Mathematics chooses the standard of the highest exactness for its language, problem statements and solutions. In most of thinkable scopes this gives complete certainty about the result. The possibility of an error in the proof has to be taken into account. In the case of average CS papers, this happens too often for math standards, but in the case of established mathematical results it is really very rare. Considering uncertainty that we have to face everywhere in our lives, this makes mathematics a powerful tool.
Its weakness immediately follows: meeting the standards of exactness essentially disables us to solve anything complicated. High school mathematics usually gives a wrong impression that everything can be solved. The high school math is a carefully chosen set of (in certain sense) simple problems and theorems that (by some lucky coincidence) can be solved and proved easily. If you took a random choice of a natural problem of the same or just slightly higher complexity, you might get stuck for the rest of your life easily.
A little bit about this and much more can be found in the book Mathematics: A Very Short Introduction; there is a Czech translation by Jiří Matoušek.
Its weakness immediately follows: meeting the standards of exactness essentially disables us to solve anything complicated. High school mathematics usually gives a wrong impression that everything can be solved. The high school math is a carefully chosen set of (in certain sense) simple problems and theorems that (by some lucky coincidence) can be solved and proved easily. If you took a random choice of a natural problem of the same or just slightly higher complexity, you might get stuck for the rest of your life easily.
A little bit about this and much more can be found in the book Mathematics: A Very Short Introduction; there is a Czech translation by Jiří Matoušek.
A case where math brings power: computers. Luckily, computation/algorithms, data transfer, networks, etc., have relatively simple and precise mathematical models.
In the case of computation/algorithms, models do not take hardware errors into account usually but they happen very rarely. In addition, many of the simpler algorithms for important problems can be mathematically analyzed, i.e. various guarantees (such as correctness or running time) can be proven.
Of course, people also have to use and struggle with more involved algorithms where it is not within compass of human to prove or disprove any reasonable claim, we hit the weakness of mathematics . Then less mathematical approach can be used such as running experiments with some real-life input data. Also much of the engineering hard work such as routine programming would not be claimed as math neither by coders nor mathematicians.
Still, as far as I can see into the world of computers, the fundamental achievements (such as cryptography, routing of data streams over internet, error correcting codes for media, optimization, page ranking, media compression and many others) are essentially mathematical problems and are solved by (often both ingenious and beautiful) math methods.
The world of computers+IT went through a rapid and overwhelming development during last decades. Is not its mathematically-exact character the profound reason for this success?
Cardinal weakness-aspect of mathematics: most of the questions concerning human - probably the most important subject to study - are out of math's scope. Human body, health, brain, psyche as well as interaction between people are in many respects way too complicated to have reasonable exact description. Even when one can be obtained (after simplifying the situation), there is no guarantee that we are able to prove any meaningful mathematical answer.
Still, as far as I can see into the world of computers, the fundamental achievements (such as cryptography, routing of data streams over internet, error correcting codes for media, optimization, page ranking, media compression and many others) are essentially mathematical problems and are solved by (often both ingenious and beautiful) math methods.
The world of computers+IT went through a rapid and overwhelming development during last decades. Is not its mathematically-exact character the profound reason for this success?
Cardinal weakness-aspect of mathematics: most of the questions concerning human - probably the most important subject to study - are out of math's scope. Human body, health, brain, psyche as well as interaction between people are in many respects way too complicated to have reasonable exact description. Even when one can be obtained (after simplifying the situation), there is no guarantee that we are able to prove any meaningful mathematical answer.
An illustration: in economy, rather widely taken assumption claims that people act in a way leading to maximization of their profit (or its mean value, in a more realistic probabilistic settings).
On the mathematical side, this leads to an exciting area of the game theory where many important results have been obtained (such as that free competition can lead, in theory, to a situation (equilibrium) far from optimal for the society). But for example questions regarding computational complexity of finding equilibria for various games are far from resolved. Moreover, if you consider in some situation more realistic scenario, where people are risk averse (they maximize the profit mean value minus variance), much less is known.
Yet we still have to completely sober up in the bitter reality of our world: the original assumption on humans as perfect maximizers can be in some important situations, such as financial crisis of past years, very far from truth. I can recommend the book "Nudge" by Richard H. Thaler and Cass R. Sunstein where this and much more is explained in a nice readable way.
One of the messages for us could roughly read: the financial crisis left us with a need for a mathematical description of "human that is sometimes perfect maximizer and sometimes behaves like a sheep".
Good luck!
