Finally, we come to the beginning.
Our discussion of Euclid and his errors was meant to elucidate the importance of where we are going, and give some deeper intuitions of what it all means.
Now we can get our math on!
Proof, Rigor, And The Sexiness Of Clarity
It helps to define (up front) the thing we are shooting for. In fact, I used the word “define” on purpose as we are about to learn one of the most important definitions in all of human discourse: mathematical proof.
The word ‘proof’ means totally different things in different contexts. So, to avoid the pitfalls of Euclid’s implicitness, we will be pedantically explicit at all times. For us, ‘proof’ will always be something with a very specific meaning directly related to how it is used in math and NOT how it is used in science – or any other field.
DEFINITION – Proof(1)
A ‘proof’ is a finite sequence of formulas such that:
1. Each formula is an ‘initial formula’, OR,
2. Follows from an earlier formula via the applications of a ‘transformational rule’.
Before we dive in, a little about the nature of Proof and why it matters…
Proof Is Power
A mathematical proof is mechanical. That is its power, its beauty, and where its connection to ethics comes from.
Unlike some faith-based religious style proofs where knowledge of something divine, or having a “calling” or an “enlightenment” experience is necessary before one is able to “understand” what is being said, or what is going on; a PROOF is completely egalitarian!
Anyone can not only understand the logic in the proof – it’s essence – but can also verify its validity for themselves. It is the ultimate expression of democracy, liberalism, and the Enlightenment ideal.
YOU have the power to KNOW and to ACT.
Few things in this world are more liberating than that.
Martin Luther believed that if he translated the Bible into the common language everyone spoke, then all people would finally understand the “word of God”. He didn’t bank the fact that each human would read into that book such remarkably different meanings!
Maybe Martin Luther should have translated the Bible into the language of mathematics. At least we’d all get our semantics right, and be on the same “page” before we began arguing about the content.
Many wars have begun over simple semantics…
The Primitive Basis Of A Formal Axiomatic System
In order to create a formal axiomatic system, you only need FOUR things. How cool is that? And they are surprisingly intuitive once you get to working with them.
ONE – A List Of Symbols
Includes all symbols to be used in the system. Common ones in a “mathy” or “logic-based” language would be ‘+’, ‘=’, and numbers like ‘1’, ‘2’, etc.
Keep in mind that the ‘numbers’ are NOT necessarily ‘numbers’ as we think of them! They can stand for ANYTHING. ‘1’ could stand for ‘pig’, and ‘2’ could stand for ‘turkey’.
This subtly will play a MAJOR role in our near future.
VISUAL: An alphabet!
TWO – Formation Rules
The legal combinations of symbols to produce formulas.
So, ‘1+1=2’ may be legal. Where as, “+ – + = 1” might not be. Who knows. You have to make sure you clarify that here. Clarity is everything.
VISUAL: spelling! how do you allow your letters to be arranged into words? In English, “WTF” is not actually a word, it stands for a whole phrase made up of 3 words!
THREE – A List Of Initial Formulas
The list we begin with. The goal is that when interpreted correctly, this list will become the axioms of our system.
VISUAL: a list of basic words, or a lexicon! (What you find in the dictionary.)
FOUR – A List Of Transformational Rules
The rules we use to create theorems out of the axioms we just created above.
VISUAL: grammar! By using the different combinations of basic words, you can create phrases, or sentences that are true or not true based up the rules you have set up.
Discussion: Artificial Languages, Meaning, And Who YOU Are
As you can see, an artificial language (aka, a formal axiomatic system) is very much like a natural language. It’s just that we’ve not only created it on purpose, but we also added a level of rigidity that doesn’t normally exist.
A word of caution! I have used as “VISUALS” the language of natural language to help make this more intuitive for you. Do NOT get too attached that. It isn’t perfect, and in fact, many others have used the natural language analogy differently than I have. (For instance, they call #2 the grammar, where I called #4 the grammar.) Building easy intuitions via analogy is what learning is all about – it is what our brains DO. And, in fact, is precisely what we building up the theory to support!
How’s that for recursive: Using analogy to explain how our brains are analogy-machines. Sneaky.
The BEST part about an artificial language is that the symbols don’t “mean” anything until we assign them meaning. This is going to prove to be our “backdoor” into understanding the nature of meaning, and how we can use that to make yourself into exactly who you want to be.
I’ll leave you with the point:
“You are not your biography” – Tony Robbins
And the corolary to that is my own saying:
“You are who you decide to be.” – Nick Horton
The meaning of ‘meaning’ is at the core of this philosophical discussion. And the nature of mathematics (and its “strange loopiness”) is going to help us greatly in understanding how to make that into a practice.
Now go lift something heavy, Nicolas Bourbaki 😉
- This definition is taken directly from A Profile Of Mathematical Logic, by Howard DeLong. A wonderful book with a very nice history of mathematics, and of the development of 20th century thought on metamathematics and logic. [↩]