Mathematics as a cultural artifact.

I have been thinking about Martin Gardner’s critique of Reuben Hersh’s ideas in an article he wrote in Eureka. I am not sure I understand Prof. Hersh’s ideas but Gardner’s comments were obviously wrong.
   I am going to assume that Hersh thinks that mathematics does not exist independent of people. Gardner says that Hersh asserts that mathematics is a “cultural artifact”.
In the first place, to think that humanity has discovered timeless, universal truths is an amazing act of human hubris.
   Since the right triangles of mathematics don’t exist the question arises: What does it mean to say there is something true about something that doesn’t exist? How do you say something true about unicorns?
   Of course mathematics is a cultural artifact. How could one think otherwise?
In my view mathematics can be likened to a computer game. The symbols of mathematics are game pieces. The rules of mathematics are the rules of the game. The symbols don’t represent real objects. They are like the trolls, ogres and mages. The problem solutions and theorems are like the gold coins, better guns and health.
   It turns out that humans can make game pieces that seem to model the approximate “real” world and then make up game rules that when followed seem to lead to a game solution of a “real” world problem. Then humans perform experiments to see how the game solution measures up to the humans’ approximation of what actually happened.
   The game gives a useful answer often enough that mathematics game rules are taught in school.
   Different cultures come up with different games with different game pieces. When someone says that the Pythagorean Theorem is true everywhere, they are wrong. It’s only true where the Earth mathematics game can be played and the Pythagorean Theorem is a secret place in the game.
    In the “real” world there are no right triangles, there are no line segments to
use as sides for triangles or points to use as vertices. A board 2 feet long doesn’t exist. These are all game pieces but they were designed to model approximations to the “real” world as humans saw the “real” world.
    The unit square is no more “real” than a dragon. Saying things about the unit square is like saying things about a dragon. What does it mean to say things about imaginary objects? Assuming that “true” and “exist” have meaning, can you make true statements about objects that don’t exist?
    I feel like I’m swimming in molasses when trying to explain why mathematics isn’t real. I don’t see why the idea is so resisted. It doesn’t require believing in the impossible.
    It is not hard to think of an environment where Earth mathematics can’t be played. I don’t think that the game can be played in the world of dolphins and whales.
    What need would whales or dolphins have to count? They surely wouldn’t count plankton one by one. Humans tend to count because there are a lot of collections whose objects are stationary enough to allow counting.
    I think that counting arose from a need. If there was no need to count, numbers would not arise. A culture that had no need of counting would have no need of numbers.
    The purpose of numbers was to convey a certain kind of information. It would be remarkable if there were only one way to convey that information and that it required human physiology.
    What does it mean to say that the Pythagorean Theorem is true everywhere? A culture may have an entirely different way to perceive the universe and have no way to think about geometry but if you, from Earth, were there would you see the truth of the theorem?
    Well, how would you check the proof of it? You would set up your game and see if you could play it. So, what if there is no way to set up the geometry game? Is the Pythagorean Theorem still true?
     On Earth there are no right triangles that satisfy the conditions of the theorem. There are no real triangles so the Pythagorean Theorem only makes sense in the world of the game.
    I read somewhere that there are 50-100 billion neurons in the brain. (I have no idea why there is that big a spread.) 10 billion of these neurons pass signals around the brain over 100 trillion synaptic connections.
    So to have a brain you need 10 billion “things” that pass messages among themselves and to, say, 100 billion other “things”. You probably need other stuff too but the 100 billion “things” is a good start.
    In the human brain the “things” are called neurons and the connections are hardwired. I don’t see why a wireless connection wouldn’t work as well if not better. This means that the “things” wouldn’t have to be localized in a skull. The size of such a “brain” would be limited only by what speed of thought was satisfactory to the “brain”. And who knows how the “brain” looks at time. The “things” could be light years apart.
    The “brain” could be 10 billion years old in which case a short interval of time would mean something quite different to the “brain” than it does to me.
How would such a “brain” look at truth?
    Humans still think that they are the center of the universe. They think heaven is the next step in human development. Maybe the urge to give a mysterious reality to right triangles is the desire for God.
     Why do humans insist on the reality of ghosts and goblins? Why do they insist that the universe was made in six days? What’s the point?
     I can see where belief in God offers the possibility of salvation and good crops. What does ∞ do for you if you believe it is real and not just a game piece?

And that’s all it is…a game piece.

And that is what mathematics is…a cultural artifact.    

Graphing

Graphing Calculators
One of the major obstacles to teaching a topic in mathematics is that the student has little or no idea why they are learning it.
When I was learning the C Language, I never understood pointers because I never knew why they were there. What problem do they solve? Evidently the guy who invented them did it to make programming easier but I didn't see why.
I struggled with epsilon-delta process until we got into 20th Century mathematics; when I saw what the process was used for the concept became trivial.
As a rule, mathematics spring topics on students like a magician pulling a rabbit out of a hat. The rabbit is obscure because why would a person keep a rabbit in a hat.
The part of a method or a theorem that the student sees is the result of lots of thought with the thought removed. It is one thing to see that the steps of a proof lead to a correct result, it is quite another see where the steps came from and why you are proving the result in the first place.
And now we come to graphs.
What are graphs for? They allow a person to use their eyes to help in the understanding of functions. Graphs are of no use to a person blind from birth. Graphs were invented by people who could see.
There are several ways to look at functions visually. Some graphs use the magnitude of the distance between two points, as can be seen, to visually represent a numerical quantity. You see that points A and B are farther apart than points C and D. Another method is to represent a number as a color. This is the method used to tell when steel is hot enough to pour. Film comes in different color temperatures.
So, the point of the graph is to see a function and the basic idea is to represent numbers in the domain and in the range by the size of the separation between two points.
But to understand graphs, the student has to understand functions. If they don't understand functions they can't see a reason for graphs.
The next time you are trying explain graphs, see how many of your listeners can tell you what a function is.
Supposing that the student understands the function, then they are ready for graphs. I suggest Chapters 3,4 and 5 of  The Calculus: An Opinion by J. Davis
I think that paper and pencil is the best way to start. I think that the student gets a feeling for drawing the picture of a function. If I want to really look at a face, I will sketch it, not take a Polaroid.
The graphing calculator is of use when you want a more accurate picture of the function like when you want a more exact number you use a calculator.

A suggested goal of teaching

Teaching Revisited The following question came up in a conversation: Is learning to ride a bike the same kind of learning as learning the quadratic formula. A distinction made was that learning to ride a bike is learning a motor skill while learning the quadratic formula is learning a mental skill. But motor skills are handled in the brain so they are mental skills as well. I find much in common between learning music, learning how to shoot a basket, learning calculus, learning anything. Learning is learning.> And once you learn something, you don't forget it, like when you've learned to ride a bicycle. That's how you can tell if you have really learned something; you don't forget it The brain directs the muscles in performing a motor skill. When a person first learns a skill they memorize how to do it and the person is mentally and consciously involved in performing the skill. After the skill has been learned the brain skips the consciousness and communicates with the muscles directly. Sometimes the brain communicates with consciousness when it solves a problem. Sometimes the brain tells the finger what note to play and the brain listens via the ears. A musician learns music and can pick up any instrument and play it. Maybe not as a virtuoso but they can play it. The musician has consolidated music into one instance.
A mathematician learns mathematics and sees what mathematics is all about. Then she can learn any chunk of mathematics. The mathematician has consolidated mathematics into one instance. When I am teaching max-min problems I see it as working the same problem over and over. I have consolidated max-min problems into one instance. The student has consolidated nothing and everything I put on the blackboard is new. They don't see one instance, they see a jillion instances. They are flooded with instances. A teacher should remember this. I don't remember how to do max-min problems, I just know how to do them. Like a musician doesn't memorize a song, he just knows it. I watched Sa Chen, the young Chinese pianist, play the piano on TV and I thought, "There is no way to memorize what she is doing." The whole sonata was one instance to her. It's more than that because she knows a lot of sonatas and other stuff too. The piano is part of one instance of music. I am considering extreme cases of consolidation and why extreme cases occur is a mystery to me. I don't know where Mozart and Gauss came from. But most people can consolidate something to some degree. A guy drove me right to the Harley-Davidson shop in Kansas City, Kansas and he had only been there once two years previous. He told me it was just something he could do. He could find his way back to any place he had ever been.
I think people have consolidations that they can only realize every so often. Every once in a while a song is consolidated and is played without thinking. You shoot a basketball and you just know you are doing everything right and it's going in the hoop. And it does. It isn't that consolidations aren't there somewhere inside us; the problem is in the extraction. A friend, Chris Barrett, pointed out this idea of consolidation to me. He defined God as that entity that reduced the universe to one instance. Anyway, we need to teach consolidating. I think, however, that changing the way things are taught would be a Harry Sheldon project. (Foundation and Second Foundation by Azimov)

Remarks About Numbers

Numbers, Sort of I would like to talk about numbers and suggest that they should be talked about in the classroom and at the dinner table. What about the number 1? It‘s a symbol that conveys information; let‘s say from Joe to Sam. The information to be passed is that Joe has a single child. Here I run into difficulty. The phrase ‘single child’ means the same as ‘one child’ so I‘m not really defining anything. How do I tell a person what ‘one’ means? If both have sight and they are close to each other, holding up a finger might work. For the two guys on sight, this works but I am faced with explaining what ‘holding up one finger’ means. Having sight and being there is a big help in two people coming to agreement on what ‘one’ means. Some kind of sensory communication is needed. (See Johnny Got His Gun by Dalton Trumbo.) Let‘s suppose that Joe and Sam have the usual five senses and that they agree that they can convey a certain kind of information by holding up their fingers. A thumb down might mean that they don‘t have any of the objects in question. They may also decide that a thumb, forefinger and pinkie convey the same information as a forefinger, middle finger and ring finger. They can work out sounds that will designate different collections of fingers. This would be useful if Sam couldn‘t see how many fingers Joe was holding up and had to yell the information across a meadow. If Sam and Joe live in different towns they can devise marks to put on a clay tablet that will stand for different collections of digits. The Romans developed a pretty good system to convey the kind of information that I‘ve been discussing. Let‘s call symbols that describe how many objects are in a collection of objects, numbers. Or maybe a number represents a property of a set. I remember 1st grade as the place where symbols were attached to small sets. Here memorization was necessary since I, at least, saw no reason why 4 denotes **** objects. Maybe it does to an Arab since it is an Arabian system. Most people, however, agree on the meaning of number to some extent. I can‘t think of any cultures that don‘t agree on what ‘1’ means. 0 works ok for a place holder. 2056 means that there are no hundreds. 0 works ok to say that there are none of something. I have zero cats means that I don‘t have any cats. These are not philosophical ideas. Nothing deep here, whatever nothing means. I think one should start with the everyday meanings of numbers and then go to more abstract concepts. When enough is known about numbers, abstraction is natural. There is an opinion that if a body of material is presented logically and consistently, a student should be able to understand it. The New Math was a result of that opinion. It is not that the New Math is inherently a bad idea but it requires people who can teach it, which requires that the teachers know it. A miss is as good as a mile. Zero, one and two are concrete but at a certain point we come to ‘many’. I take eight pills from one bottle. When I pour them into my hand, I look for four and four or five and three, not the whole eight. I can‘t glace at a class and tell if there are 17 or 18 students present; I am in the realm of many and I have to count. I was giving a test to over a hundred students and to pass the time I counted the number of students present. I counted three times and got three different answers. At the end of the test I knew there were more than a hundred. The number of students had become an abstract concept. When I grade tests I first alphabetize them, then count the number of students for each letter and then add. I am fairly confident of this number; but not absolutely confident. My point is that in The Real World most numbers are to some degree abstract. I remark that a number can be partially abstract. I can usually tell if a number is big or small and this description is concrete. When it is announced that the attendance at the Super Bowl is 104,368, this number is not correct. If it were possible to count the house, it would probably be more than 100,000. The number on a test is an example of a number that is mistakenly thought to be exact by both teacher and student. When I was a new teacher I would give points to each problem on a test and add them up for a final score. Then I would decide where to draw the A, B, C, D and F lines. I would end up with a 79 C and an 80 B. This is ridiculous. A C is a ridiculous grade. There is the C where the student works 7 out of 10 problems correctly and doesn‘t answer the other three. And there is the ‘Partial credit C’ where the student can‘t answer any questions but somehow scrapes up enough points for a C. It is more reasonable to give A or B to the student that gets it, D and F to the student who doesn‘t; no Cs. I finally stopped giving numerical grades on tests. I could tell a B paper from a C paper and so forth, so I just put the letter down. In upper division courses, which are usually of a reasonable size, I would only accept correct homework and the students could redo it until they got an A. Some of my colleagues thought I should take the number of tries into account when grading. They were wrong. The whole point of the course was to get the student to learn and giving them an A when they achieved that end makes sense to me. Prisons are punitive, schools are supposed to be educational in a classical sense. So, what‘s the point of the Sam and Joe story? Mathematics presents itself in a pristine form and the student asks, ‘Where does this stuff come from? Why do we study it?’ I tried reading the Richie/Kerrigan book, The C-Language, to teach myself, with a little help from my friends, how to program in the C-Language. It‘s quite a thin book but I‘m sure all the information needed is in it. However I was unable to extract it and got stuck on ‘pointers’. I would say what a pointer is but I am still not sure. I realized that I didn‘t know why they were introduced; I didn‘t know what problem pointers were supposed to address; evidently pointers made something that was hard to do with earlier languages, easier to do but I had no idea what that something was; and The C-Language wasn‘t telling. Another example is the epsilon-delta process. Mathematics got along without it just fine until well into the nineteenth century. What undergraduate calculus student doubts that sin x is continuous? That the limit of x2 as x approaches 2, is 4? What did the epsilon-delta process make easier? Where was it really needed? Where was it used in a practical way? The thing about the epsilon-delta process is that when you see the point of it, the concept is obvious. I tried to present the epsilon-delta process in a more reasonable way in The Calculus: An Opinion. Actually, I don ‘t really talk about the process but try to present 18th mathematics in a way that makes the epsilon-delta process a natural thing to introduce. The point of my remarks about Sam and Joe is that mathematics should be talked about from the very beginning of a student‘s introduction to mathematics. What gap in the life of early humans was filled by adding numbers to their daily experience? Sam didn‘t tell Joe, ‘Remember these symbols because sometime you might need to use them. If I have kids someday, I might want to tell you how many.’ The larger the context a new idea can be placed in, the easier it is to understand the new idea. The context of mathematics can be started in the womb. Well maybe not in the womb, but shortly after emergence from said womb. I think that adults thinking that they know the best way to grow a context is incorrect, dare I say it, stupid. Numbers arise naturally. If they didn‘t, why study them? You don‘t start with Latin when you are teaching English at the elementary level. When a language context is big enough, you fit Latin in.

Teach How To Learn

Teach How To Learn

I was troubled by what I taught in elementary courses. It seemed like I was teaching stuff and I didn’t know why I was teaching it.
A hundred years ago students could put the mathematics they learned to use right away. Now that is not so.
It used to be that the student was looking for a job for life. It took generations to build a cathedral. Now technology moves so quickly that a student can look forward to many different jobs and each one has to be learned.
Consider a radio repairman. I doubt if there are any now, they had to learn new skills. From tube radios to transistor radios and TVs to VCRs to DVDs to Blueray. Every few years there is a new technology to learn.
The appropriate skill to teach is not how to, say, differentiate but how to learn how to differentiate. This should be started in pre-school.
And it doesn’t make any difference what the skill is. They should start learning something that appeals to them. The goal is for the student to have confidence in their ability to learn, in their ability to face an unknown skill without fear.
Memorizing is not learning. The confusion between the two is probably the biggest obstacle to learning. It turns out that it is easier to test memorization than learning and since the student has been memorizing from the beginning it is a preferred way to take classes. In an unholy alliance the students and the teachers take the easy way out.
I would have my students take an oral in my office where they would have to prove a significant theorem, say, that the sum of the angles in a triangle is less than or equal to 180 degrees in hyperbolic geometry. It was too long to memorize, they had to see the flow of the theorem and see how the parts fit together naturally. And I gave them multiple tries.
I found that students didn’t know if they knew something or not. A student came in for help on implicit differentiation. It turned out that he didn’t understand what the derivative of a function was. And then it turned out that he didn’t understand what a function was. He wasn’t trying to con me; he thought he knew these things and was surprised when he didn’t.
I have many stories like this. A student sat through an entire semester of business calculus and through the semester he thought it was beginning statistics. And he thought he was doing ok in the calculus although his highest grade had been a 25. A student thought he understood max-min problems and didn’t.
Why is it that students believe they know things when they don’t?
For one thing, when you learn something, barring serious head injury, you don’t forget it. You forget things you’ve memorized. You don’t memorize how to ride a bike, you learn how to ride a bike”¦and you never forget.
Some people learn things easier than others. I was with a guy as he drove right to the Harley shop in Kansas City, Kansas and he had been there once three years previous. Arlo Guthrie never learned how to read music. When he started piano lessons, his teacher always played the piece he was supposed to learn from the sheet music. Once he heard the tune, he didn’t need the sheet music and never looked at it. Some people don’t forget tunes they’ve learned.
I was playing the guitar chords to “Don’t Think Twice, It’s Alright” while a friend played lead. At the point where it goes, “When the rooster crows at the break of dawn” he stopped us.
I had played an A instead of an A minor or something like that. He said, as if he was pointing out that the sky was blue, “If you play the A there it ruins the tune.”
I realized that he was hearing something that I wasn’t. I had memorized the chords, he understood the sequence of chords. He would always play the right chord.
I have heard that the piano prodigy and the piano non-prodigy have about the same proficiency when they are in their twenties so if a person isn’t a prodigy they can still learn to play the piano.
So how do you teach a student to know what learning is?
First they have to understand what “knowing” is. They have to have some idea of when they have learned something.
I think you have to start when they are young. Really young kids seem to be into acquiring skills and knowledge.
I t

The wrong stuff is being taught; not even in the ballpark.

What to teach? What to teach?

Every time I think about how to teach undergraduate mathematics, say, algebra through calculus, I can think of ways to teach the standard stuff but I always get hung up on the question: Why am I teaching this shit?
The first topic that should be taught in school is “learning technique”. If a student knows how to learn, the life of the teacher is much more interesting. There is a big difference between talking to someone who knows how to learn and someone who doesn’t. It’s more interesting for the student too.
So the topic that should begin in pre-school is learning how to learn. It doesn’t make any difference what they start learning because the point of the exercise is the act, the process, of learning. The child should start learning something that that isn’t unpleasant for them.
When the child is young they will probably want to learn something that most adults know how to do and are quite able to show the child how to learn it. The first things I learned were taught to me by adults and their teaching technique was quite satisfactory.
I went to Junior High in Cheyenne, Wyoming and we lived on Warren Air Force Base. As soon as we moved in my dad gave a plane geometry book, an algebra book and a college algebra book; I was instructed to learn them.
This was OK with me. The winters in Cheyenne were long and cold and learning mathematics while listening to radio plays was as good a way to spend winter evenings as any. Well, there were also the Friday night fights to go along with factoring polynomials.
But the mathematics that I learned wasn’t as important as the fact that I learned how to learn.
From that point on, school was no problem. I didn’t necessarily want to learn everything but if I wanted to, I did.
People who know how to learn do it at different rates. Actually people who don’t know how to learn do whatever it is they do at different rates.
As I think back on my teaching career, the time restraint always bothered me but I didn’t stop to examine time more closely.
The time that it takes to learn something is a statistical distribution. There are some students that can learn the next thing in an instant, there are others that take a long time before they are ready to learn something else.
Because most of the students don’t learn the mathematics, we have tests to evaluate”¦ It is not really clear what is being evaluated. My class grades gave a bi-modal distribution, those that learned how to take tests and those that didn’t. Learning the mathematics was a sufficient but not a necessary condition to pass tests.
The more I think about giving a 50 minute test every 2 weeks the more bizarre it seems.
In my upper division courses the tests were all the same: “Write down what you know”.
If the class wasn’t too large, about 15, plus or minus, I would give an oral final. They would have to give an hour talk on, say, the proof of the Heine-Borel Theorem without notes. They would get as many tries as they wanted until the day before grades had to be in.
But in larger calculus classes there isn’t enough time.
I didn’t give numerical grades, just letter grades. I couldn’t tell the difference between a 71 and a 72. It’s pointless to make such distinctions. A teacher should be able to tell the difference between and A paper and a B paper.
It’s not that there weren’t students who could do well on tests without learning the material. A student who does well on tests but doesn’t know any mathematics, has learned a skill. The student has been given a problem to solve, get through school with good grades. The problem is not “to learn mathematics”.
I think there is a better appreciation of what learning how to make a guitar or to use a lathe (no, not the computer lathes), how to turn a cartwheel. There is an understanding that something more than memorizing instructions is requires. Learning how to play a guitar is more than putting your fingers on the right frets and hitting the right strings. I have heard a man play the piano and hit all the right keys at the right time and it wasn’t music. He hadn’t learned how to play the piano.
Perhaps it is music they haven’t learned. Memorizing which notes to play doesn’t make music and memorizing how to work selected max-min problems doesn’t make mathematics.
People say that music and mathematics are related and then start talking about octaves and fifths and Pythagoras and group theory. They miss the point.
You learn how to groove on both.
I think one problem with beginning mathematics is the teachers. I think most teachers like to read and their students see this. I think that a lot of elementary school teachers do not like mathematics and that shows.
I knew a grammar school teacher who told her students that she didn’t like mathematics either but that it is something you have to learn. And she really felt that they should. As distasteful as it might be, she thought that balancing a checkbook is important.
The conversation took place some years ago.
It is my opinion that once you know how to learn, you can learn anything. Well, anything within reason.
One of the problems is the amount of material that is shoe -horned into beginning courses. In the Calculus I that I took we spent three weeks on conic sections. In the last Calculus I syllabus I taught it was about three days. What I learned about conic sections has stayed with me for over fifty years. That’s what learning something does for you.
I liken the Calculus syllabus to driving down the freeway at night at 90 mph with your dims on. You read the green signs but you miss the off-ramp. You touch on everything and don’t get to really teach anything”; no time.
The “include everything” mentality leads to books that are too big, too heavy, too poorly written.
But if the student knows how to learn, all these problems dissolve in the mist. You teach the important basics, like what a function is and what its graph is, the pros and cons of continuity, what the derivative actually is and what an integral actually is.
A student who knows how to learn can learn the technique of differentiation in a few days. I had a friend who claimed that he could teach a parrot to take derivatives. After all, there are five basic functions and five ways to combine them; you have to know the derivatives of five functions and how the derivative deals with the five ways to combine them. End of story.
Doesn’t anyone ever wonder why more than a day is spent on the derivative of a product? Could it be because the wrong things are taught and that memorization is called learning?
We spend all this time thinking about ways to teach stuff that the students should be capable of learning on their own. The fact that they are not capable is the fault of education, not the student. If I were a conspiracy theorist, I might see a conspiracy to keep knowledge from the populace.
One might have thought that as the country progressed from being a manual labor economy to a more mechanized economy, the citizens would be educated to keep up. But there is a problem with people coming to this country, legally or illegally, and are taking manual labor jobs. Why haven’t we left manual labor jobs behind for people in countries on their way up?

How Do Different Species Arise?

An Idea on Evolution and Species Differentiation


I will start with the primeval ocean full of stuff and at some moment a piece of replicating DNA appears.
The first question is: Why did the stuff in the ocean combine to make something that was self replicating? I don’t know and when looking for a book to read I never choose biochemistry. I take the fact that we are here as prima facie evidence that it did happen.
Here I am making the assumption that life started as a very simple organism. This is contrary to the creationist point of view that man was created fully made some thousands of years ago.
I don’t see why more than one piece of DNA would appear but I see no reason that they couldn’t appear in a volume of ocean or even throughout the ocean. This mass appearance seems very unlikely to me.
I will suppose that just one appears. It seems to me that there would be just one primordial piece of DNA but that is more for aesthetic reasons than logical reasons.
It doesn’t really make any difference. For my argument to hold, the pieces just had to be small. After a fairly “short time” there would be so many pieces of DNA that the original number of pieces would be seen as a point.
The piece of DNA had no predators because it hadn’t previously existed to develop any. But it could have happened that there were killer molecules. If a killer molecule accidently bumped into a piece of DNA they might have combined in such a way as to ruin the DNA’s reproductive ability. I suppose a piece of DNA could have been destroyed if it was hit by lightening. I suppose there were a lot of ways a piece of DNA could have met an unreproductive end.
The presence of life as we know it implies that the DNA population grows faster than the DNA is annihilated so after a certain amount of time, the doubling time, the DNA population effectively doubles.
Since the population is pretty homogeneous in the beginning I’m going to model the early growth process as starting with one piece of DNA and that an individual doubles after a length, T, of time; T is the doubling time. Early on T is fairly constant.
I assume that the time estimates of science are in the ballpark so I’ve got three billion years to fool around with.
How many times will the population double in a million years? If it takes a thousand years for a population to double, so for example it takes a thousand years for the first piece to double, then in a million years there will be 21000 individuals. This is on the order of 10300 individuals. If a grain of sand was 1/64 inch on a side, 1060 of them would fill the universe.
Clearly the population of pieces of DNA doesn’t double a thousand times in a million years but after a million years there must have been a lot of DNA floating around.
Now I come to my point which depends only on the fact that there is a growing population of DNA pieces and that, since the pieces are small, there are significant mutations that are replicated.
At this early time a mutation of a piece of DNA would involve a significant part of the piece and would thus change it quite a bit. One gene changing out of thousands is different than one gene changing out of a few.
My point is that the differentiation of species starts here. The mutations initiate changes that lead to different species.
When the organisms get larger, mutation is followed by natural selection. The mutations do not give rise to a new species, they change the phenotype; the feathers of Darwin’s finches get darker but they are still finches.
Different species can trace their ancestry back to one of the early mutations of DNA and hence back to the primordial piece of DNA.

I have always wondered about the origin of life. I had this picture of DNA appearing all over the primordial ocean and this seemed odd. But then I thought that, because of exponential growth and in particular doubling, you only need to start with one self-replicating piece of DNA.
I have also always been bothered by the appearance of different species. I can’t see how that could happen. Breeders can develop very different dogs but haven’t come up with a new species.
If you start with an ape, how does it mutate into a human? Does a whole bunch of apes become a whole bunch of humans? Does one ape change a little bit and some how start the path to humanity?
So apes with bigger heads are chosen for? So what?
Why can’t small heads think? Computer chips get smaller every day and “think” better. Not that I think you can model the human brain as a computer.
If I assume that species were differentiated soon after DNA appeared these problems go away.
It seems to me that a mutation of an early piece of DNA, a very small piece of DNA, could make a big difference in the end result of that piece of DNA. The end result is sensitive to initial conditions.
I don’t think that the random mutations to apes change the outcome of the succeeding apes much. The end result is a change in phenotype but not a change in species. The more complex the organism, the less sensitive it is to initial conditions. Mutation gives rise to a change in phenotype but not species.
Early DNA changes from chaotic evolution to non-chaotic as the DNA becomes more complex, like being an ape.