How to Start Thinking Mathematically

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Imagine you are a lowly clerk. You are in this rather mundane job, not necessarily because it is your chosen career path, but because you need to prove to everyone else, especially your immediate family, that you have some worth to society. In order to do this, you needed a job.

The daily work is routine. Your job is to review previously approved patents in comparison to new applications to determine if the claims that are already covered by previous patents would conflict with the new patent applications.

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Extraordinary Discoveries While Doing Boring Work

Since new patents are only granted to new ideas that have not been registered before; it is easy to see innovations. Nevertheless, to prove this you must search manually for all the previous patents that have been granted to make sure the new patent application is innovative and extends the “art” of invention, rather than simply replicating what has gone before.

Because your work is in the realm of intellectual efforts, your supervisors leave you almost untouched. In fact, if they knew how quickly you were able to see innovative ideas, they might reduce your pay. Nevertheless, you keep this intrinsic talent to yourself. To your managers, you appear to be diligently plodding along, to verify the innovative qualities of the newest patent applications.

Everyone in the patent office thinks you are doing a decent job, because your workflow goes forward in a speed that is acceptable. Nevertheless, the mental challenges of this work are way below your mindset. It is easy to do the work in an almost robotic automated fashion, because in this time before database searches, it is simply identifying any potential relationships between new ideas and old ones that is necessary for you to do a good job.

While performing these mundane duties, your mind fills with new imagined relationships. They do not come in terms of the language you are used to speaking and writing. Nor do they come in any language of a particular country. These new ideas come, as you daydream, allowing your mind to wander looking out the window of the upper story patent office. Moreover, they come in a universal language, which is the language of mathematics.

You eventually leave this job to pursue your new way of thinking. Your name is Albert Einstein.

The New Math

Not so many people have taken the time to understand the mathematical calculations that after years of thinking about it, Albert Einstein derived the most elegant and simple solution of e = mc2.

Energy equals mass times the speed of light squared.

This elegant result of complicated mathematical calculations proves to be so powerful and provable that it propels Einstein forward to international recognition and fame. Not so bad for a student kicked out of school. Yet Einstein was extremely saddened when he realized that the conversion of mass into energy is such a powerful force, when the discovery is placed in the wrong hands the science could be used to make a bomb.

Einstein was a pacifist. He escaped Germany, when the fascists came into power under the rule of Hitler, taking his brilliant intellect to America. He felt it was his moral duty to warn the then President of the United States, Franklin D. Roosevelt that his discovery and the mathematics behind it combined with the work of his scientific contemporaries could be misused as a weapon of war. He wrote incredible letters directly to the President to describe the impending disaster.

The intentions were good on behalf of Einstein, because if Hitler gained use of such technology, there was no way to stop the fascists. It was only that the United States vigorously pursued the production of the atomic bomb by thinking mathematically that thwarted Hitler’s efforts. Yet, Einstein was incredibly saddened that his mathematics lead to the attacks by atomic bomb of the United States on Japan that resulted in the deaths of tens of thousands of innocent lives in the cities of Hiroshima and Nagasaki.


Einstein was so moved by his sadness, and in some ways, he felt responsible for the real-world results of his thinking mathematically, that he spent the rest of his life lecturing around the world to promote world peace. We know Einstein for his elegant solution to the mathematical problem that shows mass and energy are interchangeable, but he would prefer we know him not for his brilliance, but for his speech about world peace, which is quoted here:

“Today, the physicists who participate in watching the most formidable and dangerous weapon of all time… cannot desist from warning and warning again: we cannot and should not slacken in our efforts to make the nations of the world and especially their governments aware of the unspeakable disaster they are certain to provoke unless they change their attitude towards each other and towards the task of shaping the future. We helped in creating this new weapon in order to prevent the enemies of mankind from achieving it ahead of us. Which, given the mentality of the Nazis, would have meant inconceivable destruction, and the enslavement of the rest of the world.”

Mathematics is Universal

What Einstein discovered is the universal language of mathematics. This language goes beyond the limitation of the written and spoken languages that exist on Earth. The symbolism used may not be understood elsewhere immediately, but the principles demonstrated by the mathematics are universal. On Earth, mathematics passes all cultural lenses of scrutiny and is interpreted directly without any cultural bias.

This is why Einstein who was a German Jew could communicate with scientists all over the world. The formula of e = mc2 is so easy to understand that an elementary student can comprehend the result of the equation. The mathematics to arrive at such as solution is very complex requiring pages and pages of proof.

Thinking mathematically requires this ability. It is the ability to see the world and the universe in its essence without being bound by cultural limitation. This is what Einstein demonstrated as his mathematical genius. He reveled in forming thought experiments, which are imagined examples of new ways to perceive something, which also considers observable facts. In this way, he extended the discussion to new ideas that had never been considered before.

This type of mathematical thinking problem-solving and proofs led Einstein and others who followed the same path to make extraordinary predictions that in many cases took decades to prove. When the proofs of Einstein’s theories did arise, scientists were both shocked and delighted in how accurate Einstein’s predictions were, when they were made decades before the proof could possibly exist. An example is Einstein’s prediction that time is elastic and that time is relative to the speed that an object is traveling in relation to the mass of nearby object. This warp of space-time was reported to be confirmed in the UK Guardian over 100 years after Einstein predicted this with his thinking about mathematical models.


Thinking mathematically is a path to greater understanding of ourselves and our place in the universe. Younger students are encouraged to look at mathematics as something far greater than simple arithmetic. Yes, one plus one equals two and multiplication tables and long division are still possible, yet this is equivalent to learning the most basic words in any language. By looking deeper and using the imagination, it is possible to use mathematical thinking in the way that Einstein used it, to discover fundamental truths that can be proven from direct experience. In this way, thinking mathematically becomes alive with robust new ideas and concepts that lead human beings to marvelous discoveries, yet always with the caution that evil people may misuse any such discoveries and this is the responsibility inherent in any new discovery.

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