Quantum Mechanics and Free Will

"In their hearts humans plan their course,

but the Lord establishes their steps."     -  Proverbs 16:9

See here for my earlier blog post on the issue of free will from a theological viewpoint.
Disclaimer: My only formal experience with quantum theory is a very introductory undergraduate course. Someone much further from an expert than I one would struggle to find.

Introduction 

We wish to determine whether or not free will exists, so let us toss a coin and let it decide for us.

The coin soars out of my hand high into the air, spinning rapidly; neither of us have any idea whether or not free will is going to exist or not.

Hang on a second, though. The fact that we can't predict the outcome of the toss is a measure of our incapacity more than it is a testament to the universe's fundamental unpredictability. If we knew more about the coin toss (exactly how heavy the coin is, how fast it is spinning etc) we surely could know which side would land face up.

This was the dominant view of physicists before quantum mechanics. All the main theories of the time (Newton, Maxwell, ...) were entirely deterministic in nature; they all said "tell me about a system at a certain time, and I'll tell you what happened before, and what will happen in the future".

If the universe right now contains enough information to predict with certainty what will happen for the rest of its course, then that includes information about everything you'll ever do. On absolute determinism, then, in what sense can free will remain except than as a pitiable illusion?

For the sake of clarity, we henceforth define an action to be free if it cannot be determined, and say that an action is willed if it is a non-arbitrary choice made by some conscious being. So we get for free that determinism definitely rules out free actions.

It is easy to see that it also rules out us willing actions too; on determinism the state of the universe before we were born determines what we do, and we weren't around to will anything then!

Though it is of course true that we could still want things to occur that do, but as Schopenhauer puts it, "we could not will what we willed".

This is contentious philosophical ground, and has severe implications for the way we view religion, truth, and morality. At the turn of the 20th century it certainly seemed that physics and the notion of free will were in conflict, if not outright contradictory.

However, a lot has changed in physics world since then. Now the debate is much more open, largely due to the development of quantum theory, and its introduction of fundamental uncertainty into physical theories of the universe. The aim of this post is to briefly discuss the theory itself, and then explore its implications regarding the plausibility of free will.

Quantum Theory

"If you think you understand quantum mechanics, you don't understand quantum mechanics" - Richard Feynmann

Little hope have I, then, of providing an adequate introduction here, but it seems I have no choice.

Quantum theory says, in brief, that God plays dice with the universe.

Back to the coin toss. Suppose you knew was that the coin toss was fair, but weren't allowed to see the result. The best thing you could say regarding the state of the coin is "According to the information I've been given, the best I can do is assume that the coin has probability 1/2 of being heads, and 1/2 of being tails".

Quantum theory says that, in actual fact, no matter what we know about the system, the best we can say is "the probability of one event is this, the probability of another event is that, ..." until we actually take a look at what has happened.

There might be many possible events, so instead of listing them like this we encode them neatly into something called a wave-function. This tells us about the system, and so might change over time as the system does, and the way it does that is governed by Schrodinger's equation.

Observing the system corresponds to operating on the wave-function. Upon observation, we know that we observe an actual event, and not probability superpositions of events. Thus at this point the probabilities must collapse so that the event that you observe has probability 1, and all the others have probability 0. Mathematically this means the wave-function instantly changes to become an eigenfunction of the Hamiltonian of the system. Take or leave the formalism, it will not be needed.

If we want to observe two facts about a system, we must do two successive measurements (observations). Sometimes the order in which we do these measurements makes a difference to the results we get. This results in Heisenberg's uncertainty principle, which says that there are some pairs of observables that we cannot know simultaneously, like position and momentum. 

A related result tells us that information about any physical system is necessarily incomplete. These are very problematic results to interpret philosophically!

Free will

"I used to be indecisive but now I am not so sure" - Tommy Cooper

We used to think that the universe was predictable. But now we are pretty convinced that it is not deterministic. The possibility of free actions arises from the murky depths of incoherence, but it is not at all clear what we know beyond that.

Quantum theory doesn't abandon determinism completely. In fact, if it did something would have gone wrong somewhere, as science itself presupposes the intelligibility of the universe. Instead, we have moved to semi-determinism; we don't know what will happen, but we do know the probabilities of various events occurring. Or at least the latter are still, in principle, calculable!

This still seems to interfere with free will. We instinctively associate probabilities with randomness, especially when the probabilities are not simply making up for our lack of knowledge.

But randomness, while allowing free actions as we defined them, doesn't allow for free willed actions so easily.

To move us along at pace, I am going to claim:

(i) God exists, and created and sustains everything else that exists.

(ii) Humans are immaterial and eternal souls, that temporarily inhabit material bodies.

(iii) A human soul has free will in that he or she can choose to what state the wave-functions collapse within regions of space-time that their brains occupy.

Rather than defend the truth of these propositions, I would like to instead defend that if free~will does exist in a way that is compatible with physical theories this is the most plausible mechanism by which it would occur. Most of us feel (perhaps by no choice of our own!) that we do have free will, and so to find such a theory would be nice.

Mainstream interpretations

“You say: I am not free. But I have raised and lowered my arm. Everyone understands that this illogical answer is an irrefutable proof of freedom.” - Leo Tolstoy

We have said that until an observation of a system is made, the wave-function evolves according to Schrodinger's equation, but that at the instant of observation the probabilities collapse to give a definite event. However, the physical significance of the evolving of the unobserved wave-function is an area of great philosophical contention.

All we really know is how to use the theory to predict things. The wave-function evolves in the sense that if we observe the system at time t=0, we can leave it until some later t=T and know (by solving the Schrodinger equation at t=T) what the probabilities of observing all possible states is.

Maybe one can cling to determinism by insisting that, upon an observation, the wave-function doesn't collapse.  Rather, the observer becomes part of the quantum system. The necessary consequence of these presuppositions is that the universe itself is a superposition of all possible universes that could have occurred, given the beginning of the universe. If you like, traditional determinism is the trivial special case of this. Free~will probably can't be salvaged from here. 

Or perhaps there never is any uncertainty to begin with. This is not an easy one to explain, and in fact it has received criticism on the grounds of its seemingly unnecessarily complicated nature. But in essence, it postulates that the universe follows a path through the space of all possible universes, guided probabilistically by Schrodinger's equation. This is compatible with God having sovereign free will, but gets tricky when considering multiple locally free agents.

Regarding these theories (for the former is the simplest!), Budziszewski remarked: "In order to avoid believing in just one God we are now asked to believe in an infinite number of universes, all of them unobservable just because they are not part of ours. The principle of inference seems to be not Occam's Razor but Occam's Beard: 'Multiply entities unnecessarily.'"

Consciousness 

"To determine by what modes or actions light produceth in our minds the phantasm of colour is not so easie" - Isaac Newton

If we expect to be able to glean free~will from an interpretation of quantum mechanics, it should be expected that it somehow physically distinguishes between human (and possibly animal) minds, and everything else. Of course, this goes against materialism and notions of complete causal closure.

When you observe a system, what you do clearly depends on what you observe. Thus we see that any quantum mechanical must be functionally causally closed. So we must consider the whole material universe. But everything in the system is decomposable into mere fundamental particles. So who is to say that a given collection of these at any given time qualifies as an observer? 

The case of no observers would lead us back to the previous theories. 

But an alternative suggestion is that human minds (or souls) are actually non-material observers.

Further, as we need an actual (and therefore observed) world to exist for a human to even be created, we must further say that there existed a non-material being (God) "collapsing wave-functions since time immemorial"!

There is an amount of time called a Planc length, which is the smallest time interval that has any physical meaning in current physical theories. Conceivably the universe quantum~collapses every planc length of time in a way controlled by your immaterial mind in various regions of your material brain, by everyone else in their respective brains, and everywhere else by God - or randomly. 

In other words, quantum indeterminacy leaves well~defined fundamental gaps in any theory of this kind. It is in these scientifically unreachable regions that consciousness could act. This should not really be seen as a scientific theory, let alone a good one.  But it is, nevertheless very interesting. 

As a Christian I am convinced that we do have some measure of free will - though as a Calvinist I inhabit somewhat of a superposition of beliefs! - and I believe that God has set up the universe so that miracles are unusual (again, a topic for a later discussion!); that is, that the universe obeys mathematical laws almost all of the time. But without the framework discussed above, the only way one could really justify belief in free will would be to claim that a tiny miracle occurs in our brains every time we make a decision!

What the consciousness interpretation of quantum mechanics does is provides a mechanism by which these mini-miracles could happen, that is consistent with our current body of physical theory, This seems to be the closest thing to a reconciliation between quantum physics and dualism that we currently have. 

Such a theory is essential if we are to ever rid ourselves of the all permeating, dogmatic reductionist philosophy plaguing modern physics, and more broadly reconcile faith and science. In fact, the tension felt here is not between faith and science. It is between faith and reductionism. These are diametrically opposed. One is true; the other false. And currently, the wrong one is dominating scientific thought.

Quantum theory isn't going to disappear any time soon, because it is probably close to representing reality. But I believe it can be reconciled with Christianity, and more specifically, with free will. 

Of course, if I want to say these things with any authority I need to dedicate my working life to studying the theory. 

That's a big commitment. For now, this blog will have to do.

Until next time!

“For you will certainly carry out God's purpose, however you act, but it makes a difference to you whether you serve like Judas or like John.”  - C.S. Lewis

God is love? - A closer look at 1 John 4:8

Sigh. The night I decided to work until 4am would be the same day that the porters decide to do a 6:30am fire alarm. Ah well, at least now I have time to write a blog. However muddled it may be...

I don't, however, have too much time on my hands, so in contrast to the style of my previous blogs, I'll just ponder the verse below for a while, and write my thoughts down in a (hopefully) semi-articulate fashion as they come. Hopefully the reader will glean something of value from this stream of consciousness!

"The one who does not love does not know God, for God is love." - 1 John 4:8

Initial thoughts

“What comes into our minds when we think about God is the most important thing about us.” - A.W. Tozer

Perhaps the most interesting question we can ask is "what is God like?", and among the most pressing questions we can have is "how can I know Him?". It is convenient, then, that this verse seems to touch on both of these in one sentence!

Looking at the first clause, we see that if we don't love we do not know God.

This doesn't quite tell us what we must do to know God, but rather is a kind of "test" by which we can tell if we don't know God. It seems rather negative, so let's phrase it positively; we can equivalently say that "If one knows God, one loves";

[Does this statement carry the same weight and authority as scripture now that I have tampered with it? The two statements really are logically equivalent, but maybe to italicize at this point would be to assert that God is somehow constrained by logic. I'd rather sidestep that whole debate for another time. In fact, we have the positive phrasing in the previous verse, which we can safely italicize!]

"Beloved, let us love one another, for love is from God; and everyone who loves is born of God and knows God." - 1 John 4:7

We see here that if we love, not only do we know God, but we also are born of God. 

This whole relationship between knowing God, and loving is wholly unobvious to me. Let us, for your entertainment if nothing else, suppose that I am not just being painfully oblivious, and try to follow the logic behind this assertion.

On the one hand, love seems to be something that one feels - or maybe more correctly, something one does - sometimes, in certain situations, in certain moods. Even ignoring (or averaging!) this explicit time dependence, it seems that the frequency at which - and extent to which - a person loves is a function of their environment, confidence, and nature. It seems to not have all that much to do with how well they know God.

Perhaps, then, we have too strong an idea of love, or too strong a notion of knowing God?

Certainly, to some extent, every human is born of God. So perhaps the verse is suggesting that because we all love, we are thus born of God, and in some sense, know God.

This is a plausible, but rather weak and unsatisfactory interpretation. Let us try again, properly.

This time, lest we be again led astray by vague concepts, let us try to define our terms well.

The words that it is crucial to unambiguise (read: "remove ambiguity from"!) are clearly know and love. 

What is love?

The root of love in this context is agape. This is the highest of the four Biblical concepts of love; it is selfless, unconditional, and sacrificial. It perfectly encapsulates the love that God has for his people, and  is most clearly displayed in the Calvary love of Christ for his Father. 

This love is of a rather strong form, and apparently we must embody it if we are to know God.

Whom, then, is the intended object of the love that those who know God possess?
If it is intended to be God, then the statement suddenly becomes more obvious. But, I fear, it does so at the cost of its intended significance. To say that "everyone who loves God knows God" seems true, but somehow vacuously. Agape love should be more substantial than this.

Maybe the issue is with the question. Perhaps this love that is to be embodied by those born of God is, rather than prescribing how they feel or act towards a specific someone or something, describes something about them. It is not so much that they love, but rather that, in some sense, they are love.

That phrasing is familiar; we have unintentionally arrived at the second clause.
Thinking about it, we should really have started with the second half. There lies the essential substance of the verse, from which everything else should be expected to flow naturally. It was not an entire waste of time to try to understand the former clause without understanding of the latter, provided we learn through the futility of the attempt that the latter is required to make sense of the former; the for is implicative, and not a superfluous connective.

We are, then, naturally lead to the following question:



In what sense is God love?

The verse, surely, should not be taken to suggest an actual equivalence between God and love. For them to actually be the same thing doesn't really make much sense. God is very clearly presented as being far less abstract than what we think of when we think of love in general. We should not think that God is love in the sense that whenever someone loves someone else in an agape (selfless) sense that God literally is the thing the person is doing, or being. And surely the creator and sustainer of the universe is not love itself? As Russell Brand might say, love doesn't have toolkit.

What, then, does this mean? We do not seek to diminish the strength of the posited relationship between God and love at all, and it is clearly supposed to be a strong one!

We can safely say that if God is love, then God is never not loving.
In particular, as God is eternal (uncaused, logically prior to none - Genesis 1:1), and immutable (unchanging in character, and will - Numbers 23:19, Hebrews 13:8) God never has been not loving.

Thus we must view love as an essential part of Gods nature. God is constrained by his own character to love. One may wonder whether God is powerful enough to stop himself from being loving. But this doesn't even make sense as a question! God can not do that because to do it would necessitate that he were no longer God.

This is reassuring, at least in the sense that in the light of this knowledge, other areas are illuminated. For example, it can seem strange that God is described as loving, and caring for, and willing to suffer for humans. After all, even the best of us are so pathetic in comparison to him.
But now we see that God literally can not help himself!

As the Casting crowns song so beautifully puts it,

"Not because of who I am
But because of what You've done.
Not because of what I've done
But because of who You are."



Implications

Suddenly bits of scripture that were completely mind-boggling before now make sense, even though a basis for understanding why it is true does not detract from the "wow factor"! We have not done anything to inspire his love for us, to awaken his affection for us, or to add to the weight of his mercy on us. He loves us to the extent that he loves us because He is that love. Thus we see:

“But God, who is rich in mercy, out of the great love with which he loved us, even when we were dead through our trespasses, made us alive together with Christ (by grace you have been saved).” - Ephesians 2:4-5

And this unconditional nature of Gods mercy is emphasized richly, again by Paul;

“But when the goodness and loving kindness of God our Savior appeared, he saved us, not because of deeds done by us in righteousness, but in virtue of his own mercy, by the washing of regeneration and renewal in the Holy Spirit.” - Titus 3:4-5

In fact, now it becomes clear that "God is love" really underpins redemptive history.

If God we not, in essence, loving, then He could simply stop loving at any point.

In fact, from where would love have arisen, if not from God's very nature?

And without love, what could compel God to send his Son to endure the cross for a fallen creation that despised Him?

We see, then, that love being part of God's essence is a doctrine that we would do well to cling to as absolutely pivotal to our understanding of God; one that illuminates the way for us to correctly understand other attributes of God, and without which we will inevitably misinterpret the intentions of God.

But let us dwell again on the idea that love is in God's essence.

If love is in God's essence, then God has never been without love.

How, then, did this love manifest itself before creation?



In the beginning

One can sidestep this by saying that there was no time before creation, as God created time.

But Genesis 1:1 says "In the beginning, God created the heavens and the earth", so it is clear that God at least existed logically prior to his creation. I would go further, and say that before He created time as we understand it God must have decided to create at some point. This pedantic nuance leads me to believe that God created metric (measurable) time, but that even "before" that, there were still sequences of events that could happen in order. Maybe God does in fact live in actual metrizable time, just in one causally independent to ours. But this is just dry philosophy at risk of distracting us (or, at least, me!) from the real things at play here.

The point that I am getting at, however badly I motivated it above, is that God must have loved before he created anything. In fact, if this were not the case, why would God create anything in the first place? Certainly not for personal gain, as God is self-sufficient (depending on nothing, completed by nothing - Colossians 1:15;20)

One can see the motivation for creation itself much more clearly by understanding that God is love, and that this love is life-giving, and naturally seeks objects to actualise its affections. Again, we see that Gods reason for creation, even with love correctly viewed as being in his essence, could not be for his own gain, in an objective sense. 

Suppose God created simply that His pre-existing love may be realized in the expression of that love towards the created entity. But then Gods expression of his attributes is dependent on this being. Thus God would not be immutable, with the creation being the measure of his imperfection.

So, then, God must have loved before creation.

The question is how?

For a person to love the same person (ie. themselves) simply can not be agape love. How could it be self-sacrificial? Indeed, if it were, it would not be beneficial for the object of the affection, and so would not be love!



The trinity

Aha! We arrive, necessarily, at a God who must consist of multiple persons.

What, then. of 1 Corinthians 8:6?

"Yet for us there is but one God, the Father, from whom all things came and for whom we live..."

I suggest that finishing the verse may be illuminating;

"...and there is but one Lord, Jesus Christ, through whom all things came and through whom we live."

And there we have it; two distinct persons, both divine, and yet indistinguishable in nature. Of course, in Christian theology we also have the third component of the trinity - the Holy Spirit.

But the trinity is for another time, it was just nice to properly motivate it from the idea that God is love.

God takes the form of a Father, of a Son, and of Spirit. Before creation, they were all engaging in endless, boundless agape love. All of creation was a creative overflow of this pre-existing love. Gods love for His creation stems from the Fathers love for the Son, and the Sons willingness to endure the cross came from his obedience to the Father, rooted in the agape love they shared. 

Any feelings or actions resembling love that we can muster up are a mere reflection of this agape love existing between the three persons of the Godhead. This explains the ubiquitous obsession that humanity has had with love for millenia. It really is a big deal! But we would do well to base our understanding of it on a correct view of Gods love for us.



Theology

Love, being part of Gods essence, is necessarily to no extent conflicting with any of his other attributes. In particular, with His Justice. This is why the cross was necessary. Because  God could not simply overlook our sins, as it were, for free. This would violate his Justice. 

The anger we feel when justice is not carried out - the idea that child molester can sometimes go uncaught, for instance - is well-founded, and morally good. It is just that, due to having dimmed consciences as a result of our own imperfections, we sometimes fail to see the same blemishes and violations of Gods holiness in ourselves, and in other people. But it is there all the same.

The world is a messed up place, but the only way we can hope to view God in light of this is by firmly grasping, first and foremost, that God is love. When we look at the world in despair, He feels the same pain far more vividly than we do. He knows the extent of the worlds problems infinitely more accurately than we could ever, and He is emotionally invested in the well-being of humanity far more deeply that we ever will be.

The Biblical view of God is not of a passive supreme being letting people suffer, but of a loving, compassionate God. A Shepherd searching for lost sheep in his flock (Luke 15:1). A husband pursuing an imperfect bride (Hosea). A mother nursing her child (Isaiah 49:15). A Father longing for his children to come home (Luke 15:11).



Filling in the gaps

From the understanding that God is by very nature love, the first section of 1 John 4:8 becomes a lot more transparent. 

Suppose we do not love, by the standard of our original verse.
This is equivalent to saying that we are not replicating the kind of love that Christ displayed for us (1 John 4:19).
Thus we are not conforming ourselves to the likeness of Christ, and are therefore not living in Gods will (1 John 2:6).
Then we are being disobedient to God, and so do not love Him (1 John 5:3)
And thus we are not serving our own good, as God work all things to the good of those who love him (Romans 8:28)

But who would not work for their own good?
There is a modern view that, somehow, to anticipate a reward for doing a good deed somehow diminishes the goodness of the deed.

This is not, in my opinion, a biblical view.
Why did Christ endure the cross? For the joy set before Him (Hebrews 12:2).
Why did Paul endure persecution? Because his present suffering he considered nothing compared to the future glory to be revealed in him (Romans 8:18).
Why are we to love our enemies? Because our reward shall be great (Luke 6:35).
Why are we to sacrifice things for Jesus? Because we will receive a hundredfold back in the age to come (Mark 10:29).

We are, after all, commanded to love our neighbors as ourselves (Mark 12:31), which seems to presuppose that we love ourselves. I argue that we all do love ourselves, in the sense that we ultimately serve our own good. We want ourselves to be happy, content, satisfied, joyful, etc. even if this takes strange forms like self-flagellation, suicide, or doing a law degree.

Thus the only conclusion to make, if we are causing our own detriment, is that we are doing so accidentally; if we do not love, we have not understood Gods love.
But love is part of Gods essence, so if we don't understand His love, we don't know Him at all!

Phew. Now that verse makes sense!
At least to me. For now.



Application

How, then, should we respond to this (hopefully somewhat) fresh understanding of Gods character?
It is clear that we are to love. But what does that look like?

Well, quite simply, it looks like Christ.
The gospels account of Christ are by far the best resource we have for understanding how to love.
To conform our lives to the image of Gods will for us is exactly to live as closely as we can a life like that of Christs, adapted for our circumstances, guided by wise friends and our own consciences.

In particular, we must make sure that we substitute nothing for love.
This is crucial because, as love is part of Gods nature, it simply has no substitute!
If we do good deeds all of our lives, but do not love, we are nothing. Works will never be sufficient;

"If I speak in the tongues of men or of angels, but do not have love, I am only a resounding gong or a clanging cymbal.  If I have the gift of prophecy and can fathom all mysteries and all knowledge, and if I have a faith that can move mountains, but do not have love, I am nothing.  If I give all I possess to the poor and give over my body to hardship that I may boast, but do not have love, I gain nothing." - 1 Corinthians 13

So I guess the next step on this journey is to focus wholeheartedly on applying principles we see in Jesus' life into our own situations. And that's something we all need to pursue, to some extent at least, on our own.

Hopefully I'll do more of these in the near future, I've certainly enjoy this one.
I'd love to chat with anyone reading this about this (or anything else tbh) at any point, feel free to grab/message me at any time!

Jeff


"For God so loved the world that he gave his one and only Son, that whoever believes in him shall not perish but have eternal life." - John 3:16

The Awesomeness of Mathematics

Maths is probably the most hated subject in UK high-schools. It also, more worryingly, appears to be the only subject that one can be proud of being bad at in academic circles. Why, then, have I and so many others volunteered to do a degree in it?!

In this post I attempt to dispel a few misconceptions about mathematics, and reveal what it is actually like to study higher level maths. Or at least; reveal what my taste of higher level maths has indicated to me. Below are seven reasons why I think that maths is awesome, compiled in such as way as to accentuate the inaccuracies of the layman's understanding of what maths entails. Enjoy!




1 - Maths is not just about numbers

"It is not the job of mathematicians to do correct arithmetical operations. It is the job of bank accountants." - Samuil Shchatunovski


It saddens me slightly to know that a lot of people think that a degree in maths is "like school maths, just more". Like we spend our degrees memorising our 147 times tables, dividing 10 digit numbers in our heads, or honing our calculator speed. I wonder what they think the tens of thousands of full time research mathematicians are paid for! Perhaps tax payers fund them to "look for bigger and bigger numbers."

Of course, maths does include some work on numbers. And in fact number theory contains many of the most beautiful results that I have encountered in mathematics. But they are not treating numbers in the way that is taught in high-school. The results of calculations of the sort that crop up in school are considered tedious, unnecessary and mostly trivial in content in higher mathematics.

Often, when ugly expressions such as 16!*2^100 crop up in work, rather than calculating it's actual value, the mathematician will either leave it as expressed to save calculation, or do a very quick mental calculation to establish that it is less than some more more manageable number - here one might write < 10^50.

A mathematicians job is to look at numbers, yes. But also to look at space and time, and structures within this realm. They create a system, observe it, spot various patterns and conjecture about precisely what rule underlies the pattern they see. They subsequently try to prove (or disprove) their conjectures. In this way they learn about and get a feel for various logical systems. One such system - which, granted, has some interesting properties - is "the whole numbers" under usual rules of addition and multiplication. But this is by no means the only (or even the main) system under inspection!




 2 - Maths is simple

"The essence of mathematics is not to make simple things complicated, but to make complicated things simple."  - S. Gudder

"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is"  - John Louis von Neumann


"That's all very well", you may say, "but when I look at the work of mathematicians it's full of nonsensical symbols. They just complicate everything when there's no need."

Remember that mathematicians have chosen to use the notation that they use. If they could make their lives easier, they would. It is, therefore, sensible to suppose that the formal language of mathematics is necessary to be able to express abstract and convoluted statements simply, precisely and unambiguously. This is indeed true. Furthermore, the language required in order to make such progress at understanding the abstract, as is seen in mathematics, even possible.

To see how the language of mathematics has evolved, let us first look at the solution to the quadratic equation used in the 7th century A.D:

"To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value."

In contrast, here is how modern mathematicians express the solution:

For those of you panicking, this solves a rearranged version of what you're probably used to! Anyway, whether you're familiar with algebra or not, you have to admit that once one is familiar with the notation the latter notation must make things immeasurably easier. The importance and necessity of the concise and precise notation used in maths is difficult to overstate. The human mind has never invented a labour saving machine as effective as algebra, and indeed no other such labour saving machine has had such a large impact on the world.

3 - Maths is creative

“The difference between the poet and the mathematician is that the poet tries to get his head into the heavens while the mathematician tries to get the heavens into his head.” - G.K. Chesterton

"Mathematics is, in its way, the poetry of logical ideas" - Albert Einstein


This is perhaps the biggest misconception of them all. Again, this thought finds its origins in the lack of creativity required in high school maths. But the very opposite is true! Mathematicians are constantly searching for methods that make things as simple and elegant as possible. We hate unnecessary clutter, and to find neat, easy proofs and results requires a lot of creativity.

In the 19th century, a maths teacher told his seven year old students to work out what the sum of the first 100 numbers was, and started making his way out of the classroom. The kids started scribbling on their blackboards; 1+2=3, 3+3=6, 3+4=10, ... They had a long way to go. But, according to legend, before the teacher had even reached the door, a certain seven year old by the name of Carl Gauss stuck his hand in the air. He had finished; on his blackboard was written "5050", and he was correct.

The other kids had maybe calculated 1+2+3+4+5+6+7=28 by that time. Those that most people would consider good mathematicians had maybe already summed the first 15 to reach 105.  How did he do the calculation so fast?

The answer? He cheated. That is to say, he did some actual maths.
To calculate the sum by hand is doable. But it's also laborious, boring and slow. And what if they finished and the teacher said "Now sum up to 10,000"? They would have no chance!

What Gauss had realised was that instead of adding like everyone else: 1+2+3+4+...+98+99+100, he could rearrange the terms and add like this instead: (1+100)+(2+99)+(3+98)+...+(50+51). Each of these pairs of numbers summed to 101, and there were 50 pairs of them. So the answer was simply 50*101=5050. What he had done was lazy and creative; far closer to mathematics than what everyone else was doing.

The real genius with this inspired approach is that there is nothing special about the number 100 here. We can just as easily sum up to 10,000; (1+10,000)+(2+9,999)+...+(5000+5001)=10,001*5000, so we reduce the problem to a simple multiplication. In fact, summing up to any even number, n, we will pair off the numbers into n/2 pairs, each summing to n+1 and so the sum is n(n+1)/2. This also works for odd numbers, though a slight adjustment to the method is needed. Perhaps you can figure it out?




4 - Maths is beautiful

"Mathematics is the music of reason" - James Sylvester

"Mathematics has beauty and romance" - Marcus du Sautoy

"It’s like asking why is Ludwig van Beethoven’s Ninth Symphony beautiful. If you don't see why, someone can't tell you. If [numbers] aren't beautiful, nothing is.” - Paul Erdos

Perceived beauty is largely due to a combination of symmetry and chaos, of patterns and of fluidity. But symmetry and patterns are inherently mathematical properties, as are chaos and fluidity. Visually beautiful things often contain lengths with the golden ratio, a particular number with many curious properties. (See http://www.instant-analysis.com/Principles/straightline.htm)

For example, music is beautiful primarily due pitch and rhythm. But rhythm is, of course, all to do with patterns in time. Similarly different notes sound pleasant together because they are chosen according to mathematical formulae that describe how well their different wave forms interact. Beauty is also present, of course, when musicians transcend the rulebook and mess around with the timing, or play notes that shouldn't work.

 But this more creative beauty necessarily presupposes the existence of basic mathematical structures; if there was no "correct time" then what would it mean to "mess with the timing"? And what would a "blues note" be without the standard scales? Furthermore, this form of creativity is a sign of more subtle rules at play, as opposed to the true absence of rules, and as such is underpinned by maths in the more direct sense, too. Similarly with any other beautiful thing. Mathematics not only provides a basis for beauty, but actually underpins and explains the human concept of beauty.


But, of course, this is not enough. Biology, one may argue, can explain the social and survival value of a concept of beauty. It does not mean that the biological systems themselves are themselves beautiful.

Indeed, it does not. It is conceivable that a very inelegant and ugly biological system could birth conditions in which beauty is desirable. But with mathematics, we are not talking about something that exists contingently as mere product of circumstance, in the same way our biosphere and evolutionary history does. We are talking about something that, as far as we can tell, exists necessarily . Something, certainly, that is intrinsic to our universe.

And thus, the intelligibility and simplicity of mathematics, combined with its enormous explanatory power gives us reason to think that this simplicity and symmetry - to which, and indeed by which, we are affected to attach elegance and beauty - is a property (and thus, in some sense, desirable) of our existence.

This is a somewhat weak argument for the idea that beauty is a mathematical property. But of course the much weaker idea that maths can be beautiful is easier to provide a convincing case for. Namely, the vast majority of people who have ever understood higher level maths have found it elegant and beautiful, And so, in the weaker subjective sense, maths can be safely assumed to be beautiful.

If you are keen to actually see some of the beauty in mathematics, you can get a taste of it by you-tubing "Mandelbrot set" and exploring from there. If you want a full dosage, I'd recommend sticking with formally taught maths long enough for it to get going properly!




5 - Maths is practical

"But, in my opinion, all things in nature occur mathematically" - Rene Descartes

"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." - Eugene Wigner


This is perhaps the single biggest misconception about mathematics; that it is void of practical application. It is one thing to think that a degree in mathematics is useless in the real world, but quite another to suggest that mathematics is itself valueless to humanity. Granted, they are both factually incorrect statements. But the former I feel somewhat sheepish arguing against. The latter view, however, is simply unacceptable. Everything requires mathematics, at least indirectly. 

For centuries maths has been invaluable to the Physics and Astronomy, and often - unbeknownst to the viewer - to the visual arts. But nowadays it really has become ubiquitous. Biologists model population growth with differential equations. Philosophers borrow propositional logic and set theory. Sociologists use hypothesis tests and statistics. Lawyers *should* use Bayesian probability. (see http://understandinguncertainty.org/court-appeal-bans-bayesian-probability-and-sherlock-holmes)

We are not talking just about arithmetic, but proper, high-level maths. Particularly in the physical sciences and in computer science, the maths required is on a similar level to that being discovered at the very frontiers of maths research! The need for maths crops up literally everywhere. The unreasonable effectiveness of mathematics in the natural world is one of the most mysterious and curious observations we have made about our universe.

Of course, such is the delight of mathematics itself that a lot of mathematicians care little for its practical application. If it didn't happen to be a solid career choice with prospects of a decent salary then I'd be doing it in the evenings as a hobby. The intrinsic beauty of maths has enough appeal without having to relate to the physical world. But it would not be an understatement to say that without mathematics being incredibly useful, we would be living in a much more primitive world.




6 - Maths is personal

"God used beautiful mathematics when he created the world" - Paul Dirac

"Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere." - W.S. Anglin


In school, maths didn't seem to be personal in the slightest. Everyone learnt the same method(s) to solve a problem, and tended to work individually. There was not collaboration in the same way that there was in more discussion based subjects like English and RE, where we would bounce ideas off each other. However, this changes at degree level - at least in my experience!

Suddenly, as aforementioned, one is expected to solve problems creatively. Two mathematicians of similar competency will likely both have the tools at their disposal needed to solve the problem; the tricky part is coming up with an idea; a strategy or game-plan that will be employed to attack the problem.

This means that one person may not come up with a successful idea for hours, whereas another may see it instantly, even though they are of similar ability; the roles may well swap for the next problem! Thus collaboration becomes absolutely key. Maths, for the vast majority of practitioners, is inherently social.

As another corollary of this, it becomes important that mathematicians are not merely "machines for turning coffee into theorems" - here I beg to differ with the great Paul Erdos! - but, rather, are individual people with different traits and ways of seeing things. It is simply false that all mathematicians are similar in nature; there are common traits and eccentricities but mathematicians more diverse in nature and in outlook than one might suspect. It is conducive to healthy collaboration to have different perspectives and tool-kits at ones disposal when solving a problem.

Lastly, and on a related note, I am familiar with the notion that nearly all maths was done hundreds of years ago, and that there really isn't very much left to discover. This has an element of truth to it; all of "the basics" seem to have been covered, in that it is now quite rare for an amateur mathematician to solve a new problem with no knowledge of higher-level maths.

But the main motivation of the notion - namely, that new maths has all but dried up - is plainly false. The number of maths papers published per year went gone up from 3,000 in the 1940s to 160,000 in the 1990s, and shows no sign of retreating any time soon. While the collective human bank of mathematical results is incredibly vast, and while the easiest route to publishing original research now involves a significant number of years of preliminary studies, we appear to only have scratched the surface. Indeed, the vast majority of the most beautiful results in mathematics may still be waiting to be found!




7 - Maths is not (needlessly) pedantic

"I never did very well in math - I could never seem to persuade the teacher that I hadn't meant my answers literally". - Calvin Trillin


A lot of discoveries in mathematics rely heavily on intuition and creativity, as has been established. But once we use these skills to get to an answer that we think is correct, logic steps in. This is where mathematicians formalise their thoughts, and check to see whether their intuition holds up to scrutiny. Mathematics of this kind has a very rigorous layout; first we establish our assumptions (called axioms) and then we logically write down "in between steps" that lead us to our conclusion (often called a theorem), where each single step is deemed to be logically true.

Sometimes this means that mathematicians question and try to prove things that every person knows to be true already. For example, everyone knows that the shortest path between two points is a straight line, right?!

Well, why is that? Can you prove it? Your response will probably be something like:
"It is obvious. Take the straight line between the two points. Now change a small section of it so that it is no longer a straight line. That made the path longer. Therefore the straight line is the shortest possible path."

But that is not an acceptable standard of proof in mathematics. Two flaws are present in this argument. The first is that it assumes the result in the proof. Why is it true that a small deformation of the straight line results in a longer path? That only is necessarily true when the result you are trying to prove is true! Secondly, even if small deformations anywhere on the path made it longer, that means it is locally optimal but it does not follow that the path is globally optimal. 

In other words, there could be some crazy path that you just haven't thought of yet which is shorter, that looks nothing like a straight line. Just because you've thought really hard and failed to think of it doesn't mean it doesn't exist! Consider standing on a rock out to sea, so that you are only knee deep in water. It doesn't matter which direction you go, the water gets deeper and deeper. However, it is incorrect to assume that you are on the highest surface in existence; infact, the mainland exists!

As it turns out, the shortest path between two points is, indeed, a straight line.  But only once we've required a "path" to have a few tight conditions so that it is, in s strict sense, "well behaved". Indeed, there are conceivable definitions of "path" in which this doesn't hold. The two short edges of a piece of paper are, we suppose, at least a length of paper away from one another. Yet in three dimensional space we can simply bend the paper and make them touch; in a sense, the shortest path between them has zero length. (See http://www.instant-analysis.com/Principles/straightline.htm)

The reason that mathematics can seem pedantic is because human intuition is far from flawless. The Earth is not flat. Angles of a triangle adding up to 180 degrees only in a very special case. And in some sense 1+2+3+4+5+... = -1/12. Using well defined assumptions and consistent schemes of logic we can be sure that our results are true. Certainty of this kind really does not exist outside of mathematics.





Hopefully this post has enlightened readers who have not pursued maths beyond school level. It really is a fascinating field of study. While I do not hold that it is better than other subjects, I do believe that it offers something truly unique to what humans can achieve; namely discovery of absolute truth via pure reason. As such I wish that everyone was open to learning about the beautiful truths present in maths, in the same way that most are open to exploring the arts, social sciences and, more recently, the natural sciences. I would recommend you take a wander in the "strange wilderness" sometime, it may just be a very different experience to what you might expect.

Jeff

Why Atheism isn't the default position

In debates concerning the issue of God, atheists often feel that they can win the argument on these grounds of the "burden of proof" alone; if the burden of proof lies entirely with the theist, atheists need not compete to explain reality convincingly. They can merely point out incompletenesses in theists arguments! Atheism, then, merrily becomes the default position in the absence of (unattainable) standards of evidence for God. Below we consider the main justifications for this viewpoint.

1 - Russell’s Teapot

"If I were to suggest that between the Earth and Mars there is a china teapot revolving about the sun in an elliptical orbit, nobody would be able to disprove my assertion provided I were careful to add that the teapot is too small to be revealed even by our most powerful telescopes.” - Bertrand Russell

The argument implicit here is that because no evidence or explanation for disbelief in an extraterrestrial teapot is required, so it is with God. But here exists a clear category error. The existence of a china teapot in space would explain nothing and require much to be explained, whereas a proof of the existence of God would explain the origins of the universe, morality, history of religion, consciousness and much more, in one fell sweep!

Furthermore, the properties of said teapot that make its existence unlikely are that it appears to be a designed object, and we think it improbable that it could have been placed there by intelligent life without our knowledge. But this is inherently a design argument, which - by definition as both an uncreated being, and outside of our universe - God is not susceptible to.

Imagine I came home one day to find my room (usually faithfully demonstrating entropy of a closed system over time) in perfect order. My bookshelves stacked and labelled alphabetically, and my desk devoid of scrap paper and chocolate wrappers. Two hypotheses occur to me, the first is the existence of Russell’s teapot and the second is that my mother became tired of my mess, cleaned my room, and subsequently left the house.

Both of these hypotheses are coherent and (temporarily) not testable. But does that mean they are equally valid? Of course not! Either could well be true or false according to the information we have, but to assert that they are therefore both carry 50/50 probability is ridiculous.

The proper way to go about things is to observe the world around us, and thus form hypotheses that both are explanatory and not superfluous, and see which theory explains the most, assuming the least. The second criterion, while less obvious than the first, is a pretty crucial scientific principle known as Occam’s Razor;

“It is pointless to do with more what can be done with fewer.” - William of Ocker

This is often used in an argument against the existence of God as follows:

2 - God of the gaps

“People think that epilepsy is divine simply because they don't have any idea what causes epilepsy. But I believe that someday we will understand what causes epilepsy, and at that moment, we will cease to believe that it's divine. And so it is with everything in the universe” - Hippocrates

This is merely a straw-man argument. A theist does not pose that God is (and only is) responsible for the complex mysteries of reality that we are yet to fathom, by merely “performing magic”. As John Lennox puts it; “God is not a ‘God of the gaps’, he is God of the whole show.”

Furthermore, theism is the only world-view that provides an explanation for the intelligibility of the universe which is the foundation of scientific investigation! The argument assumes that we can only see God’s workings through what we do not understand, and completely fails to acknowledge that, in the eyes of the theist, knowing what God has done is not a matter of scientific determination, but of exegesis.

So we see that Occam’s razor is grossly unsuitable as an offensive weapon for the atheist. But more than this, we shall presently see that it is an adequate benefactor of theism.

Currently the only non-theistic theories concerning how the universe came to be that claim to be complete involve infinite numbers of unobservable universes. As J. Budziszewski wittily puts it:

“The principle of inference seems to be not Occam's Razor but Occam's Beard: ‘Multiply entities unnecessarily.’ ”


It is no use the atheist saying “But the only complicated bit is right at the beginning; everything else is simple!” as Occam’s razor must be allowed to work on the whole system. A train that takes you 95% of the way to your destination but leaves you having to scale a mountain on foot is far worse than a car that takes you all the way at a slightly slower speed!

3 - Pascal’s Wager

Our investigation into the burden of proof addresses the question: “How much evidence must there be for theism to warrant belief?”

But there is a similar, more subtle question that arises naturally from settling this one that we should attempt to answer; “How sure must I be of the truth of Christianity to act upon it as if it were true?”

This is somewhat of a silly question, and so I am afraid we here descend slightly onto some slightly less logical terrain. Of course the answer of this has no bearing on how likely Christianity is to be true, but there are two good reasons why the answer is not just “trivially anything above 50%”.

One reason is simply that we are not completely logical creatures. A lot of the time we believe things simply because we want to. The reason we tend to accept facts when we deduce that they are true is that, most of the time, the more closely our knowledge resembles complete truth the better off we will be.

But sometimes this is not the case (at least when “better off” is seen to mean mental and physical well-being) and thus we may reject things we know to be true to better ourselves in this regard. So the answer depends on whether believing will benefit us, or indeed benefit others.

Atheists no doubt will here jump in and claim that, as being Christian means doing a lot of extra things and restricting yourself in many ways, this supports atheism as a belief system. But to refute this argument I simply quote past president of Royal College of Psychiatrists Andrew Sims:

“The advantageous effect of religious belief and spirituality on mental and physical health is one of the best kept secrets in psychiatry and medicine generally.”

And Professor of Anthropology David Sloan Wilson:

“On average, religious believers are more pro-social than non-believers, feel better about themselves, use their time more constructively, and engage in long-term planning rather than gratifying their impulsive desires. On a moment-by-moment basis, they report being more happy, active, sociable, involved and excited. Some of these differences remain even when religious and non-religious believers are matched for their degree of pro-sociality.”

Having established that this argument does not favour the atheist, let us see if it can be pushed further in the favour of theism. We look next to Pascal’s Wager, which uses game theory to suggest that, if there is a non-zero chance Christianity is true, we should act as if it is.

Pascal’s Wager	God exists	God does not exist
Belief in God	Infinite reward in heaven	Finite loss (at worst!)
Disbelief in God	Infinite loss in hell	Finite gain (at best!)

One way out of this is to respond by further categorising people as “good” and “bad” and supposing that this, and not belief, determines where one spends eternity, if Christianity exists. But even with these assumptions (which seem to argue that if Christianity is true, then Christian doctrine is false!) are granted, one is still back to the issue of religion belief being beneficial, and so again, the argument stacks against atheism.

A second proposed route of escape is offered by the new atheists;

“[Pascal’s wager] seems a cop-out, because the same could be said of Father Christmas and tooth fairies. There may be fairies at the bottom of the garden. There is no evidence for it, but you can't prove that there aren't any, so shouldn't we be agnostic with respect to fairies?” - Richard Dawkins

“So next time someone tells me they believe in God, I’ll say “Oh which one? Zeus? Hades? Jupiter? Mars? Odin? Thor? Krishna? Vishnu? Ra?…” If they say “Just God. I only believe in the one God,” I’ll point out that they are nearly as atheistic as me. I don’t believe in 2,870 gods, and they don’t believe in 2,869.” - Ricky Gervais

But note that neither variant of this argument stands in the light of our investigation into Russell’s Teapot. The difference between belief in no gods, and in one God is exactly the same as the difference between belief in no gods, and in many gods, purely in terms of explanatory power. And the same applies to fairies.

You can assign as much weight as you like to this section. Indeed, I am not sure how much value I assign to it. But the important point is that if anything this is an argument in favour of theism.

4 - Non-cognitivism

This is the final school of thought worth considering before we wrap up this post; non-cognitivism asserts that moral statements are not propositions, and thus cannot be known to be true or false, and that therefore talk of moral knowledge (or indeed of God) is meaningless.

This viewpoint does not worry me. One can think of God without considering morality; instead simply utilising the purely philosophically (well) defined idea of an uncreated sentient being as an explanation for reality. Also, whether moral statements are propositional or not is contingent on whether or not there is an objective basis for morality, which in turn is contingency on whether or not God exists.

So, in essence, the argument reduces to: “Suppose God does not exist. Then discussion of God is meaningless.”, a sentiment with which I wholeheartedly agree, without introducing any doubt into my theistic world-view!

Overall it is fairly clear to me that the burden of proof argument for atheism as a default position holds little weight. In fact, as history readily reveals, the obvious initial conclusion drawn upon observing the universe is that God does exist. This is not at all meant as evidence that God does exist, but rather as evidence that the best course of action is to evaluate whether or not God exists fairly, without defaulting to atheism until God’s existence is as good as proven! Now that we have established that, we can go on to look at the evidence either way through the correct lenses, as it were. 

Happy searching!