Famous quotes that celebrate the beauty of mathematics

In honor of the upcoming annual seminar at the Institute of Mathematics, I work at, the Second Mathematics and Applications Seminar, December 6-7, 2017, here is a small collection of famous quotes that celebrate the beauty of mathematics:

“Mathematics reveals its secrets only to those who approach it with pure love, for its own beauty.” - Archimedes

“Mathematics, rightly viewed, possesses not only truth, but supreme beauty - a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.” - Bertrand Russell

"The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas, like the colors or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics." - G. H. Hardy

"Mathematics is the most beautiful and most powerful creation of the human spirit." - Stefan Banach

"God used beautiful mathematics in creating the world." - Paul Dirac

"I enjoy mathematics so much because it has a strange kind of unearthly beauty. There is a strong feeling of pleasure, hard to describe, in thinking through an elegant proof, and even greater pleasure in discovering a proof not previously known." - Martin Gardner

"The beauty in mathematics is in its unreasonable effectiveness that allows us to observe our mental world and help us understand the nature of mathematics itself." - Josefina Alvarez [The beauty of maths is in the brain of the beholder on plus.maths.org]

You may want to add another quote?

From child's play to Sophie Germain primes and more

Having a child interested in numbers means inventing rather than only playing games with numbers. One such game took place a year ago, during a summer relaxing afternoon. All started with a recent May 2016 puzzle published on Matematika + (a web site in Macedonian that I manage), that asks to find three positive integers which sum equals to their product. At that time, it was a challenging problem for a boy with a satisfactory understanding in addition, but beginner's experience in multiplication. It took him some time to find the 1, 2, 3 solution, maybe because of the number 1 in it and its identity property for the multiplication. The next thing was his observation that if we look for two numbers with the same property (equal sum and product), then the numbers are 2 and 2. And, that was the start for the game.

The game

We started to look for four, five numbers with the same property, so we took the little whiteboard and we started to put solutions of the equal-sum-product problem for different size of \(n\)-tuples of positive integers on it. We had been looking only for at least one solution for different sizes of \(n\), starting with already discovered solutions for \(n=2\) and \(n=3\). I must admit that our approach was not a systematic one. The rediscovered identity property of the number 1 guided us to look for \(n\)-tuples with many 1s, just for easy calculation of the product. Soon, the little whiteboard looked like this:

When the whiteboard was finished, the game finished too. At least for my boy. The fact is that all these numbers were left in my head all together with many unanswered questions. Is there always a solution of the equal-sum-product problem for any \(n \in \mathbb{N}\)? How many solutions are there for an arbitrary \(n\)? How the form of the solutions or the number of the solutions depend on \(n\)?

First observations

Another glance at the whiteboard and the \(n\)-tuples containing the number \(n\), can be quite easily spotted. These multiples on the whiteboard are: \((2,2)\), \((1,2,3)\), \((1, 1, 2, 4)\), \((1, 1, 1, 2, 5)\), \((1, 1, 1, 1, 2, 6)\), \((1, 1, 1, 1, 1, 2, 7)\), \((1, 1, 1, 1, 1, 1, 2, 8)\), \((1, 1, 1, 1, 1, 1, 1, 2, 9)\), \((1, 1, 1, 1, 1, 1, 1, 1, 2, 10)\) and \((1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 12)\). Here is the picture of the whiteboard with underlined solutions of \(n\)-tuples containing the number \(n\):

In general, each of these \(n\)-tuples contains \((n-2)\) ones, the number 2 and the number \(n\) i.e. they have the form  \((1, 1, ..., 1, 2,n)\), where there are \((n-2)\) ones. Let us check. The sum of the numbers is

\(1 + 1 + ...+ 1 + 2 + n = n - 2 + 2 + n = 2n\), 

and the product of the numbers is

\(1 \cdot 1 \cdot ...\cdot 1 \cdot 2 \cdot n = 2n\).

So, the sum equals to the product, and \((1, 1, ..., 1, 2,n)\), with \((n-2)\) ones, is a solution of the equal-sum-product problem for an arbitrary \(n\). The solution of this form is called a basic solution. This means that for each \(n\) there are \(n\) numbers which sum equals their product. And it is not a new result, I've just rediscovered it here. The following references are at least three places where you can find this result and much more about the equal-sum-product problem:

(*) When the sum equals the product by Leo Kurlandchik and Andrzej Nowicki
(**) An Algorithm to Solve the Equal-Sum-Product Problem by M. A. Nyblom and C. D. Evans
(***) When Does a Sum of Positive Integers Equal Their Product? by Michael W. Ecker

Fill free to explore these references before continue reading my post. I did it too. You can find very interesting results there. For example, you can find that there are finite number of solutions of the equal-sum-product problem for each \(n\), or that for any number \(s\) there exists \(n\) for which the number of solutions is greater than \(s\), for example, for \(n=2^{2s}+1\), there are at least \(s+1\) solutions for the equal-sum-product problem. The form of these solutions can be found in (*).

Exceptional values

In (*) there is also a table with the number of solutions of the equal-sum-product problem for each \(1 \leq n \leq 100\). In this table, \(a(n)\) denotes the number of \(n\)-tuples that have equal sum and product:

I must admit that without any intention, we actually find almost all solutions for  \(2 \leq n \leq 15\). We only missed the basic solution \((1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 11)\) for \(n=11\), the nonbasic solution \((1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2)\) for \(n=12\), two of four solutions for \(n=13\) and the basic solutions for \(n=14\) and \(n=15\). Well, quite a lot.

In the references I mentioned before, you can also find that the common value of the sum and the product for any solution \(n\)-tuple is at most \(2n\), and this value is reached for the basic solution. Such statement and similar ones can lead to construction of an algorithm for finding all solutions for the equal-sum-product problem, as it is done in (**).

It is common knowledge that actually it can not exist only one research question in a scientific investigation. Almost always one question leads to another, and results that are found may be answers to other questions. In our case, searching for all solutions, their form and number, leads us to values of \(n\) for which there is only one solution, the basic solution i.e. \(a(n)=1\). These values of \(n\) are called exceptional values. From the above table, such values for \(n \geq 2\) are \(n=2, 3, 4, 6, 24\). There are researchers that have investigated this topic more deeply. In (***) there is a conjecture that the set of all exceptional values \(n \geq 2\) is finite i.e. the set of all exceptional values is \(E=\{2, 3, 4, 6, 24, 114, 174, 444\}\). This conjecture is tested up to \(n=10^{10}\) and no other exceptional values are found. But, what is the importance of the exceptional values?

Sophie Germain primes

As it is proven in (*), if \(n>2\) is an exceptional value i.e. \(a(n)=1\), then \(n-1\) is a prime number. In (**) it is proven that if \(a(n)=1\) for \(n>2\), then \(n-1\) is a Sophie Germain prime (a prime number \(p\) is a Sophie Germain prime, if \(p\) and \(2p+1\) are both primes). Let us check one exceptional value, say \(n=24\). It is true that \(n-1=23\) and  \(2(n-1)+1=47\) are both primes, so 23 is a Sophie Germain prime. Some historic note: Sophie Germain primes are named after French mathematician Sophie Germain, who used them in her investigations of Fermat's Last Theorem. Here are some of the first Sophie Germain primes:

2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233, 239, 251, 281, 293, 359, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 719, 743, 761, 809, 911, 953, ...

So, exceptional values of the equal-sum-product problem and Sophie Germain primes are somehow in a relation. On the other hand, Sophie Germain primes play a big role in cryptography, where the number \(2p+1\), associated with a Sophie Germain prime \(p\), is called a safe prime. And safe primes are related to strong primes (a prime number \(q\) is a strong prime if \(q-1\) and \(q+1\) both have some large prime factors). The both of them, safe and strong primes, are used as the factors of secret keys in some cryptosystems, because they prevent the system being broken by certain factorization algorithms. Having infinitely many Sophie Germain primes (the Sophie Germain primes conjecture, which is still unsolved), will insure efficient secret key generating procedures. Let us go back to the set E of all exceptional values. If it is possible to prove that the set E is infinite, then the infinitude of Sophie Germain primes would immediately follow and the Sophie Germain primes conjecture will be solved. Voilà! But, it can not be that easy to prove such a deep conjecture with an elementary approach, as it is said in (**).

Something more

Now, leave the cryptography, leave the Sophie Germain primes, and let us go back to the equal-sum-product problem. Recall that \(a(n)\) denotes the number of \(n\)-tuples that have equal sum and product. We have listed these values up to \(n=100\) in a table. Let us plot that list:

From the line plot, the increasing trend in values \(a(n)\) is obvious and also the increase in the dispersion of these values (there are small and big values for \(a(n)\), as \(n\) increases). The high values of \(a(n)\) are expected due to the previous mentioned property that for any number \(s\) there exists \(n\) for which \(a(n)>s\). This might be true also for the small values of \(a(n)\), i.e. we can also expect them, due to the observed increasing dispersion. 

We can also plot the distribution of the numbers \(a(n)\) for the first 100 values: 

We can see that the distribution of the values \(a(n)\) is highly unsymmetrical and positively skewed. Due to previously observed increasing trend in values of \(a(n)\) and increasing dispersion, it is expected that the distribution will remain unsymmetrical and positively skewed, but with increasing mode. This means that the small values are also expected, but with decreasing probability as \(n\) increases.   

All statements presented in the section 'Something more' are intuitively made, and are still unproven.

  

After all

The last year equal-sum-product game/problem inspired me to compose a problem for finding all five-tuples of positive integers which sum equals their product and to post it as the Problem of the Month on Matematika + for September 2017. It was nice recreational math problem for young mathematicians.

It will also be interesting to see where the next family game will take us. I can not wait.

Advertising and logic

What an interesting advert!

About two years ago, I saw this advert: 

"If Cineplexx does not exist, your wife would be a lord of the rings."

I have been looking at it, for a long period of time. I was standing there in the middle of the mall center, while people were passing by, and some of them (those that were not in a harry) looked for a moment at the point I was looking at - the advert, possibly thinking that I am fond of The Lord of the Rings, and that I am standing there planning to go to see the film.

What is "the logic" and the logic behind this advert?

While I was standing there, I was asking myself "What do the creators of the advert want to tell us?" The fact is that Cineplexx does exist (since the advertisement was placed in a mall where there is a Cineplexx cinema). So what? What are we supposed to do with the rest of the sentence, with "the wife" and "the lord of the rings"? The next days I started a conversation with some of my friends. Suddenly, this advertisement appealed to be anti-feminine for some of them, and the others thought it is romantic (I agree with the last one, but only if Cineplexx does not exist). Why some people found that the advertisement is anti-feminine? Let us see. If we think about "the wife who is a lord of the rings" as the one that is the head of the marriage, I suppose that the men (who support anti-feminism) would like this not to happen. And since Cineplexx does exist (and The Lord of the Rings is showing at Cineplexx), the wife would not be a lord of the rings (i.e. she would not be the head of the marriage) - that is how the authors of the advert believe that everybody would make the conclusion out of the given sentence and the fact that Cineplexx does exist. But is this a right way to make a conclusion? If we denote by \(p\): "Cineplexx does not exist." and by \(q\): "Your wife is a lord of the rings.", then we have \(\neg p\): "Cineplexx does exist." and \(\neg q\): "Your wife is not a lord of the rings." The last two propositions are negations of the propositions \(p\) and \(q\) respectively. Then, "the rule" that the authors would like everybody to implement is \(((p \implies q) \land \neg p) \implies \neg q\). This is behind every logic! It is the same as "If it is a dog, then it is an animal. It is not a dog. Therefore, it is not an animal." 

Later, I found that this kind of fallacy in logical thinking is called Denying the Antecedent. (In conditional statements "if \(p\) then \(q\)", \(p\) is called an antecedent, and \(p\) is a consequent.) Denying the Antecedent happens when the inference rule Modus Tollens is misunderstood, it is not being used properly. The Modus Tollens states \(((p \implies q) \land \neg q) \implies \neg p,\) which is interpreted as "p implies q and q is asserted not to be true, so therefore p must be false." It is a tautology, which means that it is always true regardless the values of \(p\) and \(q\) (true or false for each of them), so there is no place for doubt in its correctness. When we switch the places of \(\neg p\) and \(\neg q\) in Modus Tollens, we get the logical fallacy called Denying the Antecedent.

It is always interesting to explore further. I found that the logical fallacy that is mistaken for Modus Ponens is called Affirming the Consequent. Modus Ponens is the rule \(((p \implies q) \land p) \implies q\), and Affirming the Consequent refers to \(((p \implies q) \land q) \implies p\). An example of an Affirming the Consequent fallacy is, "If someone owns Apple, then he is rich. Bill Gates is rich. Therefore, Bill Gates owns Apple".

Logical fallacies within other advertisements

After identifying a logical fallacy in my advert, it was trilling to look for the others. So, here are some advertisements that I found and "the logic" behind them:

"There are some things money can't buy. For everything else there's MasterCard."

This is a False Dichotomy or False Dilemma. It says that we have only two options that we have to choose from, instead of allowing for other possibilities. In this case we can choose between things that money can't buy and the things that we can buy using the MasterCard (and nothing else). Here is another:

"Marilin Monroe says - Yes, I use Lustre-Creme Shampoo."

This fallacies are widely used in marketing, there are called Appeal to Authority. The reasoning is based on what some authority says on the subject. In this case Marilin Monroe uses Lustre-Creme shampoo, so the advertised shampoo must be good. It is true that most of our knowledge is formed by listening to authorities, books that we learn from are written by some authorities in the particular area, teachers who teach us are trained for the job they are doing, our parents give us advises based on their experience and believes that are formed, I suppose, by listening to authorities. You have to have some background knowledge about the subject or the authority, in order to spot this kind of fallacy. Now, look at this one:

"Costs less then a personal trainer."

(showing a picture of Wendy's backed potato, a small chili and a fresh salad with fat-free dressing, a meal with just 5 grams of fat)

It is a Faulty Comparison or Questionable Analogy. When reasoning by comparison, the fallacy occurs when the analogy used for the comparison is irrelevant or very weak or when there is a more relevant disanalogy. In the advertisement, the low fat meal is compared to a personal trainer, probably on the base that the both have the same effect on someones health, which is very weak argument. A personal trainer will do much more things for you, he will help you to eat and exercise properly, to lose weight, to gain self confidence, to live a healthy lifestyle. It is true that a 99 cents meal is less then a personal trainer, but is it healthy to eat the same meal through a long period of time? 

There is a huge list of fallacies in reasoning, and each of them, unfortunately, can be used in advertising campaigns and might have great negative impact on proper logical people's reasoning.

Epilogue

Two years after I saw the Cineplexx advert for the first time, I was visiting the same mall center, and ... it was there again.   

Beautiful math

The beauty can be very subjective. I may find chocolate glazed donuts beautiful, and for someone Boston cream donuts are simply irresistible. But the both of them are tasty, sweat and round. So, the both of us will agree that donuts are beautiful. We find beauty in small and big things, in material and spiritual things, in our day lives and in our night dreams. And, there are universal beauties, such as our World (a beautiful place to live), the creatures that live in it (among them - we humans) and mathematics.

I am not sure when we had met the beauty of mathematics, for the first time. Is it when we solved our first problem, or when we learned to count, or even when we chose the bigger toy to play with. However, it might happen that we become aware of this beauty much more later.

What makes us to see the beauty in mathematics? At the beginning, for sure, it is her simplicity and elegance. Lake a magic, the sentences "There are 5 apples on the table, and you take 3 apples. How many apples are on the table now?" turn into $$5 - 3 = 2.$$ Later, it is her power to turn a variety of concepts into one meaningful and solid idea, like

The sum of the angles in a triangle is equal to \(180^o\).

Then, there are plenty of helpful formulas, like

$$x_{1,2}=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$$ are solutions of the quadratic equation \(ax^2+bx+c=0\), \(a \ne 0\).

And many useful visualizations, like

Trigonometric circle
(image credit: globalspec.com) 

And many astonishing interpretations, like

The first derivative \(f'(x)\), if it exists, is the slope of the tangent line of \(f(x)\) at \(x\) i.e.

$$f'(x)=\lim_{\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x} = \lim_{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}.$$

Which can be illustrated as

The slope of the secant line PQ is the difference quotient \(\frac{\Delta y}{\Delta x}\),
and the slope of the tangent line at P is the rate of change
  of the sequence of slopes of the secant lines at P as \(\Delta x\) 
approaches to zero. (image credit: themathpage.com)

And many, many, many other beauties.

And then, when you think that you have learned mathematics enough to be able to apply it, you will enter a new room filled with surprising and unexplored beauties. You will realize the true meaning of the saying

"Mathematics is the queen and the slave of the sciences at the same time."

You will see that this new room has infinitely many doors, opened or closed, and you should not be surprised when some of the doors will return you at the very beginnings of some of the mathematical concepts, searching for the truth and only the truth. Then you are going to find the most delightful beauty of mathematics. The fact is that, the more you do mathematics, the more you realize how wonderful she is.

But you may have your own opinion about the beauty of mathematics?