Front Cover of Theoremus

Hello,

The first draft of my booklet called Theoremus is ready for review – if anyone is interested. I will gratefully thank you in the acknowledgment section for any critiques you may have of it. It is not a thick booklet – it is only 70 pages long and targets for an A5 size print. So an expert can finish reviewing it in a couple of days.

It’s aim is to coach a First Year In Maths student on how to make their proofs more rigorous and thus convincing. That is the broad theme of this booklet – the idea of being rigorous.
After all, why would teachers want you to prove theorems if they do not want you to prove them a bit more rigorously? Surely that should be preferred.

The booklet is suitable for any first year university student who will likely be required to put more solid maths in their course. For example, Comp Sci and Engineering students, like those doing Software and Systems Engineering. It should also be suitable for students of Economics, Finance, Physics, Linguistics, Chemistry and perhaps Biology. Well, of course, it should be for the Maths and Stats students too :-)

Please let me know if you have an interest in reviewing it – your suggestions will be welcomed no matter what angle they come from. Just drop me a line in the comments side of this post and I will send you the draft in PDF. Thank you in advance.

-LPC

Let $L =$ the LISP language. Let $P =$ a programming language different from $L$, i.e. $L \cap P = \oslash$. Let $t =$ time. Then $\lim_{t \to \infty} P = L$.

Proof:
Trivial, just look at the extensions they are making on Python, Ruby, Java, C# and C++11.

Hehe.

tags:

It has been said that the smallest piece of code in the world is the C code shown below for copying strings.

while (*d++ = *s++);

That is just a whopping 17 characters long! Really?

I’ll do that in Scheme/Lisp:

(set! d s)

There 9 characters long.

By I.H. we mean by induction hypothesis.

First year in maths students are baffled about mathematical induction(M.I.). It is counter intuitive and it is true, it is hard to wrap one’s head around this concept. So to prove some mathematical property holds we are given the following template:

1. First show that the property $P$ ( the property of being divisible, being prime or equal to a formula’s value, you name it etc.) is true for the base case. If the base case is $c$ then we prove that the property is true for it. Thus we work on showing $P(c)$ is true.
2. Second, we assume it is true for $k$, that is $P(k)$ is true – we take this as fact. This second step is called the induction hypothesis – I.H.
3. Lastly, we prove that $P(k+1)$ is true as well using I.H. Thus should we succeed in this final step, we have all the right to claim that we have proof that $P$ is true for all $n$.

It is I.H. that is hard to take. Why is it that a.) we should assume it and b.) why is it that if step 3 succeeds provided we use the fact of step 2, we have the right to say – Q.E.D. or say “proven as required”! What right do we have in assuming I.H.

• Firstly, M.I. is about the property of numbers (in general). Numbers obey this I. H. property. As a classic example consider a number $x$ such that $x > c$, for some number like 8. So if we have $x > 8$, can we say that the next number after $x$ will also be greater than 8? We can guess that this should be true. So let us prove it…Consider $x > 8$, let us add $1$ to both sides still maintaining the inequality. $x > 8 \Rightarrow x + 1 > 8 + 1 = 9 > 8 \therefore x + 1 > 8$.
• Secondly, we are allowed to assume I.H. for after all we can set our $n = c$, i.e., to our base case and check if the property $P(c+1)$ holds — thus by the same token we are allowed to move from the truth of $P(c)$ to the truth that $P(c+1)$. So we can assume I.H. because of this “domino effect”. The crucial bit is to show now that due to our utilisation of I.H. for an arbitrary $n = k$, we get the truth of $P(k+1)$.
• Lastly and strongly, we can take M.I. to be an axiom! Meaning, a proposition which is self-evidently (if I can use that word) true! Indeed Wikipaedia has it like this – $\forall P[P(0) \wedge \forall k \in \mathbb{N}[P(k) \Rightarrow P(k+1)]] \bold{\Rightarrow} \forall n \in \mathbb{N}[P(n)]$.
• Take a good look at this and consider the statement before the last $\Rightarrow$. If you look we have this form $A \Rightarrow B$. Remember modus ponens? It says if we have $A \Rightarrow B$ and we have $A$, deduce $B$.

Well when we are doing M.I. what we are actually doing is that we are establishing the truth of $A$ and when we succeed – voila, use this with the axiom and so conclude the property $P$ holds for all $n$.

Dr. Benjamin Levitt alerted me to the article I first heard from a colleague at the university where I work. Apparently a group of neuroscientists examined the brains of 15 mathematicians using Magnetic Resonance Imaging (MRI). They showed these scientists pictures of mathematical formulas and the activity of the brains of these subjects reacted in the same way one’s brain reacts when it experiences viewing a beautiful piece of art or listening to beautiful piece of music.

Read the University College London article here.

Thinking of this now, it is this beauty that attracted me to studying mathematics, aside from of course, being inspired by great teachers who taught me and exposed me to its beauty. Though my father was an engineer (an my uncle a PE and  has a PhD in hydrology), I do not think it was the one which moved me to be a student of maths. I think I was more influenced by enthusiastic teachers who were passionate with the subject – I mean, these teachers loved the subject and their tamed enthusiasm nevertheless showed when they wrote proofs on the blackboard.

I like to study maths because of its beauty yet I am so poor at it.

I am so addicted to this TV series, I hope it stays for a long long time. It is beautifully done and the interaction of complex character personalities is often heartwarming and fun. Whenever I watch an episode, I cannot help but get reminded of what C. S. Peirce said…

From P.J. Davis & R. Hersh, The Mathematical Experience, 1981, Penguin Books

C. S. Pierce in the middle of the nineteenth century, announced that “mathematics is the science of making necessary conclusions.” Conclusions about what? About quantity? About space?The content of mathematics is not defined by this definition; mathematics could be “about” anything as long as it is a subject that exhibits the pattern of assumption-deduction- conclusion. Sherlock Holmes remarks to Watson in  The Sign of Four that “Detection is, or ought to be, an exact science and should be treated in the same cold and unemotional manner. You have attempted to tinge it with romanticism, which produces much the same effect as if you worked a love-story or an elopement into the fifth proposition of Euclid.” Here Conan Doyle, with tongue in cheek, is  asserting that criminal detection might very well be considered  a branch of mathematics. Peirce would agree.