Curvy Mathematics

Curvy Mathematics


Common Applications of Mathematics

7 Oct 2022

Solid Clues, by  Gerald Feinberg

It is difficult for the human mind to classify and understand all the different types of complexity that occur in large systems. Two types of such complexity, however, offer some prospect for understanding through developments in applied mathematics and computer analysis. These two types of complexity are usually thought of as opposites and go under the names "chaos" and "order". Yet, there are indications that certain kinds of chaos and order are consequences of the same type of mathematical structure, known as nonlinear equations. Perhaps this is a modern version of the ancient philosophical view of the Greek philosopher Anaxagoras, who believed that opposites emerge from a formless Absolute, and eventually return to it.

The Pea and the Sun, by Leonard Wapner

The point being made is that pure mathematics, by definition, need not to extend in the real world. Some pure mathematicians will go so far as to prefer it not be applied, as application of the art in some way soils its beauty. 

Cybernetics and Society, by Norbert Wiener

Some years ago, a prominent American engineer bought an expensive player-piano. It became clear after a week or two that this purchase did not correspond to any particular interest in the music played by the piano
but rather to an overwhelming interest in the piano mechanism. For this gentleman, the player-piano was not a means of producing music, but a means of giving some inventor the chance of showing how skillful he
was at overcoming certain difficulties in the production of music. This is an estimable attitude in a secondyear high-school student. How estimable it is in one of those on whom the whole cultural future of the country
depends, I leave to the reader.

The Greeks regarded the act of discovering fire with very split emotions. On the one hand, fire was for them as for us a great benefit to all humanity. On the other, the carrying down of fire from heaven to earth was a
defiance of the Gods of Olympus, and could not but be punished by them as a piece of insolence towards their prerogatives. Thus we see the great figure of Prometheus, the fire-bearer, the prototype of the scientist; a hero but a hero damned, chained on the Caucasus with vultures gnawing at his liver. We read the ringing lines of Aeschylus in which the bound god
calls on the whole world under the sun to bear witness to what torments he suffers at the hands of the gods.

If a man with this tragic sense approaches, not fire, but another manifestation of original power, like the splitting of the atom, he will do so with fear and trembling. He will not leap in where angels fear to
tread, unless he is prepared to accept the punishment of the fallen angels. Neither will he calmly transfer to the machine made in his own image the responsibility for his choice of good and evil, without continuing to accept a full responsibility for that choice.

Yet for all but a very few, the path to these lies through the performance of perfunctory tasks which in nine cases out of ten have no compelling
reason to be performed. Heaven save us from the first novels which are written because a young man desires the prestige of being a novelist rather than because he has something to say. Heaven save us likewise from the mathematical papers which are correct and elegant but without body or spirit. Heaven save us above all from the snobbery which not only admits the possibility of this thin and perfunctory work, but which cries out in a spirit of shrinking arrogance against the competition of vigor and ideas, wherever these may be found!

4 Jan 2018

Modern Monetary System

by G. Karakatsanis, blogger at

Our monetary system creates money via debt. In the midst if the recent financial crisis which started back at 2008, the idea that the current model is no longer viable gains more supporters every single day. The perpetual financial development required to repay debt is not always possible. Debt is steadily being accumulated and its repayment becomes impossible.

The counter-proposal is that money should be minted by a public, central authority which would decide on the necessary quantity of money for the economic state to function properly. Additionally, money would be injected into this state on public expenses, free of any chargement or debt. This way the state would become more independent and the dead-end where the current economic state is lead to, would be avoided.

Supporters of the current monetary system claim that the quantitative control of money from a central authority has failed almost at all cases throughout human history. Since the Roman times, emperors who were in desperate need of extra money in order to further support their armies, were just minting more money to pay their soldiers’ wages.

Money is just a mean to exchange materials and store wealth. Whenever the amount of circulating money increases without its matching in gold or any other valuable material being increased simultaneously, then this causes circulating money to lose its value. Whenever new money is minted without any matching to gold or any other valuable resource, then money owners are being robbed as their money loses its value. Such a financial instability caused by many Roman emperors resulted in the collapse of the entire empire.

On the other other hand, nowadays, a decision to mint new money is taken every time a bank approves of a loan. The bank judges whether the new money will contribute to creating enough wealth so that the initial loan can be repaid. Obviously, the bank’s evaluation is vital as in case of a mistaken evaluation, wealth will not be enough for the initial loan to be repaid and the bank will suffer some loss.

Theoretically, the modern monetary system is far better than the Roman one. There is no general rule or law to “enforce” the minting of new money but this decision is utterly depended upon the given circumstances. The one who takes the decision ought to do one’s best, differently the consequences may be disastrous. Still, there are some vulnerabilities inside this system. The decision to grant a loan is based on the bank’s “bona fide”(good faith) that the borrower will repay the debt. If this faith fades away, then the entire system collapses. This is what happened back in 1929 when at least a decade plus a world war were required for the monetary system to recover.

There are many people who claim that such a crisis is necessary to occur occasionally. Such situations assist in distinguishing weak monetary units from the powerful ones. The weak ones will disappear and new ones will arise. Banks with a bad judgement will sustain heavy loss or they will go bankrupt. Following the financial crisis of 1929, more than 4000 banks closed in the USA during the next decade.

The depression of 1929 is slightly different than the one of 2008. Banks are not local and in vast numbers anymore. Contrary, they are significantly fewer and consist of big organizations. Bankruptcy of any of them can cause the entire monetary system across the globe to shatter. This is what happened with the collapse of Lehman Brothers in 2008. No other bank or financial institution was left to bankrupt since then. Their survival is considered of vital importance and all governments “inject” money to them derived from taxation.

Such a privileged manipulation of the banks and financial institutions cancels the biggest advantage of the current monetary system. Banks do not put any effort at all to evaluate properly loan applications. Huge flows of money are created on enormous mountains of debt. Obviously, such a debt cannot be repaid and the borrower has no consequences to afraid of as the taxpayers will cover the bank’s loss as a result of improper loan evaluation and judgement. Legislation carries the banks’ incapability to properly evaluate a loan application to the taxpayers, so they will suffer the loss.

17 Nov 2017

Programming : Commanding a Language

As we grow up, we learn to speak our mother language. We learn it by heart and we use it to communicate orally, until we go to school when we learn to write and everything is destroyed. New knowledge has penetrated. New kids on the block. We have to re-organise all we knew and all we learn. Somehow it works. Many of us face some difficulties but in the end we manage to learn to properly speak and write our mother languages.

A great, arising difficulty is the fact that our language has rules. Unfortunately, inequality governs the world we live and as it happens with state laws, language rules have exceptions too. These exceptions show the difficulty a human mind finds to revolt against habit. We tend to like a habit as we suppose we become better when repeating and someone will admire us and applaud for us. Repetition is an advantage for learning but when it becomes habit, then the line is crossed, it has already become a disadvantage.

In the past, someone chose to defy a rule, either because he/she could not follow it or because he/she did not want to comply with it and his/her charismatic personality assisted in transferring the "mutation" to future generations, making it a habit. The language of chemistry, which I am familiar with, still uses names for substances which have remained the same since the age of alchemists.

Several people are alleged to have a tendency in learning foreign languages. I suppose one should definitely learn the rules and the exceptions in order to comprehend and fluently speak any human language. There are some languages though that have no exceptions at all. Programming languages.

The idea is simple. Each programming language has a list of pre-set words, a library of defined commands in other words, each of them executing a very specific task without any deviations. A combination of these commands may create an object, just like words create a sentence. The difference is that words may have more than one meaning which depend on the mood of the speaker or the author whereas commands have a strict and clear meaning allowing no doubt about it. Thus, sentences may be interpreted in more than one ways while objects cannot be misunderstood.

Yet, people tend to face computer programming with some fear. Good command of a human language allows us to impress the others, to manipulate the language and show a virtual supremacy over the rest. Good command of a computer language just makes us faster in commanding a cold machine which will never applaud for us neither it will gaze us in awe, admiring our knowledge and experience. And this flattens our vanity.

Personally, I find incredible the fact that programming is actually a sequence of commands to add the number of times an electrical circuit is carrying electricity. It sounds so simple and it really is. Just imagine that computing is based on mathematics. Here comes the pain of mathematics again.

Mathematics consist of a language too. A sequence of symbols which helps humanity to degrade a problem into adding units, find a solution according to mathematical rules without exceptions and re-apply the solution in our reality. The entire difficulty lies in translating to/from mathematics. To be honest, mathematics will help in commanding a programming language and each programming language will definitely assist in comprehending the simplicity of mathematics.

12 Oct 2017

Insurance : The Game

The idea of minimizing risk is called insurance. Enterpreneurs are supposed to take risks. At least this is what they declare to the rest of the world. Shipping companies insure their cargo, financial corporations insure their loss probability, common people go with the flow and insure whatever they can, either because they are obliged by legislation or because they feel that insurance will repay them at all costs. In my opinion, insurance will only deceive common people's minds by allowing them believe that in case of an unfortunate event, the insurance company will take care of everything. Practically, it will cause more trouble in the case that the unfortunate event finally occurs.

Insurance companies make money. Neither they generate nor they mint money, they just wager (invest is their favourite term) with their clients’ money. In exchange to the clients’ goodwill to trust their money in the insurance broker’s hands, they repay their clients with some interest. Interest can be high or low, acceptable or unacceptable but this is utterly subjective. Personally, if I could find repayment with an interest greater than the one banks offer, I would choose it. Obviously, I am talking about pension or savings contracts. The insurance company collects, keeps and wagers the customers’ money for twenty to thirty years and returns an equal amount of money plus the agreed interest minus the agreed (and random, not always agreed) “costs”.

Just like a bank and a casino, an insurance company can be considered as “The House” too. And The House always wins. They receive money from car insurance, health insurance, fire, death, etc. Do they return all of this money back to the clients? Absolutely not. But this is the way this system has always been functioning. In other words, we pay for services that we do not always use. Legally, it is acceptable. Ethically, I cannot say. It depends on one’s background and culture.

The savings or pension programme works like this: the client deposits every year an increasing amount of money. After the agreed number of years passes, the insurance company returns an amount of money greater than the sum of all yearly installments the client has deposited.

1st year: Client deposits C

2nd year: Client deposits C + C×i = C×(1 + i)

3rd year: Client deposits C×(1 + i) + C×(1 + i)×i = C×(1 + i)×(1 + i)

4th year: Client deposits C×(1 + i)×(1 + i)×(1 + i)

Obviously, this is C×(1 + i)^3

5th year: Client deposits C×(1 + i)^4


20th year: Client deposits C×(1 + i)^19

i stands for inflation and usually varies between 1% and 5%. It is supposed to cover the expenses of the insurance company due to yearly inflation of money. In other words, if the client signs a contract for twenty years, the insurance company supposes that after twenty years the first installment of C will have a very low value. So, the client pays for that every year. But this is fair enough, the company will return every single cent of it.

Now, the insurance company sums all the yearly installments. It is boring enough(and time consuming) to make all calculations for the sum but since it is a sum of the first twenty terms of a geometric sequence, then the result is C×(1/i)×(1 + i)^19.

Well, I have made a typographical error above. Most insurance companies prefer to start the inflation from the 1st year, which means that the 20th installment is C×(1 + i)^20 and thus the above sum of all twenty installments is C×(1/i)×(1 + i)^20. In both cases the sum is considered to be a lovely amount of money for every average human being. The client will take it back in either of the cases, it is just that the second case is more expensive. Additionally, the client will receive an interest around 40%.

Usually, this amount is tax free too. Theoritically. Practically, the government does not apply any direct tax on the final yield but very tiny ones, paid per installment which are not refundable and varying between 0.05% and 2% of the installment. Unfortunately, these “costs” occur randomly and at will by all governments across the globe. It seems that all governments have at least this in common.

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