is the title of an article published in 1960 by the physicist Eugene Wigner. In the paper, Wigner observed that the mathematical structure of a physics theory often points the way to further advances in that theory and even to empirical predictions, and argued that this is not just a coincidence and therefore must reflect some larger and deeper truth about both mathematics and physics.

"It is difficult to avoid the impression that a miracle confronts us here, quite comparable in its striking nature to the miracle that the human mind can string a thousand arguments together without getting itself into contradictions, or to the two miracles of laws of nature and of the human mind's capacity to divine them."

**The miracle of mathematics in the natural sciences**

Wigner begins his paper with the belief, common to all those familiar with mathematics, that mathematical concepts have applicability far beyond the context in which they were originally developed. Based on his experience, he says

*"it is important to point out that the mathematical formulation of the physicist’s often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena."*

He then invokes the fundamental law of gravitation as an example. Originally used to model freely falling bodies on the surface of the earth, this law was extended on the basis of what Wigner terms "very scanty observations" to describe the motion of the planets, where it "has proved accurate beyond all reasonable expectations."

Another oft-cited example is Maxwell's equations, derived to model the elementary electrical and magnetic phenomena known as of the mid 19th century. These equations also describe radio waves, discovered by David Edward Hughes in 1879, around the time of James Clerk Maxwell's death. Wigner sums up his argument by saying that

Another oft-cited example is Maxwell's equations, derived to model the elementary electrical and magnetic phenomena known as of the mid 19th century. These equations also describe radio waves, discovered by David Edward Hughes in 1879, around the time of James Clerk Maxwell's death. Wigner sums up his argument by saying that

"the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it."

He concludes his paper with the same question with which he began:

What if- just supposing (and I know nothing of of Euclid’s fifth postulate!)- Humans were made in the image of a "super-being" who designed and brought into being the universe- and therefore were able to think his thoughts after him? Would it then not be "natural" that mathematics developed because of a conformity to a pattern of intelligibility that was in fact stamped on the whole universe, and us included being part of that stamp?

###

Does this explanation possibly answer what otherwise is just amazing coincidence. What if these sort of things don't just happen in mathematics but in all areas of human endeavour? Ask Doctor Michael Behe what questions he has raised about the evolution debate, what about the language of dna? Or the issues raised by Guillermo Gonzalez and Jay W. Richards in their view of astro-biology and earths place in the universe, the happy "coincidence of the Goldilocks theory, the anthropic principle?

*"The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning."*What if- just supposing (and I know nothing of of Euclid’s fifth postulate!)- Humans were made in the image of a "super-being" who designed and brought into being the universe- and therefore were able to think his thoughts after him? Would it then not be "natural" that mathematics developed because of a conformity to a pattern of intelligibility that was in fact stamped on the whole universe, and us included being part of that stamp?

###

Does mathematics fit so well because it does in fact comport with the language by which all of the creation exists?

Does mathematics fit so well because it does in fact comport with the language by which all of the creation exists?

Does this explanation possibly answer what otherwise is just amazing coincidence. What if these sort of things don't just happen in mathematics but in all areas of human endeavour? Ask Doctor Michael Behe what questions he has raised about the evolution debate, what about the language of dna? Or the issues raised by Guillermo Gonzalez and Jay W. Richards in their view of astro-biology and earths place in the universe, the happy "coincidence of the Goldilocks theory, the anthropic principle?

*Can it then be argued that mathematics is simply logic? This would explain, to some degree, its applicability:*

**mathematics would simply be the expression of fundamental eternal universal truths.**Here we must commend McInnes for his objectivity and intellectual honesty, there is a real contradiction here which he is really struggling with. Because in fact he is conceding- if mathematics is simply logic, and it is universal - it has therefore existence in its own right independent of mankind- How does it have this independence from mankind when we know that maths is strictly a mind thing? In other words it only takes place on a mental level. And how come it does apply so well to the universe if after all the universe is a completely random accidental cataclysm of circumstances? It shouldn't fit at all to the degree it does. The difference in the basic presupposition between say Einstein and McInnes is that for his worldview Einstein was a deist, that is he believed in (at the very least) the existence of a unifying, ordering entity over and above all that we call nature. Whereas McInnes is struggling to find within the closed order of nature an explanation for all the order and universality both in logic, specifically mathematical logic, how it works in our minds, is applicable to our world and does in fact transcend them both.

McInnes is struggling in fact with the skeptical mantra of the likes of Carl Sagan from whom this quote comes:

*Our posturings, our imagined self-importance, the delusion that we have some privileged position in the Universe, are challenged by this point of pale light. Our planet is a lonely speck in the great enveloping cosmic dark. In our obscurity, in all this vastness, there is no hint that help will come from elsewhere to save us from ourselves."—Pale Blue Dot: A Vision of the Human Future in Space,*

*Carl Sagan was hugely popular in his time, and in some ways resembles the crusader for naturalism of recent times- Richard Dawkins. But I think even the most sceptic reader would have to think twice about his statement-*

"in the light of Eugene Wigner's statement from above-in all this vastness, there is no hint that help will come from elsewhere to save us from ourselves."

"The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve."

To continue with McInnes interesting journey....

*Unfortunately, despite truly heroic efforts by some of histories greatest mathematicians and logicians (Frege in “Foundations of Arithmetic” Russell and Whitehead in “Principia Mathematica”) the reduction of mathematics to pure logic failed. While Russell and Whitehead managed to construct a remarkable edifice out of pure logic, introducing along the way many new concepts such as Type theory which is now of great importance in computer science, they fell short of building mathematics as we know it: their system was too weak and required extra axioms (the axiom of infinity and the axiom of reducibility) which cannot reasonably be called purely logical principles.*

Can we not, then, define our base set of rules and proceed by logic from there? This was the approach to mathematics favoured by Hilbert and his formalist school. While this approach did much to shore up the shaky foundations of pure mathematics before running afoul of GÃ¶del’s Incompleteness Theorems, it simply sidesteps our initial question. Mathematics, in this line of thought, becomes reduced to a game played with symbols on paper and an arbitrary set of rules (we ask only for consistency).

The debate as to what mathematics actually is continues, mostly in philosophic circles - No school of thought has yet given a satisfactory answer. Personally I lean towards a more recent view that mathematics is about structure and can be grounded best in a structure oriented language such as that of Category Theory. However, Category Theory does not yet provide a sufficiently rigorous foundation to make that claim truly defensible. Only time will tell.

Can we not, then, define our base set of rules and proceed by logic from there? This was the approach to mathematics favoured by Hilbert and his formalist school. While this approach did much to shore up the shaky foundations of pure mathematics before running afoul of GÃ¶del’s Incompleteness Theorems, it simply sidesteps our initial question. Mathematics, in this line of thought, becomes reduced to a game played with symbols on paper and an arbitrary set of rules (we ask only for consistency).

**Why should such an arbitrary game apply to reality?**That is not at all clear.The debate as to what mathematics actually is continues, mostly in philosophic circles - No school of thought has yet given a satisfactory answer. Personally I lean towards a more recent view that mathematics is about structure and can be grounded best in a structure oriented language such as that of Category Theory. However, Category Theory does not yet provide a sufficiently rigorous foundation to make that claim truly defensible. Only time will tell.

*Thus we began wondering what there is in mathematics that we don’t know, and conclude realising that we don’t even know what mathematics is.*

*"No school of thought has yet given a satisfactory answer."*

One wonders what would count as a satisfactory answer for McInnes- only one that proceeded from a naturalistic worldview?

A satisfactory argument

*has*been given, but it appears not to be satisfactory to many even though logically tenable. This is somewhat surprising and unfortunate- when the logicality of it appears to be not ground

*enough*for intellectuals. Positing a supernatural being is both logically and experientially justifiable. This naturalistic bias is strongly evident here. What are the reasons for it? We turn to Vern Poythress.

Vern Poythress earned a B.S. in mathematics from California Institute of Technology (1966) and a Ph.D. in mathematics from Harvard University (1970). After teaching mathematics for a year at Fresno State College (now California State University at Fresno), he became a student atWestminster Theological Seminary, where he earned an M.Div. (1974) and a Th.M. in apologetics (1974). He received an M.Litt. in New Testament from University of Cambridge(1977) and a Th.D. in New Testament from the University of Stellenbosch, Stellenbosch, South Africa (1981).

In 1974 Poythress published in The Journal of Christian Reconstruction an article entitled:

Here is a short extract from this paper:

*"The Reformational expression “total depravity” of man is used to say that no area of man’s nature and no area of his involvement with the world is left untouched or undefiled by man’s rebellion (Rom. 3). For an exposition of this, one could not do better than turn to the second book of Calvin’s Institutes. “The whole man, from the crown of the head to the sole of the foot, is so deluged, as it were, that no part remains exempt from sin, and therefore, everything which proceeds from him is imputed as sin.” Therefore, we now expect mathematical truth also to be misunderstood and perverted by fallen man."*

Remember some of the prior observations and remember also that the words in blue are actual quotes from McInnes:

Mathematics is a mind matter:*Natural numbers are, however, a very abstract concept -***they simply don’t exist in the external world outside your own head.**

*·*Mathematics is intrinsically logical:*Mathematical arguments can be very persuasive. They lead inexorably toward their conclusion; barring any mistakes in the argument,***to argue is to argue with the foundations of logic itself**While essentially logical, current understanding of logic fails to give a complete explanation of some problems including what constitutes mathematics. Some problems in mathematics are greater than logic, or beyond our grasp of logic:

**we don’t even know what mathematics is.**

Mathematics is universal:**If mathematics is simply the abstraction of our intuitions about the physical world, then why is it so universal?**Mathematics bears an astonishing relationship to the real world- it explains the world:

**mathematics is essentially the language of much of science, and is seemingly unendingly applicable to the real world.**

McInnes asks all the right questions the most important being: Why does it work?!

Being abstract it exists only in our heads, but on the other hand it is both logical and universal and it relates perfectly to the real world. (And in all the right places as Chesterton would affirm it goes wrong!) While it is inextricably logical, on certain occasions it defies logic. We didn’t invent it so much as discovered it.

Mathematics- as the art of abstraction- exists, that is certain. It exists independently of human minds as its universality and its relationship with the real world verifies, we didn’t invent it. We did not invent the world. To say that math is a man-made entity when mathematical exercises which are happening purely on a theoretical level and*then*repetitively shown to bear amazing relevance to real world problems which are beyond the scope of human influence, is irrational. Its logic confirms that it is a matter of intelligibility, that the world may be explained in its (relatively!) simple terms also testifies to this. Some of it is not explained adequately or entirely by logic.

Let's summarize even more briefly-

- Universal

Existence

Independence

Immaterial

Intelligible

Logical

Mysterious

Does that sound like the attributes of something (someone) else you’ve heard about? Go figure!

It was Einstein who said-

"The eternal mystery of the world is its comprehensibility."and also "I have deep faith that the principles of the universe will be both beautiful and simple

It was his firm belief that the universe was intelligible and his deistic outlook encouraged him to attempt to comprehend the universe in such broad swathes with such little formulas.

The real trouble with this world of ours is not that it is an unreasonable world, nor even that it is a reasonable one. The commonest kind of trouble is that it is nearly reasonable, but not quite.Life is not an illogicality; yet it is a trap for logicians. It looks just a little more mathematical and regular than it is; its exactitude is obvious, but its inexactitude is hidden; its wildness lies in wait.

The real question is are these quirks attributable to divine personality? to G.K. Chesterton, the "not quite" symmetry of the universe was one of many evidences of intelligent design.

Richard Polkinhorne who worked for many years as a theoretical elementary particle physicist, from 1968 to 1979 was Professor of Mathematical Physics in the University of Cambridge, before resigning to train for the ministry of the Church of England. He is an Anglican priest and a Fellow of the Royal Society and was knighted by the Queen in 1997.

John Polkinghorne: The Friendship of Science and Religion

In this video he shows how compatible Science is with belief in God. In fact he shows how the two disciplines overlap on certain questions and how they can indeed be informative to each other across those boundaries. In case you missed it he is a distinguished Quantum physicist. Among some of the comments he made is this:

Richard Polkinhorne who worked for many years as a theoretical elementary particle physicist, from 1968 to 1979 was Professor of Mathematical Physics in the University of Cambridge, before resigning to train for the ministry of the Church of England. He is an Anglican priest and a Fellow of the Royal Society and was knighted by the Queen in 1997.

John Polkinghorne: The Friendship of Science and Religion

In this video he shows how compatible Science is with belief in God. In fact he shows how the two disciplines overlap on certain questions and how they can indeed be informative to each other across those boundaries. In case you missed it he is a distinguished Quantum physicist. Among some of the comments he made is this:

You can say that the physical Universe is rationally transparent to us. We're able to look into its workings, not only the vast curved structure of space/time, but also into the strange counter intuitive quantum world of the atom and things smaller than the atom, the sort of physics I myself worked in. The regime which, if you know where something is- you don't know what it's doing, if you know what it's doing- you don't know where it is. That's the quantum world, Heisenbergs Uncertainty Principle, totally different from the world of everyday, but nevertheless a world we can explore and to a very large degree understand.

So, why is science possible?....Not only is science possible but it turns out that it is mathematics, it is mathematics, which is the key to unlock the secrets of the physical Universe, the actual technique in fundamental physics to look for equations which are characterized by the unmistakable quality of mathematical beauty. Not all of you perhaps, will know about mathematical beauty, some of you undoubtedly will, it's a fairly er, er, rarified form of aesthetic pleasure...but, but it is one for those of us who are privileged to speak that language, can recognize and agree about. And we've found time and again in the history of physics that it's only... only equations that have that character of mathematical beauty which will prove to be convincing, by their long term fruitfulness of results and insights that they yield. So if youve got a friend, who is a theoretical physicist, and you want to upset him or her, then just say that "that latest theory of yours, looks rather ugly and contrived to me" and they will be very upset.

I suppose the greatest theoretical physicist I have ever known personally, was somebody called Paul Dirac. You may not, all of you, heard his name, but he was one of the founding figures of quantum theory and unquestionably the greatest British theoretical physicist of the 21st century. He was not a religious man, and he was not a man of many words, and he was once asked,"What is your fundamental belief?"And because he was not a man of many words he went to a blackboard, and he wrote on the blackboard:"The Laws of Physics Are Expressed In Beautiful Equations"

*This lecture is well worth listening to in its entirety. It is a bigger picture of science and mathematics. Two of the points that he makes reference to are- quite apart from the inexplicable character of aesthetic beauty that epitomizes good formulas- that mathematics is a universal language but it was not invented so much as discovered, and it has an uncanny ability to anticipate how the Universe is put together. It is a reflection of the inherent underlying mathematical principles by which the Universe appears designed.*

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