How do mathematicians think?

How does someone who is very into math think? What techniques do you use to conveniently analyze the situations that you encounter in your daily tasks? Is there a way that anyone can get a deep understanding of math?

In this brief article we are going to answer the question ‘’How do mathematicians think?’’ We will analyze the brain-mathematical relationship, in addition, we will highlight the importance of mathematical thinking and how to develop it.

How do mathematicians think?

Mathematicians, thanks to their great development of logical-mathematical thinking, think objectively, analyzing all factors and establishing relationships to find a quick and effective solution to any type of problem.

Grigori Perelman is considered one of the greatest mathematicians of our time. In 2002 he managed to overcome a challenge that no one had yet achieved: he proved the Poincaré conjecture.

Almost a century earlier, in 1904, the French mathematician and physicist Henri Poincaré (1854-1912) presented the hypothesis that his surname bears, although he could not prove it.

So difficult was it that the Clay Mathematical Institute of Cambridge, in the USA, included it among the Seven Mathematical Problems of the Millennium. It also offered a reward of one million dollars to whoever solved it. That person was Perelman, but he declined the award.

The Poincaré conjecture deals with the so-called three-dimensional manifolds. This term refers to abstract hyper surfaces immersed in an environment that must have at least four dimensions.

Don’t worry if reading these lines has made you dizzy because you don’t understand anything: even a careful explanation of Poincaré’s hypothesis would exceed the imagination of most mortals. Not even professional mathematicians can visualize these objects. Only by detours they manage to describe them with the help of mathematical formulas.

Why do some people manage to understand complex mathematical concepts by facing them as if it were a game while others fail in calculations as simple as the return that capital produces?

In recent years, this question has increasingly occupied the interest of neurologists. One of its objectives is to check whether the differences in the handling of numbers and equations are reflected in the neural field. Is it possible that some people’s brains are better endowed in this regard?

Mathematicians’ brain

When complex statements of algebra, geometry and numerical analysis have been made to a group of mathematicians and to non-mathematicians, it has been observed that certain areas of the temporal lobe, the prefrontal cortex and the intraparietal region of the brain were activated in the former.

The origins of the human capacity for mathematics have long been debated among scientists. Some hypotheses suggest that this ability to calculate is related to language skills, but others point out that it’s based on the innate knowledge that Homo sapiens has of space, time and numbers.

Marie Amalric and Stanislas Dehaene at the Inserm-CEA Cognitive Neuroimaging Unit, south of Paris, conducted a study whose purpose was to identify whether there is a neural basis behind advanced mathematical thinking. That is, to elucidate if in our thinking organ there is an area specialized in understanding algebra and geometry.

The results revealed a series of brain areas (from both hemispheres) of the prefrontal cortex, the parietal cortex, and the inferior temporal lobe are activated only when mathematicians are faced with statements or problems of their specialty. And they match the circuits that come into play when anyone handles numbers, does addition and subtraction, or sees a mathematical formula written on paper.

In short, the detected areas are activated only when mathematicians are faced with statements or problems in their specialty. And they match the ones that do when anyone does addition and subtraction or sees a mathematical formula.

However, despite this progress, there are still many unclear questions about how language and mathematics interact in our thinking organ.

Logical Mathematical Thinking

Logical-Mathematical thinking is related to the ability to work and think in terms of numbers and the ability to use logical reasoning. The development of this thinking is key to the development of mathematical intelligence and is essential for the well-being of children and their development.

This type of intelligence goes far beyond numerical capacities, it provides important benefits such as the ability to understand concepts and establish relationships based on logic in a schematic and technical way. It implies the ability to use calculus, quantifications, propositions or hypotheses in an almost natural way.

We are all born with the ability to develop this type of intelligence. The different capacities will depend on the stimulation received. It’s important to know that these capacities can and should be trained, with adequate stimulation great benefits are achieved.

Why is it important to develop Logical-Mathematical thinking?

Mathematical logical thinking is fundamental to understanding abstract concepts, reasoning, and understanding relationships. All these skills go far beyond mathematics understood as such; the benefits of this type of thinking contribute to a healthy development in many aspects and the achievement of personal goals and achievements, and with it to personal success.

What we do when solving a mathematical problem: collect data, we separate it, observe the relationships they maintain or systematically solve its parts in a rational way. 

If we can understand mathematics and come up with logical solutions, we can prepare our minds when we have real problems. We will look for the best logic, we will see the possible solutions and we will relate the data we have to reach the conclusion.

How to think like a mathematician?

“Mathematics is difficult, but if you think mathematically everything is simplified”, as explained in the book ‘How to think like a mathematician’ by Professor Kevin Houston.

Next, we will give you some tips so that you can start thinking like a mathematician.

Question everything

One of the most beautiful things about mathematics is that everything can be proven. You don’t have to believe everything they tell you. If someone says something is true, you can ask them to prove it.

Or better, if you really want to think like a mathematician try to prove it yourself. Your reaction should always be to doubt and try to find a counterexample. Even if the result is true in the end, mental effort will help you question other claims in the future.

Write your problem in words

How can writing help me be a good mathematician? —You might be wondering. Phrases are the bricks with which we build our arguments. Mathematics uses arguments to make the proofs and prove the conjectures. It’s not about you doing your math like crazy!

Many students don’t think this is necessary; they often say, “I haven’t signed up for math to write essays,” or, “But I almost have the solution!” If you want to understand math thoroughly and think clearly, writing will force you to watch your arguments. If you are not able to describe them, perhaps it’s because you have not understood the bottom of the problem.

What if it’s the other way around?

Mathematical theorems are based on logic. They are syllogisms that ensure that if A is true, then B is also true. But if we turn the argument around, we would be saying that if B is true, then A would also be true. For example, if I say: “If I’m Spanish, then I’m European”, its inverse would be: “If I’m European, then I’m Spanish.”

A good mathematician, when he is sure that A “is necessary” for B, will always wonder if the reverse is also true. Sometimes it will be true and sometimes not, as in our previous example. If so, it will be said that B “is sufficient” for A.

Create your own world

A mathematician creates his own examples, some will be normal, some will be extreme, and some will be counterexamples. When you know the procedure for solving a type of problem, try to go further and look for similar problems that cannot be solved with that method and need improvement.

What would happen if…?

Good mathematicians like to ask themselves, “What would happen if, for example, I dispensed with this hypothesis?” By doing this experiment, you will be able to understand why a result is true or why an element of the proof is defined in that way.

New and more elegant theorems have appeared from weaker initial conditions than in the original! The idea is to always ask yourself new questions.

Undoubtedly, mathematics develops creativity, mathematical creativity is related to the ability to create ideas, solutions or questions that are novel from the perspective of the person who generates them.

How does the brain use mathematics to interpret the world?

Despite the bad reputation of mathematics, curiously, our brain is an expert when it comes to using it, although most of us don’t even know it. In fact, it does it constantly. It’s not about checking only if they have given us the change when we buy something. Its ability goes much further.

Clumsy as we think we are, our brains are especially good at calculating probabilities, according to research conducted at Princeton University and published in the “Journal of Neuroscience.”

These calculations guide our behavior on a day-to-day basis, and we use them, unknowingly, when crossing a street as when making decisions. It could be said that more than a subject they are a matter of life and death.

As the researchers found, our brains can accurately track the probability of several different explanations for what we see around us. And this ability is located in an area of the brain located behind the eyes, called the orbitofrontal cortex. Although this area has been the subject of much research, its specific functions are not entirely clear.

It seems that this area of the brain is related to the processing and regulation of affective states and behavior and is especially sensitive to reward and punishment.

It’s involved, as was already known before this research, in the detection of changes in the environment, both positive and negative, that may entail a benefit or a risk, which allows adjusting behavior quickly. And it seems to be critical in making decisions in uncertain situations.

The calculation of probability requires the use of Bayes’ theorem by the brain. It doesn’t matter if we don’t remember that this theorem allows us to find out, once an event has occurred, the probability that it was caused by another. Our brain uses it constantly.

And it did so even before the Presbyterian mathematician and minister Thomas Bayes enunciated in 1763 this famous theorem that bears his name.

“Having a brain capable of understanding that the world works differently in different situations must have been an adaptive advantage for our ancestors. And that is what the orbitofrontal cortex seems to do,” they point out. Apparently, our brain perfectly reads the mathematical language in which, in Galileo’s words, nature is written.

FAQS: How do mathematicians think?

Now that you know how to think like a mathematician, do you dare to solve them?

In this brief article we answered the question ‘’How do mathematicians think?’’ We will analyze the brain-mathematical relationship, in addition, we highlighted the importance of mathematical thinking and how to develop it.

If you have any questions or comments please let us know!

References

Jordana Cepelewicz. (2016, April 12). How Does a Mathematician’s Brain Differ from That of a Mere Mortal? Retrieved October 15, 2020, from Scientific American website: https://www.scientificamerican.com/article/how-does-a-mathematician-s-brain-differ-from-that-of-a-mere-mortal/

‌Tall, D. (Ed.). (1991). Advanced mathematical thinking (Vol. 11). Springer Science & Business Media.

Siswono, T. Y. E. (2011). Level of students creative thinking in classroom mathematics. Educational Research and Reviews, 6(7), 548-553.