An excerpt from an email I sent to a few friends last week.
I’ve been trying to decide how to best organize my ideas. The challenge is that, in my experience, mathematical knowledge isn’t linear, so there isn’t a natural organization. Therefore, I’m going to start with some interesting ideas, to show what I’m offering here, and then proceed historically. These ideas may not make sense right now. That’s ok. Just keep them in mind while you read my explanations below and hopefully it’ll start to click.
- Math is a language. It has more in common with words than arithmetic.
- The language of math can’t describe all physical phenomenon. Just like there may not be words to describe how you feel, there also may not yet be any math to describe – for example – when an earthquake is going to occur.
- The language of math can be used to describe both the real world (similar to words in a newspaper) and theoretical worlds (similar to words in a fantasy books).
- Mathematical “proofs” exist in theoretical worlds but are prized because of what they can reveal about the real world (similar to writings like “Animal Farm”).
- As a language, perhaps what distinguishes math most from words is its ability to communicate relationships with pefect precision.
As I say above, math is a language that is exceptionally good at expressing relationships. With this in mind, try to wipe your mind of all you’ve previously learned about math and focus on this relationship aspect. We can begin to organize math into “kinds” of relationships:
- linear – relationships that grow equally fast or slow
- polynomial – relationships that grow faster or slower with bigger (smaller) values
- logarithmic – relationships that grow slower and slower and slower
- asymptotic – relationships that get closer and closer to a fixed value
- factorial – relationships that grow way way way faster as they get bigger
All of the above relationships we experience on a daily basis. The relationship between how far I press my accelerator down and how fast my car goes is linear. The relationship between my savings account and a compound interest rate is polynomial. The relationship between earthquake intensity and the Richter scale is logarithmic. The relationship between how often I practice basketball layups and how consistently I can make layups is asymptotic. The relationship between the number of different choices I could have made and how much time has passed in a day is factorial.
No calculations are required here.
Now, as human’s studied these various relationships two special kinds were discovered. Inferential relationships and deductive relationships. Inferential relationships are ones where we take a few specific experiences and make general claims about the world. An example might be, every person I’ve ever met named “Mark” is a man. Therefore only men have the name Mark. Deductive relationships on the other hand are where we go from general to specific. I am coaching the women’s volleyball team, therefore the player who gets jersey 10 is going to be a woman. Inferential relationships continue to be a thorn in the side of mathematicians to this day. Deductive relationships – on the other hand – have been mastered, though they have lost a lot of their standing within the mathematical pantheon thanks to developments like Gödel’s incompleteness theorems.
Within the category of deductive relationships there is an even more special kind of relationship, equivalence. That is, hamburgers are to cheeseburgers as deduction is to equivalence. Equivalent relationships are a stricter form of deductive relationships where both sides of the relationship deductively imply the other. Interestingly, human’s primary interaction with equivalent relationships is in nature. Human relationships don’t tend to have any equivalence though they may be equivocal (ba dum cha).
In 400 B.C. Greeks began to systematically study inferential and deductive relationships within planar geometry (a mathematical fantasy world). They found many equivalencies in this world . For example, they showed that the area of a square was always equivalent to the square of one of its sides. What was so interesting about their work was that they didn’t “solve” for the area of a square: they simply showed that the relationship between the side and area always existed (again, within their fantasy world).
Similarly, when studying triangles, they showed that the sum of interior angles for any one triangle is always equivalent to interior sum of any other triangle. Perhaps what was most groundbreaking was not that this relationship existed, but that they were able to prove it existed without actually solving for the interior angles. That is, there was no arithmetic, just deductive equivalence (A was equivalent to B without any adding or dividing or subtracting).
As the Greeks built longer and longer equivalence chains, (i.e., A is equivalent to B, B is equivalent to C, C is equivalent to D and therefore A is equivalent to D), they began to find that relationships that didn’t appear to be equivalent at all actually were. So, saying that the area of a square is equivalent to the square of its side is actually equivalent to saying all triangle’s internal angles are equivalent to each other. Eventually – over several hundred years – they discovered that the fantasy world of planar geometry, as we know it today, is equivalent to saying 5 basic rules are true:
- A perfectly straight line can be drawn between any two points
- Any straight line can be extended forever and remain straight
- Given a straight line a circle can be drawn using the line segment as a radius
- All right angles are equal (note, this doesn’t say all right angles equal 90°)
- Two, non parallel line segments will intersect once and only once
They named these five basic rules axioms (as in the “atoms” of geometry). These rules seemed so basic and intuitive they didn’t try to break them down any further. These rules, and all of the geometry that you learned in high school, were recorded in a book called Euclid’s Elements. This book is considered by many historians to be the bedrock of western civilization. The reason is simple, even though the book is several thousand pages long you can, in theory, teach somebody the above five rules and they know everything in the book. Even today that is still how advanced mathematics is taught. In my grad school classes we spend the first few weeks learning the basic axioms of what we’re studying and then spend the rest of the semester applying those axioms in useful ways.
Let’s briefly go back to this idea of a “fantasy world” and proofs. When a mathematician says they have proven somethingwhat they really mean is that they have shown some statement to be equivalent to accepted axioms. These axioms define the fantasy world we are operating in. The reason this isn’t a “proof”, at least as most people think about it, is that the axioms are simply accepted and may or may not apply to a real world scenario.
To see the connection between mathematical fantasy worlds (axioms), proofs (equivalency) and reality let’s consider two scenarios. In the first, we are trying to figure out how much land we will need to plant corn to feed 500 people. For this problem the planar geometry axioms listed above fit reality well enough to help us solve the problem. This means we can use planar proofs about geometric area to make sure we get enough land. The planar fantasy world fit the real world problem. In the second, imagine we are trying to find the shortest distance between two points on the globe. Without going into too much detail, the above axioms no longer fit reality well and none of our planar proofs help us. Instead, axiom 5 needs to be changed to “there are no paralell lines”. Once this has been done, we can develop new equivalencies to help with reality.
In short, axioms are assumptions no matter how simple and intuitive they may seem, and they never apply universally. That is, if you agree with my assumptions then I have “proven” X. If you do not agree or if my assumptions simply don’t fit the problem then I haven’t proven anything. Despite this limitation axioms are still incredibly powerful for two reasons:
- They can communicate extremely complex ideas to new students very easily
- They can often be built upon and added to without breaking previous work
These properties of Axioms are the bedrock for why mathematical knowledge continues to advance. Math can, with moderate ease in relation to its complexity, be passed on to new generations and new generations can add to it with confidence that they are actually improving things.
Well there you go. A basic introduction to axioms. An interesting application of Axioms is the Turing Machine (if you’ve ever heard of it). When people say Alan Turing created computers what they really mean is he discovered the seven axioms that define all computers (seen here). That’s it, just seven. Every computer that you have in your house from your thermostat to your smartphone to your laptop is equivalent to Turing’s seven axioms.