Chapter II: Meaning and Form in Mathematics — A two-part invention
This chapter starts with how to build theorems off axioms in a formal system. For this, a very simple language with just three alphabets is used (p, q and #). you can make a string of any length that fits an axiom schema. I will let you read the book for all the gory details and play around in Wolfram for exploring the “pq#” system. For our discussion here, it is sufficient just to give 2 examples of this string that conforms to the axiom schema but also fits the definition of a theorem in that system. Look at the 2 examples and see if you can find any pattern emerging out of them and guess the theorem.
example 1: “#p##q###”
example 2: “##p###q#####”
If you answered that the first string resembles “1+2=3” and the second string resembles “2+3=5”, you will be dead right. Even though this was never explicitly mentioned while constructing the string, our brains associate the ‘inherently meaningless’ symbols to a meaningful reality. To be more precise, your brain mapped
‘P’ <-> “PLUS”;
‘q’ <-> “EQUAL TO”;
‘#’ <-> numbers.
In this case, this mapping is not accidental. The one who designed the theorem (Hofstadter, the author) explicitly attempted to represent a reality with symbols. This mapping (back and forth) between meaningless symbols and something that occurs in ‘real world’ is called Isomorphism.
The first revelation that is made is that the “perception of isomorphism” is what lends inherently meaningless symbols their meaning. Before you go further, take time to assimilate this. Once you come to terms with this revelation, the next one that is thrown at you is that while it is mind-blowing to discover the rules of the theorem and discover the isomorphism, one should anchor himself in humility because however revelatory his discovery is, there is no ‘THE’ meaning. There exists only ‘a’ meaning. For example, another person could have mapped
‘P’ <-> “EQUAL TO”;
‘q’ <-> “TAKEN AWAY FROM”;
‘#’ <-> numbers
With this substitution, the string in example 2 (“##p###q#####”) reads, “2 EQUAL TO 3 TAKEN AWAY FROM 5.” It will still make complete sense. For our perception of isomorphism to hold, the interpretation of p, q and # should resemble something in the real world. For instance, if I arbitrarily substitute
‘P’ <-> “horse”;
‘q’ <-> “happy”;
‘#’ <-> apple,
the string in example 1 (“#p##q###”) will read ‘Apple Horse Apple Apple Happy Apple Apple Apple.’ This will be a meaningless interpretation unless you are a horse happy to see an apple and you want to articulate in human language.
The important question that is posed in this chapter is “Can reality itself be modeled in a theorem?” The implication of this is that just like symbols are pushed on a piece of paper to make theorem, ‘reality’ is nothing but a formal system in which the elementary particles move around in a 3s+t (3-spacial coordinates and 1-time coordinate) system.
My random brain-droppings, interpretation and additional dimensions not covered in the book:
Though the chapter deals with formal system, I cannot but ponder over the following points in no order of importance.
1) The chapter begins with a dialogue titled two-part invention. The ‘two part’ here is a reference to the ‘Meaning and Form’ that is discussed in this chapter. It also refers to Bach’s master pieces. Bach wrote fifteen two-part inventions. My favorite in that is the Invention #8 in F-Major because our daughter performed this while she was in 3rd grade at her school talent show. You can see the video of her performance @ https://youtu.be/6yxTzhxwGn4.
2) Pikotaro turned the concept of ‘meaningless interpretation’ on its head and took meaningful symbols, namely, ‘Pen’, ‘Apple’, ‘Pineapple’ and combined them in the most insane and inherently meaningless ways and made the 2016 super viral hit PPAP. In case you lived under a rock and missed this gem, give it a listen. That’s 2.5 minutes of your life worth spent.
3) An interesting ancillary question to “can reality be modelled as a theorem?” is “can reality itself spring out from a well-defined theorem written on a paper?” Imagine your coding a theorem and somewhere in space-time continuum, a reality pops up in some corner of the universe. Now, put that in your pipe and smoke 😊
4) Outside of the formal system, the perception of isomorphism is witnessed everyday when you see human eyeballs in a kitchen sink, faces on the side of mountains and horses in clouds. In each of the case, our associating meaning to these inherently meaningless patterns happens because of our perception of isomorphism. We connect the patterns to something from our real world. This is so common that we have a name for it. This phenomenon is called Pareidolia.
5) If you wanted more examples of our seeing patterns where none exists, checkout the flight of the starlings @ https://www.youtube.com/watch?v=L_u4IcRy2dw
6) This illusion doesn’t just work on human beings. You can watch amazing school of fish form a giant ball to scare predators @ https://www.youtube.com/watch?v=15B8qN9dre4
7) Last and the most meaningful extrapolation, for me, is this. We formed a mapping from ‘pq#’ to number system because the number system and its arithmetic operations were supposed to be ‘real’. But let’s dig a bit on this.
a. The symbols ‘0’, ‘1’, …’9’ are as inherently as meaningless as ‘pq#’. These symbols are ‘arbitrary’. Only because of its consistent usage, they take meaning. We indoctrinate our kids to this abstraction from kindergarten that we don’t even think how artificial this is.
b. So, if we recurse one level, we find that even ‘0’…’9’ and the number system itself is given meaning because we are associating these arbitrary symbols and operations to an underlying ‘reality’. That underlying ‘reality’ seems to be that no matter what symbol we use, “a stick” plus “another stick” is “2 sticks.”
c. That may look like a satisfactory ending and it appears to be a quarrel on the choice of symbols. Like instead of using ‘0123…9’ advocating to use ‘௦௧௨௩…௯’. But that is not my contention.
d. What I urge you is to take a deeper look at the meaning of what is it that we mean when we say,
Is it the number of apple cells we are counting? is it the number of carbon molecules we are counting? A 240g apple contains approximately 100 million bacterial cells. So, when we add the two apples, should we discount the bacterial cells? Will the sum of any two apples in this case will yield the same result? No! we don’t nearly get to that level of complexity when we talk about the arithmetic operation. We count something called ‘apple’ which is, but a super-structure made of 100s of millions of tiny cells which in turn are made of even more molecules, so on and so forth.
e. We chose to stop at an abstraction that is convenient for us. There is no rhyme or reason for why we stop at that level. Hence, my contention is even more fundamental. My contention is not on the symbol and other trivial detail. My contention is that there is nothing inherently meaningful in what we call as ‘reality’ itself. As per chapter II, ‘isomorphisms induce meaning’. For this perception of isomorphism to happen, we are tying meaningless symbols to reality. But if we scratch the surface of reality, we find that there is no one standard reality to anchor to. What is happening is that we try to tie an inherently meaningless symbol to another set of inherently meaningless symbol which in turn is tied to even more inherently meaningless symbol and regress all this way to infinity.
f. And that’s where we are going to wrap chapter II.
References:
1. Play around more with the ‘ pq# ‘ system. Test if your string fits the axiom and/or theorem @ https://demonstrations.wolfram.com/PqSystemExplorer/
2. PPAP song @ https://www.youtube.com/watch?v=Ct6BUPvE2sM
3. Invention #8, one of the exciting 2-part inventions by Bach @ https://youtu.be/6yxTzhxwGn4
4. Flight of the starlings @ https://www.youtube.com/watch?v=L_u4IcRy2dw
5. School of Fish forming giant ball @ https://www.youtube.com/watch?v=15B8qN9dre4
