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Okay. Now we're going to move on to Hume's theory of space and time.

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Where we'll see that the theory of abstract ideas plays quite a significant role.

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So the theory of space and time comes in. Book one, part two of the Treaties.

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Now this part is very often ignored.

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If you look at books on HUME, you will often find that no mention of this is made at all, or it's treated rather dismissively.

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I actually think that's probably fairly justified.

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But you will find plenty of HUME scholars who will argue the reverse, and I'll be referring to some of them.

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Now, what you seems to be doing in this part of the treatise is applying his theory of ideas to draw

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conclusions about the nature of both our ideas of space and time and of space and time themselves.

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Now, you might think that the former is reasonable enough.

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He's looking at our ideas, investigating them, drawing conclusions about how we conceive of space and time.

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What seems a little bit more puzzling is that in this part of the treatise,

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he seems often to be drawing conclusions that go beyond that, that talk about space and time in themselves independently of our ideas.

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So the burden if the first part of part two is to argue that space and time,

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both our ideas and space and time themselves, are not infinitely divisible.

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So in the very first section entitled of the Infinite, The Visibility of our Ideas of Space and Time.

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Now again we find humour saying that something is very obvious.

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It is evident from the plainest observation that the capacity of the mind is limited and can never attain a full and adequate conception of infinity.

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Okay. Well, that seems reasonable enough, right?

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Our minds aren't infinite. If you try to imagine something that's infinite, you try to imagine the infinite sequence of numbers.

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Or the infinite number of fractions or imagine infinitely dividing a line.

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We do give up pretty quickly. It follows that the idea which we form of any finite quantity is not infinitely divisible.

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We must ultimately therefore reach a minimum.

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Now notice that there's a little bit of a an assumption here.

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HUME seems to be assuming that because we cannot go on dividing our ideas infinitely,

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we must hit rock bottom, as it were, with a particular minimum size.

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It's not absolutely clear that that's the case. And one could imagine.

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Trying to divide up one's image of something.

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And succeeding to different extents on different occasions.

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Perhaps sometimes we tire. Sometimes we feel more acute.

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And so we're able to go farther. But HUME has an argument or an illustration to back this up.

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So here's what we're supposed to do. Imagine an ink spot drawn on a board.

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And imagine retreating from it further and further and further.

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So the ink spot gets smaller and smaller. Until you just stop seeing it, it disappears.

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Then go back just a tiny bit so you can just see it.

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That is the minimum visible quantity.

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So what you will see, HUME thinks, is a coloured point, a coloured extension list point, because it will be indivisible.

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It will just be a point. You won't be able to distinguish the left side from the right side of it, but you will just see that point.

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So what we have here is a kind of visual atom.

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And we will see that HUME uses this atomic theory of our perceptions to draw quite significant conclusions.

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Now. In a recent article in Human Studies, Rolf George has made what I think is a very interesting speculation.

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The severability principle we've already met, and we'll see later Hume's various applications of it.

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But an intriguing fact about Hume's philosophy is that the severability principle is

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not mentioned at all after the treatise in the inquiry concerning human understanding.

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It doesn't come up at all. Some people claim that it's there implicitly in certain of the arguments.

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Personally, I'm not convinced. But what's also interesting is that a number of other things seem to disappear from the inquiry,

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which at least seem to be connected quite closely with the severability principle.

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For example, HUME drops the simple, complex distinction, or at least he puts much, much less emphasis on it.

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And the distinction between simple and complex ideas seems to be very much wrapped up with

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the theory we're talking about now about the simple visual atoms and atoms of other senses.

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Likewise in the inquiry, we get almost no discussion of space and time.

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We get a little bit when he talks about the sceptical arguments and we get a little footnote where

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he refers to his theory of abstract ideas and suggests that might possibly provide a solution.

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That said, there's a bit of a mystery there. We don't really know why HUME left this stuff out of the inquiry to obvious theories.

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One He wanted to make the inquiry more simple and palatable.

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He was addressing a different audience,

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an audience who wouldn't be interested in all the intricacies of infinite visibility and his discussion of space and time.

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So he saw no need to put in the severability principle.

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Maybe. That's right. But an alternative explanation is that he'd lost confidence in it.

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And Rolf George speculates a particular reason why this might be so.

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So in 1738, James Durin published an essay upon distinct and indistinct vision.

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And Georgie's hypothesis is that this, as it were, awoke David HUME from his dogmatic slumbers.

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How could it do that? Well, here we have a line and here we have a dot.

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And you will notice that the dot is wider, a greater diameter than the line is across.

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Now, imagine retreating further and further away from that until the dot cannot be seen.

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Nevertheless, you will still be able to see the line.

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Now, that's an empirical claim. What James Durian was doing was making empirical investigations into human acuity.

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But you can see that's a little bit of a problem for him.

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If you think that there are minima in our visual field of a certain fixed size that as you go smaller and smaller,

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you hit a limit like a computer pixel, just like the pixel on a computer screen.

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You cannot represent any image which is smaller than that.

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Well, at the point when that dot has disappeared, there should be no pixels at all left representing the line.

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So the line should disappear too. But it doesn't. So as I say, that's an interesting speculation.

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We don't know whether it's true. But the dates are very suggestive.

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We have this investigation being published in 1738.

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The treatise was published in 1739, a highly plausible that HUME between then and 1748 came across this, and George gives relevant evidence.

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Okay. But let's now proceed and see what he does with his theory.

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Well, having established that there are these minima.

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HUME considers. Those.

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Now, if we hit a minimum, if we actually think of a minimum possible point, it's extinction less.

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You cannot distinguish the left from the right. It's as small as anything could be.

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It follows that nothing can be more my Newt than some ideas which we form in the fantasy that the fancy remember is another name for the imagination.

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And images which appear to the senses, since these are ideas and images perfectly simple and indivisible.

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The only defect of our senses is that they give us disproportionate images of things and represent as my Newton unkempt.

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What is really great and composed of a vast number of parts.

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So take that dot that we've drawn on a board and maybe it's quite big, but we go further and further and further away until we can only just see it.

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And at that point it looks to us completely simple and on compounded.

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We just see the dot. Now we might therefore have the erroneously draw the conclusion that the thing itself that we're seeing is totally simple.

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But what we can do at that point is pull out our binoculars or our telescope and take a look.

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And we see, oh, it's bigger. So it's clear that there are a lot of light rays coming.

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And if we use instruments, we can see the thing in more detail.

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It ceases to be so simple. But what HUME wants to say is that that simple idea that we get correctly represents the smallest part of anything.

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Nothing can be smaller than that. So we get a famous image of a flea from Hooke's Micrographia, 1665, and HUME is clearly alluding to this.

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This, however, is certain that we can form ideas which shall be no greater than the smallest

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atom of the animal spirits of an insect a thousand times less than a mite.

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The animal spirits think of what goes through the nerves,

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and we ought rather to conclude that the difficulty lies in enlarging our conception so much as

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to form a just notion of a mite or even of an insect a thousand times less than a might for.

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In order to form a just notion of these animals, we must have a distinct idea representing every part of them.

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So what's going on here?

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You might think that, you know, when we look at something from a distance, as it were, we're making an error in seeing the thing as simple.

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Our idea as it were is misrepresenting it. And him saying, we think thinking that way around it is the wrong way, actually.

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We can form ideas which are adequate to the tiny parts of things because our ideas are pure and simple.

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They just aren't compounded. So they form an adequate idea of the very smallest parts of anything.

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What's actually difficult is forming an idea adequate to a whole might.

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That's vastly complex. Or even a creature.

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A thousand. Part of that sort. Thousandth part of a mind. Okay.

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So HUME has drawn here an important conclusion,

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and it's one that he's going to use now to conclude about space and time in themselves rather than just about our ideas.

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So we get that right at the beginning of treatise one, two, two.

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Wherever ideas are adequate representations of objects, the relations, contradictions and agreements of the ideas are all applicable to the objects.

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Okay, that seems reasonable. If our ideas are faithful representations of the way objects are.

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Then inevitably any conclusions that we draw from the ideas will be applicable to the objects.

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And I realise you're all wondering where the handout for this is.

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It'll come next week with stuff for next time.

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Because what didn't make up a complete end up. Okay.

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So fair enough. If we've got adequate ideas, the ideas, as I say, faithfully represent what they are ideas of.

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Then in reasoning about the ideas and drawing conclusions about those,

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those conclusions will inevitably follow to the things that the ideas represent.

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But here comes the crucial claim, the one he's been arguing for.

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But our ideas are adequate representations of the most mind parts of extension.

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And through whatever divisions and subdivisions we may suppose these parts to be arrived at.

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They can never become inferior to some ideas which we form.

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So the ideas are so simple, so un compounded that no part of extension can possibly be less than those.

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The plain consequences that whatever appears impossible and contradictory upon the comparison of

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these ideas must be really impossible and contradictory without any farther excuse or evasion.

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Now notice that HUME here is not using what might seem seem to be his conceive ability principle.

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He's arguing from inconceivable bility to impossibility.

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That's different from arguing from conceive ability to possibility.

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We saw the conceived policy principle last time. We'll be seeing lots more of the conceive ability principle.

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HUME thinks quite generally that to conceive of something distinctly.

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Implies its possibility. You cannot distinctly conceive of something that's impossible.

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Okay, so conceive ability implies possibility. But here he's saying that inconceivable bility implies impossibility.

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But he only wants to say that that applies where ideas are adequate.

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Now that's quite important. Remember humans and empiricist. He thinks our ideas are derived from impressions.

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He thinks, for example, that a blind man has no visual ideas.

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So he surely doesn't want to say that. Inconceivable, as he quite generally implies, impossibility.

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There may be all sorts of things which we cannot conceive because we don't have the ideas.

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And he said that our minds are finite. There may be all sorts of things that we can't conceive of because we're just not capable of it.

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We don't want to conclude, in general, that that implies impossibility.

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But when our ideas are adequate, that's a different matter.

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So he's already said that our ideas are adequate representations of the most minor parts of extension.

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We've seen that our ideas are not infinitely divisible and it follows that the same is true of space.

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I first take the least idea I can form of the part of extension and being certain that there is nothing more my newt than this idea.

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I conclude that whatever I discover by its means must be a real quality of extension.

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I then repeat this idea once, twice, thrice.

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So imagine that tiny little atomic idea. And now put another one next to it and another one next to that.

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And if it helps, think of them as differently coloured. You start off with a blue dot.

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Remember, it's extension lists. You can't distinguish its parts. But now you put a red dot next to it, maybe a yellow dot next to that.

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And what you do now is build up extension. The idea of extension comes to you as soon as you've got more than one of these.

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So each of our minimal ideas is indivisible and therefore not extended.

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But as soon as you put two together, you've got the minutia part of extension.

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Add another one, you've got a bit more extension. Add another one.

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You've got more. Carry on to infinity.

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Where do you get? Well, you're going to have an infinitely large extension. No way round it.

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Although each individual, indivisible atom, as it were, is unexpended.

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As soon as you put lots of them together, you get a finite extension.

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If you put an infinite number together, you will get an infinite extension.

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So he goes as far as saying that the idea of an infinite number of paths is the same idea with that of an infinite extension.

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So he's proved. To his satisfaction.

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At any rate, that space is not infinitely divisible because we've got these little ideas that are adequate to the minimalist parts of space.

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And if space were infinitely divisible, then you'd have to be able to have an infinite number of these tiny parts within a finite amount of space.

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But you can't, because as soon as you get an infinite number of these little atoms, you get an infinite extension.

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Now. If you're familiar at all with mathematics, an objection is likely to come to your mind.

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Imagine something that's finite, extended. Imagine dividing that extension into and taking half of it.

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Then divide that in two and take half. Divide that in two and take half and go on and on and on and on and on.

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You start with a half, then a quarter, then an 1860.

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On and on and on. Without stopping, apparently. So what's wrong with that?

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Why can't you divide things infinitely? Well, Jim actually addresses this objection.

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In a footnote, he distinguishes between proportional and adequate parts.

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It's a proportional part where you're dividing up again and again and again like this, adequate parts, all of equal size.

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And he just seems rather dogmatically to say, Well, that doesn't deal with my argument.

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Because nothing I've prove nothing can be inferior to those main parts we conceive.

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When I think of this idea of a simple, nothing can be smaller than that.

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So divide up as much as you like. You cannot get smaller than that.

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And if you can't get smaller than that, then an infinite number of those is going to give you an infinite extension.

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So there you are. My argument stands. Later into the intersection, HUME again comes back and deals with a potential mathematical argument.

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So there are various mathematical arguments that seem to tell in favour of infinite visibility that seem to try to prove it.

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And HUME says these can't be right. Now, he is appealing to the conceive ability principle.

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He's saying, I have this notion of space made up of all these little atoms.

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That's a conceivable picture of the way space could be.

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Since it's conceivable, it's possible. So any attempted proof that it's impossible must be fallacious.

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So he's attacking the mathematical objection to his own view and he's attacking himself.

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The argument of mathematicians that he's claimed as a positive proof of infinite visibility.

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Now, these arguments don't seem to be.

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Ideal. Particularly his argument against infinite visibility, against proportional parts.

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Because when he says My idea is as simple as can be, this atomic idea of a visual atom.

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Must correctly represent the smallest parts of space because nothing could possibly be smaller.

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Therefore, when you finally get down to the ultimate bits of space, they're going to be simple.

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My idea is simple. Therefore, the two must match.

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That must be an adequate idea. The obvious response is to say, well, I'm sorry, who if space is infinitely divisible?

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You never do. Get down to an ultimately simple part.

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So I went in claiming that the ultimate parts of space must match with this idea.

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You're begging the question you're taking for granted that you do actually get two ultimate simples.

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And that's just assuming that space isn't infinitely divisible.

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So there's a bit of a puzzle here. I mean, humans generally are pretty acute philosopher, as I've said,

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but one part two is probably, well, almost certainly the weakest part of treatise.

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But what? The arguments aren't that great. And maybe HUME just imagined himself to be better at mathematics than he really was later in life.

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He did actually produce a treatise on geometry, and he was persuaded by Lord Stanhope, a noted mathematician, not to publish it.

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Sadly, it disappeared without trace.

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It'd be great if it turned up at some point, but at any rate, when he wrote the treatise, he seems pretty confident, doesn't it?

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Is evident this is absolutely clear, etc.

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So is something going on that brought it about that he didn't see the problems that we see in his arguments?

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Well, here we reduced really to speculation. Tom Holden, a book published in 2004.

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He suggests that HUME is presupposing an actual part's metaphysic,

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whereby anything that is divisible must in advance consist of the actual parts into which it is divided.

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So contrast to different possible accounts of visibility, you've got the Aristotelian idea of potential infinities.

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So suppose I take an extension, divide it up. Divide that again.

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Again and again and again. Like, however much I divide it, I can go on dividing further.

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So in that sense, there's a potential infinity. It's like saying, give me any number.

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I can always add one. You can always go further.

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So there's a potential infinity. But that's different from saying that there's an actual infinity.

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You can say that the the line is potentially divisible without claiming that it is already divided into parts,

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that those separate parts already exist, as it were, prior to the division.

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But Tom argues that at the time the actual part, metaphysic was very strongly in the air.

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The thought that if some things divisible, the parts already have to be there.

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They have to exist prior to the division. Now that suggestion is somewhat supported by an argument that HUME uses in treaties.

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One, two, two, three. He borrows it from Nicholas.

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The militia is evident that existence in itself belongs only to unity and is never applicable to number.

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But on account of the unities of which the number is composed,

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it is therefore utterly absurd to suppose any number to exist and yet deny the existence of unities.

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And as extension is always a number and so on.

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The thought is if a group of people exist, they exist only in virtue of the existence of each one of them.

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Take any number of things. The ultimate existence are always the unities of which the group is composed.

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Now apply that to an extension. The thought would be that unless there are ultimate parts of extension.

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Nothing exists. So if if if you always can divide further, you never hit the ground.

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Then ultimately, metaphysically, there's nothing. Another possible account of what's going on is due to Don Baxter.

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He's written an article in the Cambridge Companion to HUME quite recent.

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And he suggests that HUME is pursuing a somewhat Kantian agenda.

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So what Immanuel Kant wanted to do was to say that our knowledge of space and time,

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our knowledge of space and time in the phenomenal world, in the world that we experience, not in the world as it is in itself.

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And.

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So Baxter suggestion is that Hume's aim is to find out about objects as they appear to us, by examination of the ideas that we use to represent them.

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So it's less ambitious. The thought is Hume's concern is with space and time.

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Within, as it were, the experienced manifold. And within that realm, the limits of space and time are given by the limits of our ideas.

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I'm not persuaded, but it's an interesting suggestion.

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Finally notice that HUME draws the same conclusions about time that he draws about space.

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All this reasoning, he says, takes place with regard to time, and he adds an extra argument.

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It's the essence of temporal moments to be successive.

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Time is of its nature, successive in a way that space isn't.

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So if time were infinitely divisible, you'd get coexisting moments.

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And that's not possible. So time can't be infinitely divisible.

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And then as soon as you think of motion something moving in time through space,

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you can see that if infinite visibility of time is impossible, infinite the visibility of space must be as well.

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Okay, we'll continue next time with a little bit more on space and time.

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And then we will be getting on to book one, part three, which is the most important part of the entire treatise.

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See you then.