Do “Monkey-Typewriter” Arguments Work?

There is a famous analogy that is used in discussion of highly improbable events – that of monkeys sitting at a typewriter, banging away at the keys. It is said in this parable that, given enough time, the monkey will type out Hamlet, or a Shakespeaean sonnet, or the complete works of Shakespeare, or some other work of literature, because with unlimited time, any random sequence will eventually show patterns like that. The purpose of the monkey-typewriter parable is meant to be that a highly unlikely event cannot be dismissed just because it is unlikely, because unlikely things happen given enough time. This kind of thinking is relevant with something like evolutionary theory, say, where the development of life is attributed to random genetic mutations (while those who survive are not per se chosen at random, the set of genes from which the survivors are chosen is essentially random).

Technically, this is true. But there is a problem. We don’t have infinite time available to us. The Big Bang happened a finite amount of time ago. It is almost an unimaginable amount of time for us limited people with lifetimes no more than about 100 years, but it is finite nonetheless. So, have we had enough time? Well surely about 14 billion years (the approximate age of the universe according to the mathematical calculations of the general relativity theory of Einstein) is enough, right? That is such a massive number, surely that must be enough time to account for all kinds of highly unlikely events.

This turns out to be quite mistaken. The probability calculations involved here involve what is called exponential decay, and exponential decay and the related concept of exponential growth are notorious for defying almost everyone’s mathematical intuitions – including mathematicians and scientists. Therefore, we ought to be very careful when we try to reason about any domain in which exponential growth or decay come into play. This article is meant to do exactly that. I will do some mathematical calculations to show how you’d actually compute something like the probability of a monkey typing out Hamlet.

And I suspect that the results will be surprising to most people… so I will be careful to justify the mathematics and I will try to be as generous as possible with giving the randomness every opportunity to succeed.

What Is the Probability?

This part isn’t necessary to understand the end of the post – if you don’t want to follow the mathematics, just skip this part. The formula I derive will be used with actual numbers later.

Instead of using numbers straight away, we will begin with a broader question. And we will make some simplifying assumptions to help us out – and we will simplify in the direction of making things easier for our aspiring primate poets.

It is rather difficult to analyze whether a particular letter pressed by the monkey is a ‘good’ or ‘bad’ letter, because for that we need the previous letters. And we don’t know how often the monkey will press the spacebar. To simplify things, I will then be speaking mostly about the probability that, between any two spacebars, the monkey has typed a coherent word. We will use p as a shorthand for this probability, so

p = Probability of a coherent word.

For instance, if p = 0.1 (which is the fraction 1/10), this means that one out of every ten strings of letters typed by the monkey will be an actual English word.

To be generous to our hard-working monkeys, we won’t force them to adhere to grammatical rules. That would make things more complicated. You could introduce, if you like, a probability that between any two periods is a coherent sentence. This probability would be quite low. But to be generous, we will ignore this.

The next thing we need to know is how long the literature we are trying to write is. How many words long is the book/poem that is our aim? As a stand-in for the word length, let’s use the letter L. That is,

L = Number of Words We Need.

Again, let’s be very generous to our monkeys and not require that they actually type the correct words – let’s just allow them to type the same number of words. So we don’t ask them for Hamlet, just the same number of words that appear in Hamlet. We would also like to know how quickly the monkeys type, and how much time we have before we have to send our monkey’s work to the publishers. Let’s use e, t and T to mean the following:

t = total time it takes for our monkey to type something of the correct length.

T = total amount of time they monkeys are given to finish their job.

Finally, let’s not leave our monkey all alone – monkeys needs friends. Let’s let lots of monkeys all have their own typewriters, typing away together, each racing to be the first one to have their own book. We can use M for this:

M = Number of Monkeys Trying

This will be all the information we need for a formula. What we want is the probability that some monkey eventually produces something coherent within the allotted amount of time. First things first – how many potential books will each monkey produce? The answer to this is the total time divided by the time it takes to produce on such book – this is T / t. If we have M different monkey all doing this, the total of number of documents that will be produced by all the monkeys cumulatively is MT / t. Secondly, we need to know how likely it is that each possible book will actually “make sense” – how many books will have only words with no nonsense? This is a similar question to asking how likely it is to flip heads a bunch of times in a row – we multiply the probability to itself the correct number of times. This gives us a probability of p^L that one of works produced is actually made up of words, and a probability of 1 - p^L of failure. To calculate the probability that every attempt will fail, you take the probability of failure and raise it to the power of the number of attempts. The final formula we obtain for the overall probability that we will not be publishing the first ever book written by a monkey is

(1-p^L)^{MT/t}.

This is the probability of failure for each attempt to the power of the number of attempts.

So, now we ask – what can we do with this? What kinds of values do we get out of this formula?

What Values Should We Use?

In the spirit of optimism, let’s make an effort to be as generous to our monkeys as we can. Let’s try to make all of our numbers as large as we can to help our monkey out.

First, how big can we make M? Well, we could estimate the number of monkeys that exist on earth today, or that have every existed. But that is too boring – instead, let’s go insane. Quantum physics has as part of it something called the Planck volume, which you can think of more or less as the smallest amount of space that is meaningful to talk about in a laboratory. Anything smaller is literally impossible to measure. This number is around 4.22 * 10^{-105} cubic meters. We can hardly be any more generous than to allow our monkeys and their typewriters to only take up one Planck volume worth of space. In cosmology, scientists can also approximate the total volume of the universe, so we can actually calculate how many of these micro-monkeys we can stuff into our universe. The estimated volume of the universe is 3.58 * 10^{80} cubic meters. Working out the numbers, our impossibly generous value of M will be

M = \dfrac{\text{Volume of Universe}}{\text{Planck Volume}} = \dfrac{3.58 * 10^{80}}{4.22 * 10^{-105}} \approx 8.5 * 10^{184} monkeys.

Let’s also be as generous as possible with our time constraints. We can allow our monkeys to begin typing at the moment of the Big Bang itself – which was about 14 billion years ago. Converting to seconds, we arrive at our insane value of T:

T = 4.4 * 10^{17} seconds.

Like the Planck volume, in quantum physics there is also a Planck time – the smallest amount of time that can be meaningful in a laboratory. To be as generous as we can, let’s allow t to be the Planck time, so

t = 5.39 * 10^{-44} seconds.

We can now compute the total constant MT/t from the equation earlier. From all of these numbers, we arrive at

\dfrac{MT}{t} = \dfrac{(8.5 * 10^{184})(4.4 * 10^{17})}{5.39 * 10^{-44}} \approx 6.94 * 10^{245}.

Therefore, our probability of working out a “book” of L words in our insane scenario (let’s call this B for book) is

B = (1 - p^L)^{6.94 * 10^{245}}.

The only things we have ye to choose are p and L. The value of p = 0.1 used as an example is absurdly high, as I think is be clear upon intuitive reflection. To put some numbers to this, though, let me list out a few (approximate) probabilities for shorter words

  • The probability of two random letters making a word is around 124/676 \approx 0.18, or 18%.
  • The probability of three random letters making a word is around 1292/17576 \approx 0.07, or 7%.
  • The probability of four random letters making a word is around 5454/456976 \approx 0.01, or 1%.
  • The probability of five random letters making a word is around 158390/11881376 \approx 0.01, or 1%.
  • The probability of six random letters making a word is around 22157/308915776 \approx 0.007, or 0.7%.

To give you an idea of the meaning of all of this, I counted in the first sentence of this article 120 letters and 26 words, which averages out to 4.6 letters per word. This puts us in the 1% area. I think it would be appropriate then to allow our value of p to be around 1%. In reality it would be lower – after all the probability of pressing a space bar if everything is truly random would be 1/27, which would mean our monkey would on average type 27 letters in a row between spaces. There are, well, basically no words 27 letters long, so we would probably have to sift through hundreds upon hundreds of pages to find even one coherent word. So, I think a value of p at 1% is reasonable enough.

Now, the only value we have left to choose is L. We can use various values of L to see what happens. Let’s first use the classic example of Hamlet – which is 29551 words long. Then the value of B is

B = (1 - (0.01)^{29551})^{(6.94 * 10^{245})}.

How big is this? Well, when I initially typed it into Wolfram Alpha, a very advanced online calculator, it told me the answer is 1. In other words, the probability of failure is so close to 100% that Wolfram Alpha just assumed it really was 100%. To get an idea of the actual size, then, we have to use some methods of approximation. Using something called a first-order Taylor expansion, we know that if p^L is small positive value and N is any positive number, then the approximation (1 - p^L)^N \approx 1 - N*p^L is pretty good, and in fact that if $N$ isn’t absurdly large the amount of error is virtually undectable. Therefore, our probability is about 1 - (6.94 * 10^{245})*(0.01)^{29551}, which means the probability of a success is about

(6.94 * 10^{245})*(0.01)^{29551} \approx 6.94 * 10^{-58857}.

This is a number that is something like 0.000000….000006, where the total number of zeros is about 60 thousand. It is beyond comprehension. For a slightly more concrete idea, I can convert this probability to a coin-flip scenario. These odds are about the same as flipping heads around 195518 times in a row. And keep in mind, we have been as generous as what we know about physics even allow us to be. Actually, changing the values of M, T, t won’t really do very much. It’s really the values of p, L that are the heavyweights here. But even changing these won’t quite work. If we instead use a 200 word poem, then we obtain instead the approximate probability

(6.94 * 10^{245})*0.01^{200} \approx 6.94 * 10^{-155}.

This is still really bad – this is like flipping 514 heads in a row. This stuff just doesn’t happen.

Moral of the Story

So what is the moral of the story here? You can’t just appeal to “billions of years” to account for extremely unlikely things happening. For example, as valuable as the theory of evolution is, it requires something more than totally random genetic mutations to work. Things are just too complicated here. The genetic code of even the simplest living organism – like a single-cell bacteria, is vastly less probable to find from random chance alone than is often suggested. It is a situation like typing Hamlet by banging on a keyboard – there just isn’t enough time for an appeal to chance to work. It isn’t even close to being close to enough time.

Even if the probabilities with evolution were made reasonable, that doesn’t address how the first ever living cell appeared on earth. Evolution as we know it to day by its very definition cannot explain that, since evolution only claims to explain how one type of life gives rise to another type of life. The very first cell (or very first form of life) did not come from another living thing, so evolution doesn’t apply. Randomness alone doesn’t work – the probabilities are all too low to take that seriously. Some sort of orientation is needed, but there can’t be anything like natural selection going on, because we are not in the realm of biology anymore but in mere chemistry.

So, as thinking people I think we can dismiss arguments that appeal to probabilities playing out over billions of years. Any argument of that sort is like a plane without wings – it needs a new piece to get off the ground. So if you encounter an argument of this sort, ask for more evidence.

An Ethical Exploration of Looting and Rioting

After the horrible death of George Floyd recently, there have been a lot of riots. This is something we expect, but less expected are the many instances of looting that have been done in the name of protest against this injustice. This is a much less common and much less discussed phenomena. How do we think about this?

Understanding Some Ethical Theories

First, a way-too-brief summary of the very deep area of philosophy known as ethics. Ethics is the study of moral values and moral obligations. A moral value is a value based upon which an action or behavior of a person is judged to be good or bad. Values are generally defined first by what is good (eg. love) and then that which is bad is defined as an absence of (eg. apathy) or twisting of (eg. hatred) that which is good. A moral obligation is a kind of behavior that we are required to do in light of relevant moral values. In less complicated words, moral obligations are those things which we ought to do (or ought not to do) – the key word here being ought.

Ethics is a complicated area, and there are many competing ethical theories that attempt to provide a framework for how to differentiate between right and wrong (moral obligations) ad between good and evil (moral values). While not comprehensive, the following list will summarize a few very different views of ethics (focusing primarily on the right/wrong distinction).

  • Consequentialism – This theory holds basically that an action is right exactly when the consequences of that action are good (or that the good consequences outweigh the bad consequences).
  • Deontological ethics – This theory holds that an action is right exactly when that action is aligned with a foundational set of moral rules. A common version of such a theory would be divine command theory, within which the foundational set of moral rules is based upon the character and commands of God.
  • Virtue ethics – This theory holds that an action is right exactly when it helps the person develop within themselves greater virtue.

It might seem like these are not so different – after all, I think every one of us operates in these modes of thinking at various points in time, and each of them has some face-value plausibility to it. However, debates arise because often these ethical theories contradict one another in various situations. It is then that philosophers who study ethics work to attempt to untangle the weeds, so to speak, and improve upon these frameworks to more accurately reflect what we all mean by the words good and right.

In What Way is Rioting/Looting Justified?

With a few important ethical theories laid out, we can explore how these theories speak to the heated debates today about riots and looting.

The consequentialist holds that a riot is justified (i.e. right) when the consequences of the riot bring about more benefit than harm. In the current situation, a consequentialist might place on the positive side of the balance the attention brought to the problem of racism and police brutality in our country. On the negative side, they would place things like the damage caused by riots and looting or making different political parties less willing to cooperate in the future (if they believe that will in fact be a consequence). A consequentialist would attempt to weigh these and behave accordingly. We see this approach in social media today. Those whose cite the fact that peaceful protests have not worked in solving the racism problem as justification for riots and looting are applying consequentialist ethics.

Deontological ethics, of course, depends on what foundational set of rules you are using, since different sets of rules will lead to different conclusions. Since the view of ethics that I hold most firmly to is a type of deontological ethics, I will use that view. In particular, I hold to divine command theory based upon the God of Christianity, so that when I am trying to determine whether some action is right or wrong, the first thing I want to do is reflect on teachings and commandments of the Bible. The most central teaching here is the doctrine of imago dei – that all human beings are made in the image of God and are therefore are of infinite moral value and have intrinsic rights, freedoms, and obligations towards one another. The imago dei consists attributes shared by all humans, including but not necessarily limited to emotions, thoughts, reasoning abilities, memories, and the ability to communicate using language. The inestimable value of human life, combined with proper understanding of moral commands found in Scripture, form the basis of thought for the question. Take the example of looting. There are clear commandments against stealing, and since looting is a form of stealing, it would follow that we ought not loot. However, there is also clear precedent in the Bible that there are some moral commandments that are more essential than others (see Luke 10:27 and surrounding verses). Therefore, if there is some more essential moral command that would justify these actions of looting, then looting would be justified. An example of a possible justification would be that the looting is done in protest of murder stemming from racism – and both murder and racism are clear violations of the value we possess in light of the imago dei. An adherent of divine command theory based on Christianity would then have to try to decide whether looting as a form of protest actually does line up with the full scope of commandments in the Bible, of which I have laid out a few.

The virtue ethicist would ask themselves whether engaging in riots and looting are beneficial to the character of the individual participating in them. Since these are being done in protest of the murder of an innocent man, it could be argued that these actions would increase within a person their sense of justice, which would make the action right. On the other hand, one might think that engaging in any form of violence, even in protest, causes within a person a tendency towards more violence, which would make these actions wrong. Perhaps you think this increases a persons tendency towards rage, which would make the action wrong. Or perhaps you think by identifying with people who have been killed that it will increase their compassion, which would make it right. An adherent of virtue ethics would then have to contemplate how engaging in a violent protest would affect their ‘pursuit of virtue’ so to speak, and would have to make a decision based upon those.

I hope it is clear from this discussion how different theories of ethics lead to quite different patterns of thought, and how they might even lead to different conclusions.

Personal Thoughts and Reflections

Having laid out a philosophical framework to convey how truly complicated this topic really is, I will try now to lay out some of my own reflections. As I stated earlier, I operate in my ethics from a Christian divine command theory, which is a kind of deontology. So, my reasoning about right and wrong stems from moral obligations found in commandments in the Bible and their implications. As I have also already stated, two primary considerations for the moral rightness or wrongness of these lootings, which are repeated over and over again in the Bible, are ‘do not steal’ (see Exodus 20:15, Ephesians 4:28 for examples) and ‘defend the oppressed’ (see Isaiah 1:17, Psalms 9:9, Proverbs 14:31 for examples).

Now, before going into the points I think are meaningful, I will first handle some that I think are not meaningful. As a deontologist, I reject consequentialism, and therefore I reject any justification for looting that relies on the consequences of the action. I also reject justifications that are purely internal in nature. That is, there is more to this than what motivates a person to protest in this way. In effect, this is a rejection of virtue ethics – I believe that people can do bad things for good reasons. I’m not actually saying anything good or ill of rioters at the moment – I am merely pointing out that I do not believe that those types of reasons are probably entirely invalid, and at least they are not the most important thing to consider.

On an emotional and pragmatic level, I understand exactly why people are rioting and looting. What I have perceived in terms of the reasoning of those who riot and loot might be summarized briefly as follows: “Racism is a huge problem. Furthermore, the problem is not so much racist individuals as a system that is either unwilling or unable to punish racist behavior. Therefore, our protests are not directed at individuals, but at a system. The system we are targeting with our protests (let’s say the at-large culture of the USA) places an extremely high value on monetary wealth. Therefore, protests that do financial damage are likely to shake this system into changing, and looting is then an obvious choice of method.” This seems to me to be the pattern of thought that leads to riots. Regardless of whether we agree or disagree with these conclusions, I think that this train of thought is something we can all understand. And someone operating within this train of thought does not view these instances of violence as directed at a person, and that changes the moral framework within which they process the violence.

While I empathize quite a lot with the summary I gave, I don’t fully agree with it. I have two primary issues with this train of thought as I understand it. Firstly, it is very rooted in consequentialism, since the motivation is about how to accomplish something. Secondly, I think the logic presupposes a much stronger separation between ‘systems’ and people than is realistic. The ‘system’ is not a physical thing out there that violence can be directed towards – violence against a system seems to me to be violence against a member of that system. And violence against a person, in my view, can only be morally acceptable when it is used as a form of defense against a pressing danger to you or an innocent person (eg. self-defense) or by a governing body given protective responsibilities attempting to enforce justice (eg. prison or capital punishment) or prevent a significant future danger (eg. police officers killing a mass shooter, war against the Nazis). So, I am convinced that in order for looting to be justified, it must be justified on the grounds of either the enforcement of justice or as a form of violent self-defense.

Can riots or looting be justified on these grounds? Well, enforcing justice in this way is the job of the government and not individuals (see Romans 13:1-7), and so I don’t think the enforcement of justice would work. As for self-defense, I can understand these actions being undertaken as a kind of preventative self-defense, meant to cause a cultural shift that will decrease the frequency and/or severity of these tragedies in the future. Quite honestly, I am still not quite sure whether I am on board with that ethically. On the one had, the imago dei ethic does provide reason to undertake even very extreme actions in defense of the oppressed. On the other hand, it does seem like much of what is happening falls into the category of doing the wrong thing for the right reason.

What I am sure of is that we will be far better as a country when we can get past systematic mistreatment of black people like this. It is a tragedy and is profoundly anti-Christian. The Bible explicitly teaches that we are not to bring division by treating some ethnicities as superior to others, or for that matter any other category, because people in different categories all have the image of God and are therefore all of incomparable value (eg. Galatians 3:28). I am thoroughly ashamed that so often these things are done by those who identify themselves as Christians, because the Bible does not condone such senseless evil towards anyone. In fact, the kind of evil most thoroughly condemned by Jesus is the sin of the ‘hypocrite’ – putting on an outer shell of piety and holiness while committing horrible evils and holding hatred in your heart (see Matthew 23:1-33, a scathing criticism of this kind of hypocrisy). Apart from preaching this Biblical truth and convincing people who have been led astray into racism to come back to the God of love, I am not sure what the best way is to solve this problem, or even if this problem can be solved in this broken world in which we live. But we still must try, because calling out racism wherever we find it is the right thing to do.

For Curious Readers

My most extensive ethical exploration here was done from the context of deontological ethics, and in particular divine command theory based upon the Christian Bible. A similarly lengthy discussion can be produced from other viewpoints on ethics, and similar discussions can be had for other ethical issues. If seeing these ethical theories laid out has piqued your interest, here is an exercise that I think will help you understand better the differences between these ethical theories and to explore your own ethical thinking.

Apply the foundational ideas of ethical theories to another currently discussed topic – is it morally acceptable for the government to mandate the wearing of medical masks in public? Furthermore, do individuals have a moral duty to wear masks during the pandemic in every situation, or only some situations? If only some situations, why? For any of the questions that interest you, form some responses from the perspective of a consequentialist, a deontologist, and a virtue ethicist, being sure to adequately consider all sides of an issue. Which of your responses is closest to how you actually think?

A Brief Biblical Reflection on Death

I write this reflection as a follow-up to a tribute I wrote in light of the recent death of Ravi Zacharias, a globally influential Christian preacher and apologist. The obvious thing to reflect on here is, well, death. Death is something that has happened to nearly every human who has ever lived and will happen to each and every one of us given enough time. We are all mortal. And basically everybody has had someone deeply influential in their own life die. So even the living are affected by death. And even those of us who are living and have not seen anyone we know die are probably still from time to time afraid of death. This struggle is quite possibly the most universal human experience – we all die and are all afraid of death at one point or another.

It is therefore quite important to understand that death plays in the grand scheme of the universe. Is death the end? Is human death the absolute end, or is there some kind of afterlife? If there is an afterlife, is it a reincarnation (a return to a temporary life) or is it a resurrection (a return to an eternal life)? If there is reincarnation, how exactly does that work, and is there an escape to eternity out of this world? If there is resurrection, to what state of being will we resurrect? Is there heaven or hell?

There are many questions to be asked. I do not intend to give thorough answers to all of them here. Rather, I intend to give the sort of reflections that I would give at a funeral. This distinction is of course important – I hardly think I need to say why. These reflections ought to be no less true than more philosophical reflections that I might give to someone early in life and not suffering from emotional distress, but are of quite a different flavor.

The firs thing that comes to mind is the verse in the Bible that changed the entire course of Ravi’s own life. As a teenager, Ravi found himself in a hospital bed after drinking poison in an attempt to take his own life. A local evangelist brought a Bible to him, and Ravi heard the gospel of John read to him. Ravi’s life changed when he heard the following passage:

“Yet a little while and the world will see me no more, but you will see me. Because I live, you also will live.” (John 14:19 ESV)

When Ravi heard “Because I live, you also will live” his entire life changed course. And I am sure that this verse was regularly on his mind in his last days as well. Because this verse conveys in an incredibly succinct manner the Christian hope for the afterlife. But let me lay this out clearly – because it most emphatically is not the mere claim that there is an afterlife. It is much more than that. The phrase “you also will live” on its own could be taken as a claim that there is an afterlife, but that is not all the verse says. We live “because I live” – and in context, Jesus is speaking. So, we mean that because Jesus lives, we live.

This is a very deep claim, on which one could easily write an entire book. In summary, this passage means at least this paragraph. Since humanity fell into evil, separating ourselves from the love of the God who created us, we have brought upon ourselves both spiritual and literal death. We kill one another, we spread disease (as we all know well during this quarantine), and humanity is full of examples of hatred and cruelty. Even though most of us don’t rise to this level of atrocity, we know that we fall far short of perfection, we all hold hatred in our hearts towards someone. We all cling to and are held captive by our vices. Because we have all succumbed to these evils, we all experience death in the depths of our souls – we are separated from the ultimate source of all love and all goodness. However, God Himself came down to earth to help us. No human could help themselves, so God became a man in order to help us. Jesus Christ lived a truly perfect life and through his divine power took on upon himself our own evil deeds in order to redeem us. With all this context, we can now understand the truth depth of John 14:19 – because Jesus Christ lives even though he was executed, so we shall also live with him even though our evil deeds have destroyed our souls.

I see no way a rational person can deny the immense love conveyed by this story and the incredible sense of hope and joy and Christians can have because of this story. What is even more radical than this is that this story is true. This stuff actually happened. There is a God – and that God has such incredible love that He voluntarily humbled himself by taking on human form and a human life – working as a carpenter for years and voluntarily dying by the most painful and humiliating means ever devised by mankind. And why? Because He loves us that much. Like a majestic king who sees his son drowning in a muddy lake, He runs through all the mud and dirt, putting his majesty aside without thinking twice, in order to save him whom He loves. This is the message that Christians hold to, the message that brings many Christians to tears in worship of their God – that immense love.

There are numerous passages of the Bible that convey in equally powerful words the core of this very same message from many angles and perspectives. There is no need here to delve into each of these. Were I to write at this length on each and every one of these passages, the result would not be an article but a book. The main point is this – that death is not an irreversible evil for the Christian, because we know that death has already been reversed by our Lord Jesus when he walked out of his tomb, and that he has been given all power over death. We have immense hope as Christians, and that hope is based in the historical reality of Jesus’ resurrection from the dead. Therefore, while it is entirely appropriate to mourn over death, because death is a great evil in this world, we can also rejoice because we know that the dead in Christ have gone to be with the Lord God Himself who has defeated death.

Ravi Zacharias is one of these many who are now with Jesus. I had hoped to meet him in my own lifetime on this earth, but now that cannot be. Nonetheless, one day I will meet him face to face in the presence of our Lord and thank him for all that he did for my life and for the whole church with his life and ministry. And I pray that each and every one of us who believes in Jesus will bring this message of hope and joy and redemption to another who does not yet know the Lord.

Proof by Infinite Descent

We have previously discussed proof by contradiction [1]. Here, we will be describing what can be viewed as a specialized version of this method. This method, however, is sufficiently specialized that is it worth discussing separately. As a mathematician, I find this idea absolutely brilliant. Even though it isn’t terribly difficult to explain, only an intellect of the first order with a lot of creativity could have come up with this. In fact, if you spend enough time thinking about it, this is a sort of reverse version of the method of proof by induction [2] – in fact, it is a sort of combination of the methods of induction and contradiction. We really only began to see this method in the 1600’s with the amateur mathematician Pierre de Fermat. There is a reason it took a while for this method to be used – even though the concept is nothing more than a special version of proof by contradiction, it required a lot of ingenuity to realize that there are certain kinds of problems that contradict themselves in this particular way.

You can imagine the kind of ingenuity that I am speaking of as similar to writing a beautiful poem or book. It requires a lot of time, devotion, and cleverness to write a good story. However, once the story is written down, the amount of skill required to understand the story is nowhere near the amount of skill required to write the story. The type of proof I will now explain is like that. It is not the sort of thing you’d immediately think about when you think about contradictory statements – and for this reason I consider this to be a quite enjoyable and ingenious method.

  • Suppose that we are posed with a problem for which every possible solution can be listed in order from least to greatest in an organized way.
    • For a day-to-day life example, think about the question “how much does such-and-such cost?” The ‘least’ possible answer is ‘Oh, it’s free!’ The next smallest is 1 cent, then 2 cents, and so on. When I list it out like this, I’m also not skipping anything – there isn’t a price between 1 cent and 2 cents, so we are making a genuinely complete list here.
    • For a ‘mathematical’ example, if you have an equation like x^2 + y^2 = z^2, where we assume that both x, y and z are positive whole numbers, then the value of x + y + z is a way you could ascribe an order to the solutions. The smallest possible value of x + y + z is the first solution, the next smallest is the second solution, and so on.
  • Suppose also that there really is such a thing as a first ‘possible’ solution, which may or may not actually be a solution, and that every solution only has a finite number before it.
    • In the previous example, the smallest possible value of x + y + x is 3, when x = y = z = 1. Since x, y, z > 0, the value of x + y + z could not be any smaller than 1 + 1 + 1 = 3.
    • The ‘finite number before it’ part is much like saying ‘if I choose a positive whole number, no matter which one, the number of positive whole numbers less than that one is finite.’ For instance, you could make a list of all the positive numbers less than one million – it would take a while, but you could do it.
  • Suppose that if our problem has a solution, then we can derive a ‘smaller’ solution using this solution.
    • The ‘smaller’ here is in reference to the ordering from before.

Here is the principle of infinite descent – if you can always use some hypothetical solution to a problem to derive a smaller solution to the same problem, then you could derive smaller and smaller solutions forever. Your solutions can ‘descend’ forever – hence the word ‘descent’ in the title of this article. This is not in and of itself a problem, if you had an infinite number of smaller solutions to work with, that is not a problem. But, if the context in which you are working does not allow an infinite number of potential solutions to your problem, then you eventually run out of space and contradict yourself. To see the type of contradictory statement that would result, here is an example:

There are an infinite number of whole numbers between 1 and 10.

This is obviously false – because 1, 2, 3, 4, 5, 6, 7, 8, 8, 9, and 10 are a complete list of such numbers, and this is not infinite. Here we are thinking in the style of proof by contradiction – if my starting assumptions wind up leading me to this point, then my starting assumptions were bad.

This is known today as Fermat’s method of infinite descent, or just infinite descent. It is extremely clever, and is worth reading over a few times. To repeat, the general idea is that you obtain some kind of list that, if it every begins, can be ‘decreased’ forever in some sense, but which for another reason cannot be decreased forever. Here we arrive at our contradiction, as in the methodology of proof by contradiction.

I now try to provide what I find to be a very interesting proof that uses this method.

Theorem: There is no right triangle with all three side lengths a fractional value that has an area that is the square of some fraction.

Proof: We will use the method of proof by infinite descent. For the moment, we guess that there actually is such a triangle, say with side lengths x, y, z all of which are fractions. Then since this triangle is a right triangle, it satisfies the Pythagorean theorem, so x^2 + y^2 = z^2 must be true. It also must be true that the area A = \frac{1}{2} xy of this triangle must be a perfect square, say d^2, and so we also have the equation \frac{1}{2}xy = d^2. Since all three of x, y, z are fractions, we can multiply all three numbers by the ‘least common denominator’ in order to make all of them into whole numbers that also satisfy all of the same equations, and so we can assume from the beginning that all three of x, y, z are positive whole numbers.

I have written before about right triangles with integer length sides, and we know from that discussion that we can remove all the common factors of x, y, z, and we can find some positive whole numbers a,b which have no common factors and one of which is even, that satisfy x = 2ab, y = a^2-b^2, and z = a^2+b^2 (to see why, see my posts *CITE ARTICLES*). We can use these new equations for x and y as substitutions into $\frac{1}{2} xy = d^2$ to obtain the new equation ab(a^2-b^2) = d^2, and remembering the difference-of-squares formula a^2 - b^2 = (a-b)(a+b), we can see that ab(a-b)(a+b) = d^2.

Now, this last equation contains a ton of information. Because, remember that the numbers a and b cannot have any factors in common, and for the same kind of reason that “even + odd = odd”, no two of the four of the numbers a, b, a-b, a+b have any common factors. Now, since d^2 is a perfect square, any prime factor in the equation ab(a-b)(a+b) = d^2 must show up an even number of times. Since no factors are shared by any of the four numbers on the left-hand side, only one of these four numbers can have as a factor any prime number that factors into d^2, and since everything winds up equal in the end, there must be the same number of factors on the left and the right. You can convince yourself along these lines of thought that all four of the numbers a, b, a-b, a+b must themselves be perfect squares.

In particular, we may choose some positive whole numbers r,s that satisfy the rule $r^2 = a-b$ and $s^2 = a+b$. Since one of a,b was odd and the other even, both of r^2, s^2 are odd, and so both of r,s are also odd (since only by multiplying two odds can you get an odd). Then because adding subtracting odds gives an even number, both of r-s, r+s are even. We now define some new numbers, u = \frac{s-r}{2} and v = \frac{s+r}{2}, which are whole numbers since both of the numerators are even. If we add these together, we find that u+v = \frac{s-r+s+r}{2} = \frac{2s}{2} = s, which is odd. The only way this can happen is if exactly one of the numbers u,v is odd, and the other is even.

We now want to learn a bit more about u and v. If we square each of them, and if we remember our formulas for r^2 and $s^2$ from earlier, and we use the simplifications (s-r)^2 + (s+r)^2 = 2(r^2+s^2), we realize that

u^2+v^2 = \frac{(s-r)^2+(s+r)^2}{4} = \frac{2(r^2+s^2)}{4} = \frac{2(a+b)+2(a-b)}{4} = a.

Earlier, we have already remarked that a is a perfect square. Therefore, the three numbers u,v,\sqrt{a} actually satisfy the Pythagorean equation, and the area of the resulting triangle is

\frac{1}{2} uv = \frac{1}{2}\frac{(s-r)(s+r)}{2*2} = \frac{1}{2}\frac{s^2-r^2}{4} = \frac{1}{2} \frac{(a+b) - (a-b)}{4} = \frac{1}{2} \frac{2b}{4} = \frac{b}{4}.

Remember, however, that b is a whole number and perfect square, and since 4 is a perfect square, so is b/4. So we have generated a new triangle with exactly the same property as before, that has a smaller area.

But we could do this all over again, finding smaller and smaller and smaller areas. We cannot actually do this, since you cannot have a list of positive whole number areas that gets smaller forever. Therefore, there cannot be any such triangle to begin with. We have now finished our proof.

This is a more complicated proof than some, but because it is more complicated, we can learn much more from it. Because we now know that all of the numbers we used along the way in this proof are impossibilities, we can say a great deal more than just our original statement about triangles.

For example, along the way we found three numbers a-b, a, a+b which were all perfect squares. These three form what is called an arithmetic sequence, because I can form this list by using a starting point (namely a-b) and repeatedly adding some number to form the next entry (here, we add b each time). We therefore now also know that there is no such thing as three perfect squares in an arithmetic progression.

Another thing we know, is that if a,b were actually perfect squares, we would have solved the equation z^2 = a^2 + b^2 by using perfect squares for a,b. In other words, if a = x^2 and b = y^2, we would have solved z^2 = x^4 + y^4. So we now also know that this equation is impossible if x,y,z are all positive whole numbers. This turns out to be a special case of what became a very important problem in mathematics known as Fermat’s Last Theorem, which makes the much broader claim that the equation x^n + y^n = z^n, even though it is just the Pythagorean equation when n=2, never has any solutions at all for any value of n larger than 2. What we have just done, in effect, is to say that for n = 4 there are no solutions.

This is one of my favorite methods of proof in mathematics. It isn’t used especially often, but I like the clever twists and turns in the argument. To me, it feels a lot like reading a well-written short story. I hope my readers can have a taste of that kind of feeling. If you do, then perhaps you can better understand why those of us who like mathematics feel the way we do about it.

References

[1] https://mathematicalapologist.com/2020/04/24/proof-by-contradiction/

[2] https://mathematicalapologist.com/2020/04/10/proof-by-induction/

Proof by Contrapositive

Sometimes when you are trying to solve a problem, you realize you really don’t have a lot of information to start with. One piece of good advice in problem-solving is to try to work backwards. That is, sometimes if you know what you want your solution to look like, you can backtrack to learn something about how you should be creating that solution.

One of the mathematical versions of this is proof by contradiction – this has already been discussed. But there is another that is called proof by contrapositive. And this one doesn’t involve the very strange idea of intentionally saying something false.

The method of contrapositive comes from the broader field of logic and is a method of rewriting any statement that looks like “If A is true, then B is true.” To see how this works, we will use visuals using what are normally called Venn diagrams. In case this isn’t familiar, a Venn diagram is a collection of circles used to represent some kind of situation happening. You can represent some kind of event as a circle, and to be inside the circle means the event happened, and to be outside means it did not happen.

Consider the Venn diagram below, where the blue region B should be thought of as containing the red region R, which in turn contains the green region G.

A visual for understanding the inside/outside nature of contrapositives.

We can think of the ‘if-then’ structure in terms of the ‘inside-outside’ relationship in the picture. When we said before that ‘If A, then B,’ we can equally well phrase that as ‘In every situation that A happens, then we also know B happens.’ In terms of our picture, we might say something like ‘Whenever we are inside the green circle G, we are also inside the red circle R.’ True enough – the green circle is perfectly inside the red circle. Translating this into ‘if-then’ language as we did earlier with A and B, we can say that ‘If G, then R‘ (where by G we mean ‘inside of the green circle, and similarly with R).

So this gives us a visual depiction, via Venn diagrams, of an if-then situation. In this new visual situation, it is equally easy to think about outsides of shapes rather than their insides. So, for a moment, we will think about the outside of R and G rather than their insides. When we do this, things get a little flipped around. The outside of the red circle R is the region colored blue. The outside of the green circle G are the regions colored blue and red. So, the outside of R is literally contained inside of the outside of G. Since we have already learned that this ‘contained in’ relationship can be translated into if-then, we can use this to express the idea

If outside of R, then outside of G.

We aren’t quite back to normal if-then statements yet – what exactly do we mean by ‘outside’? Well, if we remember that the inside represents something happening (or being true), while the outside represents something not happening (or being false), then we can do a little bit better.

If R is false, then G is false.

Or, we could more concisely say

If not R, then not G.

Now, one last comment is needed. We have to notice that the reason we were capable of making these kinds of statements about the outside of R and the outside of G was because G was inside of R. Since G is totally inside of R, anything outside of R couldn’t be inside of G (since that would make it inside R also). In other words, you begin to realize that the inside/outside statements are actually just different ways of saying the same thing. So the ‘if-then’ versions must also be saying the same thing.

What we have learned then is that if we have a statement if A, then B, we can equally use the statement if not B, then not A. The rule I have just described is what goes by the title of ‘contrapositive.’

This method can have the convenience that working backwards has in problem-solving. You can choose whether not B or A gives you more usable information. If A gives you plenty of information, you are working forwards towards B. If not B gives you more usable information, then use contrapositive and work backwards towards not A.

To conclude this article on proof by contrapositive, I will write an example using odd and even numbers. You can solve this problem in the ‘regular way’ if you use some information about prime numbers – but if you do contrapositive, the only thing involved is the distributive property and the definition of even and odd numbers – which are simpler than the ideas you have to use to go directly from A to B.

So, here is the proof – using contrapositive. For anyone curious about the ‘forwards’ proof (or ‘inside’ using the visual language of the Venn diagrams) I’ll give a hint at the end about how to do that.

Note: When made to match the Venn diagram earlier in the article, the green region is to be thought of as ‘x^2 is an even number’ and the red region is to be thought of as ‘x is also even.’

Theorem: If x^2 is an even number, then x is also even.

Proof: We use the method of contrapositive. That is, we will actually prove that if x is not even, then x^2 is not even. Since the opposite of even is odd, this means we want to prove that if x is odd, then x^2 is odd. Since all odd numbers have the shape x = 2y + 1 for some other whole number y, we can calculate x^2 using substitution:

x^2 = (2y+1)^2 = 4y^2 + 4y + 1 = 2(2y^2 + 2y) + 1.

Since 2y^2 + 2y is a whole number, this means that x^2 is odd. We have therefore completed our proof.

(Hint: To begin with x^2 is even and end with x is even, we know since x^2 is even that x^2 = 2y for some whole number y. You can use a proof by contradiction by assuming that x is odd, or alternatively you can use the fact that the only way to multiply two positive whole numbers to obtain 2 is 1 \times 2.)

Dealing with Loneliness

While writing this, the world has been in quarantine for more than a month now – at least it has been more than a month where I am. We are all dealing with quite a lot of loneliness. We cannot see our loved ones as much as we normally do. In a time or crisis, our natural instinct would be to gravitate towards our loved ones for comfort and strength, and yet in many instances that is precisely what we now cannot do. Especially if our loved ones are older or have other health issues that would increase risk with this horrible virus.

The overarching goal of slowing this virus down is more important (in most cases) than spending time with friends and family. And yet, even the most introverted of us need some human interaction to stay sane. And so we have an interesting dilemma. How do we strike this balance? Loneliness is bad, and this kind of pervasive and extreme loneliness and isolation will increase the rates of suicide, depression, anxiety, and many other ills. And yet to counter the loneliness would be to increase the risk of inadvertently spreading a pandemic virus. Increased social isolation will decrease the evils of death from a virus, but increase traumas caused by mental health, poverty, and other things. And the only way to decrease this list of psychological and social traumas would be to increase social exposure, and thus increase the rate of spread of this virus.

There are lots of arguments about how to “come back to normalcy” over time – and understandably so. There isn’t really an obvious answer to this, because social distancing and shutting down global economies causes horrible things just as viruses do. It is not entirely clear how to balance this. Nor do I claim to know how. But since loneliness is on the public mind and since it has been on my mind nearly every day for years now, I thought it would be worthwhile to record some of what I have been thinking about on this topic over the past few years. A lot of this is Biblically based, because I have yet to find any message that more powerfully portrays the topic. I won’t be able to cover everything, but I’ll try to give some of the ideas that have been more prominent in my thinking.

Loneliness is Not Good

I begin with a quotation out of the Bible, specifically Genesis 2:18-20. For context, this is towards the end of the period of the narrative in which the created order is introduced. The first man (Adam) is on the scene by now, but we haven’t heard anything about any women yet. God had placed man in the garden, and told man to work the garden and care for it – which in the narrative is viewed as a great positive. At this point, God is not creating anything “new” anymore – the heavens are there, the creatures of the earth and sky and sea are there, and Adam is there. The world has absolutely beautiful (as nature still is today) and God was quite happy with things. And yet, we have this narrative followed with the following three verses.

“Then the Lord God said, “It is not good that the man should be alone; I will make him a helper fit for him.” Now out of the ground the Lord God had formed every beast of the field and every bird of the heavens and brought them to the man to see what he would call them. And whatever the man called every living creature, that was its name. The man gave names to all livestock and to the birds of the heavens and to every beast of the field. But for Adam there was not found a helper fit for him.” (Genesis 2:18-20 ESV)

Put aside anything you believe about Genesis for now – none of that matters. No matter what you think of Genesis, the real message here is quite clear. At this point in Genesis, there is not yet any sin in the world. To put this in words that are clearer to people both inside and outside of the church – everything existed in the way that it was supposed to be. I think we all know in our heart that things are not now like that – there are so many broken things that I don’t want to list them, because no matter how long your list, it would be woefully inadequate. But right now, leading up to this verse, all of the evils you might think of that exist today did not yet exist. Things were beautiful.

Yet God was not satisfied. Something was missing. In verse 18, we see God Himself saying “It is not good that the man should be alone; I will make him a helper fit for him.” God went on to create Eve, and Adam was immensely joyful seeing Eve. In verse 25, we see that “the man and his wife were both naked and were not ashamed.” I find this profoundly interesting. At this point, there is no such thing as evil yet in the world. And yet, before Eve was created, the world was profoundly incomplete. This one verse gives rise to so many profound discussions – for instance, it seems to me that this passage implies that there is a real, substantial difference between men and women, but that each if of inestimable value and that there is even greater value in the interaction between the two. Quite an interesting discussion, but that is for another time.

What I want to point out here is that the very, very beginning of the Bible makes a profound statement on the value of community. Isolation is a really bad thing. Think about this – Adam had the God of the entire universe as a companion, and yet the very same God of the universe knew that this was not optimal. He knew that Adam would only truly thrive in community with both Himself and with other humans like him. So, he made Eve for Adam to have community with. And this was now truly good. God then commanded Adam and Eve to populate the entire earth – and one important reason for this is the goodness of community and society. The God of the Bible, then clearly indicates from the very beginning of the Bible that loneliness is a real problem.

Jesus Was Lonely Too!

As profound as it is that God created Eve specifically because He knew that loneliness was not the ideal for humanity, this is not for me the most profound message of the Bible about loneliness. Far, far more profound to me is the truth that God Himself voluntarily became profoundly lonely in order to reach into our broken world and help us. This was done in the person of Jesus Christ. In an article on the topic of loneliness from Desiring God, a ministry I enjoy and which does a great job of providing insight on the Bible, has the following paragraph in its opening describing loneliness.

“Loneliness is what we feel when we’re isolated from others. Loneliness often has less to do with others’ physical absence and more to do with feeling disconnected or alienated from them. Or misunderstood by them. In fact, these are far more painful species than mere absence, because we feel the isolation of being despised and rejected.” [1] (italics mine)

Part of the radical message of Christianity is the incarnation of Christ. This deserves a bit of explanation, because it is often misunderstood. In Christianity, we believe God has taught us through our Scriptures that He exists eternally as a Trinity – as ‘three minds’ or ‘three persons’ unified in one being (A very, very rough analogy to what is meant by this would be ‘Siamese twins’ that are the one hand only one physical body but on the other hand two conscious persons. I have to emphasize that this is a simplification, but for someone who has not spent time studying Christianity this is a helpful analogy to start with).

The idea of the incarnation is that one of the persons within the Trinity – usually denoted as the second person of the Trinity (though the numerical ordering is not all that important) – decided of His own free will to live a life as a human being. We now usually refer to this person as Jesus Christ, and I will do so for the rest of this discussion. Just remember – Jesus Christ had been in heaven in glory for all eternity. Jesus was part of creating the universe out of nothing. He is all-powerful, all-knowing, and all the rest.

And yet… He came down to us. God Almighty lived about 33 years on this earth – 30 of which were in total anonymity as a blue-collar laborer. The Creator of the Universe lived as an infant, then a toddler, then an adolescent, then a professional carpenter – which was a very low class of society at the time. He suffered the loneliness of being mocked for supposedly being an illegitimate child (due to the miracle of the virginal conception). He worked hard and bleed with his hands for years and years as a carpenter. He did not come from a ‘good part of town’ either – the Bible even records some people insulting Jesus because of his place of birth (John 1:46). Jesus was, in the end, betrayed by everyone who followed him and suffered crucifixion – quite probably the most humiliating and excruciating method of execution every devised by humanity. As the Prophet Isaiah spoke beforehand, “As many were astonished at you – his appearance was so marred, beyond human semblance” (Isaiah 52:14).

Not only did God live a pretty ‘low-brow’ life while He was on earth – it cannot be understated what it means for God to live a human life at all. In and of itself, that requires giving up a lot that He rightly deserves to have – heaven, worship, glory, et cetera. Christ gave up the level of intimacy that He enjoyed within the Trinity from eternity for a time to live a life as a human being. There is a lot of subtlety here on a theological and philosophical level, but nonetheless it should be intuitively clear that this was a great sacrifice on God’s part – and would involve a great deal of loneliness.

Why All This Matters

Why does this matter? What ought to be said first and foremost is that thousands and probably millions or billions of people have been lifted out of emotional torment and despair by the message of the Bible, in particular the message of Jesus Christ and his followers like the apostles Peter and Paul. For instance, you can read Paul’s experience of suffering in 2 Corinthians 11:21-33, and his spiritual and emotional perspective on his suffering in 2 Corinthians 12:7-10. In summary, the amount that Paul suffered defies comprehension, and yet Paul reports profound satisfaction and peace in life. How can this be so?

I can report something similar, although much less extreme. There was a long period of my life in which I felt deeply lonely – in fact so much so that I thought it was impossible that I would ever cease to feel lonely and abandoned. I was dealing with PTSD, depression, and ADHD – all of which prevented me from behaving and thinking in a normal way in day to day life (and ADHD still does, the other two had predominantly non-biological causes). Sometimes, I’m surprised that I am even alive today. And yet, I am. The primary reason, and in some ways the only reason, that I escaped the intense loneliness that I experienced after years of profound loneliness and self-hatred was the even more profound actions of God coming down into this world in order to empathize with us and help us to understand how deeply He loves us.

What then is the moral of all of this? There are many, I cannot list them all. Nor can I list all of the relevant Scripture – there are far too many passages that speak about loneliness and suffering. Having read the entire Bible and taken notes as I read, I would estimate many hundreds if not thousands of verses that are directly relevant to this topic in one way or another.

I’d say that you are not alone. This is among the most beautiful parts of the Christian message. We are not alone. God Himself lives within us through the Holy Spirit. Therefore, we are never literally totally alone – we always have God Himself with us. Nonetheless, we still experience feelings of loneliness and abandonment. And reasonably so – after all, I have already mentioned that the Bible affirms the importance of human relationships. Feeling lonely is a legitimate emotion – we are not designed to be alone but in intimate, personal relationship with one another. It is absolutely okay that you feel alone – these emotions make sense in the fallen world in which we live. Equally important, God has through his Word showed us that He understands our loneliness, understands our aches and pains, and walks alongside us in all of these.

We are loved. Because of my faith in the Lord Jesus, I know that I am loved. Were it not for this, to be totally honest, all of the psychological torment of my earlier life would probably consume me. At best, I would be barely functional in society – at worst, I would be dead or in a mental hospital. I don’t know which, because I was saved from this utter isolation. All I do know is that I could never have recovered from this alone. Yet God sent amazing, loving friends into my life – so loving in fact that I am very nearly crying as I write this sentence. No matter who you are – you are loved beyond your wildest dreams by the Lord Himself – and it would be the greatest honor of my life to share that amazing love with any one of you.

References

[1] https://www.desiringgod.org/articles/jesus-understands-your-loneliness

What is Apologetics?

One of the things I am have been most interested in since becoming a Christian is the intellectual discipline of apologetics. I have found my study of apologetics deeply interesting and rewarding every dimension of my life intellectual, spiritual, and even emotional. In light of the great benefit apologetics has had in my own life, I’d like to share here a brief introduction to what Christian apologetics is, and why I have found it so valuable.

What is Apologetics?

I will lay out more specifics later, but a beginning definition of the term “Christian apologetics” is the practice of intellectual discussion about evidence in favor of the objective truth of the basic building blocks of the Christian worldview – like God’s existence and Jesus’ resurrection from the dead. In apologetics, the evidence provided does not assume that you are already a Christian – rather, the purpose is to engage in reasonable discussion those who do and do not consider themselves followers of Jesus and in understanding the reasons why each person in the discussion believes as they do. The goal of the apologetic enterprise is that everybody involved come closer to understanding the truth, whatever that may be.

Apologetics in the Bible?

The place I’d like to start discussing apologetics is in the Bible. There are many places we can go, but there are two that are most commonly cited. The first of these is found in 1 Peter 3:15,

“But in your hearts honor Christ the Lord as holy, always being prepared to make a defense to anyone who asks you for a reason for the hope that is in you; yet do it with gentleness and respect” (1 Peter 3:15, ESV)

The key phrase in the English translation is ‘to make a defense,’ which is translated form the Greek verb apologia, (the Greek is \alpha \pi o \lambda o \gamma \acute{\iota} \alpha). This word is also used in legal settings – the witness would make a defense (apologia) for their innocence. So, this Biblical command to ‘always be prepared to apologia to anyone who asks you for a reason for the hope that is in you’ is a command to actually be ready to explain what you believe and why you believe it, backed up with some reasons.

We can even see in the Bible itself that this is the intention of its authors. For example, near the end of the gospel of John, we have the following comments from the author,

“Now Jesus did many other signs in the presence of the disciples, which are not written in this book; but these are written so that you may believe that Jesus is the Christ, the Son of God, and that by believing you may have life in his name.” (John 20:30-31, ESV)

Notice the claim. The purpose of the book is to provide evidence on the basis of which people come to believe that ‘Jesus is the Christ, the Son of God’. Taken in the context of the whole gospel of John, the reader is being invited to consider this book as a report from someone who actually walked the earth with Jesus, an eyewitness.

These passages are from the New Testament. We even have Old Testament passages that directly encourage reasonable dialogue. Consider Isaiah 1:18a, which reads “Come, let us reason together, says the Lord” (ESV). In the context of the book, the Lord is speaking through the prophet Isaiah to a group of people who are disloyal and in rebellion against the Lord. My personal copy of the Bible (ESV) adds a footnote that the term ‘reason’ may also be translated ‘dispute’. This is a friendly invitation to engage in back-and-forth discussion.

There are also examples upon examples of important figures in the Bible engaging in disputes with those who oppose them – figures like Ezekiel, Jesus, and Paul in particular do this. We thus have a firm Biblical foundation for approaching at least some of our discussion around Christ in this apologetic manner.

Why is Apologetics Important?

First and foremost, the discipline of apologetics is important because truth is important, and the goal of apologetics is to find what is true. I care very deeply about truth, and so I am very attracted to studying apologetics and broader theological areas so that I may learn and grow closer to understanding the truth of the way the world is.

Secondly, learning apologetics has given me great confidence in my relationship with God, particularly in times of trouble. As I write this, the world is in quarantine due to the spread of COVID-19 virus, and the social isolation we are all facing is an immense challenge on top of the emotional burden of knowing that many are dying as a consequence of this pandemic. Painful circumstances like these do not cause me to doubt whether following Jesus is worthwhile or whether it is the correct path – because my reasons for following Jesus are not merely emotional and spiritual, but also intellectual. A famous quote of the great twentieth century Christian author and apologist C.S. Lewis comes to mind:

“I believe in Christianity as I believe that the sun has risen: not only because I see it, but because by it I see everything else.” – C.S. Lewis

Because of what I have learned over several years now of learning about apologetics, I can see more clearly what is around me. I can experience the full depth of my emotions while at the same time being swayed to and fro by them. I have gained a deeper appreciation for who God is, how awesome He is, and why I have chosen to be a follower of Jesus.

Thirdly, I deeply love learning, and apologetics is not an inherently narrow discipline, but reaches across all kinds of intellectual topics. In learning apologetics, I have immensely broadened my horizons, my reading, and my intellectual interests. My love for studying apologetics has brought me to learn much more extensively about world religions, world history, modern science, world history and the historical method, philosophy, theology, ethics, literature, and more. I personally find this of the utmost value, and I have developed a lot intellectually by spending time studying more broadly than the lone field of mathematics to which I am dedicating my career.

An Apologetic Exposition of Christianity

Now that I have given some personal commentary on why I have been so moved by the apologetic enterprise, let me now give an overview of the kinds of things I have spent time learning about. I will also make a remark here that it is vital to apologetics to listen to opposing viewpoints and answer questions, which I would be more than happy to do, but here I will mainly provide a summary of the positive case that the Christian apologist can offer for the reason for the hope that is in them.

I’ll start with the biggest pull for me, the idea that really gave the initial spark to my interest in apologetics, the cosmological argument. More specifically the Kalam cosmological argument. This was initially developed in the context of medieval Islamic philosophy and has remained an interesting argument for centuries, and in light of modern cosmology and astrophysics it has come roaring to a prominent place in public discourse. The Kalam, as it is normally called, has several forms and flavors, but the most well known is a logical deduction developed in modern times by philosopher and theologian Dr. William Lane Craig and by many more after him. The most basic form of the argument consists of three statements:

(1) Everything that begins to exist has a cause.

(2) The universe began to exist.

(3) Therefore, the universe has a cause.

I will give a brief summary of how the discussion flows. Statement (1) is basically a core principle of metaphysics, which is a discipline of philosophy that studies the nature of what is real, and can also be viewed as a core principle of modern science. When things happen, we naturally want to understand why they happen and how they became the way that they are. Denying (1) would undermine the core scientific principle that we are capable of understanding why things happen the way they do. So, we all ought to accept that (1) is true. Statement (2) has multiple lines of evidence in its favor – the evidence of modern astrophysics and cosmology points entirely to some version of the Big Bang, which just is to say that the universe began to exist. There are also very strange philosophical paradoxes that emerge from the idea of physical time going backwards forever – it would amount to a claim that by the process adding one second after another, you can eventually reach a genuinely infinite number, which as a mathematician I deny is possible. If you then apply (1) to (2), then (3) follows by the laws of logic. That is, if (1) and (2) are true, then arriving at (3) is as unavoidable as 2+2=4. If you then spend some time thinking about what it means to be a ’cause of the universe’ (without going into all of the logic now) this cause is spaceless, eternal, immaterial, enormously powerful, uncaused, and personal. There is only one concept I am aware of in the history of human thought that truly fits with these properties – and that is a monotheistic God, as is worshipped in Christianity, Judaism, and Islam.

I find the Kalam deeply fascinating, and learning more deeply about the Kalam leads to a solid education in both modern physics and various areas of philosophy, particularly metaphysics. There are a huge variety of other reasons to hold that God exists and even that Christianity is true. I’ll now try to list some of these in abbreviated form, along with the intellectual disciplines that play a role in the discussions of these reasons.

Kalam Cosmological Argument: God is the best explanation of the beginning of the universe. (cosmology, metaphysics, philosophy of science)

The Resurrection of Jesus: The only plausible account of the facts of Jesus’ life that are accepted as true by majority of modern secular historians is the one offered by Jesus’ followers – that God raised Jesus from the dead. (historical studies, ancient history)

Teleological Argument: God is the best explanation of the remarkably finely tuned physical constants in the universe that allow for life to exist. (cosmology/astrophysics, philosophy of science, metaphysics)

Personal Testimony: God is the best explanation for why so many people throughout history have spiritual experiences which lead them to conclude that a deity exists. (history of religion,

Moral Argument: God is the best explanation of why there are moral values, like the evilness of torturing a child for fun, that are objectively binding. (ethics, metaethics, theology)

Argument From Evil: The existence of objective evil implies the existence of a universal moral law, which points towards God and His nature as its source. (ethics, metaethics)

Applicability of Mathematics: God is the best explanation of why there are harmonious mathematical laws that describe the universe that we are capable of understanding (mathematics, philosophy of mathematics)

Ontological Argument: If the definition/concept of God is coherent, then God is the kind of being that absolutely must exist. (ontology, modal logic)

Argument from Rationality: God is the best explanation of why the human mind is capable of abstract reasoning and accumulating knowledge of the world. (epistemology, evolutionary science, philosophy of mind)

Argument from Induction: God is the best explanation of why there are law-like patterns that hold in our universe, like the laws of physics. (metaphysics, mathematics, science)

Argument from Beauty: Beauty is not plausibly accounted for by evolutionary psychology, and the only plausible grounding for objective beauty is God. (art, literature, aesthetics)

Hopefully this sparks some interest for people. I will hopefully go much more in depth with many of these in the future, but I hope that this summary has given some good reasons that the basic beliefs held by theists, and Christians in particular, are true, interesting, and relevant to our daily lives.

Some Sources

  • “Two Dozen (Or So) Theistic Arguments,” an essay by renowned philosopher Alvin Plantinga
  • Reasonable Faith, by philosopher and theologian Dr. William Lane Craig
  • Cold Case Christianity, by cold-case detective J. Warner Wallace
  • Mere Christianity, by philosopher and novelist CS Lewis
  • The Case for Christ, by journalist Lee Strobel
  • There is a God, by the former atheist philosopher Antony Flew

Proof by Contradiction

The proof method that we will talk about here is quite different than many others. In his famous book A Mathematician’s Apology, the great mathematician G.H. Hardy made an analogy between this proof style, which we call a proof by contradiction, to a gambit in chess. So before I try to analyze what this proof method is all about, I’ll first take a moment to explain the analogy.

There is not really a need to go all the way into the rules of chess in order to explain the idea of a gambit, because the same idea applies to other games. In chess, there are various pieces, and some of the pieces are more valuable than others. So, if a sequence of moves in chess causes you to ‘lose value,’ you want to avoid that. However, there are exceptions, and the exceptions are what we mean by a gambit. Gambits in chess are strategies that involve purposefully losing valuable pieces in order to gain position that will help win the game. Think of this as taking one step back and two steps forwards. The beginning of the plan looks really bad, but things turn out for the better.

Now, how might we apply this ‘one step backwards, two steps forwards’ idea to mathematics? Since the overall goal of mathematics is to understand what is and is not true about numbers, shapes, etc, with the goal of identifying as much truth as possible, taking one step backwards is assuming that something false is actually true – which is a mathematical failure. This amounts to beginning a math problem with a statement like “since we know 1+1=3…” I think most of us will feel a sense of unease at a claim like “1+1=3,” since it is clearly wrong. In mathematics, wrong is bad, right is good.

However, there is a principle of logic that I have discussed before that enables us to make use of false ideas. This is the principle of non-contradiction. This is absolutely essential. The principle tells us that there are no contradictions – no sentence can be at once both true and false. I think everyone knows this intuitively, and in fact if you spend some time thinking about it, it is actually impossible to deny the principle of non-contradiction (if you want a brain-teaser, try to imagine an argument about whether this principle is true). This principle can be taken as foundational to all thought, and in particular the way we think about mathematics.

Here is where the style of proof by contradiction arises. Suppose that there is a statement P and that we want to know whether P is true or false. Suppose we temporarily assume that P is false, and we later discover that assuming P is false leads us to affirm some kind of contradiction. Since the principle of non-contradiction tells us that there can never be any contradictions, our temporary guess that P is false led us into a problematic situation. We can’t continue believing that P is false any more because that is contradictory, and since there are only two choices available to us, P must be true.

This strategy is what we mean by proof by contradiction. Since all claims (in the context of mathematics at least) are either true or false, anything that cannot be false must be true. I will now show how this works with one of the most famous examples of this method, the proof that “the square root of 2 is irrational.” To be clear briefly on what this means, the rational numbers are all the fractions, and an irrational number is just any number that is not a fraction. It isn’t actually clear immediately that there is any such thing as an irrational number, and the proof by contradiction is the primary mechanism by which we begin to understand that there are such things as irrational numbers and what they are like. To begin this proof, all I claim to know about “the square root of 2,” which is normally written \sqrt{2}, is that (\sqrt{2})^2 = 2.

From here, we can now prove that \sqrt{2} is irrational.

Theorem: The square root of 2 is not equal to any rational number.

Proof: Suppose that the claim is false, that is, that \sqrt{2} actually is a fraction. Then we can write down that fraction using whole numbers a and b such that

\sqrt{2} = \dfrac{a}{b}.

Since fractions can always be reduced to lowest terms, we can take for granted that a/b is in lowest terms already (which means the whole numbers a and b share no common factor). Now, we want to see what we can learn about a and b. First, we can square both sides of our first equation to obtain the new equation

2 = \dfrac{a^2}{b^2}.

Multiply both sides of this by b2 to obtain the equation 2b2 = a2. Now, 2b2 is even, since it is a multiple of 2, so a2 is also even. Multiplying a number by itself does not change its evenness/oddness, and therefore a must also be even. That means (definition of even numbers) that we can pick a new whole number c so that c is half of a, that is, 2c = a. Then a2= 4c2 must be true by squaring both a and 2c, and the equation from the beginning of the paragraph then shows is that b2 = 2c2. For the same reason as we have just used on a, the whole number b must be even as well. Since we have reasoned that a is also even, this means a and b share the common factor of 2, and so a/b is not in lowest terms. Therefore, the fraction a/b both is in lowest terms and is not in lowest terms. This is a contradictory statement, and so it must be the case that our initial starting point of \sqrt{2} = \dfrac{a}{b} is actually false. Therefore, there is no fraction equal to the square root of 2. So, our proof is now completed.

This way of thinking takes a lot of getting used to. To see more examples (and examples that are not directly math-related) the Latin term reductio ad absurdum refers to any logical process that uses this same structure – whether related to mathematics or not. Though very strange, this proof method is extremely useful, because sometimes (as is the case with the problem I have just solved) a claim that ‘there is no such-and -such’ is difficult to work with directly, whereas a claim that ‘there is a such-and-such’ gives you more information – in this case, we gained access to the numbers a and b and an equation which supposedly related them to one another, which helped us greatly.

As one learns more mathematics over time, it is very important to develop an intuition for what kinds of situations this method will work well for, as very often it makes problems enormously easier to solve.

What is Faith?

A common theme throughout many world religions, and in particular Christianity, is the notion of faith. In my experience, often the words faith and religion are used almost interchangeably, and this is understandable, for the word faith is used frequently by religious people. In fact, from time to time you will hear the phrase “the Christian faith” in place of “the Christian religion.” The word ‘faith’ is rather ubiquitous, which is why I want to talk about it… because through my reading an personal experience, I have come to realize that both non-religious and religious people tend to misunderstand the meaning of the word in some very crucial contexts.

What Faith Is Not

I have a sense that some people (both among those who agree with me on religious issues and those who do not) might be a little skeptical at this point. Someone might think I am trying to redefine the word faith… and this is understandable. This touches on a fundamental distinction I would like to make, and so I can make it now to avoid as best I can any misunderstandings. The first point I must make is this – words often have more than one definition. There are several definitions of this word I’d like to avoid. When I speak of faith for the rest of this article, the word faith is not a synonym for a religion or belief system. The word faith can be used in that way, but that is not how I want to use it here. I also want to make clear that, while the Bible and the modern church do use the word faith with this definition in mind, there are other meanings. And that isn’t even really the most important meaning of the word in the biblical context, nor in the modern Christian context.

There is another definition of the word faith I’d like to avoid, for a similar reason. Among atheists, and more broadly among non-religious people in general, the word faith arises as a sort of antonym to reason. You can find many authors, including the likes of Richard Dawkins and the New Atheist movement, who define faith as ‘belief without evidence or reason’ or even ‘belief in opposition to evidence or reason.’ I considered compiling a list of quotes to demonstrate this (which I will gladly do if anyone doubts my claim) but I think that most people who would read this far are already familiar with this idea. Again, I reject this definition of the word for the rest of this article. I reject this definition for many reasons, the chief of which is that it is completely foreign to the Bible and the Christian religion. There are a variety of New Testament and Old Testament passages I can quote to demonstrate the importance of evidence in the Christian worldview, but I will save that for another time.

Let’s Stop Demeaning Faith

There is another comment about this particular usage of the word faith. To be clear, faith can be blind. That is certainly possible, and there may well be times when some people do have blind faith in something – to give a non-controversial example, perhaps a gambling addict believing “if I just play one more time, I’ll hit that jackpot…” I think it’s fairly clear that this is an instance of a kind of faith in opposition to evidence. However, I ask of those of you who want to apply that to religions… just stop. Please. It is an insulting stereotype of religious believers that is, in my experience, basically never true. And it is certainly not true in my case.

Just in case an example is needed – imagine if I were to say that “all atheists just hate Christians and want to walk around doing immoral stuff, and that’s why they are atheists.” That would be horrible, wouldn’t it? I would never say anything like that, because it is clearly false. Now, for anyone reading this who thinks that religious people are all just holding on to blind faith… you are doing something very similar. You are essentially implying that well more than half of the world’s population are idiots who are incapable of rational thought. You are ignoring the plain fact that many of the brightest people the world has ever seen – including the likes of Isaac Newton, Galileo, and nearly every other figure in the scientific revolution for that matter – did believe in God, and in Newton’s case, the Christian God. Not to mention the obvious fact that implicitly assuming that all religious believers are stupid or ignorant is an obvious example of false stereotyping. Stop defining the word faith like this, and don’t let other people do so either.

I will give you an actual working definition of faith, the working definition that the Bible uses. So if you’ve been taken in by the silly and malevolent attacks on religion in recent years through a redefinition of the word faith, you can shake yourself free from that.

What is Real Faith?

Before I proceed, let me take a moment to pause and make very clear that I have just made a specific claim that when the Bible uses the word faith, it does not refer to the idea of ‘belief without evidence.’ I still don’t understand why a very large number of people have come to believe this – it is so obviously false when you just open a Bible. Why are the others definitions so wrong and insulting? Well, I’ll give you one very good place to start.

There is not a single place in the entire Bible that the word faith is used. Furthermore, neither Jesus nor any of his contemporaries ever used this word in their lifetimes.

This might seem wrong, but the explanation is quite simple. Jesus did not live in 21st century America. He lived in first century Palestine. English was not a language yet. Obviously, he didn’t use the word faith then, because faith is an English word and people in Jesus’ region and time spoke other languages – in this case, the dialect of Koine Greek is relevant, though there were several languages used in that region at that time. This may seem rather trivial point, but there is a reason I make this point at such length. What we think the word faith means today doesn’t matter at all when talking about the Bible, because what matters is what the author meant. When discussing Christianity and the Bible, we should focus on the intended meaning of the author. To do this, it will be helpful to go back to original languages.

The Bible has two portions – the Old Testament, which was originally written in ancient Hebrew and is held as Scripture by both Jews and Christians (and to my understanding is considered inspired by Muslims as well, with some qualifications) and the New Testament, which details the unique claims and teachings of Jesus Christ and is followed by Christians. The New Testament was originally written in Greek, and since I am a Christian and most criticism of the idea of faith where I live is directed at Christians, I choose to look at the word in Greek which today is translated as ‘faith’.

The word in the New Testament translated as ‘faith’ is \pi \iota \sigma \tau \iota \varsigma, transliterated as pistis. So, if I am reading my Bible and I want to understand what the word ‘faith’ in my Bible means, I don’t rely fundamentally on modern atheists, modern politicians, or even modern preachers for an answer. Quite frankly, that just doesn’t make sense. Instead, I rely on the ancient languages themselves. And the meaning of the word pistis in Greek connotes the ideas of faith, trust, and confidence. To be a faithful individual is to be a trustworthy individual, for instance.

One thing I’d like to notice briefly – the word pistis does not inherently include the idea of ‘against all the evidence’. Actually it is quite the opposite – the reasons for your convictions are relevant. The word pistis appears in the works of such famed people are Aristotle [1]. The article I have linked at the end of this post provides statement from classical scholars, talking about the use of the word pistis in Greek philosophy. For an example from the resource I have linked (which itself cites classical scholars), here are two quotations from this article on the meaning of pistis in Greek rhetoric.

  • Pistis is used to represent the state of mind, namely, conviction or belief, at which the auditor arrives when the correctly chosen aspects of the subject-matter are placed before him in an effective manner. . . .”
  • “In its second meaning, pistis is the word used for a methodological technique . . .. In this sense, pistis means the logical instrument used by the mind to marshal the material into a reasoning process. It is a method which gives the matter a logical form, so to speak, and thus produces that state of mind in the auditor which is called belief, pistis. . . “

If you don’t buy that, or want to learn more, then just go read a Greek lexicon and look at the citation I provided. Read the Bible for yourself with an open mind. Email me if you’d like (mathematicalapologist@gmail.com). You not only can have pistis with evidence, evidence-based pistis is an entirely normal use of that word. So if you ever hear someone trying to discredit a religious person by insulting the idea of faith on as entirely irrational, just ignore them. For followers of Jesus, this is the wrong definition, and so their feckless insult does not apply to us. I cannot speak for Muslims or others who use the term faith, though I imagine they probably agree with me. This kind of faith is exactly what we should be using, and for the most part do use, as we go around the world every day. This is a reasonable faith. True enough, you can trust in the truth of something and wind up being wrong, but you cannot dismiss someone merely because they have faith in something, whether religious or otherwise.

Don’t Be Afraid to Have Faith!

My favorite case-in-point of the things I have been saying comes from Professor John Lennox, a mathematician at Oxford. In a debate with the famous atheist Richard Dawkins, who brought up the definition of faith without evidence, Lennox responds “I presume you’ve got faith in your wife, is there any evidence for that?” To which Dawkins responds “Yes, plenty of evidence.” To which the audience responds with laughter [2]. This is the point. That usage of the word faith is exactly the same usage as what the Bible and the modern Christian mean when they use it. This is a commonsense definition, we all know it intuitively. We have faith in our spouses, our friends, our country, the human race, our political party – all kinds of things. And all of these are based on evidence of one sort or another – and yet we still use the word faith.

So now, let’s all be adults and stop demonizing people by redefining or mischaracterizing the words they use. In most cases, I imagine people making that accusation against religious believers stems from ignorance on what religions actually teach, and probably in some cases from a childish inability to listen calmly to someone who doesn’t agree with you. The only other alternative I can imagine is a conscious desire to demean, insult, and mock Christians – but I’ll always give the benefit of the doubt on that. I don’t think very many people really think that way – at least I hope they don’t. I don’t really see any way around that conclusion. We ought not treat our fellow man that way. And I would not want to accuse anyone of a different religious viewpoint of having blind faith either – I may well believe that the reasons that they have are not very good reasons, but that is very different than an accusation of blind faith. We can have discussions about whether our reasons are good ones or not, but I hope that both religious and non-religious people in our modern world can come to a place where we recognize even our opponents as people who have reasons for their convictions, and we can have civil discussions about those reasons instead of relying on insulting each other.

I’d be happy to talk with anyone who disagrees, but I honestly can’t see anything to be argued. This seems like basic common sense and human decency.

And for those of us who follow Jesus, this is crucial. We do not have to hide, because we follow and God who is not afraid of those who claim He does not exist. We actually have books and books of evidence on our side, both scientific evidence and other kinds of evidence, which I have studied for a few years in my spare time and still spend a lot of time studying to this day. I will discuss concrete evidences for Christianity in other articles, including a follow-up to this one about apologetics.

My main takeaway for those who follow Jesus is that we must do a better job of rejecting the notion of blind faith, because I strongly believe (as do many, many others) that it has no place in Christianity.

Citations

[1] “Definition and Examples of Pistis in Classical Rhetoric,” https://www.thoughtco.com/pistis-rhetoric-1691628

[2] A Short Clip of Dawkins & Lennox: https://www.youtube.com/watch?v=aFkGDK_mteQ

More Resources

Dawkins & Lennox – The God Delusion Debate: https://www.youtube.com/watch?v=zF5bPI92-5o

Standing Firm in Suffering

I sit down to write this on the morning of April 16th, 2020. We are about one month into the global-scale shutdown due to the COVID-19 virus. There has been so much disruption. Lives are being saved, we are doing what must be done… but the cost is high. There is a lot of suffering and loss in this time. People have lost loved ones, jobs, and precious time with loved ones still alive. Those with mental health issues are hurting, and perhaps even more of us now struggle with depression than before. The undercurrent of loneliness in our society so often brushed under the surface is now surging to the forefront. On top of all of that, on this day every year I pause to remember a great tragedy at my alma mater, Virginia Tech. Thirteen years ago, 32 people were suddenly and pointlessly slaughtered on our campus. This has not been forgotten in Blacksburg, and will never be forgotten.

I allow myself to hurt today. Even though I don’t enjoy this sadness, I don’t try to stop experiencing it when it comes. My soul is heavy, but I know that the heaviness is necessary. One of the most profound ways I have learned the importance of sadness is through learning more and more about the 2007 Virginia Tech Shooting. Many of my professors were on campus when it happened – my favorite professor had taught in one of the rooms where the shooting happened the previous semester. I went to memorial events every year. As powerful as all of this is, I am most impacted by one story from a professor of the response of the student body. In the immediate aftermath, news sources were doing interviews with students. Doing their job, they of course wanted to find a diverse range of responses – how was the campus responding? Were people upset with the university? As I heard the story from my professor, interviewers were looking for remarks from students who were both angry with the university, and those who were not. But, after more than 100 interviews, they gave up trying to find even one student who was angry at anybody other than the killer himself. Even a survivor from one of the rooms where the bloodbath happened had only glowing words for the Virginia Tech community at large. Within 24 hours, students had set up a memorial at the center of the campus with 32 Hokie Stones, the same stones used for nearly all buildings on the campus. In the university, nobody resorted to blaming. Instead, the community allowed themselves to mourn openly, and to this day we continue to mourn.

You can feel the reverberation of this tragedy on that campus to this day. From the very first day I set foot there, I almost felt as if the ground itself was permeated by care and empathy. While at that university, I went through a lot. I have had flashbacks that literally paralyzed me, memories so vivid that for a moment I become scared that they are happening all over again. I know what it is to see people I love going through depression and periods of struggle with self-harm, and I know the torment of actually believing that hurting myself would make me feel better. I used to think of myself as literally less than human and not worth caring about. I could go on. But the environment on that campus supported me. There, I was cared for, and eventually I healed. Today, I do not struggle in those ways any more. The main reason I got better is simple – the university culture in which I found myself deeply understands pain, and knows how to walk through pain without ignoring it. The people I met and allowed to see what I was going through did not at first try to convince me that I ought not feel that way about myself – their first response was to acknowledge the reasons I did feel that way about myself and lament the reality of what I’d been going through.

I believe firmly that developing this ability to live inside painful moments and allow them to be felt and understood is something we all must learn. This is how we become mature, caring, loving people. This is how we grow and become wise. We cannot run from the realities of pain. That will only make the pain worse.

Jesus Acknowledges the Reality of Our Suffering

My experience with Christian faith is that this principle is acknowledged and lived out. Let us take, for example, the story of the death of Lazarus. This story takes up most of John, chapter 11. Lazarus was a close friend of Jesus and his followers, and had fallen ill. Before Jesus arrived to see him and heal him, he died from his illness. When Jesus arrived where Lazarus was, here is what we see.

When Jesus saw her weeping, and the Jews who had come with her also weeping, he was deeply moved in his spirit and greatly troubled. And he said, “Where have you laid him?” They said to him, “Lord, come and see.” Jesus wept. So the Jews said, “See how he loved him!” – John 11:33-36 (ESV)

The amazing thing about this is that just a few verses later, Jesus raises Lazarus from the dead. And when you read the passage, you get the impression throughout that Jesus understood that he was going to do this. It would then at first seem odd to see Jesus hurting emotionally – if you knew your friend was going to be alive again in just a few minutes, why would you be sad? That isn’t how you’d expect Jesus to react.

And yet, Jesus wept. He was “deeply moved in his spirit and greatly troubled.” Even though Jesus knew that it wouldn’t be long until things were made right, he paused and took the time to weep. He took the time to acknowledge the real sadness of death. He took the time to allow others to see him hurting and expressing his sorrow. Think about this – God is weeping! There is no hiding from the hideous reality of death. Christ himself, Lord of all, wept at the tomb of his friend before he raised him back to life.

Surely there is a lesson here for us. When dealing with tragedy, it is not appropriate to rush into trying to fix the tragedy without first mourning. If even Jesus wept and was not ashamed, we can too. It would have been wrong for Jesus to not acknowledge the real pain that was present where he was. This is the attitude that eventually brought me out of my frequent flashbacks and dark thoughts. I learned over time to look my suffering in its eyes, so to speak, and not back down or downplay it. Instead of running from anxious memories, thereby heightening the anxiety, I learned to look at them and acknowledge the situation for what it was, including the fact that those memories are just that – memories, not events currently happening. Over time, this enabled me to heal, and has had the benefit of teaching me to help others who are suffering, in particular those facing anxiety, trauma, or depression, as I faced.

My relationship with my Lord is the only reason I have been able to do this. As we see in Hebrews 4:15-16 (ESV translation),

For we do not have a high priest who is unable to empathize with our weaknesses, but we have one who has been tempted in every way, just as we are-yet he did not sin. Let us then approach God’s throne of grace with confidence, so that we may receive mercy and find grace to help us in our time of need.

He Himself has gone through the fire and storms of life, just as we do. He can empathize with us, weep with us, and help us. But what He doesn’t do is allow us to run away from the bad things we experience. He did not run – he faced an unimaginably agonizing death by torture followed by crucifixion. He did not back down, and He can give us the strength of heart to stand firm in our own storms. I feel every day the strength He has given me, strength without which I might not have survived my storms.

This strength is available to all who come to Him with an open heart. If you are a follow of Jesus, turn to Him for your strength in this time of great trial we are now facing. If you do not follow Jesus, I encourage you to read the gospels for yourself to see the love He has for us all and to take encouragement from the example He has set. If you want to find this strength offered by Christ, I’d love to talk with you and help you move towards Him, for the Bible tells us that he is not far from us (Acts 17:27).

God bless you all, and I hope each one of us experiences growth even in this time of pain.