Mark 4: The Seed and the Soil

“There went out a sower to sow.” In his parable of the sower Jesus gives various active and passive roles: sower, seed, good soil, soil among thorns, stony ground, and waysides. This meditation on Mark 4 considers the seed as the word and the Word Christ, our receptivity to Christ, how he can enter, germinate, grow, and transform us into new creatures.

One of the Church’s greatest theologians, Thomas Aquinas (1225 – 1274) was an astonishingly prolific writer. He’s especially known for his Summa Theologiae, which is one of my first go-to theology resources.  His style was analytic and detailed. Each of the “Questions” in the Summa reads like a geometrical proof out of Euclid, each with some assertion, supporting points, counter-assertions, and detracting points, and a conclusion. It was a masterful intellectual achievement. Yet near the end of his life Aquinas had a mystical experience that seemed to lead him away from that stage of life and into another. He was no longer able to write, not out of physical incapacity but because of the greatness of his revelation. He felt his writings, great as they were, couldn’t possibly match the greatness of the revelation he had been given. The direct experience of his revelation transcended the rationality of this most rational of thinkers. That’s a sobering thought, still, I’m inclined to think of this overwhelming experience of his as a reward for all the work that he had done in a previous stage of life. His mystical stage, if we can call it that, only lasted a few months since he died shortly after. But it’s something I think about a lot. The analytical, rational stage of the adult in his younger and middle ages, succeeded by a later super-rational, mystical stage. Something about that seems quite appropriate. I approach religion and scripture in that very analytical, rational way. It’s just more natural for me right now. But I wouldn’t be surprised or at all disappointed if that changed at some point. As satisfying as the intellectual nature of theology is, the infusion of the Spirit is so much greater. I spoke in a previous episode about a life with the Holy Spirit. Those moments of spiritual elevation are invaluable.

On this subject I’d like to share a meditation Mark 4, a chapter in which we read of Jesus’s parable of the sower, a masterful parable. I’d like to focus on two aspects of it: (1) the seed and (2) the soil. When Jesus explained the parable of the sower to his disciples he said that the seed was “the word”; “The sower soweth the word.” (Mark 4:14). For readers familiar with John’s gospel this can have at least a double meaning: (1) the word of the Gospel, the words that people speak to preach the message, and (2) the Word, Logos, is also Christ himself (John 1:1-3).

Another story about Thomas Aquinas. One day while Aquinas was in prayer before the crucifix the voice of Christ called out to him and said, “You have written well of me, Thomas. What reward will you receive from me for your labor?” And Aquinas answered, “Lord, nothing except you.”

I love the Christmas hymn “O Little Town of Bethlehem”, especially this verse:

“O holy Child of Bethlehem,
descend to us, we pray,
cast out our sin and enter in,
be born in us today.”

I thought about this a lot this past Christmas. I tried to put myself in Mary’s state of mind as one who receives and carries the Lord himself within her body. She declared so much with that statement, “Behold the handmaid of the Lord” (Luke 1:38). I think it’s powerful – also for men, who probably aren’t used to thinking in this way – to think of being the mother Mary, bearing God in her body. It’s one vivid image of something that the scriptures and the ritual practices of the Church communicate in various ways, the Eucharist for example: that we are to take Christ into ourselves and allow him to transform us into new creatures.

This is how I think about the seeds in Jesus’s parables. The seed is “the word”, the message of the Gospel, as well as “the Word”, Christ himself.

In the parable of the sower, the sower plays the active role. “Behold, there went out a sower to sow” (Mark 4:3). He is the one sowing the seeds. By the time the sower passes by the soil is either ready or it isn’t. The soil is passive but its condition makes all the difference.

“And it came to pass, as he sowed, some fell by the way side, and the fowls of the air came and devoured it up. And some fell on stony ground, where it had not much earth; and immediately it sprang up, because it had no depth of earth: But when the sun was up, it was scorched; and because it had no root, it withered away. And some fell among thorns, and the thorns grew up, and choked it, and it yielded no fruit. And other fell on good ground, and did yield fruit that sprang up and increased; and brought forth, some thirty, and some sixty, and some an hundred. And he said unto them, He that hath ears to hear, let him hear.” (Mark 4:4-9)

Let’s talk first about the active role of the sower. One of the things that strikes me about moments of spiritual revelation is that they don’t happen whenever we want them to. They come as a gift of grace. That’s because they’re not manufactured. And they’re not the product of an individual. Rather, they are special encounters between us and Spirit. The Spirit, as the other person in these encounters, has to decide to participate. The philosopher Martin Buber (1878 – 1965) called this an “I-You” encounter. He contrasted this with the “I-It” experience in which a person can individually and unilaterally perceive and consider objects, ideas, and people in a way that doesn’t require another’s free participation. Basically, how we live most of the time. But our lives are sometimes interrupted by encounters of a different kind. And he says these come by “grace”:

“The You encounters me by grace–it cannot be found by seeking. But that I speak the basic word to it is a deed of my whole being, is my essential deed. The You encounters me. But I enter into a direct relationship to it. Thus the relationship is election and electing, passive and active at once: An action of the whole being must approach passivity, for it does away with all partial actions and thus with any sense of action, which always depends on limited exertions. The basic word I-You can be spoken only with one’s whole being. The concentration and fusion into a whole being can never be accomplished by me, can never be accomplished without me. I require a You to become; becoming I, I say You. All actual life is encounter.” (Martin Buter, I and Thou, 61)

What’s crucial to understand is that the Father, Son, and Holy Ghost are persons. We can’t manufacture encounters with persons on our own. It requires the full cooperation of the other person. The Holy Ghost needs to act. And that happens when he chooses. But we can act to be receptive and prepare ourselves. We are the soil and we can condition ourselves as soil to receive Christ.

Jesus interpreted the parable for his disciples in this way:

“The sower soweth the word. And these are they by the way side, where the word is sown; but when they have heard, Satan cometh immediately, and taketh away the word that was sown in their hearts. And these are they likewise which are sown on stony ground; who, when they have heard the word, immediately receive it with gladness; And have no root in themselves, and so endure but for a time: afterward, when affliction or persecution ariseth for the word’s sake, immediately they are offended. And these are they which are sown among thorns; such as hear the word, And the cares of this world, and the deceitfulness of riches, and the lusts of other things entering in, choke the word, and it becometh unfruitful. And these are they which are sown on good ground; such as hear the word, and receive it, and bring forth fruit, some thirtyfold, some sixty, and some an hundred.” (Mark 4:14-20)

A lot here to think about. One part that stands out to me at the moment is the case of the seeds sown among thorns. The thorns are “the cares of this world, and the deceitfulness of riches, and the lusts of other things”. These create unfruitful conditions. To be fruitful it is necessary to be set apart from these things. Some Christians throughout history have applied this kind of setting apart in a physical sense, actually taking up a monastic life. But I think what’s most important is to apply this existentially, to be set apart from the world in the way we live and in our way of being. Especially in the things we care about.

Following Christ is not a light matter and Jesus made this clear.

“And it came to pass, that, as they went in the way, a certain man said unto him, Lord, I will follow thee whithersoever thou goest. And Jesus said unto him, Foxes have holes, and birds of the air have nests; but the Son of man hath not where to lay his head. And he said unto another, Follow me. But he said, Lord, suffer me first to go and bury my father. Jesus said unto him, Let the dead bury their dead: but go thou and preach the kingdom of God. And another also said, Lord, I will follow thee; but let me first go bid them farewell, which are at home at my house. And Jesus said unto him, No man, having put his hand to the plough, and looking back, is fit for the kingdom of God.” (Luke 9:57-62)

Wow! Clearly the kind of life Jesus requires is quite different from the way normal people live. In thinking about these verses it makes me reflect on the things I care about and whether they enable or impede my receptivity to the Holy Spirit. Jesus warns about the cares of the world. The Greek for “care” is μέριμνα (mérimna). The corresponding verb is μεριμνάω (merimnao): to be anxious or worried about something. It’s used several times in the following passage from the Sermon on the Mount:

“Therefore I say unto you, Take no thought [μὴ μεριμνᾶτε, me merimnate] for your life, what ye shall eat, or what ye shall drink; nor yet for your body, what ye shall put on. Is not the life more than meat, and the body than raiment? Behold the fowls of the air: for they sow not, neither do they reap, nor gather into barns; yet your heavenly Father feedeth them. Are ye not much better than they? Which of you by taking thought [μεριμνῶν, merimnon] can add one cubit unto his stature? And why take ye thought [μεριμνᾶτε, merimnate] for raiment? Consider the lilies of the field, how they grow; they toil not, neither do they spin: And yet I say unto you, That even Solomon in all his glory was not arrayed like one of these. Wherefore, if God so clothe the grass of the field, which to day is, and to morrow is cast into the oven, shall he not much more clothe you, O ye of little faith? Therefore take no thought [μὴ οὖν μεριμνήσητε, me oun merimnesete], saying, What shall we eat? or, What shall we drink? or, Wherewithal shall we be clothed? (For after all these things do the Gentiles seek:) for your heavenly Father knoweth that ye have need of all these things. But seek ye first the kingdom of God, and his righteousness; and all these things shall be added unto you. Take therefore no thought [μὴ οὖν μεριμνήσητε, me oun merimnesete] for the morrow: for the morrow shall take thought [μεριμνήσει, merimnese] for the things of itself. Sufficient unto the day is the evil thereof.” (Matthew 6:25-34)

Do we need food, drink, and clothing? Yes, Jesus said as much. “Your heavenly Father knoweth that ye have need of all these things.” But he said not to seek after them. And this interesting, he says that seeking after food, drink, or clothing is what the Gentiles do. Gentiles are those who have not entered into the covenant. The Gentile way of life is a completely different way of life, and really the normal way of life. But it’s not the way of Jesus. Jesus said these cares “choke the word, and it becometh unfruitful”.

Jesus explained the good soil represents people who “hear the word, and receive it”. And again, I like to consider the double meaning in which the Word here is also Christ himself. The good soil receives Christ. Christ enters into it, germinates, and grows. Like with Mary, the God-Bearer, the Spirit can enter into us and Christ can abide in us. This reception is also mutual abiding. We abide in Christ and he abides in us. In the Trinity this kind of relation is sometimes called interpenetration and a similar kind of mutual abiding and interpenetration is happening here:

“Abide in me, and I in you. As the branch cannot bear fruit of itself, except it abide in the vine; no more can ye, except ye abide in me. I am the vine, ye are the branches: He that abideth in me, and I in him, the same bringeth forth much fruit: for without me ye can do nothing.” (John 14:4-5)

I believe this is ultimately what Christian holiness looks like. Understanding, yes. By all means. Learn the doctrine, study the principles, develop a sophisticated philosophical and theological understanding. I think that’s appropriate and good, especially for certain periods of life. But beyond that is this direct receptivity to the Spirit and this planting of Christ into the core of our being to be transformed into new creatures.

The Structure of Infinite Series

An infinite series is a sum of infinitely many numbers or terms, related in a given way and listed in a given order. They can be used to calculate the values of irrational numbers, like pi, out to trillions of decimal places. Or to calculate values of trigonometric and exponential functions. And of greatest interest, they can be used to see non-obvious relations between different areas of mathematics like integers, fractions, irrational numbers, complex numbers, geometry, trigonometric functions, and exponential functions.

I’d like to start things off with a joke. A math joke.

An infinite number of mathematicians walk into a bar. The first orders a beer, the second orders half a beer, the third orders a quarter of a beer, and so on. After the seventh order, the bartender pours two beers and says, “You fellas ought to know your limits.”

OK, I tried to start things off in a cute way. Anyway, the joke is that if they keep following that pattern to infinity the total approaches the equivalent of two beers. That is the limit of the series. In mathematical terms this is an infinite series. An infinite series is a sum of infinite terms. With the series in the joke the series is:

1 + 1/2 + 1/4 + 1/8 + 1/16 + … = 2

Each term in the series is half the previous term. And if you continue this out to infinity (whatever that means) it ends up adding up to to 2.

Infinite series can be either convergent or divergent. The series just mentioned is convergent because it adds up to a finite number. But others end up blowing up to infinity. And those are divergent.

Convergent series fascinate me. Their aesthetics remind me of cyclopean masonry, famous among the Inca, where all the stone pieces fit together perfectly, as if part of a single block. These series reveal infinite structure within numbers.

I’d like to share some of my favorite examples of infinite series. The previously mentioned series adding up to 2 is interesting. It shows infinite structure within this simple integer, 2. But I’m especially interested in infinite series for irrational numbers. Irrational numbers, like pi, e, and logarithms, have infinite, non-repeating, decimal places. How can we find the values for all these digits? This is where infinite series became extremely useful.

Pi, approximately 3.14159, is the ratio of circumference to diameter. How can we find its value? Maybe make a really big circle and keep on making measurements with more and more accurate tape measures? No, that won’t go very far. Fortunately, there are infinite series that add up to pi or ratios of pi. And what’s fascinating is that these series seem to have no obvious relation to circles, diameters, or circumferences. Here are some of those series:

1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 + … = π^2/6

1/1 – 1/3 + 1/5 – 1/7 + 1/9 – … = π/4

You look at something like that and wonder. What on Earth is pi doing there? Where did that come from? Irrational numbers, by definition, cannot be expressed as fractions. But they can be expressed as infinite sums of fractions. You can get an irrational number like pi just by adding up these simple fractions. At least, by adding them up an infinite number of times. Of course, we can’t actually do that. But we can still add up many, many terms. And with computers we can add up millions of terms to get millions of digits. But different series will converge on the accurate value for a given number of digits more quickly than others. If we’re actually trying to get as many digits as possible as quickly as possible we’ll want a quickly converging series. A great example of such a modern series for pi is the Ramanujan series:

1/π = 2 * sqrt(2) / 9801 * sum[ (4k)! * (1103 + 26930k) / ((k!)^4  * 396^(4k)), k=0,∞]

This series computes a further eight decimal places of pi with each term in the series. Extremely useful.

But back to the earlier question. What is pi doing here? How do these sums, that don’t seem to have anything to do with the geometry of circles, spit out pi?

Let’s just look at that series that adds up to pi/4. This is known as the Leibniz formula. pi/4 is the solution to arctan(1), a trigonometric function. In calculus the derivative of arctan(x) is 1/(1+x^2). 1//(1+x^2) can also be represented by a power series of the form:

1/(1+x^2) = 1 – x^2 + x^4 – x^6 + …

Since this is the derivative of arctan(x) we can integrate it and get a series that is equivalent to arctan(x).

arctan(x) = x – x^3/3 + x^5/5 – x^7/7 + …

This is quite useful. Since it’s a function we can plug in different values for x and get the result to whatever accuracy we want by expanding out the series as many terms as we want. To get the Leibniz formula we plug in 1. So arctan(1) is:

1/1 – 1/3 + 1/5 – 1/7 + 1/9 – … = arctan(1)

And we already know that arctan(1) is equal to pi/4. So this gives us a way to calculate the value of pi/4 and, by just multiplying the result by 4, the value of pi. We can calculate pi as accurately as we want by expanding the series out as many terms as we want to. Though in the case of pi, we’ll do this with a faster converging series like the Ramanujan series. But I picked the Leibniz series as an example because it’s easiest to show why it converges on pi, or specifically pi/4. You can see a little bit here how different areas of mathematics overlap: geometry relating to trigonometry, calculus, and infinite series. Steven Strogatz has made the point that infinite series are actually a great way to see the unity of all mathematics.

So that’s pi. Let’s look at some other irrational numbers that can be calculated by series. 

To calculate e, approximately 2.71828, we can use the following series:

1/0! + 1/1! + 1/2! + 1/3! + 1/4! + … = e

Or to calculate ln(2), approximately 0.693, we can use the following series:

1/1 – 1/2 + 1/3 – 1/4 + … = ln(2)

I find these similarly elegant in their simplicity. We’re just adding up fractions of integers and converging to irrational numbers. I think that’s remarkable.

We saw earlier with the arctan function that it’s also possible to write a series not only with numbers but with variables so that the series becomes a function where we can plug in different numbers. There are series for sine and cosine functions in trigonometry. That’s good because not all outputs from these functions are “nice” values that we can figure out using other principles of geometry, like the Pythagorean Theorem. What are some of these “nice” values? We’ll use radians, where pi/2 radians equals 90 degrees.

sin(0) = 0
sin(pi/2) = 1
sin(pi/3) = sqrt(3)/2
sin(pi/4) = sqrt(2)/2
sin(pi/6) = 1/2

These are angles of special triangles with angles of 45, 30, 60 degrees, where the ratios of the different side lengths work out to ratios that we can figure out using the Pythagorean Theorem. But these are special cases. If we want to calculate a trigonometric function for other values we need some other method.

Fortunately, there are infinite series for these functions. The infinite series for the sine and cosine functions are:

sin(x) = x – x^3/3! + x^5/5! – x^7/7! + …

cos(x) = 1 – x^2/2! + x^4/4! – x^6/6! + …

Again, we have this kind of surprising result where we get trigonometric functions just from the sum of fractions of integers and factorials, which don’t seemingly have much to do with each other. Where is this coming from?

These trigonometric functions are infinitely differentiable. You can take the derivative of a sine function over and over again and no matter how many times you do it the result will be either a sine function or a cosine function. Same for the cosine function. They just keep circling back on themselves when we differentiate them. These series come from their Taylor series representations, or specifically their Maclaurin series representations. The Maclaurin series for a function f(x) is:

f(0) + f’(0)/1! * x + f’’(0)/2! * x^2 + f’’’(0)/3! * x^3 + …

What happens if we apply this to sin(x)? Let’s take repeated derivatives of sin(x):

First derivative: cos(x)
Second derivative: -sin(x)
Third derivative: -cos(x)
Fourth derivative: sin(x)

So by the fourth derivative we’re back where we started. And so on. What are the values of these derivatives at 0?

sin(0) = 0
cos(0) = 1
-sin(0) = 0
-cos(0) = -1

So the terms with the second and fourth derivatives will go to 0 and disappear. The remaining terms will alternate between positive and negative. In the case of sin(x) each term in the resulting series will have x raised to odd integers divided by odd factorials, with alternating signs. The result being:

sin(x) = x – x^3/3! + x^5/5! – x^7/7! + …

And for cos(x) it will be similar but with x raised to even integers divided by even factorials, with alternating signs. The result being:

cos(x) = 1 – x^2/2! + x^4/4! – x^6/6! + …

Using these series we can now calculate sine and cosine for any value. Not just the special angles of “nice” triangles.

The Maclaurin series also gives an infinite series for another important infinitely differentiable function: the exponential function e^x. The derivative of e^x is just itself, e^x, forever and ever. So in this case the Maclaurin series is quite simple. No skips or alterations.

e^x = 1 + x + x^2/2! + x^3/3! + …

We already saw one solution to this equation where x is set equal to 1, which is simply the number e.

These three series – for sin(x), cos(x), and e^x – allow us to see an interesting relation between exponential functions and trigonometric functions. The series for e^x has all the terms from the series for both sin(x) and cos(x). But in the series for e^x all the terms are positive. Is there a way to combine these three? Yes, there is. And it will connect it all to another area of mathematics: complex numbers. Complex numbers include the imaginary number i, which is defined as the square root of -1. The number i has the following properties:

i^2 = -1
i^3 = -i
i^4 = 1
i^5 = i

And the cycle repeats from there. It turns out that if we plug ix into the series for e^x all the positive and negative sines work out to match those of the series for cos(x) and i*sin(x). With the result that:

e^(ix) = cos(x) + i * sin(x)

To make things really interesting let’s also bring pi into this and substitute pi for x. In that case, cos(π) = -1 and sin(π) = 0. So we get the equation:

e^(iπ) + 1 = 0

Steven Strogatz said of this result:

“It connects a handful of the most celebrated numbers in mathematics: 0, 1, π, i and e. Each symbolizes an entire branch of math, and in that way the equation can be seen as a glorious confluence, a testament to the unity of math. Zero represents nothingness, the void, and yet it is not the absence of number — it is the number that makes our whole system of writing numbers possible. Then there’s 1, the unit, the beginning, the bedrock of counting and numbers and, by extension, all of elementary school math. Next comes π, the symbol of circles and perfection, yet with a mysterious dark side, hinting at infinity in the cryptic pattern of its digits, never-ending, inscrutable. There’s i, the imaginary number, an icon of algebra, embodying the leaps of creative imagination that allowed number to break the shackles of mere magnitude. And finally e, the mascot of calculus, a symbol of motion and change.”

He also said, speaking of infinite series generally:

“The most compelling reason for learning about infinite series (or so I tell my students) is that they’re stunning connectors. They reveal ties between different areas of mathematics, unexpected links between everything that came before. It’s only when you get to this part of calculus that the true structure of math — all of math — finally starts to emerge.”

I think we can see that effect in some of the relations between some of my favorite infinite series that I’ve shared here.

Having looked at all this I’d like to make a couple philosophical observations.

One is on the possibility of objectivity in mathematics, mathematical realism, or mathematical platonism. Infinite series enable us to calculate digits for irrational numbers, which have infinite digits. We find the values millions and billions of decimal places out and we will always be able to keep going. Last I checked, as of 2021 pi had been calculated out to 62.8 trillion digits. What of the next 100 trillion digits? Or the next quadrillion digits? Well, I think that they are already there waiting to be calculated, whether we end up ever calculating them or not. And they always have been. Those 62.8 trillion digits that we’ve calculated so far have been there since the days of the dinosaurs and since the Big Bang. There’s a philosophical question of whether mathematical conclusions are discovered or created. You can tell I believe they’re discovered. And part of the reason for that is because of these kinds of calculations with infinite series. No matter how deep into infinity you go there’s always more there. And you don’t know what’s there until you do the calculations. You can’t decide for yourself what’s there. You have to do the work to find out and get the right answer. Roger Penrose had a similar line of thinking with the infinite structure of the Mandelbrot set.

Now, I do think there’s a certain degree of human activity in the process. Like in deciding what kinds of questions to ask. For example, geometry looks different whether you’re working in Euclidean, hyperbolic, or elliptic geometry. Answers depend on assumptions and conditions. I like a line that I heard from Alex Kontorovich: “The questions that are being asked are an invention. The answers are a discovery.”

The other philosophical question is: What does it actually mean to say that an infinite sum equals a certain value or converges to a certain value? We can never actually add up infinite terms. Nevertheless, we can see and sometimes even prove where a convergent series is headed. And this is where that concept of limits comes up. I don’t know how to answer that question. There are different ways to interpret that. Presently, the way I’m inclined to put it is this: The limits of infinite series are values toward which series tend. They never actually reach them because infinity is not actual. But the tendency of an infinite series is real, such that, as you continue to add up more terms in the series the sum will continue to get closer to the value of convergence.