Linear Algebra: Measurement
The essence of linear algebra is the matrix which can represent the coefficients on a system of equations. We can transform the matrix with operators, find the determinant of a matrix, diagonalize the matrix, put the matrix in a Jordan canonical form, find the eigenvalues and eigenbasis of the matrix, find the spaces the matrix spans such as the inner product space, and so forth. What these operations allow us to do is to solve systems of equations in a clean and systematic fashion which allows us to take measurements of the variables within these systems of equations. For instance, a system of equations is kind of like a logic puzzle where you are given clues and you have to fit your analysis to the clues to find a solution that satisfies all the clues. Abstracting the problem to a matrix allows us to solve the problem systematically and proceed in measuring the variables needed to get the values for them as measurement after all can be with a ruler from online but most likely is by a systematic series of steps and manipulation of known quantities. We can’t measure if we are not allowed to plug and chug in other words though designing the system by which we plug and chug may be involved, that is designing the apparatus to measure may be involved.
Linear algebra can be involved in finding vectors which span spaces that we can find properties of. For example we can say subspaces of spaces defined a certain way must have certain properties. We can also look for singularities and consistency to the matrices. Think of the movie the Matrix: if we abstracted away your properties and put them as numbers into a mathematical matrix and then manipulated this matrix like it was your life and then introduced the results back into your life: how would you feel? You probably wouldn’t care since the matrix manipulation is just a way of measuring your intrinsic qualities as a person. In the fact the notion of humans really being brains in a jar, subject to sensory manipulation, is a very old one at least since the Enlightenment. The key is how we are measured does not matter as what we are is what the measurement tells us not how we are measured. If I’m measured in terms of meaning for example, I should not worry about whether it is machines manipulating my brain or my body manipulating my brain as I am what is measured: I am what I mean not how I have meant as the methods to measure me would tell me. For example, perhaps being in a machine would tell me I didn’t know how I was creating meaning but that doesn’t matter as that is just a way of measurement: what is real is what I mean and I could only mean this well if I have had real experiences with real meaningful people and places for example. Does it matter if it’s your body which met another body or if it’s a machine who introduced your brain to another brain, if what you created together can be modeled by no machine but only measured by it and you are thus told you’ve created something alive and perfect: a human being. I am fairly confident I am not a brain in a jar but am actually a real live human being in a real world as the effort it takes to measure me may be high and machine-like in the requirements but what the machine has given me in measurements I know only real humans could have created; that is roughly speaking a quote by the poet Blake: to see the world in a grain of sand, and heaven in a wildflower, to hold infinite in the palm of your hand and eternity in an hour. The meaning of what he is saying to me is that only humans can perceive the world as it is directly while machines are only able to abstract and manipulate. In perceiving the world directly, humans are thus able to see how complex the world actually is compared to how simple machines make it to be, the complexity arising from the human’s ability to see through multiple levels of reality. Despite there being multiple levels of reality, however, there is only one you and you must live somewhere in that reality so there is only one reality that matters and that is yours. To have empathy is another question as we do try to care for others when we seek to understand them. What happens when humans act like machines and abstract and manipulate? I think humans cannot act like machines as when they abstract and manipulate they will see meaning in the numbers and respond to the meaning instead of solely to the numbers. I know my life is real now and I know that even if I wanted to make it a false life, I would have to act like a machine and ignore all the meaning but I am not capable of that as meaning whether it comes in a girl wearing band of flowers after she told me she will be right back, or meaning as it comes in between the lines in a math textbook, it is the same to me in the end: I love her because I can recognize her and it will be worth it in the end. I hope it will be worth it in the end.