In the last post, I discussed how Euclid made 3 huge mistakes. However, they were so subtle, so under-the-radar that it took over 2,000 years for anyone to take notice!
I also explained that filling up these “holes” in Euclid is the foundation of what we now think of as the ‘formalization of mathematics’ or the ‘axiomatic basis of mathematics’ (there are many names). And that THIS cuts to the core of the nature of our quest: the Development of Strength Theory & it’s Applications.
Consciousness itself is built from the same materials as mathematics: analogy, or what math-geeks (like me) call ‘isomorphisms’.
Before we begin working with the building blocks of Math, or what I call: the “language of love” (LoL); we need to get (a little) more detailed about the errors of Euclid’s ways. Tomorrow we will get into what can only be called PALEO MATH 🙂
Euclid Wasn’t All Bad
In fact, we was nearly perfect. His Elements was basically the Bible of Geometry for most of human history since his death, and for good reason. He introduced the level of clarity and rigor that is what all of modern mathematics is founded upon.
The architecture of the work was novel at the time, but is what all mathy-types like us would recognize as standard. You begin with definitions, then proceed logically to theorems that add richness to your subject, and you justify these conjectures with proofs.
His problem was that he was human. He didn’t have the luxury of learning from Euclid – if only he had been able to be reincarnated 2,000 years in the future. (Maybe he WAS…)
Our current ‘language of math’ was not available to him, and he was forced to use the language of his native land. He wrote in “plain” language, natural language, the kind of language a blogger might use (albiet a rather smart blogger).
He didn’t use “math-speak”.
That was his tragedy – of conspicuously Greek design. By using the wrong language, the flaws found milenia later were guaranteed from the start.
Euclid was fated to fail by his own God-like hand.
Of course, I use the word ‘fail’ for effect. He didn’t fail, he succeeded beyond measure. And his “failure” wasn’t a bad thing, but rather the generator of our success. By leaving questions in the air, he inspired levels of creativity that have opened up the entire universe.
Euclid’s 1st Error – Definitions
Because of this “wrong” language problem, Euclid’s definitions are not clear. Well… actually it is the opposite. They are TOO clear. Or both. Or neither.
See the problem?
When you use a specific word in a natural language, that word is connected to a connotation, a meaning, that people will AUTOMATICALLY associate with that word. Sometimes it is a word with only one clear meaning, other times it is a word with multiple meanings. But there is always a meaning.
In the strength world a word like ‘squat’ can have many connotations. Did I mean a ‘back squat’ or a ‘front squat’, a ‘single leg squat’ or a ‘two legged squat’? If I said a ‘back squat’ did I mean ‘high bar’, ‘low bar’, or ‘mid-bar’?
If you and I are trying to discuss something abstract, and the word squat is only a stand-in for something (but not necessarily literal), then we don’t need to KNOW what exact type of squat we mean (or whether the person is squatting to “depth” or what “depth” means, or if they are wearing Oly shoes, or wide-stance, etc.). Those issues are completely beside the point.
In order to counteract this, we try to stress that the word ‘squat’ is not to be taken too literally. Maybe a little. We didn’t mean it to be completely meaningless – but less literal than our minds are wont to take it.
Still, that is not good enough.
It’s a slippery cognitive slope. Once you step foot at the top of the slip-and-slide, you are going to end up at the bottom. Give a word a little literalism and you’ve opened the door to more literalism. It is an infinite regress, non-stop worry that we aren’t on the same page with the exactitude of our meaning of THAT word… never mind all of the OTHER words in our argument!
This is why mathematicians have now completely dispensed of ALL meaning associated with the “real” world. The symbols are COMPLETELY meaningless. Period. From there we prove what we need to prove. Later, scientists come and they figure out if the axioms of this system (or that one) fit their project well enough. If so, then the theorems derived in this system will be more (or less) predictive of real processes – directly in relation to the connection of how well the axioms fit the real situation at hand.
Euclid took a different approach. His definitions were meant to be VERY literal. A line was meant to be a line in the REAL world. A circle was a real-live circle. Etc.
The trouble is that infinite regress problem again.
- What is a ‘line’ in the real world?
- What do you mean by ‘real’?
- Space-time curves straight lines, what does that imply?
And on and on…
By attaching his definitions DIRECTLY to what he considered “real” objects on the earth, he pulled the rug out from his own arguments from the get-go.
Math is no longer done that way.
While mathematicians for thousands of years tried to patch up the holes in Euclid because they couldn’t bring themselves to let go of what they believed the WORDS ‘straight line’ to MEAN; modern mathematicians have no such attachment.
Mathematics is treated as a GAME. Tell me the rules, tell me what the words and symbols are, tell me how to manipulate them… then we’ll PLAY. No preconceived notions.
VISUAL – When I was a kid we used to play the Mad Magazine Game, which was a reversal of Monopoly. The goal was to LOSE all of your money. If you couldn’t bring yourself to imagine a person who would seriously consider doing something so inane and “unreal”, then this game would not be any fun. It WAS fun.
Euclid’s 2nd Error – Proofs
Euclid’s second issue was with Proofs. Specifically, his proofs had logical gaps in them. Eegads!
EXAMPLE – He assumes that the constructed circles in his (very) first theorem meet. How can we be sure of that?
He used diagrams and pictures to make his points (pun intended). But the use the these implicitly assumes that we’re discussing something in the physical universe, and that our particular interpretation of this physical universe is correct. What’s more, we are assuming that YOUR interpretation of the world around you is exactly the same as MINE.
The nature of proofs will be the core of my next few articles. But, suffice it to say that they require a rigor that simply didn’t exist in his day.
It’s the language problem again!
Natural languages are evolutionary creations and subject to all manor of weirdness that makes a good discussion nearly impossible.
Artificial languages like math can be anything we want them to be. So if we require ultra-clarity, we can simply build that in.
That’s EXACTLY what we will be doing tomorrow.
Euclid’s 3rd Error – Logic
Euclid’s last flaw was that he didn’t make clear what the logical rules of the game were. He was purely implicit, rather than explicit, about what those rules might be. And this leaves us all wondering what is true and what isn’t.
Because starting out with the same axioms and definitions isn’t enough. If you give different rules to the game, you’ve (by definition) changed the game, and you will end up with totally different results – different theorems, conjectures, etc.
So even if Euclid had fixed his definitions and his proofs were all consistent… without a clear explanation of what his logical rules were, we STILL can’t be sure what he MEANT.
Our goal is to patch up these holes, take wild forks in the road, and make ‘meaning’ meaningful via an increase in clarity.
Next up… Paleo Math!
Now go lift something heavy, Nick Horton