A difficult problem

Prove or disprove the existence of God.

Answering this question requires a definition of God, which we don't have of course. We could propose a temporarily definition like 'God is the entity that created life on planet Earth, a modest definition because it leaves the question of who created the universe unanswered.

Although still debated, Gödel's incompleteness theorems imply that we won't be able to build computers with a conscious mind capable of creating new mathematics or writing a program that solves all open problems in mathematics. Besides cloning and modifying what we have ( DNA ) we won't be able to create a 'machine' smarter than ourselves.

Our own existence is a paradox we don't understand. History proves that in cases like that humans are creative in inventing a God capable of fixing any problem. Despite centuries of scientific advances we still need a God to explain our existence.

Hypercomputation

Only recently lightspeed as the ultimate limit of speed got challenged. It seems they have been challenging the Turing Machine for a while too.

If you start with studying mathematics you are only three or four centuries behind on contemporary mathematics. That's quite a lot of catching up to do. - Some fields started their development in the previous century though. Like mathematical logic, a field I have been studying this year, and have written about so now and then in this blog. Part of mathematical logic is the theory of computation which showed us what can be computed and what can't. That what can be computed is what can be computed on a Turing Machine, period. - That idea is challenged however in the theory of hypercomputation. A new field in mathematics which is trying to go beyond the limits of the Turing Machine.

A book with an overview of the theory is the following.

Primitive recursive function

Normally you calculate n factorial with Factorial[n] or short n!. Mathematica handles the details of the function for you and prints the result.

In the course M381 you have to prove that functions like factorial are a primitive recursive function. This basically means that the function can be defined only in terms of itself, add one, or set to zero. A primitive recursive definition of factorial would look as follows in Mathematica.

suc[n1_] := n1 + 1
add[n1_, 0] := n1
add[n1_, n2_] := suc[add[n1, n2 - 1]]
mul[n1_, 0] := 0
mul[n1_, n2_] := add[mul[n1, n2 - 1], n1]
fac[0] := suc[0]
fac[n_] := mul[n, fac[n - 1]]

As you can see no other Mathematica functions than "+ 1" and "= 0" are used. The functions suc, add, mul, fac are defined for the first time.

For example:
In[67]:= Factorial[6]
fac[6]

Out[67]= 720

Out[68]= 720
.

Comment on "From analog to brain computing ".

Mathematicians have a tendency to regard texts which are not written using 'protocol' as irrelevant. Long ago I wrote a note to a mathematician and his reply was that I should formulate my thoughts in 'standard mathematics'. I did my very best 'to make myself clear'. It was not enough. The thing is mathematicians lose their authority when they leave familiar territory. Well, at least I received a reply. ( Although that was all he did. And I haven't given up on the problem I was working on... )

The blog received a comment, containing what I, for the moment, call 'out-of-the-box' thinking. Non 'standard mathematics' at least. I have moved the comment to this post in an attempt to share this with as much as possible readers. I will reply. But later, I have to let it work on me first.

Regarding our active internal analog math...

I'm a civil/environmental engineer by education but I've been working off and on on a theory which takes the tact that all abstract math symbols and expressions are secondary and arise from a handful of internal analog "math" artifacts and processes. This may not be a very polite thing to say to a mathematician, but I am wondering if you have impressions along the same line?

It turns out that we all get energy to think and do math and other things from the respiration reaction (organics + oxygen -> water + carbon dioxide +energy). And basically, what that means, if you remember your biology or organic chemistry, is, body-wide, within our cells is a ~steady creative flow of about 10^20 water molecules per second -- coming from the 160 kg of O2 we each respire each year. Generally, each water molecule is sort of tetrahedral in shape with two positive and two negative vertices and so, it turns out that there are at least six ways each water molecule can orient within an enfolding field when it first comes into being at a respiration site. That also means that a chain of n-molecules can form in 6^n different ways. Thus a sequence of 12 molecules could form in 6^12, or about 2 billion different ways. A chain of eighteen molecules could associate with 6^18 or 10^14 different impressions. Now, in this analog math theory, I am assuming that repeating vibrations in the environment ought to result in formation of similar stacks and chains of structurally coded water molecules being formed. This gets us a rather crude image of the vibrations of our internal and external environment forming an internal echo or representation within this active internal analog "math", or "language".

I say it's active because the 6^n stacks of water molecules are really also structurally coded hydrogen-bonding packets and such things, when they unfurl, are connected with and influential in protein-formation and protein-folding, which is to say, memory formation and muscle movement, which is to say, in our case, ALL human expression, perhaps beginning with our nearly universal actions and impressions of counting each of our ten fingers and ten toes, and the like.

Bizarre stuff, huh? Lots of little internal Turin devices writing out structural coded signals.


I'm wondering if mathematicians are taught this type of internal analog math as the basis of the abstract math symbols and expressions, or if they are given different associations or impressions, perhaps leaving it that there is just an uncanny (and unknown) relationship between much or all of nature and math?

Also, I vaguely see the similarity between 2^n binary or boolean math and the 6^n "multiple-state structural coding" that I've made up or stumbled onto. I expect the trend continues with starting with other polyhedra which have limited orientations "within enfolding fields" -- when a containing structure is added. My general hunch is the initial condition IS actually significant for us and we can immediately get to multiple states (relevant to ~quantum mechanics/quantum gravity) by starting with tetrahedron and adding the enfolding cube container, rather than the way it's done presently of beginning with the xyz-cubic framework and adding variants.

Initial conditions do matter in mathematics, don't they?

Best regards,
Ralph Frost

@frostscientific
http://magtet.com/images/phpshow.php

Thanks you, Ralph Frost.

Who am I ?

I am the square root of -1. Who am i?

and of course from GEB:

This sentence contains ten words, eighteen syllables, and sixty-four letters.

From Mathworld - Self-Recursion

From analog to brain computing

Before digital computing took over completely, analog computing was dominant for a short while. An analog computer is based on the creation of a model which represents the problem to be solved. But mathematical models of problems can be created of ( almost ) any problem and these models can be implemented on a digital computer. A digital computer is nothing more than a convenient, fast, Turing Machine or equivalent thereof, i.e. the URM or Abacus. And from Mathematical Logic ( Goedel ) we know that these systems have its limitations. It is theoretically impossible to create a program that solves all mathematical problems. - But physicists and biologists say ( and why should we disagree? ) that we -are- computer ( brain ) controlled machines.

Is that a paradox? Humans can do more than computers, we can solve mathematical problems, in fact we -created- the concept of a 'Turing Machine'. This leads us to Roger Penrose. In The Emperor's New Mind, 1999 he claims that artificial intelligence in computers is impossible. He argued that the human brain must exploit a type of physics that he described as 'non-computable'. By this he means beyond algorithmic computing, and thus digital computing.

A picture that keeps fascinating me is that of a predator bird flying high over its prey before, at a carefully -chosen- moment, it makes the dive and following kill. And this is all done with a tiny bird brain. The best comparable thing made by humans thus far is the drone. A huge flying case loaded with bombs operated by a battery of digital computers assisted by human -computers-. Although humans have created a model of a flying bird, it is operated by a human computer on the ground.



Analog computers were special purpose computers, designed to solve one specific problem. A predator bird will never be able to learn new behavior, it cannot be trained to live with chickens. Not immediately anyaway, if ´evolution´ made the bird.

Let me summarize before this turns into a rant.
- There are other models of computing than the Turing machine, i.e. analog computing, brain computing.
- Digital computing is superior over analog computing, brain computing is superior over digital computing.
- Analog and digital computing are human creations we fully understand.
- We don't understand brain computing (yet?).
- Mathematical logic and computability theory study algorithmic ( digital ) computing.

Goedels theorems are somewhat like Russell's paradox in set theory. Goedel's incompleteness theorems are statements about logic and number theory deduced in and with the rules of logic.

The Limits of Mathematics ( or: a Lisp interpreter in Mathematica )

( ... ) mathematics because it is an extremely difficult road to traverse. The terrain is extremely demanding. The amount of work and concentration required to build the foundation necessary to continue extending the framework is immense. ( ... ) - David Andrews

Mathematics, as if you have never seen a skyscraper and are traversing the streets of Manhattan. With that mindset, you can only think that people -walk- to the 60th floor... Anyway, feeling overwhelmed by the sheer size and complexity of maths is not going to help. Only people willing to teach, without ulterior selfish motives, can help. One can write a book about mathematics to impress peers, as a way to meet publication quotas or to -teach-. Like the book The Limits of Mathematics does for example. It is a clear taste of the best mathematics has to offer, an invitation to go on to the next level.

The table of contents says it all:
- Randomness in arithmetic and the decline and fall of reductionism in pure mathematics
- Elegant LISP programs
- An invitation to algorithmic information theory
- The limits of mathematics
- Appendix. LISP interpreter in Mathematica

The appendix contains the source code of a Lisp interpreter coded in Mathematica. I love that. But the book starts with a clear description of the massive changes taking place in ( the thinking about ) mathematics during the first half of the twentieth century, from Hilbert to Turing.

Goedel, Escher, Bach - Lecture 6

In the first six minutes or so Curry gives a fairly good summary of Goedel's Theorem. Unfortunately this is the summary of the previous lecture which was not recorded. It seems nothing is free, not even free video lectures because it turns out the best ( not implying the rest is good ) is missing.

After rushing through formal stuff he wastes five minutes about a three-layer stupid joke about a book he had not read.

I quick-scanned through the rest of the video. Not worth watching, really. Too bad. I looked forward to this.

The take home message of the course. All provable things are true but not necessarily al true things are provable.
Justin Curry

Now that I am mostly through all M381 stuff I am glad it included mathematical logic. I would -not- have done it as a stand-alone course. Logic is hard in the beginning, like most new subjects. It needs time to work on you. I will get back to this in the next M381 post.

Previous posts on the series:
- Lecture 1
- Lecture 2
- Lecture 3
- Lecture 4
- Lecture 5

About M381 (1)

Regular readers of this blog know that I have a sort-of rage-button like the Hulk: it is called MathCad. Thank God, I have the anti-dote almost always open and ready: Mathematica. I am not going to repeat why MathCad is a danger to your mental health, but I have to press the MathCad button at least once.

In almost all mathematics courses you can do at the Open University there is software involved. They either deliver a standard package, or ship custom software especially developed for the course ( i.e. MT365 ). The house-package of the Mathematics Department of the Open University is MathCad, version 2001. I have argued that MathCad alone is a reason -not- to choose for the Open University. What a disgrace...

( Calming down. )

They do however recognize that software, computers, tools are relevant in mathematics. Especially in Number Theory computers are used in active research. Another area where they use software in active research is: mathematical logic. Stronger: research in Number Theory is impossible without computers.

These facts are not even mentioned in M381. There are many open source tools available for Number Theory, even more for Mathematical Logic. Not a word about it in M381. One, if not -the- reason is the fact that course development in the Open University is done in a project organization. A project is created with the objective to create course X which will then be used for the next 10 or so years. It is exactly the opposite of what one would expect of a university education. It is not reasonable to expect the Open University to be at the forefront of mathematical research. Simply because other universities in the UK have that role. But it is reasonable to expect more than a static expose of 19th century Gauss number theory and early 20th century logic from Church, Turing and Goedel. In fact, the field is presented as abstract and of theoretical importance only. But Number Theory and Mathematical Logic are extremely relevant and applicable in many industries! But I did not learn that from the course and that is sad.

It took me a lot of work but I found some relevant learning tools in the fields of number theory and mathematical logic. More about those later in this blog.

(*) - I may have misunderstood the concept of 'University' in the UK. I think many universities in the UK are what we call in the Netherlands 'schools'. They deliver professionals with a degree in all fields through excellent education but they don't do research and so on. They don't add to the body of knowledge. They process and transfer knowledge. That description fits the Open University as well. - A marketing issue is that students like to have a 'university' education. And marketing people love empty heads boxes, they have a fancy word for it too: the 'packaging'. Does that make sense?

Goedel, Escher, Bach - Lecture 5

A few months ago I started to watch the MIT video lecture series on Goedel, Escher, Bach. Due to time constraints I wasn't able to complete watching the entire series. Today I continued with watching lecture 5. I have learned quite a lot on the subject through M381 and I am about to really 'get it' as far as the Goedel Incompleteness Theorems are concerned. My first reading of GEB took months and now parts of the book begin to look simple. If you don't know what I mean browse through a mathematics book you thought was hard, a few years ago. It often seems if there is 'nothing in the book'. The odd thing with Goedel ( and with all mathematics, I suppose ) is that in your mind you think you can explain it to a laymen in one or two sentences. ( It is -that- simple, I am afraid. ) The power of mathematics is that it can capture an entire knowledge tree in a single word. That word remains meaningless without understanding of all the words in the knowledge tree.

A bit about lecture 5.

Dress shows arrogance

I wonder if Justin Curry would go to a job interview in that Club Med outfit. Students are paying customers ( and a pool of cheap labor for lucrative research deals the university makes ) deserving respect from teaching staff.

Formal number theory ( as in M381 )

Justin talks about Typographic Number Theory, ( formal number theory in M381 ). For example $$\forall x ( \neg x = \mathbf{0} ( \exists y x = y') )$$ can be interpreted as

"Every x that is not equal to 0 is the successor of some y."

Leibniz was the first to propose a formal language for number theory. He asked whether it was true that an algorithm could decide if a statement in number theory was true. - Although in M381 this question is answered negatively that does not mean computers can not play a role in proving mathematical propositions. There is an abundance of ( open source ) software for proving theorems.

( Not in video: ) Isabelle a formal proof theory assistant has been used in testing an operating system kernel written in C and assembler. It not only verified that the spec was implemented correctly but it also discovered hundreds (...) of programming and design (...) errors which were not found by traditional testing methods.

Previous posts on the series:
- Lecture 1
- Lecture 2
- Lecture 3
- Lecture 4

Exercise ( logic ).

Knowledge of mathematics is not required to solve the following exercise, but it will sure help ;-)

Dr. Who asked you for a ride in the Tardis. Naturally, you couldn't decline, it might be a matter of national, if not global, importance. The Tardis landed on the Planet of Truth which is is inhabited by people who always tell the truth. A minority however decided to lie, always. It is Doctor Who's mission to seek and destroy all liars. First you must get to the Capital of the Planet. Two roads fork out. Should you go left, or right? An inhabitant approaches, greets you and gives you the privilege of asking him one Yes/No question.

Ask him where the capital is, left or right on the fork, but beware he might be a liar!

Credit follows with the answer.

Take the challenge, test your ability to think logically.

What is the dimension of color ?

Think of color, pitch, loudness, heaviness, and hotness. Each is the topic of a branch of physics.

Benoit Mandelbrot

The prototypical fractal

( I'll continue with my 'watch notes' of the GEB series later this week. But I'll stay on topic with this post. )

Imagine a recursive process which goes to a certain depth until it stops. This 'certain depth' is a natural number which can be assigned a color. This is basically how fractals like the one above are built, pixel by pixel.

Programming 3D graphics can get quite realistic as we all know from watching movies or playing computer games. What is this reality? When does a picture look real to you? It is what made Rembrandt famous, I suppose. Control over light and thus color and shading.

What I am trying to say is that one number is not enough to encode a color. This is rather counter-intuitive, I know, but only if you assume the number of colors is finite or countable infinite. How many colors are there? Finitely many? Countable infinite many? Or not countable infinite? I don't know.

Assume a scene with an object with a certain color, say lime green. This can be coded with one string, the RGB color method uses 32 CD 32 ( Hex ) for lime green. But then you have an object that looks exactly the same everywhere. Because there is no light in the scene. By adding light we have to add the exact location of the light source in the scene. Shadows must be calculated. And light will be reflected. Each pixel will have a reflection vector which has effect on the color. Does this add to the dimension of the color? What if there are other objects in the scene? They partially reflect light and thus become a light source as well.

Do you get the idea? Then what is the dimension of ( the vector needed to encode a ) color?

More:
- The Dimensions of Colour

Goedel, Escher, Bach - Lecture 4(1)

My brain ran those neural network algorithms.

Justin Curry

The reading assignment was chapter 6 'The location of meaning'. It is basically about coding and decoding. Of course Hofstadter mentioned the Rosetta Stone in this context, the key to ancient Egypt. It contained a parallel text in three languages and was deciphered in 1821 by Champollion.

A recent example in the context of chapter 6 is space archeology. Archeologists and Egyptologists were able to interpret satellite pictures of Egypt which lead to the sensational discovery of new pyramids.

Let's go to the lecture. ( This is a 1h46m lecture and will be discussed in two posts. )

Curry talked about Goedel numbering again, a method Goedel used to code strings in formal number theory to numbers. What Curry said about coding a string in formal number theory, playing with it and then code it back is only true in theory. Simply because Goedel numbers become -extremely large-. Think of numbers built from pages full of digits. ( Would Curry ever have calculated a Goedel number? This reminds me of a DBA course I attended once. The trainer talked about all the beautiful properties of the then new RMAN from Oracle as if backups could be recovered in an instant. It turned out that he never worked in the trenches of 7 x 24 administration of large databases. )

Dialog "Contracrostipunctus" on page 75 of GEB is discussed. How this dialog has meaning on several levels. The dialog refers to itself that it contains a hidden message. The concept of 'Self' is introduced here.

Starting with what does "Snow is white" mean? he builds an argument that there is an isomorphism between electrical activity in the brain and the interpretation of symbols. ( Thought reading might be possible after all, one day. Isn't it true that man can create everything he is able to envision? )

Adam and Eve

There are at least two phases in the proces of assigning meaning two a string. The first is parsing the string, the second is the interpretation of the parsed words. Interpretation depends on the context of the interpreter.

Message in a bottle

He introduces the concept of information. For example how physicists reduce complex physical behaviour to a small sequence of symbols. Like for example how a pendulum works.

Pendulum ?

Goedel, Escher, Bach - Lecture 3

A guy named Euclid.

Justin Curry

Curry briefly explaines the concepts:
- consistency
- completeness
- and geometry.

A consistent system leads to conclusions that are not contradictory in any sense. A statement is either true or false, and never both true and false.
A system is complete if everything that is true in the context of that system can be derived from the axioms.
Regarding geometry he mentioned that there are Euclidean non-Euclidean geometries.

Then he attempts to explain Goedel's Incompleteness Theorems.
1. Any system as powerful as number theory which can prove its own consistency is necessarily inconsistent.
2. Any system as powerful as number theory is necessarily incomplete.
He explains that Goedel managed to transform the idea of provability to a property of numbers by introducing his Goedel numbers.
He says that students should now have a notion of the Goedel theorems and promises that this is just a first glance at Goedel's theorem. ( Not sure if he meant he would come back at Goedel in this lecture series. )

Trying to explain Goedel

He then talks about Euclid and his postulates.
(1) Any straight line segment can be drawn joining any two points.
(2) Any straight line segment can be extended indefinitely in a straight line
(3) Given any straight line, a circle can be drawn having the segment as radius and the
(4) All right angles are congruent.
===
(5) If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles

He explains that the 5th postulate is consistent in Euclidean Geometry but not in spherical and hyperbolic geometry.

Hofstadter Dialog - Little Harmonic Labyrinth is removed from the video due to copyright concerns. It is part of what makes GEB such a difficult book. Here is part of it.

The Tortoise and Achilles are spending a day at Coney Island. After buying a couple of cotton candies, they decide to take a ride on the Ferris wheel.
Tortoise: This is my favorite ride. One seems to move so far, and yet in
reality one gets nowhere.
Achilles: I can see why it would appeal to you. Are you all strapped in?
Tortoise: Yes, I think I've got this buckle done. Well, here we go. Whee!
Achilles: You certainly are exuberant today.
Tortoise: I have good reason to be. My aunt, who is a fortune-teller, told me that a stroke of Good Fortune would befall me today. So I am tingling with anticipation.
Achilles: Don't tell me you believe in fortune-telling!
Tortoise: No . . . but they say it works even if you don't believe in it.

About 1% of the Little Harmonic Labyrinth dialog.

He explains the cardinal arithmetic, the arithmetic of infinities.
An interesting definition of infinity is that a set can be mapped to a subset of itself. I.e. the natural numbers can be bijectively mapped to the even numbers. The points on the real line can be bijectively mapped to the points on the line between 0 and 1.

Non-Euclidean geometries

Goedel, Escher, Bach - Lecture 2

It will get a little bit mathy, but that's ok.

Curran Kelleher

Lecture 2 is given by Curran Kelleher. This lecture is all about recursion and ends with a nice explanation of the Mandelbrot set.

He starts out with the traditional examples factorial:

factorial[0]:=1;
factorial[n_]:=factorial[n-1]*n; 

and fibonacci sequence:

fib[1]:=1;
fib[2]:=1;
fib[n_]:=fib[n-1]+fib[n-2];

Kelleher's factorial program

Kelleher's hand-out ( pdf ) contains examples of Java code for drawings of the Koch curve and Sierpinski triangle. Although I fast-forwarded through this part of the lecture, it may be very interesting for non-programmers.

Explaining the Fern algorithm

Complex number implemented as a class in Groovy 

At around 1:00 he starts with the topic of the Mandelbrot set. Starting with f(z) = z^2 + c he manages to give a nice explanation of the Mandelbrot set. The color of a point in the Mandelbrot set is based on the number of iterations it took f to 'escape' a circle. Since this is done on a pixel by pixel basis and one pixel may generate not one but several iterations this explains the long time it takes to generate a Mandelbrot set.

P.S.
GEB does not seem 'outdated' at all although it was written in the late seventies. A time when there were no mobile phones, no PCs, let alone laptops and the internet was still in its toddler phase.

Goedel, Escher, Bach - Lecture 1

'Understanding Goedel' is one of the major goals I set for myself.

This final unit brings together all the ideas introduced in the course. These ideas constitute the technical machinery that enables us to prove some very important theorems which answer what we have called Leibniz's and Hilbert's Questions. These theorems, Goedel's Incompleteness Theorems, are among the most profound intellectual discoveries of the the twentieth century. Thus you should not be surprised if you find this unit hard going in places.

M381 - Unit 8.

In Goedel, Escher, Bach (GEB) Hofstadter asks the question: what happens when 'things' start referencing themselves? ( Like people do who are in essence not more than a set of linked molecules. )

At last I took the time to watch video 1 of the GEB series.

Justin Curry

The teacher is Justin Curry. He started by telling that most undergraduates don't get through GEB in less than 13 weeks and that it took him seven years to get through the book. I am not sure but I think my first attempt in reading GEB was in 2007 or 2008. It took me almost six months to get through it. Which I thought was really bad. When I finished the book and still didn't understand what he was talking about I started to seriously doubt my learning abilities. I have to admit that I still don't get it but I made progress. And I am getting closer, thanks to M381 Mathematical Logic ( read: Nigel Cutland ).

Anyway, to the point: the lecture.

He starts with the concept of isomorphism. In GEB, Hofstadter explains isomorphism as a map between structures that maps parts with similar purpose to similar purpose ( my words ). This is different than the mathematical definition which states that an isomorphic map is both surjective and injective. Hofstadters definition can be understood immediately, whereas the mathematical definition needs understanding of layer upon layer upon layer. Since what is a map in mathematical sense? What does surjective mean? What does injective mean? Analyzing a mathematical sentence always creates a ( large ) tree structure.

Recursion. The concept of recursive definition. A fascinating concept which I use a lot, since I am a programmer by profession. Curry uses the example of the Fibonacci sequence 1,1,2,3,5,8,13,... and translates it to f(n) = f(n-1) + f(n-2) and the Sierpinski triangle ( fractal ).

Drawing the Sierpinski triangle

( To be continued in the next post. ) Edit: Nope. I'll make a last note about lecture 1 here and continue with lecture 2 next time.

Some remarks, tips for if you want to give it a try ( like myself ). I was not in continuous awe while watching this lecture. You know when like you are watching the latest BBC Horizon or similar. It's not like that. I don't have the feeling as if I have wasted my time, not at all. I am going to watch lecture 2 soon.

- You definitely need the 720+ pages ( 20 chapters ) book. ( Details on the course site. )
- You need to be ( somewhat ) familiar with Bach's music, or at least -know- someone who is. ( What are forums for anyway? ) To fully grasp the genius of Hofstadter's work.
- If you are a religuous person than GEB might not be for you.

There is an audio set in the lecture room. Near the end of the lecture a piece of Bach is played. Students familiar with that music could elaborate on it. Since I am ignorant to most classical music I must have missed a lot of what Hofstadter said. It might be an opportunity to start listening to some Bach, who knows what happens.,

So far for lecture 1,

Primitive recursion

I really appreciate the books on Mathematical Logic from M381. Take for example the very concise definition of primitive recursion.

Let $a$ be a natural number and let $g: \mathbf{N} \times \mathbf{N} \rightarrow \mathbf{N}$ be a function. The function $h: \mathbf{N} \rightarrow \mathbf{N}$ is said to be defined by primitive recursion from the constant $a$ and the function $g$ if
- $h(0) = a$,
- $h(n+1) = g(n, h(n))$.

Example:
- $h(0) = 1$
- $h(n+1) = g(n, h(n)) = (n+1) \times h(n)$
is the well-know faculty function.

( Open University, M381 ML-1 )

Beautiful, isn't it?