Now don't start that again...

Everyone's favorite puzzle to hate is the old and trusty, 'How do I know that my red looks like your red?'

It's an old question, and most of us have wondered it from time to time. Sure, you can I can both agree that the red apples look different from the green apples, but apart from the words you use to describe them, what does your red have in common with mine? Could the be different?

A lot of philosophers want to make this problem disappear in a poof of smoke. It's a 'pseudoproblem', a 'bewitchment of language'. It's metaphysical hogwash. There's no such thing as 'my red' or 'your red', there's no such thing as RED! Only red things of course. Nothing to see here. Move along.

But then most of us have met at least one person who is red-green colorblind, and they throw a pretty big wrench in things. They see things differently, that's for sure. But how do they see things? What does the world look like if you're colorblind? I can imagine a black and white world, But one where red and green are the same, but the other colors are still different? What could that mean? What could that be like?

A good way to go about these things is the synesthetic route. If we can get really good at describing colors in terms of other senses, and we can agree on things that we supposedly agree on (yellow is warm and happy and sweet say) then we can try to imagine what it is like to miss out on a few colors. We can get a feeling for how the colors we see sync up with the tones we hear and the things we smell and the ways we feel, and we can stretch our imaginations just a little and get into somebody else's head.

But a lot of us have rather rusty imaginations. We might just try manipulating some photography: I know what a black and white world would look like, because I have seen black and whte pictures and movies and whatnot. And if we can do one dimensional color (greyscale woooooo) and we can do three dimensional color (how many hex codes do you know?) then surely two dimensional color can't be a problem. Just lock one of your three dimensions to another (probably Red to Green, for accuracy), average them out and set both to the same value every time. Or just keep one of the values at zero all the time. Or any other way of slicing a plane out of the color cube. If you play around with it, you should at least be able to get some sense of what it might be like if You were colorblind anyway, whether or not that has any real bearing on the question at hand. But what about the colorblind person trying to imagine what the rest of us see? Some of us, after years of trying to imagine a color we have never seen, gave up and decided that we can only imagine re-combinations of things we've seen somewhere before. And so the colorblind person is just out of luck if that's how it works. But it can be fun to try every now and again anyway. Maybe you'll see a new color. You just won't be able to show anyone else, now will you?

It's thinking like that that drives empiricist and behaviorist and language centric thinkers up the walls. What the hell could it even mean to imagine a color that doesn't exist? That's just insane. Gobbledeegook. Balderdash. Humbug. But you know, tell that to the colorblind kid who's decided that they're living in a world of schizophrenics who all think they can see a color that isn't really there. And tell that to the scientists who are looking at the light spectrum outside of the normal visible range. And for that matter tell that to your skin, which senses heat, which is just electromagnetic radiation (also known as 'light') in the infrared frequency range, but can't tell the difference between purple and yellow.

So: whatever Wittgenstein or Hume might have to say about the problem not being a problem, there are an awful lot of loose ends to tie up. If only we could figure out just what our sense of the color red was, then we could maybe have some idea of how it could differ in other people, even if we still had trouble imagining it.

Ok so strap yourselves in for some fantastic Brain Science, and maybe a little lesson in calculus, we'll see.

What your nerves do is detect change. This is an important first thing to notice, since we're used to thinking that our nerves detect properties, like temperature, color, tone, etc etc. But each of your nerve cells is like a tiny little solipsist. It doesn't know that there's a world out there and it couldn't be bothered to imagine one. But certain situations will either excite the little self centered beast or relax it. After a while though they get used to the situation and forget what happened if things don't continue to change. This sounds kind of crazy, but there are lots and lots of ways to prove it to yourself. Like the ol' three bowls of water trick: one with ice, one lukewarm, and one hot. Left hand in ice, right hand in hot. Wait for a while, then stick both in the lukewarm. If you've got a certain sort of metaphysics, you should now clean up your brains. Then stare for a while at something with a nice bright color, and then look at a white surface of some sort. The after image that you see in the optically opposite color of whatever it was you looked at to begin with is your eye telling you about whats going on inside you eye. Because that's what eyes have access to: the changes that happen inside them. That those changes happen to correlate wonderfully with what's going on in the outside world is something we'll look at in a bit, but right now, just notice that the colors that you see are the result of changes going on inside you eye, just like the temperature your hands feel is about changes taking place in the hands themselves (that's why they feel different in the lukewarm water, one is getting warmed up and the other cooled down; those little solipsistic nerves don't care what the objective situation is, just their own excitability).

Well all right then Mr. Fancy Pants, nerves detect changes in themselves, so what?

First we need to get a little bit more clear: since all the nerves do is fire in a rhythm that speeds up or slows down depending on their situation, we can't even say that the nerves detect change so much as rates of change. They don't know that anything is different. They don't know anything at all, the little critters. They just fire faster, or slower, that's it. And the nerves they're connected to then, they receive signal faster or slower, and so they react in a similar fashion. A given nerve cell in the chain leading up and into you brain will be connected to lots of nerve cells, and when enough of them fire, it will fire, so if a bunch of them speed up or slow down, the one we're looking at will also speed up or slow down. Things do get a little bit more complex when we throw in that certain nerves have an inhibitory function on some of their neighbors (which just means if the inhibitor speeds up, the inhibited slows down) and that neurotransmitters of all sorts change just how this rat effects that rate, but what we really want to notice is that it is rates of change that constitute the messages making their way around your nervous system, up and down your spine and around and around your brain.

[if you've either got a really good sense about how your brain works, or don't care to learn, or find my droning on and on terribly dull, you can skip down a bit to the next set of brackets]

Now then, these pulses in a rhythm, as they make it into your brain come up through the part of your brain that evolved the earliest first and make their way up to the new primate parts of the brain last. We can roughly divide the human brain into three sections, the reptile part, the proto-mammal part and the new fancy hardware. If Antonio Damasio knows anything about the brain, each of these newer areas get all of their info from the part that's older then them, and it's this cascade that generates feeling, in the following way (Roughly. His book, The Feeling of What Happens is more than a few paragraphs long, so I won't pretend to be capturing everything here. Also: I am not exactly following his vocabulary so much as the shape of things he describes, so this isn't a repetition or explication so much as a riff off of his work): when the rhythm hits the reptile brain, it bounces around for pure cause and effect. Certain types of rhythms trigger certain immediate effects, without there being any way to say that there was really awareness involved. This is just knee-jerk response type activity, nothing like thought or feeling, or at least nothing like thought or feeling that's going to make it into a poem anyway. At this level, we might say that there is only a sort of protoself. No real memory structure, no real sense of the future or the past, no real processing even, just a lot of stuff all going on at once, and sometimes the fight or flight alarm goes off and everything goes crazy. Damasio, for his part, will say that if this is all the brain you've got, we don't wanna say you have feelings anymore than we wanna say your computer has feelings. Input goes in, output comes out. Not too exciting.

But then just above this reptile brain is the middle of the road, the proto-mammal brain stuff. This middle part of your brain is hooked up to the lower part, and it keeps track of rates of change in the reptile brain, and can then nudge it this way or that if the situation calls for it. Now we're gonna call this structure the Core Self. This bit of newer brain hanging out with that bit of older brain, it's not so immediate. It can take things in over time, and so it can change a little bit slower. Sure, it can change real fast if things get messy down in the reptile brain, but since its one step removed from everything, it can take the time to have some feelings about whats going on. (I'm doing some serious injustice to Damasio here, but really I think this is close enough to whats going on without getting into even messier details.) Now with this middle section we can start keeping track of a little bit more of the past, anticipate a little bit more of the future, have a little sense of time, maybe some memory even.

And then there's the party upstairs. The brand new cortex of complexity. This outer shell of your brain gets input from everything going on downstairs, and, far away from the world as it is, sets about trying to figure out what is going on outside the skull it's trapped in, and in the meantime make up some stories and characters and scenery to go with it. The so-called autobiographical or extended self. This is where language and mathematics and social relations of all sorts can happen. This is the Brain as we think of our Brain, at its very best, cruisin along and just trying to figure out this wild wild world by carefully monitoring and then affecting the stuff gong on down below it.

[and now the punchline:]

What's important in all of this is the recursive structure. Rates of change of rates of change of rates of change are what makes up the rhythm that governs the neurons in the upper reaches of your brain, and so now we can take a look at a little bit of calculus to get us to where we might finally have something to say about this whole my red your red problem.

[short short version: derivatives are degenerate functions and so integrals aren't really functions at all. If that sounds like English, feel free to skip this part.]

[first your basic graph stuff, if you know about slope, feel free to jump ahead]

Way back when we're learning about linear functions and equations on the Cartesian coordinate system, if we're paying attention and havn't fallen asleep on our textbook (they make great pillows), we learn about slope. Change in y over change in x, or how steep the line is. A flat, horizontal line has a slope of zero: if x is your horizontal axis and y is your vertical axis, then for a horizontal line, whatever x is, y stays the same. The slope of zero represents zero change. A straight line at 45 degree angle pointing southwest to northeast is the graph of the equation y = x. It has a slope of 1. If x increases by 1, y increases by one. Similarly, for y = 2x you'll have a steeper line, for a slope of two: if x increases by 1, y increases by 2. For 2y = x or y = x/2 you'll have a shallower line with a slope of 1/2: x has to increase by two in order for y to increase by 1. A vertical line has an infinite slope. x doesn't change at all, and y hits every value there is. So that's slope for linear functions.

And just to be sure, a line that is pointing northwest/southeast is going to have a negative slope, such as y = -x

The refresher: for any function of x that fits the form f(x) = mx + b where m and b are
any number, m will be your slope.

[now for parabolas, and introducing Derivatives!]

When we look at a function that is a curve, things get a little trickier. The thing is, the slope changes depending on where we look. Take your classic parabola, y = x^2. Right at the middle, we have the vertex, a horizontal point, and so our intuition is that the slope there is zero. We could say that if you pick to x values that are on either side of zero, say + 1 and - 1, your slope between them is going to be zero, because when you square the numbers the negatives cancel out and you end up with the same number squared every time. But to say that the slope is zero all the way across the curve seems kind of strange. While the parabola does go through each y value twice (once on the left side where x is negative and once on the right side where x is positive) it would seem strange to say that the parabola had the same slope as a horizontal line. And of course, if we shift our frame of view, the slope is going to change: if we only pay attention to the right side of the graph, we can have a slope as high as we want (though never quite exactly vertical) and on the left side we can make things as negative as we want if we ignore all that positive business. The thing with curves is: the slope is different in different places. Now, since we've got a nice function here, if you give me two values of x I can go ahead and find out what values of y they pair up with, and then I can figure out the slope between them. If I divide the difference between the x's by the difference between the y's (so long as I keep them in the same order) then I'll find the slope in that area. But what if I wanted to know the slope at a single point?

Well: either you've got to find the limit as the difference between your two x's and your two y's shrinks to zero (limits are neat, but they'll only tell you the slope at a single point) or you can take the derivative. I'll save the proofs for those things for another day, but what they're give you is the slope at a given point. For y = x^2, the derivative happens to be y = x/2. Which means that at any point along the parabola, the slope will be equal to one half of your x value. Or more simply, if x is really negative, the slope will be really negative (pointed NW/SE) and if your x value is really positive, then your slope will be really positive (NE/SW).

So taking the derivative (I'm just asking you to trust me on this one) gives you the slope at a point, or if you want, the rate of change at a point.

[now we'll define degenerate functions! Hooray!]

Now: y = x is what's called a proper function, which is to say for every y there is one and only one xand for every x there is one and only one y.

This means that y = x can pass two tests: the vertical line test and the horizontal line test.
The vertical line test is: for any given vertical line (for an given x value) there will be exactly one intersection with the graph of the equation. For each x, one and only one y.
If the equation passes the vertical line test, then it's a function. The square root fails this test: any given square root (except zero) has two possible outputs, one positive and one negative. So y = sqrt x is not a function of x.
The horizontal line test is: For any given horizontal line (for any give y value) there will be exactly one intersection with the graph of the equation. For each y, one and only one x.
If the equation passes the horizontal line test, and it's a function, it's a proper function. y = x is one of those.
If it fails the horizontal line test, then it's a degenerate function, our parabola is one of those.

Degenerate functions are the case where we can get a given output by more than one input. They're functions, so if we've got the input, we can find the output, but if we try and go back, we find that we've got options to choose from.

Now: if we think of taking the derivative of a function as a function of functions (it takes a function as input, say y = x ^2 and gives a function as output, in that case y = x/2) then it's going to turn out to be a degenerate function. For our derivative y = x/2, any function of the form y = x^2 + C is going to produce the same derivative. So the derivative tells us the rate of change, but if we try and get back from the rate of change to the original graph, we're going to find a whole field of possibilities rather than an exact picture of what produced the rate of change we knew about.

For a derivitive like x/2, this isn't too much of a big worry: we know we're looking at a parabola, we just dont know if its shiftet up or down away from the origin at all. No big deal. But for more complex derivitives, we get much more complex fields of possible functions that could produce the same derivitive, and the field of differential equations is just the study of those fields of possibility. Things get particulary hairy when we throw in more than a single variable that we're worried about.

[now to bring it all together!]

So then: If what your nerves do is change their rates of fire, which in turn changes the rates of fire of other nerves down the line, and if all your brain has to work with are these cascading pulses of changing rates of changing rates, it looks like your brain has been presented with and elaborate differentail field from which it must attempt to make some sort of sense out of. If all it were concerned with were rates of change, there would be nthing strange at all about the lukewarm bowl that feels hot and cold at the same time, and the afterimage that you see after someone takes your picure with a bright flash would be nothing to think about. But the lukewarm bowl of water has a single temperature, and the afterimage you see is an illusion. Why? Because there's a real world out there that you're engaged in and you've got to deal with it as governed by some sort of stability or you're going to foget about the tiger chasing you or where the bananas are or how to build a fire. Doing high level processing on the rates of change your nerves detect is really, really useful for survival, and we've spent an awful lot of lekking making sure that we've got some fantastic brains in our heads to do that processing.

And so the kicker: Your color red is the result of an anti-derivitive, it is a projected guess picked from a field of possibilities as a stable cause for the rates of change detected by your optic nerve when you look at something red. Because the detection of rates of change is not a direct detection of the color (its just changes in the rods and cones in your eye, nothing like actualy picking out frequencies of light really, though your eardrums do some fancy stuff with frequencies for sure) but instead a detection f rates of change, your color red is a single choice from the field of possible causes for that rate of change. Given a different set of inputs (say we take out the rods you've got that detect light in the red range) you would have a different set of rates of change to deal with, and so the predictive model would differ. You would have a different red, or no red at all. Given slight changes in the chemical makeup of your particular rods, slight differences in their reactivity to given wavelengths of light (hey, we're not all perfect you know) there would be different rates of change taking place in your eye, and so the color red would indeed look different to you.

Also it is possible (clearly) for someone to detect light outside the normal visual spectrum and have a greater range of color than normal people have, just as much as it is possible to have a narrower range of color. It is not impossible to concieve of higher dimensional color as well.

Now: Looking into a given brain we are not going to be able to pull out and look at the color red that is experienced. Of course. It may be possible to look at differences in brain reaction to the same input (hey could you both stare at the wall for a second thanks) but to get at the subjective experience of the matter is another deal entirely.

But then: how do we know that my red and your red are the same? Well, they probably arn't, is the thing, but then we're dealing with things in a similar enough way to get along and communicate. My purple and a colorblind persons purple are not going to be the same, though we might behave in perfectly similar ways towards purples, organize them in similar fashions, etc. But as my experience of is the result of a choice (whether free or otherwise, I'm going to say its a choice determined by certain structural features of the brain, but the sort that vary far more than fingerprints and in similar fashion) from a field of possibility, it is unlikely that anyone else experiences my red or my purple or my blue or my green or my yellow, and it is not nonsense to speak of such things, and it is not a miracle that even with these variations within people we can all still get along and talk for hours about whether we should paint the room this or that shade of off-off white. Though this might have everything to do with variations in our synesthetic assosiations for given colors and our feelings thereof.

This also has some really interesting implications for the scientific worldview, as the insturments we use to measure the world are not so different in function from our various nerve based sense organs and so some paradoxes of the very large and very small may have a decent explination given this framework. As far as the understanding of what our models based on experimental data really ammount to (anti-derivitives of a sort, and so choices from a field of unknown complexity).

What I really want to say here is that given what we're given (changes in rates of change) we have to do some imaginitive work in order to have a world to live in. We have to make up the things that would be causing the things that we feel, to some extent, because there is always a plurality of possibile causes for the rates of change we experience. Is the water warm or cold?

This is the solipsists conundrum, and if we stop here (as I'm about to for the night) we're left wondering just what we have in common with other people if we're trapped in this world of imaginary things that our brain has built to explain the various sensations that we have. But any worry we have here is misguided, as it requies a number of questionable assumptions:
1) that your own understanding of yourself is singular
2) that there is a hard and fast division between yourself and other people
3) that every sense works like vision

That last assumption is a pretty big one, and thinking about the tactile world and your continuity within it with other people and other things is a good way to remind yourself that not everything is like vision.
It may be useful to think about what it feels like to drag a stick through sand, or across some rocks, or to poke a tree or sword fight with the stick. Our sense of touch seems to extend through the stick right to the point, and theres good reason for saying that this situation is no more illusory than your feeling extending throughoutyour body. These are differences of degree, not kind.

But more on that next time. For now, enjoy your red, and I will enjoy mine.