Where we learn that freedom
is a measurement of independence.
I hadn’t realized how emotionally isolated I was in my relationship with mathematics till the physical limitations of 2020 were imposed on the world. I hadn’t realized how strong my feelings of adulation were for math till there were severe limits placed on what I could do with my ‘free’ time – no hanging out with friends, no wandering about in crowded places, no in-person appointments or meetings or happy hours that weren’t occupying my time. For the most part, being in the company of others was an additive, not a subtractive. It was an input, not an output.
So, instinctually, after those cobwebbed months of WTF is going on in the spring of 2020, I fed that void of input by reading more than my usual amount of math-stuff. With the long stretches of time available to go deeper on a subject and to explore broad conceptual topics, a part of me opened up in a way that was, well, all encompassing and which brought up deep heart-felt emotions. It showed me that there were others out there that not only felt as deeply gobsmacked about theoretical math as I did, but they may have actually (also) cried about it because it brought on powerful realizations, and they (also) wanted to write about it.
My daily interactions in the Before Times was the way I reciprocated energy within the confines of my personal world, my space, my (mathematical) field. Each action separate – my output then volleying with your input, and building upon each other, with intention behind each piece of info. Sometimes those relations even had moments of magnetism.
A field is one of those basic things in math. A 2D field is a fundamental algebraic function – a set – which addition, subtraction, multiplication and division is used in corresponding operations. That set is contained, as one mathematical thing.
When you explore a 3D field in mathematics, that set opens up and now has depth, though it is still contained and usually within a defined limit or boundary. Now, it’s an X,Y,Z graphical set-up – where X is height and Y is width and the almighty Z is depth.
We all live in that Z-depth. Z is where things happen right in front of you and to you and with you in different measurements of proximity to where you are, and those actions happen linearly, one after another. This field that you are existing in is a vector space.
When you exist within a field, each of the actions within it are essentially ‘fielding’. I love the simplicity of this explanation of it:
Fielding a ground ball is one process. Throwing it to first is another.FASTER: The Acceleration of Just About Everything by James Gleick
So here I was, in 2020, in my new norm of living in a redefined field,
searching and ultimately finding a way to exist in life
with a comparable amount of fulfillment and depth that I had pre-2020,
with math as the majority replacement component.
I wasn’t fielding by having conversations with myself, I was fielding by reading the written voices of many other people, and I was unconsciously exerting a lot in order to perpetuate the fielding.
Fields are considered real, but they are non-material. (They) encourage us to think of a universe that more closely resembles an ocean, filled with interpenetrating influences and invisible forces that connect.Leadership and The New Science by Margaret J. Wheatley
I got deeper and deeper into reading conceptual and theoretical math writings as 2020 progressed. Writing about mathematics conceptually as opposed to algorithmically became more known in mid-19th century, especially those written by Bernard Riemann, one of the more well-known mathematicians of that time-period, and by the end of that century the writings of (my favorite) mathematician, Henri Poincaré. But in order for me to understand these conceptual readings, I knew I needed to go back to the foundations of mathematics, so I pulled out Euclid’s Elements, specifically reviewing Euclidean Space – the OG explanation of three dimensional space in geometry.
As with any field of study that has seemed to exist since the beginning of time, mathematical discoveries and concepts born centuries ago with the Pythagoreans in 6th century BC have subsequently gone through evolutions of discovery and an expansion of their definitions as future mathematicians explored those concepts more deeply, now with the resources and knowledge available to them during their own lifetimes. Some of those mathematicians were Riemann, Poincaré, and David Hilbert, who all continued the work that Euclid had begun.
Euclidean geometry is a theory, the first in human history. A model of a theory consists of the structures in which it is satisfied, a mathematical world, a place in which a theory is at home. Euclidean geometry is satisfied in the Euclidean plane … models make theories true and whether one theory could be expressed within the alembic of another. It is this idea of re-expression or reinterpretation that Hilbert advanced in his treatise, the tool that he developed.The King of Infinite Space: Euclid and His Elements by David Berlinksi
It was when I learned of Hilbert’s 1899 publication titled The Foundations of Geometry that I realized I was feeling — truly, deeply feeling — empathy towards this mathematician. This work of his (which he continued to revise over the next 30 years) searched for the certainty in Euclid’s work; it is very detailed, complex, and analytical. Hilbert’s deep concern wasn’t antagonistic towards the correctness of Euclid’s Elements so much as being intent on wanting to confirm they still held true, centuries after they were written. And if found not true, then he wanted to evolve it.
Y’all know I’m a “but, why?” kind of gal. I want to really really understand everything there is to know about a subject that peaks my interest. I reverse engineer any concept or belief that doesn’t quite sit right with me. Now here was a very well-respected-by-his-peers mathematician who felt a need to publicly explore his own “but, why?”
The humanity he showed in his brainiac work comforted me.
I delved further into fields and spaces and their dimensions. Field theory focuses on the activities themselves that occur within a space that has a boundary. It is reasoned that the existence of that boundary itself is one of the elements that influences the activity within the space.
My goodness, there are so many types of field theories that the ones that were born pre-twentieth century are called ‘classical’ field theories – like, Newton’s Law of Universal Gravitation or Maxwell’s Equations which are then further categorized as relativistic and non-relativistic based on if they existed before the advent of Einstein’s Relativity in 1905.
And then I read this and I sat back and hit the big ol’ PAUSE button.
Electricity and magnetism is a “field theory” which means that the degrees of freedom involve functions that depend on positions in space.The Princeton Companion to Mathematics
Do you know me well enough now to be able to guess what word in that definition made me pause? Winner winner chicken dinner if you guessed it was the word FREEDOM.
I knew I now needed to go to one of the books from a heart-centered mathematician. If anyone was going to use the word freedom in their writings, it’d be one of them. And…there it is:
The dimension of a space is a number that describes in a rough, qualitative way what life is like in that space – how much freedom you have to move around.Measurement by Paul Lockhart
Though I now realize that I’d seen the word freedom used in conceptual and theoretical as well as technical math readings before, I’m guessing it was because the call of freedom was so loud IRL in 2020 that when I saw it used in The Princeton Companion to Mathematics – a humongous 900+ pages hardcover door-stop of a book – it was unexpected at that moment, right when I was attempting to escape from the real world — and it hit me so hard.
I guess timing really is everything because since 2020 I now regularly consider, and sometimes apply, and also generally conflate the everyday definition of the word with its mathematical definition. Though they are basically saying the same thing, what was illuminating for me in the mathematical definition is the value that’s placed on independence that is in relation to its measurement – it’s degree – to something else that exists in that same field.
That – THAT – takes the word freedom to a whole new level of definition for me.
As is in Relativity Theory, there’s this acknowledgement that freedom also has a quantitative measurement of independence. It also adds to the earlier commentary from Gleick on fielding being a two step process. Each step independent while also being in relation in order for there to be a completed whole.
Let me now attempt to bring us back to where I started with this, grappling with that new experience of imposed isolation where my long stretches of time are dedicated to a deeper relationship with my math books, the reflection of how much of my time prior to 2020 was actively & intentionally stair-stepping ideas with others in person, and the study of what constitutes fielding and existing with others within some sort of boundary where degrees of freedom is a relational measurement.
I’d like to offer just a whisper of a touch upon the topic of self-imposed isolation.
Becoming invisible is not the equivalent of being non-existent. It is not about denying creative individualism nor about relinquishing any of the qualities that may make us unique, original, singular. Invisibility is a strategy for attracting a mate, protecting home and habitat, hunting and defense… It is nuanced, creative, sensitive, discerning. Above all, it is powerful.How to Disappear: Notes on Invisibility in a Time of Transparency by Akiko Busch
Yes, I read some non-mathy books, too.
I’m absolutely an introvert, through-n-through and always and forever, yet I had always pushed up against it for the first, oh, 40yrs of my life. It had always been impressed upon me that introvert<extrovert, that an introvert is less liked, less successful, less interesting, less less less. One of the greatest lessons of 2020 for me was throwing those messed-up fallacies away forever as I proved them false, one by one.
Though I’ve offered here the importance and value of any relation within your own personal mathematically-defined field, it’s been just as much a revelation for me to actively own the emotional layer of an experience, which in my case is the importance and value of ‘being invisible’ when you need to be. I definitely received that gift at my intersection point of 2020 and math, and I’m thankful for it.
Thanks math, you’re the best.