(bright uplifting music) - So Michael, I've been reading Proust with my seniors.

Volume One, "Swann's Way," and I came across a passage that I think will interest you very much.

So it occurs rather early in the book.

It's just two sentences, not very long, given what Proust is capable of.

So it's when we're meeting the characters and getting to know them.

Okay, and we hear this about the main character Swann.

"Even with respect to the most insignificant things in life, none of us constitutes a material whole.

Even the very simple act that we call seeing a person we know, is in part, an intellectual one.

We fill in the physical appearance of the individual we see with all the notions we have about him, and of the total picture that we form for ourselves, these notions certainly occupy the greater part."

So you were telling me just the other day that what I just read from Proust is something that Euclid can prove.

- Yeah.

- That I wanna see.

- All right, so Euclid in his "Optics," of course, is writing a geometrical treatment of human vision.

Not color.

- Okay.

- But just the size, shape, and clarity of the things that we see.

And so the "Optics" begins with a set of definitions about, or presuppositions about the nature of our power of sight.

And it begins with a claim that what we see occurs along straight lines, or the power of our vision occurs along straight lines that extend a great greatness into space.

All the way to the point, for instance, that we can see stars.

- Okay, so he's saying that about just the power of vision, not about light- - Not about light.

And he's really not saying, for instance, that light is the source of our sight.

Probably the simplest way to think about that is light that strikes the back of our head doesn't make us see anything.

But some power that extends in straight lines from our eyes- - From our eyes.

- And I think a lot of readers of the "Optics," there haven't been a lot of readers of the optics recently, but a lot of readers of Euclid's "Optics" understand what Euclid is saying, is to say that some kind of power emanates outward from our eyes into the world.

I don't think that's of material importance in terms of reading Euclid's "Optics" and thinking about what he's showing us about sight.

But that is one way people understand it.

You know?

- Yeah.

It's something from us, not to us.

- First definition of presupposition that Euclid lays out is that the power of our vision occurs along straight lines, and then it extends a very great distance.

He says, "A big bigness out into the world."

So we can see the things that are as far away as stars.

And what he really says about this, is what are power of vision touches, along these straight lines, that's what we see.

And what it doesn't touch, we don't see.

A few other things that he actually says in these definitions is that the size of an object at least the size that it appears to be for our sight, is dependent upon the angle that the extremities of the object make with our eye.

So if you draw straight lines from our eye to the ends of any object we're looking at, that would be its front surface that's available to us, because we can't see around to the back.

That that angle that it makes determines the size that things appear.

So, you know, when somebody is far away, you can situate their head between your fingers and crush their head if you want, or something like that, at least in appearances.

So that's sort of the beginning of the rules of sight is that sight operates along straight lines.

What those straight lines touch, we see, and what they don't touch, we don't see.

So the very first- - I can accept that.

- Yeah.

- Okay.

As a starting point, - The very first proposition then, the one that you've asked me to show that Euclid proves, is that we never see any object in its entirety all at once at the same time.

He says (speaking foreign language) in Greek.

So if I considered an object to be just the side of it that's facing me, is what Euclid's talking about, this line, BC, is the object I'm looking at- - That line.

All right.

- And A is where my eye is.

Then many lines of vision, or of the power of my vision in straight lines come out from my eye or can be drawn out from my eye, and they touch this object BC.

And where they touch it, I see a part of the object.

- In this case a point.

- A point, yes.

Because these lines are straight, and because they come from one point, my eye- - And diverge.

- They diverge.

Well, if they didn't diverge, they'd just see one point.

- Yeah.

- Right?

So in order to see many points, the power of vision must diverge.

But because it diverges, there are gaps or spaces in between the lines that define the power of vision.

And anything that falls in those spaces, we don't see.

So this is a picture of one moment in time of the power of our vision operating on one object and seeing parts of it and missing or not seeing- - Most of it.

- Parts in between.

- Missing most of it.

- Well, maybe, I mean- - It looks like from here.

- Yeah, right.

It depends on how many lines of vision there are.

- Well, that's what I wanted to... That's in a way the obvious question is, isn't it?

Why not just draw more lines, indefinitely, more lines?

- I think as many as you draw, right, they still diverge from one another.

So as long as you maintain the thought that the power of vision operates along straight lines, and that what you see comes into focus at one place, one point, your eye.

- Your eye.

- Then there'll be parts that are missed.

- Okay.

All right.

- So one at one moment in time, you never encountered the entirety of the thing you're looking at.

- Even a very small thing that's close?

- Sure, actually, I think the next proofs that Euclid does in the "Optics" indicate that if this body, BC, were to be closer, it would be touched by more lines or there would be fewer gaps in between the powers of vision.

And we would see it more clearly- - More clearly.

- But not in its entirety at one time.

Now, the one thing that seems really remarkable to me about that first proof is not the math itself, which I think is pretty straightforward once you- - It's very simple.

- Accept the definitions.

- It's so simple.

And the implications seems so big.

It's startling.

- Well, and I thought it's startling, surely, you know, just to say, when we begin from perception itself by itself, we don't get continuous wholes.

That's not our experience.

That's not derived from just vision.

But Euclid adds one line of text to the end of his first proposition, and it's not a mathematical comment.

It's a comment about our opinions and the way things seem to us.

And he says, despite this fact that we don't see something in its entirety all at once, he says, "It seems to us that we do," because our eyes flicker back and forth.

And he's indicating some holding together of a series of moments putting together a number of these experiences that are missing things and giving ourselves the impression, the opinion, that we do see wholes all at once.

And that the scene that's held together is one moment in time, and that the object itself is given to us by our vision.

- I see.

So the eye might have little movements in itself, or the head and this takes a certain amount of time.

And there's, it sounds like, this is almost, like, an integration operation where, you know, here a certain distance BC, the points that are on it are integrated to give you a continuous line, just to use the language of calculus.

- Well, I think it might approximate the thought of the calculus in some ways, this thought, except for the very next propositions, like I was saying before, our propositions about variable clarity.

So even though there may be some flickering of the eyes, which I think human beings who studied these things have discovered this several times in history, including Charles Darwin's father studied this.

Even though there's that flickering, and even though Euclid is saying we're taking many snapshots from our vision and putting them together as if they were one thing, even that doesn't give us everything.

So that the next propositions indicate, you know, not that an object that's closer to us is seen in its entirety over a period of time, but that it's seen with more clarity than the same object farther away.

Not perfect clarity, but more clarity, which also means something is being thrown into the mix, which isn't just given to us by our eyes and then collected by our soul.

- That's what interests me.

So with say, with Proust, he's talking about what it takes to know another human being or to make known a character in a book.

It takes many pages, many interactions with many people as one follows him along.

And even then, you feel that character is a mystery finally.

So does that have some counterpart in Euclidean optics, what I just described, humanly?

- Well, I think the mystery might be what do we have to add since no matter, well, maybe I'd say this.

If we wait too long, we take too long of a period, it won't seem like a single vision to us, right?

So staring and moving your head over, and scanning something, that wouldn't seem like it was one thing, one experience.

But if we just focus on those experiences that seem to us as if they're one, there's still something added in there that isn't just all the things that our powers of vision have touched.

That's something that we're putting in that blurs together with the things that we have, in fact, seen.

- Yeah, so Proust in what I read, said we have intellectual notions that we're adding.

And, you know, in this case, they're prejudices that the family has about the bourgeois background of this main character.

So they think of him in a certain kind of class.

He's that kind of person.

So does that get at what you mean, that we're adding to this line, certain intellectual notions that make it a line, a straight line, rather than some other kind of mathematical object?

- Well, if I move away from the line slightly, it does seem to me the word prejudice is a pretty good word.

That is, there's a judgment that's being made before any experience, before perception has given us anything.

There's a judgment that's being made that seems to explain what we put in to complete the whole and give us the appearance, or the opinion I should say, that we have encountered a whole.

And probably those things that we put in have to be somewhat class specific.

- Right.

- You know?

So if I think I'm looking at a straight line, I put in some kind of simulacrum of straightness in between the actual parts that I've seen to straighten it out for the whole.

If I thought I was seeing something else, I'd put in things that belong to its kind.

But those things that are put in are not as vivid, I think, at least in vision.

So that explains why the things that are farther away are still blurrier for us or less clear.

- And where do those things come from?

That we're putting in or filling in the gaps, as it were, literally, or filling in gaps.

- Filling in gaps.

Yeah.

- Where does it come from?

Do we make it up?

Is it a habit?

Is it something, go back to the word prejudice, preexisting that we simply apply that's common to all minds?

- Yeah, I would think that there probably are options, if I could put it that way.

That we're not simply predetermined by, you know, structures that are born into us.

I think in the context, you began in "Swann's Way," it doesn't seem to me that those notions of class are inborn.

And since they wouldn't be applied to every unfamiliar human being, they aren't necessitated by the experience, but they're imposed upon it.

I would think that's also true even of some of our geometric approaches, depending upon what we already have come to think about objects, how we're vaguely familiar with them, what we expect from them, we put into them.

- So, but that sounds pretty radical.

That is- - Well, yeah.

So there's a late, I just mentioned a late proof in Euclid's "Optics."

Early on, Euclid shows that if you look at an object and it's moving towards you... Well, if you look at the same object at two different distances, which is a way in which you could model movement, that same object seen closer will occupy a larger angle of your vision, and so it will look bigger to you.

So he proves that it basically just draws that same line, BC, closer, and then connects the end points to show that it occupies a larger angle.

- A larger angle at A. Yep.

- Okay, so that's what it means geometrically for something to look bigger to us.

Late in the book, Euclid has a small proof where the diagram would be just exactly the same.

But he says if there's an object that's growing, it's actually changing size and getting larger.

We will think it's coming at us.

That is, we will think it's coming at us.

And why does he say something like that?

That's not math.

The math is indifferent.

The same appearance occurs whether there's something coming at us and seeming to get bigger that way, by occupying larger angles of our vision, as would appear if something were not approaching us at all, but just getting bigger and occupying larger angles in our vision.

So he says we will think it's coming at us, because it matters to us if something's coming at us.

- If it's big enough and coming at me, it might eat me.

- There it is.

Yeah.

Or even the simplest thing.

That baseball is coming at me, right?

And it's gonna hit me.

- So what you just described about things getting bigger, either because they're coming at us or they're simply increasing in size while remaining at rest, interests me a lot.

'Cause it sort of involves character in seeing things, even just things as mathematical objects.

It involves, say, courage and cowardice or, you know, things like that.

Virtues and vices.

But I thought those things ought not to enter into doing Euclidean optics or Euclidean geometry or, you know, objective science, physics.

But in a way you're saying it's there.

- Well, it's certainly there in Euclid's text.

I would probably begin by suggesting some of these kinds of comments that Euclid makes are extra-mathematical.

That they're outside the realm of what's mathematically necessary or mathematically relevant.

In other words, the object that's growing has the same logos for vision that the object that's coming at us has.

It's increasing in size, occupying a bigger angle.

Now, in I think, every proof in Euclid's "Optics," there's basically two accounts.

There's the account of how things appear to our vision, and then there's the account of where they are in linear geometrical relationships to one another, where our eye is, how far away the object is, what its size is, and those two accounts or two logoi are separate from one another.

And there's even a proof in the middle of Euclid's "Optics" to indicate that, you know, they never quite exactly have the same proportions.

So he's always telling two stories.

How things look and then how they are.

And then he seems to throw in a third story from time to time.

What we think about them, how our mind relates to these situations.

- Or how to reconcile- - Sure.

- The, you know, the fact that they don't simply overlap, the two logoi.

Yeah, this question always often comes up when we're doing freshman math, and freshmen are working through Euclid's geometry.

What does this have to do with the so-called real world?

You know, is this geometry that we're doing real?

Is it out there in nature?

Yeah.

- So there's seem to be a really funny or interesting set of understandings, the way Euclid presents the objects.

For instance, objects really have size that belongs to them, and they really are a certain place and at a certain time.

And their dimensions could be explained or described the way Euclid describes the boundaries of objects in the elements.

That is, they have places where they end and places where other things start.

All that I think, Euclid treats as the world outside of us.

Not only that, but those boundaries and those objects are whole and they are continuous- - And stable.

- And stable.

He treats them as stable.

Our vision doesn't show that.

Our vision doesn't show us all of them at once.

Not even the side that's facing us.

And it doesn't...

It's not unchanging, because if our eye moves or the object moves and changes its relationship, its size and even its shape and its clarity, how clearly we see it, all those things change.

So that's what vision initially gives us.

Euclid presents it as if we should think that's explained by a stable, consistent, continuous and world full of wholes.

- So Euclid just, he assumes, or prejudges, there is this real stable whole that is behind the appearances.

Geometry gives us, you know, laws about it.

Yeah.

- I think that is the set of his assumptions.

Yeah.

- So if this isn't too preposterous, why should one believe the geometry rather than the eye?

What one's eyes give you?

- I'm not sure that I have a completely good answer to that question, except, I mean, I can tell you I think what is a difference between the two.

Well, I think we've already mentioned it in a certain way.

The world is stable.

If we just paid attention to what vision gives us, just vision by itself, the sizes of things change all the time on the basis of our situation to them.

- Of course.

- Right?

We tend to think as we're approaching something and it gets bigger in our vision that it's not growing, right?

But how do we test that?

We go up to it maybe, and we measure it, we touch it to see if it is where we see it as being.

Things like that.

But that's no test if the world could have been changing.

'Cause by the time I get up to somebody, maybe they've grown, and when I leave them, they shrink.

And all these things.

When I move, they seem to move.

Aspects of relative motion.

Why couldn't that be the sort of dynamic and ever-changing way in which the world really is, and we suppose that on, you know, underneath that, there is something stable and consistent?

- So this would get at that third story that you were saying you could tell us.

That is what do we think about, you know, the choice or the discrepancy between the two logoi, i.e., geometry and perception.

And as we think about it, maybe we would find ourselves more amenable to going with the eye under certain scenarios.

Let me give you one.

Another literary one that comes to mind.

It's from "Hamlet."

Okay?

So it's when the ghost returns to visit Hamlet while he's talking with his mother, Gertrude.

And Hamlet sees the ghost.

Gertrude doesn't.

And Hamlet says, "Don't you see my father dressed in his habit as he lived?"

And she says, "I see nothing there."

And he says, "Nothing at all."

And she says, this is the interesting part, "Nothing at all, and all that is, I see."

So she's not only not seeing the ghost for whatever interesting reason, but she's operating under this assumption that her eyes give her the world, give her what is there.

And Hamlet knows differently.

That is, so if one goes with one's eyes, I mean, it looks like it can go both ways.

You can see ghosts, or like Hamlet, or like Gertrude could say, "There aren't any, 'cause I don't see them.

I never have."

You know, but that makes all the difference in the world of that play.

- Well, I imagine, how would I say it?

A more modest Gertrude, and I don't mean that in any punning way, who wouldn't simply trust her own vision?

Who would accept, for instance, that maybe the logic of ghosts means they appear to some and not to others?

- Or maybe they're- - Or that Hamlet sees- - Maybe they're in here.

- Right.

- They occupy a certain gap in one's, I don't know, one's perception, one's character, one's intellect.

- Well, yeah.

So there's two things.

I mean, there's the gap that is, attends each of our perceivings, so that each of us, humbly I think, should recognize, is missing something of what is there, which could be seen.

So there's a world that could be visible and is not visible to anyone in its entirety.

So each of us are missing some of that.

And we could be humble about what we're missing.

And if someone else tells us they see something there that we don't, we could possibly consider maybe within the gaps they're seeing something that is there to be seen.

The other thing I think, is the filler, what we add in, what we put in that's not gotten by vision at all, but from our starting points.

We could call them our prejudices, I think.

- Yeah, so you mentioned a word a couple times that interests me.

It goes back in a way to character.

Humility.

That this proof should make us more humble about who we are as knowers, learners.

- There's a lot of, I think, tendency, especially amongst modern students, to think that math is about things that human beings made up.

And therefore, in a way, it's subject to our whim.

(bright uplifting music)