Mathematical Apologetics

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.

A Tribute to Ravi Zacharias

Today, we have lost a giant. This morning, Ravi Zacharias passed away from his battle with cancer. He leaves behind a wife, three children, and thousands upon thousands of lives that he touched deeply with the gospel message and God’s untold love for us. I thought I’d take an opportunity to make a gesture of gratitude for all the work he did in his life, the ways that it has influenced me, and hopefully introduce to others this profound and deeply moving speaker.

Ravi’s journey with Jesus begins on a suicide bed in India. Ravi had become so hopeless and miserable that he had attempted to take his own life. While on this bed, he was given a Bible. Reading the Bible, especially John 14:19, which reads “Because I live, you also will live.” Ravi was given new hope in life, and dedicated his life to serving Jesus. He prayed that day, saying “Jesus if You are the one who gives life as it is meant to be, I want it. Please get me out of this hospital bed well, and I promise I will leave no stone unturned in my pursuit of truth.”

And that is what he did. Ravi has advanced degrees, has studied and read intensely in philosophy, theology, and other areas. Ravi has developed an extremely useful and practical system of how to think about “worldview” – by which we mean one’s overarching, big-picture framework for thinking about life and the universe. A quick skim of mine indicates a bibliography of 33 books published or edited, which cover topics of world religions, eastern and western ways of thinking, dealing with the painful moments of life, and much more. He led a life much like Billy Graham – travelling as a preacher all over the world. He has preached in places where it would be almost unthinkable for an evangelical Christian to speak – including Moscow immediately after the fall of the Berlin wall, in the Mormon Church’s Salt Lake Tabernacle, and speaking in hostile areas of the Middle East, including a one on one with a man who founded a terrorist organization. He led an incredible life.

Why I Love Listening to Ravi

Ravi is one of my favorite speakers to listen to. I still don’t quite know how to put words to this quality, but Ravi is one of the only speakers I have ever heard that can seamlessly weave deep intellectual discussion and equally deep emotional discussion into the same speech, the same story, the same passage, and even the same sentence. And he does so far better than anyone else I have ever listened to. He can quote full poems with proper emphasis and structure, off the top of his head in responding to questions he receives from his audience. He tells stories with great humor and of great emotion to bring to light the deeper ideas of whatever topic he is discussing. He can ask questions that bring to light assumptions we have subconsciously in such a gentle and gracious way. He can expand upon many of the most powerful parables in our culture and many that ought to be part of our culture. He powerfully quotes from Sir Arthur Conan Doyle, Nietzsche, Jesus, CS Lewis, Dawkins, and all manner of other important and influential figures of the past and present. He tells gripping stories from his own experience speaking around the world, a talk with a founder of a terrorist group, a mother praying that her child would be able to feel pain, and many more. The stories stick in the mind and always bring with them a poignant message.

I can hardly write enough about the greatness of a man like this – he is a unique, irreplaceable figure. I want to close my blog post with just a small taste, just one example, of Ravi’s deep insights taking stories we know well to another level, bringing to light all the underlying emotional and existential questions that are so important and yet are not often enough put into words.

A Taste of Ravi’s Insight

Just to give a taste of the depth of this man’s insight, I’ll give just one example that floored me the first time I heard it. In order to understand the point, the following encounter of Jesus with the Pharisees (a group of religious leaders that hated him):

Then the Pharisees went and plotted how to entangle him in his words. ¹⁶And they sent their disciples to him, along with the Herodians, saying, “Teacher, we know that you are true and teach the way of God truthfully, and you do not care about anyone’s opinion, for you are not swayed by appearances. ¹⁷Tell us, then, what you think. Is it lawful to pay taxes to Caesar, or not?” ¹⁸But Jesus, aware of their malice, said, “Why put me to the test, you hypocrites? ¹⁹Show me the coin for the tax.” And they brought him a denarius. ²⁰And Jesus said to them, “Whose likeness and inscription is this?” ²¹They said, “Caesar’s.” Then he said to them, “Therefore render to Caesar the things that are Caesar’s, and to God the things that are God’s.” ²²When they heard it, they marveled. And they left him and went away. (Matthew 22:15-22, ESV)

I will first walk through the ‘normal’ observations you might hear about this passage from a knowledgeable Christian. The Pharisees were trying to put Jesus into a tight spot. The Jewish people viewed the Romans and Caesar as persecutors and longed to overthrow their rule. Thus, people hated paying taxes to Caesar. And the men who collected the tax were often very corrupt and took much more than was required, and this made Caesar’s taxes even more hated. When the Pharisees asked this, they knew that if Jesus said yes, the Jewish people there would be in outrage, and since these men hate Jesus, they’d love that. But if Jesus says no, the Roman authorities would probably arrest Jesus for insurrection against Caesar, and they would love that as well. So either way, the Pharisees thought, they win.

And yet, they did not win. Jesus turned their question on them. Jesus first outwardly makes it clear that these men are trying to trap him. But even better, he answers their question in a way they didn’t expect. He asks them a follow-up question, pointing out that Caesar’s image was on the coins in their hands. Firstly, this would be an embarrassing thing to admit out loud, because Caesar was, in a sense, worshipped as a god at this time by some people, and so walking around with an image of a ‘god’ on their person might be embarrassing. But more importantly, Jesus points out that it is right to give to a person or authority that which is due to them, and therefore we ought to give to Caesar what is due to him, and to give to God what is due to God. The image of Caesar on the coin indicates that he is the source of its value (similar to the sense in which any government is the source of the value of a currency). On many other occasions, Jesus points out how the Pharisees do not give to God what is due to God (the next chapter, Matthew 23, contains a scathing criticism of these religious teachers).

I find this interesting. It is an ingenious answer by Jesus that both conveys a sound answer and levels the pride of the men who were wrongly trying to tear him down in their anger. I had heard this story many times and even studied this passage in my Bible before, and then I heard Ravi speak about it and within just a few seconds, he blew me away. What is his observation?

The leaders should have asked “and what is God’s?” And I think Jesus would have responded by saying “Whose image is on you?”

Christianity is filled to the brim with the importance of the ‘image of God’ – being made in the image of God is, roughly speaking, that which gives us the infinite value that we each possess. It is our personhood, our ability to feel and think, to love and to cherish, to create and be creative, to have relationships, and so much more. And Ravi’s insight cuts right to the core of Christianity – that God has created us and desires to redeem us from the ugliness of our shattered minds and shattered world, but the only way for this to happen is for him to renew us. We are part of the brokenness, and our brokenness disables us.

A point like this will take up no more than 20 seconds or so of one of Ravi’s sermons or talks. He is a profound speaker, and you find from his teaching nuggets of profound insight no matter where you look.

How to Learn from Ravi

He will be missed, but his books, his public talks, and the many stories of his amazing life live on. I’ll put some links here of some of my favorite talks of his, as well as links to pages of his ministry organization, Ravi Zacharias International Ministries (RZIM), which has built up many other amazing speakers over the years who will carry on his legacy.

RZIM Website: https://www.rzim.org/

RZIM YouTube: https://www.youtube.com/user/rzimmedia

The Existence of God: https://www.youtube.com/watch?v=mRRuKDXT7Kg

Why I Believe in Jesus: https://www.youtube.com/watch?v=M3kM6Rax1AU

Why I Am Not an Atheist (at Princeton): https://www.youtube.com/watch?v=d6aDoOzYN-U

Where is God in the Midst of Suffering and Injustice?: https://www.youtube.com/watch?v=bOyBe2ag1zs

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.

Easter Isn’t Make-Believe

He is risen! Happy Easter to all of my readers. This post will be publishing on Easter in 2020. Particularly with the widespread public discourse about suffering at the moment (COVID-19 pandemic), I have been contemplating a lot on suffering and the suffering of the cross in particular. In the Christian message, there is simultaneous acknowledgment of the stark reality of undeserved suffering and of an enduring, conquering hope that even death will be overcome. It is a beautiful message, and one that has carried me through many a trial, and has carried millions of others, if not billions, through their trials.

For this reason, among others, it surprises me that there is significant public discourse that aims to undermine either the source of Easter or those who celebrate it. I don’t really understand why anyone would attack the twin ideas of self-sacrificial love and of redemption from suffering and death, but nonetheless, there are some who will. To anyone who would deny the beauty of the genuine meaning of Easter, I don’t really know what to say.

But that would be quite rare assessment. Far more common than that would be a simple denial that such an event happened. That is, very many people would deny that Jesus Christ rose from the dead on that Sunday morning. When I have heard this denial before, I’ve usually heard it in one of three forms. Firstly, perhaps the person believes that miracles are impossible, and so denies that resurrection even can happen in principle. Or maybe the person simply thinks that there is no historical record that would point to this event actually happening – maybe the Bible has been altered over the years, or maybe it just doesn’t count because it is ‘biased’, or if you were feeling especially bold, you might even dare to try to claim that even the Bible doesn’t say that this happened. The words seem quite plain, this view might argue, but in reality the ‘resurrection’ was only metaphorical or spiritual in some sense, that Jesus’ body would have still been in that grave.

Lots of people believe things like this. I do not intend here to discuss the claim that miracles are impossible (though I believe I can argue that this is profoundly mistaken), but I have been thinking about the other two a lot recently. While they are interesting ideas, and while the seem reasonable, believe it or not, it is far from obvious that either of those are true. The Christian faith actually has very solid, objective historical evidence for its claim of Jesus’ resurrection. I’ll be writing much more in detail about these points in the future, but for now I’ll dive into a broad overview of just a few of the points of secular history that give evidence to the Christian view of the physical resurrection of Jesus.

Did Jesus Exist?

This point hardly even needs clarification, but we know that Jesus existed. There are some who try to argue otherwise, but they are simply wrong. I could show you, for instance, quotes of professional historians and New Testament scholars who compare claims that Jesus is a mythical figure to claims that the earth is flat and that events like the Holocaust and moon landing never happened. I could do this for days, because that’s how, well, virtually every academically respected historian on this topic feels. Just to give one example, here is a quote from the skeptical historian Bart Ehrman, who is my his own admission somewhere between agnosticism and atheism, from his book Did Jesus Exist?

“The idea that Jesus did not exist is a modern notion. It has no ancient precedents. It was made up in the eighteenth century. One might as well call it a modern myth, the myth of the mythical Jesus” ([1], p. 96).

Ehrman is a leader in his field, and specializes in the historicity of the New Testament. And in the spectrum of scholarship, he is very much on the skeptical end, by which we mean that he is likely to accept a lot less data as established history than others in his field. And yet, even as a non-Christian, he is adamant on Jesus’ existence. (For those looking for more, I will provide resources at the end of this article.)

Historians, including Ehrman himself, are equally adamant about the fact that Jesus was crucified. I’ll give one more quote from Ehrman’s book:

“Despite the enormous range of opinion, there are several points on which virtually all scholars of antiquity agree. Jesus was a Jewish man, known to be a preacher and teacher, who was crucified (a Roman form of execution) in Jerusalem during the reign of the Roman emperor Tiberius, when Pontius Pilate was the governor of Judea” ([1], p. 12).

In light of the virtual unanimity of scholars today, Christian and non-Christian alike, we can then take for granted as the simple truth of history that Jesus both lived and died on a cross.

What About the Tomb?

Surely though the idea that Jesus’ tomb was found empty is some kind of myth or fabrication, right?

Well… no. No, it isn’t. Once again, the majority of ancient historians who study this, whether Christian or non-Christian, conclude on purely historical grounds (i.e not on religious grounds) that the tomb of Jesus was found empty by some of his women followers on the Sunday after he died. There is plenty of evidence for this, too much for me to share in just one post, but I’ll try to give an overview.

First, and possibly most importantly, is a passage of the Bible from the letter Paul wrote to the church in Corinth, the one who now call 1 Corinthians. Scholars agree that Paul wrote this letter during his third missionary journey, which took place from 52-57 AD [2], which is one of the earliest sources on the Christian church we have (Note that I call it a source – that is because historians while doing their history do not treat the Bible as an infallible, holy Book, but rather as biographical writings and letters from 1st century Palestine. There is no religious bias involved in historical study of the Bible). Here is what the passage says in the ESV.

“Now I would remind you, brothers, of the gospel I preached to you, which you received, in which you stand, ²and by which you are being saved, if you hold fast to the word I preached to you—unless you believed in vain.

³For I delivered to you as of first importance what I also received: that Christ died for our sins in accordance with the Scriptures, ⁴that he was buried, that he was raised on the third day in accordance with the Scriptures, ⁵and that he appeared to Cephas, then to the twelve. ⁶Then he appeared to more than five hundred brothers at one time, most of whom are still alive, though some have fallen asleep. ⁷Then he appeared to James, then to all the apostles. ⁸Last of all, as to one untimely born, he appeared also to me. ⁹For I am the least of the apostles, unworthy to be called an apostle, because I persecuted the church of God.” (1 Corinthians 15:1-9, ESV)

Historians have recognized that verse 3 is a preamble to a creed that was used by a large portion of the early church. Creeds were common and very important, because in a primarily oral culture, where many people were illiterate, having easy-to-remember creedal statements was extremely helpful in passing on information. Furthermore, noticing that Paul said he received this, and tracing back the likely time when Paul actually received this creed and taking into account normal historical considerations, even skeptical historians conclude that verses 3-5 were already in being repeated by Christians within about 1 year of Jesus’ death! This is a goldmine for the ancient historian. Also, notice that Jesus was said to have died, and was raised. In 1st century Palestinian Jewish culture, that term raised only means one things – physical resurrection. And since physical resurrections don’t leave bodies behind, the tomb must have been empty.

There are quite a few other reasons. This next one involves the use of the criterion of embarrassment, which historians use frequently to help identify what parts of a historical source are true, and which are false. The criterion of embarrassment is basically just the observation that people never lie in order to make themselves look bad. That’s just not how people work. If someone is lying, they want to make themselves look better. Then if a historical document presents a fact that would have been embarrassing or harmful to the message of the writer, it is quite reasonable to conclude that their report is accurate.

The gospels attest that women were the first to discover the empty tomb of Jesus. It is a well known fact of ancient history that women were not considered reliable witnesses in this culture – so much so that they were not even allowed to testify in court. If the writers of the gospels were making up a story to try to convince people that Jesus rose from the dead, why would they make women the first witnesses? That just doesn’t make any sense from a psychological or cultural perspective. Therefore, it is quite reasonable to conclude that the authors of the gospels sincerely believed this and were simply reporting what happened.

Jesus’ Appearances

The reports that Jesus’ followers had experiences that they interpreted as Jesus sitting with and talking with them in bodily form are also taken to be historically accurate by the majority of scholars. To see just one reason for this, look back at the 1 Corinthians passage. Look at verses 5-8 (as a sidenote, Cephas is another name given to the apostle Peter). It is filled to the brim with appearances – to both groups (even a group of 500) and individuals (Peter, James, Paul), and to those who followed Jesus (his apostles) and those who were not yet followers of Jesus (his brother James and Paul). Remember, this creed has been dated back to within 1 year of Jesus’ death. The people who supposedly saw Jesus were most certainly still alive – Paul even says so explicitly! Paul even knew at least some of these people personally. The most reasonable explanation of the fact that an extremely early creed in the church (which would have been ubiquitous) says this is because the people it cites in fact did have some kind of experience like that.

This doesn’t necessarily mean that Jesus did rise from the dead – but certainly there were many people who had experiences that seemed quite vividly like that is what happened.

The Origins of Christianity

Jews at this time did not expect their Messiah to die – the general expectation was that the Messiah would be a military and religious leader who would overthrow Roman rule. Furthermore, Jewish belief in resurrection was, while common, not ubiquitous, and even those who did believe in bodily resurrection only thought it would happen on judgment day, to everybody at the same time. There was no precedent for the disciples to think a thing like this would happen, because it went against the expectations of their culture. And if you read the gospels, you constantly find that Jesus tried quite hard to explain this to his disciples before he died, but the disciples are presented as, well, bone-headed, not understanding even direct explanations. Jesus even rebuked Peter in remarkably harsh language for suggesting that Jesus would never be killed. The criterion of embarrassment applies again here – the writers of the New Testament would never have made the apostles look like such idiots unless that is what actually happened.

Another point – what motive did the early church have for inventing this story? This caused the Jews and Romans alike to despise them. The apostles lived lives of poverty and suffering. Christians at this time were regularly beaten, imprisoned, tortured, and executed simply for being Christians. Doesn’t look like they have any motive to be lying… Liars make terrible martyrs. Well, that is because liars who know they are lying are not martyrs. It just doesn’t happen. That’s simply not how people work.

Conclusion

I could – and someday will – provide much more in-depth presentations of evidence of this sort. There are books 800 pages long and more (I recently heard of one around 5000 pages long) that present evidence for these claims and similar ones in excruciating detail. Papers in secular, peer-reviewed academic journals argue in favor of these points, at least more often than not. There are always (or at least almost always) exceptions. However, we have very strong secular historical evidence that not only did Jesus live and die on a Roman cross, his tomb was found empty, many people alive at the time reported seeing him after his death, and his closest followers came to believe so firmly in his physical resurrection from the dead that, even under the threats humiliation and execution, they did not give up their beliefs.

Now, I don’t want to force anyone to become a Christian. And I am quite convinced from the Christian Bible that God doesn’t want to compel anyone to follow Him either. God desires relationship, and relationship by definition cannot be compelled. These evidences can be explained or just denied, if you want to do that. But to my eyes, and frankly I think to the eyes of anyone looking seriously at what history tells us, the case that Easter really happened actually looks pretty good.

If you want to learn more, I’ll put many great books and YouTube videos I have seen or read (or in some cases, have on my reading list but haven’t quite gotten to yet) that cover this topic and related questions.

Citations

[1] Ehrman, Bart D. (2012). Did Jesus Exist?: The Historical Argument for Jesus of Nazareth. HarperOne. ISBN 978-0062206442.

[2] http://www.datingthenewtestament.com/Corinthians.htm

More Resources

YouTube:

Dr. David Wood (Acts 17 Apologetics)
- Did Jesus Exist? https://www.youtube.com/watch?v=Rg7Ge7y9-JI
- Old Testament Prophecies https://www.youtube.com/watch?v=drsLYXevBsY
- The Justice/Love of the Cross https://www.youtube.com/watch?v=k_swP34cPY0
Dr. Bart Ehrman
- Did Jesus Exist? https://www.youtube.com/watch?v=hnybQxIgfPw
Dr. William Lane Craig (Reasonable Faith)
- Evidence for the Resurrection https://youtu.be/oAxPHWF8aec
Dr. Mike Licona
- Evidence for the Resurrection https://youtu.be/rDo9mldPybs
Dr. Gary Habermas
- The Resurrection Argument that Changed a Generation of Scholars https://youtu.be/ay_Db4RwZ_M
Capturing Christianity
- Two Interviews with Professional Historians
- Dr. Calum Miller & Dr. Max Baker-Hytch: https://youtu.be/0Rtmn5LLM-0
- Dr. Gary Habermas & Dr. Mike Licona: https://youtu.be/mc6XhMySxdY
Inspiring Philosophy
- Historical Evidence for the Resurrection: https://youtu.be/ay_Db4RwZ_M

Books:

The Case for Christ by Lee Strobel
Cold-Case Christianity by J. Warner Wallace
Did Jesus Exist? by Bart Ehrman
The Resurrection of the Son of God by NT Wright
Reasonable Faith by William Lane Craig
Evidence that Demands a Verdict by Josh & Sean McDowell

By the way, the first two of these are records of how intelligent atheists, namely a top-level journalist and a top-level cold case detective, looked at the objective evidence and became Christians.