I am a verbal person: A trained philosopher, an amateur writer, a reader, a pontificator, an armchair political pundit. I am the person who feels she has something worth saying. Recently I decided the best thing for me to do would be to force myself to do some serious Mathematics, which I used to be really very good at, back before the turn of the century. I decided to take my Humanities-soaked brain and make it do STEM again. I expected to refresh my memory about the joys of Mathematics. I did not expect that I would gain key philosophical insights, which is what happened.
Gil Strang's now famous 1806 Linear Algebra course is available as a free MOOC through MIT. I bought the textbook he wrote, I did the practice problems, I really and truly took the course. Being able to rewind and re-watch was golden for my "attention deficit" brain. Some of the lectures, I probably watched twenty times. If I had unlimited time, I would take 1806 again every year. I liked the course so much that I think it should be a required course as part of every college curriculum. I cannot recommend the course more highly to anyone. This is not a humble-brag: If I can do it, then so can you.
The Basic Building Blocks of Linear Algebra
Professor Gilbert Strang is a teaching genius. I am not a teaching genius. The following is a laughable oversimplification of an entire sub-field of Mathematics, which is only now hitting its stride, with Machine Learning applications. But oversimplifications have their place, especially in the formation of new ideas.
In Algebra, the basic objects are equations. In Geometry, the basic objects are shapes in a Cartesian plane. In Calculus, the basic objects are changes in time. Linear Algebra combines all of these types of objects, encoded into "The Matrix." A matrix is a system of equations. A very simple matrix can capture polynomials (equations), coordinates, and changes in time. The Matrix is itself an object (not just a collection of objects) because each system of equations (each matrix) has system-level properties.
In a system of equations, the first property to consider is whether there is any solution set to the system. In an single algebraic equation, you can always solve for one unknown, by setting the whole thing equal to zero. But with a system of equations, i.e. a Matrix, you can solve for more than one unknown. The basis for your matrix consists of all of the unique equations that comprise it. As long as the basis matrix contains more equations than unknowns, it will yield a solution (or a no-solution) to every variable of every equation in the entire system. Because a matrix is itself an object, it is like one gigantic equation, like a Rubik's cube made out of Rubik's cubes.
The most interesting property of matrices is how they facilitate game theory. Say you have a system of equations, and you don't have enough unique equations to solve for all of the unknowns. You have A+B=C, and you don't know A or B. Well, you can substitute a theoretical A, and that will give you the solution for a theoretical B. Same thing with a system of equations. With Linear Algebra, you can do trial and error at a God scale, and see how multiple other equations in the system resolve, depending upon the substitutions you make.
Matrices can be factored, i.e. split up into separate systems of equations, or subjected to principal component analysis (a type of factoring), whereby latent properties are isolated. I am not going to get into the particulars of eigenvalues and eigenvector matrices, but suffice it to say that matrixes can be distilled into sub-matrixes, which contain the "bare bones" components of other matrices, that when combined, reconstitute the original. You can see why matrix math might be important for compressing and decompressing information.
Paired with quantum processors, or just really (really really really) fast processors, Linear Algebra can give us a God's eye view into an entire domain of complex systems. As hyper objects, Matrices provide a concrete example of the mind-world interface, in which the mind cognizes something real, objective, and potentially infinite, into a finite dimension, without loss of "Truth" as a latent property.
Last, but not least, there is the "Zero Vector," which must be included in any subspace implied by a matrix. Subjectivity is objective insofar as the view from somewhere, anywhere... always implies a view from nowhere.
Basis and Metaphysics
Descartes' Meditations on First Philosophy often is reduced to the cliché, "I think therefore I am." Something like this: In order to doubt that one doubts, one must doubt, and therefore at very least it is certain that one doubts. In the popular consciousness, the philosophy of "Modernity" gave us legally irreproachable certainty, but at the price of literally the whole world. The modern subject can be certain that it is, but only when cornered within its own thoughts, engaged in a devastating psychological breakdown... The post-modern subject may avoid philosophy altogether, through a mix of endless toil and distraction, but the modern subject occasionally still wonders.
Revisiting Descartes with a Linear Algebra lens gives "Modernity" a new life. Perhaps the turns taken were not all that wrong. For the slightly more sophisticated student of René Descartes, "I think, therefore I am," was a nuanced argument about trans-dimensional objects, and assumed a frame, which would be relevant today. The Meditations were written to appeal to the masses (nice try) and so they open much like a self-help book might open today: Let me doubt everything, including my own sanity, and see what there is left.
"I think, therefore I am" is a sleight of hand. Slipped into the argument, as if through the back door of the nightclub. Doubt gives rise to perfection, and the idea of perfection must be sourced from God, because every object contains at least as much objective reality as the idea contains subjective reality, and so only something perfect can yield the idea of perfection (or error). Wut? Exactly.
Basically, Descartes imports God into the very act of scanning for error, into the very activity of doubt, through the idea of perfection. What does that sound like to you? Because to me, it sounds like he is saying that "Truth" is a trans-dimensional object; one that we are, by the way, hardwired to cognize.
The Hyper Object: Sub specie aeternitatis
In practical applications of Linear Algebra, dimensionality reduction is a money-saving practice. By simplifying high dimensionality matrices into simpler, smaller matrices, we can save computational power. Transforming a lower dimension matrix can be a fraction of the work involved in a larger one. Then, the smaller matrix can undergo transformations with a cryptographic key to reconstitute the original.
"Principal Component Analysis," and other forms of dimensionality reduction, can solve a host of philosophical problems, as well. The fact that a technique developed by hyper-technical capitalist forces yields such a perfect illustration for salvaging modernity in philosophy may just be a romantic coincidence. In any case, it is worth looking at metaphysically. What if the idea of a "trans-dimensional object" is not just as metaphor, not an epistemological term, or an onto-theological one? But in the apperception of reality itself?
With modernity, your metaphysics always imports a healthy dose of subconscious epistemology, but does it have to? In other words, before even making the distinction between the object and the subject, what if we see our own perception as a transformation? The original object is the trans-dimensional object, which contains itself and ourselves, as principal components. As we percieve what is true, we reduce the dimensionality of the trans-dimensional object, and produce a rendering of it that is true, as in truth-ful, but perhaps does not contain the whole truth.
Spinoza's Ethics comes to mind here, in all of its mathematical weirdness. Something about Geometry, and grasping objects under the light of eternity, generates standards of moral perfection. It is not quite clear how Geometry gets us there. But maybe Linear Algebra does, via the idea of "TransDimensional Object". In Spinoza, our perceptions pick up certain facets of objects that are infinitely complex, and the idea of dimensionality reduction easily squares this circle.
In terms deontological ethics, if there is such a thing, we would be perceiving an objectively real object, whose dimensionality exceeds our own. Thus, sacred texts, which provide a divine instruction manual, as it were, for how to live the good life, could well be real apprehensions of real orders, direct from God, but only reduced to a human dimension. We exist in three, four dimensions tops, within our most rudimentary imagination. But instructions coming from God, clearly would exist in more dimensions, and would by necessity be mysterious, vague, and even cryptic, while being totally truthful. This solves a lot of interreligious squabbles, at least.
The difference between imagination and faith is the key. To see the truth of any given statement, or any given value judgment, one need only see it "Sub specie aeternitatis" as "in the light of eternity," something that only faith in the infinite (or a really good computer) can deliver.
Pitfalls of Modernity: The Free Will Problem
The most transformative insight for me, as someone who tried and failed too many times to think myself out of the pitfalls of modernity, was the idea of dimensionality (and "degrees of freedom"). At a gut level, I have always found that the "free will" problem is clearly absurd, but equally intractable. Intuitively, I felt sure that human nature is so obviously underdetermined, that it does not abide by the "causal chain" like the normal objects of human perception. "Otherwise what is the point of all of this [gestures at everything]."
The best I could ever do was to articulate the fact that "freedom must be the horizon of possibility" of moral inquiry itself. A sort of "in order to dispense with philosophy, one must philosophize," which is only a late cousin to the Cartesian, "doubting doubt requires doubting. But what if I can re-frame the question of Free Will within modernity, to eliminate the very expectation of "determinism" in the "world"? That would certainly be a more interesting starting point.
But here if I borrow the idea of a hyperobject from Linear Algebra, a lot of my free will problems resolve. A hyper object is an object whose necessary and sufficient dimensions exceed the dimensions of inuitive objects. Basically, any matrix with more than four dimensions exceeds the human imagination, and yet it continues to be a rule-governed object, full of truth and properties and laws. All that nice stuff. In other words, once you let go of the idea that "reality" is the "physical universe" so much more becomes cognizable, without any of the traditional parsing. We exist in Reality, in Being-as-a-Whole, and not just the "physical universe."
Causality doesn't allow for free will? Why not? Cause of the laws of physics? Are you sure the laws of physics apply to the question of human behavior? What are words? Concepts? Thoughts? Things without geo-location? They're a hallucination inside your brain. OK. Possible, but why settle on an explanation that makes no sense and is so demented? What if you just got the object wrong? "The physical universe" and our place in it? The physical universe is just one aspect of a trans-dimensional object. A favorable explanation that makes room for divine revelation, UFO sightings, free will, and science
I have found that this type of technical language is the best language to use when talking to people who come from a modern, secular background (i.e. myself). Even the "horizon of possibility" language, couched as it is in the language of probability and statistics, seemed a bit too bullshitty to me. Rather than being trapped inside of the physical universe, the physical universe turns out to be just one facet of a multi-faceted trans-dimensional object, and to say that it only exists in three or four dimension.
The 3D or 4D (if you include time) "Physical World" is not a wrong representation of the God object, it's just dimensionally insufficient. "The World" can be equivocal, and equivocating on "the world" at the outset of any philosophical investigation is just going to yield a system with no solutions. The object has been reduced past the point of losing its basis. Necessary and sufficient vectors have been left out. Where are we? What is inside of what? We clearly are together inside some sort of something. A soup, if you want to call it that? And what sort of substance is that soup made of? Is it the type of substance that can support free will? Maybe. And "maybe" is all we need.
Understanding Linear Algebra has liberated my mind from the moronic puppet theater of my lizard brain. Within the warm embrace of academia, the "Free Will Problem" is not really a problem, or if it is one, it is a much more subtle problem than the adolescent version of the problem. But outside of academia, most people in a modern world, if they so much as attempt to explain freedom, you will find that the free will problem very much is a problem.
As recently as Wittgenstein, who is in many ways the Foucault of Analytic Philosophy, and represents the forefront of philosophical thinking in Modernity:
Ludwig Wittgenstein, in Notebooks 1914-1916:
The work of art is the object seen sub specie aeternitatis; and the good life is the world seen sub specie aeternitatis. This is the connection between art and ethics.[5]
Later, in Wittgenstein’s Tractatus Logico-Philosophicus:
6.45 To view the world sub specie aeterni is to view it as a whole—a limited whole. Feeling the world as a limited whole—it is this that is mystical.[6]
The Language of Linear Algebra
I doubt I am the first one to notice that matrix objects are cool and heady. "The Matrix" was a globally popular movie, and has spawned its own lexicon of philosophical terms."Decompression of Being" was the amphetamine-addled Sarte's metaphysics, from Being and Nothingness. As armies of STEM majors pile into Linear Algebra classes, looking for inroads into becoming engineers, in the hopes of avoiding poverty and ruin, only more and more people will adopt the language of Linear Algebra into everyday vernacular.
Philosophically, I have always felt that the best you can do is game out various ontological systems. The body, the minds, the spirit. All of these things mean something within philosophical systems. We have more unknowns than knowns. We can be confident that we all perceive an objective reality, which is real, despite being irresolvable for us. We concieve it in fewer dimensions than would be required to fully grasp it (and here I go to metaphorical language quite intentially), and therefore it will always be a system with multiple potentially correct solutions.
There is something here, which remains to be explored. There may be something new under the sun, after all.