A Bridge to the Abstract:

In the early 1700’s residents of Königsberg, a Prussian city now in Russia and called Kaliningrad, would entertain themselves with a puzzle that, at first, seems to have little to do with mathematics. The city had seven bridges that crossed the Pregel River and connected two sides of the city as well as two central land masses, one an island, and the other splitting the river into two distinct branches. The question residence would ask themselves during lazy Sunday walks was, “is it possible walk around the city in such a way as cross every bridge only one time and return home?”

When the mathematician Leonhard Euler arrived in the city he first considered this a silly problem. After thinking more about it, he presented a proof in 1735 that no such walk was possible, and in doing so he gave birth to an entire field of mathematical knowledge: topology or “rubber sheet geometry”. Euler solved this problem by ignoring the reality of bridges connected to land and rather thought abstractly about the connections between nodes and links. The location of the bridges and the distances between them didn’t matter, but their position and connections in the relationship to each other did. From there he could create a language for solving this problem or any similar problem for ever.

This jump to abstraction is typical of mathematical knowledge. The use of pure reason using the language of mathematics proves useful for solving problems in the physical world, which once inspired Eugene Winger to comment on “the unreasonable effectiveness of mathematics natural sciences”.  Within the sciences observation of the physical world is of great importance, but in mathematics the opposite is true. The senses, from a mathematical perspective, corrupt knowledge. How, then did Euler build his knowledge without needing to experience the walk himself? How did he know how to make that jump into the abstract world of links and nodes? How does a mathematician know?

Math and the WAYS OF KNOWING

Reason:

Reason comfortably sits in indisputable supremacy within the field of mathematics. From five axioms Euclid was able to fill 13 books of mathematical proofs logically deduced. Mathematics, more than any other area of knowledge, depends on deductive reasoning to build knowledge. If we look at Plato’s philosophy of a tripartite human soul split between function, emotion, and reason, and know that he valued reason as the highest function, then we can completely understand why his school, the Academy, had “Let None Ignorant of Geometry Enter Here” above the door.  

To give a quick example of this, let’s use the Pythagorean proof of irrational numbers. First, we have to define our terms (Language): a rational number is a number that can be communicated using a fraction,ab, where both a and b are integers and b is not equal to zero. We also need to assume that a and b have no common factor or the fraction could be simplified. Now let’s prove that √2 is irrational. From our definition we can say that 2 cannot be expressed as a fraction: √2 = a/b or = √2^2 = (a/b)^2. From here Pythagoras used “reductio ad absurdum”. He rearranged that statement to say a^2 = 2b^2 and said, “let’s say this is true where a and b cannot have common factors and b is not equal to zero”. If that’s true, then a2has to be even because 2b^2 is divisible by 2.  An odd number multiplied by an odd number is always odd. So a must be an even number and we can put it equal to 2k (a = 2k). If we substitute this into our initial equation we get 2k^2=2b^2 … perhaps you realized already that if 2b^2 is divisible by 2, and a^2 has to be even, which, by definition, means that it is also divisible by 2, that means that they have a common factor (2) and this contradicts our initial assumption that that 2 can be expressed as a fraction with two integers without a common factor. √2 is therefore irrational.

This is how math uses reason.

Language:

As you can tell from my explanation of reason, language is extremely important in mathematics as well. If you don’t believe me, imagine doing something as simple as long division using Roman numerals. European mathematics flourished in the Renaissance after the introduction of Arabic (or Indian) numerals, thanks in part to the influence of Leonardo Pisano (otherwise known as Fibonacci). With a new language based on only 10 symbols and value attributed to the location of the symbol, mathematics became much easier.

Mathematics also has a grammar. Think of the order of operations (PEMDAS). There are agreed-upon symbols and rules for the mathematician to enter and then communicate the abstract world of logic.

There are numerous examples of when the introduction of a new symbol to understand a mathematical concept allows for new knowledge to be constructed (X to signify the unknown quantity, i to signify imaginary numbers, et cetera). There are also plenty of examples when a new grammar is introduced and builds more knowledge quickly (graph theory, group theory, et cetera).

Faith:

First, there are philosophers who can make a convincing argument that we should question whether or not numbers exist. To build mathematical knowledge one needs to begin with the basic assumption that they do. This is taken on faith.

Additionally, mathematics often depends on what Coleridge called the “suspension of disbelief”. For many years the square root of -1 was disregarded and when Descartes called it “irrational” he meant it pejoratively. Once Mathematicians like Bombelli and Cardano suspended their disbelief and began treating i as a mathematical object, proofs began popping up with applications for pure mathematics as well as applied.

The mathematician Paul Erdős famously viewed God as a stingy protector of mathematical truths. He believed that God kept these secrets of mathematics in a book, and it was the job of the mathematician to steal them. When a mathematician came up with a particularly clever proof he would exclaim that it was “straight from the book”. Erdős believed that the language of mathematics is a universal reality rather than a construction of the human mind; that Mathematic knowledge is discovered rather than invented.

Imagination:

Since imagination is the ability to picture in one’s mind that which cannot be experienced with the senses, mathematics is particularly dependant on Imagination. One of the famously unsolvable equations of mathematics is the ability to deduce a series of steps to trisect a given angle geometrically using a straightedge and a compass. You may think I’m speaking nonsense here since you can clearly trisect any angle simply by looking at a protractor, but when a mathematician thinks about these questions she is not thinking about the physical world, but the abstract world of logic. There is no pencil sharp enough to draw a line on a protractor infinitely small, only the imagination can do that.

A great example of mathematical imagination is Edwin Abbott’s Flatland. That imagines a three dimensional shape entering a two dimensional world, which then helps us imagine four dimensional space, which cannot be perceived with the senses.

Intuition:

In his book about the history of Algebra, Unknown Quantity, John Derbyshire discusses the discovery of negative numbers and writes, “it is situations like this that make us realize how deeply unnatural mathematical thinking is. Even such a basic concept as negative numbers took centuries to clarify itself in the minds of mathematicians…”

There is something intuitive in the concept of numbers and we can even see animals demonstrate the ability to count. If an animal is confronted with competition for resources, the animal can deduce whether or not it should activate a fight or flight response based on the number of potential enemies it faces. Beyond simple arithmetic, though, intuition needs to be trained in mathematics and the language needs to be learned similar to a native language. One is not born, not even Gauss, with the ability to imagine the rules that govern the world of vector spaces.

Memory

If a mathematician can prove a theorem, that theorem is true forever. No one will ever find a counterexample to the Pythagorean Theorem (in an Euclidean space), consequently that knowledge will be remembered, passed down, and built upon by all mathematicians forevermore. Mathematical knowledge, perhaps more than any other Area of Knowledge, builds upon itself continuously and the memory of previous proofs have as much importance today as they did when first encountered. Mathematical knowledge from two thousand years ago is not studied only by historians, but by aspiring mathematicians. The knowledge is continuously and persistently relevant to the creation of new knowledge.

Sense Perception:

If I showed you three apples on a table and asked you how many there were, you could easily, without needing to explicitly count them, tell me that there are three. If I then put 3,426 apples in front of you and asked you how many, you would probably say “lots”, and then have to count them, using the symbolic representation of numbers, to get a more exact answer. Mathematics deals with this jump into the abstract. 3,426 cannot be perceived through the senses, and this is at the heart of mathematical knowledge. Field research is unnecessary in pure mathematics because it exists in a rational and abstract space.

Emotion:

The mathematician GH Hardy in his beautiful essay A Mathematician's Apology wrote of a “mathematics which is eternal because the best of it may, like the best of literature continue to cause intense emotional satisfaction to thousands of people after thousands of years.” Though innumerable school children may claim otherwise, mathematics has a common emotional appeal. Games like Drop 7 on smartphones, or Sudoku in newspapers draw millions of organic users because emotional high of identifying a logical pattern or the completing a mathematical puzzle pulls us forward. It is in pursuit of this euphoric feeling that Hardy pursued a pattern in prime numbers, or Bombelli decided to keep solving equations that included the square root of -1, or Andrew Weil to scratch away in his flat at a proof of Fermat’s Last Theorem.

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